ACTIVE NETWORK SYNTHESIS USING THE POSITIVE IMPEDANCE CONVERTER A THESIS. Presented to. The Faculty of the Graduate Division.

Transcription

1 ACTIVE NETWORK SYNTHESIS USING THE POSITIVE IMPEDANCE CONVERTER A THESIS Presented to The Faculty of the Graduate Division by Chung Duk Kim In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Electrical Engineering Georgia Institute of Technology June, 1971

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3 ACTIVE NETWORK SYNTHESIS USING THE POSITIVE IMPEDANCE CONVERTER App^s^uajl. Date approved by Chairman: /^t^y & >/ /Q7j

4 11 ACKNOWLEDGMENTS I would like to express my sincere appreciation to my advisor, Dr. Kendall L. Su, for suggesting the thesis problem, for his guidance and encouragement throughout the investigation, and for his special patience in proofreading the thesis draft. I would also like to thank Drs. B. J. Dasher and D. C. Fielder for their services as members of the reading committee. Special thanks are due to my good friend, Mr. (and now Dr.) Douglas R. Cobb for his constructive suggestions regarding the performance of the experiments and the preparation of this manuscript. Special thanks are also due to Mrs. Lydia Geeslin for her extraordinary care in typing the thesis. Finally, I wish to express my appreciation to my wife, Jae Kyung (who has stayed in Korea with our son, Hyun Sik, during the entire course of this study), for her patience, understanding, and encouragement.

5 iii TABLE OF CONTENTS ACKNOWLEDGMENTS Page ii LIST OF ILLUSTRATIONS. v SUMMARY viii Chapter I. INTRODUCTION II. DRIVING-POINT ADMITTANCE SYNTHESIS 9 Two Fundamental RC-PIC Networks for Generating Negative Admittances Driving-Point Admittance Synthesis with One PIC Driving-Point Admittance Synthesis with Two PIC's Singly-Loaded RC-PIC Networks Numerical Examples III. SYNTHESIS OF OPEN-CIRCUIT VOLTAGE TRANSFER FUNCTIONS 32 Synthesis with One PIG Synthesis with Two PIC's Numerical Examples IV. N X N SHORT-CIRCUIT ADMITTANCE MATRIX SYNTHESIS 43 Synthesis Using a Balanced Network Synthesis Using a Grounded Network Alternative Synthesis Procedures for a Restricted Glass of Admittance Matrices Numerical Examples V. N X. N VOLTAGE TRANSFER MATRIX SYNTHESIS Synthesis Using a Balanced Network Synthesis Using a Grounded Network Numerical Examples

6 iv TABLE OF CONTENTS (Concluded) Chapter Page VI. STABILITY AND SENSITIVITY CONSIDERATIONS. 87 Stability Properties of a PIC Sensitivity Consideration VII. EXPERIMENTAL RESULTS 103 Examples Discussion of Techniques and Errors VIII. CONCLUSION AND RECOMMENDATIONS APPENDICES 123 I. PROOF FOR THE POLYNOMIAL DECOMPOSITION IN (10) 124 II. NECESSARY NUMBER OF PIC's FOR THE SYNTHESIS OF ANY N X N VOLTAGE TRANSFER MATRIX 128 BIBLIOGRAPHY VITA 133

7 V LIST OF ILLUSTRATIONS Figure Page 1. Two-Port Representation Controlled Source Representation of the PIC 3 3. Realization I of the PIC 5 4. Realization II of the PIC 7 5. Two Basic RC-PIC Circuits One-Port Active RC Network I Containing One PIC One-Port Active RC Network II Containing One PIC One-Port Active RC Network Containing Two PIC's One-Port Terminated Active RC Network Containing One PIC One-Port Terminated Active RC Network Containing Two PIC's 0« Network with One PIC Realizing Y(s) = 1/s Network with Two PIC's Realizing Y(s) ~ 1/s Two-Port Active RC Network I Containing One PIC Two-Port Active RC Network II Containing One PIC Two-Port Active RC Network Containing Two PIC's Network with One PIC Realizing T(s) in (46) Network with Two PIC's Realizing T(s) in (46) 40

8 vi LIST OF ILLUSTRATIONS (Continued) Page N-Port Active RC Network Containing m PIC's Active RC Network Containing 2N PIC's 55 Simplified Active RC Network Containing N PIC's 63 Simplified Active RC Network Containing 2N PIC's 67 Double Terminated Active Device 88 PIC with a Feedback Load 88 One-Port Active RC-NIC Network 96 Two-Port Active RC-NIC Network 96 Alternative Active RC Network Containing One PIC 101 Two Forms of RC-PIC Networks Realizing a Negative Resistance Comparison of Experimental Data of Forms I and II with Predicted Magnitude Data for Y = -1/4 X Comparison of Experimental Data of Forms I and II with Predicted Phase Angle Data for Y = -1/4 X RC-PIC Network Realizing a Lossy Inductor Y(s) «10/(s ) 109 Comparison of Experimental Data with Desired Magnitude Behavior for Y(s) = 10/(s ) 110 Comparison off Experimental Data with Desired Phase Angle Behavior for Y(s) - 10/(s ).... Ill RC-PIC Network Realizing T(s) in (130) 113 Comparison of Experimental Data with Predicted Magnitude Data for T(s) in (130) 114

9 vii LIST OF ILLUSTRATIONS (Concluded) Figure 35. Comparison of Experimental Data with Predicted Phase Angle Data for T(s) in (130)....«,, 36. Zero-Pole Distribution of p(s) Page

10 viii SUMMARY This thesis is concerned with an investigation of the use of the positive impedance converter (PIC) in active RC synthesis procedures. Specifically, resistors, capacitors, and PIC's are used as network elements for the realization of arbitrary driving-point admittances and opencircuit voltage transfer functions of real rational functions in the complex frequency variable s. This research can be divided into the following five objectives: (A) To formulate methods to realize a driving-point admittance function with RC-PIC networks. (B) To formulate methods to realize an open-circuit voltage transfer function with RC-PIC networks. (C) To extend the methods in (A) and (B) to include N-port network syntheses and establish sufficient and necessary conditions for the realization of N-port RG-PIG networks. (D) To investigate the stability criteria of the PIC itself and the basic stability properties of the PIC circuit utilized in (A) and (B). (E) To investigate the sensitivity due to conversion gain changes of the PIC for the networks developed in (A) and (B). The results of the investigation on RC-PIC syntheses can be summarized in the following theorems: Theorem 1 For the realization of an arbitrary N X N matrix of real rational

11 IX functions in the complex frequency variable as a short-circuit admittance matrix of a transformerless active RC N-port network, (a) it is, in general, necessary that the network contains N PIC's; and (b) it is sufficient that the network contains N PIC's embedded in a 3N-port RC network. Theorem 2 For the realization of an arbitrary N X N matrix of real rational functions in the complex frequency variable as a short-circuit admittance matrix of a transformerless grounded active N-port RC network, it is sufficient that the network contains 2N PIC's embedded in a (4N+1)-terminal RC network. Theorem 3 An N X N matrix of real rational functions in the complex frequency variable having L simple poles on the negative real axis in the complex frequency plane and no more than L+l zeros can be realized as a shortcircuit admittance matrix of a transformerless active network having no more than N PIC's embedded in a 2N-port RC network. Theorem 4 An N X N matrix of real rational functions in the complex frequency variable having L simple poles on the negative real axis in the complex frequency plane and no more than L+l zeros can be realized as a shortcircuit admittance matrix of a transformerless grounded active network having no more than 2N PIC's embedded in a (3N+1)-terminal RC network. Theorem 5 For the realization of an arbitrary N X N matrix of real rational functions in the complex frequency variable as a voltage transfer matrix of a transformerless active RC 2N-port network, (a) it is, in general,

12 X necessary that the network contains N PIC's; and (b) it is sufficient that the network contains N PIC's embedded in a 3N-port RC network. Theorem 6 For the realization of an arbitrary N X N matrix of real rational functions in the complex frequency variable as a voltage transfer matrix of a transformerless grounded active RC 2N-port network, it is sufficient that the network contains 2N PIC's embedded in a (4N+1)-terminal RC network. The necessary parts of Theorems 1 and 5 are proved through an argument on the ranks of the given N X N matrix and the equilibrium matrix equation obtained from the active RC network that contains m PIC's. The sufficient parts of these theorems are derived by assuming a network a priori and satisfying the constraints imposed on the network through the equilibrium equations. Numerical examples are included to illustrate each of the realization procedures. The results of the investigation on stability criteria reveal that, if one port of a terminated PIC is open-circuit stable (OCS) and short-circuit stable (SCS), the other port must also be OCS and SCS. The conditions for the terminated PIC to be OCS and SCS are imposed on the dynamic gains of controlled sources in the PIC, which can easily be satisfied from practical points of view. The study of the sensitivity in the RC-PIC network shows that sensitivities of driving-point admittance and open-circuit voltage transfer functions in RC-PIC networks with respect to the current conversion gain changes in the PIC can be reduced drastically as compared to those

13 XI in RC networks with the negative impedance converters, while sensitivities with respect to the voltage conversion gain changes in the PIC may increase slightly. Validity and practicality of the realization procedures are demonstrated by actual examples, and by constructing and testing the resultant networks experimentally. Resistors and commercially available operational amplifiers are used to approximate the PIC. The test results show that the procedures developed in this research are not only correct but also practical.

14 1 CHAPTER I INTRODUCTION Numerous active devices have been defined in the past decade and some remarkable progress has been made in their applications to network 1 2 synthesis. ' An active network device is an interconnection of passive elements and active elements. To facilitate the development of the synthesis of networks using active elements, frequently an ideal device described by simple mathematical formalism is assumed and a practical circuit is then developed to approach the behavior of its idealized counterpart. Although the art of network synthesis using active devices gives great versatility to the realization of network functions that are not realizable with passive networks, active networks frequently suffer from the disadvantage of large variational changes in network parameters and, consequently, instability in their physical realization. Hence, the sensitivity due to a change in network parameters and the stability of 3 an active network become important considerations, A two-port device is often described by its chain matrix A B i E 2 : m C D h «.

15 2 where the directions of voltage and current parameters are shown in Figure 1. Two groups of active devices are the impedance converter group, which is characterized by B-C-0 in its chain parameters, and the impedance inverter group, which is characterized by A^D^O. Some of the commonly used active devices in the impedance converter group are the currentcontrolled current source (CCCS), the voltage-controlled voltage source (VCVS), and the negative impedance converter (NIC), Examples of ideal active devices belonging to the impedance inverter group are the gyrator, the negative impedance inverter (NIV), the current-controlled voltage source (CCVS), and the voltage-controlled current source (VCCS). Among these various classes of active devices, the NIC and the gyrator constitute two extremes in terms of sensitivity and stability. Because of pure activity, the NIC has high sensitivity and a likelihood of instability, but it gives great versatility in network synthesis. The gyrator, however, has low sensitivity and high stability, but it places serious requirements on network functions in synthesis, since the gyrator is a passive device. In order to overcome the weak capability of the gyrator for synthesis and the high sensitivity of the NIC network, it is natural to consider a new class of active devices, the positive impedance converter (PIC) which lies between the gyrator and the NIC in terms of activity. The PIC is an active two-port device whose parameters satisfy the following equation

16 Figure 1 Two-Port Representation o- o -o (k 1 -l)e 2 \ * «." -^ / * «> a Figure 2. Controlled Source Representation of the PIC

17 4 -I, (1) where k.. and k~ are real constants with the same algebraic sign. When k- s l/k», the device is an ideal transformer. When k- - k«, the device is a power amplifier. A PIC can be regarded as a composite of two controlled sources, one VCVS and one CCCS as in Figure 2. If port 2 is terminated in Z_» then the input impedance at port 1 becomes (k-zko'zj. 4 The concept of the PIC was introduced by Kawakami in 1958, For the realization of the PIC using other existing electronic devices, there are several known circuits. Keen and Glover 5 used the technique of factoring the chain matrix to develop the PIC's in Figure 3(a) and (b). The circuit of Figure 3(a) has the chain matrix (ftj/r^ (R 1 +R 2 +R 3 +R 4 )ZR 2 R 4 -(R 3 ZR 4 ) (2) When all resistances are made very large, matrix (2) approaches the PIC with k- and k~ negative. The circuit of Figure 3(b) realizes the PIC with k- and k«positive and has the chain matrix 1 2 (R 1 +R 2 )ZR 2 V R 4

18 AAA K., AAA R«VSAA R AAA o- (a) VSAA i < R 4 AAA- R 3 AA i o R, R, VVNA J R. 0 a (b) Figure 3«Realization I of the PIC

19 6 Cobb used a voltage attenuator and a CCCS to realize the PIC with k. and 1 k«positive as in Figure 4. This circuit has the chain matrix (R 5+ R 6 )/R 6 R 2 R 4 /R 1 R 3 (3) 7 which is the same as (1) with k- and k_ positive. Holt and Carrey, 8 9 Daniels, Cox, Su, and Woodward used other approaches to realize the PIC, Although the realization of the PIC has been considered by several authors as mentioned above, no serious investigation on the use of the PIC in synthesis has been performed until very recently. Gorski-Popiel has shown that an LC network can be realized by an RC network and the generalized positive impedance converter (GPIC) with k- of (1) a function 11 6 of the complex variable s. Antoniou and Cobb have developed synthesis procedures using RC-GPIC networks that are very insensitive to parameter variations. The state of the art shows that there is no known published work in the synthesis of active networks using resistors, capacitors, and PIC's as network elements. The objective of this research is to investigate the use of the PIC in active RC synthesis procedures. Specifically, resistors, capacitors, and PIC's are used as network elements for the realization of arbitrary driving-point admittances and open-circuit voltage transfer ratios of real rational functions in the complex variable s. Attention will be directed toward minimizing the number of PIC's used in the synthesis procedures. The minimum number of PIC's that is sufficient to realize a

20 ; > R o R«R«W\A <* AAA- n <3=- 5 R. R R 4 R 6 7 R,» + R_ > R ' *7 Q Figure 4. Realization II of the PIC

21 8 given class of network matrices will also be investigated. This sufficient condition can be derived by assuming a network a priori and satisfy ing the constraints imposed on the network through the equilibrium equations. In addition to the objective above, studies on sensitivity performance and stability properties of PIC circuits will be carried out and compared to NIC circuits.

22 9 CHAPTER II DRIVING-POINT ADMITTANCE SYNTHESIS In this chapter, several synthesis procedures for networks that contain resistors, capacitors, and PIC's as network elements and realize given driving-point admittance functions are presented. As an initial step, two fundamental RC-PIG circuits that can generate negative admittances are discussed. For driving-point function synthesis, the networks used include an RC network with one PIC and an RC network with two PIC's. Each of these networks will be discussed separately, since the RC network with one PIC is not grounded and the RC network with two PIC's is grounded. Similar to other active synthesis procedures, the realization procedures with PIC's not only give great simplicity and versatility in realizing driving-point functions, but also offer numerous attractive advantages from practical points of view. Two Fundamental RC-PIC Networks for Generating Negative Admittances Figure 5 shows two basic RC-PIC network configurations that can be used to generate negative immittances. Consider the network structure in Figure 5(a) and establish the relations E l = k l E 2 I I " " k 2 X 2 V 2 = (E 1 -E 2 )y 1

23 10 h I 1 > f**\ 1 t 1 y l PIC X 2 t E i Q> k l* k 2 r\ \J (a) y 2 p\ 1 PIC 0 k l* k 2 Q Oi v^j (b) Figure 5. Two Basic RC-PIC Circuits

24 11 But since I n = I' + s I' h = ^W V 2 (l-k^d-kp yt E. k x J l 1 Then the input admittance becomes I- (1-k )(1-k.) * - r - ^ - y i From the expression of (4) the input admittance becomes negative if (1-k )(1-k ) L- ±- > o (5) h By the definition of the PIC, k_ and k«must have the same algebraic sign and in order to satisfy the requirement in (5), it is necessary that 0 < k k < 1 or k, k 2 > 1 A similar analysis holds for the network shown in Figure 5(b). The input admittance for that circuit becomes y. m (1-k )(1-k ) X y 2 (6)

25 12 For this configuration, it is also possible to get a negative admittance if (1-kJd-kJ -i- - > o (7) k 2 which can also be satisfied by choosing k and k«in the range 0 < k r k 2 < 1 or k x, k 2 > 1 next section. The two basic circuits shown in Figure 5 will be utilized in the Driving-Point Admittance Synthesis with One PIC A possible way of realizing driving-point admittance functions can be obtained by using the network for generating negative admittances in the preceding section. When the basic RC-PIC network configuration in Figure 5(a) is cascaded with a two-port passive RC network, the structure of Figure 6 is obtained. It is assumed that network A is an RC two-port and its short-circuit parameters are denoted by y_', y,«, and y * Th e input admittance Y for the network configuration in Figure 6 is ( a ) 2 Y s y n " a (l-kpa-kp (8) y 22 k ^ *l Since the basic RC-PIC structure in Figure 5(a) is utilized in the network of Figure 6, the same assumption on k- and k«must hold, which is rewritten

26 y l Y o 0 Network PIC A (RC) k r k 2 Figure 6 e One-Port Active RC Network I Contaixiia:: 5 ; One PIC Figure 7. One-Port Active RC Network II Containing One PIC

27 14 as (l-k^d-k^ _ > 0 k i which implies 0 < k, k < 1 or k, k > 1 For a given admittance function Y(s) = gg- (9) the problem is to identify the companion network parameters, y 1, y-,0, a y 22, and y^ that satisfy the restraint imposed on the equilibrium equation in (8). It will be proved in Appendix I that a real rational function Y(s) can always be decomposed as P 1 (s) p 2 (s) - s q 1 (s) q 2 (s) Y(s) * T^) q^s) - Pl(sr^TiT" (10) where p n (s)/q n (s) and p (s)/q 0 (s) are unique subfunctions satisfying the 1 1 Li. properties of passive RC driving-point admittances. The driving-point admittance Y(s) in (10) may be rewritten 2 2 P 1 (s) - s q 1 (s) P, (s) q, (s) Y(s) * -V-r- - > %, s (11) qi( s ) Pi(s) P«(S) q-^s) q 2 (s)

28 15 Comparison of (11) with (8) yields a = a p l (s) y ll y 22 qi(s) (12) 12 S P 1 (s) - s q 1 (s) q x (s) P 2 (s) y = 1 (l-k^d-^) q 2 (s) Since P-,(s)/q 1 (s) and p«(s)/q«(s) are subfunctions from the expression of a a (10), the network parameters, y.., Yoo> an & v i in (12) are passive RC a a a driving-point admittances. The residues of y, y««, y. in a pole at s s p,, which is a zero of q 1 (s), are given as (i) a = k :D p 1 (s) d, v 5S - q i (8) s= =p i (i) 12 «+ 2 2 P 1 (s) - s q 1 ( d, 0 + P,(s) d 1, N s=sp i a? q i (s) s=p. Therefore, the residue condition is satisfied with an equal sign and a balanced realization of network A is always possible without using transformers «2 2 For an actual realization, it is necessary that [p.(s) - s q-(s)] be a perfect square. This is always possible by the method of polynomial

29 augmentation. If the factor [p,(s) - sq.(s)] is not a perfect square, both the numerator and denominator of Y(s) are multiplied by Cp.Cs)» s 2 q (s) ] and the augmented admittance Y(s) will have the form P'(s) P:(s) - s q'(s) q'(s) Y(s) - -i- ^ -± - (13) p^(s) q (s) - p (s) q^(s) where 2 2 p{(s) * P 1 (s) + s q 1 (s) p 2 ^ * P 2 ^ pl^s^+ s q 2 ^ q l ^ q (s) = 2 Pl(s) q^s) q^cs) s P-^S) < 5 2^S^ + p 2^s^ qi^s^ As a result, the short-circuit parameters of network A and y- become y ll = y 22 p*(s) + s q*(s) 2 Pl(s) q^s) a 2 2 P 1 (s) - s q-^s) y 12 " 2 Pl(s) qi(s) y l = k x P^s) P 2 (s) + s q 1 (s) q 2 ( s ) TT-kpll-k 2 ) * Pl(8) q 2 (s) + p 2 (s) qi(s) and y 1. is a rational function. Another way of realizing a given Y(s) = P(s)/Q(s) with the equation 12 of (8) is to use the method of polynomial factorization. Let m be equal to or greater than the order of Y(s). Choose an arbitrary RC admittance

30 17 function of order m and assign it to y,,, Let y, *' be denoted by 11 11, a = v 21^1 (14) K J 11 *1 q(s) By the root-loci consideration, a positive constant K can always be specified such that \ P(s) Q(s) - q(s) P(s) - R x (s) R 2 (s) (15) where R,(s) has only m distinct negative real zeros. Substituting (14) and (15) into (8) gives a y 22 " (y 12 )2 R l (s) R 2 (s) (1-k )(1-k ) ~ Q(s) q(s) k, y l (16) At this point, an assignment can be made such that R 2 (a) *12 = * K 2 W (17) Substituting (17) into (16) gives a (1-k )(1-k ) 2 Q(s) R (s) y M.. 7 i - K 2 ^y^y (18) where K«is a positive constant to be determined later. Now, the righthand side of (18) is expanded into the Foster form of admittances. All

31 18 of the positive terms are alloted to y««and all of the negative terms to [ (1-k-) (l-k 9 )/k ] y.«surplus terms may be introduced into both y 9<? and (1-k-)(l-k_) y_/k- such that all poles of y-«in (17) are also included A 3. SL in y^. The multiplication constant K is determined so thaty-.-, > ^19* and y««satisfy the residue condition for the passive RC realization of network A without transformers. When the basic RC-PIC structure in Figure 5(b) is cascaded with a two-port passive RC network, it has the form of Figure 7. The input admittance, in this case, becomes b 3. / b.2 (y l2> Y y n" b (i-kpa-kp y 22 " k 2 y 2 where y-,, y 19 j and y««are short-circuit parameters of network B, The same procedures used for the network of Figure 6 can also be applied here for the realization of a given driving-point admittance with k and k 9 1 Z interchanged. Therefore, it can be concluded that an arbitrary driving-point admittance function can be realized with the network structure of either Figure 6 or Figure 7. In general, the RC network used in this synthesis has a transformerless balanced structure. Driving-Point Admittance Synthesis with Two PIC T s The balanced network resulting from the previous realization procedures suffers from numerous practical disadvantages. It is generally desirable to have grounded network configurations for many applications*

32 19 S?lnce the grounded realization requires stronger restrictions on the short' circuit parameters, the increase in the number of PIC's is unavoidable. In this section, two PIC^s are used to formulate the procedures for the grounded realization of a given driving-point admittance. Consider the network configuration that contains one RC two-port, two RC one-ports, and two PIC f s as shown in Figure 8. In that figure, the assumptions that the conversion gains of the first PIC be positive and those of the second PIC be negative are made; namely 0 chain matrix of PIC 1:, k^ k 2 > 0 - c i chain matrix of PIC 2:, c^ c 2 > 0 0 -c. The driving-point admittance for the network of Figure 8 is n 22 L2 + ^ - y l) (*12 + C 2 k 2 y l 2 k 2 (l - k^ci - k 2 ) C l k l Yl (19) where y,, y,, and y are short-circuit network r parameters of network ^ 22 N. The parameters in (19) are now to be determined by the method of

33 20 y 2 ;. KJ *" Y Network N (RC) PIC 1 k l,k 2 PIC 2 ~ C V~ C 2 0 Figure 8. One-Port Active RC Network Containing Two PIC's

34 21 polynomial factorization. As in the previous section, choose an arbitrary RC driving-point admittance K.. p(s)/q(s) of order m that is equal to or greater than the order of the given Y(s) = P(s)/Q(s), and assign it to y n + y = K J2M y ll y l 1 q(s) (20) Then the following factorization can be obtained by choosing an appro priate positive constant K K x p(s) Q(s) - q(s) P(s) = R^s) R^s) (21) where R- (s) is of degree m and has only distinct negative real zeros. Substituting (20) and (21) into (19) yields ( y 12 + c~k[ y l) ( y l2 + C 2 k 2 y l) RR,(s) x R 2 (s) n c 2 k 2 (1 - k x )(l - k 2 ) Q(s) q(s) (22) ^k^ yl k~ " y 2 y 22 + The following assignments may be made. R 1 (S) " / I *v ^k~ y l ~ K 2 K 3 ~ife> y 12 + R 2 <s) y 12 + c 2 k 2 y x = K 2 ^ (23)

35 where K and K are positive constants to be determined later. Now, y.. 2 and y, can be found from (23) to be C 2 k 2 K 3 R l (s) -( V IX) R 2 (S). y " = - -L-. 1-LJL (24) Y V 12 / x ^ q(s) ' c 1 k 1 / c 2 Jk., 2 y = K 2 (cfc) " C 2 k J K 3 R^s) - R 2 (s) q(s) (25) Again by the root-loci consideration, the expressions for -y-. 2 and v i in (24) and (25) can be made to be realizable RC functions by choosing an appropriate value of K and the sign of K 2 /(l/c k - c^k^) to be positive, since R (s) has only distinct negative real zeros. Next, let n 2 k 2 (1 -y(l - k 2 ) 2 Q(s) y 22 + c^ yl " *~~ k^ " 7 2 ~ K 2 K 3 q(s) (26) Substituting the expression for y' of (25) into (26) gives n (1 - kj(l - k 0) y 22 "" k y C K 2' q(s) ~2 *3 0 Q(s) (27) -r^ ^lk 3 R i (s) - R 2 (s)],

36 23 The expressions for y and y can be found from the expansion of the 22 1 right-hand side of (27) into its Foster form of admittances. Further assumption on k- and k of PIC 1 is needed as (1 - k^cl - k 2 ) > 0 which implies k, k 2 > i or 0 < k x, k 2 < 1 The positive constant K_ is determined so that y_ 1, y 1?, and y«satisfy the residue condition. Because the resultant short-circuit parameters y -, -y- o» ypp> y-i» anc * Yn are realizable RC functions and the residue condition is such that network N in Figure 8 is a grounded one, this procedure yields an active network that has a common terminal in all its constituent two-ports. Singly-Loaded RC-PIC Network An alternative network for realizing an arbitrary driving-point admittance can be obtained by terminating port 2 of the PIC in either Figure 6 or Figure 8 in a one-ohm resistance. Consider the network of Figure 9, an RC-PIC network of Figure 6 with a one-ohm termination. The input admittance for that network becomes

37 24 ( *12> 2 Y =- y * - if- (28) a s < i -y* 1 -v y 22 + k k y l Comparing (28) with (8), it can be seen that the loading of the one-ohm resistance gives a positive input conductance of k 0 /k, which is added. 1 into the denominator of the input admittance without a termination. For the synthesis of any driving-point admittance Y(s) with the RC-PIC network in Figure 9, a modification of (11) is needed as shown below. Y(S) = 2 2 P 1 (s) - s q 1 (s)?-, (s) q n (s) ~~^s) ' " -± (29) q l /P^s) k 2 x /-P (s) ^v \^(s) + rj" \^7 + rj Equation (29) yields for the constituent network parameters Kl = *22 = ^ (30) y.m 12 q,(s)

38 25 *i Y a Network A (RC) PIG k,, k«< < < Resistor in Ohm Figure 9. One-Port Terminated Active RG Network Containing One PIG o Network K (RC) PIC V k : PIC -c 1, L. Q Resistor in Ohm Figure 10. One-Port Terminated Active RC Network Containing Two PIC's

39 26 Same arguments on the residue condition and polynomial augmentation for Figure 6 hold for this situation, since only the expression of y, in (30) differs from that in (12). When port 2 of the second PIC in the RC-PIC network of Figure 8 has a termination of a one-ohm resistance, it has a form in Figure 10, and its input admittance becomes y-,) (y-.o + c o k o y- V ; 12 ' c ^ JlJ Vl2 ' w 2~2 J l. Y * y u + y i " ~~^S "TTud - k 2 > (31) y n l z.,,, v i l ^(y i + l) y iq 2 tche course of procedures for the synthesis of an arbitrary driving-point admittance function with (31) is the same as in the previous section with a few changes in equations. Equations from (19) through (25) are valid for this case. Equation (26) needs to be changed to n c 2 k 2,, (1 ' V (1 - V,.2,. Q(8) y ^[ (y l + 1} " k^ y 2 " K 2 K 3 $s> which implies n y <l - V tt - k 2> 22 " k y 2 = ^T l K 2 E S) (32) 3 «k 1 l K Vc,kJ " C 2 k 2

40 27 The remaining process is to expand the right-hand side of (32) into its Poster form of admittances and to allot all the positive terms to y n 22 and all the negative terms to [(1-k-)(l-k 2 )/k-] y«. It has been shown that the discussion in the previous sections can be applied with some modification to the synthesis procedures with singlyloaded RC-PIC networks. For an unnormalized situation, the terms different from their input admittances without terminations are multiplied by their respective associated terminal conductances. Numerical Examples It is assumed that the realization of a one-henry inductor with an RC-PIC network is desired. Two approaches, one using one PIC and one using two PIC's, are taken. Admittance Synthesis with One PIC For a given admittance Y(s) = 1/s, the decomposition of the type of (10) can be made as vr^ = a)(s+l) - s(l)(l) 1{S) (s+l)(l) - (1)(1) (33) Comparison of (33) with (10) gives P 1 (s) = 1, ^(s) * 1, P 2 (s) = s+1 ^(s) «1 The above identification makes 2 2 p (s) - s q..(s) = 1-s

41 28 which is not a perfect square, and augumentation is required. Let Y(s) be multiplied by (1-s)/(1-s). Then U S ; 8(1-8) lew parameters are identified by (13) and are p[(s) = 1+8, p^(s) = 2s+l q{(s) = 2, q ( 8 ) = s+2 Desired expressions for the companion network parameters are a _ a _ s+1 y ll ~ y 22 " 2 a _ + s-1 y 12 " 2 y-i \ (2s+l) = 1 (l-k^d-kp (s+2) Through the choice of the positive sign for y 1 and k- = k = 2, the realization of the one-henry inductor with the RC-PIC network can be achieved by the network structure shown in Figure 11. Admittance Synthesis with Two PIC's Now, the given driving-point admittance, Y(s) *» 1/s, is to be realized with two PIC's. The conversion gains of the PIC's are assumed to have the following values:

42 PIC 1: PIC 2: \ Then choose n, v s+1 which makes 1p(s)Q(s) - q(s)p(s) = K ± s + so^-1) - 4 Letting 1-1 yields R L p(8)q(a) - q(s)p(s) = (s+2)(s-2) From (24) and (25), -y_ 2 and y- can be found as ^ - ( JL> -y 12 " V 54^ 4K (a+2) - (s-2)/4 K 2 s+4 ~~ r l V 54/ K 3 (s+2) - (s-2) s^4 ~ Choosing K =* - 5/47 and K 3 = 3 gives n A i -7o i Q«1^3 s yi2 = isy = 0.057

43 30 Now by (27) n y s s s\ /0.129 s s+4.).(- s Next, the following assignments may be made n n s y 22 = s, n,, y 2 = -^+^- + 0 ' 4 6 Substituting y_ in (35) into (34) gives ^ = ^ ^ " 3 Thus, the one-henry inductor can be realized by the RC network with two FIG's as shown in Figure 12.

44 31 Y(s) Ohms, Farads Figure 11. Network with One PIC Realizing Y(s) = 1/s o 1.69 Y(s).148" o i i IvNAA l I ' L > /vv s \ I I PIC \'h" 2.. ;; 0 PIC - c r- c 2 = - 2 O Ohms, Farads Figure 12. Network with Two PIC's Realizing Y(s) *» 1/s

45 32 CHAPTER III SYNTHESIS OF OPEN-CIRCUIT VOLTAGE TRANSFER FUNCTIONS In many situations, special interest lies in the realization of open-circuit voltage transfer functions. As in Chapter II, the use of positive impedance converters in RC networks relaxes the restrictions on the function to be realized as an open-circuit voltage ratio. In this chapter, several procedures using the same network configurations as in Chapter II are presented for the synthesis of any given opencircuit voltage transfer function along with some numerical examples, An open-circuit voltage transfer ratio of a real rational function in the complex frequency variable s can be represented in terms of the overall short-circuit admittance parameters as T E 2 " Y 21 (s) " E l ~ Y (s) (36> 22 The ratio of the two short-circuit admittance parameters in (36) is expressed in terms of the constituent network parameters of the specified network structure and is equated to the given open-circuit voltage transfer function. Synthesis with One PIC The network structure shown in Figure 6 is again considered with the voltage references as in Figure 13. The open-circuit voltage transfer

46 33 y l fy _, A E l o Network PIC A A (RC) k l> k 2 ( Figure 13. Two-Port Active RC Network I Containing One PIC E, Network B (RC) O- Figure 14. Two-Port Active RC Network II Containing One PIC

47 34 ratio of this circuit is given by T-> E 2 " y 12 " E l" k l y 22 " <1-V (1 " k 2 ) y l a a where y- 2 and y «are short-circuit parameters of network A. The synthesis procedure is as follows. Let T(s)» P(s)/Q(s) be the prescribed voltage transfer ratio. Choose a polynomial q(s) which has only distinct negative real zeros and satisfies the following degree requirement: a degree [q(s)] maximum [degree [P(s), Q(s)]) - 1 The given voltage ratio can now be rewritten as T(s) = ZM = g(s)/q(s) ( s) {S) Q(s) Q(s)/q(s) (38) Comparing (38) with (37) can give the following identifications: - v - ^ (39) * y 12 "' q(s) a (i-ya-y y 22 " k^ y l. " k^ ^W (40) the right-hand side of (40) is expanded into the Foster form of RC admittances and is recollected according to the sign of each tertn as 1 Q(s) Q 1 (s) Q 2 (s) k i q( s ) q-j/s) q 2 (s)

48 35 where q(s) = q,(s)*q 2 (s), and Q 1 (s)/q 1 (s) and Q 2 (s)/q (s) are realizable 1C admittances. Arbitrary RG admittances, p.. (s)/q (s) and p«(s)/q (s), may be introduced as surplus terms to give 1 Q(s) QAs) + Pl(s) P 2 (s) fv^s) Q_ 2 (s) + P 2 (s) \ q(s) q x (s) q 2 (s) ^(s) q 2 (8) (41) low, all of the positive terms of (41) can be allotted to jjt and all of the negative terms to [ (1-k ) (1-lO/k ]y as a Q x (s) + P x (s) P 2 (s) y 22 = qi(s) + itw (42) (l-k^d-k^ P x (s) Q 2 (s) + p 2 (s) yi 1 = q 1 (a) q 2 (s) Assuming (1-k )(l-k 2 )/k > 0, y in (42) is found to be y-. = 1 (l-k^cl-^) P 1 (s) Q 2 (s) + p 2 (s) q 1 (s) + ~ q 0 (s) This completes the procedure to find the network parameters of Figure 13 from a given voltage transfer ratio. The network structure in Figure 7 can also be used for the synthesis of open-circuit voltage transfer functions with the voltage references shown in Figure 14. Analysis gives, for the open-circuit voltage transfer ratio,

49 36 where y. and y are the short-circuit admittance parameters of network B. The same procedures for realizing a given voltage transfer ratio with figure 13 can be applied for the network structure in figure 14 with k and k«interchanged. In general> this procedure requires an ungrounded structure of network A or network B in figures 13 and 14. (for example, this is necessary when the given transfer function has positive real transmission zeros.) But since the residue condition on network A or network B is always met by introducing surplus terms as in (41), no transformer is required for these synthesis procedures. Synthesis with Two PIC's If a common-ground circuit is desired, the synthesis procedure in the previous section cannot realize positive real transmission zeros. In order to remove any restriction on realizability with a common grounded circuit, two PIC's are needed. The same network structure as in figure 8 can be used for this synthesis purpose with the voltage references as shown in figure 15. Assumptions on the conversion gains of PIC 1 and PTC 2 are made exactly the same as for figure 8; that is, the conversion gains of PIC 1 are positive and those of PIC 2 are negative. In figure 15, the open-circuit voltage transfer ratio between E? and E 1 becomes E!=_ == -y * - c k y i± L-Z i (43) 1 c i k i y 22 + c 2 k 2 y i " ^d-vd-v y 2 where y- 2 and y«2 are the short-circuit admittance parameters of network N. for the realization of a given voltage ratio T(s) - P(s)/Q(s), a polynomial

50 37 _l2. o Network N G PIC PIC 2 Ch (RC) k i' k ; o "V " C 2 o Figure 15. Two-Port Active RC Network Containing Two PIC's

51 38 q(s) is selected such that q(s) has only distinct negative real zeros and satisfies the degree requirement degree [q(s)] maximum [degree [P(s), Q(s)]} - 1 Then the following assignments can be made: " y 12 " c 2 k 2 7l = f (44),n +!2^y. ^V^V =_A_2i l (45) 22 + c 1 k 1 y l k x y 2 Clk q(s) ^D) Assuming (1-k-)(l-k 2 )/k > 0, the identification of y 12, y« y 1, and. y«from (44) and (45) are straightforward. Again the right-hand sides of (44) and (45) are expanded into their Foster forms of admittances. Surn n plus terms may be introduced to make -y- and y satisfy the residue condition. In this procedure, it is always possible to choose all the network parameters, y, y«, y 22, anc * "^IO' as realizable RC driving-point admittance functions. Therefore, the synthesis procedure in this section gives an active RC network with two PIC's which contain a common ground with no transformer. Numerical Examples For illustration, the open-circuit voltage transfer function T(s) - 2 S " 1 (46) s + s + 3 is chosen to be realized. Two approaches, one using one PIC and the other

52 39 using two PIC's, are utilized as follows. Voltage Ratio Synthesis with One PIC For the realization of (46), an arbitrary polynomial q(s) - s+1 is chosen. Then by (39) and (40), we have y i2 = fir (47) - y 22 " (1 - k l )(1 - k 2 ) 1 fs 2 +s + 3 k y l " In [ s+1 With prescribed conversion gains of k = L = 2, the right-hand side of (47) can be rewritten to give 2 a J l 1 (s 2 \ + s + 3 s+1 1 / 4s 7s 's " ' " ' s+1 s+1 Now, the following identifications may be made: a 1, ON 2s y 22 -j (s + 3) + ^- y l s+1 a With an arbitrary choice of y-...'" - 1, the final network realizing the given voltage ratio is shown in Figure 16, which contains an ungrounded structure

53 1/2 2 \AAA -ll- p I Ohms, Farads Figure 16. Network with One PIC Realizing T(s) in (46) 4 1/2 2 O -"i-vsaa- [ o 1/4 = 2/11 11/2 Lv\A/^ II O 4/3 PIC k 2 =2 0 PIC - i - - 2»c* =* ~2 2 4 O Ohms, Farads, Figure 17. Network with Two PIC's Realizing T(s) in (46)

54 41 foltage Ratio Synthesis with Two PIC's Two PIC's are utilized for the realization of the given voltage transfer function of (46). Assumptions on conversion gains of the PIG's are k_ ~ k = 2 and c. = c = 2 An arbitrary polynomial q(s) «s+1 is selected to give, by (44) and (45) - y i* - 4y i - i+r < 48 > y n + S + + y - I y = I (^ 3 (49) y 22 y l 2 y 2 4 I s+1 J K^} Equation (48) can be rewritten as n, _ 2s 1 " y 12 " 4y l " s+1 Identify n 2s_ " y 12 * s+1 y l = t Substituting y_ into (49) gives n I _ I (s 2 + s + 3\ 1 y 22 2 y 2 " 4 I s+1 "4 l S i+tj -4 lst +1

55 42 Let 22 \ < s + 3 > + & 1 /"lis 2 2 Vs n n With a choice of y--» -y-.«, the resultant grounded network has the form in Figure 17.

56 CHAPTER IV N X N SHORT-CIRCUIT ADMITTANCE MATRIX SYNTHESIS In the first part of this chapter, necessary and sufficient conditions on the number of PIC's that is required for the realization of an arbitrary NXN admittance matrix with a balanced RC-PIC 1 s network will be considered. The necessary conditions will be derived by considering the ranks of the given matrix and the equilibrium matrix equation obtained from the active RC network that contains m PIC's, The realization procedures will be given as a proof of the sufficient conditions. It will be concluded that N PIC's are necessary and sufficient for the synthesis of an N X N short-circuit admittance matrix with a balanced network structure. The second part of this chapter is devoted to the development of the procedures for the realization of an arbitrary N X N admittance matrix with a grounded RC-PIC's network. This leads to a constructive proof that 2N PIC's are sufficient. In the third part of this chapter, simplified network structures are assumed and the procedure that realize a restricted class of admittance matrices are given. This will reduce the number of passive elements required. Finally, some numerical examples are given to illustrate the procedures developed in this chapter. Two of the realization procedures in this chapter will utilize the matrix factorization technique, which was used by Sandberg, * Scar borough, and Cox to study several classes of active networks.

57 44 Synthesis Using a Balanced Network Short-Circuit Admittance Matrix of an N-Port Network Containing m PIC's Consider the network in Figure 18 which contains m PIC's embedded in a transformerless (N+2m)-port RC network. Let the following set of notations be adopted: E. (50) N N N+l N+l ^+2 [ N+2 N+m N+m E N+m+l I N+m+l E. =? c ^+ 1^2 I = c I N+m+2 E. 'N+2m N+2m Each of PIC's embedded in the (N+2m)-port network is assumed to have the conversion gains as shown in the figure. The constraints these PIC's impose on the network are E = k. E, c lb Z c " k 2 h (51)

58 45 it A E N-H ^EN+. "Wl ha*: h 4 CHr i f oj- (N+2m)-Port Trans former1es s RC Network < w f E N4m 4 E N4i»f] PIC Wi L Nftn+2 k(d *1 ' k(2) k 2 *E k (2) k (2 > I N+nrf-2 1» K 2 V ^j o- f r " 3^ ^EN+i ^Zm f K l ' K 2 Figure 18 0 N-Port Active RC Network Containing m PIC's

59 46 where k_ and k_ are diagonal matrices and defined by x (1) k l (2) k i k ^ (1) (2) (m) l,et the RC Of-f 2m) port be characterized by the admittance matrix [y] which is partitioned into submatrices after the N and the (N+M) rows and columns. Then a I Cy] \ E (52) y ll y l2 y y 22 y 13 E b 31 y 32 y 33 where

60 47 y ij 1» J * - t y ji 1,2,3 1,2,3 Substitution of the constraints in (51) into (52) gives y ll y 12 + k l y 13 y 21 y 22 + k l y 23 (53) "Vb L/31 y 32 + k l y 33j the following relations can be obtained from (53) h = y ll E a + (y 12 + k l y'l3 ) E b (54) h = y 21 E a + (y 22 + k l y 23> E h ~\ \ = y 31 E a + (y 32 + k l y 33 } E b Combining the second and third equations of (54) gives E b " " (k l y 33 * k 2 y 22 * y 32 + SVza^ (k 2 y 21 + y 31 } E a (55) If (55) is substituted into the first equation of (54), we get I a - [y n - ( Vl3 )(k l y 33 + k 2 y 22 + y 32 + k lv23 )^ (k 2 y 21 + y 31 )] E f

61 48 Ihe admittance matrix of the active N-port in Figure 18 is given by Y = y ll " (y 12 + k l y 13 )(k l y 33 + k 2 y 22 + y 32 + k l k 2 y 23 )_1 ' (56) (k 2 y 21 + y 31 } Necessary Number of PIC's It is assumed that a given matrix of a real rational function Y has a pole at s-s, of multiplicity k (ki= 1) where s is off the negative real axis. The coefficient matrix of the Laurent expansion of Y about this k order pole is obtained by evaluating at s-s., which must be equal to the coefficient matrix of submatrices described by (56). IF* (s-s.)^f s=s = (e-s^*? s=s i - (s-s i ) k (y x2 + k iyi3 ) (57) (k l y 33 + k 2 y 22 + y 32 + k l k 2 y 23 )_1(k 2 y 21 + y 31> S-S. 1 The fact that y--, y., and y-_ are submatrices of the admittance matrix 'll' 12* 'IS of a passive RC network implies (s - s i )ky n S=S = [0] and (y 12 + fc y ) S^S and (k 2 y n + y^) are finite. The righta-8 hand side of (57) then becomes

62 49 (s-s. )*i 8-B - - (y 12 + Vi3> S-S. 1 (S - S i )k (k l y 33 + k 2 y 22 + y 32 + k 1 k a y 2 3 ) " 1 S=S. 1 (k 2 y 2l + ^ s=s, 1 Since the rank of a matrix product cannot exceed the rank of any of its 17 constituent factors, rank ^(s-^s ) Y. Jr S rank (y 12 + N y 13^ S=S i s-s. l (58) The rank of (y- 2 + ^713) s=s i is limited by the size of its matrix which, in this case, is N X m. It may be assumed that the rank of (s -s.)^ is I. The inequality in (58) becomes s=s. l N = minimum (N, m) The aibove equation requires m g N Therefore, N PIC's are necessary for the synthesis of an N X N shortcircuit admittance matrix which has a pole of rank N and multiplicity k (k =- 1) off the negative real axis. Realization Procedure The sufficiency of a specific number of PIC's and a transformerless passive RC network for the realization of a prescribed N X N short-circuit

63 50 admittance matrix can be proved by demonstrating a realization procedure. From the previous part, N PIC's are necessary for this realization. In this part, K PIC's are hypothesized for the sufficient number of PIC's. Mth m*m, (36) is used to derive a short-circuit admittance [y] of the 3M-port passive RC network in Figure 18. The resultant admittance matrix [y] must satisfy the conditions to be RC realizable without transformers. These conditions are (1) all the diagonal terms of the matrix are passive RC driving-point admittances, and (2) when it is expanded into its Foster form of admittances, the coefficient matrices are dominant. Let the prescribed N x N short-circuit admittance mattrix Y be given by [P] Y - ^ (60) where Q is the common denominator polynomial of the elements in Y and [p] is a matrix of polynomials. Choose an appropriate N X N RC admittance matrix that satisfies the conditions in (59) and employ the notation y n = -ij- (61) in which degree [p..] = degree q = M = N L, where L - maximum [degree {M,Q}}. then I. QEP.,3 - q[p] W! ^ii " Y ^ = 1Q" (62)

64 51 where [R] has a maximum degree of Mf-L, It is well known * ' * that the matrix [R] in (62) can be written as the product [R, ][R ] of matrices, i z where [R ] and [R«] are of degrees M and L, respectively, and the determinant of [R ] has only distinct negative real zeros different from those of q. This factorization can be achieved by choosing p., in matrix [p..] of (61) sufficiently large so that the determinant of [R] has MN negative real zeros. Equation (62) then becomes y n " Y - -^5^" <«> Substituting (63) into (56) gives (y 12 + Vl3 )(k l y 33 + V22 + y 32 + k lv23 ) " 1 (k 2 y 21 + y 31 } (64) [R^fej] qq Without any loss of generality, some simplification can. be made by assuming the following: (i)? 12 -y 2I -E.«(2) all the PIG's are identical with positive conversion gains of k and k, that is, \ «^[l] k 2 = k 2 [l] where k, k > 0 Introducing the above assumptions into (64) gives

65 52 - _ ^ x_ [RJlRol k l*13 (k l*33 + k 2 y 22 + ^2 + k lv23 } y 31 = ~^Q < 65 > At this stage, let [R,] y i3 = ^ q < 66 > where K- is a constant to be determined later. Substituting (65) into ( 4) yields - k 2: - i - - K? Q CV' A <*j [R 2 3 *33 + k^?22 + k^?32 + k 2^23 " ~ q det D^J < 67) Since det [B«] has only distinct negative real zeros different from those of q, the denominator., q det [R ], can be represented as I q det [R ] = K 2Z (s+0.) (68) L L x i»i 0 < a- < a 2 <... < a_ in which K is a constant. The maximum degree of the numerator of (67) is ;. L + NL + (N-l)L ^ 2NL which is equal to the maximum degree of the denominator of (67). Hence, the quantity in (67) is regular at infinity and the right-hand side of (67) can be expanded into the Foster form of admittances and can be represented by

66 53 K' Q [RJ 1 " Adj [R ] ^, L qlt[r 2 ] " W + I [D ± 3 ^ (69) 0 < a i < a 2 <ff 2HL where [E] and [D.] are real coefficient matrices. Further simplification of this procedure can be made by assuming y 3 a diagonal matrix. Then y 23 S y 23 * y 32 (7G > lie real coefficient; matrices [E] and [D.] in (69) are now decomposed to give [E] - [E 1 ] - [E] d (71) Co,] - [D:] - [ Di ] d where [E 1 ] and [B'] are dominant matrices with all positive diagonal terms, and [E], and [D ] are diagonal matrices. Considering (67) and (69) along with (70) and (71) yields k y 33 + k7 y 22 = [E^+ 1 k 2NL 1 ^ ± : < 72 > i-1 i 2NL -?23." r?-b: {M d + I [D i ] a i^} 12 i«i I Finally, the multiplicative constant K. in (66) is adjusted so that y n and y- satisfy the condition of dominancy* Observation of (72) reveals

67 54 that, in order to fulfill the condition of dominancy on y\_ v y_«, and 723» ^ is sufficient that conversion gains of the PIC's have the values which satisfy the following constraint: (l-k^cl-kp > 0 therefore, the short-circuit admittance matrix [y] becomes a realizable RC admittance matrix that fulfills the conditions in (59). Ofeeorem 1 For the realization of an arbitrary N X N matrix of real rational functions in the complex frequency variable as a short-circuit admittance matrix of a transformerless active RC N-port network, (a) it is, in general, necessary that the network contains N PIC's; and (b) it is sufficient that the network contains N PIC's embedded in a 3N-port RC network. Synthesis Using a Grounded Network Since the balanced network resulted from the previous section suffers from numerous practical disadvantages, it is desirable to find some procedures that will realize a given admittance matrix with a grounded active RC network. More than N PIC's are expected to be required, and it is assumed that a network structure that has 2N PIC's embedded in a transformerless (4N+1)-terminal RC network as shown in Figure 19 will be used. The passive RC network of that figure must have a short-circuit admittance matrix realizable with an unbalanced network structure that satisfies the foilowing conditions: (1) the matrix is symmetric (73) (2) diagonal and off-diagonal terms are positive and

68 55 (N+l) (4N+1)- Terminal I2N+I1 Passive X2&2X Tr an s former1ess (3N) CD o m RC Network (3N44) 1. PIC 1 PIC PIG 2 PIC I ojsl (4N) PIG -1 PIG -X, Figure 19. Active RC Network Containing 2N PIC's

69 56 negative RC admittance functions, respectively (3) when the matrix is expanded into its Foster form of admittances, its coefficient matrices satisfy the dominant condition. Assumptions on conversion gains of the PIC's in Figure 19 are as follows: PIC 1; k i ; k r k 2 > 0 PIC 2; c l 0 -c ; c v c 2 > 0 To analyze the network in Figure 19, let the short-circuit admittance matrix [y] of the (4EH-1)-terminal passive RC network be partitioned into 16 N x N submatrices to give F 1 j y n I y n y 12 y 13 y 14 y 22 y 23 y 24 (74) 31 y 32 y 33 y 34 -*J ;y 41 y 42 y 43 y 44 _J where y.. = y ^ ij ji for i-1,2,3,4, and j=l,2,3,4, and the I's and E's are column matrices of port currents and voltages, respectively, each of which consists of N variables. Subscripts, a, b, c, and d represent the ports 1 through N, 1544 through 2N, 2N+1 through 3N, and 3NHKL through 4N.

70 57 the constraints imposed on the PIC's are E * -c n EL c I D E, - k n E d 1 e I s= c I - -=- T c C 2 % k i 2 d (75) Substituting (75) into (74) gives 11 (y 12 - c x y 13 - c ^ ) T 1 ~ C I - ^ I C 2 X b k 2 <1 21 '31 (y 22 " c l y 23 "" c l k iy24 } (y 32 " c l? c l k iy 3 4 ) \J (76) a L y 4i (y 42 " c l y 43 " c l\ 7 ifi> iyrbitrarily, it is assumed that r 13 - y 3 1 = y 24 = y 42 - y 23 = y 32 = [ ] (77) Rearranging (76) and substituting (77) into (76) gives C l k l- 1 ~ K - (Yoo C l -. C l k l-, ^k '2 2 y 44 + ct y 33 + cjt y 43 + TT Y 34 } 2 2 (78) ( c ^ y41 * y 21> E a Then the admittance matrix Y at ports 1 through N of the active network in Figure 19 becomes

71 58? - ^11 - (y 14 " ^ c k y 12> < ^kf y 22 + y 44 + kf y 33 < 79 > k + kt y 43 + k 2 y 34 )_1 (y 41 ' C 2 k 2 y 21 ) 1 With the network arrangement in Figure 19 and a given short-circuit admittance matrix Y, the problem is to find the short-circuit admittance submatrices in (74) so that the matrix [y] of the (4N4-1)-terminal passive network satisfies the conditions in (73). Let the prescribed N X N short-circuit matrix be denoted by Y = 1*1 Q Choose an appropriate H X N passive RC admittance matrix y and denote it by [p, J y u = -f~ (80) Subtracting Y fromy.. n gives 11 QCP..] - qm M y V Y = i JJ :.. n =* T- (81) 11 qq qq The matrix y,, chosen must satisfy the following properties (1) degree [p..] = degree q ~ M = NL and L = maximum {degree ([?], Q)} (2) [p..]/q satisfies the conditions in (73) (3) [R] can be written as the product [R,][R ] of matrices and the 1 2 diagonal terms of [R«]/q are positive RC admittance functions,

72 59 where degree [R ] = L and degree [R ] = M. the factorization of [R] into [R.,][R 0 ] can be achieved by choosing p., in (80) to be sufficiently large so that the determinant of [R] has MN distinct negative real zeros. Next, choose a transformation matrix [j] of -\ -1 real numbers such that [J] exists and each term of [j][r 2 ]/q becomes a positive RC admittance function, which is always possible since diagonal terms of [R«]/q are positive RC admittance functions. Premultiplying [R«] by an identity matrix [j]~ [j] gives for (81) _ [RJCJI'^JJCRJ MM 7ll " Y " ~ iq l IS" 2 " < 82 > where [Ri] = [R,][j]~ and [R'] = [j][r 2 ]. Assuming that all the inverses exist, (79) and (82) give c 2 k 2 k c37 y 22 + y 44 + V y 33 + k7 y 43 + V (83) "5 ^41 " ^Vzi^^^" 1^! 3 " 1 ^14 " ^*12> Now, the following identifications may be made: [R*] *«-.Wzi -**!**-?- (84) 1 - [R P 14 c^ y.o-ik, / 12 1 q where K and K are positive constants to be determined later. Expressions for -y 1, and -y-«can be found from (84) to give

73 K x,-jt K 2 [R ] L - [Rj] y i2 = - ~T~tx: q - c 2 2 " c 1 k 1 (85) K x 2 ^ rmit ^ " c 2 k 2 [R i ] y = ± 14 ", 1 c rt 2 k 2 - c-k, 1 1 Since [Rl]/q is a matrix of positive RC admittance functions, -y.- and -y../ in (85) can be made to be matrices of positive RC admittance functions by choosing proper signs and appropriate values of c-, k-, c~, k, and K_. Substituting (84) into (83) makes C 2 k k QCl] QLU c.k, y '22 + y '44 + k7 k. y 33 '33 + K k, y 43 '43 + k 2 y 342'34 = K l K 2 T" (86) The right-hand side of (86) is a diagonal matrix and all the submatrices in the left-hand side of (86) can be assumed to be diagonal. Then y 34 y 43 c 2 k k / 1\ 2 QM Zk7 y 22 + y 44 + k7 y 33 + y 34 \ k 2 + = kj K l K 2 T~ n I q q where Qu/q I and Q q /q are positive RC admittance functions. Let the re- maining submatrices be identified by

74 61. k 2 " k y 2 Q^i] 33 _ v x? ^kf y 22 + y 44 + ^ y" = K^ K' 1 Jx- ^IX- ^ q (87 > " y 34 V JS.- JS. V Q 2 [i] i + k ' k 2 + k 2 q Care must be taken in choosing the values of k and k? such that (l-k x )(l-k 2 ) > 0 is satisfied. With such a choice, the dominancy condition on y q ~, y,,, and y 3 «in (87) can always be satisfied. Finally, the value of K is adjusted such that y.^, y 12, and y-, satisfy the dominancy condition. the resultant admittance matrix [y] meets all the requirements for the realization with a (4JW-1)-terminal passive RC network, and this proves the following theorem. theorem 2 For the realization of an arbitrary N X N matrix of real rational functions in the complex frequency variable as a short-circuit admittance matrix of a transformerless grounded active N-port RC network, it is sufficient that the network contains 2N PIC's embedded in a (4N#1)- terminal RC network. Alternative Synthesis Procedures for a Restricted Class of Admittance Matrices the procedures developed in the previous two sections served to realize a general admittance matrix. Since their computations require the matrix factorization technique, these synthesis procedures are some-

75 62 what cumbersome. The network structures of Figures 18 and 19 can be simplified in certain special cases, which brings a considerable saving in the number of passive elements used in the resultant network. The admittance matrix that is considered here is assumed to have the property that it is an N X N matrix of real rational functions in the complex frequency variable s having only L simple poles on the negative real axis in the complex frequency plane and no more than L+l zeros Let this prescribed N X N matrix be denoted by v - DLL (88) q = (s+0 1 )(s+a 2 )...(s+a L ) L where degree q = L degree [p] = B i Lfl A simplified structure to be utilized for the realization of (88) is shown in Figure 20 which is obtained by shorting each of the first N ports of Figure 18 to the corresponding port of the third group of N ports. The partitioned equation of this passive 2N-port network is of the form: I a + I c y ll y 21 y l2 y 22 E, b (89) The PIG's used impose the constraints:

76 63 2N-Port Passive (N+l) (N4-2) Transformerless RC t Network (2N) /"\ \J - (1) o PIC (2N+1) k r k 2 KJ~ (2) (2N+2) k If 1* K 2 I i I O KJ (3N) rv- 00 l! 1 1 > *> Figure 20«Simplified Jkcttwm 1 Network Containing N PIC's

77 64 E a = k l E b (90) I c «-k 2 0 I, b Equations (89) and (90) yield k h - (y n + ^ y22 + V21 + IT y 12 > E" a Thus, the short-circuit admittance matrix of the active N-port network in Figure 20 becomes k 2 - Y = y ll + k^ y22 + k 2 y 21 + T y 12 < 91 > For the realization of the given N X N admittance matrix with (91), expansion of (88) denoted by into its Foster form of admittances can be used and L? = 1 [D i ] Sk- + [E] + [F] S <92) i=l i where E^], [E], and [F] are coefficient matrices. For convenience, submatrices, y and y^, are assumed to be diagonal matrices. Then, the following identifications may be made: y u + k r L y 22 + k 2 y 2i + k: y i2 L [D i ] -dr + w + m * (93) 1 l V21 + L i=l r y i2 = d I c^ i^-+ w + m s 1 1=1 i ±

78 65 (* - f) Hi = d I ( k i k 2 tv' - [D i ] } i^: 1 i=i i + fo [.]' - M} + fe W - M} s where the letters, d, od, over the equality signs denote equality between the diagonal and off-diagonal terms, respectively. In order that the resultant matrix [y] satisfy the dominant condition, which is required for a realization with a transformerless network, it is sufficient that conversion gains of the PIC's satisfy (l-k^d-k^ > 0 This requirement is simply the consequence of the equality over diagonal terms on (93). It is always possible to choose appropriate conversion gains of the PIC's such that the admittance matrix [y] of the 2N-port passive network becomes realizable without transformers. This above leads to the following theorem. Theorem 3 An N X N matrix of real rational functions in the complex frequency variable having L simple poles on the negative real axis in the complex frequency plane and no more thanl-f-1 zeros can be realized as a shortcircuit admittance matrix of a transformerless active network having no more than N PIC's embedded in a 2N-port RC network. Another simplified procedure can be obtained by utilizing the network structure in Figure 19 by shorting each of the first N ports to the

79 66 corresponding port of the last group of N ports as shown in Figure 21. This procedure yields an active network having a (3N-KL)-terminal passive RC network and 2N PIC's with a common terminal. The assumptions on conversion gains of the PIC's are the same as for Figure 19. If the shortcircuit admittance matrix [y] of the (3TM-1)-terminal passive RC network in Figure 21 is partitioned into 9 submatrices as before, the following matrix equation holds: I + I. a b 11 y 12 *13 X b y 21 y 22 y 23 E b (94) \ y 31 y 32 y 33 The constraints imposed by the PIC's are E - k n E, - -c-k- E a lb lie (95) T =<-k T - Xc T d 2 2 c 2 b Arbitrarily, it is assumed that y 23 " y 32 " [0] (96) Introducing (95) and (96) into (94) gives

80 (N+l) Figure 21«Simplified Active RC Network Containing 2N PIC's

81 68 which gives for the short-circuit admittance matrix of the active N-port network in Figure 21 Y = y ll + kj y 22 + c^ y33 + 1^ y12 + V21 (97) 1 ~., - c k y 33 " c 2 K 2 y 31 low, the given N X N matrix in (88) is to be realized with the equilibrium equation of (97). The prescribed Y is again expanded into its Foster form of admittances, and the same notation is employed as in (92). For simplicity, it is assumed that y,-, y 22» and y are diagonal matrices, and y^^ and y - are off-diagonal matrices. Comparing (88) with (97), diagonal and off-diagonal terms of (88) must be equal to the corresponding terms of (97), or k c k?u + kf y 22 + sir y 33 + IT *i2 + hhi i I ^ j ^ i =s i i (98) + [E] + [F] s kt y l2 * k 2 y 21 " ctkt y 13 ~ C 2 k 2 y 31 ^ 1 [D i ] S5T <"> 1 11 i^l 1 + [E] +.[F] S L the dominancy condition on [y] can be satisfied by choosing proper values of c^, Cji k,, and -k- and by making the appropriate decomposition of the right-hand side of (98) The off-diagonal terms of y _ and y. can be 1Z 13

82 69 made to be negative RC admittance functions, since y,«and y_- in (99) have different signs. Therefore, the resultant admittance matrix [y] is realizable with a (3N+1)-terminal RG network without transformers. This procedure is simple to apply for the realization of a restricted but very important class of admittance matrices, since it does not involve either the root-loci considerations or the matrix factorization processes. This synthesis procedure proves the following theorem. Theorem 4 An N X M matrix of real rational functions in the complex frequency variable having L simple poles on the negative real axis in the complex frequency plane and no more than L+l zeros can be realized as a shortcircuit admittance matrix of a transformerless grounded active network having no more than 2N PIC's embedded in a (3N-M)-terminal RC network. Numerical Examples The following 2x2 admittance matrix is assumed to be given s+4 s-3 s+4 (100) tis matrix will be realized first by the synthesis procedure with N PIC's and then by the one with 2N PIC's. Synthesis with N PIC's Since the given matrix has a pole at s=s-4 and the number of zeros does not exceed the number of poles by more than one, the procedure developed for Figure 20 can be applied here. If Y in (100) is expanded

83 70 into its Poster form of admittances, it has the form 1/4 1-3/4 1/2 + s-f-4 "1/4 0 7/4-1/2 By (93), let 1/4 0-1/4 0 y ll + y 22 * ( k 2 + T) k l y 12 * 0 1/2 s-*4 0-1/2 \ k l k 2 kj y 12 J (-3/4)1^-1 k k 2 +3/4 0 0 (7/4)^^ s44-7/4 0 Mith the choice of k- = k«= 10, identify 1/2 0 ~1 1/2 0 y ll ~ y 22 "* 0 1/2 shh4 I 0 1/2 (101/10)y 12 & -3/ /2 s+4-5/ / / /4 (9999/l0)y 12 tf -7/4 0 s44 403/4 0

84 thus, the resultant [y] of the 4-port passive RC network becomes 1/ [y] 0 1/ / /2 1/ / s-* / /2 L_ Synthesis with 2N PIC's Assuming the network structure of Figure 21 for the realization of the given matrix in (100), the problem is to find a set of shortcircuit parameters of the 7-terminal passive RC network. Application of (98) and (99) with the assumption k - k«c- - c 0-10 gives I Z 1 Z 1/4-1/4 0 y ll * y 22 + y 33 *" 10 y l2 1/2 + s /2 0-1 " y y s44-1 (Continued)

85 y 13 * 100y 31 -d 1/4 0 s /4 0 With proper decompositions, the submatrices are identified as 1/3 0 1/3 0 y n -y 22 y /3 s+4 0 1/ y s y s t "* y 21 ** " y 12» - t " y 31 " " y 13 These submatrices give the admittance matrix [y] that is realizable with a transformerless grounded 7-terminal RC network. The result is as follows w * , ?0,0Q (Continued)

86 f s o '«' # <J <*}<*}

87 74 CHAPTER V N x N VOLTAGE TRANSFER MATRIX SYNTHESIS A voltage transfer matrix [T] is defined by the relation between two sets of voltages as E' E 2 = [T] H T T T T J '2 J T T "IJ E I This I x J transfer matrix can be made to be a square matrix T by introducing some additional arbitrary voltage variables. Furthermore, each row can be made to have a common denominator by the augmentation of polynomials. The resultant transfer matrix becomes

88 75 E 2 = T (101) N P la P 12^1 P IN/ Q I p 2r Q 2 I E. _ P KA P N2' Q N P NN' Q N E N where N = maximum (J, I). The rational matrix in (101) can be rewritten as a product of two matrices V Q i P 2l' Q 2 P i 2 /Qi P 22' Q 2 P IN/ Q I P 2N^2 - [Q] -1 CP] (102) %i/% w m/% P NN' Q N in whifh [QO and [P] are defined by [Q] %

89 IN [P] = N "Nl N2 NN where Q's and P's are polynomials. Therefore, any voltage transfer matrix can be written as a product [Q]~ [P], where [Q] is a diagonal matrix. In this chapter, procedures for realizing an arbitrary N x N voltage transfer matrix of the form (102) will be considered. Networks used for this realization contain resistors, capacitors, and PIC's. It will be concluded that N PIC's are, in general, necessary and sufficient for the synthesis of a given N x N voltage transfer matrix with a balanced network structure, while 2N PIC's are sufficient with a grounded network structure. Synthesis Using a Balanced Network The structure in Figure 18 contains m PIC's embedded in a transformerless RC (N+2m)-port network and may be used for the realization of an arbitrary N X N voltage transfer matrix. In general, N PIC's are necessary for this realization. The proof is similar to the shortcircuit admittance matrix case and is given in Appendix II. With m = M, all the derivations of equations in Chapter IV are applicable in this case. The first N variables of voltages in Figure 18 are related to the second N variables of voltages as in (55) and it will be rewritten as

90 77 E b = " (k l y 33 + k 2 y 22 + y 32 + V^" 1 " 0^! + y 31 } E a Again it is assumed that k x = k^i] k 2 = k 2^- 1 -' transfer matrix between E and E is, then, given by cl p T = - ( kl y 33 + k 2 y 22 + y 32 + k^y^)" 1. ( k ^ + y" 31 ) (103) By (102) a given voltage matrix is made to have the form T - [Q]" 1 [P] (104) To realize (104) with (103), choose an arbitrary polynomial q(s) that has only simple negative real zeros and satisfies the following degree requirement: degree [q] ^ maximum {degree {[Q], [P]}} - 1 The given voltage transfer matrix can now be rewritten as i. {151} HI (105) Comparison of (103) with (105) gives the following identifications: k l y 33 + V22 + y 32 + k l k 2 y 23 = q (106)

91 rpi k 2 y 21 + y 31 = " " ^107^ q Since matrix [Q] is a diagonal matrix, submatrices, y,, Yoo* y?v an(^ y^y«* can ke assumed to be diagonal matrices. The right-hand side of (106) is decomposed into the difference of two diagonal matrices of positive RC admittances, and surplus terms may be introduced as [Q] [Q ; ] [Q? ] q ~ q x " q 2 [Q x ] + CP X ] CP? ] k l y 33 + k 2 y 22 = 1p-" + 17 (107) - «- 1 r [Q 2^ + ^ " Y 23 " y '+ k x k 2 I q 2 [p i^ q x J where [Q-,]/^-,, [QQIAU* [p-id/q-i* [PolAlo are diagonal matrices of positive!c admittances. By choosing appropriate surplus terms and values ofk.. and k, the dominancy condition can always be satisfied. It is necessary that the values of conversion gains satisfy the following: (l-kpci-kp > 0 Note that the choice of y... is arbitrary, since it does not appear in the expression for the voltage transfer matrix T in (103). Ungroundedness of the realized network comes from the fact that [p]/q in (107) is not, in general, a matrix of positive RC admittances. But the realization with a transformerless network is always possible,

92 79 since the dominancy condition is satisfied and the diagonal terms of the short-circuit admittance matrix [y] of the passive 3N-port RC network are positive RC admittance functions. This leads to the following theorem. Theorem 5 For the realization of an arbitrary N x N matrix of real rational functions in the complex frequency variable as a voltage transfer matrix of a transformerless active RC 2N-port network, (a) it is, in general, necessary that the network contains N PIC's; and (b) it is sufficient that the network contains N PIC's embedded in a 3N-port RC network. Synthesis Using a Grounded Network When a grounded network structure that realizes an arbitrary N xn voltage transfer matrix is desired, the network of Figure 19, which contains 2K PIC's can be utilized. In that figure, the first and second N variables of port voltages are used to formulate a desired N x N voltage transfer matrix. With the same arguments and assumptions as in Chapter IV, the two sets of voltages are related as in (78) to give \ = (^22 + 7j-? 44 + T z ^33 + Zf 2 y 43 + ~ *34> " ' ( ^ hi -? 21> f a The transfer matrix between the two sets of voltages becomes C, K, C-, Ct, C, K* > T = /v + -±-L v + -A y + L- 7 + Ji: j- 1. (109) 1 ^y22 c 0 k y 44 c 0 y 33 cjc 0 y 43 c 0 y 34 ; fekl Z. y 41 " y 21 )

93 80 A. given N x N transfer matrix can be modified with the same q(s) chosen in the previous section to give -1 f = (Mf m l q J q (110) Comparing (109) with (110) can give the following identifications: y 22 + C l k l ~ C l k l - [Q] ix y 44 + z: y 33 + ^ - y " + ~ y = '2'2 2 c 2 k 2 34 ~ " ' q (111) c 2 k 2 y 41 " y 21 q (112) From the fact that [Q]/q is a diagonal matrix, all the submatrices in (HI), Yoo> y 44 5 y 33' y 43' an<^ y34 s a s<d can l be assumed to be diagonal matrices. To choose all the submatrices from (111), the decomposition of [Q]/q and the method of surplus terms as in (108) can be used. Now, ip]/q is represented as the difference of two matrices as follows: [P] [P x ] [P 2 ] (113) where each term of [P^/q and [P 2 ]/q is a positive 1C admittance function, From (112) and (113), let 1 ȳ A1 = c 2 k 2 '41 CP 2 ] [P x ] -y 21 =

94 81 The dominancy condition on submatrices can be satisfied by choosing proper conversion gains, c., c, k 1, and k. The following requirement on k, and k~ also applies in this case: (l-k^a-kp > o One of the choices on the conversion gains is to choose both k., and k 1 2 largej and both c. and c~ small. The resultant short-circuit admittance matrix [y] of the passive (4N+1)-terminal network can be made to satisfy all the conditions on grounded realizability that are described in (73). The choice of y,, is also arbitrary. This procedure leads to the following theorem. Theorem 6 For the realization of an arbitrary N x N matrix of real rational functions in the complex frequency variable as a voltage transfer matrix of a transformerless grounded active RC 2N-port network, it is sufficient that the network contains 2N PIC's embedded in a (4N+1)-terminal RC network. Numerical Examples As an example of the voltage transfer matrix synthesis, the following relation (s-l)/s (s+l)/s E r 2 _5/(s-l) <s+2)/(s-ti s/(s-l) is to be realized with an active RC-PIG*s network. To make this matrix

95 a square matrix, a new variable E' is introduced to give Ej' (8-1)/S (s+l)/s s/s E l E f 2 = s/(s-l) (s+2)/(s-l) s/(s-l) E 2 E i i 1 1 \ which implies T = -1 r~ "1 s 0 0 s-1 s+2 s 0 s-1 0 s s+2 s 0 0 s+2 s+2 _ s+2 s+2 Choosing q(s) = s+2 gives fl2l I q. J q "s/(s+2) (s-l)/(s+2) l "(s-1)/(s+2) (s+l)/(s+2) s/(s+2) l/(s+2) 1 s/(s+2) This given voltage transfer matrix is to be realized first by the cedure with N PIC's and then by the one with 21 PIC*s. Transfer Matrix Synthesis with N PIC's! By (106) and (107) with a choice of k = k = 10,

96 s/(s+2) &22 + y 33> + 101y 23 = s/(s+2) s/(s+2)_ 10+9s/(s+2) l/2+17s/2(s+2) s/(s+2)J (s~l)/(s+2) (s+l)/(s+2) s/(s+2)~ 10y 21 + Y 31 = " s/(s+2) 1 s/(s+2) L I Assuming y^ - [0] and y^3 - -y^ l+s/(s+2) 0 0 y 22 = 0 l+s/(s+2) l+s/(s+2). p.0+9s/(s+2) 0 0 " " y 23 " y /2 +17s/2(s+2) 0 0 Q 9+10s/(s+2). (s-l)/(s+2) (s+i)/(s+2) s/(s+2)" " y 21 ~ 10 s/(s+2) : 1 s/(s+2)

97 84 Jtgain y 11 is arbitrary. Therefore, the resultant [y] becomes the follow* 11 ing: [y]» -1 X -±- 20 o 10 zl _-l zl X X 0 zl x_ t~ -1 il zl ~ M ±ii zl ~f s+2 X X X X X X zl 20 zl ^ 1 10 zl 10 ^JL 10 ^ 0 20 _-l z± zl 10 z± o! o o ~M 91 0 _ o "ifi M 91 Note: The crosses in the matrices represent arbitrary elements.

98 85 transfer Matrix Synthesis with 2N FIC's (112) By choosing ^ = k = 10 and c = c = l/lo, and from (111) and y 22 + y 33 + y 44 + ( 101 / 10 )y 3 4 = 0 (s-l)/(s+2) 0 s/(s+2) y 41 " y 21 (s-l)/(s+2) (s+l)/(s+2) s/(s+2) s/(s+2) 1 s/(a+2) Making proper decompositions of the above equations gives y 22 y s/(s+2) s/(s+2) s/(s+2) 10 y 33 ~ y 34 n s/(s+2) /2 +17s/2(s+2) s/(s+2) y 21^ 3s/2(s+2) s/(s+2) 1 (s+l)/(a+2) s/($+2) 1; s/(s+2) 1 1 " y 41 * 1/

99 86 The resultant short-circuit admittance matrix [y] of the passive 9- termlnal network satisfies the conditions in (73) and becomes realizable with a grounded network.

100 87 CHAPTER VI STABILITY AND SENSITIVITY CONSIDERATIONS Two of the most common drawbacks of active synthesis methods are the possible instability of the physical networks and their high sensitivity with respect to parameter changes. In this chapter stability properties of the PIC are investigated and attempts are made to find conditions for the stability of the PIC with any termination in terms of conversion gains. Sensitivity functions for the network with a PIC are also studied with regard to the changes of conversion gains. Effort is made to find an insensitive RC-PIC network. Stability Properties of a PIC It is well known that an NIC is a potentially unstable device in view of the stability invariant factor. Schwarz and Brownlie 19 have given proofs that, if one port of a NIC is open-circuit unstable (OCUS), then the other must be short-circuit unstable (SCUS). The stability invariant factor originally defined by Rollett is used to examine instability of a terminated two-port as in Figure 22 and has the form 2 Re(h n ) Re(h 22 ) - ^(\ 2^2l ) l b 12 h 2ll

101 88? 1 z, ll ' Active Device h ll h 12 < y 22 L h 2I h 22-J Figure 22 9 Double Terminated Active Device z l (or y,) o- ~<y -> (7) x ll vcvs u,(a) E^ At CCCS M U 2^X2! (D E Y, '22 I PIC o ^» *, Fi ure 22. PIC with a Feecraclc Load

102 89 where h s i-l>2, are hybrid parameters of the active device. In order that the network in Figure 22 be unconditionally stable, 71 ^ 1 for all values of frequencies. Potential instability condition is determined if -1 s T < 1 for all frequencies. For an ideal PIC, h == h = (1 h 12 = k l» h 21 = " l^2 which implies T\ = 1 Therefore s an ideal PIC is absolutely stable with terminations as in Figure 22, In the theoretical work on active RC synthesis procedure developed in Chapters II - V, the PIC was regarded as an ideal element having no dynamics. However, a practical PIC can be represented in terms of dynamic gains of controlled sources. Consider again the basic RC-PIC network of Figure 5(a) with the PIC replaced by the arrangement of Figure 2. This network has a PIC with a termination across the input and output of the PIC as shown in Figure 23. It was shown that the input immittances at port 1 of Figure 23 were

103 90 11 (l-k 1 ) (l-k? ) where z. (or y.) is a passive RC network, To examine the stability dynamics, let u«(s) and u«(s) be dynamic gains of the GVS and CCCS of Figure 23, respectively, and be represented by u 1 (s) K- 1 + s N 1 x (s) 1 + s D 1 (s) (115) K 2 + s N 2(s) 2^' 1 + s D 2 (s) where K, and K«are constants, and N,(s), N»(s), B,(s), and D_(s) are polynomials of the complex variable s. Since the controlled sources are assumed to be stable, [l + sd.,(s)] and [l + sd«(s)] must have all of their zeros in the left-half plane and must satisfy degree D (s) > degree N_(s) degree B_(s) > degree N«(s) From Figures 2 and 23, dynamic conversion gains of the PIC can be written in terms of u-(s) and u_(s) as k 1 (s) = u 1 (s) + 1 (116) k? (s) = u«(s) + 1 which implies

104 91 : [l-k (s)][l-k 2 (s)] = u (s) u 2^ lor generating negative immitan-ces, the nominal values of k. (s) and k (s) 1 2 must satisfy the relations: k^s), k 2^ > s^o [l-k 1 (a)][l-k 2 (s)3 s=0 > 0 The above requirements imply K + 1 > 0, K 2 +1>0 K l K,«> 0 RoWj if (115) and (116) are substituted into (114), the input immittances have the form {K t s[n 1 (s) + ^(s)]}^ + 8D 2 (s)] 2ii(s) =. ^ ^ ^ ^ ^ ^ _ ^» ^ z ( s) (117) Y ii Cs) " " "pij^tttrij^^ [K x + sn 1 (s)][k 2 + sn 2 (s)3 y i (s) It is apparent from (117) that, If- [K. + sh,(s)]i and [K«+ sl«(s)] have only left-half plane zeros, port 1 of the EXC circuit in figure 23 becomes open-circuit stable (OCS) and short-circuit stable (SGS). The input immittances at port 2 of Figure 23 are,, when an ideal PIC is used,

105 92 k _ 2 Z 22 " " '^Y^7)*(]TO' Z 1 < 118 > Y 1 z (1-k )(l-kj =5 _ ---_--i---_--- - y &»&* R«J- 1 To examine the stability dynamics at port 2, (115) and (116) are substituted into (118) to give {K s[n (s) + D (s)]][l + sd (s)] Z 22^S ; - - -j^^ z x is;.[k + sn 1 (s)][k + sk 2 (s)'j Y 22 (S) = --ppr"mt5^(^y^^-jjj^y^r-^j^j y^8) if [K t + sn-(s)] and [K + sn 0 (s)]have only left-half plane zeros., which 1 1 Z Z is the condition for port 1 being OCS and SCS, port 2 in Figure 23 becomes also OCS and SCS, Hence, it can be stated that, if one port of a terminated PIG is OCS and SCS, the other must also be OCS and SCS. When the practical PIC has a termination: between its input and output, the condition for this circuit to be OCS and SCS is that the. dynamic gains of the VCVS and CCCS in the PIC must have only left-half plane zeros. This condition can easily be satisfied from practical points of view. Sen si tlvi ty Consideratioii; One of the methods to reduce sensitivity in active networks is through the use o feedback. Since the EC-PIC networks discussed in Chapters II and III have one or more feedback loops s low sensitivity as

106 93 to the IG-KG network can be anticipated. The sensitivity function S'" v '' of a network function 0(s) due to the variation of the parameter k is usually defined by s ( s > = din (0(g)) = d 0( S> k)/0(s, k) k d In k ^^^~d r k7k"^ '"^ This expression of the sensitivity function is approximately equal to the fractional, or percentage, change in performance from a given fractional change In the independent variable of the network. 0(s) of view, it is desirable to keep the value of S, From a design point as small as possible at all frequencies* A sensitivity function can be expressed in two, the magnitude and the phase, parts, or q0(s) _ s 0(s) Z0(s) k I / k In order to make the sensitivity function low, it is necessary to keep its magnitude small at all frequencies. The independent parameters of interest in RC-PIC networks are the conversion gains of the PIC. Particulariy, the sensitivity functions of a driving-point admittance function YC ) in Figures 6 and 7, and an open-circuit voltage transfer function ffs} in Figure 13 are to be investigated: as examples of 1C*PIC networks*: The input admittance Y(s) of the network in Figure 6 was given as 2 (y-*) Y(s). y a. ^ i ^ ^ ^ _. (U9) x a 1 s 2 v.. y 22 ' ~"~ k

107 94 If the identifications in (12) are used for the realization of Y(s) = P(S)/Q(S) with (119), the definition of the sensitivity function yields 2 2 Y(*\ 1 Pi^s) - sq (s) p (s). 1 s k = - k ' r-~ (i3r) < 120 > q i q^cs) q 2 (s) 2 2 ^ fi\ i v Y(g) i Pi C s > - sc U MI \'^ ii (s) P 0 (s) '' K«if v( F 2 / 2 \ ^ qj(s) q 2 (s) ^ V u V It is obvious that the magnitudes of the sensitivity functions in (120) will be reduced ifl/q-k,) and k? /(l-k ) are made small. However, from the discussions in Chapter II, the conversion gains in the PIC were required to satisfy the relation: (l-k^u-k^) > o Because of this requirement on the conversion gains, simultaneous reductions of both sensitivity functions in (120) are not possible by controlling the values of k and k~. But in many practical realizations o the PIG,; the change of the voltage conversion constant: k is very small com- 1 pared to that of the current conversion constant k 9, since most of the practical PIC circuits use voltage attenuators to produce the desired voltage conversion. practical point of view, to keep the expression of (120), value of k. #: By this reasoning, it is more important, from a s^ s >..Y(s) k 2 as small as possible. From can be reduced by choosing a small For the purpose of comparison, one of the active RC networks that contains the NIC with conversion gains, kf, -kl; k*, kl > 0, has the form

108 95 21 in Figure 24, which is similar to the one proposed by Kinariwala, its driving-point admittance function becomes and Y(s) = y (y l2 } 2 y 22 " ^k2^kp y l Kith the same decomposition as in (12), the sensitivity functions for this KG-NIC network are Y(s) 1 k l PQ 2. 2 P l ^ " s \ ^ P 2^ q[w q 2 (s) (121) 2 2 Y(s) 1 Pi'W " sc l 1 ( s ) 2 Q q.,(s) p 2 ^ q 9 (s) The expressions of (121) are compared to the sensitivity functions for the IC-PIC network of Figure 6 as,y(s) * k ; EG-PIC **2 JL- ^ * J ^t, i RG-NIC \l~kj J(s) k- KJL»"*lr X\J (_JL_),Y(s) / **o \ V \l-k.v 1 2 RC -NIC This comparison indicates that choosing both conversion gains in the PIC to be very small gives a very slight increase in IS, Y(s) of the RC-PIG network and a. drastic decrease in Ma) of the RC-PIC network as compared to those of the RC-NIC network. For example, if k_ *» k 2 = 0.01 is chosen,

109 96 Y fv ^ Network 0 (RG) NIC k! -k 1. _ ^ 1 Figure 24 0 One-Port Active RC-NIC Network Figure.25. Two-Port Active RC-NIC Network

110 97,Y(s) \ RG*"PIC 100 ^Y(s) 99 k % RC-NIC,Y(s).ai, c? Y(s) \ EG-PIC 99 k l t 2 RC-NIC On the other hand, a study of the sensitivity functions for the network shown in Figure 7 with the decomposition of (12) yields,y(s) *k k l 2 2 x P 1 (s) - sq 1 (s) p 2 (s), k x _... _ 2 q 1 (s) q 2 ( s ) 1 -k 1 K l (122) 2 2,Y($) ^ x X P 1 ( S ) - sq^cs) P 2 ( S ) *k 2 " PQ 2 q 1 (s) q 2 ( s ) ( ) l""kr» Comparison of (121) with (122) gives ^ JL \ & J RC PIC " k«1 RC-NIC \k - 1/,Y(s) RC-PIC QY(s) k 2 EG-NIC \k - 1/ In order to keep,y(s) *k small, both conversion gains of the PIC must be chosen to be very large. For example f with k 1 ss k as 100, 1 Z,Y(s) \ EC-PIG = 122 c Y ( s ) 99 k* 1 RC-NIC (Continued)

111 98,Y(s) RG-PIC G \ * "" 99 k! 2 RC-N1C By an analogous argument, the sensitivity function of a voltage transfer function T(s) for the network in Figure 13 can be expressed as k Sfcf 0 * - PQ ( -yi2 )(y 22 " <V X) *!> (123),T(s) - pf (-y^) d-y yx When the given T(s) ~ P(s)/Q(s) is decomposed as W Q Q (s)/q(s) - Qjs)/q(s) the following identifications may be made: a _ P(s) ~ Y 12 " q(s) a V 8 > k l y 22 " ~^J" Q b (s) < 1 -V (1 -V y i = ^(st With these identifications, the sensitivity functions in (123) can be retsritten to give

112 99 X ' ~ " 2 (Q a " k ' - 1 V 1 Q q 1 (124),T(s) 1 Q q 2 1-k 2!Ehe RC-MIG network that is to be compared to the sensitivity functions in (124) has the form in Figure 25, which is similar to the one proposed by 22 Yanagisawa, sion gains of the NIC become and its sensitivity functions with respect to the conver» ot(s) _ k 1 1 z Q q a (125) I(s) k f 2 1 Qq 2 ^b Comparison of (124) with (125) yields the following relations Ms) & j> xf X.'L*- qt(s) R l EC-NIC / k l \ \k, - 1/ (126),!( ) RC-PIC qt(s) k' 2 RC-NIC / k 2 \ \ JL Kf>» / From the expression of (126), choices of very small k_ and k 0 in the PIC give a drastic decrease in,t(s) 1 z of the RC-PIC network as compared to I T (s ) I that of the RC-NIC network,, while S. ' of the RC-PIC network approaches K- I I I that of the RC-N1G network. Another way to reduce the sensitivity is to modify the network

113 100 arrangements of Figures 6 and 13 into that shown in Figure 26. It is again assumed that the change of k«is dominant over the change of k- in the 3?IC and, thus, reduction in the sensitivity due to the change of k«is of prime interest here. Analyses yield for the network in Figure 26 (j-^) 'I» y 11 k (1-k )(l-k ) i 2 «o y 22 + k y x r 1 v. - -V-- - y l k -y 12 1^0 r 2 -o k l y 22 * k 2 y 2 " (l-^xl-^) y x lor given Y(s) «P(s)/Q(s) and T(s) «P (s)/q 1 (s) a 2 (y-io) k 2 0 : " ^ PQ 1 k- U 2 y 2 (k l 1} y l J (127) S kf } * P^7 (y 12 } Ck 2 y 2 " (k l " 1) y l ] l x l It is observed from-, (127) that, with the assumptions of k- > 1 and k 9 > 0, the sensitivity functions of Y(s) and T(s) become differences of two polynomials and, hence, low sensitivities with respect to k_ can be expected. However, since (l-k-)(l-k ) is also required to be positive for these procedures, the values of k. and k«must satisfy k r k 2 > 1 for obtaining possible low sensitivities* It is notable that the admittance

114 101 I 1 Network FIG (RC) 0 1. i u <h iw-mimiili^j Figure 26. Alternative Active RC Network Containing One PIC

115 102 parameter y. does not appear in the expression of (127), and two admittance functions, y. and y 9, can be chosen to reduce the sensitivity by making proper decompositions in the denominators of the given Y(s) and T(s). It has been shown that the double terminated PIC is unconditionally stable. This is compared to the double terminated NIC whose stability 2 invariant factor is -1 or which is potentially unstable. An examination of the stability dynamics has revealed that a practical PIC circuit for generating a negative admittance becomes OCS and SCS if the dynamic gains of the controlled sources in the PIG are properly assigned. The sensitivity functions for the networks shown in Figures 6, 7, and 13 have been considered. For the RC-PIC networks in Figures 6 and 13 s the choice of very small conversion gains in the PIC gives a slight I increase in k and s T(s) Y(s) S* and *1 k l k 2 T(s) *S. as compared to the RC-NIC networks. For the RC-PIC network in 2 ' Y(s) Figure 7 $ k K can be decreased drastically by choosing large conversion gains in the PIC. The network arrangement in Figure 24 gives 2 another way of reducing the sensitivity. With the proper decompositions in the denominators of the given Y(s) and T(s), the sensitivity functions of (127) can be made small. This leads to the conclusion that the sensitivities of driving-point admittances and open-circuit voltage transfer functions in the RG-PIG networks with respect to the change of the current conversion gain in the PIC can be reduced drastically as compared to those im the RC-NIC networks.

116 103 CHAPTER VII EXPERIMENTAL RESULTS Validity and practicality of the realization procedures developed in Chapters II and III can be demonstrated by working out examples using those procedures and physically constructing and testing the resultant networks experimentally. In this chapter* three networks are constructed and tested. The PIC used for these experiments has the form of Figure 4, whose chain matrix is represented by 5 i R.JJ, 1 J The values of resistors in Figure 4 will depend on the particular example to be worked. The three networks realize, respectively, a negative admittance, a driving-point admittance, and an open-circuit voltage transfer function* The test data obtained from each example are compared with the corresponding predicted theoretical data* le 1 Examples It was shown in Chapter II that a negative impedance could be

117 104 generated by a PIG with a feedback loop. There are two basic PIC circuits for this purpose as shown in Figure 5. For the realization of Y = -1/4 X 3 10, these basic PIC circuits are shown in Figure 27. The circuit of Form 6 I of that figure was tested experimentally by Cobbj and the results were satisfactory. In the network of Form II in Figure 27, the following resistors in the PIC of Figure 4 were used: R l * R 3 " R, =* 1 kilohm R. ~ 2 kilohms 2 R_ = R- = 10 kilohms 5 6 R ohms 7 It is notable that the theoretical value of R to make k. ^ k» 2 in the / 1 2 PIC is 909 ohms, but it was necessary to trim R? from 909 ohms to 899 ohms due to the loading effect and the non-idealness of operational amplifiers. The. magnitude and: angle o the. input impedance were measured. A. comparison of the measured data with the predicted behavior, as well as the test data of Form I by Cobb, is given in Figures 28 and 29. Example 2 In this example f the impedance of a lossy inductor will be simulated. When an. inductor of 0,1 henry connected with a one-kilohm resistor in series is desired s the given, driving-point admittance to be realized is Y(s) = ML^ (128) s + 10

118 Form I Ohms Form II Ohms Figure 27. Two Forms of RC-PIC Networks Realizing a Negative Resistance

119 h«.s en. Q. A A (0 > H 4J fl8 50 OJ S3 *w o / /I A Measured Data of Form I Measured Data of Form II Calculated Data i- ;,, * ; ;.:...;1: t j L» ' J I > J LJ 1 s 1000 Frequency (Hertz) J LJL-LJ Figure 28. Comparison of Experimental Data of Forms I and II with Predicted Magnitude Data for Y = -1/4 X 10 3 o 0V

120 2.0QL. 190L. u bd. o 180L. o 170U»d 0) > T4 4J Gg 50 0) M-l O r « L- A Measured Data of Form I Measured Data of Form II Calculated Data A JL A,.:.!» J.. I l J I I M J. J l_jl Frequency (Hertz) Figure 29. Comparison of Experimental Data of Forms I and II with Predicted Phase Angle Data for Y = - 1/4 X 10^

121 108 To realize the admittance in (128) with a RC-PIC network, it is convenient that the procedure in the third section of Chapter IV be used, since the given admittance has a negative real pole. By making use of (93) with N=l and letting y 00» -y,^ and k, = k n = Zz 1Z 1 2 2, the following identification can be made: y. 10" 3 + ~2 _^ (129) 10 (s + 1(T) -y. 12 2s 10 3 (s ) " ' 22 The PIC network with the companion network parameters in (129) has the form in Figure 30 by the procedure depicted in Figure 20. The actual PIC used for this test had the same resistor values as in Example 1 except that the value of 8.^ was 890 ohms. This network was constructed, and the data obtained from it are shown in Figures 31 and 32 along with the predicted behaviors. Example 3 To demonstrate the practicality of the transfer function synthesis procedure discussed in Chapter III, an open-circuit voltage transfer function 4 vfe.\ - 10 s M T(s) ~ -5 - ( 130 ) s + (0.1) <1(T) s + KT was chosen and realized with the RC-PIC network* By referring to (37) and (38), short-circuit parameters for Figure

122 VW 0.2 Hh o- IE Y(s) IK 2K * ^AAr 10K IK o -< fic k» k «2 10K»IK, t i f899t Ohms, MiGrofarads Figure 30. RC-PIG Network Realizing a Lossy Inductor Y(s) = 10/(B ± 10 4 )

123 ^4.5 4,0 3.5 co o 3.0 0) o a «d T3 I o » U H es & os 1.5 UQ Measured Calculated 20 :: :, : I.,1. I. I L-J-J. 100 t I J I I ' I 1000 Frequency (Hertz) BBs^»»^g^BMmm^nifcnnwBaaaMBae^w«ata Figure 31. Comparison of Experimental Data with Desired Magnitude Behavior for Y(a) = 10/(s )

124 Measured Calculated Frequency (Hertz) Figure 32. Comparison of Experimental Data with Desired Phase Angle Behavior for Y(s) = 10/(s )

125 can be found with the conversion gains of L - k = 2 as 1 2 a 10~ 4 s -y 12 sf 10 4 Y 2 2 * (0.5)(10" 4 ) + (0.5)(10" 8 ) s + -15_s s -U y ) 10" 4 s s Then the desired RC-PIC network under test has the form of Figure 33. the resistor values used in the PIC were the same as in Example 1 except that R 7 had the value of 868 ohms. The theoretical and measured values of magnitude and phase angle for this example are shown in Figures 34 and 35 s respectively. Discussion of Techniques and Errors The purpose of the experiments performed in this chapter was to show the validity and practicality of the realization procedures and, therefore, no sophisticated example was employed. On a whole, the results show that the procedures developed in Chapters II and III were not only correct hut also practical. All of the elements used for construction of the RC networks had tolerance of less than approximately 10 percent from their designed value. Of these elements, all the resistors used were carbon resistors with 5 percent tolerances and all the capacitors were mylar capacitors with 10 percent tolerances. Each PIC used in this experimental work was approxi-

126 vaaa 1 Ohms, Microfarads Figure 33. RC-PIG Network Eealizing T(s) in (130)

127 ««"* 10. \ 8 U 7 w U 6 CM <w o Ci -d 4J H. C 50 qj a i» 4 U 3 Measured Calculated h t'' J Frequency (Hertz) Figure 34. Comparison of Experimental Data with Predicted Magnitude Data for T(s) in (130)

128 f L I >«0 Calculated 20 till t»» 1 till J L 100 Frequency (Hertz) 1000 Figure 35. Comparison of Experimental Data with Predicted Phase Angle Data for T(s) in (130) Ul

129 116 mated by employing one Burr-Brown Model 1525 differential input operational amplifier and one Burr-Brown Model 3024/15 DC operational amplifier. The DC gain of the Model 1525 was approximately 106 db, the input resistance was approximately 0.5 megaohm, and the output resistance was approximately 5 kilohms. For the Model 3024/15, the DC gain, input resistance, and output resistance were approximately 106 db, 0.5 megaohm, and 7 kilohms, respectively. Note that using commercially available operational amplifiers gave the non-idealness of the PIC used in these experiments. For measurements of the magnitude and phase angle of Examples 1, 2, and 3, a Hewlett-Packard 200 CD Audio Oscillator was connected to the network under test through a series resistor. A Tektronix 545-B Oscilloscope with a 1A2 dual trace plug-in was used in conjunction with a Hewlett-Packard 400 D Vacuum-tube Voltmeter to measure the magnitudes of various voltages. The phase angle measurement was done by using a 6 Hewlett-Packard Webb-Mask on another oscilloscope. Example 1 shows good agreement between the data obtained from the networks of Form I and Form II In Figure 27 and the calculated predicted data. The negative resistances simulated by a PIC with a resistive feedback agreed fairly well with the theoretical, predicted behavior. From 10 to 8,000 hertz, the magnitude and angle of the negative resistance of Form I were in error less than 5 percent. For the negative resistance of Form II, the errors were also less than 5 percent. Therefore, both networks of Form I and Form II in Figure 27 gave satisfactory results to generate negative resistances in the frequency range from 10 to 8,000 hertz. This demonstrates the versatility and usefulness of the PIC,

130 117 Figures 31 and 32 show the test data of the lossy inductor obtained by the RC-FIC network in Figure 29 in conjunction with the calcu«lated predicted behavior. As shown in these figures, the test data agree fairly well with the corresponding theoretical ones in the frequency range from 20 to 6,000 hertz. Maximum magnitude deviation from the predicted behavior is less than 5 percent in this frequency range. The test data of the phase angle have considerable deviation as the frequency increases. But this can be explained by the inaccuracy of the measurement scheme used, the noise generated in the presence of low level signals, and the non-idealness of the PIC used especially when the frequency increases. Example 3 serves to prove the validity and practicality of the synthesis procedure for voltage transfer functions. The network constructed for Example 3 was tested and found to be satisfactory. From the given transfer function, the undamped natural frequency should be 1,590 hertz theoretically. Figure 34 shows that the measured undamped frequency is 1,500 hertz. This reveals that the location error of the natural frequency is less than 6 percent. Selectivity or Q is usually defined by U) Q - - H 2c for a second-order transfer function TCs) K s s + 2o*s + (u n

131 118 where a)^ is the undamped natural frequency and K is a constant. From the given transfer function of (130) in Example 3, the theoretical Q is 10. However, the measured Q is 9 from Figures 34 and 35, which is in error by 10 percent. An examination of the curves of Example 3 will reveal that the measured phase curve is slightly displaced to the left of the theoretical phase curve. This was expected since the measured phase curve passed through 0 degree at 1,500 hertz, the measured undamped natural frequency. Although some errors were involved in each example, the results were sufficient to verify the realization procedures and demonstrate their practicality.

132 CHAPTER VIII CONCLUSION AND RECOMMENDATIONS This investigation has made use of a new active device, the positive impedance converter (PIC), which can be approximated by existing electronic devices such as operational amplifiers, in active RC synthesis. The results of the investigation can be summarized in the following theorems Theorem 1 For the realization of an arbitrary N X N matrix or real rational functions in the complex frequency variable as a short-circuit admittance matrix of a transformerless active RC N-port network, (a) it is, in general, necessary that the network contains N PIC! s; and (b) it is sufficient that the network contains N PIC's embedded in a 3N-port RC network, ffaeorem 2 For the realization of an arbitrary N X N matrix of real rational functions in the complex frequency variable as a short-circuit admittance matrix of a transformerless grounded active N-port RC network, it is sufficient that the network contains 2N PIC's embedded in a (4N+1)- terminal RC network. Theorem 3 An N X N matrix of real rational functions in the complex frequency variable having L simple poles on the negative real axis in the complex

133 120 frequency plane and no more than L+l zeros can be realized as a shortcircuit admittance matrix of a transformerless active network having no more than N PlC's embedded in a 2N-port RC network. theorem 4 An H x N matrix of real rational functions in the complex frequency variable having L simple poles on the negative real axis in the complex frequency plane and no more than L+l zeros can be realized as a shortcircuit admittance matrix of a transformerless grounded active network having no more than 2N PlC's embedded in a (3N+1)-terminal RC network. Theorem 5 For the realization of an arbitrary N X N matrix of real rational functions in the complex frequency variable as a voltage transfer matrix of a transformerless active RC 2N-port network, (a) it is, in general, necessary that the network contains N PlC's; and (b) it is sufficient that the network contains N PlC's embedded in a 3N-port RC network. Theorem 6 For the realization of an arbitrary N x N matrix of real rational functions in the complex frequency variable as a voltage transfer matrix of a transformerless grounded active RC 2N-port network, it is sufficient that the network contains 2N PIC f s embedded in a (4N+1)-terminal RC network. Each of these theorems has been proved, and a numerical illustration has been worked out. The results of the investigation on stability criteria has revealed that if one port of a terminated PIC is open-circuit stable (OCS) and

134 121 short-circuit stable (SCS), the other port must also be OCS and SGS.. The conditions for the terminated PIC to be OCS and SCS were imposed on the dynamic gains of controlled sources in the PIC, which could easily be satisfied from practical points of view. The study of the sensitivity in the RC-PIC network has shown that sensitivities of driving-point admittance and open-circuit voltage transfer functions in EC-PIC networks with respect to the current conversion gain changes in the PIC can be reduced drastically as compared to those in B networks with the negative impedance converters, while sensitivities with respect to the voltage conversion gain changes in the PIC may increase slightly. Validity and practicality of the realization procedures have been demonstrated by actual examples, and by constructing and testing the resultant networks experimentally. The test results have shown that the procedures developed in this research are not only correct, but also practical. During the course of this research, several possible extensions to this research have been suggested. One possible extension is to Increase the number of PIC's used in the synthesis to reduce the number of passive elements. The other is to investigate the controllability of element values external to PIC's in the synthesis procedure. Necessary and sufficient conditions on the number of PIC's that realizes an N X N matrix as a short-circuit admittance matrix or opencircuit voltage transfer matrix with a transformerless grounded network need to be investigated. It will be of great value if the number of PIC's

135 122 for this synthesis is found. Another possible area of investigation is to find a synthesis procedure that realizes: two short-circuit admittance parameters simultaneously in a two-port network with one or two PLC's, This type of synthesis is sometimes desirable. In summary, this investigation includes the development of new useful synthesis realizations with networks using resistors, capacitors and PIC's as basic elements. These resulting syntheses not only offer many advantages over the network syntheses using other active elements, but also are practical as the experimental results indicate.

136 APPENDICES 123

137 124 APPENDIX I PROOF FOE THE POLYNOMIAL DECOMPOSITION IN (10) A unique polynomial decomposition described in (10) will be proved by using the RC-LC transformation and considering the pole-zero distri bution of its reflection coefficient. * By a conventional RC-LG transformation, the equivalent LC admittance Y'(s) to the given Y(s) - P(s)/Q(s) [Equation (9)] can be obtained as 2 Y'(s) - - ^~ s Q(0 (A.l) The reflection coefficient p(s) is defined by t \ 1 - Y'(s) /A _ N p(s) ^ ^ ^ - ^ (A.2) Substitution of (A.l) into (A.2) makes 2 2 where A(s) - s Q(s ) + P(s ). / > B Q(s 2 ) - P(s 2 ) A(-s) /A _. P(s) - : "-5T~" 2^ * (A.3) s Q(s ) + P(s ) A(s) Observation of (A.3) yields that to every zero (pole) of p(s) in the right-half s plane there exists a pole (zero) in the left-half s plane, that is located symmetrically with respect to the origin. Typical

138 125 zero-pole distribution of p(s) in (A.3) is shown in Figure 36. Note that any zero or pole on the jcu-axis of the s plane is not allowed. Let (M^ - N^> and (M^ - N^) denote unique polynomials obtained by grouping all the right-half plane zeros and poles, respectively. Also let (M^ + N-p and (M 2 + ^) re P res^nt unique polynomials obtained by grouping together the left-half plane poles and zeros, respectively. These representations are always possible by assuming M.. and M to be even polynomials, and H 1 and. N to be odd polynomials. Then the expression for p(s) in (A.3) can be rewritten as (M x - N 1 )(M 2 + N 2 ) P^s) " 7M". + w UM - w V ( A «4 ) (M 1 + N 1 )(M 2 - N 2 ) which gives Y ' (s) -ft^s (A.5) M 1 M 2 - N 1 N 2 M rt 2N N, - M-N, Let 2 M 1 = P 1 (s ) 2 N l * S q l^s ^ 2 M 2 = P 2 (s ) N 2 «s q 2 (s 2 ) Since (M^ -f 1^) and (M 2 + N^) are Hurwitz polynomials, M /N- and M/N

139 PMBifl- 126 /K Q. \ X \ \ \ \ \ \ v, 'x * / * «*/>> * Cr <* / J ' v v. V N. O Zeros of p(s) X Poles of p(s) Figure 36. Zero-Pole Distribution of p(s)

140 127 are passive LC driving-point admittance functions. This implies that the rational functions p.(s)/q (s) and p«(s)/q (s), obtained from M-/N. and ^2^2 ky t^: functions. kc-rc transformation, are passive RC driving-point admittance Making use of the LC-RC transformation on Y'(s) gives the desired expression for Y(s) as Y(s) - *T!" Y'(VT) (A.7) PjCs) P 2 ( S ) " s ^M <i 2 ( s ) ~ P 2 ( S ) q 1 ( s ) - Pi( s ) q 2 ( s )~"~ tjniqueness of this decomposition comes from the fact that M-, M«, N-, and M~ are unique polynomials obtained from the zero-pole distribution of pcs). This completes the proof.

141 128 APPENDIX II NECESSARY NUMBER OF PIC's FOR THE SYNTHESIS OF ANY N X N VOLTAGE TRANSFER MATRIX Consider the network in Figure 18 that contains m PIC's embedded in a transformerless (N+2m)~port RC network. With the same notations of variables as in (50), the relationship between the two sets of voltages are h '" - ( Vs3 + V22 +? 32 + \V23 )_1 (k 2 y 21 + y 31> E a which is the same as (55) and is rewritten to give T - - ( ki y 33 + k^^ + y^ + k ^ ) -1 (A.8) (k 2 y n + y ) E a Note that y 33, y 22, y 32, and y 23 in (A.8) are m X m matrices, and y 21 and y 31 are m X N matrices. Now, a given N X N matrix of real rational functions T is assumed to have a pole at s=s. of multiplicity k (k 1) where s. is off the nega- 1 x. tive real axis. The coefficient matrix of the Laurent expansion of T th k- about this k order pole is obtained by evaluating (s-s.) x at s=s. s 1 1

142 129 which must be equal to the coefficient matrix of the submatrices described by(a*8). Cs - S i )fe? = "I (s " S i )k( V33 + V22 +?32 + k lv23 ) " 1 (A.9) s=s (V21 + y 3 i> s=s from the fact that y 21 and y are submatrices of the admittance matrix of a passive RC network, (k 0 y 01 + y«n ) must be finite. Then the s=s right-hand side of (A.9) can be rewritten to give (s-s.)\ 1 s-s - Cs-a.) k (k 1 y 33 + k 2 y 22 + y 32 + k^y^)" 1 (A. 10) s*=s (k 2 y 21 + y 31 } s=s Since the rank of a matrix product cannot exceed the rank of any of its 17 constituent factors, (A. 10) gives rank {<a*s i >\ } 3 rank {(k 2 y 21 + y 1) } (A. 11) s==s. s«s. In order to include the generality of the given N X N matrix, the rank of (s-s.) 15? is assumed to be N. However, the rank of (E 9 y 91 + Xa-i) S«: S, is limited by its size, m X N. The inequality of (A. 11) then becomes s=a N ~ minimum (m, N)

143 130 which requires m ^ N Therefore, I PIC's are necessary for the synthesis of an N X N voltage th transfer matrix, if the matrix possesses a k rank N off the negative real axis. (k 5 1) order pole of

144 131 BIBLIOGRAPHY 1. K. L, Su, Active Network Synthesis, McGraw-Hill Book Company, S. K. Mitra, Analysis and Synthesis of Linear Active Network, John Wiley and Sons, Inc., S. Darlington, N. T. Ming, H. J. Orchard, and H. Watanabe, "What, if any, are the important unsolved questions facing filter theorist today?,' IEEE Tran saction on Circuit Theory, CT-15, pp , December M. Kawakami, Some fundamental considerations on active fourterminal linear networks," IRE Transaction on Circuit Theory, CT-5, pp , June A. W. Keen and J, L. Glover, "ideal-transformer and converter realizations using operational amplifiers," Proceedings IEE(London), vol. 115, pp , August D. R. Cobb, "Active network synthesis using the generalized positive impedance converter," Ph. D. Thesis, Georgia Institute of Technology, A. G. Holt and J. R. Carey, "A method for obtaining analog circuits of impedance converters," IEE Trans, on Circuit Theory, CT-15, pp , December R. W. Daniels, "Gyrators, negative impedance converters, and related circuits," IEEE Trans, on Circuit Theory, CT-16, no. 2, pp , May N. W. Cox, K. L. Su, and R. P. Woodward, "A floating three-terminal nullor and the universal impedance converter," IEEE Trans, on Circuit Theory, CT-18, pp , May J. Gorski-Popiel, "The positive impedance converter...an alternate to the active gyrator," Proceedings Eleventh Midwest Symposium on Circuit Theory, pp , * A s Antoniou, "Novel RC-active network synthesis using generalizedimpedance converters," IEEE Trans, on Circuit Theory, CT-17, pp , May 1970.