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Transmission Lines - University of California, Berkeley

1. n D gt z D amp Z Y gt a no D A 5 f O a ae n D O aa a 2 5 ke lt 02 0406 10 2 3 456 810 Line Loss In dB When Matched Fig 20 4 Increase in line loss because of standing waves SWR measured at the load To determine the total loss in decibels in a line having an SWR greater than 1 first determine the matched line loss for the particular type of line length and frequency on the assumption that the line is perfectly matched For example Belden 9913 has a matched line loss of 0 49 dB 100 ft at 14 MHz Locate 0 49 dB on the horizontal axis For an SWR of 5 1 move up to the curve corresponding to this SWR The increase in loss due to SWR is 0 66 dB beyond the matched line loss values typical on this band For example con sider an 80 meter dipole cut for the middle of the band at 3 75 MHz It exhibits an SWR of about 6 1 at the 3 5 and 4 0 MHz ends of the band At 3 5 MHz 250 ft of RG 58A small diameter coax has an additional loss of 2 1 dB for this SWR giving a total line loss of 4 0 dB If larger diameter RG 213 coax is used instead the additional loss due to SWR is 1 3 dB for a total loss of 2 2 dB This is an acceptable level of loss for most 80 meter operators The loss situation gets dramatically worse as the frequency increases into the VHF and UHF regions At 1
2. Currents in Open wire Lines Outer Conductor Fig 20 1 Common types of transmission lines used by amateurs Coaxial cable or coax has a center conductor surrounded by insulation The second conductor called the shield cover the insulation and is in turn covered by the plastic outer jacket Various types are shown at A B C and D The currents in coaxial cable flow on the outside of the center conductor and the inside of the outer shield E Open wire line F G and H has two parallel conductors separated by insulation In open wire line the current flows in opposite directions on each wire I 1 Hz signal is 983 569 082 ft Changing to a more useful expression gives _ 983 6 f where wavelength in ft f frequency in MHz 1 Thus at 14 MHz the wavelength is 70 25 ft Wavelength may also be expressed in electrical degrees A full wavelength is 360 is 180 14 is 90 and so forth Waves travel slower than the speed of light in any medium denser than a vacuum or free space A transmission line may have an insu 20 2 Chapter 20 lator which slows the wave travel down The actual velocity of the wave is a function of the dielectric characteristic of that insulator We can express the variation of velocity as the velocity factor for that particular type of dielectric the fraction of the wave s ve locity of propagation in the transmission line compared to that
3. Source Cam 9 Load F HBK0137 Fig 20 9 Matching network variations A through D show L networks E is a Pi network equivalent to a pair of L networks sharing a common series inductor F is a T network equivalent to a pair of L networks sharing a common parallel inductor cuits are shown in Table 20 4 As shown in Table 20 4 the L networks can be reversed if matching does not occur in one direc tion L networks are the most common for single band antenna matching The compo nent in parallel is the shunt component so the L networks with the shunt capacitor or inductor at the input Figs 20 9A and 20 9C are shunt input networks and the others are series input networks Impedance matching circuits can use fixed value components for just one band when a particular antenna has an impedance that is too high or low or they can be made to be adjustable when matching is needed on several bands such as for matching a dipole antenna fed with open wire line Additional material by Bill Sabin W IYH on matching networks can be found on the CD ROM accompanying this book along with his program MATCH DESIGNING AN L NETWORK The L network shown in Fig 20 9A through 20 9D only requires two compo nents and is a particularly good choice of matching network for single band antennas The L network is easy to construct so that it can be mounted at or near the feed point of the antenna resulting in 1 1 SWR on the transmis
4. 1 p and _SWR 1 9B p SWR 1 The definitions in equations 8 and 9 are valid for any line length and for lines which are lossy not just lossless lines longer than 4 at the frequency in use Very often the load impedance is not exactly known since an antenna usually terminates a transmission line and the antenna impedance may be influ enced by a host of factors including its height above ground end effects from insulators and the effects of nearby conductors We may also express the reflection coefficient in terms of forward and reflected power quantities which can be easily measured using a directional RF wattmeter The reflection coefficient and SWR may be computed as l 10A Pr and 1 SWR f 10B y Pe Pr where P power in the reflected wave P power in the forward wave If a line is not matched SWR gt 1 1 the difference between the forward and reflected powers measured at any point on the line is the net power going toward the load from that point The forward power measured with a directional wattmeter often referred to as a reflected power meter or reflectometer on a mismatched line will thus always appear greater than the forward power measured on a flat line with a 1 1 SWR The software program TLW written by Dean Straw N6BV and included on the ARRL Antenna Book CD solves these com plex equations This should come as a big relief for most radio amateurs The character ist
5. Only on 14 MHz does the loss drop down to 0 9 dB where the an tenna is just past 2 A resonance From 3 8 to 28 4 MHz the open wire line has amaximum loss of only 0 6 dB Columns six and seven in Table 20 1 list the maximum RMS voltage for 1500 W of RF power on the 50 Q coax and on the 450 Q open wire line The maximum RMS voltage for 1500 W on the open wire line is extremely high at 10 950 V at 1 8 MHz The voltage for a 100 W transmitter would be reduced by a ratio of 1300 3 ae 100 This is 2829 V still high enough to cause arcing in many antenna tuners although it only occurs at specific points that are mul tiples of 2 from the load In practice the lower voltages present along the transmission line are within the operating range of most tuners although you should remain aware that high voltages may be present along the line at some points Loss of 100 ft Loss of 100 ft Max Voltage Max Voltage RG 213 Coax 450 Q Line RG 213 Coax 450 Q Line dB dB at 1500 W at 1500 W 26 0 8 8 1507 10950 5 7 0 5 1177 3231 5 9 0 2 985 2001 10 1 0 6 967 2911 0 9 0 3 344 1632 6 8 0 3 753 1600 3 2 0 1 585 828 2 6 0 2 516 1328 9 4 0 5 703 1950 Table 20 1 Modeled Data for a 100 ft Flat Top Antenna Antenna Input VSWR Freq Impedance RG 213 MHz Q Coax 1 8 4 18 j 1590 33 7 3 8 37 5 j 354 16 7 7 1 447 j 956 12 3 10 1 2010 j 2970 12 1 14 1 87 6 j 156 1 6 18 1 1800 1470 ef 21 1 461 j 1250 4 6 24 9 155 150 3 6
6. Since the volt ages at the antenna terminals are equal and opposite with reference to ground equal and opposite currents flow on the surfaces of the line and second conductor Beyond the shorting point in the direction of the trans mitter these currents combine to cancel out each other The balancing section acts like an open circuit to the antenna since it is a 4 A parallel conductor line shorted at the far end and thus has no effect on normal antenna operation This is not essential to the line balancing function of the device however and baluns of this type are sometimes made shorter than 4 4 to provide a shunt induc tive reactance required in certain matching systems such as the hairpin match described in the Antennas chapter Fig 20 17D shows a third balun in which Input 0 01 W ji 1 8 to 29 7 MHz equal and opposite voltages balanced to ground are taken from the inner conductors of the main transmission line and a 2 phas ing section Since the voltages at the balanced end are in series while the voltages at the unbalanced end are in parallel there is a 4 1 step down in impedance from the balanced to the unbalanced side This arrangement is useful for coupling between a 300 Q balanced line and a 75 Q unbalanced coaxial line 20 5 2 Transmission Line Transformers The basic transmission line transformer from which other transformers are derived is the 1 1 choke balun or current balun shown in
7. ance to a value more suitable for the tuner components In general for any type of tuner begin with the maximum reactance to ground maximum inductance or minimum capacitance and the minimum series reactance between the source and load minimum inductance or maximum capacitance The configuration that produces the minimum SWR with maximum reactance to ground and minimum series reactance will generally have the highest efficiency and broadest tuning bandwidth To reduce on the air tune up time record the settings of the tuner for each antenna and band of operation If the tuner requires read justment across the band record the settings of the tuner at several frequencies across the band Print out the results and keep it near the tuner this will allow you to adjust the tuner quickly with only a short transmission to check or fine tune the settings This also serves as adiagnostic since changes in the set ting indicate a change in the antenna system 20 4 6 Myths About SWR This is a good point to stop and mention that there are some enduring and quite misleading myths in Amateur Radio concerning SWR e Despite some claims to the contrary a high SWR does not by itself cause RFI or TVI or telephone interference While it is true that an antenna located close to such devices can cause overload and interference the SWR on the feed line to that antenna has nothing to do with it providing of course that the tuner feed
8. its characteristic impedance Matching The process of effecting an impedance match between two electrical circuits of unlike impedance One example is matching a transmission line to the feed point of an antenna Maximum power transfer to the load antenna system will occur when a matched condition exists Microstrip A transmission line made from a strip of printed circuit board conductor above a ground plane used primarily at UHF and microwave frequencies Open wire line Parallel conductor feed line with parallel insulators at regular intervals to maintain the line spacing The dielectric is principally air making it a low loss type of line Also known as ladder line or window line Output impedance The equivalent impedance of a signal source Parallel conductor line A type of transmission line that uses two parallel wires spaced from each other by insulating material Also known as open wire ladder or window line Phasing lines Sections of transmission line that are used to ensure the correct phase relationship between the elements of a driven array or between bays of an array of antennas Also used to effect impedance transformations while maintaining the desired phase Q section Term used in reference to transmission line matching transformers and phasing lines Reflection coefficient p The ratio of the reflected voltage at a given point on a transmission line to the incident voltage
9. or to transmitting equipment If the impedances are different that is a mismatch Impedance matcher See Antenna tuner Impedance matching circuit A circuit that transforms impedance from one value to another Adjustable impedance matching circuits are used at the output of transmitters and amplifiers to allow maximum power output over a wide range of load impedances Impedance transformer A transformer designed specifically for transforming impedances in RF equipment L network A combination of two reactive components used to transform or match impedances One component is connected in series between the source and load and the other shunted across either the source or the load Most L networks have one inductor and one capacitor but two inductor and two capacitor configurations are also used Ladder line see Open wire line Lambda 2 Greek symbol used to represent wavelength Line loss The power dissipated by a transmission line as heat usually expressed in decibels Load noun The component antenna or circuit to which power is delivered verb To apply a load to a circuit or a transmission line Loading The process of a transferring power from its source to a load The effect a load has on a power source Magnetic field A region through which a magnetic force will act on a magnetic object Matched line loss The line loss in a feed line terminated by a load equal to
10. to suppress the common mode currents and restore the pattern Transmission Lines 20 17 20 18 Baluns Chokes and Transformers The term balun applies to any device that transfers differential mode signals between a balanced bal system and an unbalanced un system while maintaining sym metrical energy distribution at the terminals of the balanced system The term only applies to the function of energy transfer not to how the de vice is constructed It doesn t matter whether the balanced unbalanced transition is made through transmis sion line structures flux coupled transformers or simply by blocking unbalanced current flow A common mode choke balun for example per forms the balun function by putting impedance in the path of common mode currents and is therefore a balun A current balun forces symmetrical current at the balanced terminals This is of particular importance in feeding antennas since antenna cur rents determine the antenna s radia tion pattern A voltage balun forces symmetrical voltages at the balanced terminals Voltage baluns are less effective in causing equal currents at their balanced terminals such as at an antenna s feed point An impedance transformer may or may not perform the balun function Impedance transformation chang ing the ratio of voltage and current is not required of a balun nor is it prohibited There are balanced to balanced impedance transformers
11. transformers with isolated primary and secondary windings for exam ple just as there are unbalanced to unbalanced impedance transformers autotransformer and transmission line designs A transmission line transformer is a device that performs the function of power transfer with or without impedance transforma tion by utilizing the characteristics of transmission lines Multiple devices are often com bined in a single package called a balun For example a 4 1 cur rent balun is a 1 1 current balun in series with a 4 1 impedance transformer or voltage balun Other names for baluns are common such as line isolator for a choke balun Baluns are often referred to by their construction bead balun coiled coax balun sleeve balun and so forth What is important is to separate the function power transfer between balanced and unbalanced systems from the construction Chapter 20 1 1 Balun Optional HBKO5_ 14 055 Fig 20 19 A Basic current or choke balun B Guanella 1 4 transformer C Ruthroff 4 1 unbalanced transformer D Ruthroff 1 4 balanced transformer E Ruthroff 16 1 unbalanced transformer eCoupling of noise currents on the feed line to receiving antennas eCurrents from noise sources coupling to the feed line eCoupling between different antennas via their feed lines A single choke balun at the antenna feed point may not be sufficient to reduce com mon
12. without any matching at the antenna The use of moderately high Q Loss Q 200 70 29 uH Resistance resonating inductors has yielded almost 21 dB of gain that is less loss compared to the case without the inductors The drawback of course is that the antenna is now resonant on only one frequency but it certainly is a lot more efficient on that one frequency THE QUARTER WAVE TRANSFORMER OR Q SECTION The range of impedances presented to the transmission line is usually relatively small on a typical amateur antenna such as a dipole or a Yagi when it is operated close to resonance In such antenna systems the impedance transforming properties of a 4 A section of transmission line are often utilized to match the transmission line at the antenna Fig 20 11 shows one example of this tech nique to feed an array of stacked Yagis on a single tower Each antenna is resonant and is fed in parallel with the other Yagis using equal lengths of coax to each antenna called phasing lines A stacked array is used to pro duce not only gain but also a wide vertical elevation pattern suitable for coverage of a broad geographic area See The ARRL An tenna Book for details about Yagi stacking The feed point impedance of two 50 Q Yagis fed with equal lengths of feed line con nected in parallel is 25 Q 50 2 Q three in parallel yield 16 7 Q four in parallel yield 12 5 Q The nominal SWR for a stack of four Yagis is 4 1
13. Choke Baluns The simplest construction method fora 1 1 choke balun made from coaxial feed line is simply to wind a portion of the cable into a coil see Fig 20 24 creating aninductor from the shield s outer surface This type of choke balun is simple cheap and reduces common mode current Currents on the outside of the shield encounter the coil s impedance while currents on the inside are unaffected A scramble wound flat coil like a coil of rope shows a broad resonance that easily covers three octaves making it reasonably HBK05_ 14 008 Fig 20 21 The three basic techniques for combining modules Module A Module B R A 0 Module Combiner ay Module A R Module B R B 180 Module Combiner NY Module A Zin Module 90 B Hybrid C 90 Module Combiner effective over the entire HF range If par ticular problems are encountered on a single band a coil that is resonant on that band may be added The choke baluns described in Table 20 6 were constructed to have a high impedance at the indicated frequencies as measured with an impedance meter This construction technique is not effective with open wire or twin lead line because of cou pling between adjacent turns The inductor formed by the coaxial cable s shield is self resonant due to the distributed capacitance between the turns of the coil The self resonant frequency can be found by using a dip meter Leave th
14. Fig 20 19A To construct this type of balun a length of coaxial cable or a pair of close spaced parallel wires forming a transmission line are wrapped around a ferrite rod or toroid or inserted through a number of beads The coiled feed line choke balun is discussed in the next section For the HF bands use type 75 or type 31 material Type 43 is used on the VHF bands The Zp of the line should equal the load resistance R Because of the ferrite a high impedance exists between points A and C and a virtually identical impedance between B and D This is true for parallel wire lines and itis also true for coax The ferrite affects the A to C impedance of the coax inner conductor and the B to D impedance of the outer braid equally The conductors two wires or coax braid and center wire are tightly coupled by elec tromagnetic fields and therefore constitute a good conventional transformer with a turns ratio of 1 1 The voltage from A to C is equal to and in phase with that from B to D These are called the common mode voltages CM A common mode CM current is one that has the same value and direction in both wires or on the shield and center conductor Be cause of the ferrite the CM current encounters a high impedance that acts to reduce choke the current The normal differential mode DM signal does not encounter this CM im pedance because the electromagnetic fields due to equal and opposite currents in the two condu
15. O Cc O o Q 6 turns __y RG8X y 5 turns RG8X f 3 Clamps LZ ee ee Table 20 8 3 4 5 678910 20 30 40 506070 90 Combination Ferrite and ic eile Coaxial Coil Measured Impedance Freq 7 ft 4 turns 1 Core 2 Cores MHz of RG 8X Table 20 7 1 8 5200 Transmitting Choke Designs 3 5 660 1 4 KQ Freq Band s Mix RG 8 RG 11 RG 6 RG 8X RG 58 RG 59 7 T 6kQ 3 2kQ MHz Turns Cores Turns Cores 14 960 1 1 kQ 1 4 kQ 1 8 3 8 31 7 5 toroids 7 5 toroids A 42 KO 5009 6709 8 Big clamp on 28 470 Q 3 5 7 6 5 toroids 7 4 toroids 8 Big clamp on 10 1 31 or 43 5 5 toroids 8 Big clamp on coaxial chokes wound on ferrite toroids He 6 4 toroids used low loss cores typically type 61 or 67 7 44 5 pioride 3 Big clanib on material Fig 20 31 shows that these high Q chokes are quite effective in the narrow fre 14 5 4 toroids 8 2 toroids quency range near their resonance However 4 6 toroids 5 6 Big clamp on the resonance is quite difficult to measure and itis so narrow that it typically covers only one al i 2 se a ee or two ham bands Away from resonance the SES eee choke becomes far less effective as choking 28 4 5 toroids 4 5 toroids impedance falls rapidly and its reactive com 5 Big clamp on ponent resonates with the line Air wound coaxial chokes are less effective 7 28 31 or 43 Use two chokes in series Use two chokes in series than bead
16. Transformer If the complex mechanics of reflections SWR and line losses are put aside momen tarily a transmission line can very simply be considered as an impedance transformer A certain value of load impedance consist ing of a resistance and reactance at the end of the line is transformed into another value of impedance at the input of the line The amount of transformation is determined by the electrical length of the line its character istic impedance and by the losses inherent in the line The input impedance of a real lossy transmission line is computed using the following equation Zy cosh nf Zo sinh n4 Zin Lo Zy sinh n Zo cosh n4 12 where Zin complex impedance at input of line Rin Ai Z complex load impedance at end of line R j X Zo characteristic impedance of line Ry J Xo n complex loss coefficient a j B a matched line loss attenuation con stant in nepers unit length 1 neper 8 688 dB so multiply line loss in dB per unit length by 8 688 B phase constant of line in radians unit length multiply electrical length in degrees by 27 radians 360 degrees 4 electrical length of line in same units of length as used for a Solving this equation manually is tedious since it incorporates hyperbolic cosines and sines of the complex loss coefficient but it may be solved using a traditional paper Smith Chart or a computer program The ARRL Antenna Book ha
17. an antenna This is a good point at which to say that striving for perfect balance in a line and an tenna system is not always absolutely manda tory For example ifanonresonant center fed dipole is fed with open wire line and a tuner for multiband operation the most desirable radiation pattern for general purpose com munication is actually an omnidirectional pattern A certain amount of feed line ra diation might actually help fill in otherwise undesirable nulls in the azimuthal pattern of the antenna itself Furthermore the radiation pattern of a coaxial fed dipole that is only a few tenths of a wavelength off the ground 50 ft high on the 80 meter band for example is not very directional anyway because of its severe interaction with the ground Purists may cry out in dismay but there are many thousands of coaxial fed dipoles in daily use worldwide that perform very effectively without the benefit of a balun See Fig 20 18A for a worst case compari son between a dipole with and without a balun at its feed point This is with a 1 A feed line slanted downward 45 under one side of the antenna Common mode currents are conducted and induced onto the outside of the shield of the feed line which in turn radiates The amount of pattern distortion is not particularly severe for a dipole It is debatable whether the bother and expense of installing a balun for such an antenna is worthwhile Some form of balun should be used to
18. as RG 213 offer a convenient way to lower transmitter harmonic levels Despite the fact that the exact amount of harmonic attenuation depends on the impedance often unknown into which they are working at the harmonic frequency a quarter wave stub will typically yield 20 to 25 dB of attenuation of the second harmonic when placed directly at the output of a transmitter feeding common amateur antennas Because different manufacturing runs of coax will have slightly different velocity factors a quarter wave stub is usually cut a little longer than calculated and then carefully pruned by snipping off short pieces while using an antenna analyzer to monitor the re sponse at the fundamental frequency Because the end of the coax is an open circuit while pieces are being snipped away the input of a 4 X line will show a short circuit exactly at the fundamental frequency Once the coax has been pruned to frequency a short jumper is soldered across the end and the response at the second harmonic frequency is measured Fig 20 5 shows how to connect a shorted stub to a transmission line and Fig 20 6 shows a typical frequency response The shorted quarter wave stub shows low Transmission Lines 20 7 loss at 7 MHz and at 21 MHz where it is Y4 h long It nulls 14 and 28 MHz This is useful for reducing the even harmonics of a 7 MHz transmitter It can be used for a 21 MHz transmitter as well and will reduce any spurious emissions such as p
19. at the same point The reflection coefficient is also equal to the ratio of reflected and incident currents The Greek letter rho p is used to represent reflection coefficient Reflectometer see SWR bridge Resonance 1 The condition in which a system s natural response and the frequency of an applied or emitted signal are the same 2 The frequency at which a circuit s capacitive and inductive reactances are equal and cancel Resonant frequency The frequency at which the maximum response of a circuit occurs In an antenna the resonant frequency is one at which the feed point impedance is purely resistive Return loss The absolute value of the ratio in dB of the power reflected from a load to the power delivered to the load Rise time The time it takes for a waveform to reach a maximum value Series input network A network such as a filter or impedance matching circuit in which the input current flows through a component in series with the input Shunt input network A network such as a filter or impedance matching circuit with a component connected directly across the input Skin effect The phenomenon in which ac current at high frequencies flows in a thin layer near the surface of a conductor Smith Chart A coordinate system developed by Phillip Smith to represent complex impedances graphically This chart makes it easy to perform calculations involving antenna and transmission l
20. baluns Their equivalent circuit is 10 1 28 or 1 4 turns on 5 toroids 1 6 turns on a big clamp on A 14 28 2 3 turns on 5 toroids 2 5 turns on a big clamp on aso a simple MenO paralel Tegonanee ale they must be used below resonance They are 14 28 Two 4 turn chokes 4 turns on 6 toroids or simple inexpensive and unlikely to overheat each w one big clamp on 5 turns on a big clamp on Choking impedance is purely inductive and not very great reducing their effectiveness 90 Two 3 turn chokes Effectiveness is further reduced when the in Bac Wiha Dig clamp pn ductance resonates with the line at frequen cies where the line impedance is capacitive and there is almost no resistance to damp the resonance Adding ferrite cores to a coiled coax balun is a way to increase their effectiveness The resistive component of the ferrite impedance Notes Chokes for 1 8 3 5 and 7 MHz should have closely spaced turns Chokes for 14 28 MHz should have widely spaced turns Turn diameter is not critical but 6 inches is good such as RG 303 must be used for high power applications Even with high power coax the choking impedance is often insufficient to limit current to a low enough value to prevent overheating Equally important the lower choking impedance is much less effective at rejecting noise and preventing the filling of nulls in a radiation pattern Newer bead balun designs use type 31 and 43 beads which are resona
21. device or vice versa May be in the form of a choke balun or a transformer that provides a specific impedance transformation including 1 1 Often used in antenna systems to interface a coaxial transmission line to the feed point of a balanced antenna such as a dipole Characteristic impedance The ratio of voltage to current in a matched feed line it is determined by the physical geometry and materials used to construct the feed line Also known as surge impedance since it represents the impedance electromagnetic energy encounters when entering a feed line Choke balun A balun that prevents current from flowing on the outside of a coaxial cable shield when connected to a balanced load such as an antenna Coax See coaxial cable Coaxial cable Transmission lines that have the outer shield solid or braided concentric with the same axis as the inner or center conductor The insulating material can be a gas air or nitrogen or a solid or foam insulating material Common mode current Current that flows equally and in phase on all conductors of a feed line or multiconductor cable Conductor A metal body such as tubing rod or wire that permits current to travel continuously along its length Conjugate match Creating a purely resistive impedance by connecting an impedance with an equal and opposite reactive component Current balun see Choke balun Decibel A logarithmic power ratio abbrevi
22. feed line tuned for best VSWR and left at that setting Ifa particular antenna has a minimum VSWR in the CW portion of a band and opera tion in the SSB end is desired the tuner can be used for matching and switched out when not needed Multiband operation generally requires retuning for each band in use Antenna tuners for use with balanced or open wire feed lines include a balun or link coupling circuit as seen in Fig 20 14 This allows a transmitter s unbalanced coaxial Transmission Lines 20 13 HBKO5_21 011 Balanced Load Output Fig 20 15 Antenna tuner network in T configuration This network has become popular because it has the capability of matching a wide range of impedances At A the balun transformer at the input of the antenna tuner preserves balance when feeding a balanced transmission line At B the T configuration is shown as two L networks back to back in the L network version the two L1 inductors are assumed to be adjustable with identical values output to be connected to the balanced feed line A fully balanced tuner has asymmetrical internal circuit with a tuner circuit for each side of the feed line and the balun at the input to the tuner where the impedance is close to 50 Most antenna tuners are unbalanced however with a balun located at the output of the impedance matching network connected directly to the balanced feed line At very high or very low impedances the balun s power
23. in free space The velocity factor is related to the dielectric constant of the material in use 1 ve VE 2 where VF velocity factor s dielectric constant So the wavelength in a real transmission line becomes _ 983 6 f As an example many coax cables use poly ethylene dielectric over the center conductor as the insulation The dielectric constant for polyethylene is 2 3 so the VF is 0 66 Thus wavelength in the cable is about two thirds as long as a free space wavelength The VF and other characteristics of many types of lines both coax and twin lead are shown in the table Nominal Characteristics of Commonly used Transmission Lines in the Component Data and References chapter VF 3 There are differences in VF from batch to batch of transmission line because there are some variations in dielectric constant during the manufacturing processes When high accuracy is required it is best to actually measure VF by using an antenna analyzer to measure the resonant frequency of a length of cable The antenna analyzer s user manual will describe the procedure CHARACTERISTIC IMPEDANCE A perfectly lossless transmission line may be represented by a whole series of small inductors and capacitors connected in an in finitely long line as shown in Fig 20 2 We first consider this special case because we need not consider how the line is terminated at its end since there is no end Each inducto
24. inches There is some variation in dielectric constant of coaxial cable from batch to batch or manu facturer to manufacturer so it is always best to measure the stub s fundamental resonance before proceeding CONNECTING STUBS Stubs are usually connected in the antenna feed line close to the transmitter They may also be connected on the antenna side of a switch used to select different antennas Some small differences in the null depth may occur for different positions To connect a stub to the transmission line it is necessary to insert a coaxial T as shown in 20 8 Chapter 20 HBK0141 a v Cc O Q 9 i x 10 20 Frequency MHz Fig 20 6 Frequency response with a shorted stub HBK0142 a O PTTTTTTTTP TTT TT EE RER n a l N O Response dB l oO oO io 10 Frequency MHz Fig 20 7 Frequency response with an open stub Table 20 2 Quarter Wave Stub Lengths for the HF Contesting Bands Freq Length L Cut off MHz per 100 kHz 1 8 90 ft 10 in 57 in 3 5 46 ft 9 in 15 in 7 0 23 ft 4 in 4in 14 0 11 ft 8 in 1 in 21 0 7 ft 9 in Ae in 28 0 5 ft 10 in Yin Lengths shown are for RG 213 and any similar cable assuming a 0 66 velocity factor L 163 5 f See text for other cables Fig 20 5 If a female male female T is used the male can connect directly
25. is described in a following sec tion The antenna tuner s function is to transform the impedance whatever it is at the transmitter end of the transmission line into the 50 Q required by their transmitter Remem ber that the use of an antenna tuner at the transmitter does not tune the antenna reduce SWR on the feed line or reduce feed line losses Some matching networks are built direct ly into the antenna for example the gamma and beta matches and these are discussed in the chapter on Antennas andin The ARRL Antenna Book Impedance matching net works made of fixed or adjustable compo nents can also be used at the antenna and are particularly useful for antennas that op erate on a single band Remember however that impedance can be transformed anywhere in the antenna system to match any other desired imped ance A variety of techniques can be used as described in the following sections depend ing on the circumstances An electronic circuit designed to convert impedance values is called an impedance matching network The most common im pedance matching network circuits for use in systems that use coax cable are 1 The low pass L network 2 The high pass L network 3 The low pass Pi network 4 The high pass T network Basic schematics for each of the circuits are shown in Fig 20 9 Properties of the cir 20 10 Chapter 20 Source Canin Load A Source C Source Load Source T T Load E J 1
26. mode impedance so large that the common mode current is very small DESIGN CRITERIA It can be shown mathematically and ex perience confirms that wound coax chokes having a resistive impedance at the transmit frequency of at least 5000 Q and wound with RG 8 or RG 11 size cable on five toroids are conservatively rated for 1500 W under high duty cycle conditions such as contesting or digital mode operation While chokes wound with smaller coax RG 6 RG 8X RG 59 RG 58 size are conservatively rated for dis sipation in the ferrite core the voltage and cur rent ratings of those smaller cables suggests a somewhat lower limit on their power han Fig 20 24 RF choke formed by coiling the feed line at the point of connection to the antenna The inductance of the choke isolates the antenna from the outer surface of the feed line dling Since the chokes see only the common mode voltage the only effect of high SWR on power handling of wound coax chokes is the peaks of differential current and voltage along the line established by the mismatch Experience shows that 5000 Q is also a good design goal to prevent RFI noise coup ling and pattern distortion While 500 1000 has long been accepted as sufficient to prevent pattern distortion WIHIS has correctly ob served that radiation and noise coupling from the feed line should be viewed as a form of pattern distortion that fills in the nulls of a directional antenna reduci
27. or antenna tuner tuner for short can be used The function of an antenna tuner is to transform the impedance at the input end of the transmission line whatever that may be to the 50 Q value required by the transmitter for best performance Do not forget that a tuner does not alter the SWR on the transmission line between the tuner and the antenna it only adjusts the impedance at the transmitter end of the feed line to the value for which the transmitter was designed Other names for antenna tun ers include transmatch impedance matcher matchbox or antenna coupler Since the SWR on the transmission line between the Fig 20 14 Simple antenna tuners for coupling a transmitter to a balanced line presenting a load different from the transmitter s design load impedance usually 50 A and B respectively are series and parallel tuned circuits using variable inductive coupling between coils C and D are similar but use fixed inductive coupling and a variable series capacitor C1 A series tuned circuit works well with a low impedance load the parallel circuit is better with high impedance loads several hundred ohms or more antenna and the output of the antenna tuner is rarely 1 1 some loss in the feed line due to the mismatch is unavoidable even though the SWR on the short length of line between the tuner and the transmitter is 1 1 If separate feed lines are used for different bands the tuner can be inserted in one
28. ranges and that will match various impedances A very simple matching transformer con sists of three windings connected in series as shown in Fig 20 12A The physical ar rangement of the three windings is shown in Fig 20 12B This arrangement gives the best bandwidth Fig 20 13 shows a picture of this type of transformer An IN OUT relay is included with the transformer One relay pole switches the 50 Q input port while two poles in parallel switch the 22 Q port Three 14 inch lengths of 14 AWG wire are taped together so they lie flat on the core A 6 1 mix toroid core 2 4 inch in diameter will handle full legal power The impedance ratio of this design is 1 2 25 or 22 22 to 50 Q This ratio turns out to work well for two or three 50 Q antennas in par allel Two in parallel will give an SWR of 25 22 22 or 1 125 1 Three in parallel give an SWR of 22 22 16 67 or 1 33 The unit shown in Fig 20 13 has an SWR bandwidth of 1 5 MHz to more than 30 MHz The 51 pF capacitor is connected at the low impedance side to ground and tunes out some inductive reactance This is a good way to stack two or three triband antennas If they have the same length feed lines and they all point the same way their patterns will add and some gain will result However they don t even need to be on the same tower or pointed in the same direc tion or fed with the same length lines to have some benefit Even dissimilar antennas can sometimes show a benefit
29. related topics F Witt Baluns in the Real and Complex World The ARRL Antenna Compen dium Vol 5 Newington ARRL 1996 The ARRL UHF Microwave Experimenter s Manual Newington ARRL 2000 Chapters 5 and 6 address transmission lines and impedance matching This 20 32 Chapter 20 book is out of print C Counselman W1HIS Common mode Chokes www yccc org Articles W1HIS CommonMode Chokes W1HIS2006Apr06 pdf ONLINE RESOURCES A Collection of Smith Chart References sss mag com smith html A 100 W Z Match tuner freespace virgin net geoff cottrell Z_match htm Ferrite and Powdered Iron Cores www fair rite com www amidoncorp com and www cwsbytemark com R Lewellan W7EL Baluns What They Do and How They Do It 1985 www eznec com A mateur Articles Baluns pdf Microwave oriented downloads www microwaves101 com Table of transmission line properties hf antenna com trans Times Microwave catalog www timesmicrowave com Transmission line loss factors www microwaves101 com encyclopedia transmission_loss cfm Transmission line matching with the Smith chart www odyseus nildram co uk RFMicrowave_Theory_Files SmithChartPart2 pdf Transmission line transformer theory www bytemark com products tlttheory htm Using Ferrites and Baluns www audiosystemsgroup com RFI Ham pdf Z Match tuner description members optushome com au vk6ysf vk6ysf zmatch htm
30. the antenna analyzer for a dip in the SWR or listen for a peak in the re ceived noise Return the inductor to the setting for lowest SWR or highest received noise 4a If no SWR minimum or noise peak is detected reduce the value of the capacitor closest to the transmitter in steps of about 20 and repeat 4b If still no SWR minimum or noise peak is detected return the input ca pacitor to maximum value and reduce the output capacitor value in steps of about 20 4c If still no SWR minimum or noise peak is detected return the output ca pacitor to maximum value and reduce both input and output capacitors in 20 steps 5 Once a combination of settings is found with a definite SWR minimum or noise peak 5a If you are using an antenna analyzer make small adjustments to find the combination of settings that produce minimum SWR with the maximum value of input and output capaci tance 5b If you do not have an antenna ana lyzer set the transmitter output power Transmission Lines 20 15 to about 10 W ensure that you won t cause interference identify with your call sign and transmit a steady carrier by making the same adjustments as in step 5a 5c For certain impedances the tuner may not be able to reduce the SWR to an acceptable value In this case try add ing feed line at the output of the tuner from 1 to 4 A electrical wavelengths long This will not change the feed line SWR but it may transform the imped
31. to the transmit ter while the antenna line and the stub connect tothe two females It should be noted that the T inserts a small additional length in series with the stub that lowers the resonant frequency The additional length for an Amphenol UHF T is about inch This length is negligible at 1 8 and 3 5 MHz but on the higher bands it should not be ignored MEASURING STUBS WITH ONE PORT METERS Many of the common measuring instru ments used by amateurs are one port devices meaning they have one connector at which the measurement typically VSWR is made Probably the most popular instrument for this type of work is the antenna analyzer available from a number of manufacturers To test a stub using an antenna analyzer connect the stub to the meter by itself and tune the meter for a minimum impedance value ignoring the VSWR setting It is al most impossible to get an accurate reading on the higher HF bands particularly with open stubs For example when a quarter wave open stub cut for 20 meters was nulled on an MFJ 259 SWR analyzer the frequency measured 14 650 MHz witha very broad null A recheck with a professional quality network analyzer measured 14 018 MHz Resolution on the network analyzer is about 5 kHz Running the same test on a quarter wave shorted stub gave a measurement of 28 320 MHz on the MFJ 259 and 28 398 MHz on the network analyzer These inaccuracies are typical of amateur instrumentation and
32. windings is shown at B wish to turn one antenna in the stack in a different direction and use it by itself If the load changes the amplifier must be retuned an inconvenience at best If the antenna impedance and the character istic impedance of a feed line to be matched are known the characteristic impedance needed for a quarter wave matching section of low loss cable is expressed by another sim plification of equation 12 Z JZ Zo where Z characteristic impedance needed for matching section Z antenna impedance Zo characteristic impedance of the line to which it is to be matched 20 20 12 Chapter 20 Such a matching section is called a synchro nous quarter wave transformer or a quarter wave transformer Synchronous because the match is only achieved at which the length of the matching section is exactly 4 A long Example To match a 50 Q line to a Yagi stack consisting of two antennas fed in paral lel to produce a 25 Q load the quarter wave matching section would require a character istic impedance of Z V50 x 25 35 4Q0 A transmission line with a characteristic impedance of 35 Q could be closely approxi mated by connecting two equal 1 4 sections of 75 Q cable such as RG 11A in paral lel to yield the equivalent of a 37 5 Q cable Three Yagis fed in parallel would require a 4 transformer made using a cable having a characteristic impedance of Z V16 7 x 25 28 9 O This
33. you use a tuner to make sure your transmitter is operating into its rated load resistance you can enjoy a very effective station using antennas with feed lines having high values of SWR on them For example a 450 Q open wire line con nected to the multiband dipole shown in Table 20 1 would have a 19 1 SWR on it at 3 8 MHz Yet time and again this antenna has proven to be a great performer at many installations Fortunately or unfortunately SWR is one of the few antenna and transmission line pa rameters easily measured by the average radio amateur Ease of measurement does not mean that a low SWR should become an end in itself The hours spent pruning an antenna so that the SWR is reduced from 1 5 1 down to 1 3 1 could be used in far more rewarding ways making contacts for example or studying transmission line theory 20 5 Baluns and Transmission Line Transformers Center fed dipoles and loops are balanced meaning that they are electrically and physi cally symmetrical with respect to the feed point A balanced antenna may be fed by a balanced feeder system to preserve this symmetry thereby avoiding difficulties with unbalanced currents on the line and unde sirable radiation from the transmission line itself Line radiation can be prevented by a number of devices which detune or decouple the line greatly reducing currents induced onto the feed line from the signal radiated by the antenna Many amateurs use center fed di
34. 0 1 Although open wire lines are enjoying a sort of renaissance in recent years because of their inherently lower losses in simple multiband antenna systems coaxial cables are far more prevalent because they are much more convenient to use The third major type of transmission line is the waveguide While open wire and coaxial lines are used from power line frequencies to well into the microwave region waveguides are used at microwave frequencies only Waveguides will be covered at the end of this chapter 20 1 1 Fundamentals In either coaxial or open wire line currents flowing in the two conductors travel in opposite directions as shown in Figs 20 1E and 20 11 If the physical spacing between the two parallel conductors in an open wire line S is small in terms of wavelength the phase difference be tween the currents will be very close to 180 If the two currents also have equal amplitudes the field generated by each conductor will cancel that generated by the other and the line will not radiate energy even if it is many wavelengths long The equality of amplitude and 180 phase difference of the currents in each conductor in an open wire line determine the degree of radiation cancellation If the currents are for some reason unequal or if the phase difference is not 180 the line will radiate energy How such imbalances occur and to what degree they can cause problems will be covered in more detail later In contrast to an open wire
35. 0 Q criteria for the 160 through 6 meter ham bands and several practical transmitting choke designs that are tuned or optimized for ranges of frequencies The table entries refer to the specific cores in the preceding paragraph If you construct the chokes using toroids remember to make the diameter of the turns large enough to avoid deformation of the coaxial cable Space turns evenly around the toroid to mini mize inter turn capacitance HBK0446 10000 9000 USING FERRITE BEADS The early current baluns developed by Walt Maxwell W2DU formed by stringing multiple beads in series on a length of coax to obtain the desired choking imped ance are really common mode chokes Maxwell s designs utilized 50 very small beads of type 73 material as shown in Fig 20 30 Product data sheets show that a single type 73 bead has a very low Q resonance around 20 MHz and has a predominantly resistive impedance of 10 20 Q on all HF ham bands Stringing 50 beads in series simply multiplies the im pedance of one bead by 50 so the W2DU current balun has a choking impedance of 500 1000 and because it is strongly resis tive any resonance with the feed line is minimal This is a fairly good design for moderate power levels but suitable beads are too small to fit most coax A specialty coaxial cable HBK0447 8000 7000 8000 6000 7000 7 Turns 5 cores 6000 7 burna 6 cores 6
36. 28 4 2590 j 772 6 7 Notes 1 Antenna is a 100 ft long 50 ft high center fed dipole over average ground using coaxial RG 213 or open wire transmission lines Each transmission line is 100 ft long 2 Antenna impedance computed using EZNEC 3 computer program using 499 segments and with the Real Ground model 3 Note the extremely reactive impedance levels at many frequencies but especially at 1 8 MHz If this antenna is fed directly with RG 213 coax the losses are unacceptably large on 160 meters and undesirably high on most other bands also 4 The RF voltage at 1 8 MHz for high power operation with open wire line is extremely high also and would probably result in arcing either on the line itself or more likely in the antenna tuner 20 6 Chapter 20 In general such anonresonant antenna is a proven practical multiband radiator when fed with 450 Q open wire ladder line connected to an antenna tuner A longer antenna would be preferable for more efficient 160 meter operation even with open wire line The tuner and the line itself must be capable of handling the high RF voltages and currents involved for high power operation On the other hand if such a multiband antenna is fed directly with coaxial cable the losses on most frequencies are prohibitive Coax is most suitable for an tennas whose resonant feed point impedances are close to the characteristic impedance of the feed line 20 3 The Transmission Line as Impedance
37. 46 MHz the total loss in 250 ft of RG 58A with a 6 1 SWR at the load is 21 4 dB 10 1 dB for RG 213A and 2 7 dB for s inch 50 Q hardline At VHF and UHF alow SWR is essential to keep line losses low even for the best coaxial cable The length of transmission line must be kept as short as practical at these frequencies The effect of SWR on line loss is shown graphically in Fig 20 4 The horizontal axis is the attenuation in decibels of the line when perfectly matched The vertical axis gives the additional attenuation due to SWR If long coaxial cable transmission lines are neces sary the matched loss of the coax used should be kept as low as possible meaning that the highest quality largest diameter cable should be used Transmission Lines 20 5 20 2 Choosing a Transmission Line It is no accident that coaxial cable has be come as popular as it has since it was first widely used during WWII Coax is mechani cally much easier to use than open wire line Because of the excellent shielding afforded by its outer shield coax can be run up a metal tower leg taped together with numerous other cables with virtually no interaction or cross talk between the cables At the top of a tower coax can be used with a rotatable Yagi or quad antenna without worrying about shorting or twisting the conductors which might happen with an open wire line Class 2 PVC P2 noncontaminating outer jackets are designed for long life out doo
38. 50 12 5 This level of SWR does not cause excessive line loss provided that low loss coax feed line is used However many station designers want to be able to select using relays any individual antenna in the array without having the load seen by the transmitter change Perhaps they might Q 200 Loss 70 29 WH Resistance To Tuner HBKO5_ 21 006 1 8 MHz Fig 20 10 The efficiency of the dipole in Table 20 1 can be improved at 1 8 MHz with a pair of inductors inserted symmetrically at the feed point Each inductor is assumed to have a Q of 200 By resonating the dipole in this fashion the system efficiency when fed with RG 213 coax is about 21 dB better than using this same antenna without the resonator The disadvantage is that the formerly multiband antenna can only be used on a single band Transmission Lines 20 11 M4 A 50 Q Feedline To TX Two 750 In Parallel HBKO5_ 21 007 Fig 20 11 Array of two stacked Yagis illustrating use of 4 2 matching sections At the junction of the two equal lengths of 50 Q feed line the impedance is 25 This is transformed back to 50 by the two paralleled 75 Q lines which together make a net characteristic impedance of 37 5 This is close to the 35 4 value computed by the formula HBK0138 Fig 20 12 Schematic for the impedance matching transformer described in the text The complete schematic is shown at A The physical positioning of the
39. Contents 20 1 Transmission Line Basics 20 1 1 Fundamentals 20 1 2 Matched and Mismatched Lines 20 1 3 Reflection Coefficient and SWR 20 1 4 Losses in Transmission Lines 20 2 Choosing a Transmission Line 20 3 The Transmission Line as Impedance Transformer 20 3 1 Transmission Line Stubs 20 3 2 Transmission Line Stubs as Filters 20 3 3 Project A Field Day Stub Assembly 20 4 Matching Impedances in the Antenna System 20 4 1 Conjugate Matching 20 4 2 Impedance Matching Networks 20 4 3 Matching Antenna Impedance at the Antenna 20 4 4 Matching the Line to the Transmitter 20 4 5 Adjusting Antenna Tuners 20 4 6 Myths About SWR 20 5 Baluns and Transmission Line Transformers 20 5 1 Quarter wave Baluns 20 5 2 Transmission Line Transformers 20 5 3 Coiled Coax Choke Baluns 20 5 4 Transmitting Ferrite Choke Baluns 20 6 Using Transmission Lines in Digital Circuits 20 7 Waveguides 20 7 1 Evolution of a Waveguide 20 7 2 Modes of Waveguide Propagation 20 7 3 Waveguide Dimensions 20 7 4 Coupling to a Waveguide 20 8 Glossary of Transmission Line Terms 20 9 References and Bibliography Chapter 20 Transmission Lines RF power is rarely generated right where it will be used A transmitter and the antenna it feeds are a good example The most effective antenna installation is outdoors and clear of ground and energy absorbing struc tures The transmitter however is most conveniently installed indoors where it is out of the we
40. L which is given in dB and is equal to 20 times the log of the reciprocal of the reflection coefficient RL dB 10log 6 20log p 8 f In the example above the return loss is 20 log 1 0 593 4 5 dB If there are no reflections from the load the voltage distribution along the line is constant or flat A line operating under these conditions is called either a matched or a flat line If reflections do exist a voltage standing wave pattern will result from the interaction of the forward and reflected waves along the line For a lossless transmission line at least A long the ratio of the maximum peak voltage anywhere on the line to the minimum value anywhere along the line is defined as the volt age standing wave ratio or VSWR The line must be 4 or longer for the true maximum and minimum to be created Reflections from the load also produce a standing wave pattern of currents flowing in the line The ratio of maximum to minimum current or ISWR is identical to the VSWR in a given line In amateur literature the abbreviation SWR is commonly used for standing wave ratio as the results are identical when taken from proper measurements of either current or voltage Since SWR is aratio of maximum to minimum it can never be less than one to one In other words a perfectly flat line has an SWR of 1 1 The SWR is related to the magnitude of the complex reflection coef ficient and vice versa by swr ttle 9A
41. Q L 47 H C 1 9 pF Q 20 Fig 20 32 Choke balun that includes both a coiled cable and ferrite beads at each end of the cable few inches apart on the coil as in Fig 20 32 the balun is more effective from 1 8 to 7 MHz and usable to 21 MHz If type 31 material was used the Fair Rite 2631101902 is a similar core low frequency performance would be even better The 20 turn multiple band 1 8 3 5 MHz coiled coax balun in Table 20 6 weighs pound 7 ounces The single ferrite core combination balun weighs 6 5 ounces and the two core version weighs 9 5 ounces MEASURING FERRITE CHOKE IMPEDANCE A ferrite RF choke creates a parallel reso 20 26 Chapter 20 nant circuit from inductance and resistance coupled from the core and stray capacitance resulting from interaction of the conductor that forms the choke with the permittivity of the core If the choke is made by winding turns on a core as opposed to single turn bead chokes the inter turn capacitance also becomes part of the choke s circuit These chokes are very difficult to measure for two fundamental reasons First the stray capacitance forming the parallel resonance is quite small typically 0 4 to 5 pF which is often less than the stray capacitance of the test equipment used to measure it Second most RF impedance instrumentation measures the reflection coefficient see the section Reflec tion Coefficient and SWR in a 50 Q circuit As aresult refl
42. Turns 6 cores 7 Turns 4 cores 7 Turns 5 cores 12 diam turns 6 Turns 5 cores 4 6 Turns 5 cores 5 Turns 7 cores 5 Turns 6 cores 2 6 Turns 4 cores Impedance Q 5 Turns 5 cores Impedance Q 4 4 4 Turns 7 cores F 5 Turns 5 cores 5 Turns 4 cores 7 4 Turns 6 cores 4 Turns 5 cores 5 6 7 8 910 Frequency MHz Fig 20 27 Impedance versus frequency for HF wound coax transmitting chokes using 2 4 inch toroid cores of 31 material with RG 8X coax 20 24 Chapter 20 20 y 3 Turns 9 cores 7 4 Turns 5 cores 7 3 Turns 7 cores 1000 1 30 40 506070 90 Close spacing 3 Turns 5 cores 3 4 5 6 78910 Frequency MHz Fig 20 28 Impedance versus frequency for HF wound coax transmitting chokes using toroid cores of 31 material with RG 8 coax Turns are 5 inch diameter and wide spaced unless noted HBK0448 Fig 20 29 Impedance versus frequency for HF wound coax transmitting chokes wound on big clamp r on cores of 31 7 tums 7 Pai A oe material with aoe ci Mer TF RG 8X or RG 8 al ABS coax Turns are 6 inch diameter wide spaced except as noted 8 turns RG8X Fig 20 30 W2DU bead balun consisting of 50 FB 73 2401 ferrite beads over a length of RG 303 coax See text for details Gq lt o
43. alanced Choke baluns are particularly useful for feeding asymmetri cal antennas with unbalanced coax line The common mode impedance of the choke balun varies with frequency but the line s differen tial mode impedance is unaffected Reducing common mode current on a feed line also reduces e Radiation from the feed line that can dis tort an antenna s radiation pattern Radiation from the feed line that can cause RFI to nearby devices eRF current in the shack and on power line wiring 5 Element Yagi w Slanted Feedline Dipole w Unbalanced Coax 15 Elevation 0 dB 13 06 dBi 28 Elevation 0 dB 7 31 dBi 14 100 MHz 14 100 MHz Fig 20 18 At A computer generated azimuthal responses for two 1 2 dipoles placed 0 71 high over typical ground The solid line is for a dipole with no feed line The dashed line is for an antenna with its feed line slanted 45 down to ground Current induced on the outer braid of the 1 long coax by its asymmetry with respect to the antenna causes the pattern distortion At B azimuthal response for two 5 element 20 meter Yagis placed 0 71 over average ground Again the solid line is for a Yagi without a feed line and the dashed line is for an antenna with a 45 slanted 1 long feed line The distortion in the radiated pattern is now clearly more serious than for a simple dipole A balun is needed at the feed point and most likely point preferably from the feed point
44. amed the Guanella trans former If Z of the lines equals 2R and if the load is a pure resistance of 4R then the input resistance R is independent of line length If the lines are exactly one quarter wavelength then Z 2R Z an impedance inverter where Zy and Z are complex The quality of balance can often be improved by inserting a 1 1 balun Fig 20 19A at the left end so that both ends of the 1 4 transformer are floating and a ground is at the far left side as shown The Guanella transformer can also be oper ated from a grounded right end to a floating left end The 1 1 balun at the left then allows a grounded far left end THE RUTHROFF TRANSFORMER Fig 20 19C is the Ruthroff transformer in which the input voltage V is divided into two equal in phase voltages AC and BD they are tightly coupled so the output is V 2 And because power is constant Ioup 2h 20 20 Chapter 20 and the load is R 4 There is a CM voltage V 2 between A and C and between B and D so in normal operation the core is not free of magnetic flux The input and output both return to ground so it can also be operated from right to left for a 1 4 impedance stepup The Ruthroff transformer is often used as an amplifier interstage transformer for ex ample between 200 Q and 50 Q To maintain low attenuation the line length should be much less than one fourth wavelength at the highest frequency of operation and its Z should be R 2 A balanced version
45. an antenna Also called feed line Twin lead Parallel conductor transmission line in which both conductors are completely embedded in continuous strip of insulating material Unbalanced line Feed line with one conductor at dc ground potential such as coaxial cable Universal stub system A matching network consisting of a pair of transmission line stubs that can transform any impedance to any other impedance Velocity factor velocity of propagation The speed at which an electromagnetic wave will travel through a material or feed line stated as a fraction of the speed of the wave in free space where the wave would have its maximum velocity VSWR Voltage standing wave ratio See SWR Waveguide A hollow conductor through which electromagnetic energy flows Usually used at UHF and microwave frequencies instead of coaxial cable Window line see Open wire line Transmission Lines 20 31 20 9 References and Bibliography H W Johnson and M Graham High Speed Digital Design Prentice Hall 1993 F Regier Series Section Transmission Line Impedance Matching QST Jul 1978 pp 14 16 J Sevick W2FMI Transmission Line Transformers 4th Edition Noble Pub lishing 2001 P Smith Electronic Applications of the Smith Chart Noble Publishing 1995 R D Straw Ed The ARRL Antenna Book 21st Edition Newington ARRL 2007 Chapters 24 through 28 include material on transmission lines and
46. are meant to illustrate the difficulties of using inexpensive instruments for sensitive measurements Other one port instruments that measure phase can be used to get a more accurate read ing The additional length added by the re quired T must be accounted for If the measure mentis made without the T and then with the T the average value will be close to correct MEASURING STUBS WITH TWO PORT INSTRUMENTS A two port measurement is made with a signal generator and a separate detector A T connector is attached to the generator with the stub connected to one side The other side is connected to a cable of any length that goes to the detector The detector should present a 50 Q load to the cable This is how a network analyzer is configured and it is similar to how the stub is connected in actual use If the generator is accurately calibrated the measurement can be very good There are a number of ways to do this without buying an expensive piece of lab equipment An antenna analyzer can be used as the signal generator Measurements will be quite accurate if the detector has 30 to 40 dB dy namic range Two setups were tested by the author for accuracy The first used a digital voltmeter DVM with a diode detector A germanium diode must be used for the best dynamic range Tests on open and shorted stubs at 14 MHz returned readings within 20 kHz of the network analyzer Another test was run using an oscilloscope as the detecto
47. ated dB May also represent a voltage or current ratio if the voltages or currents are measured across or through identical impedances Suffixes to the abbreviation indicate references dBi isotropic radiator dBm milliwatt dBW watt Dielectrics Various insulating materials used in antenna systems such as found in insulators and transmission lines 20 30 Chapter 20 Dielectric constant k Relative figure of merit for an insulating material used as a dielectric This property determines how much electric energy can be stored in a unit volume of the material per volt of applied potential Electric field An electric field exists in a region of space if an electrically charged object placed in the region is subjected to an electrical force Electromagnetic wave A wave of energy composed of an electric and magnetic field Feed line See transmission line Feed point The point at which a feed line is electrically connected to an antenna Feed point impedance The ratio of RF voltage to current at the feed point of an antenna Ferrite A ceramic material with magnetic properties Hardline Coaxial cable with a solid metal outer conductor to reduce losses compared to flexible cables Hardline may or may not be flexible Impedance match To adjust impedances to be equal or the case in which two impedances are equal Usually refers to the point at which a feed line is connected to an antenna
48. ather and is readily accessible A transmission line is used to convey RF energy from the transmitter to the antenna A trans mission line should transport the RF from the source to its destination with as little loss as possible This chapter written by Dean Straw N6BV and updated by George Cutsogeorge WeVUJN explores transmission line theory and applications Jim Brown K9YC contributed updated material on transmitting choke baluns 20 1 Transmission Line Basics There are three main types of transmission lines used by radio amateurs coaxial open wire and waveguide The most common type is the coaxial line usually called coax shown in vari ous forms in Fig 20 1 Coax is made up of a center conductor which may be either stranded or solid wire surrounded by a concentric outer conductor with a dielectric center insulator between the conductors The outer conductor may be braided shield wire or a metallic sheath A flexible aluminum foil or a second braided shield is employed in some coax to improve shielding over that obtainable from a standard woven shield braid If the outer conductor is made of solid aluminum or copper the coax is referred to as hardline The second type of transmission line uses parallel conductors side by side rather than the concentric ones used in coax Typical examples of such open wire lines are 300 Q TV ribbon line or twin lead and 450 Q ladder line sometimes called window line also shown in Fig 2
49. ation resistor can serve as both the voltmeter and the load Alternatively a simple RF voltmeter or scope can be used with the calibrated load impedance being pro vided by atermination resistor of known value in the frequency range of the measurement With the ferrite choke in place obtain val ues for the voltage across the load resistor Vi oap and the generator in frequency incre ments of about 5 over the range of interest recording the data in a spreadsheet If multiple chokes are being measured use the same fre quencies for all chokes so that data can be plotted and compared Using the spreadsheet solve the voltage divider equation backward to find the unknown impedance Zyl Roan Voen Vioapl Plot the data as a graph of impedance on the vertical axis vs frequency on the horizontal axis Scale both axes to display logarithmically Obtaining R L and C Values This method yields the magnitude of the impedance Z but no phase information Accuracy is greatest for large values of unknown impedance worst case 1 for 5000 Q 10 for 500 Q Accuracy can be further improved by correcting for variations in the loading of the generator by the test cir cuit Alternatively voltage at the generator output can be measured with the unknown connected and used as Vogn The voltmeter must be unterminated for this measurement In asecond spreadsheet worksheet create a new table that computes the magnitude of the im
50. ctangular waveguide can be analyzed as a parallel two conductor transmission line supported from top and bottom by an infinite number of quarter wave stubs stubs may be connected in parallel with out affecting the standing waves of voltage and current The transmission line may be supported from the top as well as the bot tom and when infinitely many supports are added they form the walls of a waveguide at its cutoff frequency Fig 20 35 illustrates how a rectangular waveguide evolves from a two wire parallel transmission line This simplified analysis also shows why the cutoff dimension is A While the operation of waveguides is usu ally described in terms of fields current does flow on the inside walls just as fields exist between the current carrying conductors of a two wire transmission line At the waveguide Table 20 9 Wavelength Formulas for Waveguide Rectangular Circular Cut off wavelength 2X 3 41R Longest wavelength trans 1 6X 3 2R mitted with little attenuation Shortest wavelength before 1 1X 2 8R next mode becomes possible RF Potential Along Guide Electric Field Intensity Magnetic Intensity Along Guide Fig 20 36 Field distribution in a rectangular waveguide The TE mode of propagation is depicted cutoff frequency the current is concentrated in the center of the walls and disperses to ward the floor and ceiling as the frequency increases Anal
51. ctors cancel each other at the ferrite so the magnetic flux in the ferrite is virtually zero See the section on Transmitting Ferrite Choke Baluns The main idea of the transmission line transformer is that although the CM imped ance may be very large the DM signal is virtu ally unopposed especially if the line length is a small fraction of a wavelength But itis very important to keep in mind that the common mode voltage across the ferrite winding that is due to this current is efficiently coupled to the center wire by conventional transformer action as mentioned before and easily veri fied This equality of CM voltages and also CM impedances reduces the conversion of a CM signal to an undesired DM signal that can interfere with the desired DM signal in both transmitters and receivers The CM current multiplied by the CM impedance due to the ferrite produces a CM voltage The CM impedance has L and C T4A Output 120 W T5 Be fon HBKO5_ 14 056 Fig 20 20 This illustrates how transmission line transformers can be used in a push pull power amplifier Transmission Lines 20 19 reactance and also R So L C and R cause a broad parallel self resonance at some fre quency The R component also produces some dissipation heat in the ferrite This dissipa tion is an excellent way to dispose of a small amount of unwanted CM power Because of the high CM impedance the two output wires of the balun in Fig 20 19A have a
52. dom known exactly it is often possible to make a close estimate of its value with computer modeling soft ware As an example Table 20 1 lists the computed characteristics versus frequency foramultiband 100 ft long center fed dipole placed 50 ft above average ground These values were computed using EZNEC 3 A nonresonant 100 ft length was chosen as an illustration of a practical size that many ra dio amateurs could fit into their backyards although nothing in particular recommends this antenna over other forms It is merely used as an example Examine Table 20 1 carefully in the follow ing discussion Columns three and four show the SWR on a50 Q RG 213 coaxial transmis sion line directly connected to the antenna followed by the total loss in 100 ft of this cable The impedance for this nonresonant 100 ft long antenna varies over a very wide range for the nine operating frequencies The SWR on a 50 coax connected directly to this antenna would be extremely high on some frequencies particularly at 1 8 MHz where the antenna is highly capacitive because it is very short of resonance The loss in 100 ft of RG 213 at 1 8 MHz is a staggering 26 dB Contrast this to the loss in 100 ft of 450 Q open wire line Here the loss at 1 8 MHz is 8 8 dB While 8 8 dB of loss is not par ticularly desirable it is about 17 dB better than the coax Note that the RG 213 coax exhibits a good deal of loss on almost all the bands due to mismatch
53. ductor in same units as b It does not matter what units are used for S HBK05_21 002 se E E EE T E E ee ere wate Fig 20 2 Equivalent of an infinitely long lossless transmission line using lumped circuit constants d a or b as long as they are the same units A line with closely spaced large conductors will have a low characteristic impedance while one with widely spaced small conductors will have a relatively high characteristic imped ance Practical open wire lines exhibit char acteristic impedances ranging from about 200 to 800 Q while coax cables have Zo values between 25 to 100 Q Except in special in stances coax used in amateur radio has an impedance of 50 or 75 Q All practical transmission lines exhibit some power loss These losses occur in the re sistance that is inherent in the conductors that make up the line and from leakage currents flowing in the dielectric material between the conductors We II next consider what happens when a real transmission line which is not infinitely long is terminated in an actual load impedance such as an antenna 20 1 2 Matched and Mismatched Lines Real transmission lines do not extend to infinity but have a definite length In use they are connected to or terminate in a load such as an antenna as illustrated in Fig 20 3A If the loadis a pure resistance whose value equals the characteristic impedance of the line the line is said to be matched To cu
54. e ends of the choke open couple the coil to the dip meter and tune for a dip This is the parallel resonant frequency and the impedance will be very high The distributed capacitance of a flat coil choke balun can be reduced or at least con Table 20 6 Coiled Coax Choke Baluns Wind the indicated length of coaxial feed line into a coil like a coil of rope and secure with electrical tape Diameter 6 8 inches The balun is most effective when the coil is near the antenna Lengths and diameter are not critical Single Band Very Effective Freq RG 213 RG 8 RG 58 MHz 3 5 22 ft 8 turns 20 ft 6 8 turns 7 22 ft 10 turns 15 ft 6 turns 10 12 ft 10 turns 10 ft 7 turns 14 10 ft 4 turns 8 ft 8 turns 21 8 ft 6 8 turns 6 ft 8 turns 28 6 ft 6 8 turns 4 ft 6 8 turns Multiple Band Freq RG 8 58 59 8X 213 MHz 3 5 30 10 ft 7 turns 3 5 10 18 ft 9 10 turns 1 8 3 5 40 ft 20 turns 14 30 8 ft 6 7 turns trolled by winding the cable as a single layer solenoid around a section of plastic pipe an empty bottle or other suitable cylinder Fig 20 25 shows how to make this type of choke balun A coil diameter of about 5 inches is rea sonable This type of construction reduces the stray capacitance between the ends of the coil For both types of coiled coaxial chokes use cable with solid insulation not foamed to minimize migration of the center conduc tor through the insulation toward the shield The diame
55. e filtered out at the receiver and must be removed atthe transmitter One stub assembly would be connected to each transmitter output and manually switched for the appropriate band Two stubs are connected as shown The two relay selector box can be switched in four ways Stub 1 is a shorted quarter wave 40 meter stub Stub 2 is an open quarter wave 40 meter stub Operation is as shown in Table 20 3 The stubs must be cut and tuned while con nected to the selector relays RG 213 may be used for any amateur power level and will provide 25 to 30 dB reduction in the nulls For power levels under 500 W or so RG 8X may be used It will provide a few dB less reduction in the nulls because of its slightly higher loss than RG 213 20 4 Matching Impedances in the Antenna System Only in a few special cases is the antenna impedance the exact value needed to match a practical transmission line In all other cases it is necessary either to operate with a mis match and accept the SWR that results or else to bring about a match between the line and the antenna When transmission lines are used with a transmitter the most common load is an antenna When a transmission line is con nected between an antenna and a receiver the receiver input circuit is the load not the antenna because the power taken from a pass ing wave is delivered to the receiver Whatever the application the conditions existing at the load and only the load deter m
56. e increased by increased conductor size so stray capacitance will be greater with larger coax Turn to turn capacitance is also increased by larger diam eter turns At low frequencies most of the inductance in a ferrite choke results from coupling to the core but some is the result of flux outside the core At higher frequencies the core has less permeability and the flux outside the core makes a greater contribution The most useful cores for wound coax chokes are the 2 4 inch OD 1 4 inch ID toroid of type 31 or 43 material and the 1 inch ID x 1 125 inch long clamp on of type 31 mate rial Seven turns of RG 8 or RG 11 size cable easily fit through these toroids with no connec tor attached and four turns fit with a PL 259 attached Four turns of most RG 8 or RG 11 size cable fit within the 1 inch ID clamp on The toroids will accept at least 14 turns of most RG 6 RG 8X or RG 59 size cables 10000 9000 Fig 20 26 Typical transmitting wound coax common mode chokes suitable for use on the HF ham bands PRACTICAL CHOKES Fig 20 26 shows typical wound coax chokes suitable for use on the HF ham bands Fig 20 27 Fig 20 28 and Fig 20 29 are graphs of the magnitude of the imped ance for HF transmitting chokes of various sizes Fourteen close spaced 3 inch diameter turns of RG 58 size cable on a 31 toroid is a very effective 300 W choke for the 160 and 80 meter bands Table 20 7 summarizes designs that meet the 500
57. e of the line and the impedance of the load at its end What actually happens to the energy re flected back down the line This energy will encounter another impedance discontinuity this time at the generator Reflected energy flows back and forth between the mismatches at the source and load After a few such jour neys the reflected wave diminishes to noth ing partly as a result of finite losses in the line but mainly because of absorption at the load In fact if the load is an antenna such absorption at the load is desirable since the energy 1s actually radiated by the antenna If a continuous RF voltage is applied to the terminals of a transmission line the voltage at any point along the line will consist of a vec tor sum of voltages the composite of waves traveling toward the load and waves traveling back toward the source generator The sum of the waves traveling toward the load is called the forward or incident wave while the sum of the waves traveling toward the generator is called the reflected wave 20 1 3 Reflection Coefficient and SWR In amismatched transmission line the ratio of the voltage in the reflected wave at any one point on the line to the voltage in the forward wave at that same point is defined as the volt age reflection coefficient This has the same value as the currentreflection coefficient The reflection coefficient is a complex quantity that is having both amplitude and phase and is general
58. ecting operator measuring the SWR at his transmit ter might well believe that everything is just fine when in truth only about 10 of the transmitter power is getting to the antenna Be suspicious of very low SWR readings for an antenna fed with a long length of coaxial cable especially if the SWR remains low across a wide frequency range Most antennas have narrow SWR bandwidths and the SWR should change across a band On the other hand if expensive inch diameter 50 Q hardline cable is used at 28 MHz the matched line loss is only 0 19 dB 100 ft For 250 ft of this Hardline the matched line loss is 0 475 dB and the additional loss due to a6 1 SWR is 0 793 dB Thus the total loss is 1 27 dB At the upper end of the HF spectrum when the transmitter and antenna are separated by a long transmission line the use of bargain coax may prove to be a very poor cost saving strategy Adding a 1500 W linear amplifier providing 8 7 dB of gain over a 200 W trans mitter to offset the loss in RG 58A com pared to hardline would cost a great deal more than higher quality coax Furthermore no transmitting amplifier can boost receiver sensitivity loss in the line has the same effect as putting an attenuator in front of the receiver At the lower end of the HF spectrum say 3 5 MHz the amount of loss in common coax lines 1s less of a problem for the range of SWR HBK05_ 21 005
59. ection based measurements have increasingly poor accuracy when the unknown impedance is more than about three times the characteristic impedance of the analyzer because the value of the unknown is computed by differencing analyzer data When the differences are small as they are for high impedances measured this way even very small errors in the raw data cause very large errors in the computed result While the software used with reflection based systems use calibration and computation methods to remove systemic errors such as fixture capaci tance from the measurement these methods have generally poor accuracy when the im pedance being measured is in the range of typical ferrite chokes The key to accurate measurement of high impedance ferrite chokes is to set up the choke as the series element Zy of a voltage divider Impedance is then measured using a well calibrated voltmeter to read the voltage across a well calibrated resistor that acts as the voltage divider s load resistor Ry gap The fundamental assumption of this measurement method is that the unknown impedance is much higher than the impedance of both the generator and the load resistor The RF generator driving the high imped ance of the voltage divider must be terminated by its calibration impedance because the gen erator s output voltage Vopn is calibrated only when working into its calibration imped ance An RF spectrum analyzer with its own internal termin
60. ed to resonance even without special efforts to make the resistive portion equal to the line s characteristic impedance the resulting SWR is often low enough to minimize ad ditional line loss due to SWR For example the multiband 100 ft long flat top antenna in Table 20 1 has a feed point impedance of 4 18 j 1590 Q at 1 8 MHz Assume that the antenna reactance is tuned out with a network con sisting of two symmetrical inductors whose reactance is j 1590 2 j 795 Q each with a Q of 200 These inductors are 70 29 uH coils in series with inherent loss resistances of 795 200 3 98 Q The total series resistance is thus 4 18 2 x 3 98 12 1 Q If placed in series with each leg of the antenna at the feed point as in Fig 20 10 the antenna reactance and inductor reactance cancel out leaving a purely resistive impedance at the antenna feed point If this tuned system is fed with 50 Q co axial cable the SWR is 50 12 1 4 13 1 and the loss in 100 ft of RG 213 cable would be 0 48 dB The antenna s radiation efficiency is the ratio of the antenna s radiation resis tance 4 18 Q to the total feed point resis tance including the matching coils 12 1 Q so efficiency is 4 18 12 1 34 6 which is equivalent to 4 6 dB of loss compared to a 100 efficient antenna Adding the 0 48 dB of loss in the line yields an overall system loss of 5 1 dB Compare this to the loss of 26 dB if the RG 213 coax is used to feed the antenna directly
61. es and TE waves are sometimes called H waves The TM and TE designations are preferred however The particular mode of transmission is iden tified by the group letters followed by subscript numbers for example TE TM and so on The number of possible modes increases with frequency for a given size of guide There is only one possible mode called the dominant mode for the lowest frequency that can be transmitted The dominant mode is the one normally used in practical applications 20 7 3 Waveguide Dimensions In rectangular guides the critical dimension shown as X in Fig 20 36C must be more than one half wavelength at the lowest frequency to be transmitted In practice the Y dimen sion is usually made about equal to 2 X to avoid the possibility of operation at other than the dominant mode Cross sectional shapes other than rectangles can be used the most important of those is the circular pipe Table 20 9 gives dominant mode wave length formulas for rectangular and circular guides X is the width of a rectangular guide and R is the radius of a circular guide 20 7 4 Coupling to Waveguides Energy may be introduced into or extracted from a waveguide or resonator by means of either the electric or magnetic field The en ergy transfer frequently takes place through a coaxial line Two methods for coupling are shown in Fig 20 37 The probe at A is sim ply a short extension of the inner conductor of the feed coaxial l
62. esonance occurs below 5 MHz is very low Q is poorly defined and blends with the circuit resonance to broaden the impedance curve The result is a dual sloped resonance curve that is curve fitting will produce somewhat different values of R L and C when matching the low frequency slope and high frequency slope When using these values in a circuit model use the values that most closely match the behavior of the choke in the frequency range of interest 20 6 Using Transmission Lines in Digital Circuits The performance of digital logic families covers a wide range of signal transition times The signal rise and fall times are most im portant when considering how to construct a circuit The operating frequency of a circuit is not the primary consideration A circuit that uses high speed logic yet runs only at a few kHz can be difficult to tame if long point to point wiring is used If the path between two points has a delay of more than the logic family rise time some form of transmission line should be considered We know that waves propagate at 300 million meters per second in air and at 0 66 times as fast in common coax cable So in about 5 ns a wave will travel 1 meter or in 1 ns it will travel 0 2 m Consider a logic family which has 2 ns rise and fall time Using the rule mentioned above if the path length exceeds 0 066 meter or about 2 6 inches we need to use a transmission line Another way to look at it is the ap
63. g wave on the line The amount of reflection depends on the difference between the load impedance and the characteristic impedance of the transmission line some limiting cases First consider the rather extreme case where the line is shorted at its end Energy flowing to the load will encounter the short at the end and the voltage at that point will go to zero while the current will rise toamaximum Since the current can t develop any power in a dead short the energy will all be reflected back toward the source generator If the short at the end of the line is replaced with an open circuit the opposite will hap pen Here the voltage will rise to maximum and the current will by definition go to zero The phase will reverse and all energy will be reflected back towards the source By the way if this sounds to you like what happens at the end of a half wavelength dipole antenna you are quite correct However in the case of an antenna energy traveling along the antenna is lost by radiation on purpose whereas a good transmission line will lose little energy to ra diation because of field cancellation between the two conductors For load impedances falling between Transmission Lines 20 3 the extremes of short and open circuit the phase and amplitude of the reflected wave will vary The amount of energy reflected and the amount of energy absorbed in the load will depend on the difference between the characteristic impedanc
64. hase noise and wideband noise which might cause interfer ence to receivers operating on 14 or 28 MHz The open circuited quarter wave stub has a low impedance at the fundamental frequen cy so it must be used at two times the fre quency for which it is cut For example a quarter wave open stub cut for 3 5 MHz will present a high impedance at 7 MHz where it is A long It will present a high impedance at those frequencies where it is a multiple of Y Xr or 7 14 and 28 MHz It would be con nected in the same manner as Fig 20 5 shows and the frequency plot is shown in Fig 20 7 This open stub can protect a receiver oper ating on 7 14 21 or28 MHz from interference by a 3 5 MHz transmitter It also has nulls at 10 5 17 5 and 24 5 MHz the 3rd 5th and 7th harmonics The length of a quarter wave stub may be calculated as follows _ VF x 983 6 Af where L length in ft VF propagation constant for the coax in use f frequency in MHz L 15 For the special case of RG 213 and any similar cable with VF 0 66 equation 15 can be simplified to 163 oe 63 5 f where L length in ft f frequency in MHz 16 Table 20 2 solves this equation for the major contesting bands where stubs are often used The third column shows how much of the stub to cut off if the desired frequency is 100 kHz higher in frequency For example To cut a stub for 14 250 MHz reduce the overall length shown by 2 5 x 1 inches or 2 5
65. he CW and SSB frequencies in one band Open and shorted stubs can be combined together to attenuate higher harmonics as well as lower frequency bands An interesting combination is the parallel connection of two 4 A stubs one open and the other shorted The shorted stub will act as an inductor and the open stub as a capacitor Their reactance will be equal and opposite forming a resonant circuit The null depth with this arrangement will be a bit better than a single quarter wave shorted stub This pres ents some possibilities when combinations of stubs are used in a band switching system Table 20 3 Stub Selector Operation See Fig 20 8 for circuit details Relay K1 Relay K2 Bands Passed Bands Nulled Position Position meters meters Open Open All None Energized Energized 80 40 20 15 10 Energized Open 40 15 20 10 Open Energized 20 10 40 15 From TX to Antenna L A RSX Selector Shorted Open HBK0136 Fig 20 8 Schematic of the Field Day stub switching relay control box Table 20 3 shows which relays should be closed for the desired operating band 20 3 3 Project A Field Day Stub Assembly Fig 20 8 shows a simple stub arrangement that can be useful in a two transmitter Field Day station The stubs reduce out of band noise produced by the transmitters that would cause interference to the other stations a common Field Day problem where the sta tions are quite close together This noise can not b
66. he RF Power Amplifiers chapter and the very considerable literature for a deeper understanding and for techniques used at different frequency ranges Fig 20 21 illustrates one example of each of the three basic types In a 0 hybrid splitter at the input the tight coupling between the two windings forces the voltages at A and B to be equal in amplitude and also equal in phase if the two modules are identical The 2R resistor between points A and B greatly reduces the transfer of power between A and B via the transformer but only if the generator resistance is closely equal to R The output combiner separates the two outputs C and D from each other in the same manner if the output load is equal to R as shown No power is lost in the 2R resistor if the module output levels are identical APPLICATIONS OF TRANSMISSION LINE TRANSFORMERS There are many transformer schemes that use the basic ideas of Fig 20 19 Several of them with their toroid winding instructions are shown in Fig 20 22 Two of the most com monly used devices are the 1 1 current balun and 4 1 impedance transformer wound on toroid cores as shown in Fig 20 23 Because of space limitations for a compre hensive treatment we suggest Jerry Sevick s books Transmission Line Transformers and Building and Using Baluns and Ununs For applications in solid state RF power ampli fiers see Sabin and Schoenike HF Radio Systems and Circuits Chapter 12 20 5 3 Coiled Coaxial
67. he second example direct feed with 300 Q low loss line does not always give the lowest loss The combination method in Example 3 provides the best solution There are other considerations as well Hanging a balun at the antenna adds stress to the wires but can be avoided Example 3 Table 20 5 Tuner Settings and Performance Example Frequency Tuner L C Total Loss Fig 20 16 MHz Type UH DF dB 1 3 8 Rev L 1 46 2308 8 53 28 4 Rev L 0 13 180 9 12 3 2 3 8 L 14 7 46 2 74 28 4 L 0 36 15 6 3 52 3 3 8 L 11 37 332 1 81 28 4 L 0 54 94 0 2 95 has some additional advantages It feeds the antenna in a symmetrical arrangement which is best to reduce noise pickup on the shield of the feed line The shorter feed line will not weigh down the antenna as much The coax back to the shack can be buried or laid on the 200 RG 213 ground and it is perfectly matched Burial of the cable will also prevent any currents from being induced on the coax shield Once in the shack the tuner is adjusted for minimum SWR per the manufacturer s instructions 100 Antenna a Q 100 Antenna aaaout fill 200 of 300 Line 100 Antenna opr 50 of 300 Q Line Transmitter 150 RG 213 HBK0143 Fig 20 16 Variations of an antenna system with different losses The examples are discussed in the text 20 4 5 Adjusting Antenna Tuners The process of adjus
68. high degree of balance throughout the amplifier Note also the various feedback and loading networks that help keep the ampli fier frequency response flat Quite often the performance of a single stage can be greatly improved by combin ing two identical modules Because the input poweris splitevenly between the two modules the drive source power can be twice as great and the output power will also be twice as great In transmitters especially this often works better than a single transistor with twice the power rating Or for the same drive and output power each module need supply only one half as much power which usually means better distortion performance Often the to tal number of stages can be reduced in this manner with resulting cost savings If the combining is performed properly using hy brid transformers the modules interact with each other much less which can avoid certain problems These are the system design impli cations of module combining Three methods are commonly used to com bine modules parallel 0 push pull 180 and quadrature 90 In RF circuit design the combining is often done with special types of hybrid transformers called splitters and combiners These are both the same type of transformer that can perform either function The splitter is at the input the combiner at the output We will only touch very briefly on these topics in this chapter and suggest that the reader consult t
69. high impedance with respect to and are therefore isolated from the generator This feature is very useful because now any point of R at the output can be grounded In a well designed balun circuit almost all of the current in one conductor returns to the generator through the other conductor de spite this ground connection Note also that the ground connection introduces some CM voltage across the balun cores and this has to be taken into account This CM voltage is a maximum if point C is grounded If point D is grounded and if all ground connections are at the same potential which they often are not the CM voltage is zero and the balun may no longer be needed In a coax balun the return current flows on the inside surface of the braid We now look briefly at a transmission line transformer that is based on the choke balun Fig 20 19B shows two identical choke baluns whose inputs are in parallel and whose outputs are in series The output voltage amplitude of each balun is identical to the common input so the two outputs add in phase equal time delay to produce twice the input voltage It is the high CM impedance that makes this volt age addition possible If the power remains constant the load current must be one half the generator current and the load resistor is 2V 0 51 4V 1 4R THE GUANELLA TRANSFORMER The CM voltage in each balun is V 2 so there is some flux in the cores The right side floats This is n
70. ics of many common types of transmission lines are included in the software so that real antenna matching problems may be easily solved Detailed instructions on using the pro gram are included with it The various ex amples in this chapter have been solved with TLW Equation 6 is also solved in a normalized form with a graphical method called the Smith Chart Matching problems can be handled much easier than solving complex arithme tic equations by using this polar chart More detailed information about using the Smith Chart is included in The ARRL Antenna Book Many references to Smith Charts and their use may be found on the Web 20 1 4 Losses in Transmission Lines A real transmission line exhibits a certain amount of loss caused by the resistance of the conductors used in the line and by dielectric losses in the line s insulators The matched line loss for a particular type and length of transmission line operated at a particular frequency is the loss when the line is termi nated in aresistance equal to its characteristic impedance The loss in a line is lowest when it is operated as a matched line Line losses increase when SWR is greater than 1 1 Each time energy flows from the generator toward the load or is reflected at the load and travels back toward the genera tor a certain amount will be lost along the line The net effect of standing waves on a transmission line is to increase the average value of current a
71. ine oriented so that it is parallel to the electric lines of force The loop shown at B is arranged to enclose some of the magnetic lines of force The point at which maximum coupling will be obtained depends on the particular mode of propaga tion in the guide or cavity the coupling will be maximum when the coupling device is in the most intense field Coupling can be varied by rotating the probe or loop through 90 When the probe is perpendicular to the electric lines the cou pling will be minimum similarly when the plane of the loop is parallel to the magnetic lines the coupling will be minimum See The ARRL Antenna Book for more information on waveguides HBKOS5_ 21 019 Fig 20 37 Coupling to waveguide and resonators The probe at A is an extension of the inner conductor of coax line At B an extension of the coax inner conductor is grounded to the waveguide to form a coupling loop Transmission Lines 20 29 20 8 Glossary of Transmission Line Terms Antenna tuner A device that matches the antenna system input impedance to the transmitter receiver or transceiver output impedance Also called an antenna matching network impedance matcher transmatch ATU matchbox Balanced line A symmetrical two conductor feed line that has uniform voltage and current distribution along its length Balun Contraction of balanced to unbalanced A device to couple a balanced load to an unbalanced feed line or
72. ine impedances and SWR Standing wave ratio SWR Sometimes called voltage standing wave ratio VSWR A measure of the impedance match between a feed line s characteristic impedance and the attached load usually an antenna VSWR is the ratio of maximum voltage to minimum voltage along the feed line or of antenna impedance to feed line impedance Stacking The technique of placing similar directive antennas atop or beside one another forming a stacked array Stacking provides more gain or directivity than a single antenna Stub A section of transmission line used to perform impedance matching or filtering Surge impedance see Characteristic impedance SWR see Standing wave ratio SWR bridge Device for measuring SWR in a transmission line Also known as an SWR meter or reflectometer TE mode Transverse electric field mode Condition in a waveguide in which the E field component of the traveling electromagnetic energy is oriented perpendicular to transverse the direction the energy is traveling in the waveguide TM mode Transverse magnetic field mode Condition in a waveguide in which the H field magnetic field component of the traveling electromagnetic energy is oriented perpendicular to transverse the direction the energy is traveling in the waveguide Transmatch See Antenna tuner Transmission line The wires or cable used to connect a transmitter or receiver to
73. ine the reflection coefficient and hence the SWR on the line Ifthe load is purely resistive and equal to the characteristic impedance of the line there will be no standing waves If the load is not purely resistive or is not equal to the line Zp there will be standing waves No adjustments can be made at the input end of the line to change the SWR at the load Neither is the SWR affected by changing the line length except as previously described when the SWR at the input of a lossy line is masked by the attenuation of the line 20 4 1 Conjugate Matching Technical literature sometimes uses the term conjugate match to describe the condi tion where the impedance seen looking to ward the load from any point on the line is the complex conjugate of the impedance seen looking toward the source This means that the resistive and reactive magnitudes of the impedances are the same but that the reac tances have opposite signs For example the complex conjugate of 20 j 75 is 20 j 75 The complex conjugate of a purely resistive impedance such as 50 j 0 Q is the same impedance 50 j 0 Q A conjugate match is necessary to achieve the maximum power gain possible from most signal sources For example if 50 ft of RG 213 is termi nated in a 72 j 34 Q antenna impedance the transmission line transforms the imped ance to 35 9 j 21 9 Q at its input The TLW program is used to calculate the imped ance at the line input To c
74. ines the energy of the fields which are propagated through it to the receiving end by means of reflections off its inner walls 20 7 1 Evolution of a Waveguide Suppose an open wire line is used to carry UHF energy from a generator to a load If the line has any appreciable length it must be well insulated from the supports in order to avoid high losses Since high quality in sulators are difficult to make for microwave frequencies it is logical to support the trans mission line with quarter wave stubs shorted at the far end The open end of such a stub presents an infinite impedance to the trans mission line provided that the shorted stub is non reactive However the shorting link has finite length and therefore some inductance This inductance can be nullified by making the RF current flow on the surface of a plate rather than through a thin wire If the plate is large enough it will prevent the magnetic lines of force from encircling the RF current An infinite number of these quarter wave 20 28 Chapter 20 Fig 20 34 Reflections in a transmission line cause stepping in the leading edge of a digital pulse at the generator A By adding a resistor in series with the gate output a step is generated at the input to the transmission line B but a full voltage step is created at the high input impedance of the receiving gate sion line at the signal source as in Fig 20 34 A resistor is added in series with the ou
75. is approximated by using a 4 A sec tion of 50 Q cable in parallel with a 1 4 sec tion of 75 Q cable yielding a net impedance of 30 Q quite close enough to the desired 28 9 Q Four Yagis fed in parallel would require a 4 A transformer made up using cable with a characteristic impedance of 25 Q easily created by using two 50 Q cables in parallel The 100 ft flat top example in the previous section with the two resonating coils has an impedance of 12 Q at the feed point Two RG 58A cables each 4 A long at 1 8 MHz 90 ft can be connected in parallel to feed this an tenna An additional 10 ft length of RG 213 can make up the required 100 ft The match will be almost perfect The disadvantage of this system is that it limits the operation to one band but the overall efficiency will be quite good Another use of 4 A transformers is in matching the impedance of full wave loop antennas to 50 Q coax For example the driven element of a quad antenna or a full wave 40 meter loop has an impedance of 100 to 150 Q Using a4 A transformer made from 62 75 or 93 Q coaxial cable would lower the line SWR to a level where losses were insignificant This use of 4 4 transformers is limited to one band at a time Additional 4 A lines need to be switched in to change bands MATCHING TRANSFORMERS There is another matching technique that uses wide band toroidal transformers Trans formers can be made that operate over very wide frequency
76. is shown in Fig 20 19D where the CM voltage is V not V 2 and transmission is from left to right only Because of the greater flux in the cores no different than a conventional transformer this is not a preferred approach although it could be used with air wound coils for example in antenna tuner circuits to couple 75 Q unbal anced to 300 Q balanced The tuner circuit could then transform 75 Q to 50 Q POWER AMPLIFIER AND COMBINER USE Fig 20 20 illustrates in skeleton form how transmission line transformers can be used ina push pull solid state power amplifier The idea is to maintain highly balanced stages so that each transistor shares equally in the amplifica tion in each stage The balance also minimizes even order harmonics so that low pass filter ing of the output is made much easier In the diagram T1 and T5 are current choke baluns that convert a grounded connection at one end to a balanced floating connection at the other end with a high impedance to ground at both wires T2 transforms the 50 Q generator to the 12 5 Q 4 1 impedance input impedance of the first stage T3 performs a similar step down transformation from the collectors of the first stage to the gates of the second stage The MOSFETs require a low impedance from gate to ground The drains of the output stage require an impedance step up from 12 5 Q to 50 Q performed by T4 Note how the choke baluns and the transformers collaborate to maintain a
77. ition to this leakage flux the core will also carry the flux associated with common mode current When atransformer as opposed to achoke is wound on a magnetic core all of the field associated with current in the windings is carried by the core Similarly all forms of voltage baluns require all of the transmitted power to couple to the ferrite core Depend ing on the characteristics of the core this can result in considerable heating and power loss Only a few ferrite core materials have loss characteristics suitable for use as the cores of high power RF transformers Type 61 material has reasonably low dissipation below about 10 MHz but its loss tangent rises rapidly above that frequency The loss tangent of type 67 material makes it useful in high power transformers to around 30 MHz Leakage flux corresponding to 30 40 of the transmitter power causes heating in the ferrite core and attenuates the transmitted signal by a dB or so At high power levels temperature rise in the core also changes its magnetic properties and in the extreme case can result in the core temporarily losing its magnetic properties A flux level high enough to make the core hot is also likely to saturate the core producing distortion harmonics splatter clicks Flux produced by common mode current can also heat the core if there is enough common mode current Dissipated power is equal to I R so it can be made very small by making the common
78. line the outer conductor in a coaxial line acts as a shield confining RF energy within the line as shown in Fig 20 1E Because of skin effect see the RF Techniques chapter current flowing in the outer conductor of a coax does so on the in ner surface of the outer conductor The fields generated by the currents flowing on the outer surface of the inner conductor and on the inner surface of the outer conductor cancel each other out just as they do in open wire line VELOCITY FACTOR In free space electrical waves travel at the speed of light or 299 792 458 meters per sec ond Converting to feet per second yields 983 569 082 The length of a wave in space may be related to frequency as wavelength A velocity frequency Thus the wavelength of a Transmission Lines 20 1 Single Shielded Braid N a D Le Conductors Center Outer iati Conductor uter insulation Dielectric Vinyl Jacket Double Shielded o E M SY ANNAN l Conductors Braid Vinyl Braid Vinyl No 2 No 1 Rigid Hardline C a gt H Conductors Inner Foam Aluminum Outer Conductor Dielectric Conductor Available with vinyl jacket Semi Flexible Hardline D I S Sal Corrugated Vinyl Jacket lelectric Copper Shield Inner Currents in Coaxial Lines Conductor E lt lt HBK0134 300 Ohm Twin Lead Polyethylene Insulation 75 Ohm Twin Lead Polyethylene Insulation 450 Ohm Ladder Line
79. line is lower than the SWR measured at the load end of the line This does not mean that the load is absorbing any more power Line loss absorbs power as it travels to the load and again on its way back to the generator so the difference between the generator output power and the power returning from the load is higher than for a lossless line Thus P P is smaller than at the load and so is the measured SWR For example RG 213 solid dielectric coax cable exhibits a matched line loss at 28 MHz of 1 14 dB per 100 ft A 250 ft length of this cable has a matched line loss of 1 14 x 250 100 2 86 dB Assume that we measure the SWR at the load as 6 1 the total mismatched line loss from equation 11 is 5 32 dB The additional loss due to the 6 1 SWR at 28 MHz is 5 32 2 86 2 46 dB The SWR at the input of the 250 ft line is only 2 2 1 be cause line loss has masked the true magnitude of SWR 6 1 at the load end of the line The losses increase if coax with a larger matched line loss is used under the same con ditions For example RG 58A coaxial cable is about one half the diameter of RG 213 and it has a matched line loss of 2 81 dB 100 ft at 28 MHz A 250 ft length of RG 58A has a total matched line loss of 7 0 dB With a 6 1 SWR at the load the additional loss due to SWR is 3 0 GB for a total loss of 10 0 dB The additional cable loss due to the mismatch reduces the SWR measured at the input of the line to 1 33 1 An unsusp
80. line or connectors are not arcing The antenna is merely doing its job which is to radiate The transmission line is doing its job which is to convey power from the transmitter to the radiator e A second myth often stated in the same breath as the first one above is that a high SWR will cause excessive radiation from a transmission line SWR has nothing to do with excessive radiation from a line Imbalances in feed lines cause radiation but such imbalances are notrelatedto SWR An asymmetric arrangement of a transmis sion line and antenna can result in current being induced on the transmission line on the shield of coax or as an imbalance of currents in an open wire line This current will radiate just as if it was on an antenna A choke balun is used on coaxial feed lines to reduce these currents as described in the section on baluns later in this chapter e A third and perhaps even more prevalent myth is that you can t get out if the SWR on your transmission line is higher than 1 5 1 or 2 1 or some other such arbitrary figure On the HF bands if you use reason able lengths of good coaxial cable or even better yet open wire line the truth is that you need not be overly concerned if the SWR at the load is kept below about 6 1 This sounds pretty radical to some amateurs who have heard horror story after horror story about SWR The fact is that if you can load up your transmitter without any arcing inside or if
81. ly designated by the Greek let ter p rho or sometimes in the professional literature as I Gamma The relationship between R the load resistance X the load reactance Z the line characteristic imped ance whose real part is R and whose reac tive part is X and the complex reflection coefficient p is a Z Zo _ Ry jX Ro jXo 6 Z Zy Ry jX Ro jXo For most transmission lines the charac teristic impedance Z is almost completely resistive meaning that Zo Ry and Xp 0 The magnitude of the complex reflection co efficient in equation 6 then simplifies to RR X pl RiR iX 7 For example if the characteristic imped ance of a coaxial line is 50 Q and the load impedance is 120 Q in series with a capaci 20 4 Chapter 20 tive reactance of 90 Q the magnitude of the reflection coefficient is 2 2 ip _ 20 ue 2 0 593 120 50 90 Note that if R in equation 6 is equal to Ry and X is 0 the reflection coefficient p is 0 This represents a matched condition where all the energy in the incident wave is trans ferred to the load On the other hand if R is 0 meaning that the load is a short circuit and has no real resistive part the reflection coeffi cientis 1 0 regardless of the value of Ro This means that all the forward power is reflected since the load is completely reactive The concept of reflection is often shown in terms of the return loss R
82. mode current everywhere along a long feed line If common mode current on the line far from the antenna feed point is a prob lem additional choke baluns can be placed at approximately 4 A intervals along the line This breaks up the line electrically into seg ments too short to act as effective antennas The chokes in this case function similarly to insulators used to divide tower guy wires into non resonant lengths 20 5 1 Quarter Wave Baluns Fig 20 17B shows a balun arrangement known as a bazooka which uses a sleeve over the transmission line The sleeve together with the outside portion of the outer coax conductor forms a shorted 4 A section of transmission line The impedance looking into the open end of such a section is very high so the end of the outer conductor of the coaxial line is effectively isolated from the part of the line below the sleeve The length is an electrical 4 2 and because of the velocity factor may be physically shorter if the insula tion between the sleeve and the line is not air The bazooka has no effect on antenna imped ance at the frequency where the A sleeve is resonant However the sleeve adds inductive shunt reactance at frequencies lower and ca pacitive shunt reactance at frequencies higher than the 4 2 resonant frequency The bazooka is mostly used at VHF where its physical size does not present a major problem Another method that gives an equivalent effect is shown at Fig 20 17C
83. nd voltage compared to the matched line case An increase in current raises I R ohmic losses in the conductors and an increase in RF voltage increases E R losses in the dielectric Line loss rises with frequency since the conductor resistance 1s related to skin effect and also because dielec tric losses rise with frequency Matched line loss ML is stated in deci bels per hundred feet at a particular frequency The matched line loss per hundred feet versus frequency for a number of common types of lines both coaxial and open wire balanced types is shown graphically and as a table in the Component Data and References chapter For example RG 213 coax cable has a matched line loss of 2 5 dB 100 ft at 100 MHz Thus 45 ft of this cable feeding a 50 Q load at 100 MHz would have a loss of Matched line loss PoR x 45 ft 100 ft 1 13dB If a line is not matched standing waves will cause additional loss beyond the inherent matched line loss for that line Total Mismatched Line Loss dB 11 where a 1 QML 10 ML the line s matched loss in dB For most types of line and for modest values of SWR the additional line loss due to SWR is of little concern As the line s loss increases or at higher frequencies the total line loss the sum of matched line loss and additional loss due to SWR can be surprisingly high at high values of SWR Because of losses in a transmission line the measured SWR at the input of the
84. ng its ability to re ject noise and interference Chokes used to break up a feed line into segments too short to interact with another antenna should have a choking impedance on the order of 1000 Q to prevent interaction with simple antennas A value closer to 5000 Q may be needed if the effects of com mon mode current on the feed line are filling the null of directional antenna HBK0622 Tiewrap Tiewrap Coax Fig 20 25 Winding a coaxial choke balun as a single layer solenoid typically increases impedance and self resonant frequency compared to a flat coil choke Transmission Lines 20 23 BUILDING WOUND COAX FERRITE CHOKES Coaxial chokes should be wound with a bend radius sufficiently large that the coax is not deformed When a line is deformed the spacing between the center conductor and the shield varies so voltage breakdown and heat ing are more likely to occur Deformation also causes a discontinuity in the impedance the resulting reflections may cause some wave form distortion and increased loss at VHF and UHF Coaxial cable has a specified mini mum bend radius Chokes wound with any large diameter cable have more stray capacitance than those wound with small diameter wire There are two sources of stray capacitance in a ferrite choke the capacitance from end to end and from turn to turn via the core and the capaci tance from turn to turn via the air dielectric Both sources of capacitance ar
85. nt around 150 MHz are inductive below resonance and have only a few tens of ohms of strongly inductive impedance on the HF bands Even with 20 of the type 31 or 43 beads in the string the choke is still resonant around 150 MHz is much less effective than a wound coaxial ferrite choke and is still inductive on the HF bands so it will be ineffective at frequencies where it resonates with the line Joe Reisert WIJR introduced the first damps the resonance of the coil and increases its useful bandwidth The combinations of ferrite and coil baluns shown in Table 20 8 demonstrate this very effectively Eight feet of RG 8X in a 5 turn coil is a great balun for 21 MHz but it is not particularly effective on other bands If one type 43 core Fair Rite 2643 167851 is inserted in the same coil of coax the balun can be used from 3 5 21 MHz If two of these cores are spaced a Transmission Lines 20 25 100000 HBK0449 50000 30000 20000 10000 7000 5000 3000 2000 f Z O D Q 1000 700 500 300 200 100 5 6 7 8910 Frequency MHz Fig 20 31 Impedance versus frequency for HF wound coax transmitting chokes wound with RG 142 coax on toroid cores of 61 material For the 1 core choke R 15 6 kQ L 25 pH C 1 4 pF Q 3 7 For the 2 core choke R 101 k
86. ome automatic tuners are designed to be installed at the antenna for example For some situations placing the tuner at the base of a tower can be particularly effective and eliminates having to climb the tower to perform maintenance on the tuner It is useful to consider the performance of the entire antenna system when deciding where to install the antenna tuner and what types of feed line to use in order to minimize system losses Here is an example using the program TLW Let s assume a flat top antenna 50 ft high and 100 ft long and not resonant on any amateur band As extreme examples we will use 3 8 and 28 4 MHz with 200 ft of transmission line There are many ways to configure this system but three examples are shown in Fig 20 16 Example 1 in Fig 20 16A shows a 200 ft run of RG 213 going to a 1 1 balun that feeds the antenna A tuner in the shack re duces the VSWR for proper matching in the transmitter Example 2 shows a similar ar rangement using 300 Q transmitting twin lead Example 3 shows a 50 ft run of 300 Q line dropping straight down to a tuner near the ground and 150 ft of RG 213 going to the shack Table 20 5 summarizes the losses and the tuner values required Some interesting conclusions can be drawn First direct feeding this antenna with coax through a balun is very lossy a poor solution If the flat top was long a resonant half wavelength dipole direct coax feed would be a good method In t
87. output capacitor C2 The first L network then transforms the parallel equivalent back into the series equivalent and resonates the reac tance with the input series capacitor Cl 20 14 Chapter 20 Note that the equivalent parallel resistance R across the shunt inductor can be a very large value for highly reactive loads meaning that the voltage developed at this point can be very high For example assume that the load impedance at 3 8 MHz presented to the an tenna tuner is Z 20 j 1000 If C2 is 300 pF then the equivalent parallel resistance across L1 is 66 326 Q If 1500 W appears across this parallel resistance a peak voltage of 14 106 V is produced a very substantial level indeed Highly reactive loads can produce very high voltages across components in a tuner The ARRL computer program TLW cal culates and shows graphically the antenna tuner values for operator selected antenna impedances transformed through lengths of various types of practical transmission lines The Station Accessories chapter includes an tenna tuner projects and The ARRL Antenna Book contains detailed information on tuner design and construction ANTENNA TUNER LOCATION The tuner is usually located near the trans mitter in order to adjust it for different bands or antennas If a tuner is in use for one par ticular band and does not need to be adjusted once set up for minimum VSWR it can be placed in a weatherproof enclosure near the antenna S
88. pedance of a parallel resonant circuit for the same range of frequencies as your choke measurements The required equations can be found in the section Parallel Circuits of Moderate to High Q of the Electrical Fun damentals chapter Set up the spreadsheet to compute resonant frequency and Q from manually entered values for R L and C The spreadsheet should also compute and plot impedance of the same range of frequencies as the measurements and with the same plotted scale as the measurements 1 Enter a value for R equal to the resonant peak of the measured impedance 2 Pick a point on the resonance curve be low the resonant frequency with approximate ly one third of the impedance at resonance and compute L for that value of inductive reactance 3 Enter a value for C that produces the same resonant frequency of the measurement 4 If necessary adjust the values of L and C until the computed curve most closely matches the measured curve The resulting values for R L and C form the equivalent circuit for the choke The val ues can then be used in circuit modeling soft ware NEC SPICE to predict the behavior of circuits using ferrite chokes Accuracy This setup can be constructed so that its stray capacitance is small but it won t be zero A first approximation of the stray ca pacitance can be obtained by substituting for the unknown a noninductive resistor whose resistance is in the same general range as the choke
89. poles or Yagis fed with unbalanced coaxial line Some method should be used for connecting the line to the antenna without upsetting the symmetry of the antenna itself This requires a circuit that will isolate the balanced load from the unbalanced line while still providing efficient power transfer Devices for doing this are called baluns a contraction for balanced to unbalanced A balanced antenna fed with balanced line such as two wire ladder line will maintain its inherent balance so long as external causes of unbalance are avoided However even they will require some sort of 20 16 Chapter 20 balun at the transmitter since modern trans mitters have unbalanced coaxial outputs If a balanced antenna is fed at the center by a coaxial feed line without a balun as indi cated in Fig 20 17A the inherent symmetry and balance is upset because one side of the radiator is connected to the shield while the other is connected to the inner conductor On the side connected to the shield current can be diverted from flowing into the antenna and instead can flow away from the antenna on the outside of the coaxial shield The field thus set up cannot be canceled by the field from the inner conductor because the fields inside the cable cannot escape through the shielding of the outer conductor Hence currents flowing on the outside of the line will be responsible for some radiation from the line just as if they were flowing on
90. preserve the pattern of an antenna that is Shorted to Coax Outer Conductor Here Zo Balanced Unbalanced Shorted Together HBKO5_ 21 012 Fig 20 17 Quarter wavelength baluns Radiator with coaxial feed A and methods of preventing unbalanced currents from flowing on the outside of the transmission line B and C The A phasing section shown at D is used for coupling to an unbalanced circuit when a 4 1 impedance ratio is desired or can be accepted HBKO5_ 21 013 Reference 5 Element purposely designed to be highly directional such as a Yagi or a quad Fig 20 18B shows the distortion that can result from common mode currents conducted and radiated back onto the feed line for a 5 element Yagi This antenna has purposely been designed for an excellent pattern but the common mode cur rents seriously distort the rearward pattern and reduce the forward gain as well A balun is highly desirable in this case Choke or current baluns force equal and op posite currents to flow in the load the antenna by creating a high common mode impedance to currents that are equal in both conductors or that flow on the outside of coaxial cable shields such as those induced by the antenna s radiated field The result of using a current balun is that currents coupled back onto the transmission line from the antenna are effec tively reduced or choked off even if the an tenna is not perfectly b
91. proximate equivalent analog bandwidth is mu 175 MHz 2 ns BW 0 35 21 rise time If we were building an analog circuit that operated at 175 MHz we would have to keep the wire lengths down to a fraction of an inch So even if our logic s clock is running at a few kHz we still need to use these short wire lengths But suppose we are building a non trivial circuit that has a number of gates and other digital blocks to interconnect In order to reach several ICs from the clock s source we will need to run wires over several inches in length It is possible to build a high speed circuit in breadboard style if small coax cable is used for interconnections However if a PC HBK0139 eral Microstrip 75 Q Microstrip 50 Q Fig 20 33 Microstrip transmission lines The approximate geometries to produce 75 A and 50 B microstrip lines with FR 4 PC board material are shown This technique is used at UHF and microwave frequencies board is designed with microstrip transmis sion line interconnections success is more likely Fig 20 33 shows the way microstrip transmission lineis made Typical dimensions are shown for 16 inch thick FR4 material and 50 Q line There are several ways that the actual circuit can be configured to assure that the desired signal reaches the receiving device input To avoid multiple reflections that would distort the signals and possibly cause false triggering the line sho
92. r with a 50 Q load on the input This test pro duced results that were essentially the same as the network analyzer A noise generator can be used in combi nation with a receiver as the detector An inexpensive noise generator kit is available from Elecraft www elecraft com Set the receiver for 2 3 kHz bandwidth and turn off the AGC An ac voltmeter connected to the audio output of the receiver will serve as a null detector The noise level into the receiver without the stub connected should be just at or below the limiting level With the stub connected the noise level in the null should drop by 25 or 30 dB Connect the UHFT to the noise generator using any necessary adapters Connect the stub to one side of the T and con nect the receiver to the other side with a short cable Tune the receiver around the expected null frequency After locating the null snip off pieces of cable until the null moves to the desired frequency Accuracy with this method is within 20 or 30 kHz of the network analyzer readings on 14 MHz stubs STUB COMBINATIONS A single stub will give 20 to 30 dB at tenuation in the null If more attenuation is needed two or more similar stubs can be combined Best results will be obtained if a short coupling cable is used to connect the two stubs rather than connecting them directly in parallel The stubs may be cut to the same frequency for maximum attenuation or to two slightly different frequencies such as t
93. r in Fig 20 2 represents the inductance of a very short section of one wire and each capacitor represents the capacitance between two such short sections The induc tance and capacitance values per unit of line depend on the size of the conductors and the spacing between them Each series inductor acts to limit the rate at which current can charge the following shunt capacitor and in so doing establishes a very important property of a transmission line its surge impedance more commonly known as its characteristic imped ance This is usually abbreviated as Zo Zo 7 2 C where L and C are the inductance and capacitance per unit length of line The characteristic impedance of an air insulated parallel conductor line neglecting the effect of the insulating spacers is given by 120 15 Zn cosh 4 0 ae d where Zo characteristic impedance S center to center distance between the conductors d diameter of conductors in the same units as S When S gt gt d the approximation Zo 276 logio 2S d may be used but for S lt 2d gives values that are significantly higher than the correct value such as is often the case when wires are twisted together to form a transmis sion line for impedance transformers The characteristic impedance of an air insulated coaxial line is given by b Zo 138 logio e 5 where Zo characteristic impedance b inside diameter of outer conductors a outside diameter of inner con
94. r installations Class 1 PVC P1 outer jackets are not recommended for outdoor installations See the table of coaxial cables in the Component Data and References chapter Coax can be buried underground especially if it is run in plastic piping with suitable drain holes so that ground water and soil chemicals cannot easily deteriorate the cable A cable with an outer jacket of polyethylene PE rather than polyvinyl chlo ride PVC is recommended for direct bury installations Open wire line must be carefully spaced away from nearby conductors by at least several times the spacing between its conductors to minimize possible electri cal imbalances between the two parallel conductors Such imbalances lead to line radiation and extra losses One popular type of open wire line is called ladder line be cause the insulators used to separate the two parallel uninsulated conductors of the line resemble the steps ofa ladder Long lengths of ladder line can twist together in the wind and short together if not properly supported MULTIBAND OPERATION WITH OPEN WIRE LINE Despite the mechanical difficulties asso ciated with open wire line there are some compelling reasons for its use especially in simple multiband antenna systems Every antenna system no matter what its physical form exhibits a definite value of impedance at the point where the transmission line is connected Although the input impedance of an antenna system is sel
95. rating may be exceeded at high transmitted power levels Automatic antenna tuners use a micro processor to adjust the value of the internal components Some models sense high values of SWR and retune automatically while oth ers require the operator to initiate a tuning operation Automatic tuners are available for low and high power operation and generally handle the same values of impedance as their manually adjusted counterparts Some solid state transmitters incorporate usually at extra cost automatic antenna tun ers so that they too can cope with practical antennas and transmission lines The range of impedances that can be matched by the built in tuners is typically rather limited however especially at lower frequencies Most built in tuners specify a maximum SWR of 3 1 that can be transformed to 1 1 THE T NETWORK Over the years radio amateurs have derived a number of circuits for use as tuners The most common form of antenna tuner in recent years is some variation of a T network as shown in Fig 20 15A Note that the choke or current balun can be used at the input or output of the tuner to match parallel lines The T network can be visualized as being two L networks back to front where the com mon element has been conceptually broken down into two inductors in parallel see Fig 20 15B The L network connected to the load transforms the output impedance R X into its parallel equivalent by means of the series
96. reate a conjugate match at the line input a matching network would have to present an impedance of 35 9 j 21 9 Q The system would then become resonant since the j 21 9 Q react ances would cancel leaving 35 9 7 0 Q Transmission Lines 20 9 A conjugate match is not the same as trans forming one impedance to another such as from 35 9 j70Qto50 70Q An additional impedance transformation network would be required for that step Conjugate matching is often used for small signal amplifiers such preamps at VHF and above to obtain the best power gain The situation with high power amplifiers is com plex and there is considerable discussion as to whether conjugate matching delivers the highest efficiency gain and power output Nevertheless conjugate matching 1s the mod el most often applied to impedance matching in antenna systems 20 4 2 Impedance Matching Networks When all of the components of an antenna system the transmitter feed line and an tenna have the same impedance all of the power generated by the transmitter is trans ferred to the antenna and SWR is 1 1 This is rarely the case however as antenna feed pointimpedances vary widely with frequency and design This requires some method of impedance matching between the various antenna system components Many amateurs use an impedence matching unit or antenna tuner between their transmit ter and the transmission line feeding the an tenna This
97. rrent traveling along the line such a load at the end of the line acts as though it were still more transmission line of the same characteristic impedance In a matched transmission line energy travels outward along the line from the source until it reaches the load where it is completely absorbed or radiated if the load is an antenna MISMATCHED LINES Assume now that the line in Fig 20 3B is terminated in an impedance Z which is not equal to Z of the transmission line The line is now a mismatched line RF energy reaching the end of a mismatched line will not be fully absorbed by the load impedance Instead part of the energy will be reflected back toward the source The amount of reflected versus absorbed energy depends on the degree of mismatch between the characteristic imped ance of the line and the load impedance con nected to its end The reason why energy is reflected at a discontinuity of impedance on a transmission line can best be understood by examining Forward Wave Ry Reflected Wave HBKO05_ 21 003 Fig 20 3 At A the coaxial transmission line is terminated with resistance equal to its Zp All power is absorbed in the load At B coaxial line is shown terminated in an impedance consisting of a resistance and a Capacitive reactance This is a mismatched line and a reflected wave will be returned back down the line toward the generator The reflected wave adds to the forward wave producing a standin
98. s a chapter detailing the use of the Smith Chart TLW software performs this transformation but without Smith Chart graphics 20 3 1 Transmission Line Stubs The impedance transformation properties of a transmission line are useful in a number of applications If the terminating resistance is zero that is a short at the end of a low loss transmission line which is less than 4 A long the input impedance consists of a reac tance which is given by a simplification of equation 12 126000 O 7 MHz Transmitter 1 4 to Antenna 3 gt Shorted Stub HBK0135 Fig 20 5 Method of attaching a stub to a feed line If the line termination is an open circuit the input reactance is given by X Zo cot 14 The input of a short less than 1 4 length of line with a short circuit as a terminating load appears as an inductance while an open circuited line appears as a capacitance This is a useful property of a transmission line since it can be used as a low loss inductor or capacitor in matching networks Such lines are often referred to as stubs A line that is electrically 4 A long is a special kind of a stub When a A line is short circuited at its load end it presents an open circuit at its input Conversely a 4 line with an open circuit at its load end presents a short circuit at its input Such a line inverts the impedance of a short or an open circui
99. s being measured then varying the frequency of the generator to find the 3 dB point where X R This test for one typical setup yielded a stray capacitance value of 0 4 pF A thin film surface mount or chip resistor will have the lowest stray reactances If a surface mount resistor is not available use a 4 watt carbon composition leaded re sistor with leads trimmed to the minimum amount necessary to make the connections Since the measured curve includes stray ca pacitance the actual capacitance of the choke will be slightly less than the computed value If you have determined the value of stray capacitance for your test setup subtract it from the computed value to get the actual capacitance You can also use this corrected value in the theoretical circuit to see how the choke will actually behave in a circuit that is without the stray capacitance of your test setup You won t see the change in your measured data only in the theoretical RLC equivalent Dual Resonances In NiZn materials 61 43 there is only circuit resonance but MnZn materials 77 78 31 have both circuit resonance and dimensional resonance See the RF Techniques chapter for a discussion of ferrite resonances The dimensional resonance of 77 and 78 material is rather high Q and clearly defined so R L and C values can often be computed for both resonances This is not practical with chokes wound on 31 cores because the dimensional r
100. sembly instructions for some transmission line transformers See text for ferrite material type Z Balanced Z Unbalanced Z Unbalanced Toroidal Core Z o Balanced 1 1 Balanced to Unbalanced Current Balun Z 4Z Balanced Balanced 4 1 Balanced to Balanced Transformer A HBKO5_21 014 B Fig 20 23 Broadband baluns A 1 1 current balun and B Guanella 4 1 impedance transformer wound on two cores which are separated Use 12 bifilar turns of 14 AWG enameled wire wound on 2 4 inch OD cores for A and B Distribute bifilar turns evenly around core See text for ferrite material type winding coaxial cable on ferrite cores will be referred to as wound coax chokes to distinguish them from the coiled coax chokes of the preceding section CHOKES ON TRANSMISSION LINES A transmission line can be wound around a ferrite core to form a common mode choke If the line is coax all of the magnetic flux associated with differential mode current is confined to the dielectric the insulating ma terial between the center conductor and the shield The external ferrite core carries only flux associated with common mode current If the line is made up of parallel wires a bifilar winding a significant fraction of the flux associated with differential current will leak outside the line to the ferrite core Leak age flux can exceed 30 of the total flux for even the most tightly spaced bifilar winding In add
101. sion line to the shack Note that L networks as well as Pi and T networks can easily be designed with the TLW software To design an L network both the source and load impedances must be known Let us assume that the source impedance Rg will be 50 Q representing the transmission line to the transmitter and that the load is an arbitrary value R First determine the circuit Q i 17A or 6 L 17 B Next select the type of L network you want from Fig 20 9 Note that the parallel component is always connected to the higher of the two impedances source or load Your choice should take into account whether either Table 20 4 Network Performance Fig 20 9 Circuit Type Section A Low pass L network B Reverse Low pass L network C High pass L network D Reverse high pass L network E Low pass Pi network F High pass T network Match Higher Harmonic or Lower Attenuation Lower Fair to good Higher Fair to good Lower No Higher No Lower and Higher Good Lower and higher No the source or load require a dc ground paral lel or shunt L and whether it is necessary to have a dc path through the network such as to power a remote antenna switch or other such device parallel or shunt C Once you have selected a network calculate the values of X and Xc XL QRs 18 and R As an example we will design an L network to match a 300 Q antenna R3 to a 50 transmission line Rs Ry gt Rg
102. so we can select either Fig 20 9B or Fig 20 9D The network in B is a low pass network and will attenuate harmonics so that is the usual choice 300 Sa 2 236 50 X 50x2 236 1120 Xc ely 1340 2 236 If the network is being designed to operate at 7 MHz the actual component values are X L 2 54 uH 27 f _ 170 pF 2 f Xc The components could be fixed value or adjustable The larger the ratio of the impedances to be transformed the higher Q becomes High values of Q 10 or more may result in imprac tically high or low component values In this case it may be easier to design the matching network as a pair of L networks back to back that accomplish the match in two steps Select an intermediate value of impedance Ryy7 the geometric mean between R and the source impedance Rint VRLRs Construct one L network that transforms R to Ryyz and a second L network that trans forms Ryyz to Rg 20 4 3 Matching Antenna Impedance at the Antenna This section describes methods by which a network can be installed at or near the antenna feed point to provide matching to a transmis sion line Having the matching system at the antenna rather than down in the shack at the end of a long transmission line does seem intuitively desirable but it is not always very practical especially in multiband antennas as discussed below RESONATING THE ANTENNA If a highly reactive antenna can be tun
103. t at the frequency for which the line is 4 A long This is also true for frequencies that are odd multiples of the 4 A frequency However for frequencies where the length of the line is k or integer multiples thereof the line will duplicate the termination at its end 20 3 2 Transmission Line Stubs as Filters The impedance transformation properties of stubs can be put to use as filters For exam ple if a shorted line is cut to be 4 A long at 7 1 MHz the impedance looking into the input of the cable will be an open circuit The line will have no effect if placed in parallel with a transmitter s output terminals However at twice the fundamental frequency 14 2 MHz that same line is now 2 A and the line looks like a short circuit The line often dubbed a quarter wave stub in this application will act as a trap for not only the second harmonic but also for higher even order harmonics such as the fourth or sixth harmonics This filtering action is extremely useful in multitransmitter situations such as Field Day emergency operations centers portable communications facilities and multioperator contest stations Transmission line stubs can operate at high power where lumped constant filters would be expensive Using stub filters reduces noise harmonics and strong funda mental signals from the closely spaced anten nas that cause overload and interference to receivers Quarter wave stubs made of good quality coax such
104. ter of the coil should be at least 10 times the cable diameter to avoid mechani cally stressing the cable 20 5 4 Transmitting Ferrite Choke Baluns A ferrite choke is simply a very low Q par allel resonant circuit tuned to the frequency where the choke should be effective Passing a conductor through most ferrite cores that is One turn produces a resonance around 150 MHz By choosing a suitable core mate rial size and shape and by adding multiple turns and varying their spacing the choke can be tuned optimized for the required frequency range Transmitting chokes differ from other common mode chokes because they must be designed to work well when the line they are choking carries high power They must also be physically larger so that the bend radius of the coax is large enough that the line is not deformed Excellent common mode chokes having very high power handling capability can be formed simply by winding multiple turns of coax through a sufficiently large fer rite core or multiple cores Chokes made by Transmission Lines 20 21 A 1 1 Phase Reversed Unbalanced Trans 9 1 Unbalanced to Unbalanced HBKO5_ 14 057 20 22 Chapter 20 1 1 Balanced to Unbalanced Trans Single Ended Hybrid Combiner R 200 O Balanced Rin 90 QO Unbalanced 4 1 Balanced to Unbalanced 4 1 Balun Cascaded 4 1 Broadband Transformers Give 16 1 Z Ratio 12 5 Ohms Indicates Phasing Fig 20 22 As
105. ting an antenna tuner is described here and results in minimum SWR to the transmitter and also minimizes power losses in the tuner circuitry If you have a commercial tuner and have the user s manual the manufacturer will likely provide a method of adjustment that you should fol low including initial settings If you do not have a user s manual first open the tuner and determine the circuit for the tuner The most common circuit for commer cial tuners is the high pass T network shown in Fig 20 9F To adjust this type of tuner 1 Set the series capacitors to maximum value This may not correspond to the highest number on the control scale verify that the capacitor s plates are fully meshed 2 Set the inductor to maximum value This corresponds to placing a switch tap or roller inductor contact so that it is electrically closest to circuit ground 3 If youhave an antenna analyzer connect it to the TRANSMITTER connector of the tuner Otherwise connect the transceiver and tune it to the desired frequency but do not transmit In the following step it is important to verify that you hear a peak in received noise before transmitting significant power through the tuner Tuners can sometimes be adjusted to present alow SWR to the transmitter while coupling little energy to the output Trans mitting into a tuner in this configuration can damage the tuner s components 4 Adjust the inductor throughout its range watching
106. tput of the sending gate raising the gate output impedance to match Z of the transmission line whichis connected to the resistor at B and has an arbitrary length Tp No load is required on the receiving end of the transmission line which is assumed to be connected to a gate with an input impedance much higher than Zp When the sending output goes high at point A generating the leading edge of a pulse with voltage V the load on the output Waveguide Inductance Cancelling Plate resistor is equal to the Zo of the transmis sion line so a voltage divider is formed and the voltage at point B initially goes to V The pulse travels down the transmission line and after Tp it completely reflects off the open end at the receiving gate The voltage at C reaches V since the direct signal and the reflected signal add together When the reflected edge of the pulse returns to point B after a round trip time of 2Tp the voltage level at B increases to V The receiving end can be terminated in Zo if a pair of resistors each equal to 2 x Zp are connected from the positive power supply to ground at point C The transmission line and gate input are connected to the resistor junc tion The main problem with this method is the steady state current required by the resis tors Some logic gates may not have adequate current output to drive this load HBKO5_ 21 017 O Open Wire Line A 4 Stub Fig 20 35 At its cutoff frequency a re
107. uld either be matched at the load end or the source end We know that matching at the load end will completely absorb the sig nal so that there are no reflections However the signal level will be reduced because of voltage division with the source impedance in the sending gate The dc levels will also shift because of the load reducing the logic noise immunity A considerable amount of power can be dissipated in the load which might overload the source gate particularly if 50 Q line is in use It is possible to put a capacitor in series with the load resistor but only if the waveform duty cycle is near 50 If not the dc average voltage will reduce the noise immunity of the receiving gate A better method is to match the transmis Transmission Lines 20 27 C HBK0140 20 7 Waveguides A waveguide is a hollow conducting tube through which microwave energy is transmit ted in the form of electromagnetic waves The tube does not carry a current in the same sense that the wires of a two conductor line do Instead it is a boundary that confines the waves to the enclosed space Skin effect on the inside walls of the waveguide confines electromagnetic energy inside the guide in much the same manner that the shield of a coaxial cable confines energy within the coax Microwave energy is injected at one end either through capacitive or inductive coupling or by radiation and is received at the other end The waveguide merely conf
108. when connected together in this fashion 20 4 4 Matching the Line to the Transmitter So far we have been concerned mainly with the measures needed to achieve acceptable amounts of loss and a low SWR when real coax lines are connected to real antennas Not only is feed line loss minimized when the SWR is kept within reasonable bounds but also the transmitter is able to deliver its rated output power at its rated level of distortion when connected to the load resistance it was designed to drive Most modern amateur transmitters use broadband untuned solid state final ampli fiers designed to work into a 50 Q load Such a transmitter very often utilizes built in pro tection circuitry that automatically reduces output power if the SWR rises to more than about 2 1 Protective circuits are needed be cause many solid state devices will willingly and almost instantly destroy themselves at tempting to deliver power into low imped ance loads Solid state devices are a lot less forgiving than vacuum tube amplifiers which can survive momentary overloads without being destroyed instantly Pi networks used in vacuum tube amplifiers typically have the ability to match a surprisingly wide range of impedances on a transmission line See the RF Power Amplifiers chapter Besides the rather limited option of using only inherently low SWR antennas to ensure that the transmitter sees the load for which it was designed an impedance matching unit
109. ysis of waveguide operation is based on the assumption that the guide material is a perfect conductor of electricity Typical dis tributions of electric and magnetic fields in a rectangular guide are shown in Fig 20 36 The intensity of the electric field is greatest as indicated by closer spacing of the lines of force in Fig 20 35B at the center along the X dimension and diminishes to zero at the end walls Zero field intensity is a necessary condition at the end walls since the existence of any electric field parallel to any wall at the surface would cause an infinite current to flow in a perfect conductor an impossible situation 20 7 2 Modes of Waveguide Propagation Fig 20 36 represents arelatively simple dis tribution of the electric and magnetic fields An infinite number of ways exist in which the fields can arrange themselves in a guide as long as there is no upper limit to the frequency to be transmitted Each field configuration is called a mode All modes may be sepa rated into two general groups One group designated TM transverse magnetic has the magnetic field entirely crosswise to the direc tion of propagation but has a component of electric field in the propagation direction The other type designated TE transverse electric has the electric field entirely crosswise to the direction of propagation but has a com ponent of magnetic field in the direction of propagation TM waves are sometimes called E wav

Transmission Lines - University of California, Berkeley

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