The General Feedback Theorem: A Final Solution for
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1. Application Notes The General Feedback Theorem A Final Solution for Feedback Systems E R David Middlebrook re you an analog or mixed signal design engineer or a reli ability engineer Are you a manager a design review committee member or a systems integration engineer Did you fall off a cliff when in your first job you discov ered that the analysis methods you learned in college simply don t work There is help available Design oriented analysis D OA don t forget the hyphen is a paradigm based on the recogni tion that design is the reverse of analysis because the answer to the analysis is the starting point for design Conventional loop or node analysis leads to a result in the form of a high entropy expression which is a ratio of sums of products of the circuit elements The more loops or nodes the greater the number of factors in each product By common experience we know that we will sink into algebraic paraly sis beyond two or three loops or nodes Such an analysis result is useless for design In contrast analysis results need to be derived in the form of low entropy expressions in which elements such as impedances are arranged in ratios and series parallel combi nations and not multiplied out into sums of products This is the most important principle of D OA whose objective is to enable a designer to work backwards from an analytic result and change element values in an informed ma2. GFT is to permit calculation of this effect which is not accounted for in the conventional model in order to determine whether or not it is significant The GFT results for the three second level TFs for the model of Figure 9 are shown in Figure 10 and the expectations are indeed borne out The results are repeated in Figure 11 with the loop gains replaced by their corresponding discrepancy factors both of which are essentially 0 dB at low frequencies Also shown is the final result for the first level TF H the closed loop gain which from 18 is the direct superposition of the Hoo D and D graphs This final result shows that the bandwidth of H is determined by the T crossover frequency as in the conventional approach and that reverse transmission wrong way through the feedback path does not have any significant effect until the much higher null loop gain crossover frequency A More Realistic Feedback Amplifier Model The more interesting model of Figure 12 includes two added capacitances for each active device This is a much more realistic model and of course library device mod els can be substituted What are our expectations for the results in com parison with those for Figure 9 Since all the extra elements are capacitances we expect the low fre 1 BU peckeoe quency properties to remain the same but the dominant pole and hence the A YNET a kn Mirene deiver al the performance youtbe heHdng Pa
3. a pole at the crossover frequency of T beyond which D will be the same as T We expect T to be noninfinite and consequently Dn to be not 0 dB because there is nonzero reverse transmission through Celebrating the Vitality of Technology PROCEEDINGS OF THE IEEE l i gt 5 al a 4 tt if i od e Sou ACK Sele No other publication keeps you in touch with the evolving world of technology better than the Proceedings of the IEEE IEEE Every issue of the Proceedings of the IEEE examines new ideas and innovative technologies to keep you up to date with developments within your field and beyond Our unique multidisciplinary approach puts today s technologies in context and our guest editors bring you the expert perspective you need to understand the impact of new discoveries on your world and your work Enrich your career and broaden your horizons Subscribe today and find out why the Proceedings of the IEEE is consistently the most highly cited general interest journal in electrical and computer engineering in the world Source IS Journal Citation Report 2002 Call 1 800 678 4333 or 1 732 981 0060 Fax 1 732 981 9667 Email customer service ieee org www ieee org proceedings the feedback path That is if the forward path through the active devices dies the input signal e can still reach the output vo by going through the feedback path in the wrong direction The principal benefit of the
4. ensuing rise time is shorter than for Figure 9 with the perhaps unexpected beneficial result that the final value is achieved sooner for Figure 12 than for Figure 9 The bottom line is that the GFT of 28 whose principal difference from 6 of the conventional approach is the presence of the null discrepancy factor Dn predicts a substantial modification of the closed loop performance H The nonidealities represented by T or Dy are always present in a realistic model of an electronic feedback system and in at least some respects can actually improve rather than degrade the performance This rare exception to Murphy s law provides added incen tive to utilize the GFT A method of finding loop gain from a voltage loop gain Ty and a current loop gain T calculated succes sively from single voltage and current injected test sig nals has been quite widely adopted since it was pro posed in 1975 The formula is April 2006 Pei z 20 A a 20 Sne T 4T or However this formula was based on the conventional block diagram of Figure 1 and does not account for nonzero reverse loop gain You No Longer Need to Measure Loop Gain Directly In the conventional approach efforts are often made to mea sure loop gain on the actual hardware to check that the phase margin is adequate although little or no thought is given to whether or not the measurement is consistent with the actual closed loop gain It is very awkward to inject even a
5. injection are included and are therefore transparent to the user Are dnti calculations even easier than ndi calculations Absolutely the more signals are nulled the more circuit ele ments do not appear in the answer This constitutes another round of the divide and conquer approach you make a greater number of less complicated calculations April 2006 200 D le i V1 Rs rmi 100 50 i iL Ix X m S W oF Vy Y i Ce R8 20p 10meg uo R2 RL 900 1k R1 100 Figure 9 An Intusoft ICAP 4 GFT template provides the injected test signals ez and jz and performs the simulations and postprocess ing required to obtain Hoo T and Ty and hence H for the GFT of 15 How to Use a Circuit Simulator Incorporating GFT Templates In any case if you re going to use a circuit simulator rather than doing symbolic analysis all you need to do is choose an injection point inside the loop at the error summing point plus select the appropriate GFT template according to whether single or dual test signal injection is required to null simultaneously the voltage and cur rent error signals The template does simu lation runs to calculate the second level TFs Hoo T and Ty in 15 and does post simulation calculations to produce the dis crepancy factors D and Dy in 18 A Simple Feedback Amplifier Example A series shunt feedback amplifier circuit model is shown in Figure 7 The forward path is a simplified
6. model of a typical inte erated circuit IC in which voltage gain is T Crossover 5 4 MHz 40 0 80 0 T Crossover 7 9 GHz 10 Meg 1G 100 Frequency in Hz 100 m 10 1k 100 k Figure 10 The expectations are borne out Hoo is flat at 20 dB and T has a single pole The null loop gain Ty is not infinite because of nonzero reverse transmission through the feedback path a feedforward path achieved in the first two stages which may be differential and current gain is achieved in the final Darlington follower stage In this first example the frequency response is deter mined by the sole capacitance Ce Each active device is repre April 2006 sented by a simple bipolar junction transistor BJT T model which has the advantage of also representing a field effect transistor FET by setting the drain current equal to the source current as is done in this example IEEE microwave magazine 59 100 m 10 1k 100 k 1G Frequency Hz 10 Meg Figure 11 T and T replaced by their corresponding discrepancy factors D T 1 T and Dy 1 Tn Tn The normal closed loop gain H HDDrn differs from Hoo not only because of D but also because of Dn although this effect is small To apply the GFT the crucial first step is to choose the test signal injection point that makes H T and Ty have the desired interpretations of ideal closed loop gain loop gain and null loop gain The error voltage is the voltage between
7. the input and the fed back voltage at the feedback divider tap point labeled vy in Figure 7 The error current is the current drawn from the feedback divider tap point labeled i in Figure 7 The test sig nals ez and jz are to be injected inside the loop so that vy and ix are the driving signals for the forward path as shown in Figure 8 Thus the test signal configuration meets the two condi tions that injection occurs at the error sum ming point and is inside the loop implement ing the general model of Figure 5 To invoke the GFT template the appropri ate dual injection icon is selected and connect ed to provide the test signals ez and jz as in Figure 9 The icon also provides the system input signal e and observes the output signal Vo because it has to adjust the test signals rel ative to the input to establish various nulls one of which is the output signal Another principle of D OA is figure out as much as you can about the answer before you 100 plunge into the analysis In this case we expect Hoo to be 1 K the rec iprocal of the feedback ratio that was initially chosen to meet the system specification because the injection configuration was specifically set up to achieve this Here 1 K Ry Ro Ry 10 gt 20 dB flat at all frequencies We expect T to be large at low frequencies and to have a sin gle pole determined by Ce Consequently D will be flat at essentially 0 dB at low frequencies with
8. T3b 1 1 Ta 1 Ffa ee 13 Therefore the key decision in applying the dissection theo rem is choosing a test signal injection point so that at least one of the second level TFs has the physical interpretation you want it to have The Dissection Theorem Can Morph into the Extra Element Theorem For example if the injection point is chosen so that uy goes into voltage across or current into a single impedance Z 7 becomes n 144 Za 1 2 H H z 14 in which Z4 Zn are respectively the driving point imped ance and the null driving point impedance seen by Z and Alz oo is the value of H when Z is infinite In this form the dissection theorem becomes the extra element theorem EET of which a useful special case is when Z is the only capacitance in an otherwise resistive circuit Then H z_ is the first level TF the gain when the capacitance C is absent zero and the pole and zero are exposed directly as 1 CRg and 1 CRy The EET will not be further discussed here because the spotlight is on another example of the choice of injec tion point The Dissection Theorem Can Morph into the GFT It is easy to see that the block diagram of Figure 4 represents 7 and is immediately recognizable as an augmented version of the conventional feedback block diagram of Figure 1 which represents 2 So where does the test signal have to be injected so that TF 1 has the physical interpretation of Hoo
9. The answer is obvious since Ho is H when the error sig nal vanishes infinite loop gain and TF1 is H when uy is nulled then uz must be injected at the error summing point so that uy represents the error signal At the same time this makes TF2 have the physical inter pretation of the loop gain T since the test signal injection point is inside the loop Finally TF3 which does not have a corresponding appear ance in Figure 1 is a new TF that can be designated Tn since by 8 it is defined as a loop gain with uo nulled Following the above procedure 7 becomes April 2006 R3 200 lO iv Rs rmi 100 50 i D t Error Voltage Error Current R1 100 uo 900 1k Figure 7 A simple feedback amplifier model with the error voltage and the error current identified 1 aaa ie 15 and in this form the dissection theorem becomes the GFT For the second level TFs to have the desired physical inter pretations TF1 Hy TF2 T TF3 Ty the test signal uz must be inside the loop and at the error summing point as illustrated in Figure 5 Then 8 becomes u Ui u T Ux u T Uy H 16 uy 0 u 0 t 0 With Hoo T and Ty calculated from 16 the result 15 is rep resented by the augmented block diagram of Figure 6 Because superficially Figure 6 differs from Figure 1 only in the presence of an additional block that contains the nonideal ities it is imp
10. al Solution From Figure 2 the Dissection Theorem was developed in terms of a single injected test signal uz but a general version can be developed in terms of any number of test signals injected at different points The EET interpretation in terms of N test signals becomes the NEET and although this is NEET it won t be discussed further here For the GFT interpretation the most useful version employs two injected test signals a voltage ez and a current jz 58 IEEE microwave magazine both injected at the error summing point This is because to make Hoo equal to 1 K the ideal closed loop gain both the error voltage vy and the error current iy have to be nulled simultaneously For dual voltage and current injected test signals at the error summing point the GFT of 7 remains exactly the same except that the definitions of the TFs are extend ed In particular Uo H ss Ui p0 19 which says that Hoo is established by the double null triple injection dnti condition that ez and jz are mutually adjusted with respect to u so that vy and iy are both nulled The defin itions of T and T are each extended to a combination of volt age and current loop gains established by ndi conditions for T and by dnti conditions for T These definitions are not dis played here because the circuit examples will be treated by use of the Intusoft ICAP 4 GFT templates in which all the calculations needed for dual test signal
11. di calculation Note that while H and TF1 are both ratios of out z TNN Injected Test Signal Linear System Figure 2 General model of a linear system with input u and output uo and an injected test signal uz April 2006 put to input their values are different because even if the input u is the same the uo for TF1 contains a component due to uz that is absent in the uo for H An ndi calculation is made after an ndi condition has been established and is always easier and simpler than an si calculation This is not an accident any element in the sys tem that supports a null signal might as well not be there and does not appear in the calculation or in the result Also you don t have to know what the relation between the two injected signals is all you have to know is the equivalent information that the null exists To emphasize this consider the way to make the null self adjusting shown in Figure 3 the imaginary infinite gain amplifier automatically nulls uy and to calculate TF1 from 8 you simply use the fact that uy is nulled and you don t have to know what uz is The method of Figure 3 can be imple mented in a circuit simulator since the nulling amplifier bandwidth is infinite even if its gain is not The TF3 in 8 is also an ndi calculation and no further explanation is necessary The different ndi condition can be established by connecting the nulling amplifier input to uo instead of to uy If y
12. ecause it contributes only via T and its own value is of no interest It is clear from 5 that D is a unique function of T which has several useful consequences First when T is large D 1 which leads to the desired result that H H Second when T is small D T and is small also leading to a significant dis crepancy between Hoo and T Third where the magnitude of T falls to one the crossover frequency marks the end of the frequency range over which T performs its useful function How T crosses over determines the degree of peaking exhibited by D dur ing its transition between one when T is large to T when T is small According to 5 peaking in D is related to the phase margin of T and translates by 6 directly to the closed loop response H Thus 5 and 6 expose the well known unique relation ship between loop gain phase margin and both the frequency and time domain responses of the closed loop gain What s Wrong with the Conventional Approach The model of Figure 1 is incomplete because it does not account for bidirectional signal transmission in the boxes In fact the boxes are drawn as arrowheads on purpose to emphasize that reverse transmission is excluded If both boxes have reverse transmission there is also a nonzero reverse loop gain and it is convenient to lump together all the properties omitted from this block diagram under the label nonidealities Consequently all analysis based on this model also igno
13. edback amplifiers containing both dif ferential and common mode loops 200 m Analog mixed signal and power supply design engineers are not the only ones to benefit from an ability to apply the powerful methods of D OA Those 100 m 20 0 n Therefore managers system integration engineers and reliability engineers who evaluate the products of other companies as well as their own also can signifi cantly increase their effectiveness by applying the meth ods of D OA Then if you hold any of the above job titles you can contribute meaningfully to design review discussions instead of just saying to the presenting designer Well it looks as though it s coming along all right carry on In fact you can improve the effectiveness of the whole project by requiring that design engineers present their results according to the principles of D OA For further information on the GFT EET and D OA in gener al see http www rdmiddlebrook com For further information on the Intusoft ICAP 4 Circuit Simulator including a GFT Template User s Manual see http www intusoft com RK 10 1k 100 k 10 Meg 1G Frequency in Hz Figure 14 The extra capacitances cause the null discrepancy factor Dy to be drastically different resulting in a huge phase lag of 450 in the normal closed loop gain H 1 1 6 64 50 1 2n 5 4 MHz 30 ns 60 0 n 100 n Time s 140 n 180 n Figure 15 For the circuit of Fig
14. he GFT because the results for a feedback system are exact and not impaired by the approximations and assumptions inherent in the con ventional single loop model Moreover it may no longer be necessary to attempt hardware measurements of loop gain which in itself is a considerable saving of time and effort What Is the Conventional Approach The well established method of analyzing a feedback sys tem begins with the familiar block diagram of Figure 1 from which the feedback ratio K and the loop gain T AK are calculated The designer s job is to set K and the for ward gain A so that the final closed loop gain H meets a specification usually with the help of circuit simulator soft wate Several iterations often aided by hardware measure ments of the loop gain may be needed before the closed loop gain meets the specification April 2006 Figure 1 The familiar single loop block diagram of a feedback sys tem The arrowhead shapes imply that the signal goes only one way Unfortunately this approach can give incorrect results stemming from the fact that the conventional block diagram of Figure 1 is an incomplete representation of the actual hard ware system Your immediate reaction to this allegation may be If I ve noticed any discrepancies between the predicted and the actual results they ve been small enough to neglect or I ve just ignored them anyway Wouldn t it be better to be able to get the exact ana
15. k diagram for the GFT is a morphed version of Figure 4 in which Ty or Ho represents the main effects of the nonidealities At first sight you might think that replacing one calcula tion by three is a step in the wrong direction On the contrary since 7 is itself a low entropy expression the influences of TF2 and TF3 in modifying TF1 are exposed which is helpful design oriented information Moreover since two of the three second level definitions of 8 are ndi calculations the dissec tion theorem replaces a single complicated calculation by three potentially simpler calculations thus implementing another of the principles of D OA divide and conquer Nevertheless these are minimum benefits and much greater benefits accrue if the second level TFs have useful physical interpretations Thus the second level TFs 8 them selves contain the useful design oriented information and you may never need to actually substitute them into 7 For example if TF2 TF3 gt 1 H x TF1 54 IEEE microwave magazine How do we determine the physical interpretations of 8 In the above discussion based on Figure 2 nothing was said about where in the system model the test signal is injected Different test signal injection points define different sets of coefficients A B C D and hence different sets of second level TFs However when a mutually consistent set is substi tuted into 7 the same H results ie el EE Herbig er p _
16. lysis results quickly and easily so that you could accurately predict the actual system performance This desirable situation is now realized through use of the GFT Let s start by reviewing in more detail the conventional approach based on Figure 1 for which the closed loop gain H the answer is given by A A H _ 1 1 AK 14 T 1 A better form is 1 AK T 1 K1 AK IFT Sipi ol where 1 l Ha 7 ideal closed loop gain 3 T AK loop gain 4 It is convenient to define a discrepancy factor D as a unique function of T 1 D discrepancy factor 5 u so that the closed loop gain H can be expressed concisely as HaHa 6 Form 2 is better because Hoo represents the specifica tion and is the only known quantity at the outset So 52 IEEE microwave magazine K 1 H is designed to meet the specification and the only hard part is designing the loop gain T so that the actual closed loop gain H meets the specification within the required tolerances That is the discrepancy factor D must be close enough to one over the specified bandwidth One of the principles of D OA is embodied in 2 and 6 namely Get the quantities you want in the answer into the statement of the problem as early as possible In this case Ho is the desired gain and D is the discrepancy between Ho and the actual answer you re going to get Equally important A is banished from the answer b
17. nner in order to make the answer come out closer to the desired result the specification D OA is the only kind of analysis worth doing since any other is a waste of time There are many methods of D OA some of them little more than shortcuts or tricks but here the spotlight is on a R David Middlebrook rdm rdmiddlebrook com is Professor Emeritus of Electrical Engineering at California Institute of Technology in Pasadena California 50 IEEE microwave magazine new approach to analysis and design of feedback systems based on the general feedback theorem GFT A typical analysis procedure followed by designers and integration and reliability engineers is to throw the whole cir cuit into a simulator and see what it does possibly including attempts to measure the loop gain as well as external proper ties The design phase may consist of little more than tweak ing and sensitivity simulations A much more efficient approach is to begin with a simple circuit in terms of device models absent capacitances and other parasitic effects and then to add these sequentially Even if you do little or no symbolic analysis successive sim ulations tell you in what ways which elements affect the result so that when you finally substitute your library process and device models you have a much better handle on where their effects originate This procedure of D OA by simulator is significantly enhanced when the simulator incorporates t
18. or D as a unique function of T it is likewise useful to intro duce the null discrepancy factor D as a unique function of Tn according to 14 T 1 a a Tn Tn Dn null discrepancy factor 17 so that the final result for the first level TF T can be written H HxDDy 18 April 2006 200 O iv 100 10me u R R 900 1k Figure 8 The crucial step choose the test signal injection point inside the loop so that vy is the error voltage and iy is the error current The importance of this result is that the familiar tenet must be modified the closed loop frequency domain and time domain responses are no longer uniquely determined by the loop gain T and its phase margin Instead these responses are modified by a noninfinite value of the null loop gain T via a nonunity value of the null discrepancy factor Dy Calculation of the second level TFs Hy T and T can be done symbolically or numerically by use of a circuit simula tor The Intusoft ICAP 4 simulator incorporates GFT tem plates that apply a voltage or current test signal set up the appropriate si and ndi conditions and perform the required calculations As a user all you have to do is choose the prop er test signal injection point which is inside the loop at the error summing point However before proceeding to a circuit example it is nec essary to make an extension of the dissection theorem The GFT for Two Injected Test Signals Is the Fin
19. ortant to emphasize the fundamental difference between the conventional approach and that based on the GFT In the conventional approach the block diagram of Figure 1 is the starting point in which reverse transmission in both boxes is ignored and the result 2 is developed from Figure 1 In the GFT approach Figure 5 is the starting point and the result 15 is developed from 16 directly from the com plete circuit without any assumptions or approximations Since 15 is represented by Figure 6 the block diagram of Figure 56 IEEE microwave magazine 6 is part of the result The boxes in Figure 6 are unidirection al and do not necessarily correspond to any separately identifi able parts of the circuit The values of these boxes expressed in terms of the second level TFs Hy T and T automatical ly incorporate any nonidealities that may be present in the actual circuit Although the augmented block diagram exhibits a loop it represents any linear system even if there is not a physical ly discernible loop An example is a Darlington follower for which the GFT affords a means of investigating the well known potential instability It is apparent that the null loop gain T contains the first order effects of nonidealities although there may also be sec ond order effects upon the loop gain T Thus the T in 2 may not be the same as the correct T in 15 Since it was convenient in 5 to introduce the discrepancy fact
20. ou re not familiar with the dissection theorem you can easily verify 7 by setting up a set of equations that represent the properties of a linear system containing two independent sources uj and uz uy Auo Buz 10 Uz Ux Uy 12 You can evaluate the TFs of 8 and 9 in terms of the A B C D coefficients For example to find TF1 set uy 0 in 11 and solve for the ratio uz u C D that nulls uy In 10 ux uz by virtue of uy 0 in 12 Thus Up 1 B uz A Finally substitute uz u C D to get TF1 6B 1 C AD Likewise find TF2 and TF3 and insert all three TFs into 7 to confirm that the result for H is the same as that obtained directly from 10 12 by setting uz 0 which makes ux uy and H C A What Is the Dissection Theorem Good for The dissection theorem is completely general the only con straint being that it applies to linear systems As with most theorems the important thing is what is it good for and how do you use it Nulling Amplifier e a Injected Test Signal Linear System Figure 3 How to set up an ndi condition uy is automatically nulled IEEE microwave magazine 53 Figure 4 This augmented block diagram with anonymous TFs represents the dissection theorem 7 Note Arrowheads Figure 5 When the test signal is injected inside the loop at the error summing point the dissection theorem of 7 becomes the GFT 15 Figure 6 This bloc
21. r n a naw signal can reach the output Also there is now nonzero reverse transmission through the forward path which in turn creates a nonzero reverse loop gain It is not necessary to separate these nonidealities because they are all automatically account ed for in the calculation of the loop gain usually little effect and in the null loop gain the major effect The quantitative results of the GFT template simulations for Figure 12 shown in Figures 13 and 14 bear out the expectations The null loop gain crossover frequency is drastically lowered and even though the magnitude of the null discrepancy factor NE fortunes loop gain crossover frequency would be lowered Therefore to enable a more meaningful comparison between the two circuits Ce in Figure 12 has been reduced sufficiently to preserve the same loop gain crossover frequen cy Nevertheless the extra capacitances create more poles and zeros so the high frequency loop gain is expected to be more complicated The major consequence of the presence of the extra capacitances is that there is now a second feedfor ward path through a string of capac itances in addition to that through the feedback path in the wrong direction through which the input April 2006 CUNN preducts AIE end heyend ts suscder PAE of eurtesture pected prociucts whet you ell notice first shout daing bushes with Terng k hey CEDAT nec ore ahnas Gee priority You ge
22. r function TF H of a linear system can be dissected into a com bination of three second level TFs TF1 TF2 and TF3 according to 1 7h H TAF 7 TA These TF symbols are intentionally anonymous so that differ ent physical significances can be assigned later The second level TFs are calculated in terms of an injected test signal uz as shown in Figure 2 The injected test signal sets up Ux going forward towards the output uo and uy going backward towards the input u such that ux uy uz in which ux Uy Uz may be all voltages or all currents and u and Uo can independently be a voltage or a current The second level TFs are defined by u ToS 2 u 0 Ux u TE 2 Ux TH i 8 Uo 0 u 0 and the first level TF H is simply the output divided by the input the gain in the absence of the test signal Uo H Ui 9 uz 0 In 8 TF2 is the ratio of the signal going backward to the signal going forward from the test signal injection point under the condition that the original input signal is zero This is a single injection si calculation the familiar method by which any TF is calculated TF1 is the ratio of the output to the input when uy is nulled a condition that is established by adjustment of the test signal uz so that its contribution to uy is exactly equal and opposite that from u This is a less familiar null double injection n
23. res the nonidealities In most textbooks and handbooks you can find whole sections that postulate a model in which each box of Figure 1 is replaced by a two port model thus retaining the bidirectionality of a real circuit However there are several disadvantages to this approach 1 Four different sets of two port models are required to represent the four possible feedback configurations shunt shunt series shunt etc and in three of the four the model itself is still inaccurate because common mode gain is ignored Even though the nonidealities are incorporated in the model how do you account for their effects upon the loop gain and the closed loop response Because the circuit elements are buried inside the two N Ner Q wa port parameters which are themselves buried in expres sions for the loop gain and closed loop gain the results are high entropy expressions and are essentially useless for design April 2006 The Dissection Theorem Is a Completely General Property of a Linear System Model The GFT sweeps away all the a priori assumptions and approximations inherent in the previously described conven tional approach and produces low entropy results directly in terms of the circuit elements This is accomplished because the GFT does not start from a block diagram model but is developed from a very basic property of a linear system the dissection theorem The dissection theorem says that any first level transfe
24. single test signal into an IC and dual or triple injection to set up ndi or dnti conditions is so burdensome that it is rarely attempted Fortunately it is no longer necessary to measure loop gain directly The GFT of 18 is exact with respect to the simulat ed model and all you have to do is mea sure the final closed loop TF H which you would do anyway to check whether deg dB it meets the specification If the simulat 360 800 ed H differs from the measured H you have to adjust the model until the two EE are the same When this is achieved i you know the model is correct and then the simulation tells you what T and T 0 0 are individually This is the culmination of the D OA process 180 40 0 More Useful Techniques 7360 T800 for Analog Engineers The GFT is completely general the only requirement being that it applies to a lin ear system model The symbol H has been used here for the first level TF pur posely to avoid the connotation of gain because it could equally well represent input or output impedance power supply rejection or indeed any TF of interest 1 40 It was mentioned earlier that the EET interpretation of the dissection theorem can be extended to any number of injected test signals leading to the NEET theorem The GFT interpretation could likewise be extended in particular to two pairs of injected voltage and current test signals which would open the door to analysis of 600 m 200 m fe
25. t tha arrears you sep oma aha each levee af service that makes doing TOUR ee ey Cac ik se is Soe AR Task pment came pT as E requirentent T wetl chow yeu hea gaai Ht con be Temps offers a flection of Ireatock products for menaira delivery es well as quick um The Rapid Roapensa Supplier ef Choice IEEE microwave magazine 61 ul xoft GFTv W y f Yi R1 100 uo RL 1k R2 900 Figure 12 A more realistic model than Figure 7 with two capacitances added per active device library models of course could be substituted a T Crossover 5 4 t T Crossover 82 MHz 100 m 10 1k 100k 10 Meg 1G Frequency in HZ Figure 13 There is now an additional feedforward path nonzero reverse transmission through the forward path and nonzero reverse loop gain causing the null loop gain crossover frequency to be much lower D is 0 dB at both ends of the frequency range its phase under goes a huge lag that is transferred directly into a correspond ingly huge phase lag in the final closed loop gain H Although the magnitude of H at high frequencies may not be of much interest in the frequency domain its corre 62 IEEE microwave magazine 100 sponding phase has a controlling effect in the time domain The step responses of the circuits of Figures 9 and 12 are shown in Figure 15 The huge phase lag of H at high frequencies for the circuit of Figure 12 causes the expected delay in the step response however the
26. ure 12 the huge phase lag of H causes the expected who review and verify designs of others also need to know how design oriented results of analysis should be presented April 2006 delay in the step response however the ensuing rise time is shorter with the perhaps unexpected beneficial result that the final value is achieved sooner At least in some respects nonidealities can actually improve performance IEEE microwave magazine 63
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