210 Calle Solana, San Dimas, CA Tel. (909) ; Fax (909)

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1 1 Crcuts and Systems Exposton THE GFT: A GENERAL YET PRACTICAL FEEDBACK THEOREM R. Dad Mddlebrook 210 Calle Solana, San Dmas, CA Tel. (909) ; Fax (909) EMal: rdm@rdmddlebrook.com

2 2 Reew Verson Crcuts and Systems Exposton THE GFT: A GENERAL YET PRACTICAL FEEDBACK THEOREM R. Dad Mddlebrook Professor Emertus of Electrcal Engneerng Calforna Insttute of Technology

3 3 ABSTRACT Feedback systems are usually desgned wth the famlar sngle-loop block dagram n mnd, that s, the major loop feedback path s desgned to hae a transmsson functon such that the specfed gan s acheed n the presence of nonnfnte major loop gan. Varous nondealtes, such as unaodable mnor loops and drect forward transmsson, make the sngle-loop block dagram progressely less useful, especally at hgher frequences. A general feedback theorem (GFT) enables the major loop to be dentfed at all frequences, and ges a unque descrpton of the closed-loop gan n terms of three component transfer functons: the deal closed-loop gan G, the loop gan T, and the null loop gan T n. The loop gan and the null loop gan are each composed of a current return rato, a oltage return rato, and an nteracton term, all of whch can easly be calculated under certan null condtons that are establshed by test sgnal njecton. An alternate form presents the closed-loop gan as a weghted sum of G, the gan wth T =, and G 0, the gan wth T = 0. The GFT defnes a natural block dagram model that s dentcal n format to the sngle-loop model that s conentonally assumed, thus prodng a lnk between general feedback theory and a detaled crcut dagram analyzed n terms of factored pole-zero transfer functons. Dfferent test sgnal njecton ponts produce dfferent loop gans, and a prncpal loop gan s defned. The GFT s llustrated on a two-stage feedback amplfer hang arous nondealtes, ncludng loadng nteractons at all ponts, drect forward transmsson, and two mnor loops. The GFT s computer frendly. Once the null njecton condtons hae been programmed, a user unfamlar wth null njecton can obtan all the results on any crcut smulator program. The symbolc results for the two-stage amplfer example are erfed numercally wth use of an Intusoft ICAP/4 smulator.

4 4 There are many reews of the arous standard approaches to modelng and analyss of feedback systems, for example [1] and [2]. At one extreme, the bottom up approach reewed n Secton 1, an effort s made to presere the concept of a sngle loop represented as a block dagram descrbng a forward path and a feedback path, and the spotlght s on the loop gan. The model can be augmented to account for loadng effects and drect forward transmsson, but encounters dffcultes when nternal mnor feedback loops are present. At the other extreme, the top down approach, the system s treated as a whole, wth or wthout a flowgraph model, and the spotlght s on the return ratos of ts arous elements; a block dagram s one of the results rather than the startng pont. Modelng a feedback system as a combnaton of two two-port models seems to combne the worst features of both extreme methods. One of the four possble connectons has to be specfed at the outset, and three of them (those hang a seres connecton at one or both ends) are ncomplete models because of falure to nclude common-mode sgnal transmsson n the forward path. In the example crcut treated n Secton 4, t s found that ths common-mode feedforward makes a greater contrbuton to the drect forward transmsson than does the feedforward through the feedback path. In addton, use of two-port models effectely establshes a frewall between the analyss results and the physcal operaton of the system, whch makes the analyss almost useless for desgn. The feedback theorem deeloped n ths paper represents a top down approach that combnes the best features of the two extreme approaches, and ts greatest beneft s that ts consttuent transfer functons can be calculated, both symbolcally and by smulaton, from the crcut dagram and therefore hae drect physcal nterpretatons. The theorem s dered on a system bass n Secton 2 as a General Network Theorem (GNT) n terms of return ratos wthout menton of feedback, but the format s mmedately dentfable as descrbng a sngle-loop model wth drect forward transmsson accounted for separately, as n the natural block dagram model of Fg. 7. Snce ths block dagram s dentcal to that n Fg. 3 or 4 postulated as the startng pont of the bottom up approach, t s conenent to rename the GNT as the General Feedback Theorem (GFT), and the correspondng nomenclature s deeloped n Secton 3. Secton 4 treats a two-stage feedback amplfer crcut that contans all the nondealtes that ultmately paralyze the bottom up approach. The GFT s used n Secton 4.1 to fnd the oltage gan of the amplfer and, snce the GFT apples to any transfer functon of a lnear system, t s used n Secton 4.2 to fnd the output mpedance of the same amplfer. All results are erfed numercally by use of a crcut smulator. Some corollares and dfferent nterpretatons and perspectes of the GFT are dscussed n Secton 5, and conclusons are summarzed n Secton The Bottom Up Approach A desgner/analyst usually desgns a feedback system wth the famlar sngleloop block dagram of Fg. 1 n mnd, that s, the major loop feedback path s desgned to hae a transmsson functon such that the specfed gan s acheed n the presence of nonnfnte major loop gan. Fgure 1 s a drect representaton of the seromechansm type of applcaton, n whch A s arously the forward gan, the open-loop gan, or the gan wthout feedback, and K s the feedback dder rato, or the fracton of the

5 5 output that s fed back to the nput. The nput and output sgnals u, u o can be any combnaton of oltage, current, torque, elocty etc. whose rato s the transfer functon of nterest, the closed-loop gan G gen by G u o A = (1) u 1 + AK As crcut complexty ncreases, arous nondealtes emerge that are not accounted for n the smple model of Fg. 1. A common approach s to augment the model to account for certan nondealtes, make approxmatons, or, as a last resort, gnore them. Fgure 1, and the correspondng format (1), suggest that the A and K blocks should be desgned separately so that the resultng closed-loop gan G meets some specfcaton. The trouble begns rght here, because the two blocks n many cases are not really separable. That s, the feedback path loads the forward path at both nput and output, so that breakng the loop at ether pont upsets the loadng and ges wrong answers for A and/or K. Attempts may be made to connce oneself that a loadng s neglgble or, f not, to noke a dummy load to smulate a dsconnected block. An alternate s to elmnate the need to calculate A separately, by a smple rearrangement of (1): G 1 AK K 1 + AK = G T 1 + T = G D (2) where G 1 (3), K T AK (4) D T (5) 1 + T The loop no longer need be broken at the nput or output of the A block, but can be broken at some other pont where the loadng s absent. Such a pont s lkely to be found nsde the forward path wheren resde the acte deces whose outputs can be modeled, at least approxmately, by controlled generators. The loop gan T can be found by applyng a test sgnal at the new break pont, and obserng the sgnal that returns around the loop to the other sde of the break. Howeer, t s preferable, for seeral reasons, not to break the loop but to nject a test sgnal u z nto the closed-loop at that same pont, as shown n Fg. 2. The u s can be all oltages or all currents. The choce s not arbtrary: they must be currents f the output of the A 1 block s a controlled current generator, oltages f the output s a controlled oltage generator. In ether case, the loop gan T s u y / u x wth u = 0. There s another beneft n rearrangng (1) nto (2). The result for the closedloop gan G s expressed n terms of two transfer functons G and T, each of whch has nddual sgnfcance: G s the lmtng alue of G when the loop gan T becomes nfnte, and so can be nterpreted as the deal closed-loop gan; how large T s determnes how close the actual G s to G, by way of the dscrepancy factor D T/(1+T). Whereas T s a sngle-njecton transfer functon, n that u z s the only ndependent source present for ts calculaton, G s a null double-njecton transfer functon made under the condton of smulated nfnte T. Wth reference to Fg. 2,

6 6 u z s assumed adjusted relate to u so that u y s nulled, whch propagates back as a nulled error sgnal at the feedback summng pont. Thus, the output s dren by u z to a alue such that the fedback sgnal exactly matches the nput sgnal wth zero error, so that G s u o / u wth u y nulled. The oerall desgn of a feedback system can be lad out n three steps: Desgn Step #1. Desgn the feedback network so that G meets the specfcaton; Desgn Step #2. Desgn the loop gan so that T s large enough that the actual G meets the specfcaton wthn the allowed tolerances; Desgn Step #3. Snce n any real system T drops below unty beyond some crossoer frequency, desgn T to be large enough up to a suffcently hgh frequency. In an ac coupled system, a dual condton occurs n the reerse frequency drecton. Usually, #3 s the only dffcult step. Although not explctly stated, Desgn Step #3 ncludes the stablty requrement. The dscrepancy factor D 1 when T >> 1, and D T when T << 1; the transton between these two ranges, whch s not necessarly a monotonc decrease, occurs at the loop gan crossoer frequency w c, the frequency at whch T = 1. If the phase margn of T s less than about 60, D begns to peak up n the neghborhood of the crossoer frequency, and n the lmt of zero phase margn the peak n D becomes nfnte, whch marks the onset of nstablty. Thus, the dscrepancy factor D must not be allowed to peak up beyond the tolerance lmt of the specfcaton, whch s a much more strngent desgn requrement than mere stablty. Whle shftng the u z test sgnal njecton pont to a controlled generator alleates the loadng problem between the A and K blocks n Fg. 1, another nondealty requres an augmentaton of the model. The use of arrowheads rather than rectangles to represent the blocks n Fgs. 1 and 2 specfcally mples that the sgnal traels n only one drecton. Ths may be a good approxmaton for the forward path, but s usually not a good approxmaton for the feedback path, whch s normally a passe network hang comparable transmssons n the two drectons. Ths nondealty can be accommodated by addton of another block G 0 as n Fg. 3. By the basc sngle-loop propertes, an nterference sgnal njected nto the loop contrbutes an addte term to the output equal to the nterference dded by the feedback factor 1+T, so by dentfyng the nterference as the nput sgnal u njected through the G 0 block, (2) s augmented as follows: T G = G 1 + T + G 1 0 (6) 1 + T The drect forward transmsson block G 0 can be ealuated as another null double njecton transfer functon of the model of Fg. 3. Ths tme zero, nstead of nfnte, loop gan T s smulated by adjustment of u z relate to u such that u x s nulled; that s, u z soaks up u y leang no dre to the output. Hence, any output u o that appears must hae come through the G 0 block, and G 0 s u o / u wth u x nulled. The augmented feedback theorem of (6) ndcates that the closed-loop gan G can be consdered as the weghted sum of the gan G wth T nfnte and the gan G 0 wth T zero. The G contrbuton domnates when the loop gan T s large (below loop-gan crossoer), and the G 0 contrbuton domnates when T s small (beyond crossoer).

7 7 There s another erson of (6), n terms of a fourth transfer functon, dentfed as the null loop gan T n, that replaces G 0 : T G = G n (7) T where T n T G (8) G 0 The null loop gan T n can be ealuated as another null double njecton transfer functon of the model of Fg. 3, namely u y / u x wth u o nulled. The physcal nterpretaton of ths condton s that u z s adjusted relate to u such that u x dres the output to a alue equal to and opposte from the alue dren through the G 0 block; hence the recprocal relatonshp expressed by (8). The presence of the augmented term (1+1/T n ) n (7) adds Step #4 to the procedure for desgn of a feedback system: Desgn Step #4. Desgn the null loop gan so that T n s large enough that the actual G meets the specfcaton wthn the allowed tolerances. Een though the physcal nterpretaton of T n may be less obous than that of G 0, ts use may be preferable for two reasons. Frst, to determne the relate mportance of the augmented term n (7), T n need merely be compared to unty; to determne the relate mportance of the augmented term n (6), G 0 has to be compared wth G T. Second, drect determnaton of T n by the null double njecton condton s usually shorter and easer than that of G 0. The aboe short dscusson summarzes detaled treatments n [3] [5], the objecte of whch s to retan the sngle-loop model of Fg. 1 despte complcatons caused by arous nondealtes. There are stll hdden lmtatons, howeer. Test sgnal u z njecton must be at an deal controlled generator, defned as an deal njecton pont : f at a controlled current generator, the loop gan T and the null loop gan T n are ealuated as current return ratos; f at a controlled oltage generator, they are ealuated as oltage return ratos. Unfortunately, n real electronc systems deal controlled generators do not exst. Controlled generators occur n the outputs of models of acte deces, but there s always a nonzero admttance connectng the generator output back to ts controllng nput. Whle ths admttance may be neglgble at low frequences, ts capacte component, representng the dran-gate capactance of an FET or the collector-base capactance of a BJT, always ultmately shorts out the generator at suffcently hgh frequences. Consequently, calculaton of the u y / u x rato does not ge the rght answers for the loop gan and the null loop gan, and the results usually break down n the frequency range surroundng loop gan crossoer, just where accurate results are needed to ealuate phase margn. The last nondealty needed to be accommodated are these nonzero admttances, whch consttute n general local, or mnor, feedback loops wthn the major loop, as shown n Fg. 4. Although an effort was made n [4] to extend the calculaton of the loop gan T to allow for a nondeal njecton pont, the result was not complete.

8 8 It s the purpose of ths paper to oercome ths remanng lmtaton, and to prode a general feedback theorem that ges exact answers for the general model of Fg. 4. Interestngly, the two formats of (6) and (7) stll apply; the prncpal dfferences are that the four transfer functons G, T, T n, and G 0 requre extended defntons, and the last three each conssts of the sum of three terms. 2. A Top Down Approach: The General Network Theorem (GNT) The eoluton of the model as descrbed by Fgs. 1 through 4 ndcates that seeral nondealtes were accommodated by patchng up the orgnal smple model. In contrast, the deelopment of the General Network Theorem (GNT) begns wth a clean slate, smply the sngle block of Fg. 5(a), n whch arbtrary nodes W and XY are dentfed. The only constrant on the content of the block s that t s a lnear crcut model. The output u o and nput u desgnate any transfer functon of nterest, H = u o / u, where H replaces G to preclude any mplcaton that the transfer functon s lmted to a gan. The followng steps are to be executed: Analyss Step 1. In Fg. 5(a), separate node XY nto two nodes and nsert between nodes X, Y, and W an deal transformer, represented by a controlled current generator and a controlled oltage generator, together wth dentfed oltages and currents as shown n Fg. 5(b). Normal alues of the controlled generators are A = 1, A = 1 so that the deal transformer s transparent and ts presence does not affect the normal system transfer functon H. Analyss Step 2. Desgnate reference alues of the two controlled generators as A = 0, A = 0, whch causes H to change to a reference alue H ref. From Fg. 5(b), zero alues of the controlled generators are equalent to both y and y beng nulled. Thus, H ref u o u A =0 A =0 = u o u y =0 H y y (9) y =0 where, to smplfy the notaton, a sgnal that s nulled s shown as a superscrpt upon the quantty beng calculated. Analyss Step 3. Renstate the controlled generators A and A a the Extra Element Theorem, and set them equal to unty. Null double njecton, the Extra Element Theorem (EET), the Two Extra Element Theorem (2EET), and the N Extra Element Theorem (NEET) are extensely dscussed n [6] [9]. The theorem correcton factor has a numerator and a denomnator, each of whch has the same form. If an extra element s an mpedance Z, each contans the rato of Z to a certan drng pont mpedance seen by Z. If the extra element s a controlled generator, such as A n Fg. 5(b), the denomnator contans the rato of A to the negate of a return rato seen by A, ealuated wth the nput u set to zero; the numerator contans the rato A to the negate of a return rato seen by A ealuated wth the output u o nulled. The controlled generators A and A can be renstated sequentally wth use of the (sngle) EET. If A s renstated frst,

9 H y = H y y 1 + A T y n 1 + A T y 9 The notaton s that T represents a oltage return rato, and subscrpt n ndcates that the rato s calculated wth the output u o nulled, rather than wth the nput u set to zero. The superscrpt y ndcates that all four parameters are ealuated wth y nulled because A s stll at ts reference alue zero. The controlled generator A can now be renstated by a second applcaton of the EET correcton factor to the reference alue H y : 1 + A H = H y T n 1 + A (11) T n whch T represents a current return rato. Snce T s to be ealuated wth A already renstated, t can be expressed n terms of ts alue T y wth A =0 by a nested applcaton of the EET: A 1 + T = T y T y 1 + A (12) T x n whch T x s the return rato for A wth the nput x of the transfer functon (10) T set to zero, and T y functon T nulled. s the return rato for A wth the output y of the transfer In the same way, T n n (11) can be expressed n terms of ts alue T y n : 1 + A T n = T y T y n n 1 + A T x n Equatons (10), (12), and (13) can be substtuted nto (11) to ge H = H y y 1 + A + A + A A T y n T y n T y n T x n 1 + A + A + A A T y T y T y T x whch s the result for H n terms solely of parameters ealuated for both A and A zero, and could hae been obtaned drectly from the 2EET. The aboe deraton could hae been done wth A renstated before A, n whch case dual ersons of (10) through (14) would hae been obtaned. Snce these dual ersons of (14) must be the same, t follows that the dual ersons of the two (13) (14)

10 10 product terms n (14) must be equal, whch allows dual defntons of the product terms: T T x T y = T y T x (15) T n T x n T y n = T y n T x n (16) It remans only to restore the controlled generators to ther normal alues A = 1, A = 1 to ge H = H y T y y n T y T n n (17) T y T y T Ths result for the frst-leel transfer functon H can be wrtten T H = H n (18) T whch s n the same format as (7), but wth altered and extended defntons of the component second-leel transfer functons H, T, and T n : H H y y (19) (20) T T y T y T (21) T n T y n T y T n n Equaton (18), together wth the extended defntons of (19) through (21), may be desgnated as a General Network Theorem (GNT), snce t apples to any network subject only to the constrant of lnearty. All the parameters of the GNT n (15) through (17), desgnated thrd-leel transfer functons, can be ealuated by approprate adjustment of the nput sgnal u and two test sgnals j z and e z njected n place of the deal transformer of Fg. 5(b), as shown n Fg. 6. For example, H y y s gen by (9). The lstng of the return ratos s: j z and e z njected: T y = y x y =0 u =0 T x = y x x =0 u =0 (22) T y = y x y =0 u =0 (23) T x = y x x =0 u =0 (26) (27) T y n = y y =0 x u o =0 (24) T y n = y x y =0 u o =0 (28) T x n = y x =0 (25) T x n = y x u o =0 x =0 x u o =0 The normal alue A = 1 may also be substtuted nto (11) to ge (29)

11 where H = H y T n (30) H = H y T H y = H y T y y n T y T = T y T y T x T n T (31) H y = H y T y y n T y (32) T = T y T y T x T n = T y T y n n (33) T n = T y T y n n (37) T x n T x n Howeer, (31) to (33) and (35) to (37) are not needed, because the second leel transfer functons n (30) and (34) can be found drectly by njecton of only one test sgnal: j z njected: e z njected: (34) (35) (36) H y = u o u y =0 T = y x u =0 (38) H y = u o u y =0 (39) T = y x u =0 (41) (42) T n = y x uo =0 (40) T n = y x uo =0 Equatons (30) and (34) for a sngle njected test sgnal are also n the same format as (18), but wth dfferent defntons of the component second-leel transfer functons H, T, and T n than for the dual test sgnal njecton case of (19) through (21). The aboe results for the GNT requre some nterpretaton and comment. Any transfer functon of a lnear network can be dssected nto two successe component leels. Equaton (18) shows that the frst-leel H can be expressed as a combnaton of a reference transfer functon H and two other transfer functons T and T n that can prosonally be called respectely the total return rato and the total null return rato, all three of whch are second-leel transfer functons ealuated pursuant to selecton of an arbtrary current and/or oltage test sgnal njecton pont. Equatons (20) and (21) show that the second-leel total return ratos can each be consdered as the trple parallel combnaton of a current rato, a oltage rato, and a product term defned by (15) and (16), all of whch are thrd-leel quanttes. (43)

12 12 The symmetry between the entre expressons (20) and (21) for T and T n may also be noted: all the return ratos comprsng T are ealuated wth the nput sgnal u set to zero; all those comprsng T n are ealuated wth the output u o nulled. Most mportant, wth respect to the thrd-leel transfer functons (9), and (22) through (29), s that all of them are to be ealuated wth at least one quantty nulled. Null njecton, dscussed n detal n [6], s a concept whose usefulness and smplcty may not be famlar, and so the meanng and sgnfcance wll be reewed here. The four T s of (22), (23) and (26), (27) all demand null double njecton calculatons upon the model of Fg. 6. Snce all four are for u = 0, j z and e z are the two njected sgnals. Snce the model s lnear, any sgnal n the system s the lnear sum of separate contrbutons from j z and e z, and a unque rato exsts between j z and e z whch causes any selected dependent sgnal n the system to be nulled. Snce all other sgnals are also the lnear sum of contrbutons from the two njected test sgnals, the unque rato that nulls a selected sgnal can be noked to fnd the rato of any two other sgnals. In prncple, ths procedure can be employed to calculate the four T s. In practce, howeer, t s not necessary to ealuate the unque rato n order nsert t n the two contrbutons to any other sgnal. Instead, the mere exstence of the null consttutes equalent nformaton that can be used drectly to ealuate any sgnal n terms of another. For example, to calculate T y for the hdden crcut of Fg. 6, start wth x and multply t by successe factors as the sgnal progresses through the crcut, makng use of the nulled y (and the zero u ) where necessary along the way. Eentually the sgnal shows up as y n the form of a multple of x, and ths multple s the desred T y. Note that the alue of nether test sgnal, nor ther rato, has been needed n ths process. Ths short cut s the key to the smplcty and ease of use of null double njecton. It s no accdent that a null double njecton calculaton s smpler and easer than the same calculaton under sngle njecton: the null propagates, so that f one sgnal s null, other sgnals may also be nulled; any crcut element that supports a nulled oltage or current s absent from the fnal result. A logcal extenson suggests that f three ndependent sgnals are present, two other sgnals can be nulled and the double-null trple njecton calculaton should be een smpler and easer, whch s n fact the case. The four T n s and the reference transfer functon H y y all demand double-null trple njecton condtons. The nput u and the njected test sgnals j z and z are all present, and for H y y, for example, y and y are to be nulled. Agan the short-cut s to start wth u and multply t by successe factors as the sgnal progresses through the crcut, makng use of the nulled y and y where necessary along the way. Eentually the sgnal shows up as u o n the form of a multple of u, and ths multple s the desred H y y. Snce the GNT (18) has the same form as (7), t also can be expressed n the form correspondng to (6): H = H T 1 + T + H T (44)

13 13 where H 0 s an alternate second-leel transfer functon related to T n by the redundancy relaton H 0 = T (45) H T n Snce 1/T n has three components, H 0 can lkewse be expressed as the sum of three components. Howeer, for some frst-leel transfer functons, H = 0 and T n = 0 so H 0 cannot be found from (45). Instead, the three components of H 0 can be ealuated ndependently n terms of three thrd-leel calculatons upon the model of Fg. 6, as follows. As dscussed n [7], there are four ersons of the 2EET, correspondng to the four combnatons of reference alues of the two extra elements, ether of whch may be chosen as zero or nfnte. Lkewse, there are three other ersons of (17), correspondng to the three other combnatons of reference alues of A and A. The combnaton chosen for (17) s reference alues A = 0, A = 0, whch s equalent to y = 0, y = 0, for whch the reference alue of H s H y y. The other three reference alues of H can be found by extracton of one term n the numerator and the correspondng term n the denomnator from (17), and combnng these two terms wth H y y : y H y x = T H y y (46) T y n H y x = T y H y y (47) T y n H x x = T H y y (48) T n The correcton factor belongng to each reference alue of H also emerges from ths process, but these are not needed here. Fnally, substtuton nto (45) of the three components of 1/T n from (21) and replacement of H from (46) through (48) leads to H 0 = H y x T + H y x T + H x T x (49) T y T y T The three reference alues of H defned by (46) through (48) are all double-null trple njecton transfer functons defned smlarly to H y y n (9): H y x u o u y =0 x =0 (50) H y x u o u y =0 x =0 (51) H x x u o u x =0 x =0 For the sngle EET, there are only two ersons correspondng to the two combnatons of reference alues of the extra element. In (30) the reference alue s A =0, whch defnes H y. By extracton of T n from the numerator and T from the denomnator of (30), the other reference alue H x for A = s H x = T T n H y (52) (53)

14 where 14 H x = u o (54) u x =0 Ths s the redundancy relaton (45) for the sngle EET, and so for sngle current njecton where H 0 = H x (55) A smlar treatment for (34), the sngle EET for the extra element A, leads to H x = T T n H y H x = u o (57) u x =0 and so for sngle oltage njecton H 0 = H x (58) Thus, accordng to whether dual or sngle test sgnal njecton s employed, H 0 n (44) can be found from (49), (55), or (58). Each reference alue of H n H 0 s a thrd-leel transfer functon, and may be calculated from Fg. 6 by the short-cut method. In summary, njecton of a test current and/or a test oltage at an arbtrary pont n a lnear crcut model as n Fg. 6 permts calculaton of a number of sngle njecton, null double njecton, and double null trple njecton transfer functons. These thrd-leel quanttes can be assembled to ge the second-leel transfer functons H, T, T n, H 0 whch n turn can be combned n (18)or (44) to ge the desred result, an oerall frst-leel transfer functon H n the absence of the njected test sgnals. Ths dde and conquer approach has seeral adantages oer the conentonal drect calculaton of H. Most mportant, the sngle hgh entropy expresson for H s replaced by seeral other expressons that can more readly be dered n low entropy form. A low entropy expresson [10] s one n whch element symbols are grouped so as to reeal ther orgn n the crcut model, and also so that ther relate contrbuton to the result s exposed by substtuton of numercal alues. For example, mpedances are grouped as ratos and seres-parallel combnatons. The short-cut method of calculatng the thrd-leel null njecton transfer functons s well-suted to buld low-entropy expressons drectly. Because the total return rato T and the total null return rato T n are each parallel combnatons of the thrd-leel quanttes, the low entropy format s retaned n the second-leel transfer functons T and T n. The greatest adantage s that substtuton of the second-leel quanttes nto the general theorem (18) or (44) need not actually be carred out: there s no beneft n throwng away the extra useful nformaton contaned n low entropy expressons by multplyng out (18) or (44) nto an amorphous rato of sums of products of the element alues. For desgn-orented analyss, the useful nformaton s contaned n H, T, T n, H 0 : these are the functons that hae to be desgned, accordng to Desgn Steps #1 through #4 descrbed aboe. (56)

15 15 It should be emphaszed that the GNT of (18) or (44) s completely general; the only constrant s that t apples to a lnear system model. The transfer functon H can be a oltage gan, current gan, transmmttance, or self-mmttance. An aspect of the generalty s that any njecton pont can be chosen for the test current and oltage. Dfferent sets of second-leel transfer functons H, T, T n, H 0 result from dfferent njecton ponts, although f any of these sets s nserted nto the general theorem, the same result for H s of course obtaned. As mentoned aboe, ths nserton s not needed, because H, T, T n, H 0 are themseles the desgn-orented results that are of nterest. The format of the GNT exposes the total return rato T and the total null return rato T n of a fcttous deal transformer nserted at an arbtrary pont n the crcut, n whch each s a parallel combnaton of a current return rato of the transformer controlled current generator, a oltage return rato of the transformer controlled oltage generator, and an nteracton term accordng to (15) and (16). The format arses as a form of the EET n whch the two transformer controlled generators are dentfed as extra elements. As already seen, only one such transformer controlled generator need be so dentfed, whch leads to a format n whch T and T n represent only a sngle current or a sngle oltage return rato, as approprate. Ths procedure can be extended n prncple to any number of nserted deal transformers, whch results n a GNT dered from the NEET n terms of total return ratos hang addtonal contrbutng return ratos. Two such transformers mght ge a GNT useful n descrbng a crcut hang both dfferental and common-mode feedback loops. A specal case of the two-transformer model dentfes the current return rato of one transformer and the oltage return rato of the other as the two extra elements, whch means that the GNT of (18) or (44) s unaltered een f separate njecton ponts are used for the test sgnals j z and e z. Another specal case dentfes ether the two controlled current generators or the two controlled oltage generators as extra elements, meanng that two test currents or two test oltages could be njected at dfferent ponts n the crcut. The GNT of (18) or (44) would stll be unaltered, except that the subscrpt notaton would hae to be changed from and to 1 and 2 or 1 and 2. More than one fcttous deal transformer wll not be further consdered n ths paper, and the dscusson wll be lmted to the test sgnal njecton model of Fg. 5(b), ncludng the specal cases n whch one of the test sgnals s zero. The three currents x, y, j z, and the three oltages x y, e z each form a phasor trangle. The term njecton pont needs to be more specfcally defned. Actually, snce the calculatons are to be done only wth respect to the njected test sgnals j z and/or e z, what s needed s an dentfcaton of the three nodes W, X, and Y n Fg. 6, n whch W s the return node for j z and the reference node for x and y. Such a group of nodes WXY wll be descrbed as a test sgnal njecton pont, or smply an njecton pont. A test sgnal njecton pont WXY s a dual njecton pont, a sngle current njecton pont, or a sngle oltage njecton pont. It was mentoned at the end of the deraton that the GNT has the same format as the expresson that descrbes the block dagram model of Fg. 3 or 4. Consequently, the same model can be shown wth the blocks labeled n terms of a partcular set of second-leel transfer functons, as shown n Fg. 7.

16 16 The sequence of steps n arrng at the model of Fg. 7 exposes the benefts of the GNT. In the conentonal approach, as reewed n Secton 1, the block dagram of Fg. 1 or 2 s assumed, and the gan expressons for the nddual blocks are guessed n terms of the crcut structure. Ths s easy when the blocks are clearly separable, but becomes progressely more uncertan n the presence of nondealtes. In contrast, wth the GNT, the expressons for the nddual blocks are part of the answer, and are unambguous; there s no guesswork or uncertanty. Smplfcatons are optonal n the symbolc analyss and not needed n the numercal analyss, but n any case are made wth full knowledge of the degree of approxmaton, not forced n order to dentfy the blocks. 3. The General Feedback Theorem (GFT) The block dagram of Fg. 7 models any transfer functon of a crcut dagram regardless of any menton of feedback, and can be consdered a natural model. Howeer, because the block dagram s the same form as n Fg. 3 or 4, t s partcularly suted to represent a system n whch there s an dentfable feedback loop, n whch case the GNT s more usefully called a General Feedback Theorem (GFT). Although the equatons are the same, some nterpretatons are more llumnatng when slanted towards conentonal sngle-loop feedback termnology. For example, n Fg. 7 t s obous that T s the loop gan and H s the deal closed-loop gan as n Fg. 3 or 4. Thus, the second-leel transfer functon T, and ts consttuent thrd-leel transfer functons, can be consdered as ether return ratos or loop gans and, by extenson, the same apples to T n. A more detaled justfcaton for ths dual nomenclature s the followng. The GFT s a specal case of the EET n whch the extra elements are the controlled current and/or oltage generators of one or more fcttous deal transformers nserted nto the system crcut model. The thrd-leel transfer functons, the T s and T n s, are return ratos of the controlled generators but, snce the normal alues of the controlled generators are unty, the T s and T n s can also be dentfed as loop gans, and are calculated by test sgnal njecton nto the system crcut model (wthout the fcttous deal transformers). It s conenent at ths pont to summarze the general network theorem results nterpreted as a General Feedback Theorem (GFT). A partcular test sgnal njecton pont produces a mutually consstent set of four second-leel transfer functons H, T, T n, H 0 whch comprse the frst-leel transfer functon H. Owng to the redundancy relaton H 0 / H = T /T n of (45), the GFT can be expressed n terms of three out of the four second-leel transfer functons: T H = H n T (59) H = H 1 + T + H 1 0 (60) 1 + T T n whch T s the loop gan (total return rato), T n s the null loop gan (total null return rato), H s the transfer functon n the lmt T =, and H 0 s the transfer functon n the lmt T = 0. In (59), G 0 s absent, and n (60) T n s absent; the other two ersons, absent H and T respectely, are not shown. For nterpretaton of the GFT, t s conenent to defne two weghtng factors so that (59) and (60) can be wrtten

17 17 H = H DD n (61) H = H D + H 0 (1 D) (62) where D T (63) D n 1 + T n (64) 1 + T T n are respectely the dscrepancy factor and the null dscrepancy factor. The term dscrepancy s chosen because D represents the consequences of nonnfnte loop gan, and D n represents all the nondealtes, that cause the actual closed-loop gan H to dffer from the deal closed-loop gan H. In (61), the nondealtes are accounted for a D n as a multpler upon the basc sngle-loop H D, and n (62) they are accounted for a H 0 as an addte term to H D. For dual test sgnal njecton of j z and e z, these second-leel transfer functons are gen by (19) through (21): Dual njecton j z and e z : H H y y (65) T T y T y T (66) T n T y n T y T n n (67) where the product terms T and T n are gen by (15) and (16): T T x T y = T y T x (68) T n T x n T y n = T y n T x n (69) The thrd-leel transfer functons are the consttuents of the second-leel transfer functons, and are calculated from the crcut model accordng to (9) and (22) through (29). For sngle test sgnal njecton of j z or e z, the second-leel transfer functons are obtaned from (30) and (34) and are respectely equal to the thrd-leel transfer functons calculated from the crcut model accordng to (38) through (43): Current njecton j z : Voltage njecton e z : H = H y (70) H = H y (73) T = T (71) T = T (74) T n = T n (72) T n = T n (75) Any njecton pont, wth ether sngle or dual test sgnal njecton, ges a mutually consstent set of second-leel transfer functons that, when nserted nto the GFT of any of the four ersons (59) through (62), produces the same result for the closed-loop transfer functon H. A desgner has the opton to choose an njecton pont so that the second-leel transfer functons defne blocks n the natural model of Fg. 7 that best match the actual physcal crcut. Ths ges the desgner maxmum nsght nto the crcut operaton and hence ges the greatest control oer performance optmzaton. There are two crtera to gude a desgner s choce of njecton pont. Frst, the whole sgnal n the forward path (n ether drecton), and none of the sgnal n the feedback path (n ether drecton), should pass through the test sgnal njecton pont. That s, the njecton pont should be nsde the major loop,

18 18 the loop that s purposely constructed by the desgner to achee certan results, yet outsde any mnor loops that may be present ether parastcally or on purpose to shape the major loop gan. Specfcally, ths means that n Fg. 5(a) sgnal transmsson n the forward path s klled f branch XY s opened, or f node W s shorted to nodes X and Y. Second, H should be equal to the desred deal closed-loop gan 1/ K, the recprocal of the feedback path transmsson from the output back to the nput. Satsfacton of these two crtera not only makes the oerall block dagram model of Fg. 7 match that of Fg. 3 or 4, but t also makes the nddual blocks match as well. Consequently, the second-leel transfer functons H, T, T n, H 0 hae the most useful desgn-orented nterpretatons n terms of the crcut model. The set of second-leel transfer functons that satsfy both crtera may be called the prncpal set, whch ncludes the prncpal loop gan T. The njecton pont that produces the prncpal set descrbes not only the places where the test sgnals are njected, but also whether dual or sngle test sgnals are needed, accordng to whether H y y, H y, or H y produces the desred H =1/ K. Of course, ths s not to suggest that other sets of second-leel transfer functons are wrong n some way they all ge the same frst-leel closed-loop gan H t s just that they are more dstant from the physcal crcut operaton, and are therefore less useful for desgn purposes. The crucal choce of test sgnal njecton pont s best llustrated by a crcut example, as n the followng secton. 4. Example: Two-Stage NonInertng Feedback Amplfer An ac small-sgnal model of an elementary two-stage BJT feedback amplfer s shown n Fg. 8, n whch the power supply lne becomes ac ground. Each BJT s represented by a core model of two elements, a controlled generator specfyng the collector current to be α tmes the emtter current, and an emtter resstance r m = α / g m, where g m s the transconductance that represents the dece gan property. Subscrpts 1 and 2 dentfy the parameters of the two transstors. Augmentng each transstor core model s a collector-base capactance C that represents the unaodable transton-layer capactance. A beneft of the core model descrbed aboe [11] s that t apples equally well to an FET. Apart from the obous change n termnology, the only substante change s n fact a smplfcaton: for an FET, α = 1, so that the base (now gate) current becomes zero. To take adantage of ths smplfcaton, both transstors wll be taken to be FETs throughout ths example, so r m s the recprocal transconductance and C s the dran-gate capactance. The GFT s used n Secton 4.1 to dere the oltage gan G, wth H G, and n Secton 4.2 to dere the output mpedance Z o, wth H Z o Voltage Gan G Some ntal desgn choces are already mplct n Fg. 8. Qualtately, the two FETs, both n common-source confguraton, consttute the forward path; negate feedback s taken from the output oltage o a the dder K = R 1 /(R 1 + R 2 ) and appled n seres opposton wth the nput oltage e. Consequently, the crcut approxmates a oltage-to-oltage amplfer wth hgh nput mpedance and low output mpedance, hang a desred oltage gan of

19 19 1/K = (R 1 + R 2 )/R 1 = 20dB, accordng to Desgn Step #1 of Secton 1. Hence, the frstleel transfer functon H s now to be dentfed as the oltage gan G o / e. The model of Fg. 8, smple though t appears, exhbts all the nondealtes preously dscussed: mutual loadng between the forward gan and feedback blocks; drect forward transmsson; and two mnor feedback loops created by the nonzero admttances across the controlled current generators. The GFT s able to ncorporate all these nondealtes, although the results for the second-leel quanttes depend upon the locaton of the njecton pont for the test sgnals j z and/or e z, and choce of the test sgnal njecton pont s crtcal to the usefulness of the results for desgn purposes. The preferred choce s an njecton pont that meets the two crtera descrbed n the preous secton, and produces the prncpal set of second-leel transfer functons. If the feedback loop s defned to be the major loop nstalled by the desgner, the njecton pont should be nsde the forward path but outsde any mnor loops, and such that all the sgnal n the forward path goes through that one pont. Thus, the only sgnal that does not go through that pont goes through the major loop feedback path. Ponts W 1 X 1 Y 1, W 1 X 2 Y 2, and W 1 X 3 Y 3 n Fg. 8 meet ths requrement. Also, the deal closed-loop gan G should be equal to the recprocal feedback dder rato 1/K=(R 1 + R 2 )/R 1, whch n turn means that both the error oltage between W 2 and Y 3 and the error current at the R 1, R 2 dder pont must be zero. Therefore, the error current and error oltage must respectely be dentfed as y and y n order to make G y y = G equal to (R 1 + R 2 )/R 1. Hence, both current and oltage test sgnals j z and e z must be njected at the X 3 Y 3 branch wth reference node W 2, as shown n Fg. 9. Snce the thrd-leel loop gans are all calculated wth e =0, t doesn t matter whether the reference node s W 1 or W 2. Therefore dual njecton of both j z and e z at W 2 X 3 Y 3 meets both crtera and produces the prncpal set of G, T, T n, G 0 n whch T s the prncpal loop gan. The consequences of other njecton pont choces wll be consdered later. Fgure 9 also shows some other modfcatons from Fg. 8: the two capactances are generalzed to mpedances, the dece α s are taken to be unty wth resultng transstor nput currents zero (f FETs) or gnored (f BJTs); and the source resstance R s s taken to be zero. The last two greatly smplfy the algebra of the GFT calculatons wthout compromsng the prncples llustrated n the example. The consequences of nonzero R s wll also be brefly consdered later. An approprate form of the GFT s therefore (59) or (60) and (65) wth G replacng H : T G = G n T (76) G = G 1 + T + G 1 0 (77) 1 + T T G = G y y (78) The scenaro for mplementaton of the GFT s as follows: Determne the thrd-leel quanttes from the crcut, and calculate the second-leel quanttes from (66) and (67). These can be used drectly for desgn purposes, or to calculate the frst-leel quantty G, the transfer functon of nterest. The mplementaton can be

20 20 realzed by symbolc analyss, by a computer crcut smulator (preferably both), or (usually wth consderably greater dffculty) by measurement on the physcal crcut. For the example of Fg. 9, symbolc analyss wll be done, but frst the results for the second-leel quanttes from a crcut smulator wll be presented and assembled nto the GFT to arre at the frst-leel oltage gan transfer functon G, whch s then compared wth the result by drect smulaton. All of the thrd-leel quanttes are multple null njecton calculatons. Smulators are not set up to make such calculatons, but ths lmtaton can easly be oercome by adaptng, not the smulator algorthm, but the crcut under test. For example, f two sgnals are to be nulled by mutual adjustment of three ndependent sources, two sources are replaced by controlled generators each dren by one of the sgnals to be nulled, and assgned ery hgh gans. Thus, there s one ndependent source, and the two nulls are automatcally created by self-adjustment of the two controlled generators; the smulator then calculates the requred transfer functon n the usual way. Intusoft s ICAP/4 IsSpce crcut smulator has a specal GFT subprogram n whch the user has only to specfy the njecton pont and the type of test sgnal njecton (current, oltage, or both), and the frequency responses of all the thrd-, second-, and frst leel transfer functons are made aalable. Partal results for the crcut of Fg. 9, prnted out by ICAP/4 s IntuScope, are shown n Fg. 10. The fnal closed-loop gan G accordng to the GFT, calculated from (76), s compared wth that obtaned by drect smulaton of o / e. It s seen that the result from the GFT s ndstngushable from that obtaned by drect smulaton. Useful nsght s acheed by examnaton of the second-leel quanttes that consttute the GFT accordng to (76). The loop gan T s large at low frequences but falls off and crosses 0dB at about 32MHz. The deal closed-loop gan G has the desgned flat alue 20dB, and the closed-loop gan G follows at low frequences but falls below G beyond the loop gan crossoer frequency. In Fg. 10, the closed-loop gan G actually rses before fallng, exhbtng almost a 6dB peak. Vewed n solaton, as t would be f the GFT were not noked, ths peakng mght be attrbuted to a low phase margn of about 30 wth consequent step response hang sgnfcant oershoot and rngng. In fact, ths does not happen, and the actual step response s ery dfferent. As can be seen from Fg. 10, loop gan crossoer occurs well nsde seeral decades of approxmately -20dB/dec slope, ndcatng a phase margn of about 90. The culprt s T n, whch crosses 0dB before T does. One of the prncpal benefts of the GFT s the nsght afforded nto how the second-leel transfer functons, such as T n, are determned by the crcut elements, so that any correcte acton can be taken. All the thrd-leel quanttes to be dered by analyss of the model of Fg. 9 are multple null njecton calculatons for whch, as already mentoned, the shortcut method of followng the sgnal from nput to output s partcularly approprate. The form of result desred for each transfer functon s a reference alue modfed by pole and zero factors, the factored pole-zero format. Especally mportant s that the reference alue and the poles and zeros be expressed n low entropy forms, that s, the element alues should appear n ratos and seres-parallel combnatons. Ths s so that each result can be traced back to ts orgn n partcular crcut elements and can be approprately modfed n an nformed way as part of the desgn process. Ths s the essence of desgn-orented analyss, the only knd of analyss worth dong [10].

21 21 The arous thrd-leel transfer functons for the model of Fg. 9 wll now be dered. Snce seeral of them nole the current transfer functon M / x, n the long run work s saed by calculaton of ths transfer functon n adance. Furthermore, snce M n turn noles the only two complex mpedances n the model, a conenent analyss tool s the 2EET n whch the two mpedances are renstated by a correcton factor upon a reference model n whch the two mpedances are absent. Ths takes adantage of the aluable specal case when all the reactances n the crcut model are desgnated as extra elements, because then the transfer functon reference alue s a constant, all the drng pont mpedances are resstances, and the entre frequency response emerges automatcally as a rato of polynomals n complex frequency s, whch can then be factored nto the poles and zeros. The releant part of Fg. 9 s shown n Fg. 11, n whch the reference alues of the two mpedances are nfnte. The load resstance R of the second stage ultmately wll be dentfed as ether R 2, R 1 + R 2, or 0 accordng to whch thrd-leel transfer functon s beng calculated. The correspondng erson of the 2EET s 1 + R n1 + R n2 R + K n1 R n2 n Z M M 1 Z 2 Z 1 Z 2 Z1,Z 2 = 1 + R d1 + R d2 R + K d1 R d2 d Z 1 Z 2 Z 1 Z 2 where the nteracton parameters K d and K n are gen by (79) K d R d1 Z 2 =0 R d1 = R d2 Z 1 =0 R d2 (80) R n1 R Z K n 2 =0 n2 Z = 1 =0 (81) R n1 R n2 From Fg. 11, t s seen mmedately that the reference alue of M s M Z1,Z 2 = = R 3 (82) r m2 To fnd the eght drng pont resstances, the procedure s to redraw Fg. 11 wth the approprate quanttes to be nulled explctly marked to zero. Snce there are many such procedures to be accomplshed, too much space would be needed to dsplay all the redrawn ersons of Fg. 11. Instead, Fg. 11 can be consdered a template from whch copes can be made to be marked up wth the approprate sgnal alues. R d1 Redraw Fg. 11 wth a test source (current or oltage) appled between termnals 1,1, and wth the nput sgnal x (denomnator of M) set to zero: call ths Fg The drng pont resstance seen by the test source s R d1 whch, because x =0 and the nput current to the second stage s zero, s R d1 = R 3 (83) R d1 Z2 =0 Redraw Fg wth termnals 2,2 short nstead of open: call ths Fg The drng pont resstance seen by the test source s : R n1 R d1 Z 2 =0 = r m2 RR 3 (84)

22 22 Redraw Fg. 11 wth a test source appled between termnals 1,1, and wth the output sgnal (numerator of M) nulled (n the presence of x ): call ths Fg Snce =0, 2 s also zero, and there s zero oltage across the test source. Therefore R n1 = 0 (85) R n1 Z2 =0 Redraw Fg wth termnals 2,2 short nstead of open: call ths Fg Snce there was no oltage across termnals 2,2 under the condtons of Fg. 11.3, the result s the same whether termnals 2,2 are open or short, hence R n1 Z 2 =0 = 0 (86) R d2 Redraw Fg. 11 wth a test current t appled between termnals 2,2, and wth the nput sgnal x set to zero: call ths Fg The test current t also flows through R 3, so 2 = ( R 3 / r m2 ) t and = 2 + t. The drng pont resstance R d2 s the sum of the oltages across R 3 and R dded by t, so R d2 = R R 3 r m2 R = RR 3 (87) r m2 RR 3 R d2 Z1 =0 Redraw Fg wth termnals 1,1 short nstead of open: call ths Fg Snce there s zero oltage across r m2, 2 =0 and R d2 Z 1 =0 = R R n2 Redraw Fg. 11 wth a test source appled between termnals 2,2, and wth the output sgnal (numerator of M) nulled (n the presence of x ): call ths Fg Snce =0, 2 flows through the test source and the oltage across t s r m2 2, so R n2 = r m2 (89) R n2 Z1 =0 Redraw Fg wth termnals 1,1 short nstead of open: call ths Fg Condtons are the same as for R n2 regardless of whether termnals 1,1 are short or open, so R n2 Z 1 =0 = r m2 (90) Wth all eght drng pont resstances determned, the redundant defntons of K d and K n can be checked. Both ersons of (80) ge K d = r m2 RR 3 R 3 (91) The frst erson of (81) ges zero oer zero and s ndetermnate, but the second erson ges K n = 1 (92) Consequently, because K d 1, the denomnator of (79) does not factor exactly, but because K n =1, the numerator does (although one factor s unty because R n1 =0). Wth nserton of the arous components nto (79), the result for M s (88)