BEN-GURION UNIVERSITY OF THE NEGEV FACULTY OF ENGINEERING SCIENCES DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

Transcription

1 BEN-GURION UNIVERSIY OF HE NEGEV FACULY OF ENGINEERING SCIENCES DEPARMEN OF ELECRICAL AND COMPUER ENGINEERING FEEDBACK AS SUPERPOSIION HESIS SUBMIED IN PARIAL FULFILLMEN OF HE REQUIREMENS FOR HE M.Sc DEGREE By: Pavel Shanin September 2013

2 BEN-GURION UNIVERSIY OF HE NEGEV FACULY OF ENGINEERING SCIENCES DEPARMEN OF ELECRICAL AND COMPUER ENGINEERING FEEDBACK AS SUPERPOSIION HESIS SUBMIED IN PARIAL FULFILLMEN OF HE REQUIREMENS FOR HE M.Sc DEGREE By: Pavel Shanin Supervied by: Prof. Eugene Paperno Author: Pavel Shanin Supervior: Prof. Eugene Paperno Supervior: Date: Date: Date: Chairman of Graduate Studie Committee: Date: September 2013

3 i Summary hi work i devoted to the analyi of linear feedback network with multiple dependent ource. Feedback circuit are widely ued in analog circuit deign, mainly due to the fact that the circuit characteritic and the cloed-loop gain value are highly independent of the operating condition and the tranitor parameter variation. Modeling of uch network i very important for exploring the effect of feedback on the network tranfer function. An analytical model enable u to apply control theory tool for analyzing the network behavior, a well a prove a better intuitive inight into it operation. Although any linear feedback network can be eaily imulated, the obtained olution provide no inight into the network functional tructure. hee feedback circuit uually comprie many tranitor; conequently, their mall ignal model would naturally include everal dependent ource. Unfortunately, the exiting theorie do not provide analytical and functional model for feedback network with multiple dependent ource. he main goal of thi work i to develop accurate analytical and functional model for the cloed loop gain of linear feedback network with multiple dependent ource. An additional goal i to tudy the effect of negative feedback on impedance een from a pair of arbitrary terminal for uch network. hi work alo eek to extend Middlebrook formula for imulating return ratio to linear network with multiple bilateral feedback loop. he work develop an accurate analytical and functional model for the cloed loop gain of linear feedback network with multiple dependent ource and any number of feedback loop. he reult i, a cloed loop gain formula containing return

4 SUMMARY ii ratio are obtained in a matrix form. hi model decribe linear feedback network with multiple dependent ource analytically, in term of return ratio. In the general cae, in uch network, each dependent ource effect, not only it own control ignal but alo the control ignal of the other dependent ource. For reaon of conitency, thi work refer to uch cro contribution, normalized to the total control ignal appearing in the dependent ource, a cro return ratio. It i hown that return ratio and cro return ratio can be combined into a general return ratio. o the bet of our knowledge, the preent work how for the firt time, how to obtain for linear network with multiple dependent ource a generalized return ratio a a function of network element. hi i achieved by uing diection and uperpoition in network analyi. Obtaining generalized return ratio, a a function of circuit component, i important for analyzing the network tability. hi alo turn a tability problem to a Nyquit plot calculation and allow applying other control theory tool. In order to examine the effect of negative feedback on impedance een from a pair of arbitrary terminal of linear feedback network, we decribed them analytically a a function of return ratio. It ha been hown that the Blackman formula can be extended to the cae of feedback network with multiple dependent ource. o reach thi aim, we reviit the proof of the Blackman formula, proved it with a imilar manner for feedback network with two dependent ource and then extended to the cae of network with multiple dependent ource. We alo howed that thi formula can be written in a convenient matrix form. A wa already mentioned finding return ratio i very important for revealing the effect of feedback on the cloed loop-gain function, impedance and tability. Conventionally, the return ratio i found by uppreing all the independent

5 SUMMARY iii network ource, aigning a fixed value to the dependent ource, and calculating the ignal that return to it controlling terminal. However, thi procedure i not uitable for imulating electronic circuit or for teting real electronic circuit. An alternative approach wa uggeted by Middlebrook and conit in connecting tet ource to the acceible terminal of a tranitor or operational amplifier. According to thi method, two partial return ratio v and i are meaured, one for a voltage injection and the other for a current injection. hen they are tranlated into the true loop gain. hi method i inaccurate at frequencie near loop gain croover when < 1. hi defect i eliminated in an improved method uggeted by Middlebrook. In the method a loop-gain meaurement i done by two imultaneou voltage and current injection at a point of arbitrary impedance ratio. According to n thi method two partial return ratio are meaured, one v i meaured when injected ignal are adjuted to null the current in a particular circuit point and the other n i n n when the voltage on a dependent ource equal zero. hen and are v i tranlated into the circuit return ratio. In pite of the original proof of thee method i baed on an ideal feedback model which doe not account for nonzero revere loop gain, our proof i baed on a generic feedback model. A a reult it i hown that two Middlebrook formula can be applied with no approximation to any linear feedback network with a ingle dependent ource. A a reult of thi work, the claical Bode and Blackman method are extended to multiple tranitor circuit, which are much wider cla of circuit. Alo the propoed theory could erve a ueful reference to do quick calculation by hand, thu, to obtain a better intuitive inight into the effect of feedback on the cloed loop network function and circuit impedance.

6 iv Keyword Analytical model, cro return ratio, dependent ource, deenitivity factor, DSF, feedback a uperpoition, functional feedback model, general feedback theorem, Linear feedback network, multiple dependent ource, multiple loop feedback amplifier, multiple loop feedback network, network theory, network tranfer matrix, null return difference matrix, null return ratio matrix, return difference, return difference matrix, return ratio, return ratio matrix, ucceful current injection, ucceful voltage injection, uperpoition of dependent ource.

7 v Acknowledgment I would like to thank my advior Prof. Eugene Paperno for hi help and upport during recent year, hi advice and for making thi work poible. He i an incredibly patient mentor and good friend throughout my time at Ben-Gurion Univerity of the Negev. Hi advice, extenive knowledge and experience in the field of circuit theory contributed a lot to the undertanding of the field. o the Fay and Bert Harbour Foundation who gave upport to me during my mater tudie. Finally, I would like to thank my family and my niece for their love and upport, who contributed to me a lot and alway tood by me.

8 vi Content 1 Introduction Linear feedback network Cloed loop gain in feedback network with a ingle dependent ource Impedance of feedback network with a ingle dependent ource Feedback network with multiple dependent ource Meaurement return ratio in Spice imulation Superpoition with dependent ource Reearch Objective Method Modeling Linear Feedback Network with Multiple Dependent Source Introduction Feedback network with a ingle dependent ource Feedback network with multiple dependent ource Feedback network with two dependent ource Feedback network with three dependent ource General formula for cloed-loop gain in feedback network with multiple dependent ource Concluion Extenion of Blackman Formula to Feedback Network with Multiple Dependent Source 3.1 Introduction Feedback network with a ingle dependent ource Feedback network with two dependent ource Feedback network with multiple dependent ource Concluion Simulating Return Ratio in Linear Feedback Network by Middlebrook' Method Introduction Return ratio and the cloed-loop gain... 71

9 vii 4.3 Meauring return ratio directly at the dependent ource terminal Meauring return ratio at the terminal of a dependent ource linked to Improved meaurement of return ratio at the terminal of a dependent ource linked to by null double injection Concluion Bibliography 87 Appendix 90 A. Finding a return ratio from SPICE imulation B. Senitivity function of return ratio... 94

10 viii Lit of abbreviation adj - adjoint matrix. CC - common collector. det determinant. DSF - deenitivity factor. GF - general feedback theorem. KCL - Kirchof current law. KVL Kirchof voltage law. ppm part per million.

11 ix able of Figure Fig. 1.1 Fig. 1.2 Fig. 1.3 Ideal feedback functional model with unilateral block...2 Feedback amplifier contain linear element...4 Finding of return ratio in linear network with one dependent ource, (a) for the voltage dependent ource, (b) for the current dependent ource...6 Fig. 1.4 A functional model for the cloed-loop gain formula....8 Fig. 1.5 (a) Common collector amplifier, (b) Small ignal model, (c) Circuit for finding of return ratio....9 Fig. 1.7 he block diagram of the general feedback configuration Fig. 1.8 Fig. 1.9 (a) Double ranitor Amplifier Circuit. (b) Small Signal Model of Double ranitor Amplifier Circuit with the value of dependent ource Ia = h fe1ib1 and Ib = h fe2ib (a) Fundamental matrix flow graph, (b) Matrix flow graph with zero input vector and break in the branch with tranmittance X...17 Fig (a) Feedback circuit with the focu on point of interet, in meaurement of loop gain by ingle voltage injection. (b) Original equivalent feedback circuit that wa ued by Middlebrook in hi proof of return ratio formula from ucceive voltage and current injection...23 Fig Succeful (a) voltage and (b) current injection at the acceible point B of arbitrary impedance ratio z2 / z Fig (a) Dependent voltage/ current - controlled voltage ource - characteritic, (b) dependent current/ voltage -controlled current ource with there - characteritic Fig (a) active circuit example, (b) example circuit after taping...27 Fig. 2.1 Finding the partial open-loop gain of a generic ingle-tranitor circuit by applying uperpoition. (a) Original network. (b) he network, where the independent ource i the only active one. (c) he network, where the equivalent independent ource, replacing the dependent ource, i the only active one...33 Fig. 2.2 Generic ingle-tranitor circuit for finding of return ratio Fig. 2.3 Fig. 2.4 Functional model of the generic ingle-tranitor circuit of Figure 2.1. he block B repreent a bidirectional network Defining the partial open-loop gain for a generic feedback network with two dependent ource...38

12 ABLE OF FIGURES x Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Functional model of a feedback network with two dependent ource. Note that AOL 1β 12 = 12 and AOL 2β 21 = Functional model of a feedback network with three dependent ource...42 Defining the partial open-loop gain for a generic feedback network with multiple dependent ource...43 Finding the partial open-loop gain of a ingle-tranitor circuit. (a) Example circuit. (b) Original equivalent mall-ignal circuit. (c) he prime circuit, where the independent ource i the only active one. (d) he double-prime circuit, where the dependent ource i the only active one. Note that the dependent ource in (d) i controlled be the ε ignal of the original circuit (b) Fig. 2.9 Double-tranitor amplifier circuit Fig Finding the partial open-loop gain of a double-tranitor circuit. (a) Original equivalent mall-ignal circuit. (b) he prime circuit, where the independent ource i the only active one. (c) he double-prime circuit, where hfei b1 the only active ource i. h feib 2 (d) he triple-prime circuit, where the only active ource i. Note that the dependent ource in (c) and (d) are controlled by the correponding ignal of the original circuit (a) Fig. 3.1 Fig. 3.2 Fig. 3.3 Finding the impedance een by a tet ource connected to arbitrary terminal of a generic feedback network with a ingle dependent ource. (a) Connecting a voltage tet ource and finding the control ignal ε. (b) Replacing the ource v t with i t uch that i t = v t Z t and finding a control ignal ε. (c) Finding of G v for an arbitrary v t. (d) Finding of G i for it a = 0 = vt Zt a = Finding the impedance een by a tet ource connected to arbitrary terminal of a generic feedback network with two dependent ource...59 Generic feedback network with two dependent ource, where v t = 1, ε 1 = 1 and 2 = ε 0. (a) Connecting a voltage tet ource. (b) Replacing the voltage tet ource with the current tet ource with keeping the ame condition of the tet ource branch...80 Fig. 4.1 Finding the return ratio for a generic linear feedback network: (a) original network, (b) and (c) uppreing the ignal ource and replacing the dependent ource with an equivalent independent one. Z o denote the output impedance of a tranitor or operational amplifier, and Z denote the ret of the total equivalent impedance een by the dependent ource when =

13 ABLE OF FIGURES xi Fig. 4.2 Fig. 4.3 Fig. 4.4 Finding the return ratio for a generic linear feedback network: (a) by current injection and (c) by voltage injection, (b) finding input and direct tranmiion G i, and D i, (d) finding input and direct tranmiion G v and D v Finding partial return ratio in a generic feedback network: (a) * * * injection of the tet current, (b) finding G i, D iy and D ix, (c) injection of the tet voltage, and (d) finding G, * v * D vy and D * vx Finding partial null return ratio in a generic linear feedback network: (a) injection of the tet current and voltage in order to null v (b) finding G, D, G, D, G, G, D, and D, G, y vy vy vy vy D vx, G vx, D vx. (c) injection of the tet current and voltage in order to null i y, and (d) the ame network a previou with a OL = iy ix iy ix vx Fig. A.1 Fig. A.2 Fig. A.3 Example circuit: (a) original circuit, (b) imulating return ratio by replacing the dependent ource with an equivalent independent one...91 Example circuit: (a) injection of the tet current, (b) injection of the tet voltage Meauring and comparing return ratio: (top and middle) amplitude and phae of the return ratio meaured by replacing the dependent ource with an equivalent independent one (the green curve) and by Middlebrook method (the red curve), (bottom) the relative difference (in ppm) between the return ratio amplitude (the green curve) and phae meaurement (the red curve) Fig. B.1 Senitivity function = ( 1) ( + + 2) to v and i i v i v

14 1 Chapter 1 Introduction Studying of a feedback in electronic circuit i of great demand. Our literature urvey how that almot every analog deign book contain chapter devoted to feedback [1]-[9]. he reaon for uch interet i that circuit with feedback are ued widely in analog deign. Alo all operational amplifier application rely on feedback concept. he ue of feedback in analog circuit give u the following advantageou: a. he open loop gain depend on a tranitor mall ignal parameter that are very enitive to the tranitor technology. On the other hand feedback i made from the paive component that can be made arbitrary cloe to the deired value. b. Circuit characteritic can be made independent of operating condition uch a upply voltage and temperature. c. Frequency repone and the gain-bandwidth trade-off can be controlled.

15 CHAPER 1. INRODUCION 2 d. Signal ditortion, which i a reult of the nonlinear nature of active device, can be ignificantly reduced. 1.1 Linear feedback network Cloed loop gain in feedback network with a ingle dependent ource he ideal feedback ytem can be repreented a a ingle-loop functional model [1] that can be een on Figure 1.1. Where a i the forward gain and f repreent a feedback ratio. It i worth to mention that the block a and f are unilateral. Figure 1.1: Ideal feedback functional model with unilateral block. From the cheme, we obtain: ( ) S = as = a S S f 0 ε 0. (1.1) hen the ideal cloed loop gain i: A CL S0 a 1 af 1 = = = = S 1+ af f 1+ af f 1+ (1.2)

16 CHAPER 1. INRODUCION 3 Where = af i the loop gain. When the loop gain become large compare to unity the cloed loop gain approache: S0 1 lim A CL = lim = S f. (1.3) We receive the cloed loop gain depend only on the feedback network, which conit from paive element that can be made arbitrary cloed to deired value. A a reult, the cloed loop gain i independent of variation of the forward gain. he independence of cloed-loop gain from the parameter of the active amplifier i the reaon of the wide ue of operational amplifier in analog circuit. In contrat to ideal feedback model, mot real analog circuit have not only bilateral feedback network but alo bilateral forward gain. herefore we introduce in chapter more deliberate feedback model, baed on return ratio. he model decribe network with a ingle dependent ource and wa firtly introduced by Bode [10]. In the model, the cloed-loop of feedback network i decribed through return ratio of a dependent ource, in the mall-ignal model of a circuit. he return ratio analyi i often eaier then two-port analyi and i topology independent [11]-[13]. It allow u to build functional model of a circuit, where each block can be decribed in term of a network element. A a reult, we can derive from it, which element of a circuit influence on the tability. In order to introduce Bode feedback model let u conider a feedback amplifier that i hown in Figure 1.2 which conit of linear element.

17 CHAPER 1. INRODUCION 4 Figure 1.2: Feedback amplifier contain linear element. a ol - i a controlled ource value. From the figure S oc = a S ε (1.4) ol S B S HS ε = 1 oc (1.5) S = ds + B S (1.6) 0 in 2 oc he term B 1, H, d and B 2 are defined by S S ε G = = S S S = 0 a = 0 oc ε ol d B o = = 2 S S S = S S = 0 a = 0 o oc oc S = 0 S S ε ol (1.7) S H = S ε oc S = 0 G i a tranfer function from the input to the controlling ignal and d i the tranfer

18 CHAPER 1. INRODUCION 5 function from the input to the output both evaluated with a ol = 0. B 2 i a tranfer function from the dependent ource to the output and H i a tranfer function from the output of the dependent ource to the controlling ignal time -1, both evaluated with the input ource et to zero. From equation (1.5), (1.6) and Soc = aol S ε we can find cloed-loop gain So aol B2 ACL = = G + d S 1+ a H in ol (1.8) According to Body [1] (ee Chapter 8) and [10] the return ratio for a dependent ource can be found by the following procedure: 1. Set all independent ource to zero. 2. Diconnect the dependent ource from the ret of the circuit, which introduce a break in the feedback loop. 3. On the ide of the break that i not connected to the dependent ource, connect an independent tet ource S t of the ame ign and type a the dependent ource. 4. Find the return ignal S r generated by the dependent ource. he return ratio for the dependent ource i S r =. St

19 CHAPER 1. INRODUCION 6 Figure 1.3: Finding of return ratio in linear network with one dependent ource, (a) for the voltage dependent ource, (b) for the current dependent ource. he term aol H in the denominator of (1.8) i equal to the return ratio. We can how it by etting S = 0, diconnecting the dependent ource from the circuit. Connecting a tet ource S t where the dependent ource wa connected. hi i the procedure that wa decribed earlier for finding the return ratio. After thee change, oc = t and (1.5) become Sε = HS (1.9) t We became that the output of the dependent ource i a return ignal Sr = aol Sε = aol HSt. he return ratio therefore S r = = aol H (1.10) St he cloed loop gain can be expreed in term of it

20 CHAPER 1. INRODUCION 7 So aol B2 ACL = = G + d S 1+ in (1.11) or A CL = So g d S = 1+ + in (1.12) g = G a B (1.13) ol 2 Definition DSF = 1 + i deenitivity factor or amount of feedback. he feedback i negative if it decreae the cloed loop gain, then DSF > 1 and the feedback i poitive if it increae the cloed loop gain, then DSF < 1. Obviouly if the feedback i abent DSF = 1. Combining term in (1.12) we become g d ACL = + d = A (1.14) With definition: g A = d + (1.15) A functional model for the cloed-loop gain formula can be een on Figure 1.4.

21 CHAPER 1. INRODUCION 8 Figure 1.4: A functional model for the cloed-loop gain formula. Example A an example to finding return ratio we conider common collector amplifier that i hown in Figure 1.5. RE ro ib = it R r + h E o ie (1.16) i r = h i fe b (1.17) i hfei r b RE ro = = = hfe i i R r + h t t E o ie (1.18)

22 CHAPER 1. INRODUCION 9 V CC CC V BB v O v R E (a) i b h ie h fe i b Small-ignal Model v v o R E r o (b) i b h ie i t h fe i b i r v =0 v o R E r o (c) Figure 1.5: (a) Common collector amplifier, (b) Small ignal model, (c) Circuit for finding of return ratio Impedance of feedback network with a ingle dependent ource Feedback affect input and output impedance of a circuit [14]. An expreion of finding the impedance at an arbitrary port x in a feedback network i called Blackman formula (1.19). In the formula impedance i expreed in term of the returned ratio. hi formula can be ued with arbitrary linear network with no more than one controlled ource. he derivation of thi formula can be found in [1].

23 CHAPER 1. INRODUCION 10 ( ) ( ) 1 + hort circuit RX = RX ( aol = 0) 1 + open circuit (1.19) Where ( 0) R a = - i an impedance of the port x with uppreed dependent X ol ource, ( hort circuit ) - i a return ratio with the port x hort circuit, ( open circuit ) - i a return ratio with the port x open circuit. he return ratio are computed with repect to the ame controlled ource. An advantage of the formula i that we can ue it with any type of feedback in linear circuit. Example A an example to finding impedance of input port we conider common collector amplifier ee Figure 1.5. ( 0 ) ( 0) R a R h h R r = = = = + (1.20) X ol X fe ie E o When input port i horted we have the ame circuit like in the previou example Figure 1.5 (c): i hfei r b RE ro ( hort circuit ) = = = hfe i i R r + h t t E o ie (1.21) ( opencircuit ) = 0 (1.22) R h (1 h )( R r ) = + + (1.23) X ie fe E o

24 CHAPER 1. INRODUCION Feedback network with multiple dependent ource he cloed-loop tranfer-function matrix of the multipleloop feedback amplifier In thi ection the analyi of feedback network with multiple dependent ource reviewed, that wa introduced by Chen in [9], (ee Chapter 32). He introduce the concept of return difference matrix for a multiplicity of controlled ource. he model i baed on repreentation of electrical network in a matrix equation form. From the olution of thi matrix equation we can extract the cloed loop gain. In order to find tranfer matrix of a network, we will tudy a general configuration of a multiple-input, multiple-output and multiple-loop feedback amplifier hown in Figure Figure 1.6: he general configuration of a multiple-input, multiple output and multiple-loop feedback amplifier.

25 CHAPER 1. INRODUCION 12 In the cheme, the input, output and feedback variable may be either current or voltage. u( ) - i n dimenional input vector and y( ) - i m dimenional output vector. u1 y1 I 1 I1 u2 y2 u M 3 y M 3 I k I r u ( ) = M =, y ( ) = M = V1 Vr+ 1 u n 2 y m 2 u M M y n 1 m 1 V( n k ) Vm u n y m X - i the matrix of order p he matrix repreent the dependent ource. q that relate the controlled and controlling variable. Φ - i a p dimenional controlling vector and Θ - i a q dimenional controlled vector of either current or voltage. he φ k element of Φ i a controlling variable while θ k element of Θ i a controlled variable of the dependent ource. Θ = X Φ (1.24) Feedback amplifier can be repreented in more convenient way ee Figure 1.7.

26 CHAPER 1. INRODUCION 13 Figure 1.7: he block diagram of the general feedback configuration. If X repreent parameter of the dependent ource Θ and Φ are of the ame dimenion. In the block diagram of feedback model Figure 1.7 vector u and θ are input and y and φ it output. Becaue the network i linear we can write matrix equation: Φ = AΘ + Bu (1.25) y = Cθ + Du (1.26) A, B, C and D are tranfer-function matrice of dimenion p q, p n, m q and m n repectively. he cloed-loop tranfer-function matrix of the multiple-loop feedback amplifier of order m n i defined by y = W ( X )u (1.27) In order to get the equation for W ( X ) we combine the equation (1.24), (1.25) and (1.26).

27 CHAPER 1. INRODUCION 14 1 Φ = ( Ι ) (1.28) p AX Bu 1 = ( Ι p ) + (1.29) y CX AX Bu Du Finally applying (1.27) we have: 1 ( ) ( ) W X = D + CX Ι AX B (1.30) p Ι p i p p identity matrix. Alternatively we can write (1.30): ( ) = + ( Ι p ) = + (( ) ( Ι p )) 1 1 = + ( ( Ι p )) = + ( ( Ι p )) = + ( ( )) = + ( ( Ιq ) ) = D + CXX ( Ιq XA) XB = D + C ( Ιq XA) XB W X D CX AX X XB D CX X X AX B D CX X X AX B D CX X X AX B D CX X X XAX B D CX X XA X B (1.31) We can ummarize: ( ) ( ) 1 W X D C XA XB = + Ι (1.32) q Where we have ue the matrix property ( AB) = B A when A, B - are non-ingular matrixe. Obviouly W ( 0) tranmiion only. = D and y = Du mean that cloed loop i compoed of a direct When X quare and noningular, equation (1.32) can be written a:

28 CHAPER 1. INRODUCION ( ) ( ( )) ( ) W X = D + C X X A XB = D + C X A X XB (1.33) 1 1 ( ) = + ( ) (1.34) W X D C X A B Example We will olve double-tranitor amplifier circuit with thi method. Figure 1.8: (a) Double ranitor Amplifier Circuit. (b) Small Signal Model of Double ranitor Amplifier Circuit with the value of dependent ource Ia = h fe1ib 1 and Ib = h fe2ib2. he following parameter value are ued for thi model. R = 1.3 kω, R = 1 kω, R = 8 kω, R = 2200 Ω C E f B h = h = 23.5 k, h = h = 287, r = r = 25 kω ie1 ie2 fe1 fe2 o1 o2

29 CHAPER 1. INRODUCION 16 hfe X = = 0 h fe (1.35) Equation (1.24) can be written a: Ia ib1 Θ = XΦ I = = b i b2 (1.36) Aume that the output voltage v o and input current I B are the output variable. he even port network N i defined by the variable Ia, Ib, ib1, ib 2, vo, IB and v. he matrix equation (1.25) and (1.26) of the network are: ib1 Ia Φ = = A + B v i I b2 b (1.37) vo Ia y = = C + D v I I B b (1.38) According to (1.27) ( ) W X w0 = w 1 ( ) y = W X u A cl vo = = w v 0 Z v 1 w in = I = (1.39) B 1 From olving the circuit in Figure 1.8 (b) where i b1 i a current on h ie1 and i b2 i a current on h ie2 we become (1.37) and (1.38) with the following value:

30 CHAPER 1. INRODUCION ib Ia v i = + b I b (1.40) 3 vo Ia v I = + B I b (1.41) Finally from the ubtitution of A, B, C and D matrix value from (1.40) and (1.41) into (1.34) we became: W ( X ) = A cl vo = = v Z in = v Ω I = w = B he return difference and null return difference matrixe In order to introduce the return difference matrix let u tudy matrix flow graph of a ytem of matrix equation (1.24), (1.25) and (1.26) from the previou ection. Figure 1.9: (a) Fundamental matrix flow graph, (b) Matrix flow graph with zero input vector and break in the branch with tranmittance X.

31 CHAPER 1. INRODUCION 18 In Figure 1.9 (b) we et the input vector u to zero, break the input of the branch with tranmittance X and apply a ignal p vector g to the right of the breaking mark. he returned ignal p vector h to the left of the breaking mark i h = AXg (1.42) AX i loop-tranmiion matrix. he return ratio matrix i: ( X ) = AX (1.43) he difference between the applied ignal g and returned ignal h i given by the following formula: ( Ι p ) g h AX g = (1.44) he return difference matrix with repect to X i: F ( X ) = Ι p AX (1.45) In another form uing (1.43) and (1.45) we have ( ) = + ( X ) (1.46) F X Ι p

32 CHAPER 1. INRODUCION 19 For the circuit from previou example the return ratio matrix i: ( ) X = AX = (1.47) Similar to the return difference matrix the null return difference matrix can be defined. In fundamental matrix flow graph on Figure 1.9 (b) input vector u i adjuted uch that the output vector y i zero: Du + CXg = 0 (1.48) u 1 = D CXg (1.49) With uch an input u and g the return ignal i 1 ( ) h Bu AXg BD CX AX g = + = + (1.50) he null return difference matrix with repect to X i: ( ) ˆ 1 Ι ( ) Ι Ι ˆ Fˆ X = g h = + X = AX + BD CX = AX (1.51) p p p he null return ratio matrix ˆ( X ) i a quare matrix: 1 ( ) = + = ˆ (1.52) ˆ X AX BD CX AX 1 = (1.53) Â A BD C

33 CHAPER 1. INRODUCION 20 Example From the double-tranitor amplifier circuit, we define vector y to be v o i output voltage, and then intead (1.41) we have: I 3 ( ) a ( ) y = vo = + v I b (1.54) From (1.54): 3 C = ( ), D = ( ) Subtituting thee into (1.53) with A and B from (1.40) we have: 3 3 ˆ A = A BD C = (1.55) he null return difference matrix (1.51) with repect to X i: ˆ = AX = ( ) Ι2 Fˆ X (1.56) Ueful impedance matrix relationhip formula For network with ingle dependent ource we have een the Blackman formula (1.19) that can be rewritten according to [9] chapter 30 a: f ( input hort circuit ) ( ) = z ( 0) (1.57) f ( input opencircuited ) z x

34 CHAPER 1. INRODUCION 21 z ( x ) - i an input impedance looking into a terminal pair and x - repreent the controlling parameter of a controlled ource in a ingle-loop feedback amplifier. When in (1.57) calar f ( ) denote the return difference. A ueful relationhip between determinant i: detw X ˆ ( ) 0 det F X ( ) = detw ( ) (1.58) detf ( X ) A proof of it can be found in [9]. he expreion (1.58) can be ued to derive an expreion of a general Blackman formula. If W ( X ) denote the impedance matrix of an n-port network. In thi cae imilar to (1.57) F ( X ) = F ( input open circuited ) and F ˆ ( X ) i the return difference matrix with repect to X for the input port-current vector I S and the output port-voltage vector V under the condition that V i identically zero. Note: In thi cae output port-voltage vector V i the voltage on the impedance port, where I S i the current. We receive F ˆ ( X ) = F ( input hort circuited ) and (1.58) can be written a: ( ) ( circuited ) detf input hort circuited det Z ( X ) = detz ( 0) detf input open (1.59) Example For double-tranitor amplifier circuit uing (1.46) we have: detf X ( ) ( X ) = det Ι + = (1.60) detfˆ ( X ) = (1.61)

35 CHAPER 1. INRODUCION 22 If v i choen a an input and v o a an output: ˆ ( ) ( ) = v detf X o w( 0) v = detf ( X ) = (1.62) w X 3 Where ( 0) = = i taken from (1.41). w d o he reult i cloed loop gain. In a imilar manner we can calculate the impedance of an arbitrary port of a network. 1.2 Meaurement return ratio in SPICE imulation Practical method of meaurement of the loop gain a a function of frequency are given by Middlebrook [15]. he method allow making experimental meaurement of a loop gain without opening the loop in circuit with one dependent ource. he original circuit i hown in Figure 1.10 (a). he point of interet in the feedback loop at which the driving ignal i repreented neither by an ideal voltage ource nor by an ideal current ource i repreented in Figure 1.10 (b). he driving ignal i repreented a Norton equivalent. Expreion for loop gain can be imply obtained by current injection at point A, ee Figure 1.10 (b). However, in the real network, the point A i not acceible. he acceible point B i with an arbitrary impedance ratio z2 / z 1. Voltage injection from a non-ideal voltage ource with impedance z v i hown in Figure 1.11 (a). he injection mut be done in uch a way that vz = vx + vy. By meaurement of the voltage v y and v x we become a ratio / = v v. v y x

36 CHAPER 1. INRODUCION 23 Figure 1.10: (a) Feedback circuit with the focu on point of interet, in meaurement of loop gain by ingle voltage injection. (b) Original equivalent feedback circuit that wa ued by Middlebrook in hi proof of return ratio formula from ucceive voltage and current injection. 1 z z v = Gm + z = + + z z z (1.63) hen a current injection from a non-ideal current ource of impedance z i i performed, that hown on Figure 1.11 (b). he injection ignal mut atify iz = ix + iy. By meaurement of the current i x and i y we become / = i i. i y x 1 z z i = Gm + z = + + z z z (1.64)

37 CHAPER 1. INRODUCION 24 rue loop gain of a circuit from Figure 1.10 (b) i given by: m ( ) = G Z Z (1.65) 1 2 Figure 1.11: Succeful (a) voltage and (b) current injection at the acceible point B of arbitrary impedance ratio z 2 / z 1. Combining (1.63) and (1.64) the loop gain i: vi 1 2 = + + (1.66) v i he diadvantage of thi method i that it can be ued only for ideal feedback model. hat mean that only for the circuit with revere one literal loop gain [16]. However, mot of electrical circuit with one dependent ource have everal bilateral

38 CHAPER 1. INRODUCION 25 feedback loop. herefore it i important to widepread thi method to uch network. In hi further work Middlebrook decribe general feedback theorem for meaurement of a cloed loop gain and it component [16]. he GF doe not tart from a functional model but i developed on a general property of a linear ytem. According to thi property, called the diection theorem, any firt level tranfer function of a linear ytem can be diected into a combination of three econd level tranfer function. he econd level tranfer function are generally calculated with circuit imulator and in ome cae can not be calculated in analytical way. herefore thi method i out of cope of our work. 1.3 Superpoition with dependent ource In thi ection we will reexamine the topic of uperpoition in network with dependent ource that will be ued in thi work. he fact i that mot widely ued circuit book ay that we can t deactivate a dependent ource when making uperpoition in a circuit [17]-[21]. hat i alo learned in mot introductory electrical engineering coure. he proof that uperpoition of dependent ource can be done analyzing active circuit i publihed in [22] and [23]. Conider the dependent ource on Figure 1.12 (a) and (b).

39 CHAPER 1. INRODUCION 26 µ v vc = or r i x m x βix ic = or g v m x Figure 1.12: (a) Dependent voltage/ current - controlled voltage ource - characteritic, (b) dependent current/ voltage -controlled current ource with there - characteritic. From circuit theory every element i defined completely by it v i characteritic. We can ee from Figure 1.12 that v i characteritic of any type of dependent ource i exactly the ame a that of the correponding type of independent one. Only the value of dependent ource depend upon another circuit variable. We can think that the dependency relation of a dependent ource i labeled on it. hen we place a piece of making tape with the ymbol v C for a dependent voltage ource or i C for a dependent current ource over that label. hi procedure i called taping and temporarily convert a dependent ource into an independent one of the ame type with unknown value v or C i C. hen the circuit i analyzed in a way one would analyze an equivalent circuit with all ource are of the independent type. After writing of circuit equation, we untape the dependent ource and expreing their value in term of the unknown. A an example let u analyze the circuit on Figure 1.13 (a). he goal i to find i.

40 CHAPER 1. INRODUCION 27 Figure 1.13: (a) active circuit example, (b) example circuit after taping. Uing uperpoition on all the ource on Figure 1.13 (b) we have vx = v + i + ic (1.67) i = v + i + ic (1.68) If we untape the dependent ource and ue (1.67) we have gm 5gm gm ic = gmvx = v + i + ic (1.69) 6 2 2

41 CHAPER 1. INRODUCION 28 gm 5gm i = v + i 3(2 g ) 2 g c m m with gm 2 (1.70) If g m = 2 there are infinite number of olution. From (1.68) and (1.70) 3gm 1 12gm + 1 i = v + 9(2 g ) 3(2 g ) m m (1.71) It can be verified that olving thi circuit without uperpoition by KCL and KVL method give the ame reult. Another argument for uperpoition of dependent ource i that when we write a matrix linear equation for the circuit G v = u, in the circuit without taping Figure 1.13 (a) a dependent ource can be een on the left hand ide of matrix equation a a function of argument of v (1.72) i v gm v 2 = v v gm v (1.72) On the other hand, when we have circuit with taping Figure 13 (b), we conidered the dependent ource a an independent with the value i c and obviouly we can apply uperpoition for uch circuit. hen equation G v = u i of the form:

42 CHAPER 1. INRODUCION i v v 2 = v + i c v v i c (1.73) When we ubtitute to the lat equation ic = gmvx = gmv1 and after rearranging we became the ame equation a in (1.72). A a concluion we can make uperpoition of dependent ource in an active circuit. 1.4 Reearch Objective he aim of the preent work i to develop exact analytical and functional model for the cloed loop gain of linear feedback network with multiple dependent ource. Additional goal i to examine the effect of negative feedback on impedance een from a pair of arbitrary terminal for uch network. Another goal i to extend Middlebrook formula for imulating return ratio to linear network with multiple bilateral feedback loop. Such network are very common among analog circuit. he method allow meaurement of return-ratio without opening the loop and changing operation point of a circuit. 1.5 Method In all proof in our work we ue generic linear circuit. Such circuit can have bilateral feedback loop and in thi way making our proof valid for general linear network. We ue uperpoition of dependent ource, decribed in thi chapter, and diection throughout network analyi in thi work. All thoe afford u to analyze

43 CHAPER 1. INRODUCION 30 active circuit in a more efficient way and to receive general cloed loop gain formula a a function of return ratio. he claical Bode procedure of finding return ratio i modified and extended by introducing of cro return ratio to much wider cla of circuit containing multiple tranitor. In order to check the above theory experimentally everal circuit were imulated in Spice. Meaurement of component of cloed loop function of thee circuit were performed. he circuit with bilateral feedback loop were alo imulated in Spice in order to illutrate the accuracy of finding a return ratio by applying Middlebrook method.

44 31 Chapter 2 Modeling Linear Feedback Network with Multiple Dependent Source 2.1 Introduction In thi ection we ugget a new approach to the decription of feedback in linear network with multiple dependent ource, which i intuitively imple and comprehenible. It involve no approximation and i applicable to linear feedback network with any number of dependent ource and any number of feedback loop. Any linear feedback network can eaily be imulated however the obtained olution provide no inight into the network functional tructure. he uggeted approach give parametric equation for cloed loop gain function that provide better intuitive inight into feedback network operation, allow building of functional model of a ytem and applying control theory tool to analyze the tability of it. Becaue the cloed loop function i expreed through the circuit parameter we can ee from it what element of a network influence the tability of it. Unfortunately, exiting literature [24]-[33] ugget exact analytical or functional modeling only for linear feedback network with a ingle dependent ource. In ome cae, feedback network with two or more dependent ource can be approximated

45 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 32 by model baed on a ingle dependent ource. It i not alway clear, however, how accurate thi approximation i. he propoed approach i baed on the 'return ratio' concept [1] and uperpoition of depended ource that wa proved in the previou ection. With the help of uperpoition we can eaily find all the partial open-loop gain and to tranlate them into the cloed loop gain. Return ratio, i decribe the contribution of dependent ource to their own control terminal. o conider contribution of each dependent ource to the control terminal of other dependent ource in a network, we introduce cro return ratio ij, i j. hi allow u to extend the canonical Body feedback model [10] to linear network with multiple dependent ource. Moreover, we how that the return ratio i and cro return ratio ij, which can be written a a return ratio matrix and can be combined into a generalized return ratio Σ. With the help of generalized return ratio we can analyze the tability of the network. Firtly we reviit the modeling of linear feedback network with a ingle dependent ource, and then develop model for linear feedback network with two and three dependent ource. Following the propoed approach, we extend the developed model to feedback network with multiple dependent ource. 2.2 Feedback network with a ingle dependent ource Let u conider a generic linear feedback network with a ingle dependent ource, a hown in Figure 2.1 (a).

46 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 33 a OL /a OL o A OL G fwd R L D (a) ' a OL =0 ' o G fwd R L D (b) " a OL =0 /a OL A OL " o R L (c) Figure 2.1: Finding the partial open-loop gain of a generic ingle-tranitor circuit by applying uperpoition. (a) Original network. (b) he network, where the independent ource i the only active one. (c) he network, where the equivalent independent ource a ε, replacing the dependent ource, i the only active one.

47 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 34 o define the partial open-loop gain we ue the uperpoition of dependent ource, the method that wa proved in the previou chapter. We firt find the contribution of the independent ource, to the control terminal of the dependent ource ε and output o ignal. o do thi, we uppre the dependent ource a hown in Figure 2.1 (b). he contribution of the independent ource to the control and output ignal are ' ε and ' o repectively. o find the contribution of the dependent ource, aol ε we uppre the independent ource and force the dependent ource to have it value from original circuit Figure 2.1 (a) the reulting circuit i on Figure 2.1 (c). he contribution of the dependent ource to the control and output ignal are '' ε and '' o repectively. We note, that the ignal in Figure 2.1 with the prime ymbol correpond to the cae, where the independent ource i the only active one, and the ignal with the double prime ymbol correpond to the cae, where the dependent ource i the only active one. he return ratio can be found more eaily, then in the procedure that wa decribed in chapter and in [10]. he original procedure of finding return ratio wa decribed in introduction chapter and i depicted on Figure 1.3. he return ratio i: r = (2.1) t

48 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 35 Figure 2.2: Generic ingle-tranitor circuit for finding of return ratio. Let u invetigate the circuit on Figure 2.2, with the independent ource et to zero, and the dependent ource i ubtituted with the independent one with the value of dependent one in the original circuit. We define a tranfer function H o to be H o '' ε =. On the other hand from the Figure 1.3: H o oc And form Figure 1.3 (a): = ε t r = aol ε and ε = Hot, r = aol Hot (2.2) On the other ide from Figure 2.2: = H a (2.3) '' ε o OL ε Combining (2.2) and (2.3) we became that r '' ε = = (2.4) t ε We now can define the following open-loop partial gain:

49 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 36 input tranition: feed forward tranmiion: direct tranmiion: open-loop gain: feedback tranition: return ratio: G ' ε (2.5) ' ' o β fwd (2.6) ε ' D Gβ o fwd (2.7) A OL '' o ε (2.8) '' ε '' o β (2.9) '' ε AOL β (2.10) ε o find the cloed-loop gain, with the help of uperpoition of ource, we combine all the contribution to obtain the original control and output ignal: = ' + '' = G (2.11) ε ε ε ε ' '' o o o β fwd ε OL = + = G + A (2.12) From (2.11) and (2.12) we can find the cloed loop: A + Gβ A A ACL = = + G = + D o ε OL fwd ε OL ε OL β fwd ε ( 1+ ) G AOL ACL = G + D 1+ (2.13) (2.14) Baed on (2.14) and Figure 2.1, a functional model of the feedback can be developed (ee Figure 2.3). It i important to note that the functional block, combining the

50 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 37 feedback and feed forward tranmiion, repreent a bidirectional feedback network. A dc G< 1 G A OL A OL o Load A OL A dc < 1 A dc < 1 A dc fwd < 1 G fwd B Figure 2.3: Functional model of the generic ingle-tranitor circuit of Figure 2.1. he block B repreent a bidirectional network. 2.3 Feedback network with multiple dependent ource Feedback network with two dependent ource A generic double-tranitor electronic circuit i hown in Figure 2.4. he contribution of all it ource to the original control and output ignal can be found by applying uperpoition: = G (2.15) ε1 1 ε1 1 ε 2 21 = G (2.16) ε 2 2 ε 2 2 ε1 12 = A + A + D (2.17) o ε1 OL1 ε 2 OL2 Where

51 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 38 '' o ε1 ε 2 ε1 OL1, AOL 2 G1, G2, 1 ε 1 ε 2 ε1 A ''' o ' ' '' ''' ε 2, 2, ε 2 '' ε 2 ε1 12, 21 ε 1 ε 2 ''. Figure 2.4: Defining the partial open-loop gain for a generic feedback network with two dependent ource. Note that the ignal with the double and triple prime ymbol correpond to the cae, where the only active ource are aol1 ε 1 and aol 2 ε 2 one, repectively. Equation (2.15) and (2.16) can be olved for the independent ource value: = G + G G ε1 (2.18) = G + G G ε 2 (2.19) Conidering (2.17) - (2.19), the cloed loop gain can be found a follow:

52 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 39 A CL o AOL1I 1 AOL 2I 2 = + + D Σ Σ (2.20) where I = G + G G (2.21) I = G + G G (2.22) Σ = (2.23) Equation (2.20) - (2.23) can be repreented a the feedback functional model hown in Figure Feedback network with three dependent ource he above analyi of linear feedback network with two dependent ource can eaily be extended to feedback network with multiple dependent ource. For example, for a generic feedback network with three dependent ource, the control and output ignal can be found a follow: = G (2.24) ε1 1 ε1 1 ε 2 21 ε 3 31 = G (2.25) ε 2 2 ε 2 2 ε1 12 ε3 32 = G (2.26) ε 3 3 ε 3 3 ε 2 23 ε1 13 = D + A + A + A (2.27) o ε1 OL1 ε 2 OL2 ε 3 OL3

53 HAPER 2. MODELING LINEAR FEEDBACK NEWORKS A dc G< A OL1 o Load A dc < 1 21 A dc < 1 A dc G< A OL2 A dc < 1 12 A dc < 1 A dc D< 1 igure 2.5: Functional model of a feedback network with two dependent ource. Note that AOL 1β 12 = 12 and AOL 2β 21 = 21.

54 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 41 From (2.24) - (2.27), the cloed loop gain can be found a follow: A CL o AOL1I 1 AOL 2I 2 AOL 3I 3 = D Σ Σ Σ (2.28) Where I I I ( ) ( 1 ) ( 1 ) = G G G ( ) ( 1 ) ( 1 ) = G G G ( ) ( 1 ) ( 1 ) = G G G ( 1 ) ( 1 ) ( 1 ) = + + Σ (2.29) (2.30) (2.31) (2.32) Equation (2.28) - (2.32) can be repreented a a feedback functional model hown in Figure General formula for cloed-loop gain in feedback network with multiple dependent ource We can derive general formula for a cloed-loop gain for network with any number of dependent ource. A generic multiple-dependent ource network i hown in Figure 2.7. For implicity the return-ratio are hown only for the econd dependent ource.

55 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 42 Figure 2.6: Functional model of a feedback network with three dependent ource.

56 HAPER 2. MODELING LINEAR FEEDBACK NEWORKS igure 2.7: Defining the partial open-loop gain for a generic feedback network with multiple dependent ource.

57 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 44 In thi ection we define cro return ratio a: km ε k = 0, a1 = 0, am 0, an = 0 ; 1 k, m n (2.33) m Matrix equation, imilar to (1.24) - (1.26) that we ued in chapter 1, for the generic network in Figure 2.7 are: Θ = XΦ (2.34) Φ = AΘ + Bu (2.35) y = CΘ + Du (2.36) Θ i a controlled vector, conit from the value of the dependent ource. Φ i a controlling vector, conit from the value of the controlling terminal of the dependent ource. We will ue the technique of uperpoition of the dependent ource, that wa decribed in introduction, in order to find the element of the matrice A, B, C and D. aol1ε 1 Θ1 Θ = M = M (2.37) aoln ε n Θ n θ 1, θ n - i a taping of dependent ource.

58 CHAPER 2. MODELING LINEAR FEEDBACK NEWORKS 45 ε1 Φ = M (2.38) εn y = v o (2.39) u = v (2.40) A,, B C and D are tranfer-function matrice of dimenion, 1, 1 n n n n and 1 1repectively. Becaue of the fact that we are intereted in tranfer function for implicity we can et all the value of the gain of the dependent ource a = 1. We have n dependent ource, o we need to examine n + 1 circuit. We denote with the prime ymbol ignal in a circuit with active and all the dependent ource are et to zero. For integer k uch that 1 k n we denote with ' k ' ymbol ignal in a circuit with only one dependent ource Θ k = a OL k k active and all the other ource are et to zero. hen we have: Prime-circuit: - i active, all the dependent ource are et to zero. ε OL ' ε1 G1 B = M = M (2.41) ' ε n G n D ' o = = [ D] (2.42) '1' - circuit: θ 1 - i active, all other ource are et to zero. We obtain the 1-t column of A matrix and the 1-t element of C matrix.