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Masters Theses Student Theses and Dissertations

1972

Sensitivity analysis in active RC bandpass circuits Sensitivity analysis in active RC bandpass circuits

Robert Bentzinger

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Recommended Citation Recommended Citation Bentzinger, Robert, "Sensitivity analysis in active RC bandpass circuits" (1972). Masters Theses. 3571. https://scholarsmine.mst.edu/masters_theses/3571

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SENSITIVITY ANALYSIS IN ACTIVE RC BANDPASS CIRCUITS

BY

ROBERT BENTZINGER, 1949-

A THESIS

Presented to the Faculty of the Graduate School of the

UNIVERSITY OF MISSOURI-ROLLA

In Partial Fulfillment of the Requirements for the Degree

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

1972

Approved by

T2885 75 pages c.l

i i rc /

ABSTRACT

This thesis is concerned with the analysis and cataloguing of

sensitivity functions of some selected active RC bandpass netv.Jorks. The sensitivity functions of primary interest are the sensitivities of

the quantities w0

and Q with respect to changes in the element values

of each circuit. An introduction is provided to give background inforMation on how

each circuit is realized. Each circuit is then analyzed and the

pertinent sensitivity functions catalogued. Some comments are made as to the approximations which are involved in several of the realizations

and the validity of any sensitivity analysis made on these approximate

realizations. The approximate realizations include some using uniformly

distributed RC networks (URC).

i i i

ACKNOWLEDGEJ1ENT

The author wishes to thank Dr. J. J. 8ourquin for his guidance, assistance, and criticism in the preparation of this thesis. The

author also wishes to thank Dr. N. G. Dillman and Prof. S. J. Pagano for reviewing this paper. A special note of thanks is made to

Dr. D. F. Dawson whose help and personal sacrifice made this paper possible.

iv

TABLE OF COrJTEI~TS

Page

ABSTRACT ~ ~ • • • ~ • o • • • • • • • • • • • • • • • • • • • • • • • • o • • • • • o • • • o • • • o • • • • • • • • • • • • i i

ACKNOWL EDGE~1ENT o ••••••• o • o •••••• o ••••• o •••• o ••••••••••• o ••••••• o • • • iii

L I S T 0 F I L L U S T RAT I 0 N S • • o • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • v i

LIST OF TABLES •••••••••••••••••••••••••••.•••..•••••.•••••.•.•••••• viii

I. INTRODUCTIOf~ .......................... •o.. .. . ....... .. . . . . . . •. . . 1

A. STATEr~ENT OF THE PROBLEi'1................................. 1

B. ACTIVE SYNTHESIS.......................................... 1

1. REALIZATIOr~S USING CONTROLLED SOURCES................ 2

2. REALIZATIOt~S USH~G OPERATIOr~AL Af1PLIFIERS............ 6

3. HEALIZATIOf~S USING SPECIAL DEVICES................... 14

II. SENSITIVITY! Q, AfiD THE BAt~DPASS FUf~CTIOfL ••••••••••••••••.•. 29

A. SENSITIVITY .•••.•••.•••..••••.•.•..••.••••••••.•.••..••.. 29

B. BANDPASS FUNCTIONS AND Q ................................. 31

C. SOt1E COt~t·1ENTS ON THE APPLICATIOn OF SEI~SITIVITY ANALYSIS • • • . . • • . . . • • . • • • • • • . • . . . • • . • . . . • • . . • . • . . . • . . . • . . . 3 3

III. CIRCUIT ANALYSIS •.••..••..•.••.•..••...•.•••.•....•.••....•.. 37

A. C0~·1t·1ENTS ON THE SENSITIVITY TABLES....................... 37

B. A BANDPASS REALIZATION USING A TWIN-TEE NETWORK ...••.••.. 52 C. BANDPASS REALIZATIOI~S USING DISTRIBUTED NETWORI<S......... 56

IV. SUt·H1ARY, COf~CLUSIOf~S, At-JD SUGGESTIOf~S FOR FURTHER vJO R K. • • • • • • • • • • • • • • • • • • • • • • • • • .. • • • • • • • • • • • .. • • • • • • • • • • • • • • • • • • 6 2

B I B L I 0 GRAPH Y • • • • • • . • • • . • • • . • • • . . . . • . • . . . • . . . • . . . • . . . • . . . • . . . • . . • • . . 6 7

VITA............................................................... 68

APPENDICES •••••.•••••.•.••.•••••.••.....•••.•..••......•...••.•••.• 69

1. DERIVATIO!i OF THE TRANSFER FUi~CTIOI~ OF FIGURE 2.......... 70

2. DERIVATION OF THE Slf~GLE FEEDBACK PATH RELATIOf~ OF FIGURE 5.................................................. 73

3. DERIVATIOri OF Ir,1PEDAf~CE INVERSION 8Y A GYRATOR........... 75

4. DERIVATION OF THE TRANSFER FUfJCTIOr~ OF FIGURE 13........... 76

5. DERIVATION OF THE TRAiiSFER FUfJCTIOt·J OF FIGURE 15 ••••••••• 78

6. FORMING TilE INDEFI!~ITE ADtUTTANCE f1ATRIX OF THE URC .••.•. 81

v

Table of Contents (continued) Page

7. ANALYSES OF THE B/\rJDPASS FUr~CTIOIL....................... 85

A. PROOF OF THE rJ1AXH1Uf1 PASSBAND GAIN AT w FOR n

'J___£ = Hs v 2 - 2 •• 1111 ............ Ill: .................. .

1 s + 2sw s + w n n

B. DERIVATIOfJ OF THE 3db FREQUENCIES (HALF-POt~ER POINTS) AND THE Q EXPRESSIOfi FOR THC l3Af~DPASS FUNCTION ...•....•.•...•.•..••••••••••.•.•....••.•....

85

86

8. DERIVATION OF THE TRAIJSFER FUI~CTIOrJ OF FIGURE 41......... 91

vi

LIST OF ILLUSTRATIONS

Figure Page

1. Ideal controlled sources..................................... 3 2. Realization of a transfer function using an RC

3 port network and a VCVS.................................... 4

3. Circuit symbol of 11 infinite 11 gain op amp and its output equation............................................ . . . 7

4. Realization of an inverting VCVS............................. 8 5. Realization of a transfer function using a single

feedback path................................................. 9

6. A general realization of a transfer function of arbitrary degree using a multiple feedback path approach. . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . • . . . . . . . . . . . . . . . . . . . . . 10

7. General multiple-feedback realization of a third order transfer function ........•............................. 12

8. Realization of an inverting integrator .•..................... 15

9. Realization of a summing device with inverting and non-inverting inputs...................................... 16

10. A general state-variable realization......................... 17

11. The ide a 1 gyrator and its transmission parameters. . . . . . . . . . . . 18

12. The ideal gyrator as a two port .............................. 19 13. Cascade synthesis using an ideal gyrator..................... 21

14. The current inverting NIC and its transmission parameters ..................................................... 23

15. Parallel-cascade INIC realization of a voltage transfer function............................................. 24

16. The uniformly distributed RC network and a lumpedmodel ................................................. 25

17. The URC two port and its y parameters ........................ 26

18. Example of a low-pass function simulated \'Jith a URC.......................................................... 27

19. Frequency response and pole-zero diagram of the bandpass function............................................. 32

20. Circuit #1 -operational amplifier circuit using multiple feedback paths ...................................... 39

21. Circuit #2 - VCVS circuit, K -?- 0 (II~ I -+ absolute value of K) .................................................. 40

22. Circuit #3- ideal gyrator circuit ........................... 41

23. Circuit #4- negative imittance converter circuit ............ 42

vii

List of Illustrations (continued)

Figure Page

24. Circuit #5 - operational amplifier circuit using multiple feedback paths ...•.......•...•.....•................ 43

25. Circuit #6- operational amplifier, VCVS, positive feedback circuit for high Q realizations ( K < 0).. • • • • • • • • • • • • • • .. .. • • • • • • .. • • • • .. • • • • .. • • • .. .. • • .. • • • • • • • • • • .. • 44

26. Circuit #7 - VCVS circuit, I< > 0................................. 45 27. Circuit #8- VCVS circuit, K > 0 ••••••••••••••••••••••••••••• 46

28. Circuit #9- VCVS circuit, K < 0 .............................. 47

29. Circuit #10- VCVS circuit, K > 0 .............................. 48 30. Circuit #11 - multiple VCVS circuit, K1 > 0,

1(2 > 0 • • • • • • • • • • • • • • • • • • • • • .. • • • • • • • • • • • . • . • . • . • . • • • • . . • . • • • • • 4 9

31. Circuit #12 - operational amplifier, state-variable realization ......................................... 50

32. Circuit #13- operational amplifier, "tvJin-tee" circuit...................................................... 51

33. The input network for the twin-tee bandpass r·ealization ...................................•.............. 53

34. The twin-tee network ...............•..•...........•.......... 54

35. Circuit #14 - operational amplifier, URC circuit...................................................... 57

36. Circuit #15 - distributed-lumped VCVS circuit, K > 0............................................................ 58

37. Circuit #16- operational amplifier, DLA realization ...•..................•........................... 59

38. Circuit #17- VCVS, DLA realization .......................... 60

39. Circuit #18- operational amplifier, DLA realization.................................................. 61

40. A first order lumped approximation of the URC ................ 64

41. Circuit # 14 v-~i th all URC • s replaced by their first order models ........................................... 65

42. VCVS realization using a 4 terminal RC net\vork. .. . . . . . . . . . . . . 72

43. Parallel branch of figure 15 ................................. 79

44. The URC as a 3-port.......................................... 82

viii

LIST OF TABLES

Table Page

I. Sensitivity functions for circuit # 1 •••.•••••••••.•••••••••• 39

II .. Sensitivity functions for circuit # 2 ••••••••••••••••••.••••• 40

I I I. Sensitivity functions for circuit # 3 ........................ 41

IV. Sensitivity functions for circuit #4 •••••••••••••••••.•••.•• 42

v. Sensitivity functions for circuit # 5 .......................... 43

VI. Sensitivity functions for circuit # 6 •••••••••••••••••••••••• 44

VII. Sensitivity functions for circuit # 7 •••••••••..•••.••••••.•• 45

VIII. Sensitivity functions for circuit # 8 .•.....•.•..•...•.•...•. 46

IX. Sensitivity functions for circuit # 9 ••••••••••.••••••••••••• 47

X. Sensitivity functions for circuit # 10 •••••..••.•...••••••... 48

XI. Sensitivity functions for circuit # 11 •..••.•••.••.•.•..••.•• 49

XII. Sensitivity functions for circuit # 12 ••.••..•...•...•....... 50

I. INTRODUCTIOil

A. STATD·1ENT OF THE PROBLH1

1

The primary purpose of this paper is to calculate and tabulate the

sensitivity functions of various active RC bandpass circuits. The impor­

tant sensitivities calculated will be those of the center frequency, w0

,

and the Q of the circuits with respect to element value changes. The

sensitivities calculated are of the classical variety. It is the tabula­

tion, not the determination of the sensitivities \·Jhich is important in

most cases since the circuits presented in this paper are scattered throughout the literature. Also, in most texts the sensitivity functions

are not given in terms of symbolic element values but are evaluated for

a certain set of element values and are thus numerical in nature. This

paper presents these functions in their symbolic form and catalogs these functions for a variety of the more common bandpass realizations. The

calculation of the sensitivities of w and Q is simplified by first cal-a

culating the sensitivity functions of the coefficients of the denominator

polynomial of the bandpass functions. The sensitivities of w0

and Q are

simple combinations of the coefficient sensitivities. Also considered in this paper are realizations including uniformly

distributed RC net\-Jorks (URC). The transfer functions of such circuits

involve hyperbolic functions of the square root of the complex frequency

variable, s. Thus it is not possible to get a transfer function which

is l"'ational in s, and the sensitivity functions of the denominator

coefficients, w , and Q are not easily calculated. Some alternatives a-are suggested as to possible approxinations of these circuits and the

resulting approximate sensitivity functions.

Approximations are also involved in the twin-tee realization Hhich

does not contain a URC. The nature of these are discussed in Chapter

I I I.

In order that it be clear how the various bandpass circuits are

realized, information on active synthesis techniques is no\J presented.

B. ACTIVE SYNTHESIS Although a part of most active synthesis techniques is simply

passive synthesis, there is a significant amount of additional material

which should be presented so that the reader may become familiar with

2

various approaches to the active synthesis problem. It is the intent of this section to present several approaches and give simple examples to

illustrate the application of each approach. These presentations VJill

be kept brief, and the reader is referred to the literature for more

detailed explanations. ~.1 Realizations Using Controlled Sources

There are four major types of ideal controlled sources. These are

as follovJs:

1) the voltage-controlled voltage source (VCVS).

2) the voltage-controlled current source (VCIS).

3) the current-controlled voltage source (I CV S).

4) the current-controlled current source (ICIS).

These are represented schematically in Figure 1 [ 1]. In practice, one

of the most nearly realizable ideal controlled sources is the VCVS. In

the following discussions, a synthesis procedure will be described which uses the VCVS as the active element. This procedure can be generalized to make use of the other ideal controlled sources giving similar results

involving different circuit parameters [1]. Also for practical consid­erations, the design procedure utilizes passive 3 port net~JOrks contain­

ing only resistors and capacitors (RC netHorks).

The basic circuit configuration is sho~·m in Figure 2, and the

transfer function of interest is

T

With the VCVS approach, the short-circuit admittance parameters (called they parameters) prove most useful. For the 3 port passive network

s hovJn, the port voltages and currents as defined in Figure 2 are related

as follows using y parameters:

I1 y 11 yl2 y13 v1

I2 y21 y22 y23 v2

I3 y31 y32 y33 v3

3

o----oo 0 0

+ +

v v -

vcvs VCIS

I R0 1 I Kl

ICVS I CIS

Figure 1. I DEAL CONTROLLED SOU RC t:S

r-o ----, I+

PASSIVE 31 1 v3 = Vx I-+ RC VI I

3-PORT -~V2=KVx: 21 r - _j

L_ /-

vcvs

Figure 2. REI\LIZ/\TIOil UF ~~ TR/\IlSFEf~ FUtiCTIOn USIIJG MJ fZC lHREE PORT NETvJORK Aim A VCVS

..,::::,

5

It can be shown that for the circuit configuration of Figure 2 the

desired voltage transfer function in terms of the y parameters and K0

,

the gain of the VCVS, is given by

V 2 _ -Koy31 VJ:"" - Y 33 + Koy 32

The details of the development are given in Appendix 1. It now remains to obtain a passive RC 3 port network withy parameters y 31 , y33 , and

y 32 to give the specified function v2;v 1 (i.e., low-pass, high-pass, etc.).

Recently in the literature several methods have been presented for

the synthesis of specified open-circuit impedance parameters, znn' for ann port tvJO-element-kind net\vork [2,3]. In this case, a passive 3

port RC neb·Jork with specified y parameters y 31 , y32 , and y 33 is needed. Since the network under consideration contains only resistors and

capacitors, it is reciprocal and therefore

For example, the parameter y31 is equal to the parameter y 13 . Thus by

specifying y 31 , y 32 , and y 33 actually 5 of the 9 elements of the short­

circuit admittance matrix for the 3 port are specified. HovJever a problem arises from the fact that 4 elements of this matrix are not known. If all elements \'/ere specified all that \·JOuld remain \·Jould be to take the inverse of the short-circuit admittance matrix (y .. ) to get

lJ the elements of the open-circuit impedance matrix (zij) and then apply a synthesis technique to get the desired realization. But all elements

of the y matrix must be specified before any of the elements of the z

matrix can be calculated. Thus some function of the complex frequency

variable, s, must be defined for each element yij describing the 3 port

to be used. In forming these functions considerations must be given as

to whether a certain set of y parameters are realizable as a passive RC

network. The selection of the unspecified y parameters seems to be a

hit and miss procedure, but a rule of thumb might be to keep these

elements of the short-circuit admittance matrix as simple as possible.

6

B.2 Realizations Using Operational Amplifiers The operational amplifier is a device of essentially infinite gain

(some 11 op amps 11 as they are called, have open-loop, i.e., no feedback, gains of 58 db) usually VJith differential inputs. The basic op amp with

defining transfer equation (neglecting frequency effects) is shown in Figure 3.

With a simple resistive feedback network, the op amp can be used to

realize a VCVS with the gain fixed by the ratio of resistances. Such

a realization as shown in Figure 4 might be used to construct the

circuits discussed in Section B.l. In this section, hov-1ever, the fact

that the gain of the op amp is infinite will be most important.

Basically t\vo classes of op amp realizations exist. These are the

multiple feedback path and the single feedback path realizations. The

latter will be discussed first. In the case of the single feedback

realization, the resistors of Figure 4 are replaced by passive RC 3

terminal neb\lorks (usually drawn as 2 port networks). The realization

is sho\'m in Figure 5, and in this case the voltage transfer function

v2;v 1 will be expressed in terms of the short-circuit transfer admit­

tances of the netv~ork. In Appendix 2 it is shovnl that for the configu­ration of Figure 5 the voltage transfer function is

Thus if a transfer function is specified (i.e., 10\,J-pass, high-pass,

etc.), synthesis of two networks with the proper transfer admittances

(usually V..Jith common denominators) completes the realization procedure.

Note here that synthesis of specific transfer admittances of passive

RC 2 port networks is considerably easier and bettet~ defined than the

synthesis of passive RC 3 ports needed in Section B.l. 1\lso, although

this realization procedure may require larger numbers of elements, it

is a more stable realization and it has, in general, lower sensitivities

due to the single feedback path.

For the case of multiple feedback paths, a general circuit for the th realization of ann order transfer function is shovJn in Figure 6 [1].

c;» z -I-0

: L

&lf-

>::)

~a.

•z

z_

0 z t +

(!)

z I-

-1-::)

a:: a..

LL

IZ

>-

z -t I

+

+

0 >

a t ~

UJ

1-

....... . M

7

8

+

I

(/)

> (_)

>

0 (!)

~

r-~

II J.J >

--·-

0:: ~

<

'+-

I L

L

a= II

0

>N

I>

0 ........... 1

-<::::( N

...........

u _

_J

<

w

z c:::

-c(

. <

::j-

(!) C

J s..... ~

rn

LL

a=

-+

>

I

PASSIVE, RC It ... 12

+

I' -I -2 .. + +

v. ' _/' + I . v, - ,. ~ I -

v2 - I I -

v2 I

Y21 --=- -v, Y12

Figure 5. REfiLIZ/\TIOiJ OF /\ TfU\IlSFLR FUilCTIOfl USIIIG i~ Slt~GLL FCtuSf\Ct;: PATii

1..0

---

--- I I +

- _,_ -

Figure 6. A GENERAL REALIZATION OF A TRANSFER FUNCTION OF ARBITRARY DEGREE USING A MULTIPLE FEEDB.4C K PATH 1~PPROACH

1-' 0

The blocks in the diagram represent admittances which are either

resistors or capacitors. The transfer function of this realization may

be expressed in general terms of these admittances and then the

individual elements selected as resistors or capacitors in order to

11

make the general function fit a specified function. In this way terms

which are constants or terms in s, s2 , s3 , up to sn can be created for

the general realization depending upon the placement of resistors and

capacitors in the general realization. An example of a general configu­

ration for realizing up to a third order transfer function is shown in

Figure 7. The transfer function for this realization is

V2- -YlY3Y7

VI- Ya(Y3+Ys+Y6+v7)(Yl+Y2+Y3+Y4) + v5v7(Yl+Y2+Y2+Y3+Y4) + v3v4v7 + v~v8

In Figure 7, sets of elements which may be designated as groups are shown inside the dotted boundaries. In general, if the maximum degree

of the realization is n, then the general configuration must contain

n-1 groups. Thus in Figure 7 since the maximum degree of the realiza­

tion is n=3, it contains 2 groups. It should be noted that for reali­

zations of degree n greater than 2, a selection of elements exists such

that a term sP where p is greater than n will appear in the denominator. However, this selection will also cause an sq term to appear in the

numerator. Dividing numerator and denominator by sq will yield the term

s(p-q) in the denominator such that p-q = n. /\lso note that the self­

admittances of nodes 1 and 2 as shown in Figure 7 appear in the denomi­

nator of the transfer function. This is a result of the general config­

uration and these self-admittance terms \'Jill be present for any arbitrary

degree realization.

There are several other realization approaches using operational

amplifiers. These techniques essentially expand the possible complexity

of the passive RC circuitry used and also include the possibility of

more than one op amp. A good example of such a technique is the state-variable approach.

Although this type of realization requires that the number of op amps

needed be not greater than n+2, where n is the order of the transfer

v,

-

/

/

I I y4

I

\

\ y2 \ \

\

' ' '

' / ' / X

I \ I \

' I

\ I

\ I

\ I \ I \I

/ ' / '

' ' Ys ',

Ys I

I

/ /

\

I I

I

GROUP I~ GR~

Ya

I

~ v2

I -

ALL Y' s .. R or C

Figure 7. GENERAL MULTIP~E-FEEOBACK REALIZATION OF A THIRD ORDER TRANSFER FUNCTION

~

N

13

function, the sensitivities are in general much lower than single active

device realizations. ~~ithin the category of the state-variable approach there are many possible choices of the actual state variables of the

system under consideration. For the case of interest here, the equation which describes the system is the transfer function. Now assume a transfer function of the form

[1]. Multiplying numerator and denominator by x/sn yields

v2 ( a0

x/ sn) + (a1x/sn-1) + ••• + (an_ 1x;s) + a x n VI- ( b

0x/ sn) + (b1x/sn-1) + ••• + (bn_ 1x;s) + b X

n

The state variables will be chosen as

X X X

s(n-1) , - X

sn s

Thus if the transfer function is of order n then there are n independent

state variables using this approach because x can be obtained through

a linear combination of the other state variables. rJow equating the numerators and denominators of the right and left sides of the equation

immediately above gives

or

bnx = V - (b x;s") - (b x;s"- 1) - ••• - (b x/s) 1 o 1 n-1

and

v2 = (a x/sn) + (a x/sn- 1) + ••• + (a 1x;s) +a x o 1 n- n

Now note that multiplication of a variable by 1/s is equivalent to

integration of the variable vJith respect to time. Thus all of the

state variables may be realized by using one summing device (with

14

inverting and non-inverting inputs) and n integrating devices. With all

n state variables realized, only another summing device is needed to

realize v2. A simple inverting integrator realization is sho\vn in

Figure 8, and a summer with inverting and non-inverting inputs is shown

in Figure 9. The schematic representation of the general state-variable

realization is shown in Figure 10 [1]. Practically, the state-variable

approach may be used to simultaneously realize a low-pass, bandpass, and

high-pass functions. This is due to the fact that in the general reali­

zation the numerator of the transfer function is assumed to be of order

equal to that of the denominator. In the case of the bandpass realiza­tion considered in this paper, it is not necessary to include the final

summer of the general state-variable realization since the numerator

is of degree one less than the denominator. Thus the realization of

all state variables will be sufficient. This will become more clear

when the bandpass state-variable realization is presented in Chapter 3. Another type of realization exists using an operational amplifier,

passive RC circuitry, and uniformly distributed RC netvvorks (URC's).

Due to the special nature of the URC this realization technique VJill

be discussed in the following section on special device realizations.

B.3 Realizations Using Special Devices Three types of special devices used in active synthesis vdll be

discussed in this section. These devices are the gyrator, the negative

immittance converter (NIC), and the uniformly distributed RC network

(URC) .. The schematic representation of the ideal gyrator is shown in

Figure 11 with its defining transmission parameters.. The term G0

is

referred to as the gyration conductance. If the gyrator is considered

as a 2 port as shown in Figure 12, the input impedance at port 1 is the

inverse of the impedance Z connected across port 2 multiplied by the 2 constant l/G0

• This is demonstrated in Appendix 3. The main use of

the gyrator discussed here will involve putting a capacitor across port

2 of the gyrator. Si nee the impedance of a capacitor is

15

+

I

0::: 0 1

--c::(

a::: (_!:')

w

1--

:z::

c.!)

:z:: ........ 1

--er::: w

>

-I~ :z::

:z::: c::(

LL

.

I 0 :z::

.. 0 1

---4

1--

~I>

~

N

........ _

J

c:::( w

0

::: . 0

0

Q)

S-

::::; O

l

LL

-> I

I

(n) v, +~ I I I

G (n) : m

I

I I

I G'(p) : Vo

-_r- v~P>+~ _j_

k Vo = r

i =I

WHERE

AND

G (p) 0 =

G(n) 0 =

~ G~p) . I I p::

m G·(n) r I i =I

Fiqure 9. REALIZATION OF A SUMMING DEVICE WITH INVERTING AND NON-INVERTING INPUTS

16

+ VI -_j_

Oo

1/bn

---- l: c;:,

an-2 an-t

Fi~ure 10. A GENER.l\L STATE-VARIABLE RE.ALIZATION

X

an

+ v2

_j_

1-l '-J

18

j_ j_

=

Figure 11. THE IDEAL GYRATOR AND ITS TRANSMISSION PARAMETERS

N

1-

a:: 0 a..

t-0:: 0 0..

-=-t

-> ft c ·-N

I

No

-

(!)

.. c ·-N

N

19

•r-

LL

.

20

the input impedance at port 1 of the gyrator v-1ill be

z. 1n = sC G2

0

This is equivalent to the impedance expression for an inductor

Z = sl L

Thus by using a gyrator of gyration conductance G0

and a capacitor of value C, the equivalent inductance

c L = G2

0

can be realized. Now using the procedures of passive RLC synthesis a transfer function can be synthesized and then all inductors replaced

by the equivalent gyrator-capacitor combination. This method has definite advantages over strictly passive RLC synthesis. It allov1s the

elimination of bulky real inductors and the relatively strong magnetic fields they create. Using a gyrator and capacitor, it is possible to

realize large inductances with good tolerance values since capacitor

tolerances are generally lower than inductor tolerances and since the

equivalent inductance depends on the capacitance and gyration conduc­

tance both of which can be controlled. An alternate approach is to interconnect passive RC networks using

a gyrator as shown in Figure 13 [1] for which the voltage transfer

function is

This is derived in Appendix 4. The problem now becomes one of se­

lecting the proper networks to synthesize the desired transfer function

[1].

Under the class of circuits referred to as NIC 1 s there are two

sub-classes. These are the VNIC (voltaqe inversion NIC) and the INIC

(current inversion NIC) so named for the quantity which they invert to

I I ... I o-+

I PRIMED

v, NETWORK

PASSIVE RC

1 ..

I

12 Go I ,, ... I 12 I ..

I UNPRIMED

v' 2 v' I NETWORK I

PASSIVE RC -

Figure 13. CASCADE SYNTHESIS USING AN IDEAL GYRATOR

v2

N ......

22

obtain the negative sign. For voltage transfer functions, the IIJIC is

most useful and is shown schematically with its defining transmission

parameters in Figure 14. The realization of a voltage transfer func­

tion is accomplished with the parallel-cascade configuration sho\vn in

Figure 15 [1] for which the voltage transfer function is

The derivation of this transfer function is given in 1\ppendix 5. There

is now a straightforward procedure to construct the passive RC netvwrks

with the proper y parameters to realize a specified voltage transfer function [1].

The URC is a relatively new device, but it has received great

attention due to its applicability to integrated circuit realizations.

A 1 so a single URC may rep 1 ace a number of resistors and capacitors in

an active RC realization and thus may save a great deal of space. The

schematic representation of the URC along vvith a lumped element model is shown in Figure 16. One disadvantage of the URC is that it is not

possible to describe it in terms of rational functions of the complex

frequency variable s. As an illustration consider the URC as the 2

port shown in Figure 17. Also in this figure, the short-circuit ad­

mittance parameters are given. The URC is mathematically described by

hyperbolic functions whose arguments are irrational functions of s.

In network synthesis, such functions are extremely hard to \Jork \Jith

directly, and so the synthesis techniques to simulate transfer func­

tions rational in s using URC's are approximation techniques. 1\n

example circuit is shown in Figure 18 [4] and it uses a single URC and

a VCVS to approximate a low-pass function. The actual transfer function

of this circuit is

v 2 - I(

\Jl- cosh e + 1((1 - cosh e)

v>~here

e /sRC

... --- -- -+ +

INIC I v

- GAIN=Kt --. -- -

I 0 --

0

Figure 14. THE CURRENT INVERTING NIC AND ITS TRANSMISSION PARAMETERS

23

It

v,

I y .. IJ

y .. IJ

INIC GAIN= K1

Figure 15. PARALLEL-CASCADE INIC REALIZATION OF A VOLTAGE TRANSFER FUNCTION

12

v2

N +:.

R,C

v, I v2 ===f>

-n-

R - TOTAL RESISTANCE

C - TOTAL CAPACITANCE

----r 0 I T ____ L

LUMPED MODEL

OF THE URC

Figure 16. THE UNIFORMLY DISTRIBUTED RC NETWORK AND A LUMPED MOQEL

N (51

26

R,C - -

+ +

II coth 8 - csch 8 VI

fi -12 -csch 8 coth 8 v2

8 =.Js"RC

Fiqure 17. THE URC TWO PORT AND ITS y PARAtv1ETERS

27

C\J I§§

+

>

c:::r:::

I I-

1---4

3 0 w

I-

c:::r::: ___J

~

::::::: 1---4

(/)

z:

0 1---4

~--

(_)

z:

=::> L

1-

(/)

(/)

c:::r::: o

._

I 3 C

)

_J

c:::r:::

CD L

l-0

,..-I

i.L.l

u!

(.) __ ..J C

L

~::...

n:: <

>< w

L-

_.J

<(

co .--t

QJ

S-

+

>

~

(J)

LJ_

and

R = total resistance of URC

C = total capacitance of Uf(C

28

A derivation of this transfer function is given in Appendix 6. A root

locus analysis [5] of this transfer function sllovJs that it has an in­

finite number of complex conjugate poles in the left half of the s

plane. However one pair of these complex conjugate poles is dominant

for a range of gain [4], K of the VCVS, and the circuit behaves

approximately as if it had only one pair of such poles. This is the

case for the low-pass function of the form

V 2 _ H

V'1 - s 2 + 2 z:;w s + w 2 n n

and so the circuit of Figure 18 approximates a low-pass transfer func­

tion. This root locus procedure is the most frequently used in tile

approximation of transfer functions rational in s. It should be noted

that synthesis procedures are not limited strictly to the use of URc•s

but may include a general class of networks referred to as distributed­

lumped-active networks [6] which use discreet elements (resistors and

capacitors) as well as URc•s for a more varied approach to the synthesis

problem. An example of each will be included in this paper.

II. SENSITIVITY, Q, AND THE BANDPASS FUNCTION

A. SENSITIVITY

29

The sensitivity of an active circuit is important to the circuit designer since it gives him a comparative tolerance measurement for the active circuit as a whole. The sensitivity measurement can be compared to the tolerance figures given for a resistor or capacitor although the sensitivity functions defined and discussed here are in­cremental in nature and are not accurate for large changes in the design values.

The sensitivity function defined here and used in the analysis of the bandpass circuits to follow is referred to as the classical sen­sitivity [1]. The classical sensitivity of a function, F, to changes in one of its variables, v, is defined as

SF = 8F/F = _Q£_ • '!_ v dV/V dV F

It can be seen from the classical sensitivity expression that it is a relative or normalized chang~ in the function for a normalized chanqe in the variable. Since F may be a function of several variables the differential quantities are expressed as partials which means that the sensitivity of the function to changes in the variable is analyzed assuming that all other variables undergo no change. In the final form, knowing F and v, it is only necessary to calculate the partial derivative of F with respect to v. At this point, an example is in order. Suppose the classical sensitivity of the function

with respect to each of its variables is desired. These sensitivities

are calculated as follows: SF 8F A

A = aA • F

therefore

Similarly

and

2AB = -c-

SF = 2AB A -c.

sF = 2 A

sF = l B

sF c = -1

30

A

Due to the simplicity of the example, several important points should be made so that the reader is not led to unfounded generalities.

Note first that for calculations of the sensitivity functions in this manner a symbolic function in terms of all variables is needed. In

certain cases this is not possible or practical and sensitivities are numerically or experimentally demonstrated. Such cases will be dis­cussed briefly later. Note also that the sensitivity expressions are

not functions of any of the variables in F in the example. This is due to the simple relationship of the variables in F and is not true in general. As will be seen later, most sensitivity functions will

depend on the design values of the particular circuit and are not con­stants but are functions of resistance, capacitance, and gain. The example does, however, demonstrate the procedure by which sensitivity functions are calculated. Also note that a variable in the denominator of the function is reflected by a negative sensitivity of the function

to the variable [2]. If the variable is in the numerator and denomin­ator of the function this may not be true.

Besides classical sensitivity, there are other classes of sensi­tivities which are useful. One such class is root sensitivity which

encompasses both poles and zeroes of a function. The root sensitivity

is not concerned with changes in the function as a whole but with changes in the poles and zeroes of a function with respect to some change in a variable [1]. Still other sensitivities use statistical measure for their definition [7]. However due to the straightforward

31

techniques by which the transfer functions of most active RC circuits may be calculated, the sensitivity functions cataloged in this paper will be of the classical variety.

B. BANDPASS FUNCTIONS AND n This paper is concerned with the realization of bandpass transfer

functions of the form

or more generally

V2 _ Hs

VI- s2 + b1s + b

2

The typical bandpass frequency response and pole-zero diagram are shown in Figure 19. In most applications of the bandpass circuit it is d~~ired to pass a narrpw band of fre~~~~~ies and attenuate all

I I ' • '" >

others below useable level outside this band~ It is very important that some expression for the 11 narrowness11 or quality of the pass band

__ ,, .. '::0 ·""• ~ •• ~ ., ~ -

be determined so that this expression ca.n be used to tell how sensitive the pass band is to changes in the circuit and to keep the pass band of the circuit within specifications. This expression is referred to as the Q of the circuit and for the bandpass function is defined

Q FREQ. of MAX. GAIN = BANDWIDTH

where BAND\tJIDTH = w2 - wl

and

w2 Upper 3 db frequency

w1

= Lower 3 db frequency

The 1 db frequencies are defined as those frequencies where the gain

of the circuit is

GAIN AT 3 db FREO. = 1 · U1AX PASS BAND GAIN)

32

IMAGINARY

*--- +jwn~ 5 PLANE

I \ I ""' I

,n I \

I \.

I a I REAL I I I I

I I Wn 1/

~--- -jwnJI-t 2

Figure 19. FREQUENCY RESPONSE AND POLE-ZERO DIAGRAM OF THE BANDPASS FUNCTION

33

If the frequency of maximum pass band gain (often referred to as the center frequency) is w

0, then the definition for Q is

Thus the smaller the bandwidth (i.e., the narrower the range of fre­quencies passed) the higher the Q of the circuit. The expression for Q immediately above is well suited for experimental procedures but does not lend itself to sensitivity analysis. An expression for Q in terms of element values of the circuit is needed. By analysis of the general

bandpass function to determine the frequency of maximum gain and the upper and lower 3 db frequencies, it is shown in Appendix 7 that

162- 1 Q = ~- 2f

Thus from the transfer function of the circuit, an expression for Q

is easily obtained and sensitivity analysis is possible.

C. SOME COMMENTS ON THE APPLICATION OF SENSITIVITY ANALYSIS Before beginning the actual analysis of some bandpass circuits, a

few comments on the applicability of sensitivity analysis and some

problems encountered are in order. First., a discussion of the limitations of sensitivity functions

is needed. These limitations are best illustrated by an example. Let

the function F be defined as in the example in Section A of this chap­

ter and further let

A = 2, B = 3, and C = 5

which gives the result

F = 2.40.

As previously calculated in Section A, the sensitivity of F with re­

spect to A is

sF = 2 A

Now let the value of A increase by 0.02. The change in F is

calculated as follows:

~ F = ( ~A ) S ~ ( ~)

(0.2)(2)(1.2)

= 0 .. 048

and the new value ofF predicted is

F 2. 40 + 0. 048

F = 2.448

By substitution of the nevJ value of!\ directly into the expression for F, the actual new value ofF is

F

F 2. 4482

34

Thus for this case the sensitivity function gives a good approximation

of the change in F. Now let A change by 1.0. Thus

~F = (~J\) s~ ( ~)

= 2.4

and the new value of F predicted is

F = 2.4 + 2.4

F = 4.8

The actual new value of F calculated is

F (3) 2 (3) 5

F = 5.4

Thus from this example it can be seen that due to the incremental

(differential) nature of the classical sensitivity, the changes pre­

dicted by the sensitivity functions are only accurate vJhen element

values change by about 1% or less.

3S

There are several realizations considered in this paper where the

application of sensitivity functions is not directly possible or

yields ambiguous results. One such realization is the bandpass realiz­

ation using an op amp and single feedback path RC networks. For the

bandpass case the feedback network is the twin-tee network shown in Figure 17 and which exhibits 11 notch" or bandstop characteristics.

However in the bandpass realization only the short-circuit transfer

admittance y12 of the twin-tee appears in the overall transfer function. Analysis shows that several assumptions are necessary to reduce the

y12 expression so that the general bandpass function can be obtained.

These assumptions may eliminate all representation of a circuit element

in the final transfer function and thus a zero sensitivity would be

indicated. However this is obviously not the case since if the element in question is chanqed, a change in the function is inevitable because

the assumptions made at the beginning would be invalidated by the

change. This problem and possible solution will be discussed in more

detail in the following pages when the specific circuit is presented . . Another problem is encountered in all realizations usinq URC 1 s.

With such realizations the general denominator of the bandpass function,

which is a quadratic function in s and has a single pair of complex

conjugate roots, is replaced by hyperbolic functions of the square root

of s. Thus the denominator of the transfer function of a URC realization

will have an infinite number of complex conjugate root pairs. As pre­

viously discussed, one of these pairs is dominant for certain circuit

conditions (usually a certain range of gain of the active device) and thus the realization simulates the bandpass response. Depending on

the realization, the placement of this dominant pair depends on the

total resistance and total capacitance of the URC (or URC 1 s) and certain lumped elements for DLA (distributed-lumped-active) realizations.

However since a bandpass expression which is rational in s is not ob­

tainable, expressions for Q and Q sensitivity must be approxi~ated or obtained experimentally. In the literature, most authors prefer to

calculate root sensitivity functions [1] since the locations of the infinite number of complex conjugate root pairs is described by an e­

quation and such sensitivity functions are more easily found. A more

detailed discussion of the URC bandpass approximation will be pre­sented in the following pages along with several URC realizations.

36

37

III. CIRCUIT ANALYSIS

A. COt•lr'·1ENTS ON THE SEf~S IT IV ITY TAI3L ES

In this chapter the sensitivity functions of Q and w are tabu-o lated for each of the selected RC bandpass circuits immediately

following the respective circuit diagram. Also each table of sensi­

tivity functions includes the sensitivity functions of the coefficients

b1 and b2 of the standard bandpass denominator. These functions are

calculated to facilitate the calculation of the Q and w sensitivity 0

functions. Since

w =fb2 0

and knov~i ng the sensitivity of b2 with respect to some element e,

sensitivity of wo \vi th respect to that element value is expressed

A 1 so s i nee Q = fb21 b1

the sensitivity of Q v;i th respect to some

element value can be expressed as [1]

Q Swo - bl s = s e e e

the

Thus the sensitivity functions of importance, namely those of (;J0

and Q,

are simple combinations of the coefficient sensitivities. All sensitivity functions tabulated in this chapter are in terms

of s•s and G's where

s 1 c

G = l R

C is Capacitance

R is Resistance

for convenience and compactness of the expressions. /\11 sensitivities

can be obtained in terms of R's and C's by multiplying the expression

in the table by -1 [1]. For example for the resistor R1 of a certain

realization

38

At the end of this chapter are presented some bandrass realizations

u s i n g d i s t r i b u ted net VJO r k s an d t II e y a re s h o vm s c ll ern a t i cal l y a l o n g \ Ji t h

the respective transfer function. Due to the nature of the UHC, sensi­

tivity analysis such as that presented in the first part of this

chapter is not possible since the transfer functions are irrational in s.

Due to the bulk of the calculations involved in obtaining tile

transfer functions and sensitivity functions for each circuit, these

calculations are not presented. The transfer functions are o!Jtained

by application of node-voltage techniques. The calculation of tile

transfer functions for the distributed netvJOrk realizations involves

the use of the 2 port y parameters or the indefinite admittance matrix

of the URC both of \vhich are contained in Appendix 6.

Gl

1..!

I G2

"'

I s1 _, w

s2

: o--------~r~--~--~ c,

v2 - sG 1s2

+

---- = --------------------------------v, s2 + sStC G1 + G2) + s 1s2G1G2

Figure 20. CIRCUIT #1 - OPERATIONAL AMPLIFIER CIRCUIT USING MULTIPLE FEEDBACK PATHS [11]

SENSITIVITY FUNCTIOUS sb1 sb2 ~ SQ ... e e

G1 1 1 1 G1

~ I !-~

G2 1 1 1 G2

~ f I-~

1 1 1 1 r - r

1 1 1

0 I I

!!!:.U. SENSITIVITY FUNCTIONS FOR CIRCUIT 11

39

40

+o-c---~1....,_ _ _. __ __, +

cl

=

Figure 21. CIRCUIT #2- VCVS CIRCUIT, K < 0 (!KI ~ABSOLUTE VALUE OF K) [1]

SENSITIVITY FUNCTIONS sD1 5b2 ~0 sO e e e e

G1 G1S1

1 1 1 GlS2 ~1~1 + ~2~1 + ~2~2 ~ ~- ""1>1

G2 G2(S1+S2)

1 1 1 G2(S1+S2) ~1~2 + ~2~1 + ~2~2 ~ ~- 61

~ ....

S1(G1+G2) ~ s1 1 1 1 S1(G1+G2) >- ~ls1 + ~zs1 + ~zsz ~ ~- 61 V)

t-z: w

G2S2 G2S2 z: 1 1 w s2 1 ....

~1s1 + c2s1 + c2s2 ~ !- D1 w

IKI 0 ~ !Kk :!{I+, ... , J s~l

TABLE I I

SENSITIVITY FUNCTIONS OF CIRCUIT #2

41

c, +oo------~1~-----P------~

+

Fiqure 22. CIRCUIT H3 - IDEAL GYRATOR CIRCUIT [1]

SENSITIVITY FUNCTIONS sbl e

sb2 e

s»o e

SQ e

G1 G1Sl G1G2 G1G2 G1G2 G1S1

~1~1 + ~2~2 2 2 2(G1G2+G2) -~ G1G2 + G 2(G1G2+G )

G2 G2S2 sb2 ~0

G1G2 G2S2 ~1~1 + l;2S2 G1 Gl 2(G1G2+G2) -~

~ ....J

G1S1 ~ sb1 1 1 ::E sl 1 '2' ! - o;:->- G1 V)

~ LAJ

G2S2 ::.:: sb1 1 1 LAJ

1 ....J s2 ! '2'- o;:-LA.I G2

G 0 2G2 G2 $W0

2 2 G G1G2 + G G1G2 + G

TABLE I II

SENSITIVITY FUNCTIONS FOR CIRCUIT 13

42

:-H A

+ c, Rt INIC ;:::: >

v, c2 >R2 v. GAIN= K 2

- -

Figure 23. CIRCUIT *4 - NEGATIVE IMITTANCE CONVERTER CIRCUIT [1]

SENSITIVITY FUr4CT IOr4S sb1 e

sb2 e ~ SQ

'e

Gl Gl (S1-K1S2)

1 1 1 G1 (SCK1S2) ~1~1 + ~2~2 - ~1!2 I I- 1)1

G2s2 - GzS2 Gz 1 1 1

:a ~1~1 + ~2$2 - kG1~2 I I- -ot ...:::

~ s1 GlS1

1 1 1 G1S1 ~1~1 + ~2$2 - kG1~2 I I- ""'01

t;

~ sz S2(G2-KG1)

1 1 1 s2(G2-KG1)

~1~1 + ~2~2 - kG1~2 I I- 1)1

K -G1S2K

0 0 G1s2K

~1$1 + ~2$2 - KG1S2 ~1~1 + ~2~2 - kG1~2

TABLE IV

SENSITIVITY FUNCTIONS FOR CIRCUIT 14

43

+

v2 -sG1s2 - = Vt s2 + sG3 (SI + S2) + G3S1S2(G1 + G2)

Figure 24. CIRCUIT 15 - OPERATIONAL AMPLIFIER CIRCUIT USING MULTIPLE FEEDBACK PATHS [9]

SENSITIVITY FUNCTIONS sbt • sD2

e ~ SQ ~.

't 0 Gl G1 wo ~ ~(1:1+1:2) SG

1

'2 0 G2 G2 wo ~ 2(61+1:2) SG

'• 2 (..;;.

I '3 1 1 1 1 !' -,

I s 1 1 s1 iii sl !1 i !2 1 t ,-~

Sz $2

1 1 1 52 !1 + !2 !' ,-~

!!!bU. SENSITIVITY FUNCTIONS FOR CIRCUIT 15

7 ~

I "' ~ ~ d

44

+

v2 sKG1s2 -= VI s2 + s(G3s2 + G3s1- KG4S2 ) + G3s1s2 <G1 + G

4 +G2)

Figure 25. CIRCUIT 16 - OPERATIONAL AMPLIFIER, VCVS, POSITIVE FEEDBACK CIRCUIT FOR HIGH Q REALIZATIONS (K < 0) [9]

sb1 e

G1 0

G2 0

G3(S1+S2) G3 t3~1 + (;3~2 - ~4~2

-ICG4S2 G4 63S1 + t3S2 - la:4~2

G3Sl s1 tls1 + tls2 - ~4s2

S2(G3-ICG4) s2 6Js1 + 63~2 - ~.sz

bl K SG

4

bl • G3Sl + G3S2 - ICG4S2 a • 61 + 62 + 64

SENSITIVITY FUffCTIOltS sb2 e s:o

G1 G1 1:1 + t2 + 1:4 2{t1+t2+t4 l

G2 Gl 1:1 + (;2 + 1;4 :!{t1+t:2+1:4l

1 1 I

G4 G4 l:1 + (;2 + (;4 :!{tl+tz+64l

1 1 I

1 1 I

0 0

TABLE VI SENSITIVITY FUNCTIONS FOR CIRCUIT 16

SQ e

wo SG

1

wo SG

2

1 G3(S1+S2) I- 61

G4 KG4S2 ra·~

1 G3Sl I-Dl

1 S2(G3-ICG4) I- 61

bl -SG

4

45

+ +

=

Figure 26. CIRCUIT I 7- VCVS CIRCUIT, K > 0 (9]

SEI~SITIVITY FUUCTior•s

s 1 e sb2 e ~ SQ

'e

G1 Gl (Sl+S2) G1 G1 Gl -Gl(S1+S2)

5, ~ 2(~1+~2, . 2 (~1+~2, 61

G2(s1+(1-K)S2) G2 G2 G2 bl G2 51 ~ 2(~1+~2, ~(i:l+i:2J - SG2

~ G3S2 1 1 G3S2

I Gl Dl 1 1 1- D'l

V)

51 (Gl+G2) Sl(Gl+G2) !E sl 1 1 1

~ 51 2' !- 61 ~ w

s2(G3+G1+(1-K)G2) 1 1 1 bl

s2 61 1 ! - ss 2

K -KG2S2

0 0 KG2S2

--,;;- --,;'}

TABLE VII

SENSITIVITY FUf~CTIOr•s FOR CIRCUIT 17

46

+oa--~J\1\----~~~_.----~--~ R1 c1

Figure 27. CIRCUIT 18 - VCVS CtRCUIT, K > 0 [9,1]

SENSITIVITY FUNCTIONS sD1 •

sD2 e ~ s~

61 s2 s2

61<51 + y:r) 1 1 1 61 (51 + r-r>

51 I ,- 51

~ ~ +.t 1 1 1 '2S2 f f- "D'1

I 1

61s1 1 1 6151 ., s1 "D'1 1 f f--,;-;-

i _, 6 61 w S2(~ + Et>

s2 1 1 1 s2 < 62 + r-t> 51 t I- 51

k K&1S2

0 0 -K&1~

(1-k)2 bl (1-K)2 bl

TAkE VIII

SENSITIVITY FUNCTIONS FOR CIRCUIT 18

47

c2

l I

0 VV" I ( + +

VI Rl cl

v2

Figure 28. CIRCUIT #9- VCVS CIRCUIT, K < 0 [10]

SENSITIVITY FUNCTIONS sDl e

sD2 e

swo e

sll e

Gl G1S2

1 1 1 G1S2 ~1~2 + ~2{~1+~2} '2" '2"--a

~ Gz Gz(S1+Sz)

1 1 1 Gz(S1+Sz)

Gz5z + Gz{~l+sz} '2" '2"- a ~

~ 1: G2Sl G2S1 V) 1 1 1- sl G1~2 + Gz{sl+sz)

1 '2" '2"--a z: ..... ffi ~

Sz(Gl+Gz) Sz(G1+G2) ..... 1 sz 1 1

~152 + Gzts1+sz> '2" !- a

K K K K K r-r r:x '2\1-"Kr - nr=rr

TABLE IX

SENSITIVITY FUNCTIONS FOR CIRCUIT #9

48

+ +

v2 sKG1s1 - = VI s2 + s[G 3{s1 + S2 }+S1{G1+G2(1-K)}]+G3s1s2 CG1+G2 )

Figure 29. CIRCUIT ,10- VCVS CIRCUIT, K > 0 [10]

SEf~SITIVITY FUUCTIOUS

sb1 e

sb2 e s~ SQ

e

Gl SlGl Gl 61 Gl S1Gl

bl ~ 2,c;rt!i2} 2 (l:1+~2' - "1>'!

Gz s1G2(1-K) Gz Gz Gz -

S1G2(1-K)

1)1 ~ 2(~1+l:2} 2(1;1+l.;2} 61

~ GJ

G3(s1+S2) 1 1 1 G3(S1+S2)

8 bl ! !- 61

~ .., s1[G3+G1+G2(1-K)] s1[G3+G1+G2(1-K)] ....

sl 1 1 1 ::::

bl ! !- b1 ..., Q ~ ...,

sz G3S2

1 1 1 G3S2

"T1 ! ~- D1

K s1G2K

0 0 s1G2K

- "'D"1 --"til

]]!!;!_!

SEHSITIVITY FUI~CTIOfiS FOR CIRCUIT 110

49

+

Figure 30. CIRCUIT #11 -MULTIPLE VCVS CIRCUIT, K1 > 0, ~ > 0 [10]

SE~SITIVITY FU~CTIO~S

sb1 sL>2 S:o SQ e e e

G1 G1Sl

1 1 1 G1s1 ~1s1 + l;2!;2 2 ! - (;1~1 + ii2~2

G2 G2S2

1 1 1 G2S2 ~1s1 + ~;2~2 ! ! - (;1~1 + l:2!:2

~ s1

b1 1 1 SQ

.....j SG ! G1 g 1 s;: V')

b1 ~ 1 1 SQ i:i:l s2 SG 2 G2 5 2 .....j .....

K1~ K1~ lJ1 K1~ K1 I - ~1~2 SK 2{1 - ~1~} - 2U - ~1~1 1

lJ1 b1 K1~ K1~ K2 SK SK :!{1 - ~1~2} - 2U - ~1~J 1 1

TABt.E .XI

SEf~S IT IV JTY FUI~CTJ 014S FOR (;I RCU IT I 11

50

+

v2 -sG1G3s1<Gs+G6 )jG6 (G3 + G4 ) ---- = ----------~~--~--~--~~~--~-------------VI s2 + s[G1G4s1<Gs + Gs)/Gs(G3+G4)]+G1G2G5S1S2/Gs

Figure 31. ~IRCUIT N12 - OPERATIONAL AMPLIFIER, STATE-VARIABLE REALIZATION (1]

SEHSITIVITY FUHCTIOHS

s01 e sb2 e

swo e

SQ .@

1 1 1 G1 1 2 - 2

1 1 G2 0 1 2 2

G3 0 0

G3 G3 - G3+G4 ~4

~ G3 G3

...J G4 0 0 - GJ+G4 ~ ~

>= V)

t: Gs 1 1 Gs Uj

Gs 1 2 2 - G5

+G6 5 Gs+Gii

..J UJ

Gs 1 1 Gs G6 -~

-1 - 2 - 2 + G5

+G6

1 1 s1 1 1 2 - 2

1 1 1 s2 0 2 2

TABLl XII SENSITIVITY FO~cffoNs FOR CIRCUIT 112

R2 R3

c2 c3

v, c4 R4

Figure 32. CIRCUIT #13- OPERATIONAL AMPLIFIER, "TWIN-TEE" CIRCUIT [1]

+

v2

CJ'I .....

B. A BAi~DPASS REALIZATIOr~ USir~G A TWIN-TEE NETlJORK

The active RC bandpass circuit using the tHin-tee net\>Jork is of the single feedback path category discussed in Chapter I ~ Section

8.2 and shown in Figure 5. The transfer function of the realization of Figure 5 is

52

In the case of the tvJin-tee realization the primed network is slw~vn in

Figure 33 and the transfer admittance parameter Y2l is given ~Y the expression

sG 1 Y21 = - s + G

1s

1

The unprimed network is the tvvin-tee sho\vn in Figure 34, and the

related transfer admittance y 12 is given by the expression

Y -12 - -s 3/(S2+s3) + s2s4 (G2+G 3)/(S2+s3) + sG 2G3s4 + G2G3G4s 2s 3s4;(s2+s 3)

[s+G4s2s 3;(s2+s3)] [s+S4 (G2+G3)]

I t i s no V.J apparent that a pro b 1 em ex i s t s vv i t h the sen s i t i v i t y an a 1 y s i s

of the active bandpass circuit using the tHin-tee feedback netvJork since

this analysis depends on a simple quadratic polynomial in the denomina­

tor. The expression for y 12 may be written in the form

as 3 + bs 2 + cs + d Y12 = - (s + r

1)(s + r

2)

From the defining equation for v2;v 1 for the single feedlJack realization

the transfer function for· the realization using the tv~in-tee feedback

network is given by the expression

v2

_ sG 1(s + r 1)(s + r 2)

V1- - (as 3 + bs 2 + cs + d)(s + G1s1)

There are several assumptions made at this point vJhicll lead to the

bandpass function defined in Chapter II Section B [1]. These assump-

tions are:

S3

z:

0

~

+

.........

I 1

-c:(

N

......... _, c:(

w

0:::

(/)

(/)

< Q_

0

-z

0 <

(

cc. w

w

1

-I z ......... 3 1

-

w

I 1-

0::: 0 I..J._

~

0::: 0 3

-1

-

a:: w

z 1

-::::> Q

_

z 1----i

w

:::r: 1

-

M

M Q)

!"-:::;

-0

:

+

>

I ·r

-I..J._

C\1 (.)

~

0::: 0 3

: 1

-­w

:z: w

w

I--I z:

........ 3 I--

w

:r: I--

54

55

1) The transfer admittance parameter y 12

of the b·d n-tee net\.;ork can be written in the form

')

_ (s + l)(sL +a's + 1) Y12- - (s + rf)(s + r2)

2) As the zeroes of the quadratic factor in the numerator of

Y12 approach the jw axis of the s plane (i.e., as the co­

efficient a• becomes small) the poles ri and r2 approach unity. Thus y 12 becomes

s2 +a's + 1 y12 ~ - s + 1

3) For the primed netvmrk, assur~e that G1s

1 = 1. Thus the

associated transmission parameter Yzl becomes

sG 1 y 21 = - s + 1

With the three assumptions above the transfer function becomes the standard bandpass form

V 2 _ sG 1

VI-- s2 +a's + 1

One problem which arises from these assumptions is the fact that

since it is assumed that G1s1 = 1, this pole (in Yz 1) \Jill cancel one

of the poles ri or r2 of y 12 in the final transfer function. Thus

there will be no terms in the denominator of the transfer function

which contain G1 or s1 and the sensitivity functions VJith respect to

these elements \'Jill in all cases be zero. But this is olJviously not

the case since if G1

and s1 changes then G1s1 ~ 1 and assumption 3

is invalidated, and the bandpass function is altered.

Another consideration is that assumption 2 requires ri and rz to

approach unity. This in turn requires that a' approach zero (i.e.,

become small). But in practice a' is not zero and therefore rl and

r2 are not unity and assumptions of pole-zero cancellation are only

approximate. Finally it should be noted that assumption 1 requires the fac­

toring of a cubic into a linear term and a quadratic term with a

l. e ad i n g co e f f i c i en t of u n i ty . From the o r i g i n a 1 ex p res s i on f o r y 12

presented earlier it can be seen that this factoring problem is not

56

a trivial one. Due to the problems encountered in making the neces­

sary assumptions and the restrictions on element values it is difficult

to obtain expressions for the sensitivity functions of this realization

and these are best found experimentally or numerically using a digital

computer and a simulation program.

C. BANDPASS REALIZATIONS USING DISTRICUTED NETWORKS The sen s i t i v i ty pro b 1 em vJi t h c i r c u i t s us i n g U R C • s has a 1 ready

been stated; that is the denominator of the transfer function is com­

posed of hyperbolic functions of the square root of the complex

frequency variable s. Thus the transfer function has an infinite

number of poles. ~~ithout a denominator which is a rational quadratic

in s, the classical sensitivity analysis of w and Q is a difficult 0

procedure at best. Some suggestions as to hov1 to attack the URC

sensitivity analysis problem are given in Chapter IV, Summary, Con­

clusions, and Suggestions for further work. The URC bandpass realiza­

tions and associated transfer functions are now presented for complete-

ness.

~ l v,

R2,C2

Hs v2 = VI s3 + als2 + a2s + a3

• SEE APPENDIX 8

Fi9ure 35. CIRCUIT #14 - r)fJER;;TIOW\L M~ 0 LIFIER, URC CIRCUIT [8]

+

v2

•

vi '-J

cl

R,C

VI Rl v2

V2 sK ~= [coshB + K(l-coshs)](s+G1s1)

B = .jsRC

Fiqure 36. CIRCUIT #15- DISTRIBUTED-LUr·~PED VCVS CIRCUIT, K > 0 [11] (.J.

c:·

R,C

+

v, v2 0 I _ _____!.__--:•-- . 0

28(1-cosh8)-RR sinh8 n v2 ---v, Rl R

R 8 { R + B sinh 8 ) n n

s =~RC

Figure 37. CIRCUIT #16 - OPERATIONAL AMPLIFIER, DLA REALIZATION [12] (.Jj

\.0

rl~R,C

v, cl

v2 KG 1S1(1- cosh e) -= V1 (cosh e + K}(s + G1S1)

e = y'sRC

Figure 38. CIRCUIT #17 - VCVS, DLA REALIZATION [4]

v2

01 0

cl

I R,C

+ v,

v2 0 ' _____ J·--~·~----0-

v2 sinh 8

- = v, sRC 1sinh 9 + 9

8 = JsRC Figure 39. CIRCUIT #18 - OPERATIONAL AMPLIFIER, OLA REALIZATION C'i

........

62

IV. SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR FURTHER WORK

Some comments on the sensitivity expressions, commparisons of the various realizations, and possible solutions to problems encountered are now presented.

As noted in Chapter II, Section A on sensitivity functions, many of the entries in the tables of the preceding chapter are constants. These sensitivities are the same value independent of the particular set of circuit design values. Other of the more complicated functions are dependent on the element values; for example many of the Q sen­sitivity expressions are of the form

where f can be a function of one or more element values. Thus in this case the possibility of zero Q sensitivity may exist if the function f can take on the value l/2. Also a majority of the w

0 sensitivity ex­

pressions are equal to l/2. This is due to the simple functional rela­tionship of the elements in the b2 coefficient of the transfer function.

In the single active device realizations a rule of thumb is that - -~-~------- ........... ,___________ ···-···-·~---~ ~··

sensitivities are lower for realizations us~~g fewe.r. feedback paths. In general of all realizations, the state-variable realization has the lo-;~;-t-·sensi_t.i~-it-i~-~- especially to active device gain [8]. The _state-

variable .. a.ppr.oach is most applicable to integrated circuit realization since th~~umber of capa~itors need~d in any realization is always

equa 1 _to -~ .. ~~-~.r.~ .. )1 js. the order of the transfer function. This is important -~.~-~ce_ capacitors take up a relatively large amount of chip area. Thus for the ~igher order realizations, the state-variable ap­p~oach may minimize the chip -area .for a given }~eal ization. It should also b-e-noted -that the s-tate-variable realization uses operational .. "" ~ . . . . ' ·-·· .... ~ . . ~ '• . . . '. . '' ' .

amplifiers considered to have infinite gain. At higher frequencies th~--~ff~~t--.. o-f' ampl ifi,er -gain roll-off_~ s more marked on the state­

vari abTe .. lransfe·r-·funcli on fhan a·n "{he transfer function of a VCVS

realization where a finite gain is required. From a pr;zti ~a 1 -s'tandpoi nt_m~~t ~f the rea 1 iza tions-- take- the

out~~-! ___ yq-ltage di~ectly off the output of the op amp( ~r VCVS (which

G3

is_~n op amp with a resistive feedback network to set the gain). Since the ~utput impe_9?nce. 9{ __ a_::typi·~-~--l .op amp _is. less than J 0 ohms, it ap­

P~?ximates an ideal voltage source (i.e., zero internaJ impedance) and therefore loading ·of the cfrcuit is n~t a major problem as in passive filter synthesii.

The value of the URC to the integrated circuit desiqner is demon­strated by a comparison of circuit #13 and circuits #16 and #18. Cir­cuit #13 is the twin-tee single feedback path realization and circuits #16 and #18 are equivalent single feedback path realizations. Note that in each case, the six lumped elements and the input capacitor of the twin-tee realization are replaced by a single URC and a single lumped element. Thus in a integrated circuit where a capacitor oc­cupies as much space as an operational amplifier, the URC realizations save much space and allow a more compact circuit. Also the feedback networks in circuits #16 and #18 can be used as passive notch filters.

Although URC bandpass realizations do not facilitate classical sensitivity analysis of w

0 and Q, there may be several possibilities to

simplify the analysis. One of these is to go to the lumped model of the URC as shown in Figure 16. A first order lumped approximation of the URC is shown in Figure 40. In some URC realizations it is possible to replace each URC with this first order model and obtain a rational bandpass transfer function. For example by considering circuit #14 to be composed of three URC's, replacing each URC by a first order model, and combining series resistors the circuit of Figure 41 is obtained. The transfer function of this circuit is of the form

which may approximate a bandpass function for a particular placement of the poles (usually controlled by the gain of the active device). If a further assumption is made that the second and third URC's from the left in circuit #14 have low total resistances~ then resistor R3 in Figure 41 may be eliminated (this assumption essentially turns these two URC 1 S into feed-through capacitors). The transfer function of Fig-

ure 41 becomes

I~ w

:r: I--

LL

0 z

: 0 .......... 1--­c:::( ::::: .......... X

c 0:::: C

L

CL

c::(

0 w

0..

:E

:=> _

j

0:::: w

0 0:::: 0 1--­(/)

0:::: .......... L

L

64

c3

cl .rll ..

R2 R3 @

R4

@

v, Rl c2

Figure 41. CIRCUIT #14 ~·JITH ALL URC's REPLACED BY THEIR FIRST ORDER MODELS

+

v2

c~ Ul

66

which is the standard second order bandpass transfer function. It is now possible to do the w

0 and Q sensitivity analysis as performed in

Chapter III of this paper. Possibly at this point a correlation could be made as to the validity of sensitivity functions obtained from the

lumped first order approximated URC circuit of Figure 41, and the acutal circuit #14 so that Fiqure 41 could be used to get ''ball park''

sensitivity measures for circuit #14. Any correlation may be crude, however, since only a first order URC model was used. Increasing the

order of the model may help although the complexity of the approximate

model circuit is greatly increased. Another approach to the URC problem may lie in the location of the

dominant pole pair of the transfer function. It may be possible to describe the location of the dominant pair in terms of the total R and total C of the URC (or URC's). If this can be done, a transfer func­

tion which is rational in s with coefficients in terms of the total R and C of the URC (s) and the gain, K, of the active device may be

written to describe this single pair of dominant poles. Then w0 and

Q sensitivities closely approximating the actual sensitivities could

be found by the procedures used in Chapter III of this paper. This problem is not trivial, however, and needs much further investigation.

1.

2.

3.

4.

5.

6.

7.

8.

9.

1 0.

11 .

12.

1 3.

BIBLIOGRAPHY

Huelsman, Laurence P. ,_Theory and Design of Active RC Circuits, New York; McGraw-H1ll Book Co., 1971.

67

Basson, D. and others, ••rhe Realization of RC n-ports 11 , IEEE Transactions on Circuit Theory, Vol. CT-12 (June, 196~247-256.

Fu, Yumin. ~~synthesis of RC Multipart Impedance Functions 11, IEEE

Transactions on Circuit Theory, vol. CT-17, no. 2 (May, 1970 264-266.

Johnson, S. P., An Investigation of the Properties of Distributed­Lumped Active Networks. Dissertation, University of Arizona, 1971.

Dorf, Richard C., Modern Control Systems, Reading, Massachusetts: Addison-Wesley Publishing Co., 1967.

Kerwin, W. J., Analysis and Synthesis of Active RC ~etworks Con­taining Distributed and Lumped Elements. Dissertation, Stanford University, 1967.

Rosenblum, Alan Louis, Multiparameter Sensitivity and Sensitivity Minimization in Active RC Networks. Dissertation, New York University, School of Engineering and Sciences, 1971.

Newcombe, Robert W. Active Integrated Network Synthesis. Englevmod Cliffs, New Jersey: Prentice-Hall, Inc.~ 1968.

Tobey, Gene E., and others, Operational Amolifiers, Design and Applications. New York: McGraw-Hill Book Co., 1971.

Seely, Ralph M. "Preferred Active Filter Forms for Bandpass Func­tions11. IEEE Journal of Solid-State Circuits, vol. SC-7, no. 4 (August, 1972), 304-306.

Haykin, S. s., Synthesis of RC Active Filter Networks. London: McGraw-Hill Book Co., 1969.

swart, P. L. and others, 11 A Voltage-Controlled Tunable Distributed RC Filter 11

• IEEE Journal of Solid-State Circuits, val. SC-7 no. 4 (August, 1972), 306-308

Watson, Terry B., Active Bandpass Filters Usino Twin-Tee Networks, Masters Thesis, University of Missouri-Rolla, 1965.

Go

VITA

The author was born on May 27, 1949 in St. Louis, Missouri. He

received his primary and secondary education in St. Louis, Missouri. He entered the University of Missouri at Rolla in September 1967 and received his Bachelor of Science Degree in Electrical Engineering in May 1971.

The author has been enrolled in the Graduate School of the

University of Missouri at Rolla since September 1971. During the fall semester 1971 and the spring semester 1972 the author has been emoloyed

by the University of Missouri at Rolla as a half-time Graduate

Teaching Assistant.

APPENDIX 1

DERIVATION OF THE TRANSFER FUNCTION OF FIGURE 2.

The y parameters for the circuit of Figure 2 are defined as

follows:

I1 y11 y12 y13 v1

I2 = y21 y22 y23 v2

I3 y31 y32 y33 v3

The condition above assumes that an ideal VCVS is used; that

input impedance is infinite. Thus

EQ. 1

However

v3 = v X

and v2 KVX

therefore v3 v2

-y

Substituting this expression for V3 into EQ. 1 yields

0 v v + y33 v = Y31 1 + Y32 2 K 3

Collecting terms in V1 and V2 qives

y33 (y32 + ~)v2 = - Y31v1

therefore

is, the

70

71

The equations applying to Figure 2 may be applicable to the four ter­

minal network shown in Figure 42 if each of the three ports of the

circuit of Fiqure 2 has a common qround terminal.

__

... C\1

+

C\1 >

I

-> I

(../)

>

u >

Ll....

72

APPENDIX 2 DERIVATION OF THE SINGLE FEEDBACK PATH RELATION OF FIGURE S

The realization if Figure 5 assumes ideal op amps~ that is,

infinite input impedance at both inputs. Thus

For the primed network

= ~1 61

and thus

I I - I = 2 - - 1

For the unprimed network

pl1 ~F

1 ~

Now equating the expressions for 12 and - r1 gives

But for the op amp

therefore

KV. 1

EQ. 1

Substituting the expression for Vi in terms of v2 into EO. l and

collecting terms of vl and v2 yields

Y22 Y11 Y21Vl = (-K- + K- yl2)V2

For the op amp, the gain, K, approaches infinity and therefore the

quantity 1/K approaches zero. Thus the equation immediately above

73

74

becomes

and the final transfer function is

APPENDIX 3

DERIVATION OF IMPEDANCE INVERSION BY A GYRATOR

The circuit for impedance inversion by a gyrator is shown in Figure 12 of this paper. The ABCD (transmission) parameters of the

ideal gyrator are

vl 0 1

~

Il Go 0

and by definition (Ohm 1 s Law)

and

Now from the transmission parameters

Rut

- vl - I2/Go Zin - 11- - GoV2

I2

G 2v 0 2

v2

-I2

and therefore the final expression for Zin is

1 z = --in ZG 2

0

75

APPENDLX 4 DERIVATION OF THE TRANSFER FUNCTION OF FIGURE 13

For the gyrator it is known that

I' v· = __1

2 G0

and -I I = G V' 2 0 1

from the transmission parameters. By using y parameters to describe the unprimed network the following matrix equations are obtained:

~ I' 2

and

=

=

pll l11 rl G1

From the configuration given in Figure 13 it can be seen that

therefore

and

I - 0 2 -

Now working toward a relation involving only v1 and v2

Usinq the gyrator relations I'

-GoVi = Y21Vl + Y22 G~

76

77

but also

V' = z11 1l l

Thus

I' -Goz111i = Y21V1 +

I 1 Y22 G

0

Collecting terms in I' 1 y'

- (_1_?_ + Go 2 11)Ii = Y21V1 Go EQ. 2

From the z parameter equations

therefore v2

I' = 1

Using this expression for I} in EQ. 2 yields

y' v2 -(__ll_+ G z ) - = Y2

1V1 G

0 o 11 z21

The final transfer function is

V2 _ Goy21z21

VI - - Y22 + Go2zll

78

APPENDIX 5 DERIVATION Of THE TRANSFER FUNCTION OF FIGURE 15

The approach to get the transfer function of this realization will be to first get the overall y parameters of the portion of the circuit of Figure 15 shown in Figure 43. These parameters will then be added to the respective yij parameters since the portion of the circuit shown above is in parallel with they .. network. Then they para-

lJ meters of the entire realization will be obtained and from these the overall transfer function can be found.

The transmission parameters of the INIC are

~ = ~ o~ v~ I 0 - - -I 1 K 2

Now for the portion of the circuit shown above

and

~ ~

EQS. 1

EOS. 2

Equating the expressions for I2 in EQS. 1 and EQS. 2 gives I

2 I v + I v - K = Y21 1 Y22 a

Now using the fact that Va = v2 (obtained from the transmission para­meters of the INIC) in the equation imnediately above i3nd solving for

12

~)+ >

C\1 ' I

r 0 -z -

-1

0 +

>

C\1. -

I 1-

...... .... ·-~

I -1

-=t

-~6+

>

19 l -I

_J I

I t_

I

--' I

I!

. (V

")

o:::::t"

Q)

S­

::::s en

•r-

L!_

79

80

Now taking the expression for 11 from the equations for the primed net-work and again using the fact that Y a = v 2

I = 1 Y]_lVl + Y]_2V2

Thus the y parameters of the network portion as previously described are

J ~· YiJ ~ yll

=

I2 Ky21 -Ky22 v2

Now adding respective y parameters of the portion just analyzed and the unprimed network gives they parameters of the overall realization.

=

From the conditions of the realization

Therefore

Solving for v21v1 the final transfer function is

v2 _ -y21 + Ky2I VI - y22 - Ky22

~ ~

81

APPENDIX 6

FORMING THE INDEFINITE ADMITTANCE MATRIX OF THE URC

The URC as a 2 port with its accompanying y parameters is shown in Figure 17. For the URC as a 3 port as shown in Figure 44, by applying Kirchhoff•s Current Law it can be seen that

In terms of the voltages defined for the 2 port, the 3 port voltaqes

are

Therefore

Ia =~ (coth9 v1 - csch9 v2)

Ib =~(-csch9 V1 + coth8 V2)

Ic =}W- (V 1 + v2)(csch9 - coth8)

However in terms of the 3 port voltages substituting these expressions for v

1 and v

2 into the equations for the 3 port currents, collecting

terms in Va' Vb' and Vc and reducing the equations to matrix form

yields

I a cothQ -csch9 cschG-cothQ v a

Ib =N -csch8 coth8 csch8-cothQ vb

Ic csch8-coth9 csch8-coth9 -2(csch9-coth8) vc

0 .. 0::

+

+

..c >

0 >

u >

I

32

I I

83

which is the indefinite admittance matrix for the URC.

A check on the result is to see if the total of all elements in any row or any column of the indefinite admittance matrix is zero. It

is seen that this is the case in the above matrix.

As an example of the usefulness of the indefinite admittance matrix of the URC consider the circuit shown below in Figure 42 [1]. The 3 terminal network shown in the above realization contains only passive lumped elements. The transfer function is the same as that derived in Appendix 1 and is repeated here for convenience.

v2 _ -Ky31 VI - y33 + Ky32

Now consider the low-pass URC circuit of Kerwin [6] shown in Fiqure 18.

This case corresponds identically to the general realization just men­

tioned. From the derivation of the indefinite admittance matrix for the URC it can be seen that port a corresponds to port 1, port c cor­

responds to port 2, and port b corresponds to port 3 of the general realization. Thus the transfer function of Figure 18 becomes

Substituting for the appropriate elements of the indefinite admittance

matrix of the URC

V2 _ -K(-csch9) VI - K(csch9 - coth~ + cothG

and applying some trigonometric relationships for the hyperbolic func­

tions the transfer function finally becomes

v2 _ K V} - cosh9 + K(l cosh9)

This result corresponds with the analysis of Kerwin's circuit by S. P.

Johnson [4]. The indefinite admittance will be used to obtain the

84

transfer functions of the URC. realizations presented in this paper.

A.

85

APPEfJDIX 7 ANALYSES OF THE BAfJDPASS FUNCTIOf~

PROOF OF THE t1AXH1U~1 PASSBAl'JD GAIN AT w n

F 0 R ~V 2 = ---:-__ H_s_ v 2 2 1 s + 2cwns + w

n

The transfer functions

K ( j w) = _v 2 ( j w) = ____,=---------l~ij_w __ -=-v1 2 2 . 2 -w + JCw w + w

n n

and thus the magnitude of the transfer function is

Now

I K(jw) I = t~AXH1Ut~ for d I f<~~w) I = 0

/A H - H

At max passband gain frequency, w0

djK(jw)j I 0 dw w=w =

0

Therefore

Collecting terms

The maximum gain is

and finally

H(w 4 + w 4) = H 2w 4 n o o

4 4 Hw = Hw n o

w = w n o

!K(jw)l MAX I K(jw) I w~w

n

= _________ H~w~--------/(wn2 - w2)2 + 4s2wn2w2

= =

I K(jw) I = _H_ MAX 2swn

(w= w ) n

B. DERIVATION FOR THE 3 db FREQUENCIES (HALF-POWER POI~!TS) AND THE 0 EXPRESSION FOR THE BANDPASS FUNCTION

By definition at the 3db frequencies, w',

86

_/2 I K(jw) I w=w ~ - 2 K(jw) I

MAX

Hw

_ ff H - 4c;:wn

;lcw 2 _ w2)2 + 4r2 2 2 n s wn w

- /Z H - 4c;:wn

w=w'

Dividing through by H and squaring both sides yields

Collecting

= w n 4

87

The equations above indicate the possibility of 4 solutions. Now taking

the square root of both sides yields

Taking the plus siqn on the right hand side

Solving for w'

2c_:w t, f4?w 2 - 4 (l} (-w 2) w' = - n- n n

2

c_:w + /r;2w 2 + w 2 wl = - n- n n

w' = - r;wn 2:. wn k2 + 1

Neglecting a negative frequency ( since /r;2 + 1 > s)

W1 = w (- s + !'c_:2 + 1) n

Now taking the minus sign

Solving

(Jj I 2 - 2 l_;W W I n

- (Jj n 2

2c_:w + /4c_: 2w 2 - 4(1)(-w 2) 1 n - n n

(Jj = ----------~~------~--2

W1 = c_:w + /c_;w 2

+ w 2

n - n n

WI = W ( S + ls2 + 1) n -

Neglecting a negative frequency

Now since

W 1 = w (r; + /£;2 + 1)

n

88

the larger of the solutions is the upper 3 db (half-power) frequency.

89

Thus the upper 3 db frequency

w2 = wn (~:; + Jt:2 + 1)

and the lower 3 db frequency is

The definition of band width of a bandpass function is

B. W. =

= 2c;w n

and the definition of Q for a bandpass circuit is

Q = CENTER FREQ B. ~1.

wn = 2z;wn

Q 1 =~

It can be seen that for the general bandpass transfer function

v2 _ Hs

VJ - s 2 + 2sw s + w 2 n n

Hs = ------

that an expression for Q can be obtained by taking the square root of

b2 and dividinq this by b1 . Thus

1 Q =-21';

90

by expressinq Q in terms of the coefficients and expressing the coef­ficients in terms of elements in the circuit (symbolic) an expression for Q in terms of elements can be obtained thus facilitating Q sen­

sitivity analysis.

APPENDIX 8 DERIVATION OF THE TRAfJSFER FUfJCTIOU OF FIGURE 41

First assume a first order model for URC as shown in Figure 40.

Novl/ replacing URC•c in above realization and combining resistors in series yields the circuit of Figure 41. Note that since zero input

current to the op amp is assumed, R4 can be deleted since no current

will flow through it. At node A

- v 2 va- K

0

1 vb V (- + sC ) = -a R3 2 R3

Va(sC 2R3 + 1) = Vb

v2 Therefore since Va = -­K

At node B

(Vb - Va) + (Vb v2)sC3 +

R3

v v (L + L + sC 3) = _A+

b R2 R3 R3

R2 + R3 va + sC3) v3 ( = -+

R2R3 R3

Vb(R2 + R3 + sC 3R2R3) v2 K R2

(Vb - Vc) 0

R2

v v2sc 3

+~ R2

v v2sc3

+_£ R,

L.

+ v2sc 3R2R3 + V cR3

91

Substituting for Vb in terms of v2

(sC2R3 + 1)

K (sC3R2R3 + R2 + R3)V2 = V2(sC3R2R3 + ~~~) + V~ Rc

(sC2R3 + l)(sC3R2R3 + R + R )V V ( VC n R R ) '2 3 2 = 2 S ' 3r'2 3 + 2 + V C I<R3

Collecting terms in v2

At node C

(sc1

+ _!_ + _l)V = ~ + v 1sc

1 R1 R2 c R2

(sC 1R1R2 + R1 + R2)Vc = VbRl + v1 sC1R1R2

Substituting for Vc and Vb in terms of v2

At this point it can be seen that the solution VJi 11

v 2 - Ks '11- 53 + a1 s 2 + a2s + a3

be of the form

This is not the standard bandpass function. If it is assumed that the 2nd and 3rd URC's from the left have low total resistances the

assumption that R3 = 0 can be made. tiow the equations become

92

I I ) v 2 f(l v 2 (sC1R1R2 + R1 + R2 )(s{R2 [c2 + c3 (1-k)]} + 1 K = --v- + v 1 sc 1 f~ 1 1< 2 (s{R2[c2 + C3 (1-K)]} + 1)(sC1R1R

2 + R1 + R

2)v

2 = R

1v

2 + v

1sc

1R

1R

2

(s2

c 1R1R/[c2 + c3(1-K)] + s{R2

(R1 + R2

)[c2

+ c3

(1-l()] + c1

H1

1(2

} +

R1 + R2 ) v2 = R1v

2 + V

1sc

1R

1R

2

v2 _ sc 1r- 1 VI- s2c1R1R

2[c2 + c

3(1-K)] + s{(r~ 1+H 2 )[c 2 + c 3 (1-I~)J + c

1r<

1l + 1

( R 1 + R2) 1 C

1R

1R

2 + R

2[c

2 + c

3(1-1()]

''! 2 _ sG2s2s3;[s3 + s2(1-l<)]

VI - __ 2 ____________ _;~~--~--~--------~G~10G-2srlrs~2~s3

s + s { S 1 ( G 1 + G 2 ) + G 2 S 2 S 3 I [ S 3 + S 2 ( 1- rn J } + S 3 + S 2 ( 1- I~ )

This is a standard 2nd order bandpass transfer function.