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AN ABSTRACT OF THE THESIS OF

CHUN-PANG CHIU for the MASTER OF SCIENCE (Name) (Degree) Electrical and in Electronics Engineering presented on /2 G (Major) (Date)

Title:ACTIVE NETWORK SYNTHESIS USING GYRATORS Abstract approved:Redacted for Privacy Leland C../Jensen

The. paper is a study of(i) the realization of the gyrator, (ii) active filter synthesis using gyrator. Several realization methods of the gyrator are summararized. A practical gyrator composed of two operational amplifiers was built and tested.The experimental results are shown.Two RC-gyrator synthesis techniques are derived and discussed.One method is to replace the inductors in a conventional LC filter with gyrators and capacitors.The other method is the application of Calahan's optimum polynomial decomposition to the synthesis procedure.The sensitivity of the two synthesis techniques are compared and discussed. A practical active filter was synthesized by the second method.The experimental results are shown. Active Network Synthesis Using Gyrators

Chung-Pang Chiu

A THESIS submitted to Oregon State University

in partial fulfillment of the requirements for the degree of Master of Science June 1970 APPROVED:

Redacted for Privacy X-4ociate Profess,,e-of Electrical and Electronics Engineering in charge of major Redacted for Privacy

HO.d of Department of Electrical and Electronics Eigineering

Redacted for Privacy Dead of trS.duate School

Date thesis is presented Typed by Clover Redfern for Chung-Pang Chiu ACKNOWLEDGMENT

The author wishes to thank Professor Leland C. Jensen who suggested the topics of this thesis and gave much assistance, instruction and encouragement during the course of this study. TABLE OF CONTENTS Chapter Page

I. INTRODUCTION 1

ILDEFINITION OF SENSITIVITY 3 1.Classical Sensitivity 3 2. Root Sensitivity 4 3. Multiparameter Sensitivity 5 4. The Relationship Between Classical Sensitivity and Root Sensitivity 6

III.POLYNOMIAL DECOMPOSITION IN ACTIVE NETWORK SYNTHESIS 9 1.RC-NIC Decomposition 9 2.RC-RL Decomposition 13

IV.THE PROPERTIES AND REALIZATIONS OF GYRATOR 26 1.The Origin of Gyrator 26 2.The Properties of the Gyrator 28 3.The Realization of Gyrators 32 3a. Realization Using Negative Impedance Elements 32 3b. Realization by Two Parallel VCCSs 36 3c. Realization by Cascading NIC and NIV 37 3d. Experimental Results of the Gyrator 40

V. ACTIVE FILTER SYNTHESIS USING GYRATOR 48 1.Direct Design Method 49 2.Single Gyrator Design Method 56

VI.SUMMARY 75

BIBLIOGRAPHY 77 LIST OF FIGURES Figure Page

3-1.Optimum zero pattern of RC-NIC decomposition. 10

3-2.Zero pattern of An(s) and Bn(s). 14

3-3.Optimum zero pattern of RC -RL decomposition. 14

4-1.The circuit symbol of an ideal gyrator. 28 4-2.An ideal gyrator with an impedance Z at the output terminal. 30 4-3.An ideal gyrator with an impedance Z° at the input terminal. 30

4-4. A realization of an ideal transformer. 30

4-5.A realization of a grounded inductor. 31

4-6. A realization of an ungrounded inductor. 31

4-7.The feedback circuit connection. 33 4-8. A realization of an ideal gyrator by s-p connection of the feedback circuit. 35 4-9. A realization of an ideal gyrator by p-s connection of the feedback circuit. 36

4-10. A realization of an ideal gyrator using two VCCS. 37 4-11. A realization of NIV by using positive and negative resistances. 39

4-12. A realization of the gyrator by cascading NIC and NIV. 40

4-13. An equivalent circuit of an ideal operational amplifier. 40

4-14. An INIC realization by using operational amplifier. 41

4-15.The NIV circuit. 42 Figure Page

4-16.The NIV circuit using operational amplifier. 43

4-17.The gyrator circuit using operational amplifiers. 43 4-18.Input impedance characteristic of the resistively terminated gyrator. 44 4-19.Input impedance characteristic of capacitively terminated gyrator. 45

4-20.Simulated R-L circuit. 46

4-21.Phase-difference pattern. 46

5-1. LC network terminated with unity resistance. 49

5-2. The network realization of Equation (5-1). 51

5-3. Replacing the inductance in Figure 5-2 by gyrators. 51

5-4. The network realization of Equation (5-7). 52

5-5. Damping ratio vs. a for direct design method. 55

5-6. Resonant frequency vs. a for direct design method. 55

5-7. A gyrator cascaded with two RC networks. 56

5-8. The RC network at the left port of the gyrator. 61

5-9. The RC network at the right port of the gyrator. 61

5-10.The realization of Equation (5-20). 62 5-11.Damping ratio vs a for single gyrator design method (A). 66 5-12,Resonant frequency vs. a for single gyrator design method (A). 66

5-13.Gain vs a for single gyrator design method (A). 67

5-14.The experimental result of RC-gyrator filter. 68 Figure Page

5-15.The realization of Equation (5-45). 71 5-16.Damping ratio vs a for single gyrator design method (B ) 73 5-17.Resonant frequency vs a for single gyrator design method (B). 73

5-18.Gain vs Rg for single gyrator design method (B). 74 LIST OF TABLES Table Page

1. Optimum RC-NIC and RC-RL decomposition forms of Butterworth polynomials. 21

2.Optimum RC-NIC and RC-RL decomposition forms of Chebyshev polynomials (with 1/2 db ripple). 22

3.Optimum RC-NIC and RC-RL decomposition forms of Chebychev polynomial (with 1 db ripple). 23

4.Optimum RC-NIC and RC-RL decomposition forms of Chebychev polynomial (with 2 db ripple). 24

5. Optimum RC-NIC and RC-RL decomposition forms of Chebychev polynomial (with 3 db ripple). 25 ACTIVE NETWORK SYNTHESIS USING GYRATORS

I. INTRODUCTION

Over the past decade, increasing interest has been shown in synthesis techniques using active elements.Conventionally, the elements used in network synthesis are resistors, capacitors, inductors and transformers.Since capacitive elements are usually cheaper, simpler and more nearly ideal elements than are inductors, synthesis techniques using only RC elements are very important. On the other hand, non-positive real functions cannot be realized by using passive elements alone.The use of active devices will overcome some of these difficulties.The most commonly used active elements are negative resistances, controlled sources, operational amplifiers, negative impedance converters and gyrators.The gyrator is one of the most useful active elements.It has the property that it can gyrate an impedance into an admittance, and vice versa.By this property, an inductor can easily be obtained by terminating a gyrator with a capacitor.It is also attractive in microminiaturization, since the fabrication of inductors in thin-film and integrated-circuit technology is the most difficult problem, especially, for low frequency application, and no practical values of inductance have been obtained at this time. The thesis will concentrate on gyrator synthesis techniques 2 with sensitivity considerations.Sensitivity has been recognized as one of the main considerations in active RC synthesis.Chapter II gives some definitions of sensitivity.In Chapter III, optimum polynomial decompositions, developed by Horowitz and Calahan, are summarized.Butterworth and Chebyshev polynomials using this technique are decomposed and tabulated for practical use.The properties and realization methods of gyrator are described in Chapter IV. An actual circuit is synthesized and tested and the results also shown. Two RC-gyrator synthesis techniques are described and compared in Chapter V. 3

II.DEFINITION OF SENSITIVITY

Sensitivity is a measure of the degree of dependence of one quantity upon the value of another quantity.In network synthesis, the sensitivity is a measure of the change in certain network functions resulting from the change of the network elements.In this chapter, some definitions of sensitivity which have been used in network synthesis are defined.

1. Classical Sensitivity

LetN(s, xl, x2, ...,xn)be a network function ofnparam-

eters. For a single parameter case, N(s, x), the sensitivity is defined as

dININ d ln N dN x SN(s, x) (2-1) x dx /x d In x dx N Since,

ln N (jw, x) = lnI N(jw, x)+ j Arg N(jw, x). (2-2)

equation (2-1) can be written as,

SxN (jw, x) = x(dM . (2-3) dx dx where

M = InI N(jw, x) I, (2-4a)

0 = Arg[N(jw, x)]. (2-4b) 4 Thus the real part of the sensitivity is the change in magnitude of the network functions, and the imaginary part of the sensitivity is the change in phase function.

LetQ(s, x)andP(s, x)be the numerator and the denominator ofN(s, x)respectively.Then

Q(s, x) (2-5) N(s, x) P(s, x)

It can be shown that

Qt Sx(S,x)= X( - ) (2-6) Q P where _aQ ap (2-7) ax 8x

2. Root Sensitivity

Q(s,x) The root sensi- Let the network functionN(s, x) P(s, x) tivity is defined as the change of the roots of N(s, x)with respect to the change in one of the network parameters.Thus the root sensitivity can be expressed as

ds. ds. Sxj(s,x) = - x (2-8) dx ix d-x If P(s, x) = A(s) + xB(s), (2-9) 5

S is then the pole sensitivity x

p. dp. xB(p.) Sxi(s, x)= x (2-10) dx PI(p.) Similarly, let Q(s, x) = C(s) + xD(s) (2-11)

z. 1 S is The zero sensitivity x

z. dz. xC(z.) 1 1 1 Sx (s, x) = x = - . (2-12) dx Q'(z.) 1

3. Multiparameter Sensitivity

LetN be the network function withnparameters, xl, x2, ...xn.

, (2-13) N = N(s, xl, x2'3' xn).

Taking the partial derivatives with respect toeach variable, xn, the total differential is aN dN dx +Ndx + + dxn (2-14) axaN 1 l ax 2 n Divide both sides by N

a ln N dN d(ln N) = Inn x.). (2-15) N a In x, 6

The multiparameter sensitivityS may be defined as a gradient vector with elements8 In N 8 Inx. Letd(ln be a vector with elementsd(ln xi. Then

dN TF = (SN)d(ln 17i) (2-16)

The multiparameter sensitivity S is then defined

Grad {(ln N) d(ln }S (2-17)

4. The Relationship Between Classical Sensitivity and Root Sensitivity

Start from the definition of classical sensitivity,

dN/N d(ln N) SN(s,x) - x dx/x d(ln x)' and define Q(s, x) C(s)+xD(s) (2-18) N(s, x) P(s,x)A(s)+xB(s) Then

Q' 1=2" D(s) B(s) SN(s,x) = x( - ) x( ). (2-19) Q P Q(s, x)P(s, x)

Replacexbyx + Ax. The poles ofN(s, x)will be determined by the root of equation

A(s) + (x+,6.x)B(s) = 0. (2-20)

Define 7 xB(s) F(s, x) -P(s, x) (2-21)

Then Equation (2-20) can be written as

Ax 1 . (2-22) F(s,x x) = 0

Assume the degree of the numerator ofF(s, x)is lower than the degree of the denominator.Then

F(s, x) = (2-23)

Substitute (2-23) into (2-22), and examine the behavior of the equation in the vicinity of thejthpole of F(s, x). Equation (2-22) may be written in the form

k. Ax 1 + -1- -0 (2-24) x 13!-13 J J where p! is the value of s which satisfies the equation.If we writep! -p. = 6.p., substitute it into Equation (2-24), rearrange J J and take the limit

A p. dp. lim _ =_ k (2-25) Gx-0 dx x j

Thus, the sensitivity ofjthzero ofN(s, x)is 8

P. dp. SxJ(s, x)-dx/x1- -k (2-26) Similarly, define k. G(s, x) -xD(s) (2-27) Q(s) s- z. 1

The sensitivity ofjthzero ofN(s, x)is

dz. 1 Sx1(s, x) k. (2-28) dx/x 1

Substitute (2-26) and (2-28) into (2-23) and (2-27) respectively.Thus

F(s, x) = (2-29)

G(s, x) = (2-30)

Substitute (2-29) and (2-30) into (2-19), the relation between classical sensitivity and root sensitivity results

P. N Six S(s, x) = - (2-31) x s - p . J i. 9

III.POLYNOMIAL DECOMPOSITION IN ACTIVE NETWORK SYNTHESIS

In the previous chapter some sensitivity definitions have been stated.However, the synthesis technique based on the sensitivity consideration is very important and will be discussed here.Most of the active RC synthesis methods are based upon the partitioning of network functions into subnetwork functions.There are two decomposition forms.

1. RC-NIC Decomposition

Several methods have been presented to realize the immittance function by RC-NIC circuit.Horowitz (7) suggested a method by which a given polynomial can be decomposed into the difference of two other polynomials in such a way as to minimize the sensitivity. Calahan (4) has also shown that the same method can minimize the root sensitivity. Let

P(s,k) = a nsn + an-1sn-1+ + a 0

= Aon An(s) - kBonBn(s)

n n

= (a.-kb.)si= a.s (3-1) i=0 i=0 10 where n/2 A (s) =n (s+a.), (3- 2a) n 1 n/2 B (s) = II(s+b.) (3-2b) n 1

a. = a- kb. (3-2c) 1 1

The sensitivity of any change of coefficientai with respect to

aiaa kb. s i k (3-3) k ak a. a. 1 1

In order to minimize the sensitivity to the change of any coef-

ficient ink, it is necessary to minimize the corresponding coeffi- P(s) cient of BonBn (s).Horowitz (7) has shown that the polynomial 2(s) 2(s). must be decomposed into two new polynomials; a andsb These polynomials have zeros arranged as shown in Figure3-1.

Figure 3-1.Optimum zero pattern of RC-NIC decomposition. 11 The optimum pattern can thus be written as

n-1 P(s) asn +an-1s + a0

2 2(s), = a(s) - Bosb (3-4) where a(s) = (s+al )(s+a 2) ... (s+an/2) (3-5a)

b(s) = (s+b2)(s+b 3) ... (s+bn/2-1) (3- 5b)

The optimum decomposition of the polynomialP(s)can be obtained by the following procedures: 1) Form the polynomialP(s2):that is the polynomial formed 2. by replacing s with s LetF(s)contain the left half-plane roots ofP(s2). Obviously, F(s)is Hurwitz 2). andF(-s)contains the right half plane roots ofP(s Thus we obtain

P(s2) = F(s)F(-s). (3-6)

2) SinceF(s)is Hurwitz,F(s)can be expressed as

F(s) = A(s) + sB(s), (3-7)

then 2) 2(s) 22(s). P(s = A - sB (3-8) 12

3) Replace s withs2in the optimum form of Equation (3-4) which yields

2) 2(s 2). P(s2)=a2(s -Bos2b (3-9)

Compare Equation (3-8) and (3-9),

A(s) =a(s2),

2). B(s) = NTSb(s 0

4) Substitute (3-10a) and (3-10b) into (3-8) and replace

withs2, the optimum form results

2 P(s) = a2(s) - Bosb(s). (3-11)

(Example): Find the RC-NIC optimum decomposition of

P(s) = s2 + p s+ v. (3-12)

Apply the procedures

2) 2 1) P(s =s4+ P1s+ .y1

=[(s2+NrV1)+ sN2\ryi-P1)]{(s2+Nrcy-s,i2Nry1-131].

2) F(s) =(s2+NrCi1)+ s(42\71-131) 13

,2, 2 3) a(s ) s+ \171., and

Bob(s2)=42Arci1-p1.

4) Finally the optimum form results

P(s) =(s+Nr\TI)2 s(2NITI-(31). (3-13)

2. RC-RL Decomposition

Since and equivalent inductor can be obtainedby terminating a gyrator with a capacitor, RC-RL decomposition canalso be applied in R C- gyrator synthesis. Let n-1 P(s) = a nsn -F an -1s 0 n/2 = lI(s+s.)(s+s.) 1 1 1 (3-14) = Aon A n (s) + kBon B n (s), where n/2 A (s) =n (s+a.) (3-15a) n 1 n/2 R (s+b.) (3-15b) Bn (s) = 1

The zeros are distributed as in Figure3-2. 14

b bl a2al Cr b b1-1- 1 2 2 2

Figure 3-2.Zero pattern of An(s and Bn(s).

The root sensitivity of Equation (3-14) is

s. ds. kBB (s.) A A (s.) l.' onni. (3-16) k I I dk/k I I dP (s.)/ds I I PI(s.) I

The sensitivity will be minimized ifAon A n (s.)is minimized.The optimum zeros pattern is shown in Figure 3-3.

al o- a-12 1. b -1 b1 2

Figure 3-3.Optimum zero pattern of RC-Ri, decomposition.

Calahan (6) shows that if,

n/2 Tr 1) Arg Si < (3-17a) 2 i =1

A unique decomposition ofP(s)is possible and has the 15 form n/2 (n/2)-1 2 P(s) s) (s +a.)+ Bon 1-11 (s+b.) (3-17b) 1 1

n/2 2) Arg Si < 12, (3-18a)

i =1

Non-unique decompositions ofP(s)are possible and have the form

n/2 n/2 P(s) = Aon (s+a.)2+ B H (s+b.)2. (3-18b) 1 on 1 1 1

For the unique decomposition case, let

n/2 2 A ri(s+a.)2[Rm(s)] (3-19a) on 1 1 and n/2 2 , , II (s+b.) =Bon to(s)]2. (3-19b) 1 Then n/2 R (s) = \FT (s+a.), (3-20a) ono 1

n/2 Q (s) = N/T3-on (s+b.), (3-20b) m 1 and 16 n/2 P(s) = II(s+s.)(s+s.) 1

=[Rm(s)]2+[Qm(s)]2

(3-21) = [Rm(s)+j0m(s)ffRm(s)-jQm(s)].

Summarizing the above equations a procedure for finding optimum RC-RL decompositions obtained: 1) From a given polynomial

n/2 P(s) = II(s-Fs.)(s-Fs.)

determineRm(s)and Qm( s ). 2) Assign

n/2 2 2, = A (s+a.) (3-22a) [R(s)]m on 1

(n /2)- 1 [Q(s)]= Bon (s+b.)2. (3-22b) 1

3) Substitute Equation (3-22a) and (3-22b) into (3-21) to get optimum RC-RL form. For non-unique decomposition, equating the given polynomial to the optimum form and comparing the corresponding coefficients, it is found that there aren+2unknowns and onlyn+1 equations. 17

Arbitrarily select the value ofAon/Bon. Infinite numbers of other decompositions can then be found.Calahan (6) also concludes that, for a given polynomial

n- P(s) = a nsn + a s +a n- 1 0 n/2 = 11 (s+s.)(s+s.) 1

= Aon An(s) + kBonBn(s). (3-23)

If n/2 Arg Si > (3-24a)

1 only RC-NIC decomposition is possible. If

n/2 Arg Si <= 2 (3-24b) 1 both RC-NIC and RC-RL decomposition are possible.The latter can always be chosen to have the lower sensitivity.

(Examples): 1) For a given polynomial

P(s) = s2+ ails + 18

Arg S1 = Arg Tr 2

The optimum RC-NIC form can be found in the example on

page 12.If /4-yi-pi2 Arg S1 = Arg TT 2 P1

Apply the RC-RL decomposition procedure:

2 2

P1 1 1 i)P(s)(s+7+j )(s+T )

= [11m(s)+jQ m(s)NRm(s)-jQm(s)].

ii)[Rm(s)]2 -),2 = (s+2

[C2 m(s)]2=Y1 Ej,14

iii)Thus the optimum RC-RL decomposition is

2 134 pl2 1 (3-25) P(s)(s+2-)+(y(y1- )

2) WhenP(s) = (s2+as+b)(s2+cs+d) (3-26) apply the angle criterion, 19

(Case 1):

2 2 2 Arg SiArg S1 + Arg S2 = ArgNob-a i =1

+ Arg - C2> 27T (3-27)

Only RC-NIC decomposition is possible, the optimum form is

P(s) = a2(s)- sb2(s)

[s2+KE+ Nrcr+N(4N/bd+ac)-2(cirb+at471)+NTEU]2

-s[s(42N/E-a+ (42NTE-a)d+ 4(2NTa-- c)b)2 (2-38)

(Case 2):

Arg Si = Arg S1 + Arg S2 = Arg 41,17-a7

RC-RL decomposition has lower sensitivity than RC-NIC case.The RC-RL optimum form is

2 2 1 Jac-(4b-a)(4d-c2),2 P(s) =ir s +cs+ c,, 4

1 r 21 2 1 F2 j 22 (a 4d-c + c 4b-a )]. + -2-1(1v4b-a +4d-c )s +-2 (3-30) 20 For practical design convenience, the decomposition of Butterworth and Chebyshev polynomials with degree from 2 to 5 have been con- structed and tabulated (Tables 1-5).These are calculated by hand; for high degree cases the use of a digital computer is needed. Table 1,Optimum RC-NIC and RC-RL decomposition forms of Butterworth polynomaisi.

Optimum RC-NIC Optimum RC-RL n P(s) P1(s) P(s) decompositions of P decompositions of P2(s) EArg .P2(s) 1 2(s) 2 2 2 Tr 2 s+ 1. 414s + 1 1 (s+1)- 0. 5858 .s (s+0. 7071)+ 0. 5 4

2 2 2 Tr 3 (s+1)(s+s+1) s + 1 (s+1)- s (s+0. 5)+ 0. 75 3

2+0. (s 7854 +1) [ (s+0. 5250)(s+1. 9050)12 [ (s+1. 2247)(s+0.0919)12 Tr 4 1 2 2 2 2 x (s +1. 8478s+1) - 2. 5167s(s+1) + 3. 3917(s+0. 8446)

2+0. (s+1)(s 6180s+1) [ (s+2. 2899s)(s+0. 4367)12 Tr 5 s + 1 107. 91 0> 2+1. 2 2 x (s 6180s+1) -3. 2205s(s+1)

N Table 2.Optimum RC -NYC and RC-RL decomposition forms of Chebyshev polynomials (with 1/2 db

Optimum RC-NIC Optimum RC-RL n P(s) = P1(s)P2(s) P 1(s) decompositions of P2(s) decompositions of P2(s) E Arg Si

2 s2+ 1. 4256s + 1. 5162 1 (s+1.2313)2- 1. 0370s (s+0.7128)2+ 1.0081 54. 62°<- 2

2 2 (s+0. 6264) (s +1.0688)- 1. 5112s (s+0.3132)+ 1.0443 3 s + O. 6264 72. 96°< 2+0. 2 x (s 6264s+1. 1424)

2+0. (s 3508s+1. 0637) [ (s+2. 1095)(s+0.2922)12 4 1 125. 22°> 2 2 x(s2+0.8466s+0. 3564) - 3.6058s(s+0. 7328)

(s+0. 3623) [ (s+0. 2661)(s+2.6421)12 0 'TT 5 s + 0. 3623 144.78> 2 2 x(s2+0.2240s+1. 0359) - 5.006(s+0. 8208) 2+0. x (s 5862s+0. 4768) Table 3.Optimum RC-NIC and RC-RL decomposition forms of Chebychev polynomial (with 1 db ripple) .

Optimum RC-NIC Optimum RC-RL P1(s)P2(s) P decompositions of P2(s) decompositions of P 2(s) Arg Si P(s) 1(s) 2 2 0 7 2 s+ 1. 097s + 1. 1025 1 (s+1.050)2- 1. 0022s (s+0.5489)+ 0.8012 S8. 49< 2

(s+0. 4942) (s+0.9971)2- 1. 5000s (s+0.2471)2+ 0.9331 0 TT 3 s + 0.4942 75. 65< 2+0. 2 x (s 49425+-0. 9942)

2 (s2+0.2790s+0. 9865) (s+2.0782)(s40.2526)2 0 7 4 1 132.37> 2 2 2 x (s+O. 6736 s+0. 2794) - 3.7153(s+0. 6977)

(s+0. 2895) (s+0.2475) 2(s +2. 6365)2 0 TT 5 s + 0.2895 154.44> 2+0. 2 x (s 1790s+0. 9882) - 5.1203(x+0.7927)2 2+0. x (s 4684s+0. 4293) Table 4.Optimum RC-NIC and RC-RL decomposition forms of Chebychev polynomial (with 2 db ripple).

Optimum RC-NIC Optimum RC-RL n P1(s)P2(s) P(s) decompositions of P2(s) decompositions of P2(s) E Arg Si P(s) 1

2 0 7T 2 s+ 0. 8038s + 0. 6366 1 (s+0.7979)2- 0. 7920s (s+0.4019)2+ 0. 4751 59. 76<-2

(s+0. 3689) (s+0.9413)2- 1. 5138s (s+0. 1844)2+ 0. 8521 0 3 (s+0. 3689) 78.69 < 2+0. 2 x (s 3688s+0. 8861)

2+0. 2 (s 2098s+0. 9285) (s+1. 9620)(s+0. 2301)2 o 7 4 1 141.20 > 2+0. 2 2 x (s 5064s+0. 2195) - 3.8191s(s+0. 6002)

2 (s+0. 2183) (s+0. 2295)(s+2.. 6522)2 o S 0.2183 160. 04 > 2+0. 2 2 x (s 1348s+0. 9522) - 5. 2753s(s+0. 766) 2+0. x (s 3532s+0. 3931) Table 5.Optimum RC-NIC and RC-RL decomposition forms of Chebychev polynomial (with 3 db ripple).

Optimum RC-NIC Optimum RC-RL n P(s) P1(s) decompositions of P2(s) decompositions of P2(s) Z Arg S, P1(s) PP) i

2 o TT 2 s+ 0. 6450s + 0. 7080 1 (s+0.8414)2- 1. 0378s (5+0.3225)2+ 0.6040 67.47< 2

(s+0. 2986) (s+0.9163)2- 1. 5340s (s+0.1493)2+ 0.8174 0 TT 3 s + 0.2896 80.63< 2 2 x (s +O. 2986s+0. 8397)

2+0. (s 1704sl-0. 9031) (s+0.2005)2(s+2. 0974)2 0 TT 4 1 148.98 > 2+0. 2 x (s 41125A0. 1959) - 3.9279s(s+0. 6135)2

(s+0.1775) (s+0.0506)2(s+1. 8385)2 o (s+0.1775) 163.22> -2 2 x (s +0. 1096s+0. 9329) - 5.3815s(s+0.7611)2 2+0. x (s 28725+0.3770) 26

IV.THE PROPERTIES AND REALIZATIONS OF GYRATOR

The gyrator, first investigated byB. D. Tellegen (15) in 1948, is a two port device in which the impedance seen at either port is the reciprocal of the impedance connected to the other port.By this property a current is gyrated into a voltage, an impedance into an admittance, and vice versa.In network synthesis, there are many advantages such as size and weight reduction to be realized from the elimination of inductors.Since the fabrication of the inductors in integrated circuits is not feasible, the gyrator will play an important role in the integrated inductor simulation.

1. The Origin of Gyrator

In order to create useful systems, besides the four known network elements- -resistor, inductor, capacitor and transformer, we shall consider another similar element.By careful study, we find n-ports network composed of these elements have the properties of: A) the relationships between the voltages and currents of the terminals is formed by a system of ordinary differential equations, with B) constant coefficients, C) the n-port is passive:it candeliver no energy, and D) reciprocity holds. 27 Dropping any one of the first three properties will cause the system to become complicated. An n-port network possessing the first three properties but lacking the fourth may be very similar to the n-port network composed of four elements. We shall find a new type of network to realize these n-ports in which the first three properties hold but which violates the reciprocity theorem.This requirement has no significance for a one port element, such asC, R,and L,so we have to look for a new two port element.The ideal transformer

i sa type for which ilvl i2v2 as in the ideal case, energy can neither be dissipated nor stored.The equations for the ideal transformer are

i= -ni (4-1a) 1 2,

v2 = nv1, (4- lb) which satisfies the reciprocity relation.Another two-port element which satisfies iv+ iv2 = 0, but violates reciprocity is de- 1 1 2 scribed by v1 = -Ri2, (4-2a)

v2 = Rif, (4-2b) and is named "ideal gyrator." 28 2. The Properties of the Gyrator

As stated previously, an ideal gyrator is a two port device whose terminal characteristics are described by

NT = -Ri (4-2a) 1 2,

v2 = Rif. (4 -2b)

The circuit symbol as shown in Figure 4-1 where HRH is called the "gyration resistance. n i2

Figure 4-1.The circuit symbol of an ideal gyrator.

From Equation (4-2), the ideal gyrator can be described as the following matrices,

0 R Impedance matrix [Z] = (4-4) R 0

1-. Admittance matrix [Y] = (4-5) 1 0 R

0 R Transmission matrix [T] = (4-6) 1 R 0 29 The following properties are derived from Equation (4-2). A) An ideal gyrator is a passive and lossless device.

B) By connecting an impedance Z at thc output terminal, an impedanceZ' =R2 at the input terminal can

be found. This can be proven as follows:Figure 4-2 is an ideal gyrator with an impedance Z at the output terminals. Its matrix can be expressed as

0 R 0 R (4-7)

1 1 Z R 0 0 1 R R

Figure 4-3 is an ideal gyrator with an impedanceZ'between the input terminal.Its transmission matrix is:

1 0 0 0 R (4-8)

1 1 R Z' 1 R 0 R Z'

2 Comparing these two matrices, ifZ' , the two networks are equivalent. C) An ideal transformer with unity turns ratio can be realized by two identical ideal gyrators in cascade.That is 30

Figure 4-2.An ideal gyrator with an impedance Z at the output terminal.

Figure 4-3.An ideal gyrator with an impedance Z' at the input terminal.

R R'I 1 0 (4-9)

0 U 0 1

Schematically, the realization at an idealtransformer is shown in Figure 4-4. R R

Figure 4-4. A realization of an ideal transformer.

D) A grounded inductor can be realized by an ideal gyrator terminated with a capacitor as shown in Figure4-5. 31

L =R2C

Figure 4-5. A realization of a grounded inductor.

E) An ungrounded inductor can be obtained by two gyrators and a capacitor as shown in Figure 4-6. 1 : 1

vl v2

Figure 4-6. A realization of an ungrounded inductor. 32

3. The Realization of Gyrators

Since gyrators hold so many useful properties, it is very inter- esting to investigate its actual circuit realization.Briefly, the realization methods can be divided into three groups: A) Realization by inseration of negative impedance elements to compensate for the residual positive input and output im- pedances. B) Realization by two parallel VCCSs (voltage controlled current sources). C) Realization by cascade NIC (negative impedance converter) and NIV (negative impedance inverter).

3a.Realization Using Negative Impedance Elements

Consider the feedback circuit of Figure 4-7.This circuit con- sists of an amplifier with input impedanceZ1, output impedance

Z2, and open circuit gaink. 33 1.1 V k L.

(a) Series-parallel connection (b) Parallel-series connection

Figure 4-7.The feedback circuit connection.

For the series-parallel(s-p)and parallel-series(p-s) connections, we have the following impedance and admittance matrices: Series-parallel: a -Z2 [z]s-p= (4-10) Z1 -a Z2

Parallel-series: - -zi [Z]p-s (4-11) Z - a a 34 Se rie s - parallel:

Yl Yl [y] (4-12) s-p a'-Y2 a

Parallel-series:

Y 2 [Y]p-s (4-13) a' -Y1 Y2 where

a = Z 1(1-k)+ Z2, 2

(4-14) a' = Y2(1-k)+ Y1,

1 Y= , and Y = . 1 Z2 2 Z1 1

If the output is terminated by impedance ZL, then the input impedance of the series-parallel and parallel-series connections are: Series - parallel

Z1 Z2 Z. = (4-15) in Z+ZL 2 Parallel-series

Z(Z +ZL) Zin- 1 (4-16) 2

By inspection of Equation (4-10), (4-12) and (4-15), if the following conditions are satisfied, then a gyrator can be formed. 35

i)Ifa = 0,then K = 2andZ1 = Z2 = R,and

ii)ifZL>>Z2, and iii)if a series circuit is connected to a negative impedance

(equal to Z2)at the output or a parallel circuit is con-

nected to a negative admittance (equal toY1)at the input. Similarly, by inspection of Equations (4-11), (4-13) and (4-16), if the following conditions are satisfied, then a gyrator can be formed.

i)a = 0,thenK = 2andZ1 = Z2 = R

ii)ZL<

toZ1)at the input, or a parallel circuit is connected to a

negative admittance (equal to Y2)at the output. The equivalent circuits are shown in the following figures.

=Z_ R

Z1 =Z2 R

k = 2 v2 0

Figure 4-8.A realization of an ideal gyrator by s-p connection of the feedback circuit. 36

-R =Z R

= Z R

V =2 1

Figure 4-9. A realization of an ideal gyrator by- s connection of the feedback circuit.

3b.Realization by Two Parallel VCCSs

The admittance matrix of the gyrator was defined as:

1-S 0 R [Y ] = (4-17) 1 R 0

The admittance matrix can be partitioned into two submatrices , 1 0 0 0 R [Y] = (4-18) 1 0 0 0 at. R 37 Each part of the matrices can be represented by a VCCS with opposing polarities.Thus a gyrator can be realized by paralleling two VCCS with opposite polarity.The circuit is shown in Figure 4-10.

0 4 vi OGv1

ICII)Gv 2

Figure 4-10. A realization of an ideal gyrator using two VCCS.

3c.Realization by Cascading NIC and NIV

Negative impedance converter (NIC) is a two port device in which the impedance seen at either port is the negative of the impedance connected to the other port.Practical NICs have two classes. One of them is the class in which the NIC reverses the current flow at one of the ports with respect to the direction of current flow at the other.Such a device is referred to as a current-inversion negative impedance converter (INIC).The following equations apply to it:

v= v2, 1

1 i = i , 1 k 2 wherekis the gain of the NIC.Its transmission matrix is 38

1 0 [T] (4-20) nic 1 0 k

The other class of NICs operates by inverting the voltage polarity while leaving the direction of current flow unchanged.The voltage inversion negative impedance converter (VNIC) is described by the following equations:

1 v2' (4-21a)

"ii = -i2. (4-21b)

Its transmission matrix is

1 0 k [7] = (4-22) vnic 0 1

An ideal negative impedance inverter is a two port device in which the input impedanzeZiis proportional to the negative of the load admittance, YL. In other words,

2 R2 = -Z.RYL = (4-23) ZL

The necessary and sufficient conditions for the two port device to be an ideal NIV circuit, in terms of the Z parameters, are 39

Z11 = Z22 = 0, (4- 24a)

2 Z (4-24b) 12Z21 = R.

If we chooseZ12 = Z21 = ±R, then its transmission matrix can be expressed as

0 ±R

. = (4-25) [T]niv 1 ±R 0

The circuit can be realized by using one negative resistance and two positive resistances, or two negative resistances with one positive resistance as shown in Figure 4-11:

Figure 4-11.A realization of NIV by using positive and negative resistances.

When a negative impedance converter with gaink = 1is cascaded with a negative impedance inverter, the transmission matrix can be represented as 40

±1 0 0 ±R 0 (4- 26) 1 0 -1 0 ±R 0 ±R

Thus the gyrator can be realized by cascading an NIC and an NIV:

Figure 4-12. A realization of the gyrator by cascading NIC and NIV.

3d.Experimental Results of the Gyrator

The design of an actual gyrator circuit is based on cascading an INIC and an NIV.The INIC and NIV are realized by using operational amplifiers. An ideal operational amplifier is an ideal voltage amplifier of very low output impedance, very high input impedance and very high gain, with the property that the output voltage is proportional to the difference in the voltages applied to the two input terminals.An equivalent circuit is shown in Figure 4-13:

Tv+ in I

A =00 Figure 4-13.An equivalent circuit of an ideal operational amplifier. 41 The two summing point constraints are very important and defined as:

i)No current flows into either input terminal of the ideal operational. ii) When negative feedback is applied around the ideal operational amplifier, the voltage between the input terminals approaches zero. These two statements are used to analyze various circuits.The INIC and NIV are realized using the two constraints.As an example, consider the circuit of Figure 4-14.By directly applying the two constraints, the following relations result. R1

Figure 4-14.An INIC realization by using operational amplifier.

1 0 v2 (4-27) R2 0 -I R 2 1

By comparing the transmission matrix with Equation (4-20),it 42 is obvious that the circuit is an INIC withk = R1 /R2. For the NIV circuit consider the circuit of Figure 4-15.

R2 Figure 4-15.The NIV circuit.

This is a circuit replacing a current controlled current source by a voltage controlled voltage source in an NIV circuit which was given by Lundry (10).Analyzing the circuit, the transmission matrix is found to be

Rz(G+1) 1

(4-28) [T]NIV R2 G -1 GR1 GR1

When I G I >> 1, the transmission matrix reduces to

0 R2

(4-29) [T]NIV 1 0 R1

Since the operational amplifier is a VC VS with the gain IG I>> 1, the NIV can be formed by replacing the VCVS by the operational amplifier. 43 The resulting circuit is shown in Figure 4-16.Clearly, the gyrator can be easily formed by using two operational amplifiers. R1

Figure 4-16.The NIV circuit using operational amplifier.

An actual gyrator circuit was made and tested. Two NEXUS SQ -1 Oa operational amplifers, with a dc gain of 100,000 and a 2MHz cut off frequency, were used in the experiment.Figure 4-17 shows the circuit. INIC (k = 1) NIV 1000Q r 10000 1 'NAN -Wr- a

1000St 2 Li\AA. _A 100052

L _ _ _J L------I

Figure 4-17.The gyrator circuit using operational amplifier.

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Figures 4-18 and 4-19 are plots of the impedance, Zin, VS the frequency, f, when the gyrator is terminated with a resistance RL or a capacitance.The curves show that the input impedance agrees closely with theoretical results at frequencies between 250 Hertz to 2500 Hertz. Oscillograms 1 through 4 show the phase difference between a resistance and the simulated inductorL(see Figures 4-20 and 4-21) at the frequencyf = 500, 1000, 2000; 2500 Hertz.

R = 1 KS2 R= 1 KO g J. R=1 KS2

f = 500, 1000, 2000, 2500 Hertz L = R2C= 0. 02 h g Figure 4-20.Simulated R-circuit.

Phase difference0 = sin-1 ba

Figure 4-21.Phase-difference pattern. 4 .1

Oscillogram # 1 Oscillogram # 2 f = 500 c/s f = 1000 c/s Phase difference (theoretical) 6 = 86. 42o = 82.84° (e:Te.rirnental) 87o 8 z 82°

_..noraill111111111nr.,-

Oscillogram # 3 Oscillogram # 4 f = ..J3100 c/s f 2500 c/s Phase difference (theoretical)= 75.45o 8= 7i,7° (expe rime atal)74 70° 0 g 60° 48

V.ACTIVE FILTER SYNTHESIS USING GYRATOR

In the design of low frequency circuits, inductance, if required, is usually needed in large values.Consequently, the physical realization of these inductances becomes very impractical, because of size and cost limitations, and because of resistive losses.In addi- tion, when precise specifications are to be attained, conventional circuits with inductances can not be used, because of the inherent resistance of inductors.On the other hand, capacitances of large values can be obtained with low loss and for a reasonable cost.Thus, the use of networks containing only resistors and capacitors are to be considered.The natural frequencies of passive RC networks are re- stricted to the negative real axis of the complex frequency plane, and the natural frequencies must be simple,i. e. ,of the first order.It is therefore not possible to obtain the complex natural frequencies. However, the situation can be improved by using active elements. The most frequently encountered active elements are controlled sources, negative impedance converters and gyrators.Each active element had its advantages and disadvantages.The realization of the gyrator has been discussed in Chapter IV.In this chapter we shall apply the sensitivity minimization techniques, which have been described in Chapter III,to RC-gyrator synthesis. 49

1. Direct Design Method

The simplest and most direct way to design active filters using gyrators is to first design the conventional L-C filter, andthen replace all the inductors with gyrators and capacitors.As an example, a 3-pole maximumally-flat low passtransfer function can be realized using the following procedures:

i)Find the pole locations. .2 37 Tr s =e =-0. 5+ j0. 866 1

s =ejlr -1+ j0 2 .4 13Tr s3 = e =-0. 5- j0.866.

ii)Construct the transfer function

H Z 21 (s +0. 5 -j0. 866)(s+1)(s+0.5+j0. 866)

H (5-1) 3+2s 2+2s+1 s

iii) Design the network as an LC filter loaded with unityresist-

ance.

Figure 5-1.I,C network terminated with unity resistance. 50

The transfer impedance Z21 can be expressed by open circuit parameters:

z 21 Z12 (5-2) 1+z22

iv) Assignz21andz22

H 3+2s H s Z12 3 2 2+1 s +2s+2s+1 2s 1+ s3+2s z12 1+z22

Thus,

H (5-3a) z12 3+2s s

and

2s2+1 z22 (5-3b) s3+2s

v) Synthesize the network. Since all zeros ofz12lie ats =oo, the desired network is obtained through the continued-fraction development ofz22

z22 (5. 4) 3 1

2 4 1 3s +3 s 2 51 The realized network is shown in Figure 5-2. L =4h

Figure 5-2.The realized network of Equation (5-1). vi)Denormalize the element values in terms of the given speci- fication.

Suppose we wish to increase the impedance level tokz, and scale the frequency tokf, the denormalized valuesR*, L*and

C*are changed to

R* = kzR (5-5a)

k L* = L (5-5b) kz f

C (5-5c) C* -kzkf

vii) Replace the inductor by two gyrators and a capacitor

g g

Z22 Figure 5-3.Replacing the inductance in Figure 5-2 by gyrators. 52 Where

3 1 C (5-6a) 1 2 kzkf

1 1 C (5-6b) 2 2 kzkf

kz L 4 1 Cs=R2= (5-6c) 3 R2 kf g g

The active filter design using this direct method is quite sim- 2C ple, the simulated inductor isL = R . If, under certain condi- gs tionsR drifts, then an accurateLcannot be attained; there - g fore, the characteristic of the transfer function will be changed. An example of a second degree low pass filter will be used to examine this change.The transfer function is

1 G (5-7) 12 2-1-4s+1 s

Applying the previous procedure, the network can be found as follows:

Figure 5-4.The realized network of Equation (5-7). 53 where L = 2 (5-8a)

C= 1 (5-8b)

R =1 (5-8c) and R2 C (5-8d) s L

Denormalize each circuit element by changing the impedance

:a Then the element level toRzand the frequency scale to o. values become 2Rz (5-9a) 0

1 (5-9b) R oz and R . (5-9c)

If the gyration resistanceR is varied toR'the simulated inductance is changed to

R' 2 (5-10) R g where a. (5-11) g

The denormalized transfer function of Figure 5-4 becomes 54

1 LC G (s) 12 2 1 1 S + S + CR LC

2 wo (5-12) 2+ 2 s 4co s+co 0 0

When the gyration resistance is changed the simulated inductance changes toL'.The transfer function thus becomes

G 12(s) - 4.) LA) 0 o2 s2+2 a(7 )+(7 )

2 con (5-13) 2 2 s +2n(k) n+ (')n

The dc gain remains unity, but the damping ratio and the resonant frequencywo are changed to and con, respec- tiyely.

o w = (5-14a) n a 0( g and (5-14b) n= a g Figure 5-5 shows the percent change in damping ratio with percent change inR .Figure 5-6 shows the percent change in resonant frequency with percent change inR 55

0.5 1.0 1.5 2.0 a= R.' /R g g Figure 5-5. Damping ratio vs a for direct design method.

1 W nw 0

0.5w 0

0 0 0.5 1.0 1.5 2.0 a. = R' /R g g Figure 5-6.Resonant frequency vs a for direct design method. 56 From Figures 5-5 and 5-6, we observe that when the gyration resistance is increased by 10%, the damping ratio and resonant frequencywn vary 10% and 9%, respectively.This will be corn- pared with the other methods later.

2. Single Gyrator Design Method

Although the direct design method is quite simple,it has the disadvantages that pole locations of the transfer function are sensi- tive to the change of gyration resistance, and the network needs more than one gyrator.There is a better method using a single gyrator cascaded by passive networks at both ports as shown in Figure 5-7

(13).

Figure 5-7. A gyrator cascaded with two RC networks.

Let network Iin Figure 5-7 be defined by its y-parameters (I) (II) Y and the networkIIbe defined by itsz-parameterz.. . Then the open-circuit voltage transfer function for the overall network is 57

V(s) N(s) Ry(s)z Gy21(s)z21(s) 2 g12 12(s) G 12(s) - _ _ V (s)D(s) 2 1 (s) y22(s)+G2z11(s) z11(s)+RgY22 (5- 15) Divide both denominator and numerator by an arbitrary polynomial 2y Q(s) into the form of z (s)+R (s). and decomposeD(s)/0(s) 11 g22 Thus 2 D D(s) D(s)Q(s)+RD(s)Q D(s) 2(s) 1 2 1 g 1 2(s) R2 . (5-16) Q(s) g Q 1(s) Q Q 2(+0 1(s)Q 2(s)

Therefore D 2(s) D 1(s) z = Y (5-17) 11 Q2(s) ' 22 Q 1(s)

Q(s) = Q1(s)Q2(s), (5-18a)

2 D(s) = D(s)Q RD(s)Q (5-18b) 2 1(s) + g 1 2(s).

The decomposition may be achieved by using optimum polynomial techniques proposed by Calahan which have been summarized in Chapter III.The first way, denoted by (A),is the optimum non- unique decomposition.The second, denoted latter by (B), is the optimum unique decomposition.

(A).The optimum non-unique decomposition has the form of

n/2 n/2 D(s) = Aon (s+a.)2+ Bon II(s+b.)2. (5-19) 1 1 1 58 Consider a low pass filter with the transfer function of

2 H ou.) o N(s) G 12(s) = (5-20) 2 D(s) s+o) 0 0

D(s)has an optimum decomposition of the form

2 2 2 2. D(s) = s+ 2w s+co= A(s+a)+ B(s+b) (5. 20') o o Oil on

By equating the corresponding coefficients

Aon + Bon = 1, (5-21a)

Aona + Bonb co (5-21b) o, and 2 2 Aa+ Bb= co. (5-21c) on on o2

There are four unknowns and only three equations available, so we shall make some assumptions.By calculations we find that

1 A (5-22) on 1 +B /A n on Arbitrarily select

B /A =n2, (5-23) onon then

1 n2 Aon and B = (5-24) 2 on ' l+n l2+n 59 Substituting Equations (5-23) and (5-24) into Equations (5-21a, b and c) yields

a = (5-25a)

b = w(-1,1142) (5-25b) o n

From Equations (5-20) and (5-18b)

D(s)Q Aon(s+a)2, (5-26) 2 1(s) =

1 D(s)Q(s) = [13 +b)2]. (5-27) 1 2 2 on(s g Assign Bon D= k (s+b), (5-28a) 1 o 2 g

1 Q (s+b), (5-28b) 2 k

D2 = Aon(s+a), (5-28c) and (s+a), (5-28d) wherekois impedance scaling factor.

Then

D 2(s) s+a (5-29a) zl 1 Q ko"-l'ons+b 2(s) and D 1(s) koBon s+b (5-29b) y22 Q(s) 2 s+a 1 g 60

andz12can be found from the numerator: Y12

2 Ho coo ko N(s ) = -Rg y(s)z (5-30) Q(s) (s+a)(s+b) 12 12(s). Now assign X1 -y 12(s) = ko s+a and X2 z 12(s) = ko s+b where 2 Hcoo o XX2 (5-32) 1 2 k oRg

From Equations (5-29b) and (5-31a), the RC network at the left port of the gyrator can be realized and shown as in Figure 5-8.The

continued-fractional expansion ofy22takes the form of Figure

5-8.

1 (5-33) Y22 2 g 1 ko Bon ko Bon s + 1 2 R (a-b) R g g a-b ko Bon b 61 Ral

v C v2 1 a

1 Y22

Figure 5-8.The RC network at the left port of the gyrator. where 2 2 R(a-b) R g R = R g al kBb0 on ' a2 kBo on (5-34) ko Bon bBon C X1 a 2 2 R (a-b) R g g Similarly, the RC network at the right of the gyrator is realized by Equations (5-29a) and (5-31b), which yields Equation (5-35) and Fig- ure 5-9.

1 (5-35) zll 1 kAon + o 1 1 s + ko Aon(a-b) b[koAon(a-b)]1

R bl

z 11

Figure 5-9.The RC network at the right port of the gyrator. 62 where

1 Rbl = ko Aon R [1cA(a-b)], ' b2 b oon (5-36) 1 C = X2 = Bon(a-b) . b kA(a-b)o on

The complete network is shown in Figure 5-10.

R Ral Ra2 g Rbl

Figure 5-10.The realization of Equation (5-20).

Substituting the values ofa,b, Aon and BonintoRal' Rat'Rbl, Rb2, Ca and Cbyields

2 2 22 R 2 2 R (l+n ) g ,.//1.-- Rai Ra2 -1+n g k 1 / 2 ' ko' n3 o ---Nil-t n2 n

1 1-2 = ko R = 2 ' 1k (5-37) 1+n b2 no 1 2- n ko n3 1 1 Ca Cb - n 2 2 2 I-7' (1 +n )Rgcoo v14 ko oN1-7

It is instructive to examine the characteristic change of the transfer function due to the change of gyration resistance.The 63 transfer function of the network designed by this method (Figure 5-10) is

Ry(s)z g12 12(s) G (s) 12 2 z11(s)+Rg Y22(s) 1 k b(a-b)AonB Rg o on 2 2 2 s (A )+a B on+B)+2s(aAon on +bBon Aon +bon

2, 1 n o Rko\li_ g n2+1 2 52+4ws+w O o

H w2 o o (5-38) 2 2 s+4co s+w 0 0 where _,2) Ho ji (5-39) R1 o n 2 g n +1

When the gyration resistance changes from R toR' letting RI a = as before, we obtain g

aRgY12(s)z12(s) G 12(s) = , 2 2 zWs)+aRgY 22(s)

1 kobBonAon(a b) R g 2 A aA Aon a on 2 s2( )+2( on+aBa on a +bBa)s+on a +aBbon 64

1 kobBonAon(a-b) R g 2 2 Y s +aY2 (L.)os+wo Y3

1 1 k bB A(a-b) Rg Y1 0 on on Y2 Y3 2 2 )w s +() os+(Y o 1 1

2 717-[-Ti(wok '41--(--1d1-e) 2 1 n +1 Y2 Y3 s 2+ (Y1 )4wos+ (Y1 )wo2

2 Hnwn (5-40) 2 2 s nws+wn n

and the resonant fre- The dc gainHo, the damping ratio o' Hn, and wn,respectively.Thus quency woare changed to n

0 1 1 H H = H (5-41a) n o 2 Y o Y3 1 wn

Y2 (5-41b) Niyy 5 3 1 and

3 w , (5-41c) Wn Y1 o where Aon 22 l+na Y = + aBon = (5-42a) a a(n2 +1) 65

1 2 Y2 = 1 (a k +abk2) [g+nll- g+n2(g- 4142)] gwoa 1 2+1) a(n (5-42b) and kl 1 2 2) Y3 ( aa +ak 2b 2 wo

1 2 2(1-n 2 {(1-a)[g )+2ngF-T1]+n2+a2}. (5-42c) 2+1)a (n

Substituting Equations (5-42a, b, c) into Equations (5-41a, b, c) yields

2 -a2 )+2ng47:g7]+n2+a2 w a (1-n (5-43a) n o l+na2 2

+ +n2a2(g n (5-43b) n 2)+n 2+a2 qi(l+n2a 2)[g2(1-n2)+2ng ](1-a and 2+1)a (n Hn = Ho (5-43c) (1-a2)[ 2(1-n2)+2ng Ni1-g21+n2+a2

Figure 5-11 shows the percent change in the damping ratiogn,with percent change inR Figure 5-12 shows the percent change in resonant frequency wn,with percent change inR .Figure 5-13 shows the percent change in the dc gain Hn,with percent change in R Figures 5-11, 5-12, and 5-13 show the percent change ofgn, wn, Hn due to the percent change in the gyration resistanceR .This method is based on minimum sensitivity polynomial decompsotion. This method should have a lower sensitivity than the direct design 66

+241.- 2+4a2(-. -1S1-0) 1.5 = :174:7774,\-7:77777;77777:17:4

(n2= 4)

20 20 = 0. 8

20 % 20 = 0. 6 0.5 20 20 = O. 4

0 0 0.5 1.0 1.5 2.0 a = /11 g g Figure 5-11. Damping ratio vs a for single gyrator design method (A).

1.5w 0

1.0w wn

= 0. 8 = 0. 6

0.5woPow = 0. 4

w = w (4J1- 2 1+4a 0 0 0.5 1.0 1.5 2.0 a = R' /R g g Figure 5-12. Resonant frequency vs a for single gyrator design method (A). 67 method.This can be seen by comparing Figures 5-5 and 5-6 to Fig- ures 5-11 and 5-12.Using this method, the experimental results as indicated in Figure 5-14, which follows show quite good agreement with the theoretical case.

= 0. 4 = 0. 6

1.5 H0

Hn = 0. 8

1.0 H 0

0.5 H 0

0 0.5 1.0 1.5 2.0 a= /Rg g

Figure 5-13. Gain vs a for single gyrator design method (A). 2)2 Kg2 (l+n Nil- = 625 Q Ral = ko n3 n

2 R H2n R = = 62. 5 E2 a2 ko n

k Rbl = 02= 4000 l+n

2 R - 33. 33KO b2=n ko

ko Ca - n3 - 1. 274 p.f 2 (1+n2)2Rgw o

1 Cb n - 0. 0199 k ow o

Select n2= 4, = 0. 6, ko = 2 x104, f = 1000 cis, = 6280 cis,R = 100052 0 g

Ho = R 1-2 (--1,\/1-2) - 1. 28 =2. 144 db Gain o g n +1

2 Ho w o G 12(s) 2 2 s+2(0. 6)w s+w 0 0

Figure 5-14.The experimental result of RC-gyrator filter. - (X) 10. _ NM 111111111_1_1111M 11 11 III 4 111' I ; 1 t,11 I , , ! ! ! nil ! ! ; ; . : I-; 1 -1-4 1-,- _1_4 , , , Ft -I , , ,-: I 1 --1-2, b 1 _,._t_i_: 4!H:._,_.,_!!_i__La -- Rat 1-4-_ : ._; _r; ! 1 ! I 1 . 7. , - _ ,rt , 1;, - L 1_ ' 'ELIH ' I , 1- i 1 I t 101i3. ' t ; ' ,Ir 0',-', , 0 1-, I , 5.0 MIMEO ' ' ;, ; : , 1

II I

-a i- 2.5 =,...----c -i __ 1 i

3 , I , 1 ' I i I Li ; .

0 -, i 4-:--r-,

-I- ,

, . _ _ , ! : t-i- , .., 1 ,

, 1 : ; ;; -2. 5 : ! !! -:

, 1 ,- , _I 1_rii_i__LH : 11 ; 7- LI,_[ I 141 I '

! , :1 _

I 5.0 I I III

___ ,

H! ----r : , -- ,_ ! 1, ' '-,71711'

_I I I . .

; 1 4. -Ht-111; f -7---"--, -. -1- ,.- ' 1 L {

I _I -"'''±-1 H1 It:1i I'. -II -1-4- hI I 1 '

. . , r 1 I I I , I , 1 I 771- ; -7.5 t 714,. ; 1

I , ... ; 1 1 1 , ; - ; , , Imo- _ _.. , I 1 ; lid , '-'-- ,Ii 1 1 J-1-1 --I ; , -' , , __I _, , ' , t ,, p- ; ; ; !, _ _... ! , ' ; r 1 ; i ---1,_ I LL: __1 ' 1 _., :I i' --ri T -i-; ! , I-, r 1.,--.- H-i I -7 1_, , I ! . ! -10.0

1 , : -F -,--4-I.H- , ,1-,-1--1 , ' i 1- - , H.- , TT 1rI I ! IIIIII'1 T 1- --r-t' ,,, ., -i , I IliI, i 1II I I ,1 1 i ; , ' , 1 1 i -I ,.-i 4 ,-, r- I; , rii-t i t4-1- -H-ii, I _i__, _,, ! , 1 1 ii 4- -L-I : I I 1 - 1--,, , , .-Li ,, -:. 11[,H_:. ri , 1 ' 1 _,_LI__, L % 1-, , I I t-1111 I -t I-1

I , ,

1 --4 1-1Ii I.: I --- --7

1 jr-I--, 1 - ---I.I I. I -"Ti I : : I , , _-__ ! t ..,-4,-,-1 i.-; ..- , : , , -'T I 1 , I , 1 I , , H1 ! ' : i L ' I-- _ .. 1 I I 1 i 11 1 1-1,1i I I ii-rfi-'4' II 111!1 I 1 1 I I, H.!I,Hil i 1 _15.0 illWit 1 tai ri1 I ttT-1---TilI II , 5 1.0 2.0 3.0 5.0 10.0 f(Khz) Figure 5-14.The experimental result of RC-gyrator filter. 69

(B).The optimum unique decomposition has the form

n/2 (n/2)-1 2 2 D(s) = II(s+a.)+ B II (s+b.) . (5-44) 1 on i i=1 i=1

Again, consider the low pass filter with damping ratio and

resonant frequencycoo. It has the form

2 H 0 0 (5-45) G 12(s) - 2 s2+ncos+u) 0 0

Recall Equation (5-15),

N(s)/Q(s) Gy21z21 - (5-46) G 12(s) 2 D(s)/Q(s) v+G z 22 11

Decompose the denominator into the form of

2 D D (s)Q(s)+G D(s)Q D 1(s) 1 1 2(s) D(s) 2(s)+ G (5-47) Q(s) Q N(s) Q 2(s) 1 1(s)Q 2(s)

Therefore, D 2(s) D 1(s) (5-48) y 22(s)= z 11(s) = Q 2(s) Q1(s) where

Q(s) Q1(s)Q2(s), (5-49a)

2 D(s) = D(s)Q 1(s) + GD (5-49b) 2 1(s)Q 2(s).

The optimum form ofD(s)can be found in the example on page18: 70

2 2 2 2 D(s) = s+ g w s +w2= (s-qwo)+ wo )- (5-50) o o Thus

)2, (5-5Ia) D2(s)Q1(s) = (s-qwo

D1(s)Q2(s) = 1, (5-51b) and 2 G = w 1- . (5-51c) 0 Assign

D 2(s) = s + (5-52a)

Q1(s) = s + (5-52b)

D 1(s) = 1, (5-52c) and Q2(s) =1. (5-52d) Then y22 = s + (5-53a) and 1 (5-53b) Z11 s+wo are found from the numerator 2 N(s) oo = -Gy z How (5-54) Q(s) 21 21 s-qw o Thus -y21 = HI (5-55a) H2 2 andH1 H2 = How() . (5-55b) Z21 s-qw 0 71

The RC network to the left of the gyrator can be obtained byy22 and-y21. Similarly, the network to the right of the gyrator can be

found byz11 and z12' The complete network is shown in Figure

5- 15, R g

Figure 5-15.The realization of Equation (5-45). where

1 1 R - , Cl = 1, C2 = 1, 2 gwo and

2 G = w t o

The transfer function of Figure 5-15 in terms ofR1, R2, C1, C2 and G, can be written as

R1C1C2 G12(s) - G2+ R1R 2 1 1 1 2 s+s( + )+ Cl C RIC' R2C2 2

G (5-56) s 2+2;w s+w2 0 0

The gyration resistance and the gyration conductance are changed 72 from R toR' and G toG', respectively. g g Let R' a - g G (5-57) R GI g

Then the transfer functionG12(s)is changed to

G a o) G 12(s) 2 2 s2+4ws+(G)-q2w o a o

2 Hnwn _ . (5-58) s2 +4rins+wn2

Thus, when the gyration resistance or the gyration conductance is changed, the damping ratio ,, resonant frequency woand gain

Hoare changed to

2 22 2 2 iw =j(G) -qw =wo 1-q(a-1), (5-59a) n a o a

ewo a (5-59b) n w n 41+2(a2-1) and a Hn (5-59c) -2)-Fa22Ho ' respectively.

Figure 5-16 shows the percent change in damping ratio n' with percent change inR Figure 5-16 shows the percent change in g 73

1.5 4

20 1 = 0. 8 n 1.0 = 0. 6

= 0. 4

0.5

N1- 2(a 2-1)

0 0 0.5 1.0 1.5 2.0 a - Rg /R g Figure 5-16. Damping ratio vs a for single gyrator design method.(B).

1.5w 0

1. Ow 0 w = 0. 8 n = 0. 6 = 0. 4 0.5w 0 w 2 2-1) = (a wn 2a

0 0.5 1.0 1.5 2.0 a = RI /R g g Figure 5-17. Resonant frequency vs a for single gyrator design method (B). 74 resonant frequencywn, with percent change inR .Figure5-18 shows the percent change in dc gain, Hn,with respect change in

R Compare Figure 5-16 and5-15to Figure5-11and5-12and Figure 5-5 and 5-6.This comparison shows that the single gyrator design method always has lower sensitivity than the direct design method.

1.5Ho

= 0.4

Hn -20 1.0Ho = 0.6 -20

= 0.8 0.5Ho a Hn - 2 2 2 (1- )+a

0.5 1.0 1.5 2.0 a = R' /R-0- g g

Figure 5-18 Gain vs R for single gyrator design method (B). g 75

VI. SUMMARY

Two methods of active filter synthesis were derived.The first was replacing inductors in theconventional LC filter with gyrators and capacitors.The second was using a single gyrator cascaded by two RC two port networks.The gyrator is realized by cascading an ideal negative impedance inverter (NIV) and a negativeimpedance converter (NIC).Each NIV and NIC is realized by using a single operational amplifier.Therefore, the gyrator can be realized by using two operational amplifiers.It is to be noted that the gyration resistance can easily be adjusted by adjusting the passiveelements of the NIV. In RC-gyrator synthesis, although the realization ofthe gyrator is much more complicated than that of the NIC or ofcontrolled sources, it has the disadvantageof not being able to provide large amounts of gain.However, it does have the following advantages. First, from its lossless nature, the gyrator can neverbe unstable. Second, since a capacitor in general has a higher qualityfactor than an inductor, gyration using a capacitorproduces a better inductor than that now available.Third,it provides lower sensitivity than comparable realization using controlled sources or NICs.These advantages make the gyrator realization veryattractive. The applications of integrated circuit technology in active RC 76 network synthesis have received much attention recently, especially in microminiaturization.In active RC integration network design, the designer may no longer select passive elements and interconnect them to achieve a network, but rather must make the passive com- ponents simultaneously with the active elements in the integration procedure.In integrated circuit technology the initial tolerances of the resistors and capacitors are still within certain limits.There- fore, the sensitivity minimization problem is not a single parameter sensitivity problem but a multiparameter sensitivity problem The sensitivity minimization of the polynomial decomposition applied in this thesis is due only to the change in active elements.In this thesis, only the single-parameter sensitivity was considered.Multi- parameter sensitivity of integratable active RC networks is suggested as a topic of future study. 77

BIBLIOGRAPHY

1.Antoniou, A.Gyrator using operational amplifiers.Electron Letters 3f350-352.1967.

2.Burr-Brown Research Corporation.Handbook of operational amplifier active RC networks.2d ed.Tucson, 1965.104 p.

3.Bogert, B. P.Some gyrator and impedance inverter circuits. Proceedings of the Institute of Radio Engineering 43:793-796. 1955.

4.Calahan, D. A.Notes on the Horowitz optimization procedure. IRE transactions on circuit theory.CT7:252-254.1960.

5. Restrictions of the natural frequencies of an RC-RL network.Journal of the Franklin Institute 272:112-133. 1961.

6. Sensitivity minimization in active RC synthesis. IRE transactions on circuit theory CT9:38-42.1962.

7.Horowitz,I. M.Optimization of negative impedance conversion methods of active RC synthesis.IRE transactions on circuit theory CT6:296 -303.1959.

8.Huelsman, L. P.Circuits, matrices and linear vector spaces. New York, McGraw-Hill, 1963.281 p.

9 Theory and design of active RC circuits.New York, McGraw-Hill, 1968.297 p.

10.Lundry, W. Ralph.Negative impedance circuits--some basic relation and limitations.IRE transactions on circuit theory CT4132-139.1957.

11.Mitra, Sanjit, K.Synthesing active filters.IEEE Spectrum. 6-1:47-63.1969.

12.Morse, A. Stephen.The use of operational amplifiers in active network theory.Proceedings of National Electronics Confer- ence 29:748-752.1964. 78

13.Shenoi, B. A.Practical realization of gyrator circuit and RC gyrator filters.IEEE transactions on circuit theory 12:374-380. 1965.

14.Su, K. L.Active network synthesis.New York, McGraw-Hill, 1965.369 p.

15.Tellegen, B. D. H.The gyrator, A new network element. Philips Research Reports 3:81-101.1948.

16.Van Valkenburg, M. E.Introduction to modern network synthesis. New York, Wiley, 1960.498 p.