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AC Theory - David Knight. Radio, electrical and electronic articles.

1. If we want to add two impedances graphically we simply place the beginning of the second against the end of the first and draw a new line from the beginning of the first to the end of the second Thus we get a new impedance with a new magnitude and a R new direction This might all seem rather unnecessary in view of the simple addition rule given earlier but the meaning of vector addition is hopefully obvious when it is visualised in this way Whatever the method used in performing the arithmetic however the point in doing it as we shall see is that it allows X1 X2 us to keep track of the relationship between the voltage applied across an impedance and the corresponding current Vector Addition R2 23 7 Balanced Vector Equations It is here that we must observe the principal property of the equals symbol which is that if a given type of mathematical object lies on one side of it then exactly the same type of object must lie on the other Thus now that we know that impedances are vectors we must re write Ohm s law in such a way that equality is never violated There are numerous ways in which that can be done but for the moment we will examine three possibilities V IZ V IZ I V Z It is by no means obvious that all of these expressions must be true but as we shall see they all are when interpreted correctly Since impedance is a vector then either voltage is a vector or current is a vector or presuming that t
2. 21 Imaginary resonance It was observed in section 4 that the series resonance formula fy 1 2nV LC always gives two solutions for the resonant frequency one positive and one negative The parallel resonance formula 15 2 does the same but presents us with a further conceptual challenge in that it also allows imaginary solutions If we inspect the formula 1 L C Ry 2nVLC L C R we can observe that if either R1 or Re should become larger than L C but not both at the same time then the quantity inside the right most square root bracket will become negative Once again there were no restrictions on the validity of the arguments which went into deriving the formula and so imaginary resonance is possible and must have a physical meaning The answer to this conundrum can be obtained by considering the parallel resonator as two separate impedances connected across a generator see diagram right Real resonance implies a condition where the current from the generator is in phase with the voltage it produces i e it occurs at a frequency where the resonator constitutes a resistive load The output current I is the vector sum of the currents in the two branches of the resonator 1 I I Ic and so real resonance occurs when I Ic is real Real resonance can only occur however if the current in one branch can become large enough for its imaginary component to cancel the imaginary component of the current in the other b
3. RR R R2 R X R2X j XX Xi X2 X Ro X gt R Z 14 1 RIR X X or R R2 X2 RAR2 X i XRX X R2 X Z 14 1a Ri R Xi XP The real part of expression 14 1 is R and the imaginary part is X and so we can write Ri Ro R R2 R1X2 R2X X1X2 Xi X2 X Ro X2R R and X Ri Roy X X Ri Roy X X Or alternatively using expression 14 1a R Ro2 X Ro R2 X X RX XRX R and X Ri R27 E2 X X27 Ri F R27 Xi X2 The formula and variants given above for impedances in parallel while not exactly memorable has the advantage of being completely general First note that if we put X 0 and X 0 then all of the reactive terms vanish and we are left with the formula for resistors in parallel i e R R R 2 Ri R2 Similarly if we put Ri R2 0 we end up with the parallel reactance formula X X X Xi X2 More usefully however we can put only X 0 and find out what happens when a resistance is placed in parallel with an impedance and we can put R 0 and find out what happens when a pure reactance is placed in parallel with an impedance The latter operation is of particular importance in the matter of devising and analysing antenna matching networks Dimensional consistency The solution to the parallel impedance problem is our first example of what might be called a messy mathematical derivation As such it is fairly t
4. b J VC a b VC a b N N2 ll ll 24 2 Magnitude reciprocal theorem The magnitude of the reciprocal of a complex number is equal to the reciprocal of its magnitude 1 N 1 N 24 2 Proof Let N 1 j0 1 Now Ni No Nil No Therefore 1 N gt 1 No L No 1 No 59 24 3 Magnitude product theorem The magnitude of the product of two complex numbers is equal to the product of their magnitudes IN N IN N2 24 3 Proof Let N a jb and N a2 jb2 Then N N a jb a2 jb gt a bib j aib2 F arb IN N V aia2 bbz aib2 azb V aia2 bib aibz a2b1 V ai2 ax b2 b a b 2 V ar bi a b V a b V a b2 N No 24 4 Scaling theorem The magnitude of the product of a scalar and a complex number is equal to the product of the scalar and the magnitude SN s N 24 4 Proof Let s be a scalar and N a jb sN sa jsb ISN V say sb s a b sV a b s N i e a scalar can be factored out of or multiplied into a magnitude bracket in the same way that it can be done with any other type of bracket 24 5 Drop dimension theorem A phasor with a phase angle of 0 or 180 transforms as a scalar N IN 0 N N N 180 INI i Proof A phasor pointing at 0 can be represented as a complex number wit
5. j is called the imaginary operator because it operates on a number in such a way as to make it impossible to add it to a real number Once j or 1i was discovered mathematicians went on to find general solutions for cubic equations and quartic equations i e equations involving x and x and it was proved that no other type of imaginary operator was required This means that a numbers can be reduced to the sum of a real part and an imaginary part and expressed in the general form x atjb with the proviso that sometimes b 0 and the number is purely real and sometimes a 0 and the number is purely imaginary Thus it is not so much that complex numbers are peculiar but that real numbers are a special class of complex numbers which just happen to have the imaginary part equal to zero Once it was understood that numbers are in general complex the next step was to work out what that meant The clue comes from our earlier discussion of vectors Firstly we may observe that all real numbers must lie on a line stretching between o and Secondly we may observe that j causes imaginary numbers to exist in a dimension separate from real numbers Therefore the effect of j is to rotate the number line through 90 Thirdly we may observe that the numbers 0 and 0 j0 are the same so that the real and imaginary number lines must cross at 0 The upshot is that complex numbers i e all numbers can be represented as points in a plane which is t
6. 41 5 Thus a tightly coupled output transformer scales the source impedance according to the square of the turns ratio a generator with a low output impedance being converted into a generator with a high output impedance by means of a step up Ns gt Np transformer and vice versa The broadband output transformer of a fairly typical 100W short wave radio transmitter Kenwood TS430s is shown on the right The transformer core is a block of ferrite with two hollow channels passing through it known colloquially as a pig nose The primary winding consists of two short lengths of copper or brass tubing passing through the core and connected together at one end by a strip of copper laminate board The secondary winding is a length of PTFE coated multi strand silver plated copper wire threaded through the copper tubes the reason for the choice of materials is explained in another article To make a complete turn around the core a conductor must pass through one hole and back through the other As shown below diagrammatically the copper tubes form a centre 20 Components and Materials Www g3ynh info 94 tapped single turn with the DC power supply B connected to the centre tap and the other ends connected to the collectors of the RF power transistors a matched pair of 2SC2290s The transformer in the photograph has four turns and so increases the amplifier output impedance by a factor of 16 There is something more t
7. R2 and X X X gt One point in treating impedances as vectors is that it enables us to draw diagrams in order to visualise what is going on We can do this by representing an impedance as a line in a plane with a particular length and orientation In this sense a vector diagram is like a navigation chart with the distances in this case measured in Ohms Mathematicians calls such maps spaces by analogy with ordinary space and a space in which distance is measured in Ohms is called impedance space Now observe that although the R and X parts of an impedance exist in different dimensions they both exist in the same space because they are connected by the fact that they are measured using the same units i e Ohms We may therefore deduce that the difference between a space and a graph is that all of the axes in a space must be labelled in the same units whereas the axes of a graph can have different units e g temperature vs time You may of course have heard of four dimensional space time which appears to disobey the rule just stated but in fact the unit of the fourth physical dimension is not time but the speed of light multiplied by time i e ct The units of ct are metres per second x seconds i e metres and so Einsteinian space has four dimensions with units of length Working in impedance space if we adopt the standard convention that resistance increases to the right and reactance X increases upwards we can obtain the lin
8. RINV RN 1 RRN 1 Ny Ro RiNv R2 RINV RN 1 RiRa Ny 1 Nv R2 RINV RNy R R2 RiNy ROR R2 Ny 1 R2 RiNy OR OR2 D N OR2 N D OR D Ro RiNy Ri Ny 1 RRN 1 R2 RiNvy RNV R2Ny R2 RNV aR JOR RyNy 1 Ny Ro RINV GR Ny D N ENy N D ENy D R2 R Nv R R RiR2 Ny z 1 R R2 RiNy al RR RR R2 RiNy OR ONy RiR2 R2 R1 Ro RiNvy The error function in this case is o V ARg ORi ori ORg OR2 or2 Re ONv on The derivatives all share a common denominator D and so on writing the expression in full a factor 1 D can be removed from the square root bracket Hence o 1 Ro RiNvy Vf R Nv Noni R Ny 1 Nv Jone RiRa Ro Ri ow In the previous section we determined R 23 3Q from the following measurements R 29 640 340 R2 75 140 70 Ny 1 364 0 04 These give D R RiNy 1205 8673 ORR RANK 1 D 2052 9636 1205 8673 1 7025 OR OR2 RPN 1 Ny D 435 0100 1205 8673 0 3607 OR ONy RiR2 R2 Ri D 101144 68 1205 8673 83 8771 o V ORg OR1 x Opi ARe OR2 x On Rg ONv x on V 1 7025 x 0 34 0 3607 x 0 77 83 8771 x 0 04 V 0 57892 0 2525 3 3551 90 Note that the error contributions in the expression above are very close to the averages of the de
9. Vret and 16 When Slide Rules Ruled Cliff Stoll Scientific American May 2006 p68 75 68 N dB 20Logi0 I ret Which is all very clever but leaves people struggling to decide whether they should use 10Log or 20Log and so leads to lots of mistakes So remember a ratio in Bels is the Log of a power ratio and a deciBel is a tenth of a Bel that s where the 10 comes from By Joule s law the square of a voltage or current magnitude ratio is also analogous to a power ratio and the squaring can be obtained by doubling the logarithm that s where the 20 comes from In using deciBels the basic approach is to consider the power levels at two points in a circuit or power transmission system and thereby define the gain It is also useful however to express power in relation to some external reference or standard and this leads to an extension of the notation some commonly encountered variants being as follows Unit Definition ae ee Cea dBm dB relative to ImW in 50Q 223 6mV 4 472mA dBu dB relative to ImW in 600Q 774 6mV 1 291mA dBW dB relative to 1W dBV dB relative to 1V 1V in old audio publications and service manuals dBm may be used to mean dB relative to 1mW in 6009 By extending the definition in this way the dB notation may be used to express an absolute power rather than a relative power and if a reference resistance is specified an absolute voltage or current as well For example if the line out
10. because it encourages the reader to re derive all of the mathematics in order to find out what the writer was trying to say 9 Voltage Magnification amp Q One of the curious properties of the series LCR network discussed above is that the voltages across the reactances can be much larger than the applied voltage Take for example an impedance consisting of a 1Q resistance in series with an inductance having X 100Q and a capacitance having Xc 100Q all connected across a generator giving 1V output In this case the system is resonant because X Xc 0 and so the voltage across the total reactance Vx 0 and the phase angle p 0 Because the two reactances have cancelled each other out the impedance looks like a pure 1Q resistance but there is a current of 1A flowing through each reactance and so each has a voltage of 100V across it This voltage magnification 100 1 is also by definition the Q or quality of the series tuned circuit formed by L C and R i e Q X1 R and also Q X R or Xc R The smaller the series resistance the better the quality In the case of an impedance such as an antenna of course we cannot get inside it and measure the voltages across the individual components and a simple series LCR combination is nowhere near complicated enough to account for the way in which antenna impedance varies with frequency When maximising the power delivered to an antenna however we frequently need to place the antenna i
11. p21 27 the energy flows in an electrical circuit and indeed how it flows into space from a radio antenna The explanation for the principle of continuity comes once again from the work of Einstein this time from his investigation of the photoelectric effect It also follows logically from Maxwell s equations the issue being that there must be a reason for the rate frequency at which energy swaps between the fields in a propagating electromagnetic wave On the assumption that energy is the simplest thing in the Universe the only possible governing factor is the amount of energy being transported Hence it was bound to be discovered that light is made up of particles i e discrete units not solid objects each oscillating at a frequency dictated by the amount of energy it contains The particles of course are nowadays called photons and the relationship between energy and frequency is known to be a direct proportionality E hf where h is Plank s constant and has a value of 6 62606896 10 Joule seconds This introduces another strange concept known as wave particle duality which is sometimes claimed to be a paradox but is actually nothing of the sort For a glib explanation we can say that it would only be paradoxical if a batch of propagating energy was not made up of discrete units because then the energy would have no way of knowing its own frequency and so would not be able to form a wave For a more formal way of th
12. 0 0149384 Ryo 150008 0059 4 410223156 Ryo 34 0137 KQ The only problem with equation 16 1 is that it is very cumbersome and resistant to simplification We might therefore be inclined to look for some simplifying assumptions and the most obvious of these is to note that since X is very nearly equal to Xc we might as well assume the term X Xc to be zero This also implies that X X XcX L C hence equation 16 1 becomes Ryo RRc L C Rit Rc Le Ryo RRc L C Ri Rc 16 2 44 Notice that this formula has lost all of its reactance terms which is very convenient If we apply it to our example data where L C 71428 5702 we obtain Ryo 2 0 1 2 1 71428 57 2 1 0 095 34013 61 Q Ryo 34 0137 KQ The approximation is almost exact for components of moderate Q Also we may observe that the term R Rc Ri Rc is much smaller than the term L C Ri Rc and given that we are unlikely to know the component resistances very accurately we might as well drop the first term Hence the appropriate formula for calculating the dynamic resistance is Ryo L C Ri Rc 16 3 This equation is an excellent approximation for the dynamic resistance but strangely it is not the one offered in most textbooks The usual approximation is that in addition to Xi Xc being zero the ESR of the capacitor is assumed to be zero This causes all of the terms containing RC in equation 16 1 to disappear and gives
13. 10Log P Po 0 SEREEN EEEE E ERNER EER 3 Series Resonator 6 Power vs frequency 10 15 1 1 2 1 4 f fo 16 1 8 2 So having shown how the shape of the frequency response function varies with resonant Q we will now derive an expression for the relationship between Q and bandwidth the bandwidth being defined as the interval between the upper and lower half power points The procedure is to write a general expression for the current and solve it for the frequencies at which I I V2 Notice that the word frequencies is plural the I expression will be a quadratic equation To As we have already determined I V Z and Io V R Hence at To the 3dB bandwidth limits 2 T V Z I V2 V RV2 Thus the bandwidth limits occur at the frequencies where Z RV2 Ne y R X RV2 9 __ _ __ R2 X2 2R 2 f fo f X2 R2 which taking the square root of both sides and noting that there are two possibilities from so doing gives 72 X R Now writing X explicitly we obtain the expression R 2nfL 1 2nfC which we must solve for f We may proceed by putting the right hand side onto a common denominator i e by multiplying top and bottom of the 27fL term by 22fC R 2af LC 1 22fC L e 2nfCR 2nfPLC 1 This rearranges to 20fPLC 27fCR 1 0 Which is a quadratic equation in the form af bf c 0 with a 42 LC b 2aCR and c 1 Notice however that this particular
14. 26 0 01 0 22 6 55 emia 0 94 Notice in the graph above that as the load resistance R is increased and becomes greater than source resistance R the power delivered to the load tails off gently The reason for this behaviour is that as the current drawn from the generator reduces the output voltage increases and so the system possesses a self regulating property when lightly loaded When the load is twice the source resistance the power delivered is still 89 of the maximum possible a droop in output of only 0 51dB The major advantage of light loading however is seen in the transfer efficiency When a conjugate match is achieved the efficiency is only 50 but it rises to 67 2 3 when R 2Rg and 80 4 5 when R 4R This means that light loading when compared to conjugate matching gives a reduction in generator dissipation and power input for a given power output In radio practice of course the generator is a radio transmitter and if the transmitter is designed for light loading it can have smaller heat sinks and reduced battery or mains power consumption in comparison to a transmitter designed for conjugate matching Consequently the figure often referred to as the output impedance of a radio transmitter often 50Q is usually nothing of the sort it is instead and should be called the preferred load resistance or alternatively the design load resistance The preferred load resistance of a broadband transistor power amplifier is
15. General statement of Joule s Law In section 3 we gave Joule s law in its standard form P PR This now that we know that I should be interpreted as a phasor pointing at 0 or 180 proves to be a correctly balanced vector equation but it is only so by an accident of notation and as we shall see shortly the restriction on the phase of the current is unnecessary and limits the scope of the formula Joule s law even in its standard form is a more fundamental statement than P IV because the squaring of the current prevents the direction of the current from having any effect on the direction of the power It therefore tells us that power is positive when resistance is positive i e the dissipation of energy is a uni directional process The direction in question is that of entropy the general spreading out and cooling down of the Universe which is associated with the irreversibility of time Thus assuming that an impedance is by definition a strictly passive network our relationship with impedance space is skewed in that we are not allowed to venture into the regions where resistance is negative One far reaching consequence is that we cannot devise electrical networks which will give an output before receiving an input i e we cannot build circuits which violate causality There are however non linear passive electronic devices which have a negative resistance characteristic such as the Esaki diode or tunnel diode but this is only
16. In this way we see smooth waves in the collective behaviour of many particles Heaviside knew nothing of this but the photon theory explains his continuity principle by identifying the energy carrier The fields extend throughout space because they represent the propensity to exchange energy and the photons can turn up anywhere that the fields have finite intensity but the photons themselves undergo no additional exchange processes during transit Our ideas on the flow of electromagnetic energy are nowadays associated with John Henry Poynting who formalised the continuity principle in a theorem which bears his name Heaviside however had already been using the principle for some time and had a far more elegant derivation lodged with his publisher at the time of Poynting s first public presentation Poynting also did not interpret his findings correctly whereas Heaviside had no such trouble and so it was the latter who first described the underlying mechanism What follows may come as a shock to those who have been taught the lies to children version It turns out that the inside of a good conductor is the one part of a circuit where the transmission of electrical energy does not take place We can understand Heaviside s explanation by using Faraday s lines of force which provide a way of visualising the electric and magnetic fields Some field patterns relevant to to the workings of circuits are shown below 3 Oliver Heaviside Paul
17. J Nahin 2nd edition paperback John Hopkins University Press 2002 ISBN 0 8018 6909 9 Ch 7 Tech note 3 p129 131 The left hand diagram represents an electric field as it might exist between two charged spheres or between two electrical conductors seen in cross section Recall that like charges repel and opposite charges attract and so a positively charged particle will be repelled by the electrode and attracted to the electrode Hence depending on the starting point the arrows show the direction in which a positive charge will be accelerated and the lines show the path which will be followed There are of course an infinite number of possible starting positions and so the field has an infinite number of lines but a sparse representation is sufficient to give the general idea The curvature of the lines arises because the mutual force between pairs of charged bodies is governed by an inverse square law i e the attraction or repulsion is strong when the bodies are close but falls off rapidly with distance Hence a particle close to one electrode will have a trajectory almost perpendicular to the surface but the field line becomes curved further away because the particle is then influenced by both electrodes The middle diagram shows the lines of the magnetic field surrounding a wire when a current of positive charges is flowing away from the observer The wire is shown in cross section and the cross within its boundary repre
18. Lorentzian to the electrical resonance curve we may first note that the Lorentzian is always exactly symmetric about xo and that xo can be set to zero We have noted before section 30 that the electrical resonance curve is skewed when plotted against linear frequency but becomes symmetric to a good approximation when the Q is high We also noted that the resonance curve can be made perfectly symmetric by plotting it on a logarithmic frequency scale in which case since the logarithm of unit frequency is zero the curve can also be symmetric about Log f 0 In fact natural resonance processes have such high Q that they appear symmetric on linear logarithmic and even reciprocal wavelength scales but to find the relationship between the Lorentzian and the electrical curve it is obvious that we must identify the x axis as corresponding to logarithmic frequency i e x Log f where the base a can be chosen arbitrarily Here we will use Naperian logarithms because it will allow us to use the series expansion of e to solve the problem Hence we choose x Log f which means that f and fo e Substituting these identities into the electrical resonance curve 32 2 we obtain 1 P Po 1 Qof e e e e 7 but from the rules of logarithms discussed in section 28 e e e and e e e eGo Hence 1 P Po 1 Qo er e P qi X Xy The quantity e e is related to a function known as
19. P This puts the bandwidth function into a form most similar to a curve known as the Lorentzian line shape function next section but a further simplification is possible by multiplying both numerator and denominator by Qo 1 P Po 32 2 1 Qol f fo fo J which demonstrates in the clearest possible way that the bandwidth of an LC resonator is dictated entirely by Qo and fo 76 33 Lorentzian line shape function The electrical resonance curve is closely related to a simple mathematical function known as the Lorentzian or Cauchy h y line shape function which has the general form h w y 33 1 h 2w gt lt w X Xo 2 where h is the peak height and w is called the half width The expression can also be written y 1 gt X 33 2 0 Xo ho 1 x xo wP which is the form most similar to equation 32 2 The Lorentzian is regarded as the characteristic signature of natural electromagnetic resonance processes In particular the peaks in molecular and atomic spectra in the microwave optical x ray and gamma ray regions are all of this form when displayed on a linear amplitude y axis scale The curve is called a line shape function because the narrow spikes which occur when dense spectra are drawn by a chart recorder or otherwise displayed are traditionally known as ines It is only when the frequency scale is expanded that the individual peaks resolve into Lorentzians In comparing the
20. Q We can of course adjust the source and load resistances using transformers and as we shall see shortly we can replace the resonator coil with a transformer so that the inductor and the transformer become one and the same Before we look at such coupling schemes however we must draw attention to a particularly misleading inference of the formula which is that high Q can be obtained by making the ratio L C as small as possible This suggestion has appeared in at least one amateur radio publication but it is a fallacy If the reactive components are of reasonable quality the parallel form L C ratio L C is only slightly different from the series form L C ratio and as we showed in section 21 imaginary resonance can occur if the L C ratio becomes too low The imaginary resonance condition is entirely a function of the series loss resistances of the coil and the capacitor It is nothing to do with the source and load resistances because avoidance of imaginary resonance is a matter of ensuring that the 90 component of the coil current at resonance is sufficiently large to cancel the 90 component of the capacitor current or vice versa but in practice coils are more lossy than capacitors Consequently the design procedure for a parallel resonator BPF is to make the L C ratio large enough to obtain a good strong resonance without making the inductance so large that stray capacitance and coil self capacitance prevent the target maximum freq
21. So it is with the dot product in general vector theory but in phasor theory it transpires that there is also a meaning to the negative solution Notice that when computing power from the dot product we don t actually specify the acute angle Strictly the angle which must be used is the phase angle and the power will be negative if the phase angle should happen to be obtuse Noting that P I R it should be apparent that the magnitude of the phase angle will be greater than 90 and the power will be negative for an impedance which has a negative resistive component Negative power dissipation does not occur in nature because it violates the principle of conservation of energy It can occur in circuit analysis however when a network which has been defined as passive turns out to be active The point is that power can flow out of an impedance if the network inside it should happen to include a generator One situation in which negative resistance can be encountered is when modelling antenna systems with multiple feed points Due to the coupling between the different parts of the antenna it sometimes occurs that more power flows out of one of the ports than flows in and the computed input impedance then has a negative resistive component The same can happen in any network with multiple ports Normally the situation is avoided by defining the port as active in advance but when modelling a complicated antenna there is generally no analytical
22. Some authors e g Hartshorn use Y G jB as is done here and others e g Langford Smith use Y G jB The alternative definition gives B X R X and thus Bi 1 27fL and Bc 2afC In the next section we analyse the parallel resonator bandpass filter and determine the relationship between resonant frequency bandwidth and Q In the author s first attempt at the derivation the definition Y G jB was used and the formula which resulted had it that either Q is negative or fo is negative The change to Y G jB fixed the problem and so since Q is by definition positive when loss resistance is positive the other definition is wrong according to the convention that frequency is positive We may also note a reflection symmetry in the correct choice in that we have X 21fL in impedance space and Bc 2afC in admittance space etc i e inductive reactance and capacitive susceptance are positive capacitive reactance and inductive susceptance are negative 45 Parallel resonator BPF The reader may have noticed that having determined the relationship between bandwidth and Q for a series resonator we did not immediately do the same for a parallel resonator but instead digressed into the subjects of source impedance and impedance transformation There was a very good reason for doing so as we shall soon see which is that there is no satisfactory design procedure for parallel resonant bandpass filters if the source and load impedances c
23. The expression P PR is known as Joule s law and is a statement of the fundamental relationship between electricity and thermodynamics One important point to note about the standard power and resistance formulae however is that they are all derived from experiments with DC electricity They represent incomplete statements of Ohm s law and Joule s law because they can only be applied to AC circuits when the load on the generator is a pure resistance Later on we will show how to state these laws in a completely general way but some groundwork will be required before that can be done Notice incidentally the correspondence between the power law P VI and the earlier given definition of the Poynting vector P E x H The former is a dimensionally reduced version of the latter as can be seen by noting that the unit of electrical field strength is Volts per metre and the unit of magnetic field strength is Amperes per metre Also even though we do not need to know how to perform 3 dimensional vector multiplication it is not difficult to understand that any type of multiplication also multiplies the units of measurement Hence the unit of the Poynting vector is Watts per metre squared which is a measure of illumination i e the delivery of electrical power is a matter of illuminating the receiving object with electromagnetic energy In the case of inductors and capacitors the entries in the left hand column tell us that they also obey Ohm s law
24. Vo as V V Rs Rp jXcp jXip Re Rs The bandwidth function is the magnitude of this expression but with all of the components represented as impedances anyone attempting to expand and simplify it or isolate part of it as the load will have a hard time keeping track of all of the intermediate terms We will therefore convert it into an admittance problem using the relationship V Zi Zp Z ff Za Y i Y Y Y Hence 1 Gs Gp jBo jBry V V 1 Gp Gs Where G stands for conductance and B for susceptance and Gs 1 Rs Gp 1 Rp Bop 1 Xcp and B 1 X1 The expression above can be re written Gs Gp V V Gs Gp jBop a jBip and the magnitude is VE Gs Gr V Vo VU Gs Gr Bop Bry i e Gs T Gp IV Vo VE Gs Gr Bop Bry This can be plotted against frequency by substituting Bc 2afCp and B 1 2afLp but we will not bother to do so here because it is identical in appearance to the graph of Ij I0 for a series resonator given in section 29 We will instead go on to determine the half power points by noting that whatever proportion of the parallel resistance Rp is designated as the load power will always be delivered to it in proportion to V so the half power points occur when V V V2 Hence at the half power points we have 103 Gs T Gp 1 V2 VE Gs Gr Bop Bry Which upo
25. a factor of 16 is a secondary consideration and is of no great concern unless the source impedance begins to approach the design load resistance the latter situation being associated with low transfer efficiency and poor load regulation as discussed earlier It follows that to keep the output impedance as low as possible a step up ratio just sufficient to provide the required output voltage is optimal The actual output impedance R of the TS430s transmitter measured at the antenna socket see the example at the end of section 38 is about 23Q measured 23 343 4Q at 1 9MHz for a design load resistance of 50Q The output impedance of the power amplifier Rg is therefore approximately 23 16 1 4Q Dye and Granberg give an approximate formula for calculating the output impedance of a transistor power amplifier below 100MHz as Rg Ve V eat Poutmax where Vee is the supply voltage and Poutmaw is the maximum power available from the amplifier If we assume a Saturation voltage Vsat of about 1V this gives Rg 12 8 100 1 64Q Multiplying this by 16 gives Rg 26 2Q which is within lo of the measurement without taking any of the circuitry between the power amplifier and the antenna socket into account 21 Radio Frequency Transistors Norm Dye and Helge Granberg Motorola inc Butterworth Heinemann Newton MA 1993 ISBN 0 7506 9059 3 Output impedance of a power amplifier p118 95 Notice incidentally that the powe
26. be controlled manually by the expedient of providing it with control knobs and switches instead of motors and relays This approach replaces the automatic control system with a human being but makes no allowance for the fact that humans in general have little aptitude for the task Here we monitor the load magnitude and phase using bridge circuits which are the subject of a separate article The bridges produce error signals which tell their respective control systems which way to go in event that the error exceeds a certain preset 22 Solenoids D W Knight www g3ynh info 98 threshold Not shown on the diagram but necessary to make the system work are limit switches two for each variable device These tell the control system when a motorised device has hit one of its end stops so that the change over relay can be switched and the motor direction reversed in the case of the impedance transformer so that the switch over from coil to capacitor or vice versa can be made in the case of the series reactance network and as protection against motor burn out in the event that the load is outside the matching range The control systems for magnitude and phase are shown as being completely separate which they are except in respect of common signals such as the request for a tuning carrier or an instruction to reduce power which they might send to the transmitter on detecting a matching error The independence of the two systems is possible becaus
27. circuit In resolving the potential for discrepancy we must first recognise that circuit diagrams fall into two categories those which are used for production engineering and end up in service manuals and theoretical diagrams used by designers As we will see in this and subsequent articles a theoretical diagram is actually a type of mathematical statement which can be extended to describe the behaviour of a physical circuit to an almost arbitrary degree of precision A production diagram on the other hand is just a record of the interconnections in a set of manufactured sub assemblies resistors transistors coils etc As has already been implied for equipment operating at audio frequencies and below there may be a great deal of similarity between the diagram used by the design engineer and the diagram in the service manual but for well designed radio equipment this will not necessarily be the case On the subject of circuit diagrams it will be noted that the North American or Japanese preferred zig zag line symbol for resistance is used here This convention is adopted or in the author s case was never un adopted because the rectangular box symbol was already in use by theoreticians long before European standards apparently intended solely for the convenience of draughtsmen were put forward In this and all of the other documents produced by this author the box symbol is 13 used strictly to represent a generalised electrical
28. context as though it has one less dimension A two dimensional vector which drops a dimension in this way of course becomes a one dimensional vector i e a scalar Hence whenever a voltage or current appearing in a mathematical expression is written as a scalar the symbol can be and as we shall see later must be interpreted to mean that the corresponding vector is lying along the 0 axis A vector which transforms as a scalar in some specific context is called a pseudoscalar A pseudoscalar has the property that when its space co ordinates are reflected with respect to the origin 0 0 it changes sign whereas a true scalar remains unchanged Hence voltages and currents become pseudoscalars when we choose their directions to be 0 or 180 Another electrical pseudoscalar is resistance a special kind of impedance which can be treated as a scalar but which becomes negative if the co ordinates of impedance space are reversed If we choose the current in Ohm s law to be our reference vector and set its phase angle to 0 it becomes a pseudoscalar of value equal to its extent in the 0 direction i e it is identifiable as the quantity I Cos0 or I Thus in the relationship V IZ we can recognise I as the reference vector against which the phase of V will be determined T I 0 I and V IZ CGII Z The pseudoscalar current I is therefore equal to the current magnitude I the latter being the quantity registered by an
29. does to the output impedance of a generator Here we will call the apparent source impedance as seen from the secondary side of the transformer Z with Z as the actual generator output impedance The relationship between Z and Z is perhaps guessable but to derive it mathematically requires a trick which is that of defining an equivalent circuit with all of the source resistance moved to the secondary side of the transformer A suitable approach to the derivation is then to write expressions for the voltage V across the load using both the original and the equivalent circuits and then equate the two expressions 93 fe N ll For the left hand circuit let us define Z as the load impedance seen by the generator its relationship to to the load Z being given by equation 41 3 above Z Z Np N s The voltage V is then the output of a potential divider formed by Z in series with Z i e Vi V Z Z Z and V is related to the load voltage V by the turns ratio ie V V Ns Np Hence V Ns Np Vz Z Zg Z V Ns Np Va 1 Z Zz 41 4 For the right hand circuit V is the output of a potential divider formed by Z and Z V V Z Z Z V V 1 Z Z where Ve Ns Np Vg Hence V Ns Np Vs 1 Z Ze Equating this to expression 41 4 gives 1 Z Z 1 Z Z i e Z Z4 Z Z but by rearrangement of equation 41 4 Z Z Ns Np hence Zs Zg Ns Np
30. equation will have four solutions rather than the usual two because the b term has a t symbol attached to it The reason for that is that there are both positive and negative frequency solutions for each of the band edges To obtain all four of these frequencies we apply the general solution for quadratic equations 12 3 f b vV b 4ac 2a f 42nCR 4V 2nCRY 4x42 LC 2 42 LC and using the substitution C C C to obtain a cancellation of C from all but one term f 4CR V CR 4LC7 C 4nLC f ER V R2 4L C V 4aL In order to determine which are the positive frequency solutions among these four possibilities observe that V R2 4L C is always larger than R Hence the upper positive bandwidth limit is V R 4L C R 4nL and the lower positive bandwidth limit is V R2 4L C R 4aL and the bandwidth is fy f V R2 4L C R V R 4L C R 4aL L e fw R 27L a 29 3 Now recall that the resonant Q can be defined as Qo X1 R 27foL R Hence Qo fo R 2nL R Hence fy f Qo 29 4 This is the classic expression for the bandwidth of an LC resonator and is exact when the resistance in the circuit remains constant with frequency We have of course already observed that the loss resistances of inductors and capacitors vary with frequency but it transpires that this will make practically no difference to the accuracy of the expression under most circumsta
31. for complex arithmetic is removed if all you want to know is a magnitude i e if the left hand side of an equation is a magnitude then all of the phasors on the right can be replaced by their magnitudes This observation simplifies some problems enormously since failure to apply the magnitude ratio and product theorems when the situation allows results in unwitting repetition of the working used in the proofs in sections 24 1 to 24 3 Shown below are some of the possible interpretations of Ohm s law which stem from the discussion in this chapter There is no need to memorise these formulae because they are all derived from the statement V IZ What they show is that all manner of complicated arguments involving phasor diagrams are in fact trivial and can be deduced by inspection of the master equation V IZ Z V 1 VIF IP l V Z VZ ZP V I R jX R jX V I I V R jX V R jX R X V IZ Z V 1 VI IP I V Z VZ ZP V IZ Z V I I V Z VZ ZP IV I Z IZ V E II V Z V 1 Z Z V 1 T V Z V I Z IZ V Ji I V JZ V IR R V I I V R V DR R V CD D V R Where V I and Z are phasors V I and Z are complex conjugates V I and Z are magnitudes V and I are phasors pointing at 0 V and I are negative values of V and I and 63 thus are phasors pointing at 180 and un bold Z is not normally used because an impedance pointing at 0 already has the symbol R 26
32. has a small s In old textbooks and papers the unit of admittance is often given as the Mho Ohm spelt backwards but in either case the actual dimensions are in reciprocal Ohms i e Q or Q7 A pure resistance of 50Q therefore corresponds to a conductance of 1 50 Siemens i e 20 milli Siemens or 20mS A pure reactance X 100Q corresponds to a susceptance B 10mS and so on Siemens incidentally is a name like Jones The singular of Jones is not Jone and so Siemens keeps its final s in both singular and plural forms one Siemens several Siemens The plural Siemenses is not recommended but is a lot less embarrassing than the quasi singular Siemen The double slash product was previously defined as 17 5 a b ab a b We can demonstrate that addition is the reciprocal space counterpart of the double slash operator by transforming the parallel impedance formula i e if Z Z Z Zi Zn then Y Zi Z2 Zi Zo If we let Y 1 Z and Y 2 1 Z then Y Y Y 1 1 Y Which rearranges to 100 Y Y Y i e when two networks are placed in parallel their admittances are added Recall that the formula for resistances in parallel R R R2 R R2 is a rearrangement of the expression 1 R 1 R 1 R2 It should now be apparent that what the formula really says is G G G The formula for impedances in parallel is of course a rearrangement of 1 Z 1 Z 1 Zp and this expression can
33. i e they always move at right angles to each other This means that impedance cannot be represented by an ordinary number i e a one dimensional quantity lying on a line between o and o it must be represented by a point on a two dimensional plane which is another way of saying that Z can be plotted as a point on a graph of R against X With regard then to solving problems involving impedance it so happens that we are spoilt for choice because there are no less than two appropriate branches of mathematics namely vectors and complex numbers The vector approach traditionally preferred by engineers is that of making sketches or graphs and using trigonometry to work out the actual numbers whereas the complex number approach is algebraic in that it allows equations involving two dimensional objects to be written down and re arranged Both approaches are equivalent however and sometimes one can clarify the other and so we will adopt a notation and a way of thinking which enables us to switch freely between them 6 Vectors amp Scalars A vector is by definition a mathematical object which must be described by two or more independently variable numbers Impedances as we have noted fall into this category and so vectors can be used to describe them One very useful property of vectors is that they can be mixed with ordinary numbers and manipulated using the normal rules of arithmetic provided that the rules are generalised to accommodate
34. in the sense that there is a region in the graph of I vs V where the current goes down as the voltage is increased Thus negative resistance devices can go from a particular level of power dissipation to a lower level as the applied voltage is increased but they can never achieve a state of negative power dissipation In order to generalise Joule s law completely we must write it in a way which allows the current phasor I to adopt an arbitrary phase but which gives an explicitly scalar result The obvious candidate expression is P IPR 26 1 It transpires that this is the definitive statement of Joule s law as we shall demonstrate but first it is interesting to note a parallel between it and energy laws in the wider context The square magnitude theorem 24 6 tells us that I I I and so we can write expression 26 1 as P II R or more to the point P I RI 26 2 Mathematical structures of this type occur everywhere in physics Quantum mechanical energy equations for example are of the same form and in that context the vector which occupies the position if I is known as the eigenvector or state vector and also the wavefunction If I describes the state of the system the implication is that all we have to do is define I in order to determine the energy If the analogy holds then the expression above is universally true upon provision of a definition for I This is not difficult to demonstrate because all electrical power transmi
35. is always just as far from zero frequency as it is from infinite frequency i e we need an infinity to the left of the resonance and an infinity to the right and both infinities must be of the same type Such a requirement is satisfied by and indeed is one of the principal properties of the logarithmic function x Log f the choice of base being arbitrary Hence the peak can be said to be symmetric about its logarithmic centre frequency Xo Log fo Since frequency can be scaled arbitrarily without affecting the shape of the bandwidth function units of Hz are not mandatory this matter can be proved numerically by plotting P Po against Log f with fo 1 and noting that the function is symmetric about xo Log 1 0 for any value of Qo It should also be noted that there is an infinity of scales from microscopic to macroscopic and it is often more natural to think in logarithmic dimensions than in linear ones In the case of frequency this can be seen by considering the classic representation of the electromagnetic spectrum as illustrated below 74 The Electromagnetic Spectrum Frequency f Hz 4 10 100 1K 10K 100K 1M 10M 100M 1G 10G 1006 10 10 1d 10 1d 1d 1d 10 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Log f VLF LF MF HE ET Micro Infrared ai UV gt waves Visible ee eee Radio X rays _ Light y rays There is no theoretical minimum frequency on the logarithmic scale although the lo
36. is positive As was mentioned earlier AC analysis is the art of representing circuits as networks of interconnected resistances inductances and capacitances Hence we have the task of developing the general rules of combination for those elements What we have to combine is summarised in the table below which gives the basic electrical formulae much as they appear in numerous textbooks 5 The RMS Average D W Knight Available from g3ynh info 15 is ee ee V IR Toga i WAV J Ohm s Law R R R ER R Ri Re R Ri R2 _RiR2_ Ri Rz2 pen ml XL 2rfL et La hs V IXL Li e EF L L1 L2 EF Lit L2 dee alee Xo 1 27fC o LL Lo c crc V IXc LT C2 es C1C2 C C1 C2 py z C1 C2 Observe that only the entries in the left hand column contain fundamental scientific information The uppermost entry is a statement of Ohm s law which is that the electrical current I is proportional to the voltage applied across the ends of a conductor the constant of proportionality being known as the resistance The entry below it is the power law which represents the observation that a conductor resistor heats up as a consequence of an electric current i e it dissipates energy and the power consumed i e the energy delivered per unit of time is the product of the applied voltage and the current i e P VI Watts Also by using the substitutions I V R and V IR we obtain two alternative power laws P V R and P R
37. network It is also in particular and in keeping with long standing practice used to represent a generalised two terminal linear network called an tmpedance a mathematical construct with which we are about to become very familiar In the following sections of this article we will derive the basic AC theory which deals with notionally discrete resistances inductances and capacitances these being known as ideal components As will be shown the behaviour of networks of these components can be determined by starting with a handful of empirical electrical formulae i e formulae determined by experiment and then using the properties of the Poynting vector to determine the rules of combination We will not incidentally need to know how to carry out 3D vector multiplication explicitly we simply need to know that for multiplication operations of any type when the two quantities being multiplied have the same algebraic sign the answer is positive and when they have opposite signs the answer is negative This will result in an extendible theory of linear networks extension being a matter of incorporating circuit modules black boxes called component models or equivalent circuits which have ports defined as networks of ideal components These modules were mentioned previously as a way of dealing with non linear devices but the approach can be applied to any electrical device which requires the use of a more sophisticated theory such a
38. not exist therefore what will be obtained is a network which has a voltage current phase relationship always on one side or the other of zero degrees only approaching 0 at zero or infinite frequency It was mentioned earlier that resonant circuits used in HF radio applications tend to have large L C ratios often greater than 100007 In the case of parallel resonance one reason for this policy should now be apparent i e we need to obtain a high characteristic resistance Ro V L C in order to ensure that the circuit will function properly with practically realisable inductors A parallel resonator with an L C ratio of 100Q for example will not work if the RF resistance of the inductive branch is greater than 10Q at the expected resonant frequency and it is by no means impossible for a practical inductor to exceed such a limit We may conclude from this discussion that a parallel tuned circuit will only resonate usefully if V L C is made larger than the resistance in either of the branches The qualification usefully must be applied however because if the resistance in both branches is allowed to become larger than the critical value then both the numerator and the denominator of the term inside the square root bracket will become negative and so the term itself will be positive Thus there will be a real resonance but the current in both of the branches will be feeble and so the resonance will also be feeble and of little practica
39. note that component Q can be expressed as Qeomp Rotoss X the higher the parallel loss resistance the higher the Q Magnitude of an impedance in parallel form The magnitude of an impedance in its series form is given by 6 1 Z WR X gt Substituting for R and X using expression 18 1 we obtain IZ VE Ry XP Xp Re Ry Xp V R X HE Ry XY We can take the square root of the R X term and so factor it out of the square root part of the expression provided that we only use the positive result magnitudes are always positive Hence Z Rp Xp V R 2 X 2 18 2 A convenient rearrangement of this expression can be obtained by forcibly factoring X from the denominator IZ Rp Xo XV ERe X2 11 Now since R and R X are always positive we can drop the magnitude brackets to obtain IZ R V R X 2 1 18 3 This form is particularly useful for frequency response calculations because it allows the reactance contribution to be treated as a correction factor 1 V R X P 1 which goes to unity 1 when the reactance is large in comparison to the resistance 19 Series to Parallel transformation From expression 18 1 in the previous section we have R R X R 2 X 9 1 and R X X R R X 19 2 gt Rp jXp Obtaining the series to parallel transformation is a matter of using these two equations to obtain equations for R a
40. of energy transferred by collisions with the atoms of the conductor comes nowhere near to the amount transferred by the electromagnetic field In fact the collision energy is merely the resistive loss which occurs in imperfect conductors The electrons do not carry the electrical energy but they do extract a small tax for the service they provide in guiding it along the outside surfaces So how are we now to think of electric current The current which conveys energy would seem to be best described as a displacement current at least in the sense that it is not carried by electrons It merely becomes correlated with the number of electrons passing a given point in a conductor per second when the generator frequency is low For historical reasons however i e because Maxwell 9 modified the definition of current instead of replacing it we think of current as the sum of conduction and displacement currents This turns out to be a reasonable approach because there are many situations in which conduction current is important Electronics for example is the art of controlling electricity by controlling the conduction current Charged particles are also involved in the workings of chemical energy sources batteries and fuel cells and electrochemical processes in general e g electro plating For the vast range of AC electrical problems however including the design of electronic circuits conduction current is a source of misconceptions and needs t
41. one about electricity being electrons flowing through wires is an intellectual dead end Electricity is actually an invisible form of light Specifically it is electromagnetic energy of very long wavelength in comparison to visible light which is why we can build devices called radio transmitters which cause electricity to propagate off into space Hence we will never understand high frequency electricity by counting electrons and we must first refine our ideas of voltage and current by thinking about certain mysterious entities known as fields The term field has a vernacular meaning region of influence and this is the route by which it came into the language of physics In scientific parlance however a field is more rigorously defined as a quantity which can take on different values and possibly also different directions of maximum action at different points in space and time The geometric field idea can be used say to describe the 3D temperature gradient around a hot object or the average velocities of molecules in a flowing liquid but the fields which are the most perplexing and which ultimately reveal the deepest secrets of the Universe are those which appear to produce action at a distance Of these so called force fields the gravitational electric and magnetic are the most familiar and of course it is the latter two which concern us here The beginning of what is loosely called modern physics can be traced to a
42. ordinary AC ammeter an device ignorant of phase It is however not identical to the magnitude because it can be negative in principle even if not usually in practice i e if for some reason the reference phase is chosen to be 180 then I I 180 I An AC ammeter must be considered to register magnitude I rather than pseudoscalar current I because swapping the connections makes no difference to the reading i e the instrument can never give a negative indication and putting a minus sign in front of each of the numbers on the scale won t help because then it will never be able to give a positive indication We can however equate the meter reading I with the reference vector I if we want to know the phase of the voltage relative to 0 A similar logic applies in the case of the relationship I V Z where on the correct assumption that the reciprocal of a vector i e 1 Z is also a vector we can identify the pseudoscalar voltage V as the reference vector against which the phase of I will be determined V V V 0 V V is of course the quantity registered by an ordinary AC voltmeter and we can equate it to V if we want to know the phase of the current relative to 0 So what we have seen here is that if one of a set of voltage or current vectors is replaced by its magnitude it becomes a reference vector pointing at 0 We may also deduce the converse which is that if a vector should happen to be pointing a
43. particular application there is really no resolution to the issue We must therefore eschew vague concepts like usefulness in favour of a definition on which everyone can agree and so it is universally accepted that bandwidth unless stated otherwise is defined in terms of what are known as the half power points i e the upper and lower frequency points at which the power delivered by the system for a constant input has fallen to half of that which is delivered at the frequency at which the maximum response occurs The half power points are chosen as we shall see because they have special mathematical significance and for simple networks at least knowledge of where they lie provides a complete definition of the frequency response function of the system Now if we call the power delivered at the frequency of maximum response Pmax then the power delivered at the half power points is P Prnax 2 and P Pimax 2 We can express this ratio in deciBels using the general definition Ratio in dB 10Logio P Prer where Pret is the reference power level against which power P is being compared l e 10Logi0 4 3 010299957 Hence the half power points are also known as the 3dB points and the frequency interval between the lower point and the upper point is also often called the 3dB bandwidth It is a good idea to be specific in this way because when the term bandwidth is used without qualification there is alwa
44. phasors and thus armed we are in a position to attack the parallel impedance problem 39 14 Impedances in Parallel If Z Rit jX and Z2 R2 jX2 what is the impedance Z R jX which results from placing Z in parallel with Z Z Zp Rit jX Ro jX Z Zi Zp Ri Ro j Xi Xp RR X1X gt j RX F XiR2 L Ri R2 j Xi X Now multiply numerator and denominator by the complex conjugate of the denominator RiRo XX j RiX2 XiR2 Ri Ro j Xi X2 J CRi Ro X1 Xa TE CRi Ro j Xi X and multiply out the terms in the denominator to show that it is now real RiRo Xi X2 j RiX2 XiR2 Ri Re j Xi X J L RERY E X X F The terms in the numerator are now multiplied out and rearranged so as to separate the real and imaginary parts i e the numerator is put into the form a jb as follows RiRo X1X2 RitR2 RiX2tXiRa X X2 j R X2 XR R R2 R R2 X1X X X2 L Ri R2 Xi X F Simplification of this expression involves multiplying out the brackets and crossing out any pairs of terms which are equal and opposite R R gt R R2 R X X1 R2 j Ri X2 X R X 2X X 1X22 Ri R X1 X F Which leaves us with R R RPR HRX RoX j R X Ry X1 X 7X2 X Xi J L Ri Ro Xi X Y This solution can be written in various ways depending on preference e g 40
45. provides an introduction to the subject of AC circuit analysis with particular emphasis on radio frequency applications It was developed as part of a collection of writings on the subject of radio frequency impedance matching and measurement which were first made available via the Internet in 2005 and were grouped under the working title From Transmitter to Antenna Its purpose was and still is to widen the audience for the other articles by providing essential background material but it can just as well be read by those who have a more general interest The approach adopted is that of starting with the basic laws of DC electricity and expanding them to deal with AC The modified laws are then used to derive and explore results which are normally accepted without proof thereby explaining the origins of various standard formulae and demonstrating the general method by which linear circuit design equations are obtained The level of treatment is one which does not demand a high level of mathematical skill at the outset because the required techniques are introduced as the narrative progresses Hence the discussion should be accessible to anyone who has some knowledge of basic algebra and is reasonably familiar with circuit diagrams and electrical terminology Apart from providing a conventional introduction to AC theory however there is also a subtext This relates to the author s concerns as a scientist one being that there appears to be an al
46. signs of the circuit parameters The outcome is an internally consistent theory of circuits which produces results which are correct in both magnitude and phase In this way we eliminate the need for the so called physical considerations traditionally used to resolve ambiguities and we discover which of the two commonly used definitions of admittance is actually the correct one Some of course will ask why should we bother to learn phasor analysis when we can model circuits using SPICE In fact the use of SPICE is highly recommended but simulation is essentially a way of checking existing design work It does not offer a systematic approach to the business of optimising circuits or inventing new ones The techniques discussed here on the other hand allow the equations describing the behaviour of a circuit to be written down explicitly We might then for example separate out the terms describing unwanted behaviour with a view to making alterations which will eliminate them Such is the basis for the development of precision measuring instruments and all manner of other high performance circuitry David Knight June 2012 1 Field electricity When AC theory is introduced and especially when there is a bias towards radio frequencies the very first new idea required by many people at least is a correct understanding of the word electricity The teaching of basic science often involves what are known as lies to children and the
47. solution for the input impedance of a port i e it is not possible to find a soluble algebraic expression and the computation involves so called numerical methods Hence the existence of an active element in a port cannot always be foreseen It follows that when writing computer programs for network analysis it is important to code for the possibility of negative resistance rather than terminating with an error message when it occurs or worse still ignoring the sign There is nothing intrinsically wrong with it it is just an unconventional way of defining an active network and it should not be taken to constitute a fault or inconsistency in the mathematics 9 RMS watt or not Lawrence Woolf Electronics World Dec 1998 p1043 1045 Why Vrms x IrMs is not RMS power 35 12 Complex Numbers Although the graphical phasor diagram approach outlined in the previous sections is suitable for problems involving phasor addition and scaling i e series networks it is somewhat less tractable for solving problems involving phasor multiplication and division one of particular importance being that of how to analyse networks involving impedances in parallel In section 3 we derived an expression for resistances in parallel and also by inference an expression for reactances in parallel i e R R R2 Ri R2 X X X Xi X3 It should come as no surprise that if we repeat the exercise with impedances instead of reactances or
48. the hyperbolic sine Sinh pronounced shine which is defined as Sinh x e e 2 Hence 1 P Po 33 3 1 2 Qo Sinh x xo P The function e can be expanded as an infinite series 2 3 41 5 where an exclamation mark indicates a factorial number the factorials being defined as Factorial O I 2 3 4 n n 1 Value 1 1 2x1 3x2x1 4x3x2x1 n m 1 n 2 x x n 1 xn It follows that the series for e is 2 3 4 5 and that by subtracting one series from the other we can obtain a series for e e 2Sinh x 2x3 2x5 2x 2x 2Sinh x 2x f f FS chs wake at 3 5 7 9 or x x x x Sinh x x 4 Ping Grace aes 3 5 7 9 Now notice that when the magnitude of x is somewhat less than 1 the magnitudes of the terms in which x is raised to a high power become very small and so we can make the approximation Sinh x x when x lt 1 and Sinh x x when x lt lt 1 means approximately equal to lt lt means much less than Substituting this into equation 33 3 we get 1 P Po 1 2 Qo x Xo which is a Lorentzian with w 1 2Qo Hence the electrical resonance curve is Lorentzian when X Xo lt lt 1 The electrical resonance curve is of course an electromagnetic resonance curve and like any spectral line is Lorentzian when the Q of the resonance is reasonably large 78 34 Maximum power transfer So far we have consi
49. the load power and the power wasted in the generator as a measure of the power transmission efficiency We can define efficiency as Transmission efficiency Power delivered Total power generated and here we will give it the symbol n Greek lower case eta Thus n P P P Now substituting the definitions of P 34 1 and P 34 2 into this expression we get n WR WR V Rg l3 i e n R R Rg Shown plotted below for comparison are P the power delivered to the load P the power dissipated as heat in the generator P P the total power generated and n the ratio of power delivered to power generated Power Transfer efficiency 0 Rg 2Rg 3Rg 4Rg Load resistance The tabulated results below show the various power levels as a proportion of the maximum deliverable power Pmax and are applicable to any power factor corrected generator load system 81 Load Total Power Power Loss Load Power Load power Efficiency R Rg P Px Pmax Pe Pmax P Pmax dB R R Rg 0 oniinn 4Q0 gassasvessmuvsenets 4 00 0 00 00 0 00 1 16 aseaseasesassaseans 3760 animen 3 54 0 22 6 55 0 06 TS aaseasessastenaive S00 sepmana 3 16 0 40 4 03 0 11 VA aati 3201 dps 2 56 0 64 1 94 0 20 V2 namens LOI RE 1 78 ap 0 89 0 51 s033 O 2 00 oaen 1 00 1 00 0 00 sais 0 50 an nn fr 1 33 0 44 ave 0 89 S51 sesse 0 67 4 vee 0 80 0 16 0 64 S194 onai 0 80 8 0 44 0 05 0 40 4 03 cisi 0 89 16 0
50. to an alternative expression i e XR XX XR XX Tang RRE Xc Re Riv X becomes XiLRC L C Xc XcR L C Xi Tang Ri Re Xc RR X17 which rearranges to Xi Re L C Xc Rx L C Tang 22 1 Ri Rc Xe Re Ri X17 Which since the L C ratio is a fixed parameter for the resonant circuit somewhat simplifies calculation We will now use the expression above to evaluate the effect of resistance in a fairly representative parallel resonator For this example we will use an inductance of 1H and a capacitance of 100pF This combination gives an L C ratio of 100000 and hence a critical resistance Ro W L C 1002 The ideal resonant frequency i e the resonant frequency when R Rc is fos 1 2nV L C 15 9154943 1MHz i e 21fo 100M radians sec and at this frequency Xi Xc V L C 100Q Shown below is a set of graphs of the I V phase relationship for our example resonator with various values of Rc and R between 1Q and V L C These graphs were produced using the Open Office Calc spreadsheet program available free from OpenOffice org the procedure being to create columns for frequency X and Xc and use the calculated reactance values in the Arctangent inverse tangent of equation 22 1 given above Note that spreadsheets often give the results of inverse trigonometric functions in radians and so it is necessary to multiply the expression by 180 m 57 29
51. total reactance This also means that if the source impedance is purely resistive then maximum power transfer occurs when the magnitude of the load impedance is equal to the source resistance Observe also that when the unity power factor condition X Xg is satisfied the X X term disappears and maximum power transfer occurs when R R Thus the overall maximum power transfer condition occurs when R R and X Xg i e when Z Z The condition obtained when the load impedance is the complex conjugate of the source impedance is known as a conjugate match 80 We can address the most common misconception regarding impedance Py matching by stating that although unity power factor X Xg is always desirable is it is not necessary and not always desirable that the load resistance should be equal to the source resistance The reason can be understood by considering the poor generator which must dissipate power A in its internal resistance and will therefore get hot If we assume that power factor correction will normally be carried out then there is no need to consider the reactances in the system and we can analyse the power dissipated in the generator using the case where both the source impedance and the load are purely resistive Thus P PR where the current is as defined earlier I V R Rg Hence Pe V Rg R Rg 34 2 We will plot this function shortly but when doing so it will be interesting to use the comparison between
52. two equations Zo Zi Vo Vo Vo Zi CVel Ve 1 waeess 36 3 From equation 35 2 given above Vo V Z Z2 Zp and by considering Z as part of the potential divider itself V yV Z Zu Z2 Zp Hence Vo Vo Zi Zo Zi Z Za 1 Z 1 29 1 22 J 0 2 1 Z2 J 1 1 Z Z 1 Z2 J V Ve 1 T Zi Z2 Zu Substituting this into 36 3 gives Vo Zi 1 Z Z Z 1 L e Zo Z Z2 The output impedance of a potential divider is the parallel combination of the component impedances Note that the output impedance of the main generator is part of Z2 If however as is often the case this output impedance is small in comparison to the total Z then it can be neglected 84 37 Th venin s Theorem The current in any impedance Z connected to a network consisting of any number of impedances and generators is the same as though the impedance were connected to a single generator having an output impedance equal to the impedance seen looking back into the network when all generators are replaced by their output impedances and an output voltage equal to the voltage which appears at the terminals when Z is disconnected Th venin s theorem pronounced Tae ven in arises from the observation that the output impedance of a generator is effectively in parallel with its load and so the output impedance of an active network is its impedance when all of its generator
53. usually higher than the output impedance and attempting to provide such an amplifier with a conjugate match will result in excessive internal dissipation overheating and possibly catastrophic failure Fortunately most modern amplifiers are provided with protection circuitry to prevent over dissipation and this circuitry gives the transmitter a loading characteristic which makes it appear that the source resistance is higher that it really is This loading characteristic will be different from the power transfer curve derived above because it is caused by the action of non linear circuit elements level detectors etc and so the load resistance which corresponds to the middle of the permitted operation window is known as the pseudo output impedance or if you like the pseudo source resistance The diagram below shows what the power transfer curve might look like with the operation window centred on twice the source resistance unprotected transfer function shown dotted 82 Load resistance R 0 Rg 2 Rg 3 Rg 4 Rg Notice that the protection circuitry also operates when the load resistance is higher than the preferred value This is not usually necessary for the protection push pull transistor power amplifiers the most common type of output stage in modern practice but it helps to ensure that any harmonic suppression filter after the amplifier will function correctly and it occurs because the load impedance is traditionally detected u
54. when connected to a generator of alternating voltage but insofar as we can construct them without inadvertently including resistance they consume no power The reason why the ideal versions of these components cannot dissipate energy is that they have no resistance by definition i e they cannot convert energy into heat or work Instead over the course of a generator cycle the amount of energy which flows into the component is exactly equal to the amount which 16 flows out This property forces a 90 phase difference between the voltage and current waveforms a 4 cycle or quadrature offset being that which causes the Poynting vector to reverse its direction four times per cycle Hence although the average or steady state power consumption is zero the instantaneous power flow is alternating at twice the generator frequency In the case of an inductor the AC resistance or reactance X measured in Ohms is directly proportional to the inductance in Henrys and to the frequency f in Hertz i e cycles per second of the applied voltage The quantity 2af is known as the angular frequency i e the frequency in radians per second where 27 radians corresponds to 360 and is often given the symbol Greek lower case omega In the case of a capacitor the reactance Xc is inversely proportional to the angular frequency and also inversely proportional to the capacitance in Farads Note also that capacitive reactance is shown as bein
55. 1 364 22 730 0 571 0571 29 600 75 800 1 364 23 054 0 248 0 248 0 2526 29 600 74 400 1 364 23 559 0 258 0 258 29 600 75 100 1 404 26 775 3 474 3 474 3 3590 29 600 75 100 1 324 20 057 3 244 3 244 Esd gt gt 3 4179 Note that the formula is somewhat non linear in its behaviour because the deviations caused by incrementing and decrementing a variable are not exactly equal and opposite The correct way to allow for this effect is to take the average of the deviation magnitude RMS for each case Therefore the estimated standard deviation in Rg is o V 0 579 0 253 3 359 3 418 Hence R 23 3 43 40 Notice that the major contributor to the uncertainty in Rg in this case is the uncertainty in Ny We cannot ignore the effect of the resistance uncertainties however because if we repeat the experiment with more closely spaced values for R and R we will find that their contributions to the uncertainty increase dramatically Analytical approach to error analysis While the error analysis technique just described is perfectly respectable those who write computer programs will generally prefer an analytical approach The derivation of an error function from a formula requires the use of calculus Those who are unfamiliar with calculus may proceed to the next section without losing track of the narrative The analytical form of an error function is obtained from the observation that an error in a variable is transmitted th
56. 577951 to get the result in degrees there are 27 radians in 360 i e Q 57 2957795 1 Arctan Xi Rc L C Xc Rv2 L C RuREe Xc RA Ri2 X17 The plotted curves below were created using the spreadsheet chart tool see accompanying spreadsheet file par_res_ph ods 54 90 60 4 Capacitive Current Phase f fos Effect of resistance on the I V phase relationship of a parallel resonator L C 10000 Q Of the curves shown only the example with R 1 and Rc 1 constitutes a good healthy resonance The choice of 1Q in each of the branches incidentally was made simply so that the resonant frequency would coincide with fos Any curve with at total resistance Rp Rc 2Q will have an almost identical appearance The Q of the resonance as will be explained later is 50 in this case i e Qo X1 Ri Rc which is fairly high and so the phase of the current lags the voltage by nearly 90 at frequencies a few percent below the resonant frequency and leads it by nearly 90 at frequencies a few percent above Hence the circuit provides the generator with a nearly pure inductive load below resonance and a nearly pure capacitive load above In the case where R 50 and R 50 the Q of the resonance is 1 A large resistive component is present in the impedance at all frequencies and so the I V phase difference never approaches 90 in either direction The curves for R 50 and Rc 1 and R 1 and Rc 50 are included t
57. AC electrical theory An introduction to phasors impedance and admittance with emphasis on radio frequencies By David W Knight Version 0 11 5 June 2012 D W Knight 2005 2012 Ottery St Mary Devon England Please check the author s website to ensure that you have the most recent versions of this article and its associated documents http www g3ynh info Table of Contents PECL ACC sco texca rice ena n R 2 1 Field SlCr City cisscissccaesstecsnpssiceteaiiocassecaeacs 3 2 Circuit analysis overview 10 3 Basic electrical Torn ae siccsssccossscazecssxneniss 14 4 PCC SOMAMOE 4 ocicreaccnadeeseccuateanonccnsuesceaveastexeees 17 5 Impedance Resistance Reactance 19 6 Vectors amp ScalarS ccsccssccssscssetsssseesenes 20 7 Balanced Vector EquationS c00c06 23 B Pha SOTS ciani sae 25 9 Voltage Magnification amp Q eee 28 10 Power Factor amp Scalar Product 32 11 Phasor dot ProduCtiscdscscsticepeivdealconions 34 12 Complex Numbet s c cccccceseeeeseees 35 13 Complex arithimetie scsisesiscpssiseiseasttneseasciza 38 14 Impedances in Parallel eee 39 15 Parallel PESOMANCE sscicseconnesssccsesneacsoeatsenates 41 16 Dynamic RESIStAnCe c dsaicsiccceinaenecsvncens 43 17 Double slash notation ccceeeeee 44 18 Parallel to Series transformation 46 19 Series to Parallel transformation 47 20 Parallel resonator i
58. MHz the coil has a loss resistance of 2Q and the capacitor has an equivalent series resistance ESR of 0 1Q Thus L C 71428 57 Ri 4 and Rc 0 01 Hence the correction factor is V 71428 57 4 71428 57 0 01 0 99994414 The precise resonant frequency to the nearest 1Hz is therefore 0 999944 14 178649 14 178253MHz The quantity L C is called the L C Ratio of the tuned circuit and it has units of Ohms squared Note that 43 L C XL Xc X Xel It will turn out that the L C ratio is an important parameter of resonant circuits Also there is some precedent for referring to the square root of the L C ratio as the characteristic resistance of the tuned circuit by analogy with the characteristic impedance of a lossless transmission line which is Ro V L C where L is inductance per unit of length and C is capacitance per unit of length but the lengths cancel and so the characteristic resistance of an ideal transmission line is the square root of its L C ratio In the example given above the resonant frequency differed from the ideal case by only 0 0028 or 396Hz the reason being that the L C ratio was very large in comparison to the squares of the loss resistances In HF radio applications the L C ratios of tuned circuits are generally in the order of several tens of thousands of Q whereas the value tolerances of radio components are seldom better than 1 and often considerably worse In order to obtain an exact resonant frequ
59. MHz this figure is 1900 2 7 704 It is extremely difficult to make a lumped inductor with a Q of greater than about 400 so on 160m the Q limit can be avoided by controlling the length of the coil If on the other hand the whip is of length comparable to or shorter than the coil then most of the radiation occurs from the coil In that case the lumped component description fails completely and the system is best described as a quarter wave transmission line resonator In the transmission line regime the Q and hence the voltage magnification can become enormous and the usable input power is limited by the tendency for the air around the top of the coil to ionise and become electrically conductive Coils operated at or slightly below the quarter wave transmission line resonance frequency are used for artificial lightning experiments in which context they are known as Tesla coils In particular the voltage magnifier coil connected in series with the output from a step up transformer is known as the Extra Coil The transmission line properties of coils will be discussed in a separate article While on the subject of MF and HF mobile antennas when forced to use a very short whip it is possible to increase the antenna capacitance artificially and hence reduce the reactance by adding a capacitance hat to the antenna some prongs sticking out sideways symmetrically or if there s a risk that you might poke someone s eye out an aluminium disk R
60. R2 2nfL 1 2nfC P 29 1 and since P Po I Io P Po R R 2afL 1 2nfC 29 2 Graphs of both of these functions are shown below the procedure used for generating them being to choose a value for fo and an L C ratio and then calculate using the Open Office Calc spreadsheet program a set of points at closely spaced intervals for various different values of Qo see accompanying file ser_res ods The initial choices are arbitrary since as we are about to show the shape of the curve obtained depends entirely on Qo In this case the author chose fo 10MHz and L C 10 i e C L 10 Values for L and C were then obtained by solving the resonance formula for L i e 107 1 2nV LC 1 2nV L2 10 100 2aL L 100 22x 107 1 59154943 WH C L 10 159 154943pF Since V L C 1009Q is also the value of X and Xc at resonance resonant Q values of 100 10 and 1 correspond to total resistances R of 1Q 10Q and 100Q respectively Notice in the graphs below how the squaring pushes the curve of P Py downwards in comparison to I Ip Notice also that the half power level is 1 V2 0 7071 for I Io and for P Po and that the deciBel scales on the right differ accordingly 71 1 0 5 20Log I Io 0 0 9 08 he ieee aie Semi oC nena sian casi 3 0 6 0 5 6 i Series Resonator i Current vs frequency AD 0 2 E 15 0 1 20 30 0 T T T T T T T T 0 0 2 0 4 0 6 0 8 1 1 2 1 4 fifo 16 18 2
61. V and Vo by treating the circuit as a potential divider thus noting that Xc Xp gt at resonance i e when f fo we get Vo Vg Re Rst Rp and if we choose the generator voltage as our phase reference we can drop dimensions Rp QQQQ 24 Radio Frequency Measurements by Bridge and Resonance Methods L Hartshorn Principal Scientific Officer British National Physical Laboratory Chapman amp Hall 1940 Vol X of Monographs on Electrical Engineering ed H P Young 3rd imp 1942 Ch I section 3 Defines Admittance as Y G jB hence BL 1 a L and BC aC 25 Radio Designer s Handbook Ed Fritz Langford Smith 4th edition 4th impression with addenda Iliffe Publ 1957 A later reprint exists 1967 ISBN 0 7506 36351 Section 4 6 v p153 Defines inductive susceptance as positive and capacitive susceptance as negative hence Y G jB This is contrary to the more convincing derivation given by Hartshorn above 102 Vo Vg Rp Rs Rp We will also avail ourselves of a useful property of the potential divider formula 35 4 which is that if we multiply it by a unit quantity consisting of the source resistance divided by itself i e Rs Rs it becomes a double slash product Vo Ve Re Rs Rs Similarly for the output voltage in general V V Re jXcp jXtp Rs Re jXcp jX and using the associative rule 17 4 y Ve Rs Rp jXcp jX1 Rs So we can write the ratio V
62. X R2 X1 which allows us to extract the L C ratio Io V VE L C R2 X2 RX 47 2 Now let us define the unloaded Q of the resonator as the ratio of the circulating current to the through current Qou Io To Which can be expanded using equations 47 1 and 47 2 L C Qou RRP tX RRX RAKE RI Xv and rearranged Rc Xc Ri X_17 Qos V L C Rc Ry X17 Ri Rc Xe F The rightmost square root bracket is simply the reciprocal of R as defined in equation 46 4 hence Qou V L C R and we have proved that the current magnification definition for unloaded Q is identical to that obtained on the assumption that Q is the magnitude of the resonant frequency divided by the bandwidth of the resonator 108 The only residual issue is that of why the exact expression for R is as given by equation 46 5 R Ri V XdXi Re V XU Xo rather than simply R R Rc This however can be understood by noting that the real current flowing through the resonator will be very slightly biased in favour of the branch with the lowest resistance This difference is very small for practical resonators of moderate unloaded Q and may normally be ignored 48 Controlling loaded Q As determined earlier 45 1 the Q of a parallel resonator can be given as Qo Rsre Rpo Rroaa VW Ly C where with a slight change from the previous notation Rsre is the output resistance of
63. XYV X9 46 5 strictly R but resistance is positive allowing us to ignore the negative solution and so Qon VW L C Ru V X0 X1 Re V X X0 This is an exact solution provided that Ri and Rc do not vary and once again may be assumed exact for normal engineering purposes because R and Rc will not vary significantly in the vicinity of the resonant frequency Note however that X Xc to an extremely good approximation when the L C ratio is reasonably large and this relationship is exact when R Rc Hence for most practical purposes Qu WL C Ri Rc 46 6 Which means that the unloaded Q of the parallel resonator is the same as that of the series resonator it is the square root of the L C ratio divided by the total series resistance In other words we can estimate the unloaded Q of the parallel resonator by considering it to be a series resonator connected as a loop 47 Current magnification There is another way to determine the unloaded Q of a parallel resonator which stems from the observation that just as a series resonator exhibits the phenomenon of voltage magnification the parallel resonator exhibits current magnification In effect the parallel resonator is a series resonator connected in a different way because its characteristics at resonance are principally determined by a large circulating current and the current it draws from the generator is small in comparison Q times smaller than the cir
64. accurate technique when correctly applied but it has limitations and idiosyncrasies of which the practitioner needs to be aware A peculiarity which is often introduced without comment is that AC generators of the analytical variety are considered to produce sinusoidal outputs Many practical generators e g mechanical alternators radio transmitters do indeed produce something approximating a voltage or current sine wave but the reason goes somewhat deeper than that If we take for example a moving coil microphone which is a type of generator which produces electricity from air pressure variations we will find that its output in response to say the sound of the human voice is extremely complicated A technique known as Fourier analysis however shows that all waveforms can be built up by adding together sinusoidal waves of different frequencies and physical investigation shows that these separate frequency components actually exist A set of one or more frequency components is known as a signal It transpires that no new frequencies will be added to a signal when it is processed e g passed from the input to the output of an electrical network provided that the materials encountered in the transmission path behave in a linear manner In general a material is said to be linear when its change in response to some force is proportional to the intensity of that force i e the graph of change versus force is a straight line and the c
65. actical inductors and capacitors always have some internal resistance A measurement made across any part of the total series resistance will however always produce a voltage which is in phase with I A device which measures current by sampling the voltage across a resistance is of course an ammeter It was stated earlier that capacitive reactance is defined as a negative quantity in order to make AC theory consistent with trigonometry The convention we follow is of course that which says that the phase angle of a vector increases as it rotates in the anti clockwise direction Hence the choice of Xc as the negative reactance stems from the fact that voltage lags i e peaks 90 later 28 than current for a capacitor whereas voltage leads peaks 90 ahead of current for an inductor This can be remembered by considering what happens when a capacitor in series with a resistor is connected to a battery a large inrush of current precedes the build up of the voltage across the capacitor terminals If the capacitor is replaced by a coil the opposite happens the build up of current is delayed by a back voltage produced by the growing magnetic field There is no need for convoluted reasoning in AC theory however just remembering the sign of Xc takes care of everything Note however that many technical articles follow the hallowed tradition of treating Xc as negative in some statements and positive in others This is done as an aid to comprehension
66. ad R is disconnected It should also be obvious by inspection that no power is delivered when R 0 short circuit and also that no power is delivered when R is disconnected i e when Ro Hence we expect a peak in power output at some intermediate value of R and we can obtain this value in relation to Rg by determining the relationship between P and R and plotting it as a graph In the circuit on the right the power delivered to the load is I Rg PZPR Where 1 V R R gt P Hence P V R R R y 34 1 This function is plotted below for constant V and shows that maximum power output occurs when The result is of course well known but it is by no means the whole story and its interpretation is subject to various common misconceptions We can settle all of these issues by deriving the complete maximum power transfer condition see box below This requires the use of calculus which will not be explained here but those unfamiliar with the technique may still avail themselves of the result Power transferred to load 79 The maximum power transfer theorem Zo In the circuit shown on the right the power P delivered to the load Z is Anr P JIP R I Ro jXo where I V Z Ze Hence P VPR Z Zo VER R Rg j X X P VP R R Rg K X P There are two maximum power transfer conditions to be obtained here one being the value of load reactance and the other being the value of load resista
67. ad generator voltage 17 How Big is a Bad SWR Bob Pearson G4FHU Rad Com March 1993 p64 65 April 1993 p62 63 The greatest danger for push pull transistor amplifiers lies in low load resistance which for a given value of SWR is more pernicious than residual reactance SWR is a poor matching criterion because it does not indicate whether the magnitude of the load impedance is too high harmless or too low harmful 83 If the impedances are pure resistances the formula above reverts to Vout Vin Ri Ri Ro 35 3 where R1 is the resistance across which Vout is said to appear Alternatively multiplying by R2 R2 Vout Vin Ri R R2 35 4 36 Output impedance of potential divider The output impedance of a network is defined as that impedance which when placed in series with a hypothetical perfect generator accounts for the drop in output voltage which occurs when a load is connected Shown below is a representation of a potential divider network loaded with an impedance Z Ifthe load is removed the output voltage is Vo but when the load is connected the output drops to a new voltage V the single inverted comma is pronounced prime This situation is modelled on the right as a perfect generator with an output V in series with an impedance Z the latter being the output impedance we wish to define Using the definitions given in the diagram Zo Vo Vo i where IL V Zu Hence combining these
68. alculated when necessary but more to the point is just a resistance with an obvious definition While straightforward however the use of the notation involves a subtlety which lies in the distinction between physical and mathematical objects In describing a test procedure for example we might put an entry in a table Test load 68Q 100pF The item 68Q 100pF is a physical object a capacitor in parallel with a resistor but it is not a complete mathematical statement of impedance and cannot be treated as an impedance in any calculation In order to turn the parallel combination into a mathematical object we must ensure that the quantities on either side of the symbol are of the same type and that they are expressed in the same units In this case we can fix 45 the problem by noting that if the report is to have any useful meaning a test frequency must be stated somewhere If that frequency is say 14MHz then the reactance of the capacitor becomes 1 2nfC 113 7Q and its impedance assuming that losses are negligible is 0 j113 7Q Hence we can re state the test load as 68 j114 Q This is the same as saying 68 j0 0 j114 Q and is of course a complete statement of the load impedance in the form Z Z which can be converted into the R jX form if so desired A particular logic emerges from these observations and it is important to be aware of it 17 1 A resistance is an impedance Resistances and impedances a
69. an the magnitude of either IX or IXc The relationship between the phase angle obtained from a phasor diagram and the waveforms which can be observed using a two channel oscilloscope is shown below where gt means greater than and lt means less than 2T Inductive X Xc gt 0 Capacitive XL Xc lt 0 V leads I I leads V Here we have obtained a waveform which is exactly in phase with the current by measuring the voltage across the resistive component bottom trace When this is compared against the waveform of the total voltage V using the upward zero crossing as an arbitrary reference point we Resonant XL Xc 0 find that V is advanced in time i e leading relative to I when the impedance is inductive X X0 and V is retarded lagging relative to I when the impedance is capacitive X1 Xc lt 0 If we call the time difference observed on the oscilloscope At where A is upper case Delta a symbol normally used to mean the difference in then the ratio of At to the time of a complete cycle is the same as the ratio of the phase angle to a complete circle The time per cycle also known as the period of the waveform is of course the reciprocal of the frequency 1 f hence if is measured in radians At 1 f o 27 i e At 9 2zf Note incidentally that it is impossible neglecting the use of superconductors to make a series LCR network from which all of the resistance can be isolated because pr
70. annot be controlled This situation prevails because in order to use the resonator as a filter we need methods for injecting energy into it and extracting energy from it and the impedances presented by these input and output networks affect the Q The prototype band pass filter is shown on the right The Rs generator and load coupling scheme used is not the only one possible but all other schemes are equivalent to this one after suitable transformation Here we inject energy via a source A resistance Rs which is the sum of the generator output resistance and any additional resistance placed in series with it Rp is the parallel combination of the resonator dynamic resistance and any load resistance which might be placed across it Notice that we have provided the model with source and load resistances rather than impedances We are at liberty to do so without affecting the generality of the analysis because any reactive components in the source and load impedances will turn out to be effectively in parallel with the resonator This means that these additional reactances will modify the effective values of Xcp and Xz i e they will change the resonant frequency but they will not affect the general circuit behaviour provided that they do not exhibit any self resonances in the analysis frequency range If we define Vo as the output voltage at resonance then the bandwidth function is V Vo plotted against frequency We can write expressions for
71. ant regardless of any motion on the part of the observer This led to the Special and General Theories of Relativity which overturned all 19th Century notions of space and time He also gave us the explicit unification of electricity and magnetism by showing that electromagnetic induction is a relativistic phenomenon Most readers will be aware that an electro mechanical generator works by moving a coil of wire relative to a strong magnetic field The changing magnetic field as seen from the coil s viewpoint gives rise to an electric field which manifests itself as a voltage across the ends of the coil Einstein tells us that the magnetic field does not so much create an electric field it is an electric field when seen from a moving frame of reference Likewise an electric field is a magnetic field when viewed by a moving observer This means that generators and by a converse principle electric motors make use of relativistic effects when they convert energy between its electrical and mechanical forms Heaviside s extended version of Maxwell s equations was background to the work of Paul Dirac who later went on to predict the existence of anti matter Heaviside s most important work however was carried out before the advent of radio as a technology and was primarily related to the problems of long distance electrical communication telegraphs and telephones It is Heaviside who gave us the correct picture of electricity by way of another corollar
72. ard 69 16942 2 2 Sample mean and standard deviation Area under the Gaussian distribution Table C 2 p308 87 This process can be extended to find the magnitude of a vector in an arbitrary number of dimensions we can t make perspective drawings in more than three dimensions but there is no restriction on the number of dimensions that a vector can have Hence U U2 U2 U3 0U 2 Now note that this formula says to find the overall uncertainty calculate the sum of the squares of the uncertainty contributions and take the square root The uncertainty contributions are not the same as the uncertainties in the measurements made Imagine that an unknown quantity x is given by a formula f which is a mathematical function involving measurable quantities variables m m2 m3 etc We can express this situation by writing X fm m Ms ical and we can determine x by plugging m m2 m3 etc into the formula We can also determine the uncertainty contribution due to any one of the variables by changing it and noting the change which occurs in x The obvious amount by which to change the variable is its standard deviation hence X 0x f mitoi M2 M3 a Here we have assumed that a positive change in m will cause a positive change in x This might not be the case but since we intend to add the contributions from changes in each of the variables as orthogonal vectors it makes no difference either way Now restoring m to its or
73. ared to a reference energy per unit of time and since the unit of time s will cancel provided that they are the same seconds are very popular a power ratio is also an energy ratio Hence N dB 10Logi0 E Ever Newton s laws of motion tell us that the kinetic energy of a moving body is given by E mv 2 where m is the mass and v is the velocity so energy is proportional to velocity squared as well as to voltage squared and current squared Hence a speed in dBmph is given by 20Log v so 30mph becomes 29 5dBmph and 70mph becomes 36 9dBmph 69 Now having upgraded all of our road signs to be in keeping with the preferred notation for Government standards documents we are only left with the problem of how to measure money in deciBels Here we may note that currency names are often derived from weights of silver but there has been some devaluation since Roman times and that Newton s and Einstein s laws tell us that mass is proportional to energy Thus we can deduce that the 10Log formula is the correct one in this case We might have solved this conundrum without recourse to physics however by recalling the famous old saying money is power 29 Bandwidth of a series resonator When used to make filters for the RF and IF amplifiers of radio receivers high Q resonant circuits provide better selectivity than low Q resonant circuits Since good selectivity is synonymous with narrow bandwidth there is evidently a relationship
74. ative to the true power P PR and the magnitude of the apparent power is the diagonal of the phasor diagram shown below From this we can determine a correction factor for the I V power formula particularly by observing that the cosine adjacent hypotenuse of the phase angle is P VIj i e P V I Coso or after factoring out the pseudoscalar I P V I Cos or since I I P V I Cos Notice that Cos is zero for p 90 no power is delivered to a pure reactance and Cos 1 for 0 real and apparent power are the same for a pure resistance Be aware also that the formula above appears in standard textbooks as 33 P V I Cos but unfortunately there is nothing we can do to salvage this traditional version We will prove later that the un bold symbols V and I when used in an equation must be interpreted as phasors pointing at 0 or 180 because that is the only way in which we can incorporate DC and AC into the same theory V and I however can only point in the same direction when 0 The standard formula is therefore internally inconsistent a mathematical oxymoron The best that can be said for it is that there is little choice but to assume V and I to be magnitudes in this instance since the expression is nonsense otherwise The quantity V I Cos is known as the scalar product or dot product of the two vectors and is defined in the same way for all vectors regardless of th
75. aving the latter dimensions i e a voltage divided by a current may legitimately replace a quantity measured in Ohms Similarly the reactance laws X 2nfL and Xc 1 2zfC tell us that Ohms can also be replaced with Henrys x radians second or by 1 Farads x radians second Thus we should not be confused by structures such as Z R j 2nfL 1 2nfC The bracket after the j is internally consistent and represents a quantity measured in Ohms 15 Parallel resonance In an earlier section we said that it is not possible to calculate the exact resonant frequency of a parallel tuned circuit nor the impedance which it presents at resonance without taking the resistances of the coil and capacitor into account Now of course having derived a general equation for impedances in parallel we Re RL are in a position to rectify that omission The network we need to analyse is shown on the right where Rc is the so called equivalent series resistance ESR of the capacitor and R is the loss resistance of the coil which we previously defined as xX XL R X1 Q here Q is given the subscript L to indicate that it is the Q of the coil not the overall Q of the tuned circuit For the purposes of this discussion we will assume that both Rc and Ry are predominantly due to the RF resistance of the wires and other conducting materials used to make the components and for reasons which are explained in subsequent articles are considerab
76. aw to include DC We do this by noting that if a circuit has any series capacitive reactance then as the frequency goes to zero Xc goes to infinity hence the magnitude of the impedance goes to infinity and the impedance becomes an open circuit The only type of reactance which gives DC continuity is of course inductive reactance and at f 0 X 0 Thus Z R X drops a dimension and becomes Z R 0 R Hence we can write V IR for DC or for pure resistance and AC but since both V and I are then in phase they can drop dimensions also Thus we obtain V IR if we drop dimensions at 0 but more to the point we are also at liberty to drop dimensions at p 180 and obtain the perfectly valid alternative CM COR i e we have a theory which covers all aspects of AC electricity and also allows us to have the negative voltages and currents required for the analysis of DC circuits This is why we must insist that the un bold symbols V and I are not magnitudes they are pseudoscalars or if you prefer complex numbers in the form a j0 which can point in either a positive or a negative direction It is only resistance which can never be negative in a passive network and that is for physical rather than for mathematical reasons An additional interpretation of Ohm s law is also given to us by the magnitude ratio and product theorems 24 1 24 3 These allow that if V IZ then V E Z i Z and all possible rearrangements This says that the need
77. be extended to cover any number of impedances in parallel by adding more terms i e 1 Z 1 2 1 22 1 Z3 1 Z This is a sum of admittances and may be re written as Y Y Y2 Y 3 ye a N Y We can express this result using the double slash notation WC Zi Zo Z fl I Za Yi Y2 Y Y where Yx 1 Zx k being any subscript The admittance representation of an electrical circuit is no less authoritative than the impedance representation and is no more difficult to use Admittances are phasors and all of the phasor techniques we have developed in this chapter will work on them It is however helpful to remember that a numerically large admittance corresponds to a small impedance and vice versa Reciprocal space counterparts Impedance space Admittance space Impedance Z R jX 1 Y Admittance Y G jB 1 Z Resistance R G G B Conductance G R R X7 Reactance X B G B Susceptance B X R X Pure resistance R 1 G Pure conductance G 1 R Pure reactance X 1 B Pure susceptance B 1 X Inductive reactance X 2fL Inductive susceptance B 1 27fL Capacitive reactance Xc 1 2fC Capacitive susceptance Bc 2afC operator operator operator operator Straight line Circle Circle Straight line 101 One further issue of which the reader will need to be aware is that two different definitions of admittance appear in the electrical and electronic literature
78. between bandwidth and Q and as we shall see this relationship is a particularly simple one if we define the bandwidth as the interval between the half power 3 01dB points The question we must address next therefore is what is this power to which we must relate the bandwidth The answer in the case of an amplifier driving a resistive load is obvious it is the power dissipated in the load In the case of a resonant circuit however there may or may not be a load in the normal sense and we are left with the uncomfortable conclusion that bandwidth must be defined in terms of the power which would be dissipated in the load if there were a load We can crack this riddle for the series resonant case by considering the circuit shown below This is the circuit of a simple band pass filter It L C has an input voltage an output voltage and a load I Re Rc resistance and the bandwidth is very clearly the frequency interval between the points where the Vin Vout Road power in the load is half of that which occurs at the frequency of maximum response for constant Vin It is also however a series resonant circuit and the Q at resonance can be defined as Qo Xo_ R Xoc R where R is the total resistance i e R Rr Rct Rioaa and the subscript 0 has been added to the reactances as a reminder that Qo is defined in terms of their values at the resonant frequency fo Now we can easily write an expression for the current which flows from the g
79. but the real power of the transformation lies in the fact that phasor problems requiring the double slash product in one space become problems of addition in the other 23 Impedance matching D W Knight www g3ynh info 99 Converting an impedance into an admittance is simply a matter of taking the reciprocal Admittance is usually given the symbol Y and here we put it in bold because it is complex hence Y 1 Z Now if Z R jX this gives Y 1 R jX which can be put into the a jb form by multiplying the numerator and denominator by the complex conjugate of the denominator i e R jx Y I R JX R jX hence R jx Y R X This expression can be written Y G jB where the real part of the admittance G is called the conductance and the imaginary part B is called the susceptance of the network under consideration From the above we can extract definitions for conductance and susceptance which are Conductance G R R X Susceptance B X R X Now observe that when the impedance of a network is purely resistive the conductance is 1 R and G 1 R is the definition of conductance in DC electrical theory When an impedance is purely reactive the susceptance B 1 X susceptance has no DC counterpart Admittance conductance and susceptance of course have units and the modern unit in this case is the Siemens which is given the dimension symbol capital S as opposed to the second which
80. carries the coils and enables them to be connected to the active circuit via very short leads see below mn iid te l 7 Motorised bandchanger turret from 1958 vintage Marconi AD307 aviation transmitter In the matter of making a resistance tuned parallel resonator therefore our rough calculations and observations of what others have been able to achieve in practice seem to indicate an approximately 2 1 rule of thumb for the upper limit of the frequency range What this means in this instance however is that we should not try to push the resonator to much more than about twice its ideal case resonant frequency so if we consider our example 1H in parallel with 100pF resonator which resonates at about 16MHz when the resistance in both branches is equal we might reasonably expect to be able to tune it from 0 32MHz This although not infinite is nevertheless a phenomenal tuning range but unfortunately there is a catch The problem is that if we use a variable resistor of value equal to V L C the Q of the circuit will be approximately 1 We will investigate the relationship between Q and bandwidth shortly but we can pre empt those findings by stating that such a circuit will be completely useless as a band bass filter such as might be used to provide a radio receiver with selectivity We might therefore consider raising the circuit Q to 10 by reducing the value of the variable resistor in our example L C 100Q resonator so t
81. cesses and note the gains in dB positive or negative for each of those processes we can find the overall gain of the system simply by adding all of the individual stage gains together So much for the basics but now we arrive at the point which causes greatest difficulty A quantity in dB implies a logarithmic power ratio It can however also be taken to represent a voltage ratio or a current ratio but the definition must be modified in that case The reason why the definition can be extended to embrace current and voltage ratios is that power is a function of the voltage across and also the current through an impedance Hence we can substitute for power using the general power laws derived earlier i e P V R R X and P I R To obtain the voltage ratio formula we write N dB 10Logio V Vret which reduces to N dB 1OLogiof V Vve and if we adopt the convention that the impedance against which the two voltages are compared is a resistance N dB 10Logio V Veer J A similar argument applies for the current ratio formula N dB 10Logio A Inet Now everything would be fine of we left the quantity inside the logarithm brackets as the square of a voltage or current ratio but everyone who teaches the subject will insist on performing a simplification which is to note that a number can be squared by doubling its logarithm Hence we get rid of the power of 2 by writing N dB 20Log10 V
82. circuit The first problem however is that the spurious self resonances of the network are of higher Q than the desired resonance and if nothing is done about them the circuit will probably oscillate at somewhere around the self resonance frequency of the inductive branch We might however decide to use an amplifier which is too slow to oscillate at VHF or implement some kind of low pass filter in order to ensure that the system has no gain in the troublesome self resonance region Thus by some artifice we might actually get the oscillator to submit to our will at which point it will deliver its final insult by producing a signal which appears to be modulated by a hissing noise This ill mannered behaviour will occur because an oscillator is effectively a generator of filtered white noise No oscillator produces a pure sine wave A practical RF oscillator always produces a band of noise centred on a selected resonance of the frequency determining network with a width determined by the network Q and the amplifier gain The usual objective in oscillator design for radio applications is therefore to obtain as high a Q as possible so that the desired output is a spike in the amplitude vs frequency domain sufficiently narrow to be regarded as a sine wave Thus the resistively tuned parallel LC resonator is of little practical appeal in situations demanding spectral purity Its significance to this discussion lies instead in the fact that the circuit idea
83. cting c from both sides so that ax bx c and then divide both sides by a so that x bx a c a eae EEL We now need to find a substitution for the term x bx a such that x is on its own We can do that by observing that x bx a looks similar to part of the expansion of a quantity in the form x p where p is just an arbitrarily chosen symbol i e xt p x 2px p cease 12 2 To use this substitution we equate the term 2px in equation 12 2 with the term bx a in equation 12 1 i e we put p b 2a and rewrite equation 12 2 thus x b 2a P x bx a b 4a which can be rearranged by subtracting b 4a from both sides to give x bx a x b 2a P b 4a Substituting this into expression 12 1 gives x b 2a P b 4a c a and adding b 4a to both sides gives x b 2a P b 4a c a We then put the terms of the right hand side onto a common denominator thus x b 2a P b 4ac 4a Now we can take the square root of both sides to get x on its own but note that when a square root 36 is taken there are two possibilities because qq is the same as q x q i e V q 4 Hence x b 2a V b 4ac 4a b 4ac 2a finally we subtract b 2a from both sides to obtain x b v b 4ac 2a 12 3 which is of course the standard school formula for solving quadratic equations The formula 12 3 look
84. ction components In the process of adjusting a reactance in series with the antenna to achieve a resistive input impedance any phase shift due to the transformer is automatically taken into account Consequently it is possible to keep the inductance small which helps in the avoidance of self resonance problems at the high end of the operating frequency range 43 Prototype Z matching network An antenna matching system based loosely on the Collins 180L 3 is depicted in the diagram below The only major difference is that the transition from step down to step up is accomplished by means of a change over relay This increases the transformation range in comparison to the overwind method but also increases the complexity of the control system Error signals This is the prototype of all antenna tuners in the sense that it approaches the impedance matching problem in the simplest possible way The object of the exercise in every case is to transform the impedance in its two dimensions magnitude and phase and the most direct approach is to do so using one device which only affects the magnitude and one device which only affects the phase The magnitude correcting engine is the variable auto transformer and the phase correcting engine is a series reactance relays being provided to insert a series coil in the event that the antenna is capacitive or a series capacitor in the event that the antenna is inductive Such a matching unit can of course
85. culating current in fact O00 In the diagram on the right the current I flowing into the A R Load resonator is I Ic where Ic V Rc jXc V Rc jXc RX and 107 I V Ry jX1 V RL JXL R 4X1 but at reasonance I is real which means that the imaginary parts of I and Ic add up to zero and the total current at resonance becomes Rc Ri Ilb V Rc Xc RX V and o are now in phase and will be treated as real Putting the expression onto a common denominator yields Ro Ri2 X1 RRX l V 47 1 ReX R X 2 where the term inside the square brackets is the reciprocal of the dynamic resistance see equation 20 5 i e Io V Rpo Now the current circulating in the resonator can be determined from either branch as the total current flowing in the branch less the current drawn from the generator Hence the circulating current is simply the imaginary part of the current in the branch The resonant condition and the concept of circulation also implies that the circulating current is of the same magnitude for both branches but of opposite sign Hence if we call the circulating current Io then taking the imaginary parts of the expressions for Ic and I above we have Io jVXi RX jVXc Rc Xc and the magnitudes are Tol VX_ Ri 2 X 7 V Xc Rc Xc We can also create a definition involving both branches by taking the geometric mean Iq V Vf XcX1 RE
86. d treat it as a scalar If we also define Z R jX1 and Zc Rct jXc then Z Zi Zc Zi Zc hence I V Zi Zc I Zi Zc Now we noted earlier that the phase angle of a complex expression a jb is given by Arctan b a so in order to obtain the I V phase difference we first write I V Z Zc Z Zc then split the right hand side of the equation into its real and imaginary parts divide the imaginary by the real and take the inverse tangent Expanding the expression above we get I Rit jXit Ret jXc V Ri jXi Re jXc and multiplying out the terms in the denominator gives I Ri Re j Xi Xo V R Rc X Xc Ra j RXc XiRc Now we multiply numerator and denominator by the complex conjugate of the denominator I Ru Re j X Xc J RRc XXe j RiXc XiRo V RiRce XXe R Xc XiRe then multiply out the numerator crossing out equal and opposite terms to get I Ri Re2 X2 R Ri2 X2 jI X R2Z XZ Xi R2 X2 V RRc XiXcY RiXc XiRc This is in the form a jb so the phase angle is given by Xi Re Xe XR Xv Tang Ri Re Xe RR X17 Now there is no need to rearrange this formula any further in order to use it but since we are analysing the phenomenon of parallel resonance it is interesting to recall that X Xc L C If we 53 multiply out the numerator we will obtain two terms which contain X Xc and this leads
87. dered generators to be sources of constant RMS voltage In reality in the absence of a control system to keep it constant the output voltage of a generator will droop as the output current is increased This means that the generator has an internal impedance which is somehow distributed throughout its wiring and component parts but which will be seen from outside as though there is a single impedance in series with an otherwise perfect generator This impedance is known as the source impedance or the generator s output impedance and must often be taken into account when carrying out circuit analysis In particular it is necessary to include the source impedance explicitly when determining the characteristics of the parallel resonator bandpass filter but there are various connotations relating to power transmission in general which must be addressed The basic matter is that of the effect that the load impedance has on the amount of power delivered to the load and is encapsulated in a set of relationships known as the maximum power transfer theorem For the special case of a generator with a purely resistive output impedance and a purely resistive load we can obtain the maximum power transfer condition using a graphical method The circuit to be considered is shown below where Rg is the generator output resistance and R is the load V is the off load generator voltage i e it is the voltage which will be seen at the generator terminals when the lo
88. e using very short wires It is assumed that the source impedance is purely resistive this being reasonable in the case of a transistor RF amplifier but very unreasonable in the case of a tuned valve tube RF amplifier In order to avoid interference from any protection circuitry the test should Rg be carried out at a low power level lt 10 of maximum output The voltmeter should be capable of measurement at the generator frequency and should have a high input resistance An oscilloscope with a high impedance probe is suitable but ordinary multimeters do not work at radio frequencies Only the voltage ratio needs to be determined accurately the absolute voltages are immaterial Let the output voltages be V when R is connected and Vz when R is connected The source and load resistances form a potential divider Hence using equation 35 3 Vi VR Rg Ri and V2 V R2 Rg R2 Rearranging both of these expressions to get V on its own and then equating them gives V Vi Rg oy R Ri V Rg R2 R2 R2Vi Rg F Ri RiV2 Rg F R2 Rg RV Ri V2 RR V2 Vi 85 Rg RiR2 V2 Vi RV R V2 If say V is factored out of the numerator and denominator a form is obtained which makes it clear that only the voltage ratio is needed Rg R R2 V2 Vi 1 R2 R V V 38 1 A respectable difference between the two load resistors is necessary in order to minimise the effect of measurement errors but too large a de
89. e the two chosen matching criteria are independent i e the two matching processes can proceed simultaneously without altering the outcome The system can even adjust itself when presented with a speech SSB signal but will reach a solution fastest when the error signals are continuously available One desirable property of this matching system and of any properly designed matching system is that it corrects for the defects of its own components In this case when the phase control system adds series inductance for example the increasing resistance of the coil will increase the impedance magnitude seen at the input but the magnitude control system will simply back off to compensate Similarly the inductance of the impedance transformer will cause a positive phase shift but phase control system will back off in the capacitive direction to compensate While the simple magnitude phase matching system is entirely practical however it has never been particularly popular The reason is that it is difficult to design an efficient and resonance free variable broadband transformer The required transformations can just as well be obtained using only variable capacitors and inductors and this subject is examined in detail in a separate article 44 Admittance conductance susceptance The linear circuit analysis technique demonstrated so far consists of breaking the circuit down into two terminal networks and treating those networks as impedances T
90. e and vice versa 24 11 About these Theorems The theorems given above do not appear in standard engineering textbooks Therefore it is legitimate to ask Why have they been stated here when everyone else manages without them The answer to the question is this By sticking to the mathematical rules particularly by ensuring that we always use properly balanced vector equations and by using any simplifications which can be proved in a general way we eliminate the need for phasor diagrams Essentially we can let the algebra do all of the reasoning We can still use phasor diagrams for the purpose of explaining what is going on but they become merely illustrative and make no difference whatsoever to the outcome of a problem solving exercise The traditional role of the phasor diagram has been to help in resolving the ambiguities caused by unrigorous mathematical definitions But the mathematics is self consistent If the problem is defined correctly the hand waving becomes unnecessary 62 25 Generalisation of Ohm s Law We have already arrived at a general statement of Ohm s law in section 7 by observing that it can be written as a phasor equation V IZ We have also observed that we are at liberty to treat either I or V as a scalar equal in value to its own magnitude in order to learn the phase of the other relative to it The drop dimension theorem 24 5 gives our justification for doing so but also allows us to generalise our phasor Ohm s l
91. e following designations R2 75 1Q Ri 29 6Q the voltage ratio V2 V was 1 364 0 04 Using equation 38 1 the source resistance Rg was calculated to be 23 3Q An error analysis see next section gave an estimated standard deviation of 3 4Q i e Rg 23 3 3 4Q Note incidentally that this determination assumes that the output impedance does not change with power output level Given that power transistors are non linear devices this may not be the case 39 Error analysis While it would be inappropriate here to delve too deeply into the subject of scientific data analysis the reader should nevertheless be aware that all physical measurements are meaningless unless they have some kind of error window or confidence interval associated with them This is not a serious problem when taking a reading with say a multimeter because assuming that the instrument has been calibrated the manual will say what the measurement accuracy is A digital multimeter for example might have a quoted accuracy of 0 8 1 digit i e 1 in the last decimal place for its resistance ranges so if we obtain a resistance reading from this instrument of say 75 1Q the actual measurement will have a confidence interval of 0 6 0 1 i e the reading should be recorded as 75 1 0 7Q Scientists and engineers normally equate error boundaries stated in this way with the estimated standard deviation ESD of the measurement where on the assumption that errors are scatter
92. e number of dimensions AsB A B Coso It is the component shadow length of B when projected onto the direction of A multiplied by the length of A and vice versa i e A and B are interchangeable Had we attacked the DC power formula P IV with a foreknowledge of vector theory we would have failed it on the grounds of dimensional inconsistency P has too many dimensions and deduced that the scalar product is required i e P Ve I V I Coso 10 1 Instead we attacked the problem backwards and discovered the definition of the scalar product instead Note however that there is a subtle difference between the general vector dot product and the phasor dot product which will be discussed shortly In the context of impedance Coso is known as the power factor PF and is of particular interest to electricity generating companies which prefer their customers to place pure resistances across the supply so that they do not have to run their generators into reactive loads Thus if a load such as an electric motor is inductive as well as resistive a suitable capacitor placed across it or in series with it can be used to cancel the reactance and bring the power factor to unity i e 0 and Cosg 1 This brings the apparent power into coincidence with the actual power consumed and has the effect of minimising the consumer s electricity bill as well as minimising the stress on the generators and power transmission equipment Thus powe
93. e of a transformer can be transferred to the other side in an equivalent circuit by the act of multiplying it by the turns ratio squared So we might represent the inductance of a transformer as a separate inductance L in parallel with the primary side of an ideal transformer of otherwise infinite inductance or we might represent it as an inductance L in parallel with the secondary side The 96 transformation rule 33 3 tells us that j2nf L j Ns Np 2xf L i e L Ns Np L This is a remarkable result because not only does it give us the basis for constructing equivalent circuits to serve as models for real transformers it also tells us something about inductors The expression can only be true if the inductance of the coil is proportional to the square of the number of turns in it We can see why by considering the two 1 N auto transformer equivalent circuits shown below In the left hand circuit the inductance of the transformer is referred to the primary side and for reasons of convention is given the symbol A In the right hand circuit the inductance is referred to the secondary side and is given the symbol L From the foregoing discussion we can immediately write the relationship between L and A L N AL We can also interpret L as the inductance of the whole coil and A as the inductance of one turn of the coil A is known as the inductance factor and depends on the physical dimensions of the coil and the
94. e representing an impedance by plotting a point then moving right by a distance R R and upwards by a distance X or downwards if X is negative K and plotting another point The length of the line which joins the two points is called the magnitude or modulus of Z and is written Z and pronounced mod Z The magnitude is always positive by definition and is obtained by using Pythagoras theorem the square on the hypotenuse of a right R angled triangle is equal to the sum of the squares of the other two sides Hence OY Z V R2 X 6 1 Notice also that the definition of magnitude has a meaning for Tang X R ordinary numbers because they can be regarded as a one dimensional Cosg R Z vectors Hence if s is a scalar Sing X Z Is V s i e the effect of taking the magnitude of an ordinary number is simply to remove the sign or The direction of Z is given by the angle lower case phi it makes with the horizontal resistance axis which is the angle whose tangent is X R i e 22 X R Tano Hence o Arctan X R 6 2 Arctan is sometimes written Tan Note that can be positive or negative and in particular if we adopt 90 the standard trigonometric convention that a positive angle is obtained X by going anti clockwise from zero see diagram right will be k positive for an impedance with an inductive reactance and negative for an impedance with a capacitive reacta
95. ed randomly according to a normal or Gaussian distribution a standard deviation represents a region where we have a 68 confidence that the true result will lie The standard deviation is usually given the symbol o Greek lower case sigma and so if we obtain a measurement xo we have 68 confidence that the true answer lies between x o and xto From the properties of the Gaussian error distribution also we have a 95 5 confidence that the true answer lies between x 20 and x 2o and a 99 7 confidence that the true answer lies between x 36 86 and x 3o ref The use of standard deviations rather than brick wall tolerances reflects the reality that there is always a finite probability that the true result will lie outside the stated error range We can only ever have absolute confidence that the magnitude of the true answer lies somewhere between zero and infinity but we expect only 3 measurements in every 1000 to fall outside x 30 It is always advisable to try to write down an ESD for every measurement made This is a reasonably straightforward matter where direct measurements are involved but a difficulty arises in situations where several measurements are made and then put into a formula in order to obtain the required result The problem is that of working out how much influence the deviation of a particular variable has on the overall result and how to add the various deviations together in order to arrive at the overall ESD It is th
96. educing the antenna reactance in this way reduces the amount of loading inductance required and hence allows the coil to be wound with thicker wire for a given size less resistance Placing the hat at the top of the z antenna moreover increases the current in the vertical section and actually increases the radiation resistance slightly every little helps 41 Basic impedance transformer In the previous section it was implied that the generator output impedance could be adjusted but we have yet to offer any method for doing so There are numerous options in this respect but for the present purpose it will be sufficient to have just one the transformer Transformers are discussed in detail in a separate article but here we will avail ourselves of the properties of straightforward but unfortunately mythical circuit models known as the perfect transformer and the ideal transformer An ideal transformer has no losses and its voltage ratio is the same as its turns ratio In truth well designed transformers can have power transfer efficiencies of more than 98 within a certain band of frequencies and so the myth of the ideal transformer is not so far from reality A perfect transformer is an ideal transformer which has large winding reactances so that the off load input current is negligible which means that its current ratio is the inverse of its voltage ratio Here 19 Electromagnetic Induction D W Knight available from www g3ynh in
97. eing the only version which Neper himself would not have recognised It arises from the observation that if two numbers are each expressed as a base number raised to a power then the numbers can be multiplied simply by adding the powers i e B x BP pa This is obvious when the powers also known as exponents are whole numbers e g 10 x 10 108 but it works just the same when they are not Also we can perform division just as easily by noting that B 2 B Be Consequently in the 17th and 18th Centuries long before the advent of affordable calculating machines great effort was made to produce tables allowing difficult multiplications and divisions to be performed by looking up the exponent which when used to raise a common base represents a particular number These exponents are known as logarithms and are defined as follows 15 Admiralty Handbook of Wireless Telegraphy B R 230 Vol II HMSO London 1938 Appendix A The DeciBel and the Neper Appendix available from http www g3ynh info 67 If n B then a Log n if n equals B to the power of a then a is the log to the base B of n If a logarithm is written without the base subscript then base 10 is usually implied i e Log means Logio although some older documents deviate from this convention Naperian natural logarithms which crop up frequently in physics use Euler s number e as the base e 2 71828 and can be written either Loge or In
98. en there must be a live phasor or an expression with j in it e a live phasor on the other side 38 Euler s Formula For those familiar with exponents note that Cos jSin e This equation is known as Euler s formula and defines the relationship between algebra and trigonometry where e is sometimes referred to as Euler s number and is to more decimal places then you ll probably ever need 2 718 281 828 13 Complex arithmetic Complex numbers can be added in the same way as vectors 1 e Ry jX1 R2 jX2 Ri R2 j X X2 and they can be scaled in the same way as vectors i e s R jX sR jsX it is traditional to move j to the beginning of the term it operates on to make its presence more obvious The real power of the representation however comes from the fact that we know immediately how to perform multiplication involving complex numbers because although expressions having non zero real and imaginary parts cannot be reduced to a single number we can deal with the multiplication cross terms by observing that j 1 Hence Ri jXi R2 jX RR jXiR2 jeoRi j XX2 RiR2 X1X2 j RiX2 AP XiR2 Thus we can multiply two complex numbers and always obtain a result which can be re arranged into the form a jb This outcome demonstrates also that the ordinary algebraic product of two phasors AB is another phasor and is not the same as the dot scalar product AeB The ordinar
99. ence electrical energy cannot flow inside a good conductor This understanding incidentally gives rise to a semantic difficulty regarding whether there is a difference between electricity and electrical energy It is hard to justify the preservation of different meanings for the two terms and yet people will persist in saying that electricity flows through conductors We can sidestep the issue by saying that electricity flows along the wires but that does little to rectify the basic misconception The general consensus now seems to be that unqualified use of the word electricity should be avoided altogether in any rigorous scientific context The electricity for which the utility company demands payment however is definitely of the Heaviside rather than the electrons in wires variety Now that we have established the location of the electrical energy it must be added that a small amount does flow into but not through practical conductors This is because metal presuming that the temperature is too high for it to be superconducting always has some resistance The inflowing energy is of course lost from the fields and converted into heat The mechanism of energy delivery can be understood once again using the Poynting vector and it explains not only unwanted losses but also what happens in relation to devices which are deliberately made resistive so that they can absorb large amounts of energy We start by imagining a small particle
100. ency it is necessary to make either the coil or the capacitor adjustable and the required adjustment range will easily swallow any deviation caused by using the ideal case formula fo 1 2nV LC We may therefore conclude that in normal circumstances the assumption of zero losses may be perfectly acceptable when calculating the resonant frequency of a parallel tuned circuit but as we shall see in the next section it is not acceptable when calculating the impedance at resonance 16 Dynamic Resistance For an ideal parallel tuned circuit i e Ri 0 and Rc 0 the impedance becomes infinite at resonance This of course does not happen in practice but provided that the loss resistances of the components are small it does rise to a high value Since we have defined resonance as the frequency at which the reactance is cancelled this impedance is also purely resistive and it is known as the dynamic resistance of the parallel tuned circuit Here we will give it the symbol Ryo effective parallel resistance when f fo It is of course given by the real part of equation 14 1 the parallel impedance formula given earlier i e RiRc RitRe HRX ReXi Ryo 16 1 RitRc Xi Xc In the example from the previous section we had Rp 2Q Rc 0 1Q L 3 WH C 42pF fo 14 178253MHz Xc 267 26871120 and X 267 2537728Q If we apply the above formula to these data we obtain Ryo 0 42 2 71432 56399 0 1 71424 57907 2 1
101. enerator because the impedance connected across the generator is simply Z R j Xit Xc and so taking the current as a reference phasor and using Ohm s law and the magnitude ratio theorem 24 1 we obtain T I Vin Z We can also state that the power delivered to the load is Proad P Rroaa and that maximum power will occur at the resonant frequency because the total reactance will then be zero and the magnitude of the total impedance Z will be at a minimum Hence we will call the maximum load power Poroaa and the power at the bandwidth limits will be Proad Poroaa 2 i e P 2 where Io is the current at resonance Hence the bandwidth limits lie at the points where P 2 i e where 1 1 2 70 So the load resistance having served to allow us to define the bandwidth has promptly vanished and the bandwidth becomes the interval between the points where the current has fallen to 1 V2 of its value at resonance Furthermore we can observe that we will always obtain this result regardless of which resistance we define as the load Ri Rc and Rioas are only symbols and since the corresponding resistances are connected in series we can swap their designations at will We can also consider any combination of these resistances to be the load including the total resistance R and this will always cancel and tell us that the half power points occur when I I V2 Thus to define the bandwidth of a series resona
102. energy at one or more new frequencies From a circuit analysis point of view anything which absorbs energy is a resistance and anything which which creates a new frequency component is a generator Hence we can put the behaviour of an alien device into a metaphorical black box with one or more two terminal connections called ports which look to the outside world like networks of basic circuit elements The rule by which energy disappears into a resistance inside the box and reappears from one or more generators inside the box is called the transfer function The fact that anything with electrical connections and a known transfer function can be incorporated into circuit theory confers enormous power upon the method As mentioned previously however circuit analysis does have its limitations It is after all not a general theory but a projection or degenerate form of electromagnetic theory Naturally a price is paid for the simplification and it is instructive to consider what that is So look at a circuit diagram and try to find where the lengths of the wires and the physical dimensions of the components are written That information is conspicuous by its absence because a circuit diagram 11 is a purely topological representation like the famous London Underground map It was a curious and usually unremarked discovery of the early circuit experimenters that it doesn t matter how the equipment is laid out or whether a component
103. ep philosophical implications of this strange quantity That is unfortunate because it doesn t actually exist Maxwell coined the term because he initially imagined it as distortions of the ther the latter being an elastic medium supposed to permeate all space and thereby explain the paradoxical phenomenon of action at a distance The 19 Century luminiferous ther has now gone the way of the Earth centred Universe and good riddance and Science has come to explain all electromagnetic phenomena in terms of fields and particles It will do no harm to think of displacement current as a convenient fudge which allows us to extend the laws of DC electricity to higher frequencies and thereby avoid having to subject every problem to the full electromagnetic treatment Hence displacement current is that which has to be evoked because electromagnetic energy doesn t always follow the wires It is not a physical current It is just a quantity which corrects for the difference between the magnetomotive force and the conduction current Magnetomotive force or MMF is incidentally not completely synonymous with current Were it so we would gladly drop the misleading concept current altogether but unfortunately we are stuck with it The reason is that MMF and current are only identical in circuits formed of a single conducting loop When the circuit is composed of overlapping loops disposed in such a way that adjacent conductors carry current in t
104. equency SRF even when there is nothing whatsoever connected to it and it is part of the HF resonator design procedure to ensure that the SRF is outside the frequency range of interest A physically small resonator coil suitable for radio receiver applications might have a self capacitance of about 1pF Let us suppose therefore that this applies to the 1H coil from the previous example This amount of unavoidable capacitance places an upper limit on the maximum attainable resonant frequency somewhere very roughly around 1 2aV LC 160MHz Stray capacitance between the connecting wires will reduce this frequency so if we construct the circuit carefully we should expect the inductive branch to self resonate somewhere in a range from about 40 to 160MHz All electrical conductors have inductance a coil is simply a structure designed to enhance inductance by causing the magnetic fields developed by adjacent turns to add together Hence the wires and plates involved in making up the capacitive branch of the resonator will constitute an additional series inductance which we can model to a good approximation by imagining a small inductor Lc in series with the capacitor For the 100pF capacitor of our previous example it will be very difficult to get this self inductance to be less than about 10nH so we might place the upper limit for the series self resonance of the capacitive branch somewhere very roughly around 1 2V LcC 160MHz Inductance of the c
105. equivalent circuit of this antenna is shown on the right with one extra resistance added that being the true source resistance Rg With this additional piece of information it becomes possible to calculate the Q and hence the bandwidth of this system If we take the same example component values as were used before we have Zi Rp jX 7 5 j3000 Za Ra Ry jXa 2 5 j3000 giving an input resistance of 10Q for a whip length of about 0 07 The whole system is of course a series resonator and we can define the circuit Q as Qo X1 Reotat Xal Rtotai Since bandwidth is proportional to fo problems of excessive Q are likely to occur at low operating frequencies so let us see what happens if this antenna is built to operate on say 1 9MHz with the generator source resistance adjusted to be 5Q This will make the total series resistance 15Q and with X 3000Q the Q will be 3000 15 200 The 3dB bandwidth of the antenna will therefore be fo Q 9 5KHz This is wide enough to accommodate a communications SSB signal 2 7KHz bandwidth but there is very little latitude for incidental detuning and the antenna will exhibit a small variation of input impedance depending on the modulation frequency Light loading of the generator will help to offset these problems because it will create a situation where transient detuning forces the generator load system closer to its maximum power transfer point although detuning won t increase the amount o
106. erefore fortuitous that we have been engaged in the study of vectors because it turns out that this is a problem of vector addition and magnitudes If two or more measurements are made in such a way that the outcome of one has no influence on the outcome of any of the others the measurement errors are said to be uncorrelated An example of uncorrelated errors is that of readings taken from two separate instruments where an error or inaccuracy in the reading of one instrument is not related to any error or inaccuracy in the reading of the other On the other hand the errors in two measurements made using the same instrument may be correlated in the sense that if the instrument always reads too high or too low it will introduce errors in the same direction in both cases If measurement errors are correlated then it means that there is some systematic design interpretation or calibration defect in the measuring process but if we believe that the measurements have been made to the best of our abilities with the equipment available then it is usually sensible to assume that any measurement errors are uncorrelated Now if the errors in two or more measurements are uncorrelated this means that a deviation from the true value in one measured quantity can occur without influencing the deviations in any of the other quantities If we determine a quantity by applying a formula to a set of measurements each measurement will contribute a random error to t
107. es UHF however the struggle to adapt the lumped component representation will become increasingly difficult and the need to resort to Maxwell s equations or at least to standard solutions obtained from the scientific literature will become more and more apparent While on the subject of scale incidentally note that the wavelength range of visible light runs from 0 7 to 0 4 microns where 1 micron lum 10 m If we were to represent the interaction of visible light and matter using circuit diagrams the circuits would have to be built on the molecular scale around Inm 10 m Hence visible light has no measurable tendency to be guided by ordinary electrical circuitry but it does have the convenient habit of reflecting from the components so that we can see them We can now draw together two threads from the preceding discussion Representing a circuit diagrammatically begins with a basic assumption that either the circuit is infinitesimally small or that the speed of light is infinite Either way it means that every part of the circuit is assumed initially to be in instantaneous communication with every other part It is also assumed that the electrical energy always follows the wires whereas it is actually distributed in the fields surrounding the circuit In both cases in the absence of corrective measures this can result in disagreement between the behaviour calculated from circuit theory and the measured performance of the actual
108. es the reactance goes to zero i e the combination behaves like a short circuit neglecting resistance when X Xc 0 In the case of an inductor and a capacitor in parallel since the term X Xc is on the bottom of a fraction it would appear that the reactance goes to infinity i e the combination behaves like an open circuit when Xi Xc 0 A complete open circuit does not appear in practice however because in the parallel case it transpires that we are not at liberty to neglect the resistances of the coil and the capacitor We therefore cannot calculate the exact resonant frequency of a practical parallel tuned circuit nor the resistance which remains when the reactance has been cancelled until we have developed a more comprehensive theory and so that is another matter which we must leave until later We can say however that the exact resonant frequency of a series tuned circuit and the approximate resonant frequency of a typical parallel tuned circuit occurs when XL Xc L e 2n fo L 1 27 fo C Now if we rearrange this equation to put both instances of f on one side we have fo 1 40 LC and taking the square root gives fo 1 2r VLC 4 1 pronounced f nought equals one over two pi root LC or in rhyme One over two pi root LC gives the resonant frequency Equation 4 1 is of course is the standard resonance formula but before accepting it we should note that because it contains a square root every combi
109. expression for the real part of two impedances in parallel as derived above RR R R2 R 1X RX R pas Ri Ro X X The truth of this statement is not immediately obvious but a check of dimensional consistency can very quickly tell us if it is capable of being true In this case the denominator the bottom part of the fraction has two brackets each containing quantities having the units of resistance Ohms Hence the terms R R2 and X X2 have dimensions of Q and the overall dimensions of the denominator are Q In the case of the numerator there are three terms to be added each having the dimensions of 3 and the overall dimensions of the numerator are Q Dividing the dimensions of the numerator by the dimensions of the denominator we obtain 03 Q Q and so the equation is dimensionally consistent and represents a quantity which can be expressed in Ohms It is also possible to test the dimensional consistency of equations involving mixed units The point here is that units have aliases which are composites of other units and so we can check any equation provided that we know the relationships between the units used In the context of circuit analysis these relationships are easily obtained because they are embedded in the basic formulae from which the mathematical argument is constructed Ohm s law V I Z for example tells us that Ohms are equivalent to Volts divided by Amperes and so a quantity h
110. f arithmetic and linear electrical devices in parallel is obeyed 1 e Zi Zn Z3 Zi Zo Z3 17 5 Double slash product definition The notation implies a specialised kind of phasor multiplication which we might call the double slash product or the parallel product of a pair of phasors Since its use in conjunction with parallel capacitors is pointless we will adopt the following strict mathematical definition a b ab a b 46 18 Parallel to Series transformation In the discussion so far we have adopted the habit of representing every impedance as a resistance in series with a reactance It makes good sense to do so in most circumstances because it allows the impedance to be written directly in the form R jX There are many situations however in which circuit analysis can be simplified by representing an impedance as a resistance in parallel with a reactance The two possible representations are equally valid but it should be obvious from the parallel impedance equation 14 1 derived earlier that the parallel representation for a particular impedance requires a different combination of resistance and reactance to that of the series representation In the next two sections we will explore the relationships between the two representations beginning with the transformation of an impedance from its parallel to its series form To derive this transformation we simply regard the parallel elements as two separate
111. f power delivered light loading will give better regulation than a conjugate match Note incidentally that the antenna discussed above is not physically small when designed for operation in the 160m band The wavelength at 1 9MHz is c f 157 8m and so a 0 074 rod will be 11m long and consequently far too large for mobile use To make a mobile antenna the rod must 91 be shortened and this will reduce the antenna capacitance and hence increase the reactance and sadly for efficiency will cause the radiation resistance to fall The larger antenna reactance will necessitate a larger loading reactance and although this will bring more resistance with it the increase in reactance will be greater than the increase in total resistance and the Q will rise A point can be reached where serious curtailment of the modulation bandwidth occurs although for this system it it not predictable using lumped component theory The coil can be regarded as a lumped component provided that the whip is long enough to ensure that most of the radiation occurs from the whip rather than from the coil If that condition applies then the Q of the antenna system can never be larger than the Q of the loading coil because the total series resistance will always be that of the coil plus a little extra The maximum tolerable Q causing some but not serious audio degradation occurs when the antenna system bandwidth is the same as the audio bandwidth and for SSB on 1 9
112. fo 92 we will assume that a tightly coupled transformer is ideal when operating within its pass band on the understanding that it requires a more advanced analysis to determine what the pass band is Such a transformer is also approximately perfect when used as part of a low impedance electrical network A transformer loaded with an impedance Z is represented on I I the right Here Np is the number of turns in the primary 3 z generator side winding and Ns is the number of turns in the ZV N secondary load side winding The dots next to the windings P P indicate either the start or the finish it doesn t matter how this is designated as long as it is done consistently and it it assumed that both coils are wound in the same sense clockwise or anticlockwise when looking at a particular end of the coil The dotted line between the coils indicates that the transformer is wound on a magnetic core the purpose of which in this instance is to produce a very tight magnetic coupling between the windings If all of the magnetic field from the primary winding is captured by the core and linked to the secondary winding i e if there is no magnetic leakage and if the coils and the core have no heating losses then all of the power delivered by the generator is transferred to the load Also if the inductive reactance of the windings is much larger than the magnitudes of the impedances seen on either side and the capacitance of both windings is
113. g between the coils Note also that the capacitor formulae take on the opposite forms of their resistance counterparts this being due to the reciprocal inverse relationship between capacitance and capacitive reactance 4 Resonance The combination rules discussed above allow us to deal with resistors or capacitors or inductors in series and parallel but for reasons which will become clear in the following sections they do not provide a method for dealing with combinations of resistance and reactance if we try to add resistance to reactance directly our calculations will not agree with our measurements We can however deal with combinations of inductive and capacitive reactance provided that we observe the convention that capacitive reactance is negative We may therefore add to our repertoire of standard formulae by writing general expressions for pure reactances in series and parallel i e Reactances in series Reactances in parallel X X X X X X2 Xi Xp 18 Now since inductive reactance is positive and increases with frequency and capacitive reactance is negative and decreases with frequency if an inductance is placed in series or parallel with a capacitance there will occur a frequency at which the two reactances cancel That frequency of course is the resonant frequency of the combined reactances A resonant frequency is usually denoted by the symbol fo f nought In the case of an inductor and a capacitor in seri
114. g negative because it transpires that when capacitors and inductors are connected to form resonant circuits the reactance of the inductor in some sense cancels the reactance of the capacitor This means that one of the types of reactance has to be considered to be negative and as will be explained later we choose it to be the capacitive variety in order to be consistent with the conventions of trigonometry The other entries in the table are derived from the formulae in the left hand column using only Ohm s law and a basic electrical rule known as Kirchhoff s first law pronounced kir khov Kirchhoff s law tells us that the sum of all the currents flowing into a given point in a circuit is equal to the sum of all the currents flowing out This law was originally regarded as proof of the principle of conservation of charge in DC circuits what goes in must come out but it also turns out to be true of current in the general magnetomotive sense provided that we use the correct rules of addition to be determined shortly in circuits involving both resistance and reactance The entries in the middle and right hand columns are of course the well known series and parallel combination formulae for passive electrical components These expressions may all be regarded as examples of simple mathematical models in this case in the sense that a single component can serve to represent a combination of several components Of these the formula for re
115. gh X Xc is very nearly zero at resonance and can therefore be deleted from the denominator without making much difference The result is Xi X R R This expression can be simplified by observing that everything inside the square root bracket is squared but in doing so we must be mindful of a common fallacy The square root of the square of a number is not the number itself A square root always has two solutions one positive one negative and if only one of the solutions can be true additional information is required for selection of the correct one In this case we know that Ryo must be positive if the network is passive and so we accept the positive square roots but note that in section 6 we defined the positive square root of a square as a magnitude 1 e X2 X This rule must be strictly applied because simply deleting the superscripts and the square root symbol would have given us a negative value for R o because Xc is negative Hence Rp Xx Xc Ri Ro We have noted before that X1 Xc L C hence po 50 Ryo LIC Rit Ro which we have seen before as equation 16 3 While it is instructive to attack a derivation from several directions and verify that all approaches lead to the same conclusion the point of the parallel impedance representation is that it often makes problems easier to solve The parallel resonator is a good example because the parallel representation gives a direct se
116. h a positive real part and a zero imaginary part A phasor pointing at 180 can be represented as a complex number with a negative real part and a zero imaginary part Hence if N a j0 then N a and IN V 07 al Hence N N 4 N where N is a pseudoscalar equal in value and sign to the real part of N This may appear trivial but it shows that our assumption that a phasor which has dropped a 60 dimension can be treated as a scalar is universal rather than a special interpretation of a particular phasor expression A further implication however is that N is not identical to the magnitude of N because magnitudes are always positive whereas N can be positive or negative We can force N to become equal to N by stipulating that 0 We can also drop a dimension i e set the imaginary part to zero by choosing 180 but in that case we get N N Thus the alleged scalar which results from dropping a dimension is not a magnitude but it is a quantity which is equal in magnitude to a magnitude and if 0 it is positive This may seem a pedantic distinction but the point in making it is that if we restrict the scope of our phasor algebra through erroneous interpretation we lose the ability to include DC electricity in our theory and we lose the ability to explore exotic ideas such as negative resistance The pseudoscalar we obtain by dropping a dimension can be negative even if usually it isn t 24 6 Square magnit
117. h it took some years before that point was fully accepted They are instead stores of and agents for the transfer of pure energy The liberation of electricity is however subject to a strict condition which is that the energy exists by virtue of continuous transfer between the two fields according to the laws of induction i e a decaying electric field gives rise to a magnetic field and a decaying magnetic field gives rise to an electric field and so the energy swaps backwards and forwards The mechanism only works if the energy is propagating through space in a direction at right angles to the crossed fields with a velocity given by the expression v 1 N us where u Greek lower case mu is the magnetic permeability and e epsilon is the electric permittivity of the surrounding medium Permeability u is a constant of proportionality obtained from the force of magnetic attraction or repulsion which occurs between wires carrying an electric current Permittivity is a constant obtained from the relationship between the physical dimensions of a capacitor and its capacitance Maxwell found that the best available measurements for the permeability and permittivity of vacuum lo and o mu nought and epsilon nought gave a propagation velocity for free energy 4 c 1 V uo 0 which turned out to be the same as the speed of light Thus he was able to confirm a suggestion put forward by Michael Faraday some years before which is that l
118. hange is reversible Resistance of course obeys a straight line law called Ohm s law The AC voltage versus current laws laws governing inductance and capacitance are also linear Thus the basic circuit elements combine to make linear networks which lack the ability to produce new signal components This means overall that a linear network treats each frequency component as if it exists in isolation We can therefore quantify its behaviour one frequency at a time which is why the basic generator of circuit analysis produces only a single frequency The response of a circuit to more complicated waveforms can always be built up when required by adding the results of analyses carried out at the component frequencies For many purposes however the focus of interest when several frequencies are involved is the frequency response which is just a stepwise application of the one frequency at a time approach Despite the simplicity offered by single frequency analysis there are of course numerous electrical and electronic components which behave in a non linear fashion Semiconductor devices diodes transistors etc are an obvious example but there are also materials which change their characteristics according to field strength This might seem to place a limitation on linear network theory but actually there is a straightforward solution A non linear device is one which accepts energy at one or more excitation frequencies and converts it into
119. hat the total resistance in both branches adds up to 10Q In this way as we shall see we sacrifice some of the tuning range in order to obtain a poor but possibly useful Q To find the tuning range which results we can use the full parallel resonance formula fo 1 2aV L C VI LIC RZV LIC R 57 L e fo fos VI L C R2 LIC R2 Now if for the sake of simplicity we assume that all of the resistance is in the inductive branch at the low frequency limit and in the capacitive branch at the high frequency limit the correction factor V L C Ri2 L C Rc2 becomes 0 99499 when R 10 and 1 00504 when R 10 So for our 1H in parallel with 100pF resonator with its ideal case resonance of 15 915MHz we obtain a tuning range of 15 836 to 15 996MHz a spectacular 0 5 with a Q of 10 Of course if we dispense with the variable resistor and use a variable capacitor or inductor instead we can easily obtain a tuning range of more than 2 1 while sustaining a Q of around 50 So much for the resistance tuned resonator as a variable band pass filter but perhaps we can use it as the frequency determining device in an oscillator That was certainly the suggestion in the circuit idea article from whence it came An oscillator is effectively an amplifier with some of its output fed back into its input via a frequency selective network and we can easily design an amplifier with sufficient gain to overcome the losses of the candidate
120. he product of two vectors is also a vector both current and voltage are vectors In fact both voltages and currents are vectors because they each have associated with them a magnitude a frequency and a phase the phase being defined as the time at which a chosen event in the wave cycle occurs e g the time of zero crossing from negative to positive in the illustration below The generator frequency is not an independent variable in the definition of impedance because it already appears in the reactance X 2afL Xc 1 2fC and so we may deduce that the direction of the impedance vector constitutes phase information i e it gives us the time difference in degrees or radians between corresponding events in the voltage and current waveforms Hence is known as the phase angle and can be converted into a time difference in seconds by dividing a complete cycle of the waveform into 360 or 27 radians and noting that the time per cycle or period of the waveform is 1 f time cycle 1 f 90 m 2 0 180 0 360 lt 1 f gt d an 2m n O x 27 3r Radians 3n 2 360 180 0 180 360 540 Degrees 270 Now note that since I and V are vectors we can write them in polar or rectangular forms using the transformations 6 3 given earlier i e I E o I I Cos I Sing VC IVI 9 VC V Cos V Sing In general it is natural to think of currents and voltages in their polar forms but the rectangular form is
121. he result but there is just as much chance that the error due to one measurement will partly cancel the error due to another as there is that a pair of error contributions will both increase or decrease the result Therefore it will be unduly pessimistic to add the uncertainty contributions of the individual measurements directly Instead we should allow for the independence of the uncertainty contributions by regarding each one as a vector pointing in a direction which is at right angles orthogonal to all of the others In effect by virtue of its randomness each uncertainty contribution exists in its own dimension and we may identify its magnitude as its length in that dimension It follows that the overall uncertainty is the length i e the magnitude of the vector which results from the addition of a set of orthogonal uncertainty vectors This situation is represented in the diagram below where U U2 and U are the uncertainty contributions to the determined value of an unknown and U is the overall uncertainty in the result We can easily find U by successive application of Pythagoras theorem as follows Let the magnitude of the vector sum of U and U be Un Un V U ae U Then U is the magnitude of the vector sum of Uj and U U V U Us but U UP U Hence U V U U U 2 18 Data Reduction and Error Analysis for the Physical Sciences Philip R Bevington McGraw Hill 1969 Library of Congress cat c
122. he same as saying that the number a jb can be plotted as a point on a graph of a vs b That graph is of course number space and maps in this space are known as Argand diagrams 37 We must observe at this point that jB complex numbers are so like impedances that had they been b discovered by electrical engineers they might well have been named after Com plex impedances Naturally since complex Numbers numbers are the general class of numbers to which all numbers belong they are essential for solving all kinds of mathematical problems but nowhere is the association so direct f and so profound that all we have to do Real to convert an impedance into a Imagina complex number is to write Numbers e areas Z R jx This says that impedance is a quantity jB with a real part R and an imaginary part X The original terms real and imaginary are also perfectly appropriate because the apparent power P IV x dissipated in a resistance is indeed real while the apparent power P I Vx dissipated in a pure reactance is entirely imaginary Thus it is hard to make a logical distinction between the two statements impedances can be represented by complex numbers and impedances are complex numbers It follows also from the relationships implicit in Ohm s law that if impedances can be treated as complex numbers then so too can voltages and currents This does not mean that these objects have somehow ceased to be vector
123. he same direction the MMF is increased due to a phenomenon called magnetic flux linkage Such overlapping structures are of course known as coils or inductors and have the property that they allow the amount of magnetic energy which can be stored in a given volume of space to be magnified Still for the greater non overlapping part of an electrical circuit current and MMF are practically the same and to a good approximation we can dispense with the details and treat coils as separate objects having a single magnetic concentrating property called inductance Certainly it is very useful to know how to calculate inductance from the number of turns and the physical dimensions of a coil but it is a matter which can separated from the general business of designing electrical systems 4 The term Ether has however come back into favour in the discussion of the properties of the quantum vacuum see The Lightness of Being Frank Wilczek 2008 Penguin edn ISBN 9780141043142 especially ch 8 10 2 Circuit analysis overview The basic theory of electrical circuits is known as lumped component analysis The verb to analyse incidentally means to break down into simpler or more fundamental parts and in this case analysis is the art of describing and predicting the behaviour of circuits by treating them as networks of interconnected resistances capacitances inductances and generators This turns out to be an extraordinarily
124. here are ways of dealing with such problems and in the example case it is to represent the wires as inductances with some capacitance between them but it is important to understand that the truth of circuit diagrams is contingent upon unspecified factors Knowing the difference between representation and reality is the art as opposed to the science of circuit analysis No one would want to apply the full electromagnetic theory to routine circuit problems and indeed success in the solving of Maxwell s equations for some particular class of problems is often regarded as a scientific event Hence electricity is primarily associated with circuits rather than fields and waves and being good at it requires a level of understanding which is difficult to formalise Experience comes with time but we can at least invite entry to the Guild by offering a straightforward rule of thumb A light wave in vacuum completes one cycle of variation of its electric and magnetic fields upon travelling a distance given by the expression AK c f where f is the frequency c 299792458 metres second is the speed of light and Greek lambda is the wavelength Due to the essentially refractive nature of circuits the apparent velocity of signal propagation through an electrical network is never exactly c but it rarely deviates from c by more than a few Hence we can easily obtain an idea of the phase errors which will accumulate as a result of constructing a circ
125. his approach has allowed us to attack a wide range of problems but it results in extremely messy algebra when impedances in parallel are involved Ultimately we need a way of dealing with arbitrarily large numbers of impedances in parallel just as we can already deal with any number of impedances in series and it transpires that this can be achieved by defining the properties of our component two terminal networks not in terms of impedance but in terms of the reciprocal of impedance this being called admittance By so doing we move the problem out of what we so far think of as its natural space impedance space and into what is known as its reciprocal space and the re definition trivial though it is in the case of phasors is known as a reciprocal space transformation The reciprocal space transformation is another mathematical invention of James Clerk Maxwell Its most far reaching application is in the field of X ray crystallography it being the means by which the X ray diffraction patterns of crystals are traced back to the internal arrangement of atoms Here however we need only a simplified version because the problems we wish to solve are strictly two dimensional The reciprocal of impedance space is known as admittance space A pair of two dimensional reciprocal spaces has the property that straight lines in one appear as circles in the other a correspondence which is used extensively in the article Impedance Matching cited earlier
126. ical that the definition should simultaneously involve both inductance and capacitance If so then why not write Qo V Xoc Xo R which is the same as multiplying the two standard definitions and taking the positive square root i e taking the geometric mean of the two definitions Of course Xoc XoL 2nfoL 2nfoC L C i e the L C ratio hence o V L IC R or 31 1 Qo Ro R Here is a definition of resonant Q which properly involves all of the components and as an added bonus in calculation does not involve the resonant frequency 75 32 Bandwidth in terms of Q A bandwidth function for the series resonator was derived earlier and given as equation 29 2 P Po R R 2nfL 1 2nfC Now that we have a sensible definition for Qo however we can see that it can be used as a substitution for R in the expression above i e from 31 1 R V L C Qo Hence L C Qo L C Q 21fL 1 2n fC P P Now if we forcibly factorise the quantity L C from the right hand term in the denominator we obtain Qo Q V C L 2mfL 1 2nfC P Po which noting that L VL VL and VC C 1 VC simplifies to 1 Qo 1 Q 22V LC 1 f22V LC 2 P Po We cans substitute for the quantity 2nV LC by noting that the standard series resonance formula can be rearranged thus 2nV LC 1 fo Hence 1 Qo P Po 32 1 1 Qo fo fo
127. ice which is a source of energy a battery or generator or by devices which can store energy inductors and capacitors this being a matter which we will explore in detail later When we think of the Poynting vector in relation to a complete electrical system we are really thinking of the average of a large number of microscopic J energy transfer processes In the case of a simple circuit consisting only of a generator and a resistive load the Poynting vector is directed along the wires from generator to load and the direction is the same on both sides of the generator Should we examine the average energy flow close to a wire however we will see that the direction is tilted very slightly towards the wire on account of the distributed resistive losses So now we have the basic field picture of electricity but there remain a few issues which need to be explained Particularly we need to look again at electric current and the matter of why it is defined in terms of moving charges As it says in every school physics textbook an Ampere is a current of one Coulomb per second and since the charge of an electron is 1 6021892x 10 Coulombs an Amp flowing from to corresponds to 1 1 6021892x10 6 241460122x10 electrons per second flowing from to This is correct assuming that the current is due to electrons but only in the special limiting case when the frequency of the electromagnetic energy being transferred tends to zer
128. ight is composed of electromagnetic waves Maxwell had also shown of course that electrical energy is a form of light and that older ideas derived from DC experiments were no longer tenable Maxwell died in 1879 at the age of 46 only six years after the publication of his great treatise on electricity and magnetism Thus it was left to others to explore the ramifications of his work In the latter part of the 19th Century there were two great interpreters of Maxwell s electromagnetic theory Oliver Heaviside and Heinrich Hertz both of whom were brilliant mathematicians in their own right These two scientists independently cleared up Maxwell s notation and reduced a nest of algebraic clutter to a set of four equations which describe the fields The four Maxwell s equations which we know today are actually a variant of the form preferred by Heaviside extra terms which are zero for the Universe in its present state are nowadays usually deleted The climax of Hertz s work was the creation and detection of Maxwellian waves under laboratory conditions which means that Hertz is the father of radio telecommunications and also the inventor of the first radio antennas His clarification of Maxwell s theory was also the basis of the work of one Albert Einstein a Zurich patent examiner with a habit of daydreaming about objects in relative motion Einstein realised that Maxwell s separation of light and matter implies that the speed of light is const
129. iginal value we determine the uncertainty contribution due to m2 X 0x2 f m Moto m3 a and so on If we work through all of the variables in this way and determine their error contributions we can obtain an estimate of the standard deviation of x by summing the squares of the contributions and taking the square root 6 V 0x1 0x2 0x3 Om Note that there are a number of assumptions inherent in this procedure firstly as discussed before that the uncertainties are uncorrelated and secondly that we have assumed that the function fis linear for changes in any of the variables The latter condition is normally true to a good approximation for small changes and the effect of any non linearity is mitigated by the fact that the object of the exercise is to obtain an estimate Example The output resistance Rg of an RF amplifier was determined by loading the output with two different resistances and noting the change in the output voltage with all other conditions held constant The applicable formula is equation 38 1 Rg R R Nv 1 R2 R Nv Where Ny is the ratio of the output voltages Ny V Vi The voltage measurements were made using an oscilloscope and it was considered that each measurement had an uncertainty of about 2 It was also considered that these uncertainties were uncorrelated because they were incurred by different operations one operation being to set the transmitter carrier level and osc
130. illoscope Y shift until the waveform just touched the top and bottom of the measuring graticule with the higher value resistor connected the other being to read the height on the graticule with the lower value resistor connected The overall uncertainty of the voltage ratio measurement was therefore taken to be the square root of the sum of the squares of the two voltage measurements i e 22 22 2 8 which was rounded to 3 in view of the approximate nature of the estimate The actual voltage ratio was 1 364 and 3 of 1 364 is 0 04 Hence Ny 1 364 0 04 The resistances were measured using a multimeter known to read correctly within 0 1Q against a standard resistance of 100 0Q The stated accuracy of the instrument was 0 8 1 digit The 88 measured resistances were R 29 6Q and R2 75 1Q Hence R 29 6 0 340 R 75 140 70 The output impedance Rg was calculated from the formula 38 1 using a spreadsheet program and determined to be 23 3Q The output impedance was also calculated with each of the measured values individually incremented and decremented by an amount equal to its estimated standard deviation and the resulting deviation in Rg was noted The spreadsheet Rg_meas ods is shown below D3 lS a3 B3 C3 1 63 A3 C3 OS en E aa ee an G an Output resistance of TS430s R1 R2 Nv Re Deviation RMS dev Average 29 600 75 100 1 364 23 301 0 000 0 000 29 940 75 100 1 364 23 888 0 587 0 587 0 5789 29 260 75 100
131. impedances R j0 and 0 jX and apply the formula for R impedances in parallel 1 e Z Z2 ZiZ Zi Zp Hence Rp jXp gt R jX R jX jX L e R jX JX R Rp jX and R and X are simply the real and imaginary parts of the right hand side of this expression once it has been put into the form a jb We proceed as usual by multiplying the top numerator and bottom denominator by the complex conjugate of the denominator thus j Rp Xp Rp JX R jX Rp jXp Rp jX Which rearranges to R X j XR R jX 18 1 R tX Hence for the series representation R X X R R and X R X R X Further pieces of information which we can extract from the parallel to series transformation and which will be useful later are the phase angle magnitude and Q of an impedance in its parallel form Phase angle and Q of an impedance in parallel form The phase angle for an impedance in its series form was given earlier as expression 6 2 Arctan X R By using expression 18 1 above we can substitute for X and R to obtain Arctan X Ry R Xp i e Arctan R X which also tells us that X R R X i e the ratio of resistance to reactance of an impedance in its series form is the inverse of the ratio for the impedance in its parallel form Also since we know 47 that X Rross is an expression for the Q of an electrical component we may further
132. important for understanding what happens when the phase angle is either 0 or 180 Taking a current vector as an example IC I 0 IC I Cos0 I Sin0 1 I 0 and I I 180 I I Cos180 f Sin180 I I 0 When a two dimensional vector lies along the 0 direction either pointing with it or in opposition its extent in one of its spatial i e rectangular form dimensions is zero and as in our interpretation of negative frequency given in section 4 the minus symbol is associated with a 180 phase shift or phase reversal 24 So now that we know that both current and voltage are vectors we must conclude that V IZ is the general statement of Ohm s law It transpires however that we may admit the validity of the other possibilities I V Z and V IZ under certain circumstances The point is that in AC theory we are usually interested not in the absolute phases of the voltages and currents i e the phases relative to some external reference but in their phases relative to each other This means that we are often at liberty to choose the direction of one of the vectors in order to learn the directions of the others relative to it The direction chosen for this special reference vector is in principle arbitrary but a simplification occurs if we choose it to be either 0 or 180 because Sing goes to zero in either case and a vector which is zero in one of its spatial dimensions behaves in this
133. inking about this issue however note that an electromagnetic wave is defined in relation to a route through a field The wave like nature of the energy flow is detected by inserting probes measuring devices into the field and building up a picture by intercepting photons From this we infer that photons travel as waves even though we can only discern that by adding together the small packets of energy delivered by them This incidentally raises a general point in relation to scientific observation which is that there are no paradoxes in nature Paradoxes exist only in the mind of the observer and result from attempts to interpret information using faulty starting assumptions When we detect waves we do so on the assumption that our probes measure field strength This is very convenient because it allows us solve problems using field theory but when the meaning of an observation is in doubt or when discrepancies begin to accrue it is important to remember that all measurements are ultimately purely dependent on what can be inferred from the absorption and emission of energy We rarely need to think about the granularity of light when working at electrical frequencies because the amount of energy in each photon is exceedingly small The wave property is instead the most dominant feature and is often reinforced by a behaviour called coherence which is the ability of identical photons traversing the same set of paths to synchronise their fields
134. ion equations 19 3 i e Rep a Rc T Xe Rc 20 1 Xcp Re T Xc Xc 20 2 Rip Ri ae X17 Ri 20 3 Xip Rz T X XL 20 4 Using the appropriate transformations 20 2 and 20 4 the resonance condition X Xcp becomes Ry X12 Xp Re Xe Xc which can be rearranged to Xc Ri X17 X Re Xc 0 and then to XcXi Xc X_ XcR X Rc 0 We have seen this expression before as equation 15 1 and so the derivation may continue as in section 15 to give the parallel resonance formula 15 2 fo 1 2nV LC VI L C R2ZV LIC Rd The dynamic resistance R o is given by Ryo R Rep Hence using the transformations 20 1 and 20 3 we have Ri X17 Ri J Re Xe Re J Ryo Re Xi Ri Re Xc Re J which simplifies to Ri X17 Re Xe Rpo 20 5 R Ri XA R Re Xe 49 Thus we obtain another formula for the dynamic resistance of a parallel resonator and it is interesting to compare it with equation 16 1 which was our original derivation here we show it rearranged slightly Re Rv X17 Ri Re Xc Ryo 20 6 Ri Rc Xi Xc The two formulae are radically different in appearance but it is easy to verify by plugging in the numbers from the example in section 16 that they both give exactly the same answer This leaves the issue of which one of them is the best simplification and the answer i
135. ion arises because the bandwidth of the resonant circuit is infinitely narrow when R 0 i e the upper and lower band limits become coincident with fo in the limit that R 0 It is therefore an essential property of a correct expression for the mid band frequency but it does give us a simplification for equation 30 1 Squaring 30 1 gives fm R 4L C 4aL R 4aL fo but from expressions 29 3 and 29 4 given earlier fo Q R 2aL hence fin fo 2Q fo f 1 1 2Q fm fy V 1 2QP 30 2 The positive mid point frequency is always very slightly above the resonant frequency for a practical resonator but becomes coincident with fo when Q i e R 0 This skewing of the bandwidth function can be said to arise because fy is always closer to zero than it is to infinity the function spreads out on the high frequency side because there is more room The difference between the mid band frequency and the resonant frequency is however very small being 0 125 12 5KHz at 1OMHz for a Q of 10 and 0 00125 125 Hz at 1OMHz for a Q of 100 Hence the bandwidth function can be considered to be approximately symmetric for moderate Q Although the bandwidth function is not symmetric about its peak if we choose frequency as the horizontal x axis it can be made symmetrical if we instead plot it against an appropriately chosen function of frequency In particular we need a frequency function such that any resonance peak
136. is a plausible fallacy and that its appearance in an electronics publication did not generate a flurry of letters pointing out its flaws We might comment at this point that it is necessary to build a circuit and try it before recommending it to others but that is no help in finding out what went wrong if the circuit should fail to work as expected There is a great deal of difference between a resonance and a useful resonance and a practical circuit component operated at radio frequencies does not bear description as a pure inductance capacitance or resistance Wide range resistance tuned LC oscillators operating at low frequencies have nevertheless been built but the theory of operation depends on more than the simple observation that resistance appears in the parallel resonance formula Such circuits were used many years ago for resistance to frequency converters in scientific instrumentation applications but are nowadays rendered obsolete by simple and spectrally noisy RC oscillators such as can be implemented using CMOS logic inverters or the 555 timer IC 12 Theory and Application of Resistance Tuning C Brunetti and E Weiss Proc IRE June 1941 p333 344 58 24 Phasor theorems Early in this chapter we observed that the standard electrical formulae represent incomplete statements of Ohm s Law and Joule s Law We then went on to generalise Ohm s law but have yet to state all of its implications and we repaired the VI
137. is large or small or how long the wires are provided that they are a lot more conductive than any of the designated resistances This of course ceases to be true as the frequency is increased and this breakdown of DC theory is partly due to the finite speed of light Consider a sine wave generator connected by means of relatively long wires to a resistive load If we measure the voltage difference between any two points in the circuit we obtain a quantity which is proportional to the total electric field existing between those points In this case the field is associated with electromagnetic energy propagating from the generator to the load Since it takes a finite time for the energy to make the journey this means that the voltage measured across the generator will not be identical to the voltage measured across the load If we presume that the resistive loss in the wires is negligible the main difference will be in the relative phases of the two sine waves 1 e if we take some reference point on the waveform such as the zero crossing point on going from negative to positive we will find that the load waveform is delayed relative to the generator waveform This will not be noticeable if the measurements are made using an ordinary AC voltmeter but the time difference can certainly be demonstrated using a dual channel oscilloscope and the same effect will give rise to performance deviations in more complicated i e phase critical circuits T
138. is wrong with the equation in a purely abstract way by noting that I and V are phasors whereas P is scalar Now we will fix the problem by finding a method of vector multiplication which produces a scalar The first step in doing so is to refer to the product of the magnitudes I V as the apparent power Papparent I V The true power on the other hand is the power dissipated in the resistive part of the impedance which can be determined from the magnitude of the current i e using a properly balanced version of the DC formula P IPR and if we choose I as a 0 reference vector P PR Earlier in this chapter we showed how an impedance phasor diagram can be scaled by a reference phasor I to obtain a voltage phasor diagram 1 e every resistance or reactance in the diagram is multiplied by I The phasor diagram below has been scaled by P to obtain a power phasor diagram Here we should be aware that the phrase power phasor is an oxymoron i e a contradiction in terms like encrypted broadcast because average power is scalar but apparent power is not power and we can think of it as a vector In particular having set the phase of the current to be 0 the phase of the apparent power is given by the expression Papparent I Vv and since V I Z R X then Papparent P Z R X which gives the definition of apparent power as Papparent I R PX Thus the phase of V relative to I is the phase of the apparent power rel
139. ith the Poynting vector spending equal amounts of time in the two possible flow directions Hence for a resistance when the instantaneous voltage is positive the current is positive and when the voltage is negative the current is negative 1 e the voltage and current waveforms are perfectly in phase For the Poynting vector to alternate and give zero average power delivery however there must be a 4 cycle difference between the voltage and current waveforms It is the 0 difference in the resistive case and the 90 difference in the reactive case which gives rise to a condition of mathematical independence or othogonality which we can exploit to obtain a generalised form of Ohm s law Once we have that generalisation the rules of combination follow and give rise to a complete and internally consistent AC theory For DC circuits we can write Ohm s law as V IR For AC circuits therefore we must suspect that we can write something along the lines of V IZ as long as we recognise that impedance Z the generalised attribute of objects which obey Ohm s law must be represented by some composite quantity containing two distinct elements R and X In circumstances such as this it is traditional to see if anyone has developed a branch of mathematics which suits the problem and the clue regarding where to look lies in the independence of R and X If two quantities are completely independent they must in some sense exist in different dimensions
140. ive zero output voltage or current if it were spinning and where it will give an initially positive output if its shaft is turned 19 anti clockwise see illustration below Now if the shaft is turned clockwise the output will initially go negative It follows that the difference between the positive and negative frequency outputs is that while the voltage or current associated with one is positive the voltage associated with the other is always negative and vice versa Hence changing the sign of a frequency has the effect of shifting the phase of the associated waveform by 180 Incidentally for anyone who might insist on taking the direction of rotation analogy too seriously it is of course obvious that if the generator is an electronic oscillator the concept of rotation is meaningless In that case the negative frequency solution can be obtained by swapping the connections as it can with any generator or resonator 5 Impedance Resistance Reactance The basic electrical laws discussed earlier tell us that resistors consume power when connected across a generator but that perfect inductors and capacitors do not The combination formulae then tell us how to deal with resistances or reactances in series and parallel but they do not tell us how to deal with combinations of resistance and reactance This is a serious limitation which can only be overcome by introducing the generalised concept of impedance i e the theory of two ter
141. l use One final significance of the characteristic resistance which is worth remembering is that it is equal to the magnitudes of the reactances in the circuit at the ideal case resonant frequency i e the resonant frequency when the resistance in both branches is equal This frequency as was mentioned earlier is given by the series resonance formula i e fos E 2nV L or in radians sec 2nfos WL Now if we call the inductive reactance at this frequency Xros then Xios 2tfos L L LO and since any number is the square of its own square root Xios W L C Similarly for the capacitive reactance Xcos 1 2rfos C V LC C Xe W L C 52 22 Phase analysis We can visualise the phase relationship between voltage and current in a parallel resonant circuit by deriving an expression for the I V phase angle and plotting it as a graph against frequency for various values of included resistance This is only one of the many situations in which graphs of phase vs frequency are instructive and so this section will serve as a general introduction to the technique of phase analysis as well as a specific investigation of the parallel resonator The circuit to be analysed is shown on the right and we can use Ohm s law straight away to write an expression for the current I V Z where Z is the parallel combination of the impedances in the two branches of the resonator and we choose the phase of V to be 0 an
142. late a hypothetical resistance R which represents the parallel combination of R and Ro we have V IR 1 1L R We can eliminate I and I by using equation 3 1 above i e V Ri and L V R hence V V R V R2 R The voltage can then be factored out and cancelled to give 1 R1 C R2 R and dividing each side of the equation by R gives 1 R 1 R 1 Ra This is one form of the standard expression for resistors in parallel and a little rearrangement will give us the other Inverting the expression above gives R 1 Ri 1 Rz J We then arrange the terms inside the square brackets to have a common denominator multiply the 1 R term by R R and multiply the 1 R2 term by Rj R i e R 1 T Ro RiR2 Ri RiR2 hence R 1 Ri R2 R i R2 which upon inversion gives R R R2 Ri R2 The formulae for inductors and capacitors in series and parallel may also be derived by using exactly the same approach as was used above the only difference being that inductive reactance X 27fL or capacitive reactance Xc 1 2mfC is substituted in place of resistance The 2af factors and any minus signs disappear by cancellation leaving formulae involving only inductance or capacitance Note incidentally that the inductors in the illustrations in the previous table are shown orientated at right angles to each other this being done as a reminder that the formulae are only true when there is no magnetic couplin
143. load impedance of 52Q An interesting feature of the transformer is that it achieves a continuous transition from step down to step up by having an overwind see diagram right i e the brush contact at one end of the coil goes to a centre tap and the end of the coil is left unconnected The coil has 28 turns and the input tap is at 14 turns so a maximum impedance step up of approximately 4 1 is obtainable Np 97 The disadvantage of the Collins transformer is that the coil does not have a magnetic core The stray magnetic fields will therefore induce currents eddy currents in the surrounding metalwork and give rise to resistive losses The open magnetic circuit also implies that the impedance transformation obtained will not be exactly proportional to the square of the turns ratio and due to the absence of a magnetic core the inductance might appear on first consideration to be rather low The inductance for the whole coil estimated using Wheeler s Formula is about 20uH giving only about 5uH when referred to the primary side This will give rise to significant phase shift at lower frequencies the inductive reactance seen by the transmitter at 2MHz being something around 2nx2x10 x5x10 63Q It transpires however that the choice of primary reactance about equal to the target input impedance at the lowest operating frequency is sensible because in addition to the impedance transformer the antenna tuner also has power factor corre
144. ly larger than the DC resistances For the types of components used in HF antenna matching applications Rc will be of the order of 0 10 and Rr typically a few ohms 10 Components and Materials www g3ynh info 42 In the general electronic literature several different definitions are used for the resonant frequency of a parallel tuned circuit the alternatives being the frequency at which the impedance of the circuit has its largest magnitude and the frequency at which X Xc Here however we will adopt the most straightforward definition which is the frequency at which the impedance is purely resistive also known as the unity power factor frequency We can find this frequency by setting the imaginary part equal to zero in equation 14 1 above i e X XcX Xc X_ XcR X Rc Re R Xc XY 0 Where the subscripts 1 and 2 have been changed to C and L as befits the current problem Now notice that to make the reactance equal to zero we only need to make the numerator of this expression equal to zero i e we can ignore the denominator Hence XcX Xc X1 tXcR X Rc 0 PE 15 1 We now need to make the frequency dependence of this expression explicit by using the substitutions Xc 1 2afoC and X 2afoL i e 27foL C 2nfoL 1 20fC Ri 2afoC 2nfoL Rc 0 The resonant frequency can now be found by re arranging this expression to get fy on its own Also since we know that the se
145. magnitude of the current By so doing all of the quantities have been turned into voltages and so the diagram has become a voltage phasor diagram IX 0 IXc i ee IXL Xc With regard to the physical phenomena represented here observe that since R C and L are in series they must all carry the same current We can deduce the magnitudes of the voltages across across the three components using Ohm s law i e Vi IX1 Vc IXc and Vr IR the latter being written as a scalar because it is in phase with I and therefore pointing at 0 We also know the relative phases of these voltages because they are all linked to the phase of the common current i e the voltage IR across the resistance is in phase with the current the voltage IX across the inductance is at 90 relative to the current and the voltage IXc across the capacitance is at 90 relative to the current We can therefore add these three voltages as vectors to obtain the magnitude of the generator voltage and its phase relative to the phase of the current although in the diagram the voltages across the two reactances have been added first to produce the more diagrammatically convenient quantity X IXc this being the voltage across the total reactance in the system Note that the voltages across the two reactances always tend to cancel because there is a fixed 180 phase difference between them and so the magnitude of the voltage across the total reactance is always smaller th
146. minal devices which obey Ohm s law but do not necessarily consume all of the electrical power delivered to them A concept which needs to be formalised at this point is that of a linear passive two terminal network An electrical device is linear if its graph of voltage versus current is a straight line i e if it obeys Ohm s law and it is passive if it contains no sources of energy The general term network is used because although the theory we are about to develop covers simple devices like capacitors and resistors it also covers any combination actual or hypothetical of resistances and reactances in series and parallel connected to a single pair of terminals A network can be hypothetical in the sense that it behaves in the same way as i e serves as a model for an actual two terminal device For example when an antenna system is connected as a load to a radio transmitter we can treat it as a hypothetical network of resistances and reactances An antenna incidentally is not completely passive because it also receives radio signals but we can model the receiving case by considering it to be exactly the same network as in the transmitting case but with one or more generators connected in series with it Any linear passive two terminal network can be regarded as an impedance This means that its electrical behaviour at a particular frequency can be explained by invoking two and only two mutually independent properties namely resis
147. most universal public misconception regarding the nature of electricity and the other being a lack of mathematical rigour in the way in which phasor techniques are commonly used Both of these issues can and should be addressed at this stage in the development of working knowledge and so the accompanying discussion attempts to do that As all experienced engineers and physicists know our understanding of electricity comes from Maxwell s equations The problem for those who wish to teach electrical subjects however is that the electromagnetic field approach requires advanced mathematics and does not lead directly to the practicalities of circuit design Hence it is sensible to hold back on the more abstract ideas until they become unavoidable but that leaves the problem of how to dispel the notion that electricity is synonymous with electrons flowing through wires The author s solution is to provide an extended preamble which gives a purely qualitative explanation of electricity in terms of fields and is intended to leave the reader with the same mental picture as will be held by those who are familiar with Maxwell s theory On the matter of mathematical rigour it is not the intention to wrap the subject in formalism but merely to eliminate certain bad practices To this end we pay particular attention to the definitions and properties of the mathematical objects involved and develop a way of working which identifies and preserves the algebraic
148. mpedance in series with another impedance in such a way as to create a pure resistance into which the transmitter can deliver all of its power We would of course like to cancel the reactance of the antenna by placing a pure opposite reactance in series with it but pure reactance is unattainable and so our compensating or conjugate reactance always brings some extra loss resistance with it In this case although the voltage appearing across the terminals of the combined impedance may be very low the voltage across the antenna terminals can be enormous and we must choose the voltage ratings of our matching network components accordingly For an illustration of the voltage magnification effect consider the short vertical antenna system depicted in the diagram below loading coil JRL antenna Rr Va ground plane 29 In order to avoid misconceptions it is important to be aware that the vertical rod itself is not the antenna To use the rod as a radiator we must apply a voltage to the pair of terminals formed by it and the ground plane and so the antenna is the combination of the rod and the ground plane The input impedance of an electrically short less than a quarter wavelength long i e lt A 4 vertical antenna looks predominantly like a very small capacitor which is essentially the capacitance which exists between the vertical section and the ground A small capacitor has a large negative reactance recall X 1 2fC and s
149. n of the Squares this being a mathematical trick to find an equivalent constant DC voltage or current which gives the same heating effect as an alternating voltage or current The voltages and currents referred to in the theory of impedance must be RMS values by definition because that is the only way in which Ohm s law can be generalised to include both DC and AC DC becomes a special case of AC with f 0 The need for an RMS average arises because the ordinary average of a sinusoidal alternating voltage is zero the voltage spends as much time being positive as it does negative If however we square the instantaneous voltage we obtain a function which is proportional to the power it will deliver to a resistance If we average that power function i e take the mean of the squares and then take the square root we will obtain the equivalent direct voltage i e the constant voltage which will deliver the same amount of power to a given resistance The RMS average of a sine wave is the instantaneous peak value divided by the square root of 2 i e Vams Vox V2 and Vp Vrusx V2 Thus if we calculate a voltage of 3000VRMS across the antenna terminals the maximum instantaneous peak voltage will be 3000x1 4142 4247V and it is this higher figure which must be used in calculating the voltage ratings of the components used The final piece of information we can extract from the vertical antenna example under discussion is the efficiency of the sys
150. n parallel form 48 21 Imaginary TESOMANCE scssctionssasesecevesavsssues 50 22 Phase ANALYSIS sccsvcssicxandseateevssedicaaasdcanel nie 52 23 Resistance tuned LC resonator 55 24 Phasor theorems 58 25 Generalisation of Ohm s Law 62 26 General statement of Joule s Law 63 27 Bana Wid Win resintere 65 28 deciBels amp logarithms cceeeeeee 66 29 Bandwidth of a series resonator 69 30 Logarithmic MequenCy ciissssvsvseseidausvnese 73 31 A proper definition for resonant Q 74 32 Bandwidth in terms of Q cece 75 33 Lorentzian line shape function 76 34 Maximum power transfet 0 06 78 35 The potential divider icsesisssssssvecsatosedestsies 82 36 Output impedance of potential divider 83 37 Th venin s TREGTE NM cssaccsceraeaeesacasearcnde 84 38 Measuring source resistance 84 39 Error ANALYSIS scsstisatessdicascsaaientusesivnsaancess 85 40 Antenna system Qo eeeeeeceseeeestteeeees 90 41 Basic impedance transformet 91 42 Auto transformetrs cccceeseeeeeeeeeeeeees 95 43 Prototype Z matching network 97 44 Admittance conductance susceptance 98 45 Parallel resonator BPPF c ccee 101 46 Unloaded Q of parallel resonator 104 47 Current magnification 106 48 Controlling loaded Qu eeeeeeeeeees 108 Preface This document
151. n squaring gives IVI Vo Gs Gp Gs Gp ae Vo Gs Gp Bop Bip V2 and upon inversion gives Bep Bip 0 F1 2 f fo f Gs Gp i e Bcp Bip Gs Gr 1 and taking the square root Bcp Bip Gs Gp 1 Thus Brp Bep Gs Gr and if we define the sum Gs Gp as Gg i e the conductance which determines the Q Bip Bep Go Now using the substitutions Bc 22fC and B 1 27fL we obtain 1 2nfL 2afC Go and by factoring out 1 27fL from the left hand side and re arranging 2nfL Go 1 2af L C L e 47 L C 2rL Go f 1 0 This is a quadratic equation in f with a 47 L C b 2nL Gg and c 1 It has four solutions as was the case for the series resonator section 29 these being the upper and lower bandwidth limits for positive and negative frequencies To solve it we apply the standard formula f b V b 4ac 2a Hence f 2nL Go V 2nL Go 4 4107L C 2 42 L Cp and using the substitution L L L to obtain cancellation of Lp from all but one term f 1 Gq EVICL Ga 4 LyMLp Cy 40 Cy i e f Go V Ge 4Cy Lp 40C Now since V Go 4C L will always be larger than Go we can identify the positive frequency upper bandwith limit as Go 4C L Go 4C and the positive frequency lower bandwidth limit as V Gq 4C Lp Go Any and the ba
152. n systems using convenient numbers It carries over seamlessly into the field of radio firstly because modulated radio signals are converted into sound and secondly radio receivers also use automatic gain control systems and so also have a roughly logarithmic response Common logarithms are used because the notation hails from the time when log tables were used for multiplication The deciBel is preferred over the Bel because it transpires that 1dB 25 9 is close to the minimum change in audio power which can be detected by the human ear this being 10 20 i e 0 4 0 8dB depending on the waveform and the listener Hence there is rarely a need to express power ratios in dB to a greater accuracy than to the nearest whole number An attempt to introduce a log ratio notation based on natural Naperian logarithms resulted in an alternative unit the Neper unit symbol Np i e N Np 2 Loge P Pres giving 1 Np 8 686dB but its use never became widespread in English speaking countries The use of log tables for multiplication has of course died out from the school curriculum and this leaves modern students unprepared for the introduction of deciBels and other logarithmic functions A little revision of the subject will therefore not go amiss The technique of multiplication using logarithms was introduced in 1614 by the Scottish mathematician and astronomer Jhone Neper spelt variously but nowadays usually written as John Napier this b
153. n this case is that it is equation 20 6 We can tell by looking at the power or degree of the numerator and denominator of each equation Observe first that all of the quantities involved in the expressions are measured on Ohms Hence the numerator of 20 5 has dimensions of Q4 and the denominator has dimensions of In equation 20 6 however the numerator has dimensions of Q and the denominator Q Hence the numerator of 20 6 is of lower degree than that of 20 5 and the denominators likewise This means that 20 5 can be simplified further and ultimately transformed into 20 6 although for anyone who cares to try it the manipulations required are laborious and require the use of equation 15 1 as a substitution Something more tractable happens however when we multiply equations 20 5 and 20 6 and take the square root to obtain a new expression for Ryo i e we take the geometric mean of the two formulae In this case the denominator of 20 5 cancels the numerator of 20 6 and we obtain R X R X Ri Re Xi Xc 20 7 po Here we can make the following simplifying assumptions 1 Since Xr is normally much greater than R in radio circuits R1 can be deleted from the numerator without making much difference 2 Since X is also usually much greater than Rc Rc can be deleted from the numerator without making much difference 3 If the Qs of the resonator components are reasonably hi
154. nation of L and C has two resonant frequencies associated with it Every equation involving a square root has two solutions because the square root of a number is by definition a quantity which when multiplied by itself gives the number in question When two negative numbers are multiplied the result is a positive number Hence if x is a positive number we must note not only that xX x XxX but also that x x x x hence V x x So the resonance formula stated explicitly becomes fy 1 2nV LC and there are two solutions numerically identical but of opposite sign By convention we usually assume the positive result but since there were no restrictions on the validity of the arguments used in deriving the formula the negative frequency solution must exist and must mean something AC electrical theory as we delve more deeply into it will present us with various little conundrums often involving square roots and although negative frequency is one of the most trivial we will be ill prepared for the others if we simply let it pass The negative frequency solution arises because a sinusoidal waveform is derived from circular motion and there are two possibilities for this clockwise and anti clockwise This does not mean that the positive and negative frequencies are identical however as the following argument will illustrate Consider an alternator mechanical AC generator stopped at a position where it would g
155. nature of any magnetic core material It may be interpreted either as the inductance of a one turn coil or as the inductance of an auto transformer referred across a one turn tap A has the units of inductance Henrys but is more informatively given units of inductance turn Henrys per turn squared Continuously variable auto transformer One of the drawbacks of ferrite or iron cored transformers as impedance matching devices is that the the transformation ratio can only be altered in a stepwise fashion by changing windings or tappings one turn at a time or half a turn if the core has two holes If the turns in the coils are few as tends to be the case in radio frequency applications then the steps available can be very coarse indeed It is however possible to make a continuously variable inductor or auto transformer by rotating a coil about its axis and tapping into it with a rolling contact the coil end connections being made by slipping contacts known for historical reasons as brushes Such a device is known colloquially as a roller coaster and an example is shown in the photograph on the right This is the motor driven variable impedance transformer from a 1957 vintage Collins 180L 3A automatic HF antenna tuner The tuner is designed to match end fed wire Marconi antennas of 14 to 40 metres in length over a frequency range of 2 to 25MHz and is for use with transmitters with an output of up to 150W and a preferred
156. nce 180 0 Notice also that Z and taken together provide a complete characterisation of a two dimensional vector and so give us an alternative way of recording its properties The form introduced Xc earlier Z R X is known as the rectangular form because it contains a 90 list of values in dimensions chosen to be at right angles to each other The alternative Z Z is known as the polar form because it uses polar co ordinates distance and bearing The polar form uses different units in its two dimensions Ohms degrees or radians whereas the rectangular form has the same units in both dimensions Ohms Ohms There is no ambiguity between the rectangular and polar forms because the list in brackets is optional and a vector has the same properties regardless of how it is defined Also if a specific vector quantity is to be noted by putting actual numbers into the brackets a degrees symbol next to the angle will indicate that the polar form is intended We can now regard equations 6 1 and 6 2 as the transformations which take a two dimensional vector from the rectangular to the polar form The reverse transformations are obtained from the standard trigonometric relations Cosp R Z and Sing X Z i e R Z Cosp and X Z Sing The full set of transformations is summarised in the following table Rectangular form Polar form Z R X Z V R X Arctan X R 6 3 Z Z Cos Z Sing Z Z 9
157. nce For changes in either of these variables there will be a peak in the graph of power versus the variable and the peak will of course occur at the point where the gradient of the curve is zero Hence for the reactance condition maximum power transfer occurs when 0P OX 0 and for the resistance condition maximum power transmission occurs where OP OR 0 where is known as partial d or curly d and indicates a partial differential i e differentiation of one variable with respect to another is carried out with all other variables held constant In order to carry out these differentiations on the expression above we can use the quotient rule If y N D then dy dx DdN dx NdD dx D Hence if we let N V R and D R Rg X X 4 RAR 2RR tX X 2XX then ON OX 0 OD OX 2X42X ON OR V and 0D OR 2R 2Rg Hence OP OX 0 VPR 2X 2X D 2 V R X X D therefore OP OX 0 when X X maximum power transfer occurs when the power factor is 1 and P AR VP R R X X 7 VPR 2R 2R D VP X X R R 2R D VPL Re X X R D therefore OP OR 0 when R X X R 0 i e 6P OR 0 when R R 2 X X Notice that this latter maximum power transfer condition is a magnitude it is the same as R Re i X X i e maximum power transfer occurs when the load resistance is equal to the magnitude of the impedance formed by the source resistance and the
158. nces Resistance makes only a small contribution to the overall shape of the bandwidth function because it only makes a significant contribution to the magnitude of the impedance and hence to the current when the frequency is close to resonance Far from resonance the impedance magnitude is dominated by the reactive component unless the Q of the resonator is very low The physical laws governing the various processes which contribute to the loss resistance moreover are smoothly varying functions of frequency unless some additional system resonance is encountered in the region of interest and so the loss resistance component will not normally vary significantly over a small frequency interval Consequently for a reasonably high Q the relationship Bandwidth f Q is sufficiently accurate to be presumed exact for all normal engineering purposes 73 30 Logarithmic frequency One additional matter which might be of interest is that although some authors refer to fo as the centre frequency the frequency interval between the half power points is not symmetrical about fo We can find the mid point or median frequency by taking the average of the upper and lower band limits i e fm 2 V R2 4L C R R 4L C RJ 2x4aL 2 R 4L C 2 4aL fm V R2 4L C 4aL 30 1 This quantity is only equal to fo when R 0 i e noting that VL L 1 NL fm gt V 4L C 4nL 1 22VLC fo This limiting condit
159. nd X This would prove to be J a somewhat tricky problem had we not noticed from the preceding derivation of the magnitude equation 18 2 that Z R X R X R X The right hand side of this equation can also be obtained by multiplying expression 19 1 by R or by multiplying expression 19 2 by X Hence RR R X and XX R xX ote R R X2 R 19 3a and X R X X 19 3b 48 20 Parallel resonator in parallel form Having derived the series to parallel transformation we are now in a position to analyse the parallel resonator in a different way The outcome should be mathematically unsurprising because we are bound to obtain the same results as before but the technique will give us a new way of thinking about the circuit Rc Ri Xcp Rcp RLp Xc XL The symbol means is by definition equal to The symbol means in parallel with As the diagram above illustrates the parallel impedance representation allows us to visualise the circuit as an ideal parallel resonator with a resistance connected across it This separates the reactive and the resistive parts of the problem and tells us immediately that unity power factor resonance occurs when X Xcp and that the dynamic resistance is given by the value of Rip Rcp at fo We can of course relate the parallel impedance form of the resonator to the series impedance form by using the transformations given in the previous sect
160. nd so we can now forget about forces on charged bodies and magnets and think about electromagnetic energy Recall that Maxwell discovered that light travels with its electric field oscillating at right angles to its magnetic field the direction of propagation being at right angles to both fields Heaviside and Poynting now tell us that the transport of energy in electrical circuits occurs in exactly the same way An arbitrarily chosen point in an electric or magnetic field has both intensity i e magnitude or strength and a direction of action Such points are known as vectors and a region of space filled with vectors is called a vector field There is also a mathematical operation called vector multiplication which can be applied at points where two fields cross to produce a new vector at right angles to the original two Notional directions were assigned to the electric and magnetic vectors in the discussion above and the adopted convention is of course one which gives the 7 correct direction of energy transport when the fields are combined using the vector product or cross product as it is also known The electric field is usually given the symbol E and the magnetic field the symbol H The cross product is then written P ExH where P is known as the Poynting vector and gives the intensity and average direction of the energy flow at some point in the combined electric and magnetic i e electromagnetic field Now co
161. ndwidth is fw f Go 22C This is the admittance counterpart of the result obtained at this stage in the derivation of the Q ofa 104 series resonator equation 29 3 and so we will deduce that the bandwidth of the parallel resonator BPF is fo Qo and use this deduction to find a definition for Q fo Qo G 2TCp 2rfoCp Bcpo Qo Go Qo Bopo Go Now let Ro 1 Go where Rg is the resistance which determines the Q Also observe that Bcpo 1 Xcpo and at resonance Xc o Xipo Hence Qo Ro Xcpo Ro Xpo or in keeping with the definition of resonant Q given in section 31 Qo Ro V L C 45 1 Ro is simply the parallel combination of the source resistance the load resistance and the dynamic resistance of the resonator 1 e Ro Rs Ryo Reoaa This result gives us the theoretical information we need in order to be able to design parallel resonant bandpasss filters Firstly we may observe that the source and load impedances are effectively in parallel with the resonator which is why any minor source and load reactances can be lumped with the resonator reactances and cause only a detuning effect if such reactances are very large however they will cause a significant change in the dynamic resistance and the problem is best re analysed from scratch The source load and dynamic resistances however are critical in determining the Q and we need to obtain high values for all of them in order to obtain a high
162. not give maximum Q but it does cause the coil to radiate to some extent some of its loss resistance is actually radiation resistance The 6W fluorescent tube was added for this demonstration but the neon bulb at the tip was always used as a tuning aid The generator in the photograph is a Kenwood TS430s HF transceiver with its mains power supply and the AMU is an MFJ989C T network The input to the antenna is resistive when the length is adjusted correctly about 25Q mainly due to the coil and the AMU was used to transform this resistance to 50Q as required by the generator Those wishing to reproduce this demonstration should note that apart from the mains lead there is no proper ground plane for the set up and the author had to tune up wearing rubber gloves in order to avoid getting burnt fingers Mounting the antenna on a car is safer 32 10 Power Factor amp Scalar Product In the preceding discussion we observed that reactance acts as an impediment to the delivery of power into an impedance and that the applied voltage must be increased in order to overcome it This means that the DC power formula P IV if we interpret it to mean the product of the current and voltage magnitudes is not generally true for impedances because except in the special case that X 0 i e when the impedance is a pure resistance it will give a result which is larger than the actual power delivered As mentioned earlier we can deduce what
163. nsider the electromagnetic field around a wire in a circuit as shown on the right Electric field lines emerge E P perpendicular to the surface magnetic field lines encircle the conduction current and there are an infinite number of points at which they cross at right angles Thus assuming that the E and H fields are related in accordance with the continuity principle we can work out the direction in which energy is travelling and also the rate of flow at any location The key to the right of the diagram gives the direction of the Poynting vector in relation to the E and H fields It follows that if the electric field is strictly perpendicular to the wire then the Poynting vector lies parallel to the wire and the fields as they are depicted have it running away from the observer Notice also that all of the propagating energy is on the outside of the wire albeit in greatest concentration close to the surface where the magnetic field is strongest and it transpires that if the wire is a perfect conductor there is none on the inside at all It requires both an electric field and a magnetic field for the transportation of energy A perfect conductor however is a material which by definition cannot sustain an electric field This can be understood by noting that the electrical resistance between any two points within the body of a perfectly conducting object is zero in which case there can be no voltage difference and so no electric field H
164. nt circuit we do not need to designate any resistance as a load we need only to consider the current So it transpires that we can choose any resistance in a series network and analyse the power dissipated in it to determine the bandwidth and since we are interested here in the relationship between bandwidth and Q the obvious resistance to choose is R the total resistance We can always isolate a portion of R to determine the power delivered to it or the voltage across it if we so wish that is a trivial matter of proportions but for a general analysis the problem simplifies to that of understanding the behaviour of the simple series LCR network shown below The first part of the analysis is to determine the frequency response function for this circuit and plot it as a graph to see what it looks like A good function to plot for this purpose is the ratio P Po vs frequency because this ratio has a value of 1 at fo and is also in the correct form for conversion into deciBels The power ratio is equal to the square of the current ratio P Po P I I Ib because P PR and Polo R Hence we will start by obtaining an expression for the current ratio The general expression for the current is 1 I V 2Z I where Z R j Xi Xc At the resonant frequency however the impedance is purely resistive so Ip V R L Hence 1 To JVI IZ V R V C R Z R VR XiAXcP R which by writing the reactances explicitly gives I Ib R
165. ntion without which the notation will appear very cumbersome which is that whenever we refer to a current or a voltage without mentioning phase we mean magnitude i e the observable quantity which can be measured with a two terminal meter In other words a measurement taken from a voltmeter may be written in isolation say Vou 27V but as soon as it is inserted into a formula with other vectors it acquires a phase even if we don t need to know what that is and must then be identified as Voutl So mindful of the warning that reference vectors and magnitudes are not quite the same thing the expression V IZ now tells us that if we multiply an impedance by the magnitude of the current passing through it we will obtain a vector representing the magnitude of the applied voltage and its phase relative to the phase of the current This is an extremely useful result and stems from the fact that the vector representation has captured the physics of the situation exactly In effect having observed that resistance and reactance act independently on the current and that inductive and capacitive reactances act in opposition we have elected to represent pure inductive reactance as a vector pointing at 90 pure resistance as a vector pointing at 0 and pure capacitive reactance as a vector pointing at 90 Thus we have satisfied the requirement that the Poynting vector must alternate for reactance but not for resistance and we have incorporated it in
166. o Rep Rip Rep Rip and an expansion in terms of the series forms of the impedances has already been given as equation 20 5 Rc Xe Rre X1 Ryo Ri Re Xe Re Ri Xi The unloaded Q is defined as Qou Ryo VW Xcp Xt and from the series to parallel transformation equation 19 3b we have Xcp Rc alg Xe Xc EE 46 2 and Xip Ri a X1 Xi cei 46 3 Putting all of this together we have R2 X2 Re X2 RiRe X RAR 2 X J Qou VE Re X2 Re Xi2 Xe Xi and noting that V Xc Xi V L C this rearranges to V L C VE R2 X2 Re Xi Ri Re Xe Re Ri Xx J So at this point we have extracted V L C as required by equation 46 1 and the resistance by which V L C must be divided in order to obtain Qou is 106 R R2 X2 R R7 X R 46 4 R6 X Ri X7 Which upon expanding the numerator gives R Re X Re R X 2RiRc R RAXA RAX The simplification we require here comes from noting that the terms Rc Xc and R X1 occur in the expressions for Xcp and X p given above equations 46 2 and 46 3 and that at resonance Xcp X1p Hence Rc Xc Xc Rv X12 Xt i e Rc Xc Ri X12 Xc X_ and Re X17 Re Xc X Xc Hence R R Xc X_1 Rce Ki Xc 2RiRe which can be factorised R Ri V Xo X1 Re V X Xo Hence R R V Xc X1 Re V
167. o In other words it is only strictly true for DC electricity As the frequency increases the correspondence between conduction current and effective current becomes progressively less accurate which is why Maxwell invented displacement current The role of the electrons in conduction is actually an optical one They interact with the electromagnetic field in such a way as to increase the amount of energy which can be stored in the region of space immediately surrounding the conductor This creates a duct through which the energy prefers to flow in a manner analogous to the way in which mirages are sometimes seen in hot deserts and over the horizon VHF radio communication becomes possible on hot days Ducting occurs when the refractive index of the medium increases with distance from the surface i e a light ray which tries to move away is bent towards the parallel direction The existence of electrons was hypothesised and indeed the name was coined some time before J J Thomson identified them as the current carriers in cathode ray tubes 1897 It must then have seemed to many that the electrical fluid theory was confirmed but the discovery was actually its nemesis Upon estimating the number of free electrons in a given volume say of copper it turns out that the average velocity of propagation of an electron current through a solid medium is of the order of a few millimetres per second The mass of the electron is also so small that the amount
168. o be distinguished from current in the general electromagnetic sense We shall proceed by thinking in terms of a type of current called current which may or may not be correlated with the movements of charged particles but for most of the time we don t care whether it is or not This might seem to imply that we have turned current into an abstract idea but actually we have simply unloaded some unnecessary baggage Many readers will be aware that voltage is sometimes referred to as electromotive force EMF Its unit of measurement is not a pure force in the Newtonian sense but it is proportional to the force exerted on a charged particle and so the habit of referring to it as a force is loosely and widely accepted Likewise current is proportional to the force exerted on a magnet in the field and its unit the Ampere is the measure of magnetomotive force The electric field in a circuit is everywhere proportional to the driving voltage Likewise the magnetic field is everywhere proportional to the effective current and the two fields between them set the intensity of the Poynting vector Also it has to be said that AC ammeters are calibrated according to the amount of energy delivered and so read the true or magnetomotive quantity Before we move on it is perhaps worth making a few observations on the use of the term displacement current There are occasional non peer reviewed publications which agonise about the supposedly de
169. o show that the resonant frequency the point where the curve crosses the zero phase difference axis moves to low frequency when R exceeds Rc and vice versa The curves for R 100 and Rc 1 and R 1 and Rc 100 show that the resonant frequency goes to zero when Ri L C and goes to infinity when R V L C These results seem to indicate that the parallel resonator is infinitely tunable by means of a variable resistor a proposition which warrants careful examination 55 23 Resistance tuned LC resonator The parallel resonator shown on the right was offered as a circuit idea in Electronics World it being pointed out in the article that the tuning range is 0 to oo if R V L C and the Q of the circuit is stable because the total resistance is C constant Both of these claims are true within the scope of the model but there are R a couple of fatal flaws in the concept and we will address them lest people should 1 start to believe that the circuit will work The author of the article was perhaps a little unsure of the 0 to claim and so concluded that a variable resistor can give a much wider frequency range i than a variable capacitor or inductor We however can straight away dispense L z with the infinite upper limit by drawing the circuit model on the right We CL C might describe the original circuit as what you try to build whereas this Ri Rc makes some attempt to simulate what you actually get R A capacitor is simpl
170. o the use of an output c Dr i transformer than impedance transformation however the B principal issue being that the transmitter discussed above uses a 13 8V power supply and yet must deliver 100W into a 50Q load The required output power and target Ferrite load impedance defines the output voltage as Out V PR V 100x50 70 7V RMS i e 70 7x2V2 200V peak to peak p p A simplified version of the power amplifier circuit is shown below and we can deduce the i minimum allowable transformer step up ratio by examining it OS Low Pass C Output Filter bw Transformer This is a so called push pull amplifier circuit in which one transistor provides the positive half cycle of the output waveform and the other transistor provides the negative half cycle When a bipolar transistor is turned hard on its collector voltage does not go to zero but stops at some saturation voltage which is usually around 1V Also it is not a good idea to drive the transistors close to saturation because this will lead to considerable distortion of the output waveform Therefore we must assume that the output stage can produce positive and negative half cycles of no more than about 12 5V across half of the primary winding i e 25V per transistor across the whole winding hence 50V p p To obtain 200V p p 70 7V RMS therefore a voltage step up ratio of 1 4 is required The fact that this transformation increases the source impedance by
171. o we need to place a large inductive reactance X 2zfL in series with the antenna to make the whole thing look like a resistor If we now fill in some of the details about the antenna and the loading coil we are in a position to calculate the voltages across the antenna terminals and the loading coil for a given generator power and also the overall efficiency of the complete antenna system i e the proportion of the applied power which is actually radiated Any mechanism which dissipates energy i e consumes power must look electrically like a resistance The resistive part of the antenna input impedance is shown to contain two components Ra and Rr Ra is the electrical losses of the antenna due mainly to the RF resistance of the metal conductors used to make the rod and the ground plane and the dielectric losses RF heating of any insulating materials used R is the radiation resistance of the antenna i e a resistive component associated with energy radiated into space Both R and R are in some sense distributed over the whole antenna but they appear as a single resistive component Rat R in the antenna input impedance Take for example an antenna with a radiation resistance of 2Q and an input reactance of 3000Q These are the approximate values to be expected for vertical section and ground plane radial lengths of about 7 of the wavelength at the frequency of operation i e 0 07 graphs for estimation of radiation resistance for shor
172. obtained from a voltmeter If the impedance has no reactive component or if the frequency approaches 0Hz DC the general formulae above revert to their standard textbook forms P PR P V R P VI where V and I are the readings from AC or DC instruments and can be positive or negative but if V is negative then I must be negative presuming that the resistance to which the power is being delivered is positive 27 Bandwidth We are often interested the way in which the gain or loss of a network or circuit varies over a particular band of frequencies We will introduce this type of analysis shortly in connection with resonant networks but before doing so it is necessary to define the term bandwidth Most readers will be aware that an amplifier will generally show a fall off of gain at low frequencies this often being due to the increasing magnitudes of the reactances of coupling capacitors in series with the signal path and it will also show a fall off of gain at high frequencies this being due to a variety of factors including the falling magnitudes of the reactances of any stray capacitances in parallel with the signal path Consequently amplifiers and indeed many other types of circuit usually show a hump like frequency response and will only pass signals usefully over a particular frequency range The problem in defining bandwidth therefore lies in the definition of what we mean by useful and since this will vary according to the
173. of resistance such as an infinitesimal resistive region in an otherwise perfectly conducting wire We know of course that resistance is distributed throughout conducting materials but the continuity principle allows us to break energy transfer processes down into separate components which can later be combined to give the overall picture When a current flows along a conductor a resistive region gives rise to a voltage drop or potential difference Hence an electric field exists between a point just upstream and and a point just downstream of the obstacle The diagram on the right shows the interaction between the magnetic field encircling the wire and a H single electric field line other lines are left to the imagination Using the sense of the Poynting vector given earlier we see that energy flows into the resistance from both the upstream and downstream sides Now mentally rotate the diagram about the wire axis and it becomes a spherical wave front converging onto the point resistance A further important observation arises when we consider what happens when the direction of the current is reversed In that case the directions of the electric and the magnetic vectors are both reversed and so energy continues to flow into the resistance The fields explain the strange fact that the direction of energy flow cannot be reversed by swapping the polarity of the power supply It can be reversed however by replacing the resistance with a dev
174. of the applied voltage Now let us check that this is consistent with the conventional approach Here we use the power factor V I scalar product rule 10 1 P Vel V I Coso Now using the diagram on the right we can see that Cos adjacent hypotenuse is R V R2 X3 Hence P V I R VR X3 26 4 From Ohm s law we know that I V R4jX and using the magnitude ratio theorem 24 1 we obtain IH V R iX L e I V WR X Now substituting this into expression 26 4 we have P VPR R X Which is the same as equation 26 3 and so demonstrates that the power factor rule is already embedded in Joule s law when we write the latter as a properly balanced and un restricted vector equation Notice also that however we manipulate the power law the average direction of power flow is always dictated by the sign of the resistance We can incidentally also obtain equation 26 3 by using the series to parallel transformation discussed in section 19 If we write the impedance in parallel form then the power is simply given by the square of the voltage magnitude divided by the equivalent parallel resistance Thus if Z R jX R jX Then P V R where R R X R Le P VP R R2 X 65 The universal steady state electrical power laws can be summarised as follows P IPR P VPR R X P Vel V I Coso where I is the reading obtained from an ammeter and V is the reading
175. oint which should be noted on the subject of power is that power flowing from a generator to a resistance is by convention positive In DC circuits this means that if the voltage applied to a resistance is taken to be positive then the current in the resistance must also be taken to be positive and if the voltage is taken to be negative then the current is negative Also note that in AC circuits power is calculated from RMS voltages and currents This means that it is already an average or steady state quantity and there is therefore never a need to compute the RMS value it can be done out of mathematical curiosity but it is numerically not the same as 34 Vams Irms Cos see ref Thus the term RMS Watts commonly seen in the Hi Fi literature is nonsense and should be avoided 11 Phasor dot product When the angle between two vectors is taken for the purpose of computing the dot product there are actually two possible choices the acute angle lt 90 and the obtuse angle gt 90 These are shown in the diagram on the right as and q Now using the trigonometric identity Cos 180 Coso we can see that there are two possible solutions to AsB A B Coso which are numerically identical but of opposite sign Recall from the earlier discussion that there are also two possible solutions to the taking of a magnitude because it involves a square root but that by convention we take the positive answer
176. om DC to the AC steady state is a matter of replacing the battery with a sine wave generator and then adopting definitions for voltage and current such that the established DC laws continue to be true insofar as that is possible The necessary choice is to use RMS values of voltage and current where the RMS which stands for the square Root of the Mean of the Squares is an average chosen so that AC and DC electricity both have the same heating or long term energy delivery effect By using RMS values we conscript various formulae and diagrammatic conventions into AC service except that due to the alternation of the power supply it no longer makes any sense to write the symbols and against the generator terminals and it no longer makes any sense to equate current to the flow of electrons The solution is to ensure that the polarities of voltages and currents are defined in a way which gives the correct sense to the Poynting vector which is what the DC conventions achieve by rote instead of reason in any case An accepted usage which is adopted here is to draw an arrow across the generator terminals to indicate the direction in which electric potential is assumed to I increase Current then flows away from the high potential terminal If both current and voltage are then taken to be positive or both negative if you like it amounts to R the same thing the chosen convention is that energy or power flowing away from a generator
177. onnecting wires will reduce this frequency so we should expect the capacitive branch also to self resonate somewhere in a range from about 40 to 160MHz In the days before synthesisers simple single conversion short wave radio receivers used 11 An unusual tuned circuit S Chekcheyev Electronics World Jan 2004 p41 56 wide range VFOs and variable capacitance tuning For full short wave coverage however it was necessary to provide the receiver with a band switch one reason being that it was very difficult to obtain a tuning range of much greater than about an octave without changing coils The larger the coil the larger the self capacitance and so the band switch selects progressively smaller coils with progressively shorter connecting wires as the frequency is increased Winding a set of candidate coils and checking them for self resonance will quickly indicate to the designer that a set of frequency ranges like 1 2 2 4 4 8 8 16 and 16 32MHz is easily achievable but trying to reduce the number of bands to four e g 1 2 35 2 35 5 52 5 52 13 13 30 6 requires careful construction and reducing the number to three 1 3 16 3 16 10 10 31 6 is very difficult using a conventional rotary switch This does not mean that a three band solution cannot be obtained but it falls close to the borderline at which it becomes preferable to use an elaborate low capacitance technique such a turret bandchanger i e a rotating turret which
178. paration of the resistive and reactive parts of the problem A further and very important point however is that we do not use the parallel representation with a view to converting it into the series form at the earliest opportunity It is simply another way of expressing impedance and it is no less authoritative than the series form Hence if we have data for an inductor or capacitor in series form we can transform it into the parallel form and use it like that The parallel form may seem less authoritative than the series form because the expression for R equation 19 3a has reactance in it and so explicitly varies with frequency In reality however the resistive component in the series form also varies with frequency due to a variety of frequency dependent losses such as skin effect and dielectric absorption see Components and Materials capacitive and inductive coupling to resistive materials in the vicinity of the component and of course our old friend radiation Thus when solving problems using simple circuit models we need to be aware that resistances inserted to represent losses are expected to vary with frequency regardless of representation Thus the practical problem of finding the dynamic resistance of a parallel resonator becomes that of measuring the impedances of the components at a frequency reasonably close to the desired resonance transforming the losses into parallel resistances and taking the parallel combination of those
179. perience them as real by touching the junction between the rod and the loading coil not recommended In fact the electric field strength at the top of the loading coil is so great that a neon lamp or a small fluorescent tube held there will light without any wires connected to it see photograph below The actual voltage appearing across the coil Vi can be obtained by using Pythagoras Theorem i e Vi IW K2 R12 IW 30002 7 52 3000 01V and the voltage across the antenna terminals 1 e the voltage between the bottom of the rod and the ground plane is Val IW X 2 Rat Re 2 IV 30002 2 52 3000 001V These voltages are barely different from the voltages across the theoretical pure reactances and reflect the fact that reactance dominates the impedances of both the coil and the antenna but despite the reactive input impedance of the antenna we have nevertheless turned it into an effective radiator One way to look at this is to say that by resonating the antenna with a loading reactance we exploit the voltage magnification of the resulting tuned circuit in order to force power into a reactive load We could of course do the same by sheer brute force but that would involve using a generator with an output of just over 3000V to get a measly 1A into the antenna One further point to note here is that the voltages calculated by the above methods are all RMS voltages RMS as mentioned earlier stands for square Root of the Mea
180. power law by introducing the scalar product but have yet to analyse Joule s law We also introduced the the idea that if a phasor is pointing at 0 or 180 it can be treated as a scalar a trick which obviously works but for which we offered no convincing mathematical proof All of these discomforts arise because of a narrative expediency which is that of delaying the introduction of complex numbers until after that of vectors We will now resolve all of the residual issues with the aid of a handful of simple theorems which require complicated geometrical arguments if they are to be proved using vectors but are easy to prove using complex numbers These theorems incidentally are also true for real numbers which are effectively one dimensional vectors 24 1 Magnitude ratio theorem The magnitude of the ratio of two complex numbers is equal to the ratio of their magnitudes INi No Nil No 24 1 Proof Let N a jb and N a2 jb2 Then N N a jb az jb2 Now multiply numerator and denominator by the complex conjugate of the denominator N N a jb az jbo an bo m aiaz bib j bia aiby a b Now a jb V a b3 therefore N N y aia bib y T bia b2 y a b Y V a1a2 b1b2 bia2 b2ai a2 b VI aiaz b1b2 b1a2 b2a1 a 2 b Vf ar a b F b22 a b y a b27 V a b a bo VI a bi a
181. put level from an audio recorder is specified as 10dBu then the output voltage is obtained by rearranging the expression 10 20Log Vout Veer where Ver V 0 001 x 600 774 6mV Hence Vout Vret 10 Veer V10 244 9mV RMS The dBW notation was brought into European Amateur Radio documents some years ago this being the preference in the field of broadcast and professional radio engineering Thus a 400W transmitter for example becomes a 10Log 400 26dBW transmitter and it is possible to determine the effective radiated power ERP of a radio installation by adding the transmitter power in dBW to the negative gain in dB of the antenna feeder and the gain in dB of the antenna This is all very well of course but it does beg the question why for a group of spectrum users generally only equipped to measure voltage and resistance to a reasonable accuracy is it necessary to state power restrictions in a way which requires a knowledge of exponential functions in order to work out what they mean It would seem equally logical to state road speed restrictions in dBmph or dBkm h and so for the sake of any bureaucrats who might read this we will also address the question do speed ratios require the 10Log or the 20Log formula This question perhaps surprisingly is not meaningless and can be answered by noting that power is equivalent to energy delivered per unit of time A power ratio is thus an energy per unit of time comp
182. r amplifier is shown as feeding into a low pass filter LPF before connection to the antenna system Such a filter is always necessary with a broad band transistor power amplifier because such amplifiers produce relatively high levels of harmonics The push pull configuration actually cancels even harmonics but there are still high levels of odd harmonics 3rd 5th 7th etc which must be removed in engineering the first harmonic is the same as the fundamental In section 34 we noted that the power amplifier protection circuitry operates when the load impedance is too high as well as when it is too low This is not usually necessary for the protection of the amplifier but the LPF may not provide the required degree of harmonic attenuation when incorrectly terminated and so the protection circuitry helps to keep spurious emissions within acceptable limits if the load impedance is too high 42 Auto transformers Auto transformer self transformer is just another name for a tapped inductor The transformation rules for tightly coupled auto transformers are identical to those for tightly coupled transformers with separate windings The significant functional difference between the two types of transformer is that an auto transformer does not provide DC isolation between source and load A more subtle difference is that a transformer with separate windings by judicious use of electrostatic shielding can be made in such a way that the coupling bet
183. r factor correction in relation to electricity distribution is equivalent to the business of bringing an antenna system into resonance The reactance cancelling step in antenna matching and the insertion of a loading coil into a vertical antenna can both perfectly well be regarded as a forms of power factor correction Now since power can only be delivered to the resistive part of an impedance only that part of voltage multiplied by current which corresponds to true power i e the IPR component can be measured in Watts The reason is that power the amount of energy delivered or work done in unit time establishes the relationship between electricity and thermodynamics and the connection is through energy dissipation It is therefore the convention in electrical engineering to express apparent power in volt amps VA and only true power in Watts Many readers will already be aware that mains transformers and portable electric generators for example are rated in VA the implication being that to get the full power output without over stressing the device it is necessary to make the apparent power in VA equal to the true power in Watts i e to provide the transformer or generator with a resistive load Since maximum power output will be associated with a particular value of load resistance it transpires that all generators not just radio transmitters require impedance matching if the maximum allowable output is to be obtained One further p
184. ranch Notice that if we allow Ry to 51 become extremely large then practically no current will flow in the inductive branch and the circuit will not resonate The same argument applies of course to the capacitive branch The parallel resonance formula can therefore be seen to tell us that true i e real resonance cannot occur if the resistance in either branch rises above a certain critical value that value being the square root of the L C ratio the characteristic resistance i e Ro V L C If the resistance in one branch rises above V L C then the current in that branch will always be too feeble to bring the system into resonance What happens instead is that the phase of the total current I can approach and move away from the phase of the generator voltage as the frequency is varied but it is never able to reach it The resonant frequency is simply the imaginary frequency of closest approach and it does not exist on the real frequency line It is imaginary because the combined impedance of the two branches can never become real 1 e resistive by cancellation Note however by inspecting the circuit that the combined impedance does become resistive at zero and infinite frequencies but that this is not due to cancellation At OHz X 0 and Xc o means approaches or tends towards so the impedance is simply R and at infinite frequency X and X 0 so the impedance is Rc Ifa real resonant frequency does
185. re the same type of object A resistance in parallel with an impedance is an impedance A resistance is simply an impedance which happens to have its imaginary part equal to zero This means incidentally that the preferred pseudoscalar symbol for Z is usually R rather than Z 17 2 A reactance is not an impedance The statement Z 68 114 has a completely different meaning to the statement Z 68 j114 the first is a resistance in parallel with a resistance the second is a resistance in parallel with a reactance Mathematically a reactance cannot be combined directly with an impedance but a reactance can be converted into an impedance by multiplying it by j Looking at this another way impedance and reactance have reference directions which are 90 apart To make them compatible it is necessary to rotate one of them through 90 17 3 Scalability is preserved When the double slash notation is used to create a mathematical object i e the same type of phasor exists on both sides of the symbol it has the useful property that a common factor can be multiplied in or divided out of the parallel object i e sZ sZp Zi Z2 Proof SZ sZ2 sZy Zp Z Z2 s Zi Z2 Zi Z s Z Z2 17 4 The associative rule The double slash notation can be extended to represent any number of impedances in parallel Zi Zo I Zs I Za 1 Z 1 22 1 23 1 Zn and the associative rule o
186. resistances applying ordinary arithmetical operations to the phasors without knowing what they mean we end up with the expression Z Z Z Zi Z The problem now is that of how to interpret this equation a somewhat inconvenient matter if we continue to define phasors as comma separated lists but it transpires that there is a short cut due to the fact that we are only dealing with two dimensional vectors which is that such vectors can be treated as complex numbers Complex numbers were first discovered as a necessary evil in solving quadratic equations i e equations which can be written in the form ax bx c 0 They were once described as the work of the Devil but in fact they merely indicate that ordinary numbers are not the whole story Those who studied quadratic equations at school but never got as far as complex numbers may be surprised to learn that all of the examples they were given were deliberately chosen so as not to involve complex numbers and that education systems in general expend more effort trying to protect students from the knowledge of complex numbers than they expend trying to teach the subject A derivation of the general solution for all quadratic equations is shown in the box below General Solution for Quadratic Equations Obtaining the general solution to all quadratic equations is a matter of re arranging the general form ax bx c 0 so that x is all alone on one side of the equation We can start by subtra
187. ries resonance formula is an approximation for the expression we are about to derive we expect the result to look like the series resonance formula with an additional correction term or factor We can begin by multiplying out the first two brackets Hence 2mfoL C L 2nfoC R 2foC 2afoL Rc 0 Now we will put all of the terms containing 27f on one side and the terms containing 1 27f on the other 2nfo LRe L C 1 2rfo RL C L C Then multiply both sides by 27f and divide both sides by LRc L7 C 2nfo R C L C LRe L7 C and factor out 1 LC from the right hand side 2nf 1 LC R L C Re L C Here we will also multiply top and bottom by 1 to put the L C terms first L C generally being much larger than the resistance squared terms hence 21f 1 LC L C R L C Re which rearranges to L C Ri 15 2 L C R re In VLC Thus we find that the resonant frequency of a parallel tuned circuit is the same as that for a series tuned circuit except for a correction factor V E L C Ri2 LIC R2 which is usually close to unity Notice that this factor is equal to 1 if Ri and Rc are zero and also that the factor is 1 when R Rce Example A 3uH coil is connected in parallel with a 42pF capacitor The approximate resonant frequency is 1 2nN LC 1 2nV 3x 1042x1077 14 178649MHz In the region of 14
188. rise to a considerable simplification viz Ryo Ri Xe R i e Ryo Xe Ri If we apply this formula to our example data we obtain Ryo 71432 56399 2 35 7163KQ In this case the deviation from the true value is 1702 6Q or 5 which may be a reasonable approximation for many purposes but needs to be treated with caution Also the failure to eliminate reactance from the formula makes computation more difficult 17 Double slash notation In geometry the expression AB CD means the line drawn from point A to point B lies in parallel with the line drawn from point C to point D Hence by existing convention the symbol means in parallel with In electrical engineering of course we are frequently interested in circuits in which components are connected in parallel and so we can usefully adapt the double slash notation to have a non geometric meaning We can for example re state our basic parallel component formulae as follows R Ro Ry R2 Ri R X X2 X X2 X X2 Z Zo Z Zo Zi Zp L L2 L L2 L Ll and possibly but best avoided Ci C C C This convention is often convenient because it saves the bother of having to define a temporary variable to represent the parallel combination i e instead of writing Let R represent the parallel combination of R and R and then having to remember what R is we simply work with the quantity R R2 which can be expanded or c
189. rough a formula according to the rate of change of the formula with respect to the variable Thus the error contribution from a variable is the partial derivative of the formula with respect to the variable multiplied by the deviation in the variable Hence if x m M Ms and the ESDs of the measured quantities are 0 62 63 etc the contribution which the variable m makes to the ESD of x is given by Ox of Om O1 and so on strictly we should take the modulus of the derivative because standard deviations are by definition positive but it does not matter in this case because orthogonal addition involves squaring 89 of the error contributions Hence the analytical form of the error function is o V ef 6m o Af Amz o2 Cf Em3 o3 Example The output impedance of a generator is obtained from the formula Rg R R Nv 1 R2 R Nv Differentiation of this function requires the use of the quotient rule If y N D then dy dx DdN dx NdD dx D where in this case the numerator is N R R2 Nv 1 R RoNy E R R gt and the denominator is D R2 R Ny Differentiating the numerator with respect to each of the variables gives ON OR R2 Ny 1 ON OR2 R Nv S 1 ON ONy z RR and differentiating the denominator with respect to each of the variables gives OD OR Ny OD OR2 1 OD ONy Ri Using these results we obtain OR OR D ON OR N OD OR D R2
190. s i e phase vectors or carriers of phase and diagrams involving them as Phasor Diagrams The special properties of phasors as distinct from vectors in general are as follows 26 The phase co ordinate is defined in relation to other phasors rather than to an absolute time reference Time is measured in degrees or radians relative to one cycle of the frequency at which the analysis is being carried out A phasor deemed to be pointing at 0 may be replaced by its magnitude and a phasor deemed to be pointing at 180 may be replaced by the negative of its magnitude Phasors are strictly two dimensional i e the vector cross product which produces a new vector at right angles to the original two has no meaning for phasors Shown below is a phasor diagram illustrating what happens when an impedance consisting of a resistance an inductance and a capacitance in series is connected across a generator We can easily deduce the total impedance by inspection in this case but notionally it is obtained by regarding the individual series elements as phasors Zp R 0 Z 0 X1 and Z 0 Xc and adding them together Thus Z R 0 Zi 0 X1 Zc 0 Xc ZR X1 Xc We can draw the resultant phasor Z by moving along by a distance R and moving up by a distance X Xc or down if X Xc is negative but notice that in the diagram the resistances and reactances have all been scaled by a reference phasor I which is equal to the
191. s are replaced by their internal impedances This is entirely logical when we think of the generator as an ideal generator in series with an impedance because the ideal generator part of the model is a short circuit with regard to any power reflected back into the network Th venin s theorem simply takes this observation to its logical conclusion by allowing that the network can be represented as a single generator which may be characterised completely by knowing its output impedance and its off load voltage and its output spectrum but in a linear network we can treat each frequency component separately and so need only consider sine waves Notice that in the preceding section we could have obtained the output impedance of the potential divider directly by using Th venin s theorem i e with the generator replaced by a short circuit it is obvious that the impedance looking back into the network is Z Zp This is an extremely useful trick but even more useful analytically is the technique of replacing a complicated network with a single generator and a series impedance i e the output impedance The replacement network is known as the Th venin equivalent 38 Measuring source resistance In the test setup shown below the output voltage of a generator is measured with two different load resistances all other variables being kept constant and the circuit being constructed in such a way as to minimise stray capacitance and inductance i
192. s electromagnetics or which is not well described by treating it as a discrete ideal component It will be noticed that we have talked of resistances capacitances and inductances rather than resistors capacitors and inductors There may be no great difference between the two conceptions from the point of view the audio engineer but for the radio engineer ideal components and practical components are not the same Whether a practical component can be regarded as an ideal component is a matter of internal dimensions and wavelength Some capacitors for example are made by rolling up long lengths of metallised plastic film giving an assembly which behaves quite like a pure capacitance at low frequencies but turns into an inductance at radio frequencies thus radio engineers have a certain fondness for capacitors which have small electrodes Similarly an inductor is made by coiling up a length of wire and its property of pure inductance is modified by the resistance of the wire and by the time it takes for an electromagnetic wave to propagate along it Even the humble resistor is not perfect and in general every practical two terminal device is replaced by an equivalent circuit which may sometimes correspond to a discrete ideal linear component at low frequencies but at high frequencies will always mutate into a network of resistances capacitances and inductances The subject of component models is developed in detail in later article
193. s however far from it The complex number form is just another two dimensional vector representation which complements the rectangular and polar forms we have already met In fact it is merely a version of the rectangular form in which the 90 difference between the dimensions is imposed by the j operator and a vector always behaves in the same way regardless of how it is defined This minor change makes a huge difference however because it allows a phasor to be written as an ordinary algebraic sum An expression with j in it might not seem ordinary of course but it is so in the sense that the existence of j is required by the rules of common arithmetic and so j is by definition subject to those rules The complex form of a phasor makes the rectangular form effectively redundant The transformations from the complex to the polar form are given below and are very similar to the transformations given earlier in table 6 3 Complex form Polar form Z R jx Z R X Arctan X R 12 4 Z Z Cose jSing Z Z Notice also that j can be regarded as a phasor operator because its effect on an algebraic expression is to turn that expression into a phasor another good reason for writing j in bold Hence in the matter of writing properly balanced vector equations we may note that if a live phasor i e one which has not been turned into a scalar by taking a magnitude or a scalar product exists one one side of the symbol th
194. s in this series but for now the important point is that circuits designed on the assumption of ideal behaviour may require extra work if they are to be realised in practice Neglecting possible but usually minor non linearities the difference between ideal and practical components can always be attributed to a combination of three factors time delays displacement currents and the finite resistance of conductors at ordinary temperatures The same can also be said of wiring and layout Hence an accurate theoretical model for a practical electrical system may also require components to represent spurious effects in the circuit at large In the case of extreme time delays i e long connecting wires we can introduce a two port module called a transmission line which is a solution of Maxwell s equations in a box Minor effects can be accounted for by including stray or parasitic resistances capacitances and inductances here and there In the absence constructional details and related corrections however it is implicit in any theoretical circuit that the components will be grouped on a scale which is small in comparison to wavelength that interconnection resistance is negligible in comparison to specified resistance and that there may be a need for shielding to prevent energy from turning up in places where it is not wanted Having cautioned against the pitfalls however we should also caution against over modelling Even at radio freq
195. s innocuous enough but what happens when 4ac is larger than b In that case the solution for x has a term containing the square root of a negative number i e a number which is negative when multiplied by itself even though the basic rules of arithmetic demand that when a number is squared the answer must always be positive Take for example the seemingly innocent quadratic equation x x 1 0 In this case a 1 b 1 and c 1 and the solution is x 1 2 V 3 2 The best simplification we can manage is to factor out the square root of 1 i e x 0 5 0 866V 1 Thus there are two solutions x 0 5 0 866V 1 and x 0 5 0 866V 1 both of which contain a part which is a real number and a part which is not a real number That which is not real is imaginary and so the oddball quantity V 1 was given the symbol i by Leonhard Euler 1707 1783 and this symbol is still used by mathematicians When it became apparent to scientists researching into electricity that this branch of mathematics might be useful however the symbol i had already been allocated to represent current and so the next letter in the alphabet j was allocated for use in conjunction with electrical problems here we will write the symbol in bold to make it easier to spot Thus we can write the unsimplifiable solution to the previous example as x 0 5 0 866j That which is not simplifiable is complex and so in this case x is a complex number
196. sents the flow direction as the tail fins of a receding dart The right hand diagram shows the field when the current is flowing towards the observer the dot being interpreted as the point of an approaching dart Notice that we use a convention established before the discovery of the electron by J J Thomson in 1897 which is that current flows from to Electrons flow the other way but the continued use of conventional current makes no difference to the theory and serves to preserve the intelligibility of past scientific literature In the case of a magnetic field the arrows drawn on the lines of force show the direction in which a compass will point when placed in proximity to the wire presuming that the current is large enough to overcome the Earth s magnetic field The clockwise rotation of the field lines when the conventional current is flowing away is known as Maxwell s corkscrew rule This rule derives from the convention that the field lines around a bar magnet or compass needle emerge from the North seeking pole and return to the South seeking pole Magnetic bodies repel when their force lines are in opposition North pole to North pole and attract when their force lines point in the same direction North pole to South pole Hence a compass needle is repelled by the field lines coming towards it and attracted to the field lines going away from it The Maxwellian fields have the same geometric properties as Faraday lines a
197. sing a bridge circuit often called a reflectometer or SWR bridge but really an impedance bridge balanced for a particular value of resistance An interesting discussion of the conditions which provoke transistor failure is given by Bob Pearson If the protection circuitry is correctly designed and adjusted the pseudo output impedance should be the same as the preferred load resistance When determining the effect of source impedance on the Q of antenna systems and bandpass filters however it is the true output impedance not the preferred load resistance which must be used Unfortunately this quantity is often impossible to obtain from the manufacturer s data but as we shall see shortly it can be measured with the aid of two dummy load resistors of different value 35 The potential divider The potential divider is the simplest three terminal electrical network We have made some use of its properties already but there comes a point when it is useful to characterise it formally Here we will do so for the general case which is that of defining the voltage at the intersection of two impedances Referring to the diagram Vout I Zi where since Vin I Z Z2 T Vin Zi Zz hence Vout Vin Zi Zi Z2 35 1 and if we multiply the right hand side by Z Za Vout Vin Z Zn Zp 35 2 Note that Z is the sum of the source impedance and any additional impedance placed in series with the generator Vin is the off lo
198. single deduction made by James Clerk Maxwell in the latter part of the 19th Century Maxwell collected the details of every known scientific result concerning electricity and magnetism and lent his phenomenal mathematical skill to the problem of finding a single theory This led him to discover inconsistencies in the laws of induction i e those laws which govern the effects of time varying electric and magnetic fields which would nowadays be interpreted as violations of the principle of conservation of charge Electrons had not been discovered at the time and electric charge was thought to be some kind of fluid but whatever it was the physical ideas of the day did not permit it to disappear from one place and reappear in another He fixed the problem by making a bold and unprecedented step which was to postulate the existence of a new kind of electric current not associated with flowing charges which he called displacement current Speculation is one thing but Maxwell had a test for his theory With the inclusion of displacement current the modified laws still allow the existence of electricity when all of the terms relating to physical matter are deleted This free electricity has to be in the form an oscillating electric field combined with an oscillating magnetic field with the directions of action of the fields disposed at right angles It had turned out that the fields do not represent mysterious action at a distance after all althoug
199. sistances in series is the simplest of all and tells us that whenever we encounter two resistors in series we can treat them as a single resistor with a value equal to the sum of the two resistances That this statement is derived from existing physical laws can be seen by applying some basic techniques of circuit analysis to the circuit shown below Resistors in Series To analyse this circuit we first observe that as a requirement of Kirchhoff s I first law the current in the two resistors must be the same Ohm s law then s tells us that V IR and V2 I R2 Now since voltage is analogous to pressure common sense otherwise known as Kirchhoff s second law tells Ri V1 us that the total pressure drop is equal to the sum of the pressure drops A J across the two resistors i e V V V R2 V2 Putting these ideas together we have eeeseses ssSsSY f V I R R I Ri R Now if we postulate a hypothetical resistance R which represents the series combination of R and Ro it must be possible to replace R and R with this resistance and obtain the same current for a given voltage i e V IR I1 R R Hence R Ry R 17 Resistors in Parallel In the case of two resistors in parallel the voltage across the two resistors is the same Hence Ohm s law tells us that I V L R lL R TETE ET 3 1 I and Kirchhoffs first law tells us that 2 I l h amp R2 Now if we postu
200. ssion problems boil down to the matter of delivering power to an impedance 13 The Art of Electronics Paul Horowitz W1HFA and Winfield Hill 2nd edition 1989 Cambridge University Press ISBN 0 521 37095 7 Tunnel Esaki diode p14 15 amp p1060 Back diode p891 893 14 Physical Electronics C L Hemenway R W Henry M Caulton Wiley amp Sons New York 2nd edn 1967 Library of Congress cat card no 67 23327 Section 14 6 The tunnel diode p290 294 64 Consider the system shown on the right In this case V and I are not I necessarily in phase but we can easily obtain an expression for I in terms of V and since we now appear to have a version of Joule s law which allows I to l j xX point in any direction there is no need to impose a restriction on any of the A phasors involved Thus we can write gt p I V R jX B Now putting the reciprocal impedance into the a jb form by multiplying numerator and denominator by the complex conjugate of the denominator we obtain I V R jX R X and using the conjugate product theorem 24 7 I V R jX R X Now we can insert these definitions into equation 26 2 P I RI thus P VVFR R 2 X and using the square magnitude theorem 24 6 we obtain P VPR R X 26 3 Thus without any convoluted discussion about phases or reference phasors we have obtained in a few lines of algebra a general expression for the power dissipated in an impedance in terms
201. t 0 by virtue of a choice made elsewhere then it too can be replaced by its magnitude It is however important to understand that there is a difference between a vector which has dropped a dimension and a magnitude because there will be 6 Elementary Particles Enrico Fermi Silliman memorial lecture series Yale University Press 1951 Definition of pseudoscalar p9 25 many circumstances in which we will want to use the magnitudes of vectors which are free to point in any direction In particular we will need this distinction later in order to generalise Joule s law It will however become apparent that adoption of the convention that vectors written as scalars i e un bold are pointing at 0 or 180 preserves the meaning of most of the DC and pure resistance only formulae which appear in standard textbooks The correspondence arises because whenever a vector is written as a scalar a statement is made to the effect that the phase of that vector except for the sign can be ignored A DC formula works for AC when the circuit contains only pure resistance because in that case rotating one vector to point at 0 rotates all of the others to point at 0 and so they can all drop a dimension Hence V IZ becomes V IR for example One consequence of all of this is that in formulae we should avoid writing voltages and currents as scalars unless we really mean them to be pointing at 0 or 180 We must however permit a common conve
202. t antennas are given in the references listed below If we are very careful about the materials used in the antenna system we might keep the loss component Ra down to about 0 5Q so that the input impedance of the antenna will look like 2 50 of resistance and 3000Q of reactance To cancel the antenna reactance X 3000Q we need to place a coil having X 3000Q in series with it Such a loading coil is normally placed out with the antenna mainly because coils inside metal boxes have more losses than coils mounted in wide open spaces but even so the coil will not be perfect and will have a distributed RF resistance which looks like another resistive component in the antenna input impedance The amount of coil resistance is given by the Q of the coil which is the ratio of reactance to loss resistance i e Q X1 Ry A well made loading coil might have a Q of about 400 and so Ri X1 Q 7 5Q With the reactance of the antenna now cancelled by the coil the input impedance of the whole antenna system now looks like a pure resistance of 2 5 7 5 10Q Suppose we now decide to deliver 10 Watts 10W from a generator transmitter into this 10Q resistance Knowledge of the power level enables us to calculate the antenna current and hence the voltages which appear between the various terminals but a word of caution is in order before using the standard power formulae for this purpose The expressions P IV P R and P V R are all deeply s
203. tance and reactance Resistance R is that property of the network which enables it to dissipate i e consume or dispose of energy and reactance X is that property which enables it to store energy It is also a special property of our Universe that while energy dissipation is on average a one way process the electromagnetic energy storage mechanisms come in the form of a complementary pair This means that true resistance is always 20 positive a statement which we will qualify later but there are two opposing types of reactance which of course we know as inductive reactance X 2afL and capacitive reactance Xc 1 2mfC Inductive reactance arises through the storage of energy in a magnetic field and capacitive reactance through the storage of energy in an electric field When inductance capacitance and resistance are combined within the same two terminal black box the opposing reactances will always tend to cancel out to some degree and so the two types of reactance make only a single contribution to the impedance at any particular frequency There is however no way in which resistance and reactance can be combined to form a single numerical quantity because the physical processes they represent turn out to be mutually exclusive A natural distinction arises between resistance and reactance because perfect energy dissipation implies that the Poynting vector never reverses whereas perfect storage and return implies alternation w
204. tem In this case all we have to do is note that the total input resistance was 10Q whereas the radiation resistance was 2Q This gives an efficiency of 2 10 or 20 i e 2W radiated for 10W in 20W radiated for 100W in This incidentally is not a disaster and represents a good figure for a short loaded vertical around 10W of SSB radiated from a reasonably high location being sufficient for worldwide short wave communication in suitable atmospheric conditions 31 Voltage magnification in action For the antenna in the illustration on the right the frequency of operation is 1 84MHz and the physical length of the antenna assembly is 1 45m from the bottom of the rubber mounting base to the top of the neon lamp soldered to the tip The section above the loading coil is 0 76m long 0 00472 The clamp holding the fluorescent tube is made from acrylic resin Perspex and there is no electrical connection between the tube and the whip The glow from both lamps is visible at an input level of 1W but since the photograph was taken on a bright summer s day albeit in the shade the power input to the antenna matching network AMU was turned up to 100W to overcome the daylight The antenna is one of the author s old 160m mobile whips from the early 1970s Itis not an optimal design but it gave useful service a range of several miles using 1W o AM despite having an efficiency of considerably less than 1 The long thin shape of the coil does
205. the energy injection network the source Rpo is the dynamic resistance of the resonator and Rroaa is the input resistance of the network to which energy is being delivered The inductance and capacitance are defined in their parallel forms so that the dynamic resistance can be treated as a separate parallel element To be continued
206. the latter pronounced line and being short for log Naperian Hence working in base 10 if m 10 and n 10 then mxn 10 All we have to do to perform the multiplication is look up the logarithms a and b add them together then look up the quantity a b in a table of anti logarithms to find the required mxn The anti log of a number x is simply 10 As an alternative to using tables the same operations can be achieved by using two identical engraved logarithmic scales and sliding one relative to the other the device for so doing invented by William Oughtred in 1622 being known as a slide rule We no longer need to use logarithms for everyday multiplication but we do need to memorise some of their basic properties in order to use logarithmic units with confidence The first and most fundamental point is that any number raised to the power of zero is one i e B 1 always regardless of B Hence Log 1 0 always regardless of base Now if we represent power gain in deciBels i e N dB 10Lo0gi0 P Pret then if P Prer N dB 10Logio 1 0 i e a system which neither amplifies nor attenuates a signal has a gain of OdB Also we find that if P is greater than Pres then N dB is greater than zero and vice versa Hence a positive quantity in dB represents gain and a negative quantity represents loss i e negative gain Finally the additive property of logarithms allows that if we subject a signal to a number of pro
207. them Because vectors are different from ordinary numbers however it is helpful to note each one as a letter in a bold typeface or in handwriting by putting a little arrow above the symbol and an optional comma separated list in brackets may be included to denote its extent in its various dimensions Thus we can represent an impedance as Z R X by 21 which we mean that Z is characterised by an amount R in the resistance dimension and an amount X in the reactance dimension In the context of vectors ordinary numbers are known as scalars because the effect of multiplying a vector by a scalar is to scale it i e magnify or shrink it without otherwise changing it Thus if s is a scalar we can write sZ R X Z sR sX Note also a widely used mathematical notation which is to use an apostrophe or prime to indicate that an object has been modified We can immediately deduce a rule for adding vectors by observing that two quantities will only add together if they exist in the same dimension you can t increase the length of an object by adding to its width Thus if we want to add two impedance vectors Z Ri X1 and Z R2 X2 i e find out what happens when the impedances are placed in series all we have to do is add the R parts and the X parts separately to find the new impedance Z Ri R2 Xi X2 This operation is indicated by the symbol just as in ordinary arithmetic i e if ZR X Z Ri X1 ZR X2 then R R
208. to the definition of impedance itself Now however it follows that when some mixture of resistance and reactance is connected across a generator the angle for the voltage current phase difference will lie at some intermediate value and the use of vectors allows this angle the phase angle to be determined from simple geometry More to the point a phase angle of 0 implies that an impedance will absorb all of the power delivered to it and a phase angle of 90 implies that an impedance will not accept any power Thus we can observe that the phase angle represents not only the relationship between voltage and current for an impedance but also the effectiveness with which power can be delivered to it 8 Phasors As we have just shown one of the interpretations of Ohm s law is that if an impedance vector is scaled by a current magnitude it is transformed into a voltage vector Since the act of scaling a vector does not change its direction it transpires that both the impedance vector and the voltage vector contain the same phase information and that this information is conserved after multiplication by a scalar Put in plain language this means that although the current through an impedance will change according to Ohm s law as the applied voltage is changed the V I phase relationship will not change provided that the frequency is held constant It is for this reason that vectors used in impedance related applications are known as Phasor
209. true that a b The magnitude operation discards the directional information of a vector and the sign information of a scalar Note for example that although la a al a any one of the quantities inside magnitude brackets is definitely not identical to any of the others The magnitude retains only the length of the object In so doing however it does retain the unit of measurement i e a magnitude is a length in impedance space or voltage space or current space etc and so has the units of the space in which it exists 24 10 Magnitude Equivalence As noted above for any complex number Z Z Z Z Z When designing electrical circuits it is not unusual to meet situations in which the magnitude of a voltage or current needs to be determined but the phase is unimportant As the theorems above show when only the magnitude is needed all of the impedances involved in the calculation can be replaced by their magnitudes provided that the impedances are factors i e multipliers or divisors not terms in a summation What is less obvious however is that an impedance Z enclosed between magnitude brackets can then be replaced by one of the alternatives having the same magnitude namely Z Z and Z This principle of Magnitude Equivalence allows us to deduce alternative networks which will produce the same outcome In particular it allows us to identify situations in which inductance can be replaced by capacitanc
210. ude theorem The product of a complex number and its complex conjugate is the square of the complex number s magnitude N N NP 24 6 Proof Let N a jb and N a jb N N a b but IN V a b3 therefore N N N Hence the product of a complex number and its complex conjugate is a true scalar It is also literally a scalar product Recall that the definition of a scalar product is aeb a b Coso but if a and b are identical then 0 and Cos 1 Hence NeN N N N 24 7 Conjugate product theorem The complex conjugate of the product of two complex numbers is the product of the complex conjugates N N2 Ni N2 24 7 Proof Let N a jb and N a2 jb2 Then N No a jb a2 jb2 a bib j aib2 arb Therefore N N2 aa bib j aib2 arb ai a2 jb2 jbi a2 jb2 a jb az jb2 N N 61 24 8 In phase quotient theorem If two phasors are in phase their ratio can be treated as a scalar Ni iNi No Nol Nil IN 24 8 proof Using the polar to complex transformation 12 4 Ni Ni Ni Cos jSing N2 No P N2 Cos jSing where the phase angle is the same in both cases Therefore N N2 Ni NJ N No 24 9 Magnitude Caveat The mathematical operation of taking a magnitude destroys information Specifically it is important to be aware that if a b then it is not necessarily
211. uencies there need not always be a great deal of difference between the idealised and the practical representations This is because firstly circuits will often work adequately when built according to the assumption that practical components are nearly ideal and secondly nominal 14 component values are subject to manufacturing tolerances which means that accurate performance can only be achieved by making some components adjustable Adjustment not only serves to correct the nominal component value to the required value it can also absorb deficiencies in the original analysis Hence it is important to understand that a simple approach will often do the trick and extreme attention to subtleties is usually only needed when attempting to achieve the most exacting standards of radio frequency RF performance 3 Basic electrical formulae The theory which will be developed in the sections to follow is of a type classed as steady state analysis This does not mean that nothing changes but that everything which does change is assumed to do so sinusoidally i e there are no sudden or transient events We can always represent more complicated phenomena if necessary by adding together sine waves of different frequencies or we can look to other bodies of theory better suited to systems which undergo non periodic i e non cyclical change but for a large range of problems the steady state approach is all that is needed The translation fr
212. uency from being reached and then to make R even larger by minimising loss resistances in order to obtain a useful working Q 46 Unloaded Q of parallel resonator The expression for Q obtained in the previous section is sometimes referred to as the loaded Q of the resonator because it is the Q which results when the source and load impedances are taken into account We may also imagine that the resonator has an unloaded Q which is that which obtains when the source and load are disconnected It is not immediately obvious why we should wish to employ such a concept because it is impossible to use the resonator without coupling to it in some way but it is nevertheless useful because it sets an upper limit on the Q which can be obtained in a practical circuit It is obviously obtained by substituting the dynamic resistance in place of Rg in 105 equation 45 1 i e Qou Rpo V L C but it would be a lot more useful intuitively if we could express it in terms of the coil and capacitor impedances in their series R jX forms We can do so by using the series to parallel transformation section 19 and using the definition of Q from section 31 as precedent we expect a result in the form Qou V L C R 46 1 the point being to find out what is meant by R in this case The translation from parallel to series form is indicated in the set of equivalent circuits shown below Here we identify Ryo as Rcp Rip 1 6 Ry
213. uit on a particular physical scale A given amount of phase error does not necessarily translate into the same error in some other quantity but if we confine ourselves to thinking of the order of the error i e its magnitude thereabouts it is a fairly good guide On that basis we can answer the question If I build the circuit as drawn how well will its performance agree with my analysis The scale on which attention to physical detail layout wire lengths component size etc will be required for a given level of agreement is shown for various frequencies in the table below using the mental arithmetic approximation c 3x108 m s 12 Frequency Wavelength Construction scale for a given accuracy f A c f 10 1 0 1 300KHz 1Km 100m 10m Im 3MHz 100m 10m lm 10cm 30MHz 10m Im 10cm lcm 300MHz Im 10cm lcm Imm 3GHz 10cm lcm Imm 0 1mm Assuming that most signal processing circuits are constructed on a scale of about 10cm we can see that there will be no serious discrepancies between analysis and practical results for frequencies up to about 3MHz ignoring displacement currents for the time being Beyond that we are definitely into the realm of radio frequency engineering where layout is important but note that this does not mean that analysis will fail Rather as was alluded to earlier it requires a modified approach where some physical variables have to be turned into theoretical circuit components As we approach ultra high frequenci
214. uspect because if we try to convert them into vector expressions simply by changing the voltages and currents into vectors then the equations which result will be nonsense because power energy per unit of time is scalar strictly pseudoscalar as we will see later We need to develop some additional ideas on the subject of vectors before this matter can be fully resolved but the standard expressions will balance if all of the vectors involved can drop a dimension Hence we can concede that the expressions are true for voltage and current magnitudes provided that the generator is driving a purely resistive load a somewhat restrictive condition but we happen to satisfy it here Thus using the standard formulae we can work out that the current in the antenna will be 7 Efficiency of Short Antennas Stan Gibilisco W1GV Ham Radio Sept 1982 p18 21 Graphs of radiation resistance vs electrical length for short verticals and dipoles Efficiency calculations 8 How long is a piece of wire J J Wiseman Electronics and Wireless World April 1985 p24 25 Discussion of the efficiency or lack thereof of electrically short verticals The effect of top loading 30 I 1 V P R V 10 10 1A and the voltage at the generator will be IV V V PR V 10x10 10V Note however that the current will result in a voltage of 3000V IX across both reactances and although these voltages are cancelled at the generator we can certainly ex
215. very small then the secondary voltage will be in phase with the primary voltage and the secondary current will be in anti phase with the primary current i e as a current appears to flow into the primary a current appears to flow out of the secondary If the number of turns in the secondary winding is greater than the number of turns in the primary then Vs will be larger than Vp and vice versa and the voltage transformation will be in proportion to the turns ratio i e Vs Vp Ns Np sens 41 1 It follows that if the power produced by the generator is transferred to the load without loss then the VI product will be conserved which means that if the voltage is stepped up then the current will be stepped down to keep VI very nearly constant and vice versa This implies that the transformer performs on the current the inverse of the transformation it performs on the voltage i e interpreting the currents in the sense of the arrows in the diagram above Is IpNp Ns 41 2 Now by definition the impedance looking into the transformer primary is Z Vp Ip which gives using 41 1 and 41 2 as substitutions Z Vs N Ns I Ns Np and since Z Vs Is Z Z Np Ns 41 3 Thus to a reasonably good approximation a tightly coupled transformer having relatively large winding reactances scales an impedance according to the square of the turns ratio Now let us consider the problem in reverse and see what a transformer
216. viation from the preferred load resistance is likely to provoke a transistor power amplifier s protection circuitry For a transmitter designed to operate into a 50Q load 25 and 100Q dummy load resistors correspond to the upper and lower 2 1 SWR points and should give an easily discernible output voltage difference A 25Q resistor can be had by connecting two 50Q dummy load resistors in parallel with a coaxial T piece 100Q coaxial resistors are less readily available but an old fashioned 75Q load will do instead In the calculation the actual resistance of the load measured with an accurate resistance meter should be used rather than the nominal value stamped on the resistor Example The output voltage of a Kenwood TS430S 100W HF transmitter was measured with two different dummy loads The measurement frequency was 1 9MHz and the test power level was very approximately 1W One load was a 75Q nominal coaxial resistor measuring 75 1 0 7Q the other was the combination of this resistor and a 50Q nominal coaxial resistor in parallel with it the combination measuring 29 6 0 3Q The voltage ratio was measured using an oscilloscope with a 10MQ x10 probe The resistors and the probe were attached directly to the antenna socket using coaxial T pieces no cables The measurement was made by attaching and removing the 50Q resistor from the T piece with the transmitter running and noting the change in the peak to peak excursion of the output waveform Using th
217. viations calculated by the incremental spreadsheet method used previously Finally we have o 3 414 Rg 23 3 43 40 A spreadsheet version of this calculation which can be used as a template is given on sheet 2 of the accompanying file Rg_meas ods 40 Antenna system Q In previous sections we showed that conjugate matching is not necessarily a good idea and that radio transmitter manufacturers do not necessarily design power amplifiers to work into a conjugate load Besides the obvious advantages in terms of output regulation and efficiency however there is a further reason to load a radio transmitter lightly in cases where the input impedance of an antenna system changes rapidly with frequency or is subject to variable environmental factors Recall from section 34 that the true maximum power transfer condition occurs when R V Re X X4 Consequently if the antenna system is subject to disturbances which can cause a reactive component to appear after matching has been carried out then the best average maximum power output will be obtained when the load resistance is somewhat higher than the source resistance Possible causes of transient residual reactance are many and various including changing physical environment of mobile and portable transmitters wind rain component heating birds etc and as is easily forgotten modulation In section 9 we discussed an electrically short inductively loaded vertical antenna system The
218. ween the primary and secondary windings is almost entirely magnetic An auto transformer will always exhibit some stray capacitive and resistive potential divider coupling and so if its inductance is part of a filter circuit the filter may exhibit poor attenuation of signals outside its passband The step up and step down auto transformer configurations are shown below Also shown is the somewhat redundant 1 1 auto transformer otherwise known as an inductor the point in including it being to draw attention to the inductive reactance which every transformer places in parallel with its load Zo Step down It should be obvious by inspection of the 1 1 auto transformer circuit that the voltage current relationship for the load seen by the generator is given by VA jXL Z If the coil has losses moreover we can represent these as a resistance Ri say in series with the coil VI RL jX_ Z We can also transform the impedance of the coil into its parallel form see section 19 in which case the load on the generator becomes V T Rip jX1p Z The implication is that unless the magnitude if the inductive reactance is very much larger than the magnitude of the load impedance the transformer will not preserve the load phase relationship If the load is reactive the parallel loss component will also alter the load phase relationship slightly In the previous section we introduced the idea that an impedance located on one sid
219. west electromagnetic frequency which can be encountered in practice is the reciprocal of the age of the universe about 1 13 72x10 x365 242199x24x60 2 31 lt 10 Hz by current reckoning Practical DC electrical systems come nowhere close to that because even the best stop working after a few years Hence zero frequency is impossible we employ the concept merely as a mathematical convenience for the purpose of circuit analysis Note incidentally that some writers have been moved to claim that there is a flaw in Maxwell s equations because electrical formulae tend to produce infinities when frequency is set to zero This is a fallacy of course the infinities being merely a reflection of the fact that zero frequency is not a property of the Universe A practical consequence however is that annoying divide by zero errors occur when putting zero frequency into for example frequency response calculations The solution when calculating a frequency response which needs to appear as though it starts from 0Hz is to input a very low frequency instead of zero For radio frequency calculations starting from 1Hz instead of 0Hz will usually do the trick 31 A proper definition for resonant Q Now that we have established that Qo is an important circuit parameter we will take the opportunity to have another look at at its definition The point is that there is something horribly unsatisfying about writing Qo Xo R OR Qo Xo R It seems more log
220. y product is known as the complex product or the phasor product and is also not the same as the cross product used in general vector theory The statement j 1 incidentally is the same as saying that rotation of a number through 90 followed by another rotation through 90 has the effect of reversing its original direction i e multiplying it by 1 We now have part of the solution of how to interpret the expression Z ZZ Zi Z One further trick is required in order to cope with the division part of the problem however and this comes from noticing what happens when the complex number a jb is multiplied by the complex number a jb a jb a jb a j b a b a jb is called the complex conjugate of a jb and vice versa An asterisk is normally used to denote the complex conjugate of a number e g if Z R jX then Z R jX Z is pronounced Z star When a number is multiplied by its complex conjugate the result is always real Thus if j appears in the denominator the bottom part of a fraction we can multiply both the numerator the top part and the denominator by the complex conjugate of the denominator Multiplying both the top and bottom of a fraction by the same number makes no difference to the value but the operation makes the denominator real so that the fraction can then be rearranged into a form which looks once again like a jb We now have a complete set of definitions for mathematical operations involving
221. y of Maxwell s theory called the principle of continuity of energy not to be confused with the principle of conservation of energy The principle of continuity dictates that energy cannot simply disappear from one location and reappear in another it must in some sense make a journey This incidentally is not the same as imagining that energy follows a specific route because we can only explain phenomena such as optical diffraction and remain consistent with quantum theory if we allow that even the very smallest quantity of energy can follow a multiplicity of paths during flight Nevertheless it retains a form of integrity it is conserved which means that electric and magnetic fields from different energy sources cannot combine to make electromagnetic radiation It is intriguing to note that were such combination possible every stray field would interact and the Universe would explode perhaps to expand to a state in which such interaction can no longer occur The continuity principle allows us to break reality down into separate energy transfer processes and so without it we would not be able to understand the Universe On a more immediate level however it tells us exactly how 1 Hertz the Discoverer of Electric Waves Julian Blanchard Bell System Technical Journal July 1938 Vol 17 No 3 p326 337 Available from http bstj bell labs com 2 Is the Universe leaking energy Tamara M Davis Scientific American July 2010
222. y two pieces of electrically conducting material in f proximity The conductors do not have to be plates Capacitance appears whenever two conductors have the ability to be at different relative voltages i e capacitance is made by not shorting things together and so there will always be some stray capacitance across the coil This precludes resonance at infinite frequency but in fact a coil behaves as though it has considerably more parallel capacitance than simple consideration of strays would predict The reason is that it takes a finite amount of time for an electromagnetic wave to make its helical journey along the wire in the coil and the resulting phase shift has to be represented by placing a hypothetical capacitance the coil s se f capacitance C in parallel with the the idealised pure inductance Self capacitance is dependent on the length of the winding wire and the effective velocity for a wave travelling along it This propagation velocity the so called phase velocity is frequency dependent but most radio coils are operated in a regime where the velocity is changing in such a way that the self capacitance appears to take on a definite value In this regime the apparent self capacitance turns out to depend only on the external dimensions of the coil the turn to turn spacing and the number of turns are practically irrelevant The inclusion of self capacitance in the model allows for the fact that the coil has a self resonant fr
223. ypical of circuit analysis problems which involve no difficult logical steps but tend to expand into large numbers of terms many pairs of which subsequently turn out to be equal and opposite and so cancel Thus the problem expands alarmingly and then contracts again into one or more relatively simple expressions It can be difficult to keep track of the various parts of the equation when carrying out such manipulations which means that mistakes are likely to occur There is however a simple reality check which identifies invalid terms and gives an immediate indication of the likely correctness of the result This is the test of dimensional consistency which with a certain amount of practice can be carried out at a glance The rules are as follows If two quantities are to be added together or subtracted it must be possible to express them in the same units It would make no sense to add a distance in metres to a temperature in C It would also make no sense to add a distance in metres to a distance in centimetres but in that case the distance in centimetres can be divided by 100 to convert it into metres and then the addition can be performed 41 It follows that if a or a symbol appears anywhere in an equation the dimensions of the quantities on either side of that symbol must be the same An equation which supposedly represents a certain quantity must have dimensions appropriate to that quantity Take for example the
224. ys the fear that it may involve some non standard definition 66 28 deciBels amp logarithms Having just found cause to mention the dB notation we should perhaps clarify some of the aspects of its use which can be a source of confusion Firstly a deciBel is a tenth of a Bel hence the small d and the capital B in dB The Bel named after Alexander Graham Bell is an internationally accepted unit equal to 10 transmission units TU the TU now the dB being a logarithmic relative signal measurement method introduced by AT amp T in 1923 A ratio in Bels is the logarithm of a power ratio defined in the simplest possible way using base 10 i e common logarithms hence N B Logio P Pret Thus a ratio in deciBels is N dB 10L0g10 P Pret The practice of expressing audio power ratios on a logarithmic scale was developed because human hearing has an automatic gain control system This makes our hearing response logarithmic with respect to loudness and thus enables us to extract information from sound over a vast range of sound pressures it also makes our hearing asymmetric if a loudspeaker produces a loud sound which is asymmetric about the ambient air pressure axis there will be a change in perceived quality if the amplifier connections are reversed The use of logarithmic scales is also favoured because it allows us to represent the vast gains of electronic amplifiers and the vast ranges of power encountered in communicatio

AC Theory - David Knight. Radio, electrical and electronic articles.

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