GFT Template User`s Manual v.1.0
Contents
1. is defined by what test signals are employed and where they are injected into the circuit model A particular test signal configuration produces a corresponding set of second level TFs H T T H p which when inserted into the GFT 17 or 18 gives the first level TF H Although all test signal configurations give the same H the sets HoT T H o may be different The objective of establishing the block diagram of Fig 3 is to interpret each block in terms of distinct circuit functions which in turn can be identified in terms of circuit elements If you make an arbitrary choice of injection configuration the resulting set of H_ T H P T won t mean anything and so will be useless for informed design oriented analysis even though if you insert it into the GFT you get the correct answer For example the result for T may be completely unrecognizable as the loop gain you thought you had built into the circuit Therefore the crucial step in applying the GFT is to choose a test signal injection configuration that does produce the set H T T H 0 with the desired interpretation In many cases the desired result is that Ho be equal to 1 K the reciprocal of the feedback network 12 transmission and the corresponding HoT T H 0 set is referred to as the principal set which contains the principal loop gain T Other sets containing different T s are not wrong they just aren t useful for the same purpose So how do we2. hb ddbo hdb ero hndb2 Plot hdb ddbo hdbzero hdb2 100 10k 1Meg 100Meg 10G frequency in hertz Fig 21 Dissection of the output impedance H into the product of two components H 0 and 1 D according to 24 The H term is zero The vertical axis is dB relative to 1 ohm and at 100Hz H is displayed on Intuscope as 79 13dB 9 047k in complete agreement with the expected value Because of the high loop gain the closed loop value at low frequencies is much smaller displayed on Intuscope as 7 51dB gt 2 4 ohm 4 2 Example 3 Emitter source follower In some circuits the desired H _ requires that only i orv be nulled not both An example is the emitter source follower of Fig 22 which is ready made in spice8 circuits gft figure22 dwg 26 The inductance typically arises from the output impedance of a driver emitter source follower in a Darlington configuration This circuit is well known to be potentially unstable for certain values of load capacitance CL L1 150n Fig 22 A source follower circuit with 1 K 1 The error voltage x appears between input and output so nulling n is sufficient to set up the desired H 1 K 1 and the single injection icon GFTve is appropriate 27 Even though we don t usually think of this as a feedback circuit the GFT of 17 or 18 gives the gain H in terms of a loop gain T from which stability can be assessed As usual the crucia
3. H H Ho 1 T CTT in which a new quantity H _ is introduced 1 H 4 K 4 The seemingly trivial substitution of H_ for 1 K is actually very valuable because it enables an interpretation of 3 that discloses much useful information Equation 3 says that the closed loop gain H of a feedback system can be expressed as the weighted sum of two components one containing H _ which would be the gain if the loop gain T were infinite and the other containing H 0 which would be the gain if T were zero Thus Ho H T gt 5 Hj H T gt 0 6 can be considered as definitions It follows that H is dominated by H __ when T is greater than 1 and is dominated by H 0 when T is less than 1 The objective of designing a feedback system is to make T large enough that H is dominated by H and the H 0 contribution is negligible over the useful frequency range A further interpretation of the expectedly dominant term in 3 is that H _ is the ideal closed loop gain and T aaa 7 is the discrepancy factor because it is the factor by which the actual closed loop gain H deviates from the ideal closed loop gain H Another useful version of 3 arises from combining the two terms into one jp sA iaa BE hr l 14 7 lta or H H n a 8 1 8 where Tn _ Hoo T ie 9 In the derivation of the GFT in 1 T is identified as the null loop gain which in turn suggests the definition of a null discrepancy fa
4. choose an injection configuration so that H _ equals 1 K This is simply an extension of the process set up in Fig 5 that led to 13 The starting point is an extension of the block diagram of Fig 3 to incorporate two things identification of the generalized signals u as being currents or voltages or both second recognition that in general the A and K boxes are each bidirectional and so are drawn as rectangles instead of arrowheads as in Fig 9 Fig 9 Selection of a test signal injection configuration that sets up H_ 1 K Step 1 Identify the feedback network in an equivalent circuit model The input and output signals have been rewritten as V and v 50 that H v y is a voltage to voltage gain You may wish to visualize the two blocks as containing the circuit of Fig 2 but to make the discussion more concrete only the feedback divider will be shown explicitly as in Fig 10 13 V error voltage error current Fig 10 Step 2 The error signal u has to be identified as both a voltage 5 and a current i The criterion is that the test signal is to be adjusted so that the error signal is nulled which makes the fedback signal exactly equal to the input signal which in turn sets up H H In Fig 10 the fedback signal is RI RI R2 v so for this to be equal to the input voltage Vp the error signal must be identified as the voltage i shown in Fig 10 However it is not sufficient for i to be nulled For the fed
5. level TFs required for the Single voltage test signal configuration are the same except with v in place of i Regardless of which test signal configuration is employed the result is a set of second level TFs H_ 1 Tp H 0 The GFT Template then does various postprocessing calculations that define some additional quantities useful in presenting various forms of the final result for H These include a pseudo loop gain T into which both T and T are absorbed and three discrepancy factors each of which is uniquely determined by T Tp Ty The postprocessing calculations can be viewed in spice8 script GFT scp Finally the GFT produces three versions of H with numerical designators to indicate which form of the GFT is its origin ie Tn Hy Ho Ft HDD 1 T Ao H T H H D HoD a eea Vg i o T 3 H Ho Dp 1 Tp where DNs Tp 1 7 T T T 1 D Dy 14 T aan T 39 A 19 A 20 A 21 A 22 A 24 1 7 D Tp T A 25 TEP n D n A 26 Reminder Different test signal configurations produce different sets of the second level TFs H_ T T H 0 Even the same test sources for example the Coupled voltage current icon of Fig A 2 produces different sets for different connection nodes WXY emphasized by writing A 1 as I H H e Ha jee PE SU T T Although the constituent TFs may be different in each set when they are assembled into A 27
6. ly T Vv an L a A 5 and similarly for T gs T iy and T xe iy T ni _Vy i A 6 ug Vy 0 and similarly forT T andT_ NI_VX nv_ly NV_1IX 32 The interpretations are as follows Although there are three independent sources use and j in Fig A 2 only u appears explicitly The constraint upu 0 in A5 means that u is set to zero and e and j are mutually adjusted so as to null the dependent signal v Ti wy is the ratio if L under the null double injection ndi condition of nulled vi The constraint uu 50 in A6 means that ui is restored and all three independent sources Up and j are mutually adjusted so as to null the two dependent signals u and v Taw is the ratio i i under the double null triple injection dnti condition of nulled u and v All four T s are ndi calculations all four Tn s and all four H s are dnti calculations There is lengthy discussion in 1 4 to support the contention that ndi and dnti symbolic calculations are easy to make in fact easier than familiar single injection calculations All are single path calculations you never have to know what the independent source mutual adjustments are you just follow the starting signal to the ending signal incorporating along the way the condition that certain signals are nulled Often if one signal is nulled others are too so the null propagates Any element that supports a nulled signal does not contribute to the result whi
7. of a feedback system 1 This model predicts that the closed loop gain H u Ju is given by 1 AK 1 where A is the forward open loop gain K is the feedback ratio and T is the loop gain defined by T AK 2 Consider the simplified model of a typical two stage transistor feedback amplifier shown in Fig 2 Fig 2 A simple equivalent circuit model of a two stage voltage to voltage feedback amplifier We can see immediately that this model is not completely represented by the block diagram of Fig 1 Equation 1 says that if A goes to zero H goes to zero however in Fig 2 if Bm2 and C go to zero the output u does not go to zero because the input u can reach the output by going through the feedback path in the wrong direction This nonideality of the real circuit is not accounted for in Fig 1 because the blocks are defined to be unidirectional which is the reason they are drawn as arrowheads rather than as rectangular boxes However this nonideality can be accommodated by augmenting the block diagram with another block as in Fig 3 where H o represents the direct forward transmission Fig 3 The block diagram of Fig 1 augmented by a block that represents direct forward transmission It is identical to the natural block diagram that is the result of application of the GFT to an equivalent circuit model Figure 3 predicts the presence of an added term in the expression for the closed loop gain H T 1
8. 0 60 0 6 Z 180 80 0 100 4 O O 270 120 140 5 10k 100 frequency in hertz Fig 25 Increase of the load capacitance CL from 50p to 250p causes a 14dB drop in T in the neighborhood of crossover but no change in T in the same neighborhood The circuit is now stable The T graph is 250 50 5 14dB lower and the crossover frequency decreases to 42MHz where the phase margin is 33 and the gain margin is 7 0dB so that stability is achieved There is still peaking in H but less than originally 30 APPENDIX What the GFT is and how the GFT Template works The General Feedback Theorem GFT is a general network theorem that applies to any transfer function TF of a linear system model The purpose of the GFT is to dissect H the first level TF of interest into several second level TFs HI T and H 9 8 that insight can be gained in to how the circuit elements contribute to the result The GFT is derived in 1 as an application of the Extra Element Theorem EET and has two basic versions 1 n H H 1 A 1 1 T H H H A 2 aiir er une There is a redundancy relation between the four second level TFs Hoat H T A 3 The natural block diagram of Fig A 1 represents either of the two basic versions of the GFT in which H u Ju is the first level TF of interest The input and output signals u and u can represent any combination of voltag
9. GFT Template User s Manual v 1 0 by Dr R D Middlebrook http www rdmiddlebrook com The ICAP 4 GFT Template is an implementation of the General Feedback Theorem GFT presented in 1 in which the GFT is derived by application of the Extra Element Theorem EET developed in 2 Many examples of useful applications of the EET are available in a book 3 and extensive background material is presented in a series of Video CDs 4 1 R D Middlebrook The GFT A General Yet Practical Feedback Theorem to be published 2 R D Middlebrook Null Double Injection and the Extra Element Theorem IEEE Trans on Education vol 32 no 3 Aug 1989 pp 167 180 3 Vatch Vorp rian Fast Analytical Techniques for Electrical and Electronic Circuits Cambridge Univ Press 2002 4 R D Middlebrook Structured Analog Design Video CDs Ardem Associates http www ardem com to be published CONTENTS 1 What we know already 2 Injected test signals 3 The General Feedback Theorem GFT 3 1 Example 1 2 stage voltage to voltage feedback amplifier 4 Manipulations of the natural block diagram 4 1 Example 2 Output impedance of the 2 stage amplifier 4 2 Example 3 Emitter source follower APPENDIX What the GFT is and how the GFT Template works 1 What we know already Electronic feedback systems are usually designed with the familiar single loop block diagram of Fig 1 in mind Fig 1 Conventional single loop block diagram
10. H meets the specification within the allowed tolerances Design Step 3 Since in any real system T drops below unity beyond some crossover frequency design T to be large enough up to a sufficiently high frequency In an ac coupled system a dual condition occurs in the reverse frequency direction Usually 3 is the most difficult step Design Step 4 Design the null loop gain so that T is large enough or the direct forward transmission HO is small enough that the actual H meets the specification within the allowed tolerances Although not explicitly stated Design Step 3 includes the stability requirement The discrepancy factor D 1 when T gt gt 1 and D T when T lt lt 1 the transition between these two ranges which is not necessarily a monotonic decrease occurs at the loop gain crossover frequency wc the frequency at which T 1 If the phase margin of T is less than about 60 D begins to peak up in the neighborhood of the crossover frequency and in the limit of zero phase margin the peak in D becomes infinite which marks the onset of instability Therefore the discrepancy factor D must not be allowed to peak up beyond the tolerance limit of the specification which is a much more stringent design requirement than mere stability 2 Injected test signals Although the second level TFs are derived quantities defined in terms of the blocks in Fig 3 in looking ahead to evaluating them on a circuit diagram w
11. However the other second level TFs are different and in particular H _ is no longer flat at all frequencies On the other hand if the GFTv or Loop Gain icon is inserted in Fig 19 at W 1X5Y 5 none of the second level TFs is the same because the test signal injection is now inside a minor loop 4 1 Example 2 Output impedance of the 2 stage amplifier In the PartBrowser GFT Sub Types besides the GFTv there are three others GFTi GFTg and GFTh for use when the first level TF H of interest is a current gain a transadmittance or a transimpedance The procedure for use is exactly the same as for the voltage gain There are other uses for the Sub Types which is to calculate TFs other than the one that describes the gain of the system For example the circuit of Fig 13 represents a voltage to voltage amplifier but you may also be interested in its output impedance Zo Output impedance is a special case of transimpedance and the GFTh icon can be employed with the output signal still being the output voltage but the input signal is an independent current driven into the output terminals Figure 20 shows the first level TF H set up to represent the output impedance Zo which is ready made in spice8 circuits gft figure20 dwg 24 v10 Fig 20 Replacement of the GFT Template icon GFTv in the model of Fig 13 by GFTh for calculation of the output impedance as a self impedance H whose output is the voltage uu and whos
12. a closed loop response Tp An H H D p P 21 where 5 is a pseudo loop gain defined by equating 21 with the GFT of 17 jp 2 n T er Ta and Tp Dp P14 23 Graphs of ly and D are shown in Fig 18 hb3 tdp ddbp hodbinf Plott hdb3 tdbp ddbp hdbinfin unknown 100 10k 1Meg 100Meg 10G frequency in hertz Fig 18 The closed loop gain H dissected into the product of two constituents H _ and DT T The pseudo loop gain Le has poor gain and phase margins that do not imply imminent instability 21 The reason we want to dissect the closed loop gain H into its constituent parts is to be able to adjust the circuit element values in order to meet the specifications and optimize the design This is usually an iterative procedure lubricated by both symbolic and numerical analysis as done for Example 1 of Fig 13 here and in 1 It is obviously advantageous to have the second level TFs expressed in terms of as few as possible intuitive single path signal flows This is the case for H T and T and so the GFT form H 1 of 17 has the closest relation to the equivalent circuit elements However H involves certain loop gains as well as single path signal flows from u tou see Appendix and so the GFT form H gt of 18 is not quite so closely related to the circuit elements The GFT form H 3 of 21 which describes the apparently simpler block diagram of Fi
13. alculation in which u is adjusted in the presence of u to null the output signal u as in Fig 8 Uo 0 nulled Fig 8 Null double injection calculation of the null loop gain T by mutual adjustment of the injected signal u and the input signal u to null the output signal uy The null loop gain T is then uy or al a 16 The interpretation is as follows The output u is the sum of Au H Tu and H gtr If the output is nulled H Tu H oui But there is no fedback signal so u 50 and H Tu H oy which leads to 9 and so justifies the interpretation of Tas the null loop gain All the familiar properties of feedback are expressed in 11 or 12 which relate to the block diagram of Fig 3 The first level TF the closed loop gain H is expressed in terms of three of the four second level TFs which can be determined from 13 through 16 by injection of a test signal u that sums with the error signal The next step is to relate the block diagram of Fig 3 or 4 to an actual equivalent circuit model of the system under consideration This is where difficulties begin to emerge the conventional approach attempts to identify the three blocks as containing distinct parts of the circuit but this is only true in simple cases Usually interactions between blocks loading impose approximations upon the resulting block gains especially at high frequencies which is just where accurate results are most desired This approach is effectively d
14. back voltage to be R1 R1 R2 v there must be no current drawn from the divider tap Therefore the error signal must also be identified as the current i shown in Fig 10 The bottom line is that in a circuit such as Fig 10 to set up H _ to be equal to R1 R2 R1 both the error voltage i and the error current i must be nulled This requires that the injected test signal u also be both an injected current j and an injected voltage e as in Fig 11 14 error i voltage error current Fig 11 Step 3 The injected test signal u_has to be both a voltage e_ and a current j both of which have to be mutually adjusted relative to u to null both error signals i and iy This is a double null triple injection condition The result is Hao 2 1 iy vy 0 tyVy 20 where the second version is merely a shorthand version of the first and the test signal injection conditions have been purposely set up to make H __ equal to the required R1 R2 R1 We see that a test signal may be both a current and a voltage source and that both test sources need to be adjusted relative to a given input signal in order to null both an error voltage and an error current This is a condition of double null triple injection in which three independent sources are mutually adjusted to null two dependent signals This is just as easy as a null double 15 injection calculation You don t need to know what the values of the sources are you need to
15. ch is therefore not only easier to obtain but simpler than if a null does not exist The GFT Template simulates null conditions by making the null self adjusting Only one independent source is retained and the others are replaced by dependent sources driven by the signals to be nulled For example the dnti simulation Hwy can be implemented as in Fig A 3 in which u is the retained independent source The higher the nulling gain the better the resulting null The default value is 1G Fig A 3 Typical GFT Template algorithm for creation of self adjusting nulls by use of high gain nulling amplifiers 33 In principle it doesn t matter which independent signal is retained or which signal is nulled by which nulling gain Also because of the happy circumstance that the nulling gains have infinite bandwidth the nulling polarity doesn t matter and no system instability is introduced The GFT Template simulates the twelve third level TFs in succession without interruption by invoking the alter feature to sequence the required ndi and dnti conditions All twelve TFs along with the first level TF H simulated with no test signal injection are available for magnitude and phase display in Intuscope The GFT Template then calculates the second level TFs HoT T H 0 according to the GFT definitions derived in 1 and makes them also available for display in Intuscope Ho Hi y A 7 be l r 1 i 1 nee T T Vy Ty
16. ctor 1 Damian 10 n In summary a first level transfer function TF such as the closed loop gain H can be expressed in terms of three out of four second level TFs H T a and H 9 Because of the redundancy relation 9 the four versions each omits one of the four second level TFs The two versions useful for the present purposes respectively omit T and H p and are T 1 H Ha Ho Hp D Ho 1 D o a o D 11a 11b e n HShe aD 12a 12b ee Although H g represents nonidealities physically identified in Fig 3 their alternative representation by T has two advantages First in 12 D appears as a multiplier factor within H and therefore for the nonidealities to be negligible D must be close to 1 or equivalently T must be much larger than 1 on the other hand in 11 H g appears in an additive term and to be negligible H g must be much smaller than H _T which is a more complicated comparison Second in symbolic circuit analysis T is usually a much simpler and shorter calculation than is H p The job of a designer as distinct from that of an analyst is to construct hardware that meets certain specifications within certain tolerances If you are designing a feedback amplifier you effectively proceed through four steps Design Step 1 Design the feedback network K 1 H _ so that H _ meets the specification Design Step 2 Design the loop gain so that T is large enough that the actual
17. e input is the current uu The injection configuration is the same because we want T still to be the principal loop gain and the only differences from Fig 12 are that the icon is GFTh instead of GFTv don t forget to rename it xgft and uu is connected to uu with the original uu grounded The zero voltage source V6 is there to preserve the distinction between the uu and uu terminals The H _ for Fig 20 is zero as can be seen by doing the null double injection calculation for the output impedance with i and v nulled if m is nulled there is no voltage across R1 if i is also nulled there is no current in R2 and so vL is also zero Thus e and j have been adjusted so that the entire independent driving current u goes backwards into the forward path leaving none to go down the feedback divider The result H_ H T 0 is consistent with the basic interpretation that voltage feedback reduces an amplifier s output impedance so infinite loop gain would reduce the output impedance to zero 25 Consequently H 1 of 17 cannot be used but H j of 18 is available and reduces to 1 H gt Ho HoD 24 RR 00 24 in which H 17 H T 0 is interpreted to be the open loop output impedance At low frequencies by inspection of Fig 20 this is seen to be Ho R2 R1 rml 9 048k 25 The dissected constituents are shown in Fig 21 in which the calculated H E is seen to be identical to the directly measured H
18. e all infinite which can be seen by following 1_VX V_IX NI_VX NV_1X arguments similar to that in Example 3 Consequently all four simulations give incorrect results and Ta is not equal to T and Tii is not equal to Ty 2S demanded by A 11 through A 14 However none matters because the T and Tai contributions to T and T in A 8 and A 9 are negligible 36
19. e find it useful to see how they could be determined individually by direct virtual measurements on the block diagram of Fig 3 We want to do this by signal injection without disturbing the circuit configuration and therefore without disturbing the circuit determinant The most important second level TF and also the simplest is the ideal closed loop gain H The actual closed loop gain falls short of H __ because the error signal which is the difference between the input signal u and the fedback signal Ku is not zero when the loop gain T is not infinite A test signal u_can be injected so as to add to the error signal as in Fig 4 error signal feedback signal Ku injected signal Fig 4 Injection of a test signal u So as to add to the error signal in the block diagram of Fig 3 The injected signal can be adjusted so as to null the error signal u which makes Ku exactly equal to u so that the closed loop gain is 1 K which is H_ Thus H _ can be directly measured under the null double injection condition shown in Fig 5 u 0 nulled i u 0 U y u L u H_u o K i i Fig 5 Null double injection calculation of the ideal closed loop gain H _ by mutual adjustment of the injected signal u and the input signal u to null the error signal Mi which simulates infinite loop gain The result can be written Y o3 It s worth going through this process again Initially u 0 and t
20. ealt a coup de gr ce by the presence of nonzero admittances across active device dependent generators which introduce not only local feedback paths but also an additional bidirectional signal path in the forward gain block A Such nonzero 10 admittances result from C 1 and C 3 in Fig 2 which create considerable analytical difficulties even in such a simple circuit The General Feedback Theorem GFT avoids all these difficulties by bypassing the initial step of trying to relate each block individually to distinct parts of the circuit Instead the blocks in Fig 3 are evaluated from 13 through 16 by test signals injected into the complete circuit whose determinant remains undisturbed at all times 3 The General Feedback Theorem GFT In the bottom up approach to feedback analysis described above the starting point is a block diagram intended to preserve the concept of a single loop describing a forward path and a feedback path The accuracy of the results depends on how well the blocks represent the actual circuit configuration In contrast the GFT provides a top down approach in which the starting point is the actual equivalent circuit model and a block diagram is the result This block diagram is exactly the same as in Fig 3 which therefore may be called a natural block diagram because it represents any transfer function of any linear circuit model regardless of whether there is any identifiable feedback pat
21. es currents torque angular velocity etc Fig A 1 The natural block diagram for any TF H dissected according to the GFT into three out of four second level TFs H_ T T H 0 that are calculated directly from the circuit model 31 The natural block diagram obviously resembles a single loop feedback system and is one of the most common applications of the GFT The value of the GFT lies in the relationship between the second level TFs HT Tp H 0 and the circuit elements This is determined by the results of certain third level calculations on the circuit model in the presence of one or more injected test signals The test signal configuration is defined by the number and type of injected signals and where they are injected Different test signal configurations produce different sets of H_ T T H 0 The GFT Template is limited to two test signals and the typical configuration of Fig A 2 shows a voltage e and a current j injected at the same nodes WXY although in other configurations these nodes are not necessarily the same for each test signal Fig A 2 One example of injection of test signals e and j for determination of the third level null double injection and double null triple injection TFs needed to calculate the second level TFs The third level TFs simulated for the test signal configuration of Fig A 2 are o Hi v us A 4 Ly Vy 0 and similarly forH H andH vyix yvx ixvx
22. g 17 is even further removed from intuitive interpretation in terms of circuit elements which is not surprising considering the tortuous definition 22 of the pseudo loop gain T Thus although in Fig 18 the results for H and H 3 are correct it is hard to relate the shapes of T and D to the physical operation of the circuit Although the ICAP 4 GFT Template makes it very easy and fast to graph all sorts of TFs you have to be very clear as to what are the meanings of the second level TFs you are calculating ICAP 4 Intuscope shows that the loop gain T displayed in Fig 14 has a very docile phase margin of 87 and a gain margin of 52dB far from instability on the other hand the pseudo loop gain T displayed in Fig 18 has a much smaller phase margin of 45 and a gain margin of only 0 2dB which might be taken incorrectly as a red flag that the system is on the brink of instability The conclusion is that you are probably better off to deal with the natural block diagram of Fig 3 rather than the deceptively simpler one of Fig 17 A block diagram with fewer blocks implies that the blocks are described by more complicated expressions which can be counterproductive in design oriented analysis Indeed the opposite is one of the principles of Design Oriented Analysis divide and conquer or make more simpler blocks rather than fewer complicated blocks In the SpiceNet PartBrowser there is a GFT Sub Type called Loop Gain which is a short c
23. ge source and the GFT provides extensions of 13 through 16 by which the four second level TFs can be derived from 12 third level measurements made under various single null double and double null triple injection conditions established on the circuit model These extension formulas are needed if you want to do symbolic analysis of the circuit model However if you want only numerical results you can ignore these formulas the ICAP 4 GFT Template does it all for you so the extension formulas are not displayed here but are listed in full in the Appendix The only decisions you have to make are which is the appropriate test signal injection configuration and how to connect it to the circuit After the program has been run the second level TFs are available for display in Intuscope as are the discrepancy factors D from 7 and D from 10 and the calculated closed loop gains H 1 from 17 and H y from 18 The derived results H 1 and H 2 should be the same and as a final check both should be the same as H measured directly on the circuit without any injected test signals which is also available in Intuscope The GFT is completely general the only condition being that it applies to a linear system model Any number of injected test currents and or voltages can be employed at the same or different places in the circuit model The GFT Template incorporates up to two injected test signals in any combination A test signal configuration
24. h or not Since the two approaches have the same block diagram they also have the same relations 11 and 12 between H the first level TF and the four second level TFs H T TH P repeated here T H H Tp H DD 1 T 1 Hy H Hy H D H D 18a 18b 14 T 1 T The designators 1 and 2 are used to distinguish the results calculated from different arrangements of the second level TFs or different dissections of the first level TF H and henceforth the unadorned H will be reserved for the first level TF measured directly with no test signal injection as would be done in a normal ICAP 4 calculation The order of the two forms is reversed because H is the form that first emerges from the GFT derivation and therefore has the most direct relationship to the equivalent circuit model Consequently H 1 is designated the natural form of the GFT and H and others still to be introduced are designated as derived forms Likewise the redundancy relation 9 is also reversed Hy _T H T co n 19 11 In the previous section it was shown how the four second level TFs could in principle be evaluated by injection of a test signal u into the block diagram The difficulties arise when we try to implement these calculations on the actual circuit diagram instead of on the block diagram In particular is u a current or a voltage source The answer is that in general u is both a current and a volta
25. he system is in the normal closed loop condition with a given input signal u SO that the output is u Hu Asu is increased the output rises causing more signal to be fed back and decreasing the error signal U When u has been reduced to zero the output has risen to Hou the value it would have if the loop gain were infinite Note that H _ is determined under a condition of simulated infinite loop gain in accordance with 5 when the error signal u is nulled u which is then equal to u drives the output to the value necessary to make Ku equal to u Thus the forward gain appears to be infinite although the output is actually being driven by the hidden test signal u Null double injection is a very powerful tool for numerical as well as for symbolic analysis In general two independent signals are mutually adjusted in order to null a designated dependent signal The purpose is to determine a special value of a certain TF that exists under this null condition In the present case the purpose is to determine the special value of H defined as H_ that exists under the condition of nulled error signal u There are two important features of null double injection One is that the system determinant is not disturbed because only system signals are manipulated The other is that determination of 7 the required answer the TF under the specified null condition is VERY EASY because we don t need to know the value of the injected test s
26. ignal we need to know only that the resulting null exists In the present case we use the existence of the null condition u 0 to conclude that the fedback signal exactly equals the input signal which immediately sets up the required result 13 If you are not familiar with the term null double injection you may be surprised to learn that you ve actually been doing the corresponding calculation all along The conventional procedure for Design Step 1 is to assume that the error signal is vanishingly small which makes the fedback signal approximately equal to the input signal and hence makes the closed loop gain approximately equal to 1 K But this is exactly the same calculation that is the culmination of the null double injection procedure described above So you ve actually been doing the null double injection calculation all along by assuming infinite loop gain instead of simulating infinite loop gain the result of which is H _ and not H If as is commonly the case K is a resistance divider H _ is flat to infinite frequency even though the actual H falls off because of the declining discrepancy factor D T 1 T At increasing frequencies the injected test signal u has to be increased in order to maintain the output Hou flat in spite of decreasing loop gain All four second level TFs can be determined by various manipulations of the two injected signals in Fig 3 without disturbing the circuit determinant After H the next m
27. iy Tiy l z 1 P l m 1 Re Ta Ini_v Tavi Div T T Ho H PHa 0 v i T iV om i Vy To A 10 where lw li v iv i A 11 Tyi T _i Ti_v A 12 Triv Thi_v tv is A 13 Tyi F yy i nivy A 14 T and re do not appear in A 7 through A 10 because I and Ty should be equal and Taw and Tvi should be equal so A 12 and A 14 exist merely as computation checks Also H 0 is calculated from the redundancy relation A 3 and should be equal to H 0 calculated from A 10 The GFT Template Parts Browser under GFT lists four Subtypes one for each of the four first level TFs voltage gain current gain transimpedance and transadmittance Each of these has an icon that requires connection to the input and output and a selection from six combinations of injected test signals The one shown in Fig A 2 represents Coupled voltage current injection Dual voltage current is the same except that the WXY connections are different for the two test sources Dual voltage voltage and Dual current current have two like sources with different WXY connections There are also icons for Single voltage and Single current injection 34 The third level TFs required for the Single current test signal configuration are directly equal to the corresponding second level TFs u A Hj a i 0 i T Tj gt Ix u O i ay lam Thri ma X lug o Ho Hy Hj tli 0 A 15 A 16 A 17 A 18 The third
28. know only that the two nulls exist because this is equivalent information and is sufficient to deliver the required answer The procedure for using the GFT Template will be illustrated below through a series of examples the first of which is taken from 1 3 1 Example 1 2 stage voltage to voltage feedback amplifier The circuit of Fig 2 is redrawn in Fig 12 with a slight modification Each FET model is replaced with an equivalent model that represents either an FET or a BJT depending respectively on whether a 1 or a lt 1 error voltage error current Fig 12 The equivalent circuit model of Fig 2 redrawn to resemble the model of Fig 11 with the error signals r and i identified 16 Draw this circuit in the SpiceNet window and go to Parts PartBrowser Part Types GFT Under Sub Types select the one that corresponds to the required transfer function H in this circuit which is Voltage Gain voltage to voltage We already know that the principal set of second level TFs H T H p T the set with H made equal to 1 K is produced by both current and voltage test signal injection in the configuration of Fig 11 so in the Voltage Gain Parts List select GFTv coupled voltage current injection Place the icon in the SpiceNet circuit diagram In the Subcircuit properties window re name the Ref Des as xgft Connect the necessary wires either directly or by use of Continuations as in Fig 13 Note that the u pair i
29. l question is what test signal injection configuration to adopt The purpose of an emitter source follower is to provide unity voltage gain at all frequencies so H_ 1 However because of finite transistor transconductance gm and various capacitances the discrepancy factors D and maybe D are not unity and the actual voltage gain H is less than H For an emitter source follower the error voltage is the difference between the input and output voltage because the fedback voltage is equal to the output voltage for H 1 Therefore the error voltage i is identified as in Fig 22 and the test voltage must be injected between Y and X so as to drive the transistor with the voltage Ve Adjustment of the injected voltage to null A is sufficient to raise the output voltage to be equal to the input voltage and so i need not be nulled In fact it had better not be nulled because i is the only current that produces the output voltage The appropriate GFT icon is GFTve providing single voltage injection to a circuit in which the first level TF H is a voltage to voltage gain and is already installed in Fig 22 hdb habinf e 2 20 0 Plot hdb hdbinf in unknown 60 0 100 140 10k 1Meg 100Meg 10G frequency in hertz Fig 23 Some of the Intuscope results for Fig 22 Check that H 1 K 1 as expected The closed loop gain H exhibits a considerable peak After you run the GFT Template a
30. n the icon replaces the normal input signal source This is because the GFT Template has to adjust the input signal together with the nulling conditions in order to measure the 12 third level TFs The zero voltage source V3 is there merely to preserve the distinction between the nodes uu and W The element values are those from the example in 1 with o 1 and Rs 1n essentially zero The model of Fig 13 is ready made in spice8 circuits gft figure13 dwg Fig 13 Insertion of the GFT Template icon GFTv into the model of Fig 12 Running the program simulates twelve second level TFs and calculates the four second level TFs H_ T T H P together with the simulation of the normal closed loop first level TF H 17 You are now ready to run the GFT Template Open Intuscope and go back to the SpiceNet window Click Actions gt ICAPS Select Config 1 Setup 1 and in the Mode area select GFT Make sure the Script box is checked in the Data Reduction area Click Simulate Selections and answer Yes to the query The simulation now runs Go to Intuscope In the Add Waveforms window select result in the Type box in the Source area All the TFs appear in the Y Axis box and can be displayed in dB and phase on the Graph Alternatively all the TFs in 17 and 18 can be displayed in both dB and phase by clicking Calculator gt GFT Some of the results are shown in Fig 14 The first result to check is that H _ is flat at 20dB at all freque
31. ncies confirming that H _ 1 K R1 R2 R1 and that the H_ T TH g et is indeed the principal set so the proper test signal injection configuration was in fact chosen If H__ is not equal to the expected result something is wrong and all the results should be ignored until the problem is fixed Next note that five other TFs are displayed but only three are visible This confirms that the derived first level TF H 1 and H 9 calculated from 17 and 18 respectively are indistinguishable from the directly measured H This Figure appears in 1 together with interpretation and discussion of the agreement with the symbolic results Ord Qtcd Gtdbn nio hd2 GE hobint 2 Plot1 hdb fb tdbn hdb1 kdb2 hdginf o gt ro do oO o 100 10k 1Meg 100Meg 10G frequency in hertz Fig 14 Some of the Intuscope results for Fig 13 Check that H_ 1 K R1 R2 R1 as expected The calculated closed loop gain H and H are indistinguishable from the directly simulated H The loop gain T crosses over on a single 20dB dec slope The null loop gain T is not negligibly large at all frequencies 18 At first sight you might think that the apparent peaking in H results from inadequate phase margin but this is not the case The correct explanation is that the crossover frequency of L is lower than that of T which causes D to increase before D decreases as seen in the constituent
32. ost important second level TF is the loop gain T From the definition 2 the loop gain is that experienced by a signal such as u that starts around the loop and arrives as a signal u at the other side of the injected test signal u under the condition that the input signal u is zero as illustrated in Fig 6 Fig 6 Single injection calculation of the loop gain T 8 The input signal u is not nulled it is set to zero and so the result emerges from a single injection not a null double injection calculation y a pe 14 To measure the direct forward transmission gain H p we invoke the interpretation 6 that H 0 is the gain when the loop gain T is zero Just as H _ is measured by simulating infinite loop gain H 0 is measured by simulating zero loop gain in Fig 4 This is done by adjusting the injected test signal u in the presence of u to null u This is shown in Fig 7 whose interpretation is that if u is zero the only way the input signal u can reach the output u is through the H 0 block Fig 7 Null double injection calculation of the direct forward transmission gain H 0 by mutual adjustment of the injected signal u and the input signal u to null uo which simulates zero loop gain So the direct forward transmission TF emerges from another null double injection calculation as 15 It may not be immediately obvious but the null loop gain T can also be found from Fig 4 by yet another null double injection c
33. s always the first result to check is that H _ is in fact the intended value Figure 23 shows that indeed H_ OdB at all frequencies but that H peaks up before falling off This is a red flag indicating inadequate phase margin so we want to look at the magnitude and phase of the loop gain T as in Fig 24 28 pt abt 36 0 48 0 z z 108 amp 24 0 SS Ar ae O E a gt sc 180 s 0 ay 5 O O 252 24 0 324 48 0 10k 1Meg 100Meg 10G frequency in hertz Fig 24 The loop gain T for the source follower of Fig 22 has negative phase and gain margins and so is unstable ICAP 4 Intuscope automatically presents the phase and magnitude margins as M 15 and TM 7 0dB These are both negative and the circuit is actually unstable One of the main benefits of the GFT is that it enables you to relate the numerical results to the circuit structure and element values one of the important steps in the design oriented analysis strategy So let s examine the second level TFs T and T We can dismiss T quickly because it s infinite The condition in Fig 22 to set up the null loop gain T is that there is a given input voltage uu and the injected voltage is adjusted to null the output voltage uu which makes v uu Also i 0 and i 5 1 70 so there is no current driving the transistor and so v 0 Hence TN 0s The condition in Fig 22 to set up the loop gain T is that the input
34. s of H shown in Fig 15 hdb1 hobid ddbn Plott hdb1 hdbinfd ddbn in unknown 100 10k 1Meg 100Meg 10G frequency in hertz Fig 15 The closed loop gain H dissected into the product of two constituents HD and D according to 17b The constituents H D and H oPo of H 3 in Fig 16 confirm that the nonidealities whether represented by H or by T_ take over above the null loop gain crossover frequency p y H orbyT 19 hd2 hdbinfd habzerodo Plott hdb2 hdbinfd hdbzerodoin unknown 100 10k 1Meg 100Meg 10G frequency in hertz Fig 16 The closed loop gain H dissected into the sum of two constituents H_D and H gl D according to 18b 4 Manipulations of the natural block diagram Equations 17 and 18 describe the closed loop gain H dissected into two different sets of constituents and correspond to the natural block diagram of Fig 3 Other dissections are possible a common one being the elimination of the H 0 block by modifying one or both of the remaining blocks in such a way as to preserve the same answer for H One possible choice is to keep the K block the same and incorporate the H 5 block in terms of T into a modified forward gain block as shown in Fig 17 Fig 17 Absorption of the block H 0 into a pseudo forward gain block that creates a pseudo loop gain I that contains both T and T 20 This model has
35. they all miraculously produce the same H if everything works properly see Caveats below Therefore even though the GFT Template does all the work you are responsible for choosing an injection configuration such that H T T H 0 have useful interpretations with respect to circuit performance otherwise they are useless in design oriented circuit analysis and you might as well resort to the try it and see approach Choice of injection configuration is the crucial step in implementing the GFT which is why it is emphasized in the preceding Examples Caveats Some second level TFs HoT Ts H o may be zero or infinite Because the GFT Template algorithm employs nulling gains that are necessarily finite a zero or infinite result cannot be produced although the incorrect Intuscope graph may superficially look legitimate Fortunately it is easy to determine in advance whether a certain TF is zero or infinite and then you can avoid using a form of the result that depends upon that incorrect TF Example 2 exhibits a case in which H _ 0 and the reason is described The version A 20 for H p has to be used and A 19 for H 1 gives a wrong answer because both H _ and T which is actually infinite are simulated incorrectly Likewise the simulation of H 0 from A 10 is wrong and does not agree with the calculation from A 3 as both should be zero In Example 3 T is infinite for the reason described and so D In Example 1T T T T_ ar
36. ut to obtaining only the T result 22 Fig 19 The model of Fig 13 showing other possible injection configurations WXY and an optional icon GFTs that provides calculations only of T Figure 19 shows the same circuit as Example 1 but with the Loop Gain icon GFTs alternative to the GFTv icon There are no input and output signal connections to the Loop Gain icon so don t forget to set uu 0 by connecting uu to ground The injection configuration is the same as that in Fig 13 if WXY are identified as W 7X3 you have to break the X 3Y connection in order to insert either the GFTv or the Loop Gain icon of course To run this configuration select GFTs instead of GFT in the Simulation Control window and in IntuScope select Calculator GFT T The dB and phase of T are displayed plus the phase and gain margins are explicitly identified The results for T are the same as when the GFTv icon is employed 23 As already mentioned different injection configurations produce different sets of second level TFs HT 1 H 0 although in some cases T may be the same For example if the GFTv or the Loop Gain icon is inserted in Fig 19 at W XY xat WXY T remains the same as for the original insertion point W X 3Y 3 because the test signals e and j are being injected inside the major loop but outside the minor loops in such a way that the major loop is broken if either e is opened or j is shorted Thus T is still the principal loop gain
37. uui is set to zero and the injected voltage establishes Vi and Vy For single voltage injection ii so Tov WV ZZ where Z and Z are respectively the impedances that the injected voltage source looks into at the X and Y nodes At zero frequency T R1 rm 2 700 16 169 44 5dB in agreement with Fig 24 As frequency increases T goes into a single negative slope 20dB dec at the RICL corner and eventually displays a double negative slope due to L1 and CL with an asymptotic phase of 29 180 In between however T encounters a rather complicated transition consisting of a quadratic pair of poles at the L1Cd resonance of 49MHz and a zero at the transistor ft 1 2m7Cdrm 140MHz Thus there is a region of triple negative slope between 49MHz and 140MHz that causes the phase lag to exceed 180 The T crossover frequency of 85MHz is within this range and the phase margin is negative The knowledge that THZ Z provides insight into ways to make the circuit stable For example beyond the R CL corner ZV sCL and Z is independent of CL therefore T is inversely proportional to CL but T stays the same Therefore increasing CL lowers T and moves the crossover frequency lower such that the corresponding phase lag is less Figure 25 shows the results of increasing CL from 50p to 250p g pht Qdbt hnb pht 2 CO dbz hdb 2 c e S 5 90 0 40 0 amp 20 0 Z C D ra 5 05 0 3 20 0 ss N a X 90 0 X 40
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