Methods of Design-Oriented Analysis The GFT: A Final Solution for Feedback Systems

Transcription

1 v.1, 5/11/05 Methods of Design-Oriented Analysis The GFT: A Final Soltion for Feedback Systems R. David Middlebrook Are yo an analog or mixed signal design engineer or a reliability engineer? Are yo a manager, a design review committee member, or a systems integration engineer? Did yo fall off a cliff when in yor first job yo discovered that the analysis methods yo 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 recognition that Design is the Reverse of Analysis, becase the answer to the analysis is the starting point for design. Conventional loop or node analysis leads to a reslt in the form of a high entropy expression which is a ratio of sms of prodcts of the circit elements. The more loops or nodes, the greater the nmber of factors in each prodct. By common experience, we know that we will sink into algebraic paralysis beyond two or three loops or nodes. Sch an analysis reslt is seless for design. In contrast, analysis reslts need to be derived in the form of low entropy expressions in which elements, sch as impedances, are arranged in ratios and seriesparallel combinations, and not mltiplied ot into sms of prodcts. This is the most important principle of Design Oriented Analysis, whose objective is to enable a designer to work backwards from an analytic reslt and change element vales in an informed manner in order to make the answer come ot closer to the desired reslt (the Specification). Design Oriented Analysis 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 shortcts or tricks, bt here the spotlight is on a new approach to analysis and design of feedback systems, based on the General Feedback Theorem (GFT). A typical analysis procedre followed by designers, and integration and reliability engineers, is to throw the whole circit into a simlator and see what it does, possibly inclding attempts to measre the loop gain as well as external properties. The design phase may consist of little more than tweaking and sensitivity simlations. A mch more efficient approach is to begin with a simple circit in terms of device models absent capacitances and other parasitic effects, and then to add these seqentially. Even if yo do little or no symbolic analysis, sccessive simlations tell yo in what ways which elements affect the reslt, so that when yo finally sbstitte yor library process and device models yo have a mch better handle on where their effects originate. This procedre of D OA by simlator is significantly enhanced when the simlator incorporates the GFT, becase the reslts for a feedback system are exact and

2 not impaired by the approximations and assmptions inherent in the conventional single loop model. Moreover, it may no longer be necessary to attempt hardware measrements of loop gain, which in itself is a considerable saving of time and effort. What is the conventional approach? A The well established method of analyzing a feedback system begins with the familiar block diagram of Fig. 1, from which the feedback ratio K and the loop gain T =A K are calclated. The designer s job is to set K and the forward gain A so that the final closed loop gain H meets a specification, sally with the help of circit simlator software. Several iterations, often aided by hardware measrements of the loop gain, may be needed before the closed loop gain meets the specification. Unfortnately, this approach can give incorrect reslts, stemming from the fact that the conventional block diagram of Fig. 1 is an incomplete representation of the actal hardware system. Yor immediate reaction to this allegation may be: If I ve noticed any discrepancies between the predicted and the actal reslts, they ve been small enogh to neglect, or I ve jst ignored them anyway. Woldn t it be better to be able to get the exact analysis reslts, qickly and e asily, so that yo cold accrately predict the actal system performance? This desirable sitation is now realized throgh se of the General Feedback Theorem (GFT). Let s start by reviewing in more detail the conventional approach based on Fig. 1, for which the closed loop gain H (the answer ) is given by A A H = = (1) 1 AK 1 T A better form is 1 AK T 1 H = = H = H = H D K 1 AK 1 T 1 1 (2) T where 1 H ideal closed loop gain K (3) T AK loop gain (4) I t is convenient to define a discrepancy factor D as a niqe fnction of T, T 1 D discrepancy factor (5) 1 T 1 1 T so that the closed loop gain H can be expressed concisely as H = H D (6) Format (2) is be tter becase H represents the specification, and is the only known qantity at the otset. So, K =1/ H is designed to meet the specification and the only hard part is designing the loop g ain T so that the actal closed loop gain H meets the specification within the reqired tolerances. That is, the discrepancy factor D mst be close enogh to 1 over the specified bandwidth The form (2) or (6) is also better becase it embodies one of the Principles of Design Oriented Analysis, namely Get the qantities yo want in the answer into the

3 statement of the problem as early as possible. In this case, H is the desired gain, and D is the discrepancy between H and the actal answer yo re going to get. Eqally important, A is banished from the answer, becase it contribtes only via T, and its own vale is of no interest. It is clear from the simple relation (5) that D is a niqe fnction of T, which has several sefl conseqences. First, when T is large, D 1 which leads to the desired reslt that H H. Second, when T is small, D T and is small also, leading to a significant discrepancy between H and T. Third, wher e the magnitde o f T falls to 1, the crossover freqency, marks the end of the freqency range over which T performs its sefl fnction. HowT crosses over d etermines the degree of peaking exhibited by D dring its transition between 1 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 th e closed loop response H. T hs, (5) and ( 6) expose the well known niqe relationship between loop gain phase margin and both the freqency and time domain responses of the closed loop gain. What s wrong with the conventional approach? The model of Fig. 1 is incomplete, becase it does not accont for bidirectional signal transmission in the boxes. In fact, the boxes are drawn as arrowheads on prpose to emphasize that reverse transmission is exclded. If both boxes have reverse transmission, there is also a nonzero reverse loop gain, and it is convenient to lmp together all the properties omitted from this block diagram nder the label nonidealities. Conseqently, all analysis based on this model also ignores the nonidealities. In most textbooks and handbooks yo can find whole sections that postlate a model in which each box of Fig. 1 is replaced by a two port model, ths retaining the bidirectionality of a real circit. However, there are several disadvantages to this approach: (1) For different sets of two port models are reqired to represent the for possible feedback configrations, shnt shnt, series shnt, etc., and in three of the for the model itself is still inaccrate becase common mode gain is ignored; (2) Even thogh the nonidealities are incorporated in the model, how do yo accont for its effect pon the loop gain and the closed loop response? (3) Becase the circit elements are bried inside the two port parameters, which are themselves bried in expressions for the loop gain and closed loop gain, the reslts are high entropy expressions and are essentially seless for design. The Dissection Theorem is a completely general property of a linear system model The General Feedback Theorem (GFT) sweeps away all the a priori assmptions and approximations inherent in the above described conventional approach, and prodces low entropy reslts directly in terms of the circit elements. This is accomplished becase the GFT does not start from a block diagram model, bt is developed from a very basic property of a linear system, the Dissection Theorem.

4 The Dissection Theorem says that any first level transfer fnction (TF) H of a linear system can be dissected into a combination of three second level transfer fnctions TF 1, TF 2, and TF 3 according to 1 1 H = TF1 TF 3 (7) 1 1 TF2 These TF symbols are intentionally anonymos so that different physical significances can be assigned later. The second level TFs are calclated in terms of an injected test signal z, as shown in Fig. 2. The in jected test signal sets p x going forward towards the otpt o, and y going backward towards the inpt i, sch that x y = z, in which x, y, z may be all voltages or all crrents, an d i and o can independently be a voltage or a crrent. Th e second level TFs are defined by TF1 o y y TF2 TF3 i 0 x = = 0 x = 0 y i o and the first level TF H is simply the otpt divided by the inpt (the gain ) in the absence of the test signal: H o (9) i z = 0 The definitions (8) reqire some interpretation. The TF 2 is the ratio of the signal going backward to the signal going forward from the test signal injection point, nder the con dition that the original inpt signal is zero. This is a single injection (si) calclation, the familiar method by which any TF is calclated. The TF 1 is the ratio of the otpt to the inpt when y is nlled, a condition that is established by adjstment of the test signal z so that its contribtion to y is exactly eqal and opposite that from i. This is a less familiar nll doble injection (ndi) calclation. Note that while H and TF 1 are both ratios of otpt to inpt, their vales are different becase, even if the inpt i is the same, the o for TF 1 contains a component de to z that is absent in the o for H. An ndi calclation is made after an ndi condition has been establish ed, and is always easier and simpler than an si calclation. This is not an accident: any element in the system that spports a nll signal might as well not be there, and so does not appear in the calclation or in the reslt. Also, yo don t have to know what the relation is between the two injected signals is; all yo have to know is the eqivalent information that the nll exists. To emphasize this, consider the way to make the nll self adjsting shown in Fig. 3: the imaginary infinite gain amplifier atomatically nlls y, and to calclate TF 1 from (8) yo simply se the fact that y is nlled, and yo don t have to know what z is. The method of Fig. 3 can be implemented in a circit simlator, since the nlling amplifier bandwidth is infinite even if its gain is not. (8)

5 The TF 3 in (8) is als o an ndi calclation, and no frther explanation is necessary. The (different) ndi condition can be established by connecting the nlling amplifier inpt to o instead of to y. If yo re not familiar with the Dissection Theorem, yo can easily verify (7) by setting p a set of eqations that represent the properties of a linear system containing two independent sorces i and z : x = Ao B z (10) y = C i D z (11) z = x y Yo can evalate the TFs of (8) and (9) in terms of the A, B, C, D coefficients. For example, to find TF1, set y = 0 in (11) and solve for the ratio z / i = C/ D that nlls y. In (10), x = z by virte of y = 0 in (12). Ths, o = (1 B) z /A. Finally, sbstitte z / i = C/ D to get TF 1 = (B 1)C/AD. Likewise, find TF2 and TF 3, and insert all thr ee TFs into (7) to confirm that the reslt fo r H is the same as that obtained directly fro m (10) (12) by setting z = 0, which makes x = y and H = C/ A. (12) What is the Dissection Theorem good for? The Dissection Theorem is completely general, the only constraint being that it applies to linear systems. As with most theorems, the important thing is what is it good for, and how do yo se it? At first sight, yo might think that replacing one calclation by three is a step in the wrong direction. On the contrary, since (7) is itself a low entropy expression, the inflences of TF 2 and TF 3 in modifying TF 1 are exposed, which is helpfl designoriented information. Moreover, since two of the three second level definitions of (8) are ndi calclations, t he Dissection Theorem replaces a single complicated calclation by three potentially simpler calclations, ths implementing another of the Principles of Design Oriented Analysis, Divide and Conqer. Nevertheless, these are minimm benefits and mch greater benefits accre if the second level TFs have sefl physical interpretations. Ths, the second level TFs (8) themselves contain the sefl design oriented information and yo may never need to actally sbstitte them into (7). For example, if TF 2, TF 3 >> 1, H TF 1. How do we determine the physical interpretations of (8)? In the above discssion based on Fig. 2, nothing was said abot where in the sys tem model the tes t 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 mtally consistent set is sbstitted into (7), the same H reslts: H = TF1a TF 3 a = TF1 b TF 3 b =... (13) TF2a TF2b Therefore, the key decision in applying the Dissection Theorem is choosing a test signal injection point so that at least one of the second level TFs has the physical interpretation yo want it to have.

6 The Dissection Theorem can morph into the Extra Element Theorem (EET) For example, if the injection point is chosen so that y goes into (voltage across, o r crrent into) a single impedance Z, (7) becomes Z n 1 H = H Z Z = (14) Zd 1 Z in which Z d, Z n are respectively the driving point impedance and the nll drivingoint impedance seen by Z, and H Z= is the vale of H when Z is infinite. In this p form the Dissection Theorem becomes the Extra Element Theorem (EET), of which a sefl special ca se is when Z is the only capacitance in an otherwise resistive circit. Then, H Z = is the first level TF (the ga in ) when the capacitancec is absent (zero), and the pole and zero are exposed directly as 1/ CR d and 1/ CR n. The EET will not be frther discssed here, becase the spotlight is on another example of the choice of injection point. The Dissection Theorem can morph into the General Feedback Theorem (GFT) It is easy to see that the block diagram of Fig. 4 represents (7), and is immediately recognizable as an agmented version of the conventional feedback block diagram of Fig. 1, which represents (2). So, where does the test signal have to be injected so that TF1 has the physical interpretation of H? The answer is obvios: Since H is H when the error signal vanishes (infinite loop gain), and TF 1 is H when y is nlled, then z mst be injected at the error smming point so that y represents the error signal. At the same time, this makes TF 2 have the physical interpretation of the loop gain T, since the test signal injection point is inside the loop. Finally TF 3, whi ch does not have a corresponding appearance in Fig. 1, is a new T F that can be designated T n, since b y (8) it is defined as a loop gain with o nlled. Following the above procedre, (7) becomes 1 1 T H = H n (15) 1 1 T and in this form the Dissection Theorem becomes the General Feedback Theorem. For the second level TFs to have the desired physical interpretations TF 1 = H, TF 2 = T, TF3 = T n, the test signal z mst be inside the loop and at the error smming point, as illstrated in Fig. 5. Eqations (8) then become y o y H T Tn (16) i 0 x = = 0 x = 0 y i o With H, T, and T n calclated from (16), the reslt (15) is represented by the agmented block diagram of Fig. 6.

7 Beca se sperficially Fig. 6 differs from Fig. 1 only in the presence of an additional block that contains the nonidealities, it is important to emphasize the fndamental difference between the conventional approach and that based on the GFT. In the conventional approach, the block diagram of Fig. 1 is the starting point, in which reverse transmission in both boxes is ignored, and the reslt (2) is developed from Fig. 1. In the GFT approach, Fig. 5 is the starting point, and the reslt (15) is developed from (16) directly from the complete circit withot any assmptions or approximations. Since ( 15) is represented by Fig. 6, the block diagram of Fig. 6 is part of the reslt. The boxes in Fig. 6 are nidirectional, and do not necessarily correspond to any separately identifiable parts of the circit. The vales of these boxes, expressed in terms of the second level TFs H, T, and T n, atomatically incorporate any nonidealities that may be present in the actal circit. Althog h the agmented block diagram exhibits a loop, it represents any linear system even if there is not a physically discernible loop. An example is a Darlington Follower, for which the GFT affords a means of investigating the wellknown potential instability. It is apparent that the nll loop gain T n contains the first order effects of nonidealities, althogh there may also be second order effects pon the loop gain T. Ths, the T in (2) may not be the same as the correct T in (15). Since it was convenient in (5) to introdce the discrepancy factor D as a niq e fnction of T, it is likewise sefl to introdce the nll discrep ancy factor D n as a niqe fnction of T n, according to 1 T 1 D n n 1 nll discrepancy factor (17) Tn Tn s o that the final reslt for the first level TF T can be written H = H DD n (18) The importance of this reslt is that the familiar tenet mst be modified: the cl osed loop freqency domain and time domain responses are no longer niqely determined by th e loop gain T and its phase margin. Instead, these responses are modified by a noninfinite vale of the nll loop gain T n via a nonnity vale of the nll discrepancy factor D n. Calclation of the second level T Fs H, T, and T n can be done symbolically, or nmerica lly by se of a circit simlator. The Intsoft ICAP/4 simlator incorporates GFT Templates that apply a voltage or crrent test signa l, set p the appropriate si and ndi conditions, and perform the reqired calclations. As a ser, all yo have to do is choose the proper test signal injection point, which is inside the loop at the error smming point. However, before proceeding to a circit example, it is necessary to make an extension of the Dissection Theorem. The GFT for two injected test signals is the Final Soltion

8 From Fig. 2, the Dissection Theorem was developed in terms of a single injected test signal z, bt a general version can be developed in terms of any nmber of test s ignals injected at different points. The Extra Element Theorem interpretation in terms of N test signals becomes the N Extra Element Theorem, and althogh this is NEET it won t be discssed frther here. For the General Feedback Theorem interpretation, the most sefl version employs two injected test signals, a voltage e z and a crrent j z, both injected at the error smming point. This is becase, in order to make H eqal to 1/ K, the ideal closed loop gain, both the error voltage v y and the error crrent i y have to be nlled simltaneosly. For dal voltage and crrent injected test signals at the erro r smming point, the GFT of (7) remains exactly the same, except that the definitions of the TFs are extended. In particlar, H o i vy, iy= 0 w hich says that H is established by the doble nll triple injection (dnti) condition that e z and j z are mtally adjsted with respect to i so that v y and i y are both nlled. The definitions o f T and T n are each extended to a combination of voltage and crrent loop gai ns established by ndi conditions for T, and by dnti conditions for T n. These definitions are not displayed here becase the circit examples will be treated by se of the Intsoft ICAP/4 GFT Templates, in whic h all the calclations needed for dal test signal injection are inclded and are therefore transparent to the ser. Are dnti calclations even easier than ndi calclations? Absoltely: the more signals are nlled, the more circit elements do not appear in the answer. This constittes another rond of the Divide and Conqer approach: yo make a greater nmber of less complicated calclations. How to se a circit simlator incorporating GFT Templates In any case, if yo re going to se a circit simlator rather than doing symbolic analysis, all yo need to do is choose an injection point inside the loop at the error smming point, pls select the appropriate GFT Template according to whether single or dal test signal injection is reqired to nll simltaneosly the voltage and crrent error signals. The Template does simlation rns to calclate the second level TFs H, T, and T n in (15), and does post simlation calclations to prodce the discrepancy factors D and D n in (18). A simpl e feedback amplifier example A series shnt feedback amplifier circit model is shown in Fig. 7. The forward path is a simplified model of a typical IC in which voltage gain is achieved in the first two stages, which may be differential, and crrent gain is achieved in the final darlington follower stage. In this first example, the freqency response is determined by (19)

9 the sole capacitancec c. Each active device is represented by a simple BJT T model, which has the advantage of also representing an FET by setting the drain crrent eqal to the sorce crrent, as is done in this example. To apply the GFT, the crcial first step is to choose the test signal injection point that makes H, T, and T n have the desired interpretations of ideal closed loop gain, loop gain, and nll loop gain. The error voltage is the voltage between the inpt and the fed back voltage at the feedback divider tap point, lab eled v y in Fig. 7. The error crrent is the crrent drawn from the feedback divider tap point, labeled i y in Fig. 7. The test signals e z and j z are to be injected inside the loop so that vx and i x are the driving signals for the forward path, as shown in Fig. 8. Ths, the test signal co nfigration meets the two conditions that injection occrs at the error smming point, and is inside the loop, implementing the general model of Fig. 5. To invoke the GFT Template, the appropriate dal injection icon is selected and connected to provide the test signals e z and j z, as in Fig. 9. The icon also provides the system inpt signal e i and observes the otpt signal v o, becase it has to adjst the test signals relative to the inpt to establish varios nlls, one of which is the otpt signal. Another Principle of Design Oriented Analysis is Figre ot as mch as yo can abot the answer before yo plnge into the analysis. In this case, we expect H to be 1/ K, the reciprocal of the feedback ratio which was initially chosen to meet the system specification, becase the injection configration was specifically set p to achieve this. Her e, 1/ K = ( R1 R2) / R1 = 10 20dB, flat at all freqencies. We expect T to be large at low freqen cies and to have a single pole determined by C c. Conseqentl y, D will be flat at essentially 0dB at low freqencies, with a pole at the crossover freqency of T, beyond which D will be the same as T. We expect T n to be noninfinite, and conseqently D n to be not 0dB, becase there is nonzero reverse transmission throgh t he feedback path. That is, if the forward path throgh the active devices dies, the inpt signal e i can still reach the otpt vo by going throgh the feedback path in the wrong direction. The principal benefit of the GFT is to permit calclation of this effect, which is not acconted for in the conventional model, in order to determine whether or not it is significant. The GFT reslts for the three second level TFs for the model of Fig. 9 are shown in Fig. 10, and the expectations are indeed borne ot. The reslts are repeated in Fig. 11 with the loop gains replaced by their corresponding discrepancy factors, both of which are essentially 0dB at low freqencies. Also shown is the final reslt for the first level TF H, the closed loop gain, which from (18) is the direct sperposition of the H, D, and D n graphs. This final reslt shows that the bandwidth of H is determined by the T crossover freqency as in the conventional approach, and that reverse transmission

10 ( wrong way ) throgh the feedback path does not have any significant effect ntil the mch higher nll loop gain crossover freqency. A more realistic feedback amplifier model The more interesting model of Fig. 12 incldes two added capacitances for each active device. This is a mch more realistic model, and of corse library device models c an be sbstitted. What are or expectations for the reslts in comparison with those for Fig. 9? Since all the extra elements are capacitances, we expect the low freqency properties to remain the same, bt the dominant pole and hence the loop gain crossover freqency wold be lowered. Therefore, to enable a more meaningfl comparison between the two circits, Cc in Fig. 12 has been redced sfficiently to preserve the same loop gain crossover freqency. Nevertheless, the extra capacitances create more poles and zeros, so the high freqency loop gain is expected to be more complicated. The major conseqence of the presence of the extra capacitances is that there is now a second feedforward path (throgh a string of capacitances), in addition to that throgh the feedback path in the wrong direction, throgh which the inpt signal can reach the otpt; also, there is now nonzero reverse transmission throgh the forward path, which in trn creates a nonzero reverse loop gain. It is not necessary to separate these nonidealities, becase they are all atomatically acconted for in the calclation of the loop gain (sally little effect) and in the nll loop gain (the major effect). The qantitative reslts of the GFT Template simlations for Fig. 12 shown in Figs. 13 and 14 bear ot the expectations. The nll loop gain crossover freqency is drastically lowered, and even thogh the magnitde of the nll discrepancy factor Dn is 0dB at both ends of the freqency range, its phase ndergoes a hge lag that is transferred directly into a correspondingly hge phase lag in the final closed loop gain H. Althogh the magnitde of H at high freqencies may not be of mch interest in the freqency domain, its corresponding phase has a controlling effect in the time domain. The step responses of the circits of Fig. 9 and Fig. 12 are shown in Fig. 15. The hge phase lag of H at high freqencies for the circit of Fig. 12 cases the expected delay in the step response; however, the ensing rise time is shorter than for Fig. 9, with the perhaps nexpected beneficial reslt that the final vale is achieved sooner for Fig. 12 than for Fig. 9. The bottom line is that the GFT of (28), whose principal difference from (6) of the conventional approach is the presence of the nll discrepancy factor D n, predicts a sbstantial modification of the closed loop performance H. The nonidealities represented by T n or D n are always present in a realistic model of an electronic feedback system, and in at least some respects, can actally improve rather than degrade the performance. Th is rare exception to Mrphy s Law provides added incentive to tilize the GFT.

11 A method of finding loop gain from a voltage loop gain T v and a crrent loop gain T i calclated sccessively from single voltage and crrent injected test signals has been qite widely adopted since it was proposed in The formla is TT 1 or v i = T = (20) 1 T 1 Tv 1 Ti 2 Tv Ti However, this formla was based on the conventional block diagram of Fig. 1, and does not accont for nonzero reverse loop gain. Yo no longer need to measre loop gain directly In the conventional approach, efforts are often made to measre loop gain on the actal hardware in order to check that the phase margin is adeqate, althogh little or no thoght is given to whether or not the measrement is consistent with the actal closed loop gain. It is very awkward to inject even a single test signal into an IC, and dal or triple injection to set p ndi or dnti conditions is so brdensome that it is rarely attempted. Fortnately, it is no longer necessary to measre loop gain directly. The GFT of (18) is exact with respect to the simlated model, and all yo have to do is measre the final closed loop TF H, which yo wold do anyway to check whether it meets the specification. If the simlated H differs from the measred H, yo have to adjst the model ntil the two are the same. When this is achieved, yo know the model is correct and then the simlation tells yo what T and T n are individally. This is the clmination of the D OA process. More sefl techniqes for analog engineers The GFT is completely general, the only reqirement being that it applies to a linear system model. The symbol H has been sed here for the first level TF prposely to avoid the connotation of gain, becase it cold eqally well represent inpt or otpt impedance, power spply rejection, or indeed any transfer fnction of interest. It was mentioned above that the EET interpretation of the Dissection Theorem can be extended to any nmber of injected test signals, leading to the NEET Theorem. The GFT interpretation cold likewise be extended, in particlar to two pairs of injected voltage and crrent test signals, which wold open the door to analysis of feedback amplifiers containing both differential and common mode loops. Analog, mixed signal, and power spply design engineers are not the only ones to benefit from an ability to apply the powerfl methods of Design Oriented Analysis. Those who review and verify designs of others also need to know how design oriented reslts of analysis shold be presented. Therefore, managers, system integration engineers and reliability engineers who evalate the prodcts of other companies as well as their own, also can significantly increase their effectiveness by applying the methods of D OA. Then, if yo hold any of the above job titles, yo can contribte meaningflly to design review discssions, instead of jst saying to the presenting designer Well, it

12 looks as thogh it s coming along all right; carry on! In fact, yo can improve the effectiveness of the whole project by reqiring that design engineers present their reslts according to the Principles of Design Oriented Analysis. For frther information on the GFT, EET, and Design Oriented Analysis in general, see For frther information on the Intsoft ICAP/4 Circit Simlator, inclding a GFT Template User s Manal, see

13 i - A o = H i T = AK K Fig.1. The familiar single loop block diagram of a feedback system. The arrow head shapes imply that the signal goes only one way. i < > y x z injected test signal linear system o Fig.2. General model of a linear system with inpt i and otpt o, and an injected test signal z.

14 nlling amplifier i < y > infinite x gain z injected test signal o linear system Fig.3. How to set p a nll doble injection (ndi) condition: y is atomatically nlled. i TF1 TF2 TF4 TF4 TF3 TF3 TF 1 TF 2 = H o i TF2 1 TF1 Fig.4. This agmented block diagram with anonymos transfer fnctions represents the Dissection Theorem (7).

15 i y x < < Note boxes, o = H i < z not arrowheads Fig.5. When the test signal is injected inside the loop at the error smming point, the Dissection Theorem of (7) becomes the General Feedback Theorem (15). H 0 H 0 H T T n i T n H T o = H i T 1 H Fig.6. This block diagram for the General Feedback Theorem is a morphed version of Fig. 4, in which Tn (or H0 ) represents the main effects of the nonidealities.

16 V2 R3 200 rm2 200 i(v3) i(v4) i(v2) rm3 50 rm4 10 i Rs 100 V1 i(v1) rm1 50 Cc R8 20p 10meg V3 V4 o > R2 900 RL 1k error voltage error crrent R1 100 Fig.7. A simple feedback amplifier model, with the error voltage and the error crrent identified.

17 V2 R 200 rm 200 i(v3 i(v4 i(v2 rm 50 rm 10 R 100 V1 i x v < x W v y i(v1 rm 50 X > Y i y R 100 R 10me C 20p R 900 V3 V4 R 1 Fig.8. The crcial step: Choose the test signal injection point inside the loop so that v y is the error voltage and i y is the error crrent.

18 V2 R3 200 rm2 200 i(v3) i(v4) i(v2) rm3 50 rm4 10 i W Rs 100 V1 i x v < x v y i(v1) rm1 50 X i > Y i y Ui GFTv - Y Iy - - Ez Vy W Ei Vo Uo - Ix X Vx Jz - o xgft GFTv R1 100 Cc R8 20p 10meg R2 900 V3 V4 o RL 1k Fig.9. An Intsoft ICAP/4 GFT Template provides the injected test signals ez and j z, and performs the simlations and post processing reqired to obtain H, T, and T n, and hence H, for the GFT of (15).

19 db H T crossover 5.4MHz T T n crossover 7.9GHz 100m 10 1k 100k 10Meg 1G 100 freqency in hertz Fig.10. The expectations are borne ot: H is flat at 20dB, and T has a single pole. The nll loop gain T n is not infinite, becase of nonzero reverse transmission throgh the feedback path (a feedforward path). db 80.0 T n D H D n H = H DD n 100m 10 1k 100k 10Meg 1G 100 freqency in hertz Fig.11. T and T n replaced by their corresponding discrepancy factors D= T/(1 T) and Dn = (1 Tn) / T n. The normal closed loop gain H = H D D n differs from H not only becase of D, bt also becase of D n, althogh this effect is small.

20 R3 200 V2 C2d 16p rm2 200 C3t 5p i(v3) C4t 50p i(v4) i W Rs 100 i x v < x v y Ct1 5p C1d 30p V1 i(v1) rm1 50 X i > Y i y Ui GFTv - Y Iy - - Ez Vy W Ei Vo Uo - Ix X Vx Jz - o xgft GFTv R1 100 Ct2 5p i(v2) Cc R8 5p 10meg C3d 30p R2 900 V3 rm3 50 rm4 10 C4d 160p V4 o RL 1k Fig.12. A more realistic model than Fig. 7, with two capacitances added per active device (library models of corse cold be sbstitted).

21 db H T T crossover 5.4MHz Tn T n crossover 82MHz 100m 10 1k 100k 10Meg 1G freqency in hertz Fig.13. There is now an additional feedforward path, nonzero reverse transmission throgh the forward path, and nonzero reverse loop gain, casing the nll loop gain crossover freqency to be mch lower. deg db hdbinf, ddb, ddbn, hdb in nknown H D D n H H = H DD n 100m 10 1k 100k 10Meg 1G freqency in hertz Fig.14. The extra capacitances case the nll discrepancy factor D n to be drastically o different, reslting in a hge phase lag of 450 in the normal closed loop gain H.

22 m 200m -200m (1 1 / e) = 64% 50% 1/(2 π *5.4MHz) = 30ns 20.0n 60.0n 100n 140n 180n time in seconds Fig.15. For the circit of Fig. 12, the hge phase lag of H cases the expected delay in the step response; however, the ensing rise time is shorter, with the perhaps nexpected beneficial reslt that the final vale is achieved sooner. At least in some respects, nonidealities can actally improve performance!