A SWENABMBLE APPROACH TO
Tihfi'E-VARYIRG EQUWALENT KETWDRKS

flea-sis for Hue Degree of P11. 5.

MiCHlBfiN STATE UHWEBSITY
PEYER J. GRAHAM
1.968

   
  
 

u:- _

Michr .
ho a e

UHXVCAQ

tnciifi

  

This is to certify that the
thesis entitled

A STATE-VARIABLE APPROACH TO
TIME -VARYING EQUIVALENT NETWORKS

presented by

Peter J. Graham

has been accepted towards fulfillment
of the requirements for

degree m Electrical Engineering

\MQKVL

Major professor

Ph. D.

Date April 18, 1968

0-169

ABSTRACT
A STATE-VARIABLE APPROACH TO
TIME-VARYINC EQUIVALENT NETWORKS

by Peter J. Graham

Let two networks be equivalent with respect to some
network function F(s). These networks are said to belong to
a set of continuously equivalent networks if the components of
either of the networks can be changed continuously in a manner
that will ultimately yield the other network such that the
continuum of networks produced in the process all possess the
same network function F(s). The state-variable characterization
has been used to introduce a class of continuously equivalent
networks in the first chapter of this dissertation. A set of
differential equations describing how the components should be
varied has been proved sufficient to assure equivalence.

Suppose, now, that the components of such a set of
continuously equivalent networks are time-varied in the
manner prescribed by these differential equations. ‘Will the
equivalence network function be time-invariant? This question
is answered affirmatively in the form of a theorem in Chapter II,
with the interesting result that while the network function is
time-invariant, it is different from.the function of the
associated continuously equivalent fixed networks. Furthermore,
it is observed that such an invariant function of a time-varying

passive network can have properties which are not realizable by

invariant passive networks. The remainder of the thesis is
devoted to exploiting this fact.

Practical solutions of the equivalence differential
equations are defined and developed in the third chapter.
Chapter IV deals with the particular case of passive time-
varying RC networks. It is shown that complex poles can be
realized by such networks with component variations which are
all periodic and non-negative. Numerical examples are given
in Chapter V. Time domain solutions for equivalent time-varying
and fixed networks are compared in digital computer print-out

form in the Appendices.

A STATE-VARIABLE APPROACH TO

TIME-VARXING EQUIVALENT NETWORKS

By

Peter J. Graham

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1968

ACKNOWLEDGMENTS

Appreciation is extended to Dr. James Resh for his
insistence on mathematical rigor, to Dr. D. A. Calahan
for his uncanny intuition, to Dr. J. 8. Frame for his
careful and constructive review of the completed
dissertation, and, above all, to my wife, Marie, for
her patient encouragement through the years.

11

TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS .......................................... ii
LIST OF TABLES ........................................... v
LIST OF ILLUSTRATIONS .................................... vi

LIST OF APPENDICES O0.0......OOOOOOOOOOOOO0.00.00.00.00... Vii

Chapter

I. 000...... ..... 0......0.00......OOOOOOOOOOOOOOOOOOO 1

Introduction

Definition of a Class K of Linear Networks
Derivation of Equivalence Constraints
Example

II. OOOOOOOOOOOOOOOOOOOOOOO0.0.0.000...OOOOOOOOOIOOOOO 17

Introduction

Theorem on Equivalence of Time-varying and
Fixed Networks

Lemma 1

Lemma 2

Lemma 3

Proof of Theorem 3

III. OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO000...... 28

Introduction

Solution of Second-Order Example
Constraints on the Coefficient Matrices
Periodic Solutions

Summary

Iv. 00.0.0.0...OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 49

Introduction

A Fourth-Order Time-Varying GC Network

Properties of the Equivalent Fourth-Order
Invariant Network

A Linear, Reciprocal, Passive, Time-Varying
Realization of the Fourth-Order Network

A Sixth-Order GC Time-Varying Network

iii

TABLE OF CONTENTS-~Continued

Chapter Page

V. ........... . ..... .................. ........ .... 82

Introduction

Synthesis of a Fourth-Order Time-Varying
Network

Synthesis of a Sixth-Order Time-Varying
Network

Conclusion

LIST OF REFERENCES 0......00.0.0.0...00.000000000000000. 91

iv

Table

5-1.

LIST OF TABLES

Fourier Series Coefficients for Typical Entries
of the Sixth-Order'éflt) Matrix ...............

Page

88

Figure

1-1.

h-l.

LIST OF ILLUSTRATIONS

A Continuously Equivalent Network Example ......

Scheme for Realizing any Dominant Matrix
as a Short-Circuit Conductance MMtrix ........

Realization of the Fourth-Order Time-
Varying Equivalent Network Example ..... ......

Time-Invariant Network Equivalent to
the Fourth-Order Example ................ .....

Time-Invariant Network Equivalent to
the $1Xth-order Example OOOOOOOOOOOOOOOOOOOOO.

Frequency Response of the Fourth-Order

Example 00......OOOOOOOOOOOOOOCOOOOOI00.0.0...

vi

Page

13

67

7O

71

8O

86

LIST OF APPENDICES

Appendix Page
A 0.0.0.000...0.0.0....OOOOOOOOIOOOOOOOOOOOIOOOOC0.0. 94

B OOOOOOOOOOOOOOOIOOOOOOOOOOOOOOOOOOO0.000.000.0000. 101

vii

CHAPTER I

CONTINUOUSLY EQUIVALENT TIME-INVARIANT NETWORKS

1. INTRODUCTION

The concept of networks which are equivalent with respect to
a selected network function has existed for many years [1,2].
Until recently the relationship between such networks has been
in the form of congruent transformation matrices which carry
some Specific description, the nodal matrix for example, of one
network into the same description of the other. While such a
tool would seem to possess great potential in synthesis problems,
useful applications were few and the subject did not attract
much attention.

In 1962, J. D. Schoeffler [3,4] introduced transformations
which were continuous functions of a real parameter. His approach
to the problem was the following.

Given a network with a desired transfer function, the
established theory of equivalent networks was used to generate
another network.with the same transfer function.whose elements were
different from the original by an incremental amount . Taking the
limit as the increments approached zero generated differential
equations whose solutions gave the element values of a set of
equivalent networks as continuous functions of an independent
variable, designated the equivalence parameter. These were dubbed,
appropriately, continuously equivalent networks.

Schoeffler's differential equations were derived in the form

gear) - shame) + manage) .

dx
1

2
I‘s,x) being the nxn admittance matrix of a network having n
independent node pairs. For many network forms the solutions
could be interpreted at the component level, and they proved
to be very useful in computer aided sensitivity minimization
techniques [4,5].

The existence of continuously equivalent networks leads
naturally to the investigation of the properties of the time-
varying networks which result from making the equivalence
parameter a function of time. A reasonable conjecture is
that a network function which is independent of the equiva-
lence parameter 2 over a set of continuously equivalent fixed
networks will be a time-invariant function of the associated time-
varying network. A derivation of the element differential
equations in a form applicable to time-varying networks was
accomplished by Calahsn [6] by starting with a state-variable
network model. He observed that such time-varying networks did
indeed have invariant network functions, and that these functions
were different from the corresponding functions of the associated
fixed network created by stopping the element variation at some
instant t.

It has been demonstrated by several authors that certain
tune-varying networks can exhibit invariant network functions
which have properties not inherent in the associated fixed
network. Franks and Sandberg [7], for example, showed that a
passive RC network with sinusoidal modulators at the input and
output realizes a time-invariant transfer function with complex

poles. A similar result accomplished using another modulation

3
scheme (referred to as "outphasing using quadrature modulation")
was recently reported by Saraga [8]. Both these approaches deal
with a particular problem and the implementation does not involve
the direct time-variation of passive R, L, or C elements.

An example of a time-varying two-port network having an
invariant Open-circuit impedance matrix is given in a corres-
pondence by Anderson and Newcomb [9]. This network, consisting
of fixed capacitors and ideal three-winding transformers with
time-varying turns ratios, is non-reciprocal and has a conjugate
pair of j-axis poles. In another correspondence [10] the same
authors show that a similar arrangement is equivalent to a
gyrator. From this they conclude that any invariant passive
network can be realized using resistors, capacitors and time-
vsrying transformers. It is worth noting that the realization
requires negative capacitances.

Another example of the equivalence of a time-varying and
fixed network appears in a paper by Desoer and wong [11], where
the conditions required of an invariant RLC one-port to be
equivalent to a one ohm resistance are extended to include cases
where the elements are non-linear and timedvarying. An equally
specialized situation is described by Silvennan and Meadows [12]
where a theorem on the controllability of time-varying networks
is used to generate networks having transfer impedances which are
zero for all t.

All of the procedures described in these references suffer

in one way or another from a lack of generality. In [7 - 10]

the forms of the time-varying networks are fairly rigidly

4

prescribed, while in [11] and [12] the equivalent fixed networks
are restricted to having a single property. It will be seen in
the next section that the continuously equivalent approach is
not applicable to all network configurations; nevertheless it
is sufficiently general to suggest that the concept might be
usefully exploited.

Calahan's work was published as a correspondence and
dealt only with the derivation of the equations along with a
few remarks regarding their potential usefulness. No attempt
was made to prove the condition that the element functions
satisfy the differential equations (henceforth referred to as
the constraint equations) sufficient for the existence of an
invariant network function. In a Ph.D. dissertation at the
University of Illinois Sen [13] examined the significance of
these equations from the viewpoint "given a network with an un-
controllably varying element, what minimum number of elements must
be varied in a controlled manner in order to counter the effect of
the uncontrolled element?". It should be noted that Sen did not
include a proof of the sufficiency of the constraint equations.

The constraint equations are re-derived in the following
pages in the form of a theorem in order to put them on a sound
basis. They are then applied to the question "What useful
properties can be realized by the invariant network function of a
linear, passive time-varying network which are not inherent in the
associated fixed network?".

In the first chapter the constraint equations for invariant

networks are derived and are shown to be sufficient for a certain

5
class of continuously equivalent networks. The second chapter
shows that the constraint equations for time-varying networks
can be meaningfully discussed only in terms of the equations
for invariant networks. Chapter III is concerned with the
physical realizability of the component functions whichcare.solutions
of the time-varying network constraint equations, while in Chapter IV
linear, passive time-varying CC networks are shown to be practical
and to have desirable characteristics not realizable with invariant

passive CC networks. Numerical examples are presented in Chapter V.

2. DEFINITION OF A CLASS K 0F LINEAR NETWORKS

In order to accommodate some basic assumptions required to
prove the sufficiency of the constraint equations their deri-
vation will be valid only for a restricted class of networks.
This class will be defined as satisfying a set of conditions
which are sufficient to insure that the state-variable equations
of the networks will have certain properties. These equations

must be expressible in the form

magma - gauze) - rm (H)
at

where the statedvector1§(z,t) contains the inductor currents and
the capacitor voltages. ‘we will require in addition that:

1. 13(2) be diagonal and non-singular

2. Zfit) be independent of the equivalence parameter 2.

Property 1 will be satisfied if the network has no mutually
coupled components, and if there are neither voltage source and/or

capacitor loops nor current source and/or inductor cutsets. Under

6
these circumstances it can also be shown that each diagonal entry
of 13(2) will be the value of one of the inductances or capacitances
in a one to one correspondence. This will become clear during the

following discussion of property 2 which is presented as a theorem.

Theorem 1

If the tapology of a network having property 1 is such that
each independent voltage source is in a cutset whose other elements
are all inductors, and each independent current source is in a
loop whose other elements are all capacitors, then the forcing
function vector zfit) of the state-variable equations (1-1) of
the network will be independent of the values of the passive
components, and therefore independent of the equivalence

parameter 2.

Proof: The following procedure can be used to derive equations
(1-1) of any network having property 1.

(a) Replace the inductors by ideal current sources (hence-
forth referred to as L-sources) and the capacitors by
ideal voltage sources (C-sources).

(b) Solve the resulting network (now containing resis-
tances and sources only) for the voltages across the
L-sources and the current through the C-sources. The

solutions can be expressed as

[gig] ' [iii ‘32:] [$12] + [it] (1-2)

pa

58

CE

7
(c) Replace each element of 2L and EC by the appropriate

Lg; and Cd! terms respectively. The left side of (1-2)

dt dt

then becomes

1. 9. “d .14.

9 9. d" V ’

thus generating equation (l-l), with
142) 9, 511(2) 5“”)

22(2) " , 5(2) I and
9 3(2) 531(2) 59(2)

(2.0
guzst) " %

26w»

In general the 1 vector will be a function of the equivalence
parameter 2. If, however, each independent voltage source is in
series with an inductor and each independent current source is
paralleled by a capacitor the series and parallel combinations
can be replaced by L-sources and C-sources respectively. With
all the independent sources absorbed in this manner, the re-
sulting network, consisting only of resistances, L-sources and

C-sources is solved to yield equations in the form

EH": "ZHZJ

Step (c) now requires the substitution

er
er

8
where XV and 11 are the independent voltage and current sources
and are therefore clearly independent of 2.

Finally we note that if the independent voltage sources are
in inductor-voltage source cutsets and the independent current
sources are in capacitor-current source loOps, the voltage and
current shift can be used to transform the network to a form
where each voltage source is in series with an inductor and
each current source is paralleled by a capacitor. Since this
transformation does not alter the state equations the theorem

is proved.

Corollary The forcing function vector will have a zero entry
in each row corresponding to the row location in §(z) of an
inductor which is not included in any voltage source-inductor
cutset or a capacitor not included in any current source-
capacitor loop. Such inductors and capacitors will not have
associated shifted sources in the modified network. Substituting
for their L-source voltages and C-source currents will therefore
introduce only Lg; and Cg!_type terms into their corresponding
equations, thus :zntribuging nothing to those rows of the zfit)
vector.

Finally, networks having state-variable equations (1-1)
with properties 1 and 2 will be defined as belonging to the

class R if they are controllable, and if their output variables

are linear combinations of state variables.

3. DERIVATION OF EQUIVALENCE CONSTRAINTS

If a network belongs to class K it can be shown to be a

9
member of some set of continuously equivalent networks provided
the equivalence is defined with respect to a sufficiently small
number of driving—point and/or transfer functions. This state-
ment will be justified with a theorem followed by its interpret-

ation.

Theorem
Let a network belonging to class K have the state equations

5(z)§_x___(z.t) - 5(z)x_(z,c) -g_(c) (1-1)
a:

where z and t are real independent variables.

Let.§(z) be an nxn matrix having a column of zeros corres-
ponding to each non-zero row of the vector 1(t).

If the components of the network are functions of the
variable 2 such that the matrices §(z) and 5(z) satisfy the

differential equations

15(2) = g(z)u(z) " ~1‘3(z)g(z) (1-3)
dz
gm - §(2)A(2) - 5(z)g(2) . (1-4)
dz

then any solution vector §(z,t) of (1-1) will satisfy the

relationship

5§(2.t) - 3(2)§(2.t) . (1-5)
3';

Proof: From the given properties of fifiz) and Zfit) ,

9(2)Z(t) E Q .

Also

('1
~31
m

whez

Sho:

be C:

so

wh:

Q0!

10

Also, since 1(t) is independent of z,

510:) E 9 ,
82

It follows, therefore, that

_a__ Lu(z)3_X(z. t) - A(z)X(z. t)J
5” at

= m2) L13(2)§§(2.t) -A(2)§_(2.t)]. (1-6)
at

Expanding (1-6) and collecting terms yields

du- gg+na_ ax -(dé “§$+A3 X-Q (1-7)
at E 5;

where the arguments of the functions have been dropped to
shorten the equations.

Upon the substitution of (1-3) and (1-4), equation (l-7)
becomes

Q23; 'WBE-Qfi 'éQE-Q.
oz 52' 32'

Assume that the derivatives of‘g are continuous in z and t

so that §_(a§> can be replaced by a _(6X). Then,
a” a: at 32

Ha ax - 0.x) - _(ax - _)-o (1-8)
at 3—:
Equation (1-8) holds for all §12,t) in response to all Xfit).
It now must be shown that the only solution to (1-8)
which is consistent for all 1(t) is 2% - 9.}; - 9_. For

32
convenience, let

B§ ' 3E - 212,11).
5?

Fi

he

(h.

11
First note that if the network is in the zero state then
{(2,t) - 9_ since _X_ - Q, and since a change in component values
has no effect on the state of a network in the zero state.
Suppose now that there exists some value of 5, say
£(zo.to). such that g(z°,t°) ,‘ 9. For t > to, {(2°,t) will
decay asymptotically to the zero state along the trajectory

defined by

£(zo.t) = exPle"(zo)§(zo)(t-to) ]£(zo.to) (1-9)

It follows, therefore, that

H U20,” H 5 E > 0 .
for all finite t > to.

On the other hand, since the network is controllable a
1(t) can be found which will drive the network from §(2°,t°)
into the zero state, and therefore 2(2o,t°) to Q, in finite
time. Since 20 and to can take any values, this contradiction
proves that 2(z,t) is identically zero. q.e.d.

This theorem has the following interpretation with regard

h

to continuously equivalent networks. Let the pc row of the

matrix 9(2) be identically zero. If the state vector satisfies
gym) - g<z>zs<z.c>
62
then the pth row of §(z,t) must be independent of 2. Thus,
the set of networks having state-variable equations whose §(z)
and ~5(2) matrices satisfy (1-3) and (1-4) respectively will be
continuously equivalent with respect to the transfer functions

whose outputs are all xp(t) and whose inputs are the independent

54

"a

12

sources of the network. This equivalence can be extended to
include additional state variables as outputs by making the
appropriate rows of 3(2) identically zero. The total number
of transfer functions whidh can be made independent of 2 is
clearly bounded by the limiting case of 9(2) - g(z) - 9 which
reduces the set to a single network. The most useful case
will usually be equivalence with respect to a single transfer
function or driving-point imittance. Then 3(2) and §(2) will

have one identically zero row and column respectively.

4. EXAMPLE
The network illustrated in Figure 1 belongs to class K.

With the state vector

é-[iJ

where i is the inductor current and v the capacitor voltage,

the state-variable equations are

L(Z) 0 d 1(2,t)
0 6(2)] dt v(z,t)]
-R(2) -l 1(2,t) e01)
- +
l -G(2) v(z,t) 0 (1-10)
Choosing

- ad I
9 o o a g o o ’

equations (1-3) and (1-4) have respective solutions

13

R(z) L(z)

 

 

 

9-01) C) C(z) omit “9

 

 

 

Fig. 1-1. A continuously equivalent network example.

an

C8

SL‘

it

14
L(0)e‘°z o
2

5(2) ' o 0(0)e°

and

[440)?"2 -1 ]
A I
‘"(z) 1 -c(0)e°z

The differential equation for the voltage across the

capacitor is

g:!.+r[R§2) + 6(2) g!.+ R‘2)G(2)+l 1v? . (1_11)
ata L(2) 0(2) dt L(2)C(2) L(:)C(2)
Substituting

L(2) - Loe-°z
C(2) - Coeoz
R(2) - Rae-oz

C(2) - Ggeaz

into (l-ll) yields

5111+ 5+9; 511+ 1 (R°G°+1)v- e

at“ L, c, dt LOCO LOC° (1'12)

showing that the characteristic polynomial is independent of 2.
Since the forcing function is also independent of z, v(t) is a
function of t only.

The forcing function of the differential equation for the

inductor current on the other hand is readily found to be

1 (gas, + coe)€°z.
LOCO dt

Thus, the state variable i(2,t) depends on 2.

15
It is clear that in order for any of the state variables
to be independent of the component changes the variable 2 must
not appear in the characteristic polynomial. This condition
is assured by equations (1-3) and (1-4). To demonstrate, first

note that

M" = -' PM“ - =."[e: - 5912:" .

32' 3?

Therefore

dt" = 922" - 3’13.
d2

The characteristic polynomial is associated with the matrix
5'15. Observe that

dew - dale. + was
a. z; 3-;

Therefore
51%-?) " 302.19) ' Q5199 (1-13)
d2

The solution of (1-13) is given by

E1(2)§(z) - 3(2.zo)§'l(zo)5(zo)fi"(2.20) (1-14)
where

gfi(2.zo) - 9(z)a(2.zo); 3010.20) = .I, .
d2

This is easily verified by substituting (1-14) into (1-13).
Equation (1-14) shows that the matrix ~ltg"1(z)‘A‘(z) is similar

for all 2 to §f1(2o)éxz°), and it is well known that similar

16
matrices have the same characteristic polynomials. Consequently

these polynomials are independent of 2.

CHAPTER II

EQUIVALENT TIME-VARYING NETWORKS

1. INTRODUCTION

The application of continuously equivalent network theory
to time-varying networks was first proposed by Calahsn. Suppose,
for example, that one component of a network is unavoidably
varying with time. Is it possible to compensate for this by
appropriately time-varying the other components such that a
chosen transfer function is time-invariant? Some insight is
gained toward answering this question by replacing the
equivalence parameter 2 by the time variable t in the example
of Chapter I.

When the components of the network of Figure l are functions

of time the differential equation for the capacitor voltage is
V + £%-(LC+2CL+GL+RC)v +-£%-(LC+LG+LC+LG+RC+RC+l)v=e (2-1)

where the "dots" denote differentiation with respect to time.

With L - L,e'°t
c - Coe 0‘
R _ Roe-at
G - Gee at
equation (2-1) reduces to
v +-ifE;-(R°C°+G°L°+OL°C°)v + i:%;-(R°Go+l+oRoco)v-e (2-2)

This is a constant coefficient differential equation, but it is

17

C0

V2

18
not the differential equation obtained for Figure l with the
component values fixed at Lo, Co, R0 and CD. It is, rather,
the differential equation for the network with a conductance
value of C°+OC°, as a comparison of (2-2) with (1~12) will

verify.

2. THEOREM ON THE EQUIVALENCE OF TIME-VARYINC AND FIXED NETWORKS

While the substitution of t for 2 appears to be a simple
and fairly obvious step, it can safely be said that the result
is by no means obvious. The relationship between the fixed
and time-varying networks must be very carefully established.
Although the equivalence constraints on 5(t) and 5(t) of the
time-varying network are similar to equations (1-3) and (1-4),
they cannot be derived using a method paralleling Theorem 2 in
Chapter I. The main complication is the presence of only one
independent variable in the time-varying equations; there is
no independent equivalence parameter 2.

In this section the main theorem relating time-varying
networks to continuously equivalent fixed networks will be
stated.‘ Three supporting lemmas will be introduced in

subsequent sections before proving the theorem.

Theorem 3
Let a time-varying network belonging to class K be

described by the state-variable equations

gamma] - A(t)§_(t) - gt). (2-3)
dt

If the matrices §(t) and 5(t) satisfy the constraints

19

3.120) = fi<t>§<t> - yoga) (2-4)
dt .
350:) - mow) - 50090:) + gamma) (2-5)
dt dt

where 3(t) and §(t) are nxn matrices, §(t) having zero

columns correSponding to the non-zero rows of Z(t), andlg(t)
being such that ab’i - O, i - l...n; then, for all 1(t), the
pth entry of §(t) can be found by solving the time-invariant

equations

wage-.0 - Qua) - g(z)g(z)]§(z,t) = 10:). {2-6)
at

Further, the matrices of (2-6) satisfy

3313(2) - more) - n<z>g<z> <2-7)
d2

6L5“) -§(2)5(2)J = 512) Ll:(2)-§(2)§(z)]
- Q(2)-§(2)§(2)]9(2) (2-8)

so that equation (2-6) corresponds to a set of continuously

equivalent time-invariant networks.

3. LEMMA 1

The base upon which the proof of the theorem is established
is that the substitution of t for 2 in the continuously
equivalent fixed network equations is to yield the time-varying
network equations. This leads to the problem of introducing a
vector function _F_(2,t) such that F(t,t) - _X_(t). A little
thought will reveal that there is an infinity of {(2,t) which

will have this property. There is only one of these, however,

20
which will satisfy the condition on {(z,t) proved sufficient

for the existence of continuously equivalent fixed networks

in Theorem 2; that is

9-2-(zst) ' 9(2)£(2,t)
82

This uniqueness is established in the following lemma along

with the means for deriving the function fromԤ(t).

Lemma 1: Given the n-row vector functionԤ(t) and an nxn
matrix 30), there exists a unique vector function [(z,t)

of the real independent variables 2 and t such that

£(t.t) - £02) (2'9)
and gym) -g(z).l:(2.t) . (2-10)
32

Proof: Solve (2-10) for £(z,t).

yet) - a<z.z.>§(z..c> (2-11)

where

gfl(zszo) - 9(2)Q(2920); 3(zOszo) ‘3 ‘1, 6
d2

Then
[(t.t) - fi(t.zo).F_(ze.t).

Therefore, let
3(tvzo)§.(z02c) - yr).

solve for £(2o,t) - a" (t,zo)§(t), and use it in (2-11) to get
[(2.0 - ,a(z,:t‘,),a"1 (t,z,)§(t) .

Since‘g is a fundamental matrix it has the properties

21
e,“(t.2o) g<zo.t)
and 3(2,2°)§(2°,t) fi(2,t).
Therefore
§(z,t) - g(z,t)_x(t) (2-12)
From the derivation, [(z,t) clearly satisfies conditions

(2-9) and (2-10). Furthermore, since a given matrix‘g(z) has

a unique fundamental matrix the lemma is proved.

4. LEMMA 2

To clarify the roles played by the g and g matrices in the
time-varying case it is necessary to establish the effect of the
identically zero rows or columns on the corresponding fundamental
matrices. This requires a rather cumbersome sequence of steps

which is best presented as a lemma.

Lenrna 2: Let $(x,x°) be the fundamental matrix of an nxn matrix

30:); that is

336.2%) = 2(X)S*(X.Xo). 2(xo.xo) = 5.
dx

If the kth {on } of g (x) is identically zero, then the
0

entries of the kth {:3: of ij,x°) are all zero with the

exception of the main diagonal entry which is 1.
Proof: Consider the equation

dyx) - £00500 . (2-13)
3;

Equation (2-13) can be integrated to form the integral equation

22
EC!) - 3(xo) + I 30050.)». . (2-14)

x0
An iterative method [15] of solving (2-14) is to repeatedly
substitute the right side of (2-14) for £00 in the integral.
For example, the first iteration is

got) - 30:.) + I 110.) Lee.) + f g(§)§(§)d§1dx
xo xo

" 50‘s) {I g(>.)d>.]E(xo) + I 30.)}: g(g)§(g)d§. (2-15)
1‘9 xo 0

Let £0) '] (')d>..
x0

Then, continuing this process and separating the terms as in

(2-15) the solution can be written
em - (.1. + §® + §EE§®1 + Amuse] }+ .. Ebro)

Thus, the fundamental matrix of ~11(x) is given by the series
We.) = .I. + as) + §[&é®l + E{&§[E§Q’)J} + .. (246)

Now observe that :

row

th
(1) If the k colum

of 3(x) is identically zero, then

the kth r2: } of SQ?) is identically zero.

(ii) If the kth {22: of 30:) is identically zero, the

th row of the product formed by {:52J-multiplying any

k col

conformal matrix by 3 will be identically zero.

It follows from these two observations that if the

kth {row of P is identically zero, then the kth {row
col "‘ col

of every term in the series (2-16) will be zero except for

23

the identity matrix. Therefore the only non-zero entry in

the kth {row of the fundamental matrix QKx,x°) of P
colum ‘* v

is Wkk - l. q.e.d.
As an immediate consequence of Lemma 2 and equation (2-12),
if row k of the matrix g;is identically zero then the kth entry

of F(2,t) will equal xp(t), the kth entry of £(t).

5. LEMMA 3

The next step is to introduce the function

yz’t) . 33(2sz + wager)

+ §(2)B§_(2.t) - 5(2)£(2.t) (2-17)
at

where {(z,t) is derived from §(t) using Lemma 1.
From the chain rule of differentiation in the calculus, it

follows that

§§(Z)§(Z.t) + E(Z)§_§(Z.t) + §(Z)_B_§_(Z.t) = dLP}(t)2£(t)]
[dz 62 5t ]z=t at

and consequently

Emu) - gramme] - more) - :(t). (2-18)
dt

In general, §(z,t) is a function of 2 and t, and is not

equal to Zfit). If, however,

§g(z.t) ' fi(2)§_(2,t) (2-19)
32

then, from Lemma 1 and equation (2-18)

Emu) - 9(z.t)¥_(t> ‘ . (240)

5!

I?“

24

where

dg(2,2o) = g(z)g(2,zo); $(Zo:Zo) ‘ 3
d2 '

Finally, if‘fl(z) has zero columns corresponding to the
non-zero rows of 1(t), Lemma 2 and equation (2-20) combine to

show that

§(2,t) - 2(t) for all 2. (2‘21)

For any given value of 2, equation (2-21) represents a fixed
component system having the same input 1(t) as the time-varying
system.

It remains, therefore, to establish the conditions for which

§(z,t) satisfies equation (2-19).

Lemma 3: Given the function

he») = d§(2)£(2.t) + gcz)az(z.c>
E E

+ E(z)5i<z.t) - 5(z)§_(z,t), (2-17)

at

where
gym) - g<z>_1:<z,c). (2-10)
32

If
gm - sense) - 31(z)g(z) <2-7)
d2

and
193(2) - 2(2)A(2) - 5(2)9(2) + d£(2)§(2) (2-8)
d2 3;

then

gym) = 5(2)§(2.t)
32

25
Proof: Substitute (2-10) into (2217); then

§(z.t> = §(2)BE(Z.t) - Lye-dye~32<z>g<z>3§<z¢>
"a? 5‘2

which, upon substitution of (2-7) becomes

E(Z.t) = §(2)g(2.t) ‘ LA,(Z)-E(Z)§(Z)]E(Zst) (2'22)
at

For the remainder of the proof the arguments of the
functions will be dropped to shorten the notation. Now form

the partial derivative of §_with reSpect to 2.

as. - as g; + E9821) - [ere - deg - eds]: - [easier <2-23>
d2

37 dz 3t 32 3t d2 d2

Assuming continuity of the partial derivatives, the order of

differentiation with respect to 2 and t can be interchanged;

6 eye eyes.
2 3t 3t 32 3t
The substitution of this result and equations (2-7) and (2-8)

reduces (2-23) to

_a__s - g[_n____ar- Lia-egg] ,

which, from equation (2-22) is seen to be

32; - g1} q.e.d.
32

6. PROOF OF THEOREM 3
The proof is essentially a summation of the preceeding

three sections.

26

Given

gramme] - more) = :(t)
dt

with

go» = g(t)y,(t) - muse)
dt

and
gm - .B,<c),A,(t:) - 50030:) + gg(t)§(t) .
dt dt
where‘g has a zero column for each non-zero row of 1(t); then
from.Lemmas 2 and 3 we can construct for all 1(t) the equations
§(2)6.Ii(2.t) - Q(2)-§(2)5(z)lz(z.t) = 1(t) .
at
whose solution vector satisfies

Hut) = 3E“)
and

3212.0 - 3(2)_Ii(2.t)
52'

With 9(2) such that 0.91 8 0, i = l...n, it follows from

the above conditions and Lemma 2 that the pth

row of F(z,t) is
xp(t), the pth row of §(t). q.e.d.

Perhaps the importance of Lemma 3 in establishing the
equivalence of the time-varying and the fixed networks should
be emphasized. Suppose §(2,t) were a function of 2 and t, say
fi(2,t) = Xl(z,t). This would still represent a fixed system

h

whose response F(z,t) has xp(t) for its pt entry (since

fl(t,t) - 1(t) and afi(z,t) - 92(2,t) ). This response, however
32

27
is to an input different from X(t), so the fixed and time-
varying networks would not be equivalent with reSpect to any
driving-point or transfer function.
To show that

sees) = tees) - was . (2-2.)
d2

and, therefore, that (2-6) represents a set of continuously
equivalent networks, expand the derivative and substitute (2-7)

for fig . Upon cancellation of some terms the result reduces

dz
to equation (2-8).

CHAPTER III

PRACTICAL SOLUTION OF THE CONSTRAINT EQUATIONS

1. INTRODUCTION-

This chapter will consider the problems associated with
solving the constraint equations derived for the time-varying
matricesgflt) and ~A”(t). The equations are repeated here for

convenient reference.

d“

53“.) = more) - s<c>s<t> ‘3'”
d An. - H

EL“) WU”) - gown-moms] -

ER) @0023“) 190:) (3-2)

To save needless repetition, 3(t) and ~{i(t) will henceforth
be referred to as component matrices, and g(t) and.§(t) will be
called coefficient matrices.

A practical solution of these equations will be defined as
having the following characteristics:

1. It can be found analytically in closed form.

2. It must be such that, as time inoreases, the time-
varying network is not required to grow new elements
(including mutual coupling). This condition will be

,described as requiring the network to retain its
initial topology for all t.
3. The component variations corresponding to the solution

must be periodic and non-negative for all t.

28

29
While these requirements greatly reduce the number of
solutions to be considered, they represent a realistic approach
to the problem.
Since (3-1) and (3-2) are linear, homogeneous, time-varying
differential equations, their solutions can be written in closed
form [14] in terms of the fundamental matrices g and Q of 3(t)

andԤ(t) respectively. Thus,

he) - uic.c.>h<c.>g“<c.to> (3-3)
and

5(t)-g(t)g(t) = 9(t.ts)£¢(to)-fi(to)u(to)le,"(t.to) (3-4)
where

3““) - 9(C)fi(tsto); name.) = i,
and

d

gm.) -= momma); $9010.30) ,1,

With 9(t) and g(t) given, ~450;) can be solved for explicitly
by using (3-3) to eliminate 5(t) from (3-4). Then
~Mt) - m(t.to)g(co)a"(c.to)
+ [g(t)g(t,t°) - 9(t.to)§(to)E(to)g"(t,to)
This can also be expressed as

Mt) - m(t.to)g(to)§,"‘(t.to)

330mm) + 93(t,t,) ,1 _
+ [dt dto Jflfiom (t.to) (3 5)

since

9mm.)

dto - -g(t,t°)§(t°) .

30

If §(t) is such that g(t,t°) = 9(t-to), then

22(tsto) + 93(tsto) 3 9,,
dt dto

and (3-5) becomes

50:) - g(t-t,)g(t,)g'1(t-to) . (3-6)

This will be the case when‘g is a constant matrix.

While (3-3) and (3-5) are indeed solutions of (3-1) and
(3-2), they are not useful in general because the fundamental
matrices in most cases cannot be found analytically. The only
exceptions occur when,g(t) and'§(t) satisfy

30930:.) - g<m>g<tn

fi<t1)g(tg) - E(ts)fi(t;)

(3'7)

for all t, and t,.

It is important to emphasize the extent to which a time-
varying equivalent network differs from its associated con-
tinuously equivalent fixed networks. The equations satisfied
by the component matrices in each case are deceptively similar.
In fact, corresponding to equation (1-13) in Chapter I we

can write

-, _
it”! <9.st = se‘1e-e91- Le'1e-22919.

which has the solution

5'10) [50) £03210») = 9,(t.to)[§°"(§,o-§oys)]a'1(t.to) .

where

d
$202,110) = g(t)fl(t,tc)3 fl(to’t0) = I .

31
In the fixed case of Chapter I, where,fi was a function of
the independent variable 2, it was valid to remark that the
eigenvalues were preserved through the similarity transformation.
WithIfi a function of time this statement has no meaning. To
illustrate, compare the time-varying differential equations for

the voltage and current of Figure 1.

LC; + (LC+2LC+LG+RC)v + (LC+LG+LC+LG+RC+RG+1)V = e

Lci + (2LC+LC+LG+RC)i + (citic+ic+éc+sé+ss+1)i = cé + (G+C)e

Using the same functions for L, C, R and G which were
substituted into (2-1) to get (2-2), the above two equations
become

Loco; + (R°C°+G°L°-OL°C°)v + (R°G°+l+OR°C°)v a e
and

L°C°I + (R°C°+G°L°-OL°C°)i + (R°G°+l-OL°G°)i=[Coé+(l+O)e]e°t

Thus, even though in this particular case the homogeneous
equations have constant coefficients, they are different
equations. It would appear therefore that the free response
of this time-varying network has separate modes for inductor
current and capacitor voltage, and that there is no single
output where these modes can be observed simultaneously. The
associated continuously equivalent networks have only the mode
associated with the selected invariant output; that is, the
capacitor voltage. An interesting conjecture is that the nth-
order differential equation for any of the state variables

of time-varying equivalent networks will have constant coefficients.

32

Equation (3-7) is true for constant matrices and for
diagonal time-varying matrices. It is also true for any matrix
which can be expressed as the sum of a diagonal matrix whose
entries are identical functions and either a constant skew-
symmetric matrix or a constant symmetric matrix with diagonal
entries all zero. Finally, either of the above constant matrices
can be replaced by time-varying matrices if all the non-zero
entries are identical functions (except for sign in the skew-
symmetric case).

Solutions exist, of course, for,g(t) and.g(t) not falling
into any of these categories. The fundamental matrices must be
found numerically, however, which makes it difficult, if not
impossible, to state any general conclusions regarding their

associated equivalent networks.

2. SOLUTION OF THE SECOND-ORDER EXAMPLE

A major consideration when looking into the practical
solution of the constraint equations is the realizability of the
resulting time-varying network. In particular, this requires the
topology of the network to remain unaltered for all t. To
illustrate how this reduces the arbitrariness of the coefficient
matrices we use the formal solution of the second-order example
referred to in Chapters I and II.

The topology of the network requires §(t) and 5(t) to main-
tain the form

L(t) 0 -R(t)

210:) - . 50:) =
0 C(t) 1 -G(t)

I
H

33
for all t. To obtain an analytical solution the coefficient
matrices must be either constant or diagonal. (Note that
symmetrical forms are not possible for secondaorder matrices
having zero rows or columns).

For the constant matrix case, let

[p q] 0 r
S:‘ 9 and ,fi = ,

Using standard techniques the fundamental matrices are

found to be

ep<t-to> a<eP<t'to> - 1)
fl(tgto) 3 P
O 1
1 £(e8(t-t0) - 1)
and 9(t,t°) - s .
0 e3(t'to)

The notation will be compressed with no loss in generality by
choosing to = O, and letting L(t°) - L0 and C(to) - Co. Then,
from (3-3)

50:) - s P

1 nest-1) L. e'Pt -s<1-e'Pt)
0 eat 0 co 0

1
e"Pt -g;°(1-e'Pt) + £90 (est-1)
8

P
O CoeSt

In order to retain the original network form,

-ng(1-e'pt) + 390(eSt-l) = o (3-8)
p s

34

for all t >»O. This will be true if
Sis
r a and s a -p. (3-9)

Using these conditions and (3-5)
1 g_o(1-e pt) e'pt -g(1-e-Pt)
P

I
H

A“) ' Pcoe

It is the off-diagonal terms of this rather cumbersome matrix

which are of immediate interest. In particular
A31(t) = 3-2pto
Thus, A.1(t) = l for all t requires that p = 0. Then

A12“) = -1 + qt(Ro " 60.1.10 ' (Niles) .
Co Co

In order that A13(t) = ml for all t, therefore, q must be zero.
With p=q=0 the network is time-invariant so the result is trivial.

Returning to equation (3-8) it will be seen that an
alternative to conditions (3— —9) is q=r=0. This leaves the choice
of p and 3 open insofar as 5(t) is concerned. Under these

conditions, 5(t) becomes

1 o ’-R, -1 e'Pt o
50:) =

0 e8‘.- 1 -c o 1

«Roe-pt -l

n .

e(s-p)t -G°e8t

35
Thus, Afit) will retain its initial form for all t if s=p=o.
This corresponds to the example used at the beginning of
Chapter II.
To complete the formal solution of the example let the

coefficient matrices be diagonal and time-varying; that is,

let
190:) O 0 0
g(t) s and g(t) . .
O 0 0 s(t)
t
Then i p(A)dk
e 0
s(t) -
0 l
and _ _
1 0
90:) -
Is().)d).
.0 e .

 

 

From the conclusions of the constant coefficient matrix
example it is clear that s(t) and p(t) must be equal. Thus,
let s(t) - p(t) - O(t). Then

Lee 0
gm -

 

 

The advantages of periodic non-negative component variation

are obvious; therefore let

36
L(t) - L°(1 +msinwt), o in < 1 .

Then we need
t

I C(Dd). " -1n(1 + msinwt),
o .
from which
a -ggcosgt _
a“) l + msinwt ’ (3 98)
... .(s . ___.,_ . E
l -I- msinwt

To evaluate 50:) from (3-5), first note that with

”511' " -'.

 

1 o
9(tsto) "' t
[omen ’
0 etc
A o o ‘
glide). + £10111 .. I omd). .
O

t
0 [00) «mm 1e

Substituting (3-98) for 0(t) and letting to - 0, this becomes

 

 

O O
Qfiato) + 22(tsto) -
dt dto

mel-coggt)
t° " ° ° (1+1nsinwt)
Now use this in (3-5) to find

'-R° (l+msinmt) -1 ]

5e) -
l _L-t meO (l-coswt)
L 1+1nsitwt (insith,

 

37
If the conductance is to remain non-negative for all t, A3.(t)§0

for all t > O, requiring that

CO i meo (l - coggt ) for all t.
1 +-msimwt

The maximum of the right side of the above inequality occurs at
t-to such that

coswto = ma-l and sinwt = -2m
m3 +1 the +1

at which time the inequality becomes

Go : ZmeO
l-ma .

The minimum.conductance in any of the equivalent fixed

networks, however, is given by

Gmin '3 CC + 0(0)C° . Go "’ Mo ,

which demonstrates that it is not possible with this particular
component variation to obtain a time-varying network whose
equivalent fixed network has a negative conductance without
driving the conductance of the time-varying network to negative

values during part of the cycle.

Before leaving this example a few remarks are in order
regarding the possibility of an invariant driving-point
admittance. The solution details are very similar to the

transfer function case with whatever differences which arise

38
being due to making the first row of gflt) identically zero in
place of the second. It turns out that the constraint equations
cannot be satisfied in this case unless s(t) and g(t) are
identically zero; that is, there is no manner of varying the
components of Figure 1 that will maintain an invariant driving-
point admittance.

While conclusions are not possible from a single example,
there is some indication that invariant driving-point immittances
will require stronger constraints than invariant transfer
functions. Since driving-point functions have the same
denominators as transfer functions and usually more complicated

numerators, the result has some intuitive appeal.

3. CONSTRAINTS ON THE COEFFICIENT‘MATRICES

The experience with the example in the previous section
suggests that general properties might be found, which, when
possessed by s(t) and s(t), will insure that the associated
timedvarying network will retain its initial topology. These
properties will be generated for the defined class of networks
from the characteristics of the ~{i(t) and 5(t) matrices.

Since 5(t) is diagonal for all t, {i(t) must also be

diagonal for all t. Therefore

[.33 ' 219]” " 0 for i 5‘ j, from which

n
131 [31km ' “11.0:er - o for .11 1 e j,

39
Since;§ is diagonal for all t this simplifies to
For i-j it is readily seen that

fiii " (Bii ' “11)”11 - (3'11)

Further conditions are imposed upon the coefficient
matrices by requiring that the structure of the Aflt) matrix
be preserved. Since,é(t) can take a variety of forms, however,

these conditions must be found for particular categories.

I Symetric ~.{\,(t)

In RL networks,é(t) is the negative of a resistance matrix;
in CC networks it is the negative of a conductance matrix. In
either case, Afit) is symmetric for all t, which requires that

A(t) - AT(t), and, consequently

showtime-swast-
Rearranging terms, this becomes, since Alia symmetric

mere-emf) +3E'Efir'9°

This equation is satisfied if

g, - -§T (3-12)
and 3" - 5'155 . (3-13)

Substitute (3-12) into (3-10). Then

9i; ' ‘53 831. 1 7‘ J . (3-14)
“11

40
The (i,j)ch entry of (3-13) is

g .‘Mii '

Combining (3-14) and (3-15)
Bijdaji + Bjidaij '5 0 .

Therefore let

Bijaji = "kij’ where k is a real constant.

With (3-14) substituted into (3-16),

13 “11

5:1 I k3 “11

It is useful to observe from (3-12) that

.33,“ 1535 t
dt

from.which it follows that

19:

dt

1 . aria? .

Transposing this equation yields

dQ.1)T - EQ'1)T
dt

which shows that, as a consequence of (3-12), 3:1 a

(3-15)

(3-16)

In sumary, M“) will be diagonal and Alt) symetrical for

all t if

s - 29"”
a" 291

(3-12)

(3-17)

41

‘M
91 -1:1 .11
J J ij
(3-18)
Mi
5 --k .11
ji ij M11
and E11 3 2311M.“- . (3'11)

II Hybrid ~Ajt)

The most general form of,é(t) which will be considered is
that associated with RLC passive time-varying networks. It is
assumed that the state vector is ordered so that the first p
rows are inductor currents and the last n-p rows are capacitor
voltages; (these, of course, could be reversed). This will
allow the problem to be treated with only two partitions. All
matrices are functions of t, so the arguments are dropped with

this understanding. Thus, let

Au An 21:: Q.

A " . 5 " .
Au Ass .0. Has
SI: Sea 31: Ru

9 ' and g I .
9m 20s 3s: be

T
An " AM. a Ass '9." s and Au ' ‘53; (3-19)

42

The partitioned form of the constraint on 5(t) expands

into four equations:

An " £119.11 ' £119“ + Esta: " 518%: + 31115:: (3'20)
es. - eases: - Aliases + sates. - the. + has... 0-21)
.5... - are... + he... - ens... - the. + he» <3-22)
As: ' 391511 + 3-89.91 ' $019211 ' Ange: + 321311 . (3'23)

When (3-20) is subjected to the condition.é31 8 531, we

get, with the help of (3-19),
T T T
(fin +9u>éu " 9.1: (3.11 +9“) +948Q13 " 301)

T o T
+ (Ru " 900$: +3112!“ " 24.11211 " 9°

This is satisfied for all t if

EM. " '31:: . (3'24)
&2 ' g. s (3'25)
h. - set: . (3...)

Equations (3-24) and (3-26) are identical to the conditions
required of g and g in the symetrical Alt) case. They can be
combined with (3-10) to show that the entries of jh; are given

by (3-18).

From the symmetry of (3-20) and (3-21) it immediately follows

that

T . . -
962 " '38:: a 9:. ' .51.: and ~ass ' Essfigsnsf a

saw

43
and that Eh! is similarily given by (3-18).

Equations (3-22) and (3-23) are combined by setting

‘T
518 +'éfll ' O '

with the result that

T T T
(fin + Sums: 4‘ (fits " Sofie.“ + Anifls: " 9:9)
I 0 e
’ 518(flas + See) *‘ihsflba +'!h1£bl “.9-
The conditions already introduced are sufficient to make this

zero for all t if we add

312 ' .zhléfllgg (3'27)

Thus, the 811 in the off-diagonal partitions must simultaneously

satisfy

‘M
"11
Comparing these with (3-14) and (3-15), it readily follows

that

$15 ' k“ 11:1 and Bji - kij gil- . (3-28)
1

The derivation of (3-28) assumes that the entries in the
off-diagonal partitions of,é(t) are variable. In most cases
at least some of the entries will be constant ratios, requiring
that the right sides of (3-22) and (3-23) for such entries be
zero. This imposes much stronger constraints on the corres-
ponding,g.andl§ entries. Sufficient conditions in this case

would be 9“ - 9, «R13 - Q, 9, - g and g“ - 9, along with

Shiéis ' Alas-a

44

and

1538581 '.éalSni-
It is worth noting that the substitution of (3-19) and

(3-24) into the second of the above equations yields the first,

showing that the conditions are consistent.

4. PERIODIC SOLUTIONS

Sufficient conditions for periodic, non-negative variations
of the resistive components must be discussed for particular
examples, since there is no general interpretation of the
entries of the,é(t) matrix in terms of individual components.
More general conclusions are available, however, regarding
conditions which are necessary under some circumstances.
These will be considered in this section.

Since 5(t) for our defined class of networks consists of
individual component functions as diagonal entries, it follows

from equation (3-11)

“ii ' (511 " “iimii (3'11)

that if 011 and 511 are constants, the L1(t) and C1(t) functions
will all be exponential. For periodically varying inductances
and capacitances, therefore, 011 and 611 must be suitable
functions of time. The only alternative is for all of these
main diagonal entries of the coefficient matrices to be zero,

in which case the gmatrix will be constant. Although this
might appear to be too restrictive, it turns out to quite useful

as will be illustrated in the examples treated in Chapter IV.

45

It can be seen from equation (3-5) that for Aft) to be
periodic it is necessary that Q(t,t°) and g(t,t°) be periodic.
This, in turn, imposes necessary conditions on the coefficient
matrices. If g. and Q are functions of time they must be
periodic. This is readily shown. ‘With

E - ea .
if fi,is periodic, then,§ will be periodic. Therefore, since

9:' 9-1 a
it is the product of two periodic matrices, and as such must
be either periodic or constant.

If g and Q are constant matrices their eigenvalues must
occur in conjugate pairs on the imaginary axis for g and Q to be
periodic. This will be the case if they are similar to a skew-
hermitian matrix having the following properties:

(i) Each element is either real, imaginary or zero,

(ii) The rows and columns can be ordered such that a partitioning

Ell Ens

 

 

Es: fies,

h

exists, where E11 and 233 are skew-symmetric and,§,, - g}, is
purely imaginary.
It is interesting, and most fortuitous, that these conditions

are compatible with those already dictated by the topology

46
preserving requirement. To show this consider first the
symetrical 5(t) case. With the coefficient matrices constant
we have seen that their main diagonals must be zero for a
satisfactory 2; matrix. Also, from (3-14), the off-diagonal
entries of g are related by

913 - 32—:- 611 (3-14)

Assuming j to be real. the combination of these two results
leads to the observation that Elfi g is a skew-symmetric
matrix, and 3 is therefore similar to a skew-smetric matrix.
The same can be said of 9 since, from (3-12), 9 . -§T. It
should be noted that since 5 is diagonal it need not be constant
in order to properly define a similar transformation, provided
all the non-zero entries are, except for multiplying constants,
identical functions. This follows immediately from (3-14).
When ~A,_(t) is hybrid, 9 and 3 must take the
form described on page 42. It has been shown that the off-

diagonal partitions of g in this case should satisfy

a
Bij .1133 $31 -

For imaginary eigenvalues, therefore, these entries must be

imaginary because they are of the same sign.

5. SUHIIARY
It is remarkable how compatible are the conditions
imposed by the three properties required of a practical

solution of the constraint equations. Since the next chapter

47

deals with specific examples, the following emery of the

conclusions of the present chapter will provide a useful

reference.

In order that the solutions of the constraints be obtainable

analytically, preserve network topology and be periodic:

1.

If ~A'(t:) is symetric and g(t) is constant:

(a)
(b)
0:)

15-13 g is skew-symmetric,
T
9 ‘ 'fi
-1 T
3 ‘ 9 -

If ~Alt) is symmetric and g(t) is a function of time:

(a)

(b)
(e)
(d)

~15(t) must have periodic functions on the main-
diagonal which are identical within a multiplying
constant.

a - :3?

s"- :3

The 3 matrix must be the sum of two matrices,

one constant and similar to a skew-symmetric
matrix, the other diagonal, time-varying, and

with identical non-zero entries.

If ~A’,(t) is hybrid with all off-diagonal partition

entries time-varying:

(a)

(b)

The main diagonal partitions must satisfy the
same conditions outlined in 1 and 2.
The off-diagonal partition entries of 3 must be

purely imaginary and related by

1"ii
51 -_"'Bi -
.1 u“ .1

48

(c) The off-diagonal partition entries of,g.are

related to those of 3 by g - 5T.

If,é(t) is hybrid and has some fixed off-diagonal
partition entries, the corresponding entries of the

coefficient matrices must be zero.

CHAPTER IV

EXAMPLES WITH PERIODICALLY VARYING COMPONENTS

1. INTRODUCTION

HMving shown that there are practical solutions to the
constraint equations, the remaining question is whether the
corresponding networks have useful properties. Some thought
will reveal that the structure with the best possibilities is
the GC network. The reasoning goes something like this.

Consider first a network of fixed capacitances and time-
varying resistances. This is a symmetric ~A,(t) constant 5 case,
so the conditions for a practical solution include that
[3: 1% should be a skew-symetric matrix. It follows,
therefore, since 5 is diagonal, that

rat‘s/m - as
is also skew-symmetric.

The fixed network to which the time-varying network is
equivalent has for its 95? matrix

A:- ' Ute) ' g °
at is therefore not symmetric; in fact as 3 becomes large 5f
approaches a hybrid matrix. This indicates, and it turns out
to be so, that g'léf can have complex eigenvalues. Thus it
appears that a time-varying passive CC network can be used to
realize a time-invariant transfer function with complex poles.
The same remarks apply to RL networks but these are probably

not so important as CO.

49

50

2. A FOURTHeORDER.TIME~VARYING CC NETWORK
The minimum order of a network which will fit the
category described in the previous section is four. This
can be deduced from the following observations:
(1) Row p of g and column q of g must be zero, p i‘ q,
in order to have an invariant transfer function.
(ii) Since 3 a 8T, row q of g. and column p of 8 must
also be zero. Thus non-zero g and 8 must be at
least third-order.
(iii) Such a third-order matrix would have only a single
non-zero entry, and it would be on the main diagonal.
Therefore, for,§ to be similar to a skew-symmetric

matrix it must be at least fourth-order.

The fourth-order example will first be considered with,fi a
time-varying matrix since this can be easily reduced to the
fixed case by making the time function identically zero. The
procedure used will be to choose E,appropriately, find its
fundamental matrix Q, and using 3:1 = 91" determine g and ‘A' from

(3-3) and (3-5). Thus choose

 

 

(o o o o]
0 f(t) W 0
g -
0 «It/fink!11 f(t) 0
L o o o 0‘,

Note that

20950:.) - £(ts)§,(t1) (4-1)

51

and, therefore, the solution of

e - es <4-2)

can be written as

I go...

9“ t to) 3 Eta

To compress the notation, let

ii

Mj] P s

and drop the argument of f(t).

The characteristic equation for,§ is
)6 + 2.5),a + (Parana -- o ,

which factors into
fine)” + k‘] = o .

To find 9(t) it is only necessary to work with the non-zero

portion oflg,
f kp

s"-
-k/p f

which has the characteristic equation
(M15)a + k“ s o.

The eigenvalues are
X“ “fijk .

which we use to obtain a modal matrix

 

NIH

 

_j/p

 

 

~J/pt

52

and its inverse

Now form the matrix product

NIH

 

_j/p

j[f(u)+jk]du
l] _eto 7 r1 -jp-
t
J[f(u)-jk]du
-j/p. -0 et° . .1 JP.

 

 

choose to a 0, and let

t
Jf(u)du s(t); s(O) = 0 .

O

 

 

 

In these terms the eXpansion of the above product is

es02)

coskt

-sinkt

P

psinkt

coskt

Therefore, the fundamental matrix for,§ is

90:) -

p-

1

0

 

0

e3(t)coskt

-e8(t)sinkt

P
0

0 0
pe3(t)sinkt O

e3(t)coskt O

 

O l

. (4-3)

53

An alternative approach to solving equation (4-2) is due

to the following theorem by Frame [18]:

Let,§(t) -‘3(§,t),‘§ a constant, diagonable, nxn matrix,
where_q(l,t) is continuous in t and analytic in a simply
connected open region of the complex k-plane that contains
all the distinct eigenvalues l1 of,§. Letlgi denote the
constituent idempotent matrices of K. Then the homogeneous

v

matrix differential equation

(31% ‘R. 9(0)=L s

has the solution

t
(X . )d
E .12: (El 3 i u “>19.

The non-zero sub-matrix of'g can be written as

E' ' f(t).; + k5 = st.t) .

where

TL 0
LP .

 

 

The eigenvalues of,§ are k1 - ij, and the constituent idem-

potents are

1 'JP1 1 JP

NIH

El ' %' and 5D .

 

 

 

 

-'o|1_..

54

Therefore
2 I[f(u) + kK1]du ‘
I
gg(t) -:23 e ~')
181 K1
f(u)du 1 ‘JP 1 jp
I e ejkt e‘jkt
— + .—
2 2
_j_ 1 j 1
P p
coskt sinkt
e es(t) p
-lsinkt coskt
P

The solution is just as easily obtained if kt is replaced

by any continuous function k(t). Indeed, let

1:
k(t) - Jr(u)du ,

o
and

q(l.t) - f(t) + Ar(t) .

then the submflrix of 9 becomes

9’0) - e30) “8““) Psink(t)
-lsink(t) Cosk(c)
p

This extension cannot be exploited, however, since equations
(3-16) and (3-28) have established that for the,é(t) matrix to
be symmetric for all t the off-diagonal elements of,§,must be
constant.

Returning, therefore, to equation (4-3),

55

let e8“) - h(t); h(O) = 1, and form ~130:) - g(t)§(0)g1'(t).

Letting 1
M1 0 0 O
0 Mg 0 0
5(0) -
0 0 M5 0
Lo 0 O I“)

 

 

and observing that p3 8 E3, , we find that

u,
"M, o 'o o)
o h3(t)Mg o o
50:) -
o o ha(t)M. o .
_o o o u,_ t

 

 

2if(u)du
Clearly the function f(t) must be so chosen that e is
periodic.

Next form the Afit) matrix from (3-4); that is

50:) - marrow-monomers) + more) .
As one might expect the details of this expansion are quite
cumbersome. It is worth noting, however, that
”o o o o)
o h’<c)f<0)n. h‘(c)kpns 0
0 -h’(t)kr_1., h3(t)£(0)u, 0

P
0 0 O Q

o<t>§<0)5(0)§(t) -

 

 

while

56

 

 

'0 o o o
o h3(t)£(t)u, h’(t)kpu, 0
50350:) - ,
0 -h3(t)k§g_ h3(t)£(t)u, 0
p
[0 o o o.

and, therefore, -Q(t)§(0)§(0)gr(t) + §(t)g(t) is given by

 

"o o o o1
o h3(t)u.[£(t)-£(0)] o o
o o h2 0:)M.[f(t)-f(0)] o

_o o 0 Q,

 

When 5(0) is symmetric, Q(t)A(O)gT(t) is symetric, from
which it follows that ~Alt) is symmetric. Since the expanded
form of 5(t) covers so much space it will be given by listing

the entries.

An. ('1) ' Au

A,,(t) - A“ (t) = h(t) [At-coskt + pausinkt] (4-4a)

A,,(t) - A,,(t) a h(t)[A,,toskt - Algsinkt] (4-4b)

Au“) " Au (t) " A14 p

A“ (t) - A“ (t) - h(t) [Aucoskt + pAusinkt] (4-4c)

A,,(t) - A,,(t) - h(t)[A,,coskt - Agisinkt] (4-4d)
P

A“ (t) - h2 (t( A” cosakt + paA”sinakt

+ 2pA.,sinktcoskt + u,[£(t)-£(0)J) (4-4e)
A,,(t) a h2(t)(A,,cos3kt +-e!£sin?kt
p2
- ZA‘alsinktcoskt + }fi[f(t)I-f(0) :1) (ti-hf)

P

57
a a a
A,,(t) -—— A“(t) = h (t)[A,,(cos kt-sin kt)
+ (pAaa - A33)sinktcoskt] (4-4g)
P
We are now prepared to discern whether anything is

accomplished by time-varying the 5 matrix. To do this, examine

Of of the equivalent fixed network.

PA“ A1” Ala A“1
5(0)-§<0)y,(0) _ Ala Ass-Hsfw) Ass-WEE A“
A1: A. 3+Wfi§ A“ 41. f (O) A“
”A“ A“ A“ A,”

 

 

Note that the hybrid nature of éf is induced by the factor
k/NEM; which is independent of f(t). Time-varying;§ affects only
the 2,2 and 3,3 entries of Cf: Recognizing 5(0) as the negative
of a conductance matrix, however, indicates that there is a
possibility that a suitable choice of f(O) will introduce
negative conductance terms into the equivalent network. This
choice of f(O) must be compatible with periodic, non-negative
g(t) to be significant.

As h(t) is defined it must be periodic and strictly greater
than zero, with h(O) a l. The simplest function having these

properties is probably

h(t) - l-msinwt , 0 e m < 1 .

The corresponding g(t) is ln(1-msinwt), from'which

f(t) - -wmco§gt .
l-msiuut

58

With this choice of f(t), (4-4e) becomes

A33 (1:) = (I’mSinLUt)2[Aag C0821“: + pzA'asinakt

+ pAaasinZkt +~M§wm l-cosgt-sinmt
l-msiunt

and [5(0) -§(0)£‘I,(0) 122 = Ass + wml‘h (4'5)

The conditions about to be investigated must be simul-
taneously satisfied by the 3,3 position of,é(t). The best
possibility of this will exist if the 2,2 and 3,3 positions are
equally weighted. Therefore, let

Ass 3 PaAss-

Equivalently, A32 - A33; that is, the time constants are equal.

11.x,

Then

A,a(t) = (l~msinwt)3[A‘a + pA..sin2kt

 

+ Hanan l-coswt-sinwt]
1-msinwt ’

Now let t - 5,, so that
4

Asa<23> ' (1+m)a Ass +'mes lh/Z +'PAas] ,

1+'m ’ (4-6)

and consider the following argument:

(a) A" < o, A” > o, and n, > o.

59

(b) A.,(t) must be less than zero for all t since it is
the negative of a main diagonal entry of a conductance
matrix.

(c) The only advantage to time-varying the 31 matrix is
measured by the degree to which the term mm}!- makes
the right side of (4-5) less negative.

(d) Since

M>l forallO<m<1,

1+ 39.

[2

it is seen from (4-6) that in order to keep ~{g(t)
negative, A33 must be made more negative because of
the time-varying of g by an amount exceeding the
positive contribution of M in (4-5). The closer m
is to 1, that is the greater the variation of g, the
less desirable become the characteristics of the 5f
matrix (from the viewpoint that we are looking for an

5f not realizable with a passive fixed GC network).

While this particular choice of f (t) does not prove that the
best results will be obtained with a fixed 1‘; matrix the evidence
points quite strongly in this direction, especially since any
periodic function for ~g(t) will be expandable into terms
similar to h(t).

The foregoing represents a considerable effort to demonstrate
a negative result. It has been worthwhile in that we can now turn
our attention to networks with fixed 2; matrices feeling fairly

secure that nothing useful is being overlooked.

60

Next to be investigated is the realizability of equations (4-4)
with h(t) - 1. Note first that the off-diagonal terms of g(t) may
be allowed to alternate in sign provided the matrix stays dominant
for all t. Whether this happens depends on the initial values
5(0). While it is possible to start with a dominant ~13(0) and
have g(t) become non-dominant at a later time, it will be
demonstrated that it is also possible to obtain useful results
with a matrix which is always dominant.

To do so, simplify equations (4-4) by interrelating the

values of 5(0) by:

 

 

A: s " PA: a
A“ " PA“
Asa " PaAas - (4‘7)
Then
’ A; 1 [211 s cos (kt-Hz?)
flA; .cos (ktflj) A” + pA” s in2kt
4
g(t) '-
f—ZA; acos (kt ~55) Aa.cosZkt
_A1 ‘ m“ cos (kt-Hit)
m: 3608 (kt 'g) A1‘ 7
As .cosZkt M" cos (kt-Hz!)
P
mucos (kt -2) A“ _ (4'78)

It is sufficient for dominance that the difference between

the absolute value of a diagonal term and the sum of the absolute

61
values of all the other entries in the row exceeds zero for all t.
Since each main diagonal entry has a term A11 which appears in no
other entries the condition for dominance can be fulfilled by making
the magnitude of these terms sufficiently large. It should also

be observed that since

 

 

A11 A12 All A14

A1 a Asa Asa 'W “ans Ase
éf -

A13 A,.+k/fi;fi; Asa Ase

.Aie Ase Ase Ate.

its sssymmetry is unaffected by the magnitude of the A11.

3. PROPERTIES OF THE EQUIVALENT FOURTH-ORDER INVARIANT NETWORK
As was noted at the start of section 2, the requirement
3 - ET for a practical solution of the constraint equations
leads both 3 and g to have their 1th and jth rows and columns
identically zero in order that the transfer impedance 211(s)
be invariant. This additionally assures the invariance of the
open circuit driving-point impedances 211(8) and ij(s) so that
such a time-varying network is in fact equivalent to a fixed
two-port network. To gage the potential usefulness of such
two-ports we will examine the pole and zero locations of the
fourth-order example.
From.the main theorem it is known that the time-varying
network is equivalent to a fixed network whose state equations

are

293.“) ' mortals“) + 10:) (4—8)

62
Since our interest lies in the open circuit impedance
functions, zero initial conditions are assumed and equation

(4-8) is Laplace transformed and solved for g(s).
5(a) - [an - 5(0) + £15141“) (4-9)

With the first and fourth columns of‘g zero, 118) can be
allowed to have non-zero first and fourth rows. Similarily
since the first and fourth rows of g are zero the first and
fourth entries of g(s) are independent of the component
variations. The first and fourth rows of (4-9) are therefore

of interest, and are written in the form

It; (8) Zn (8) 2;. (8) 71(8)
X. (s) 2.; (s) z“ (8) y. (a)
where
zijm .. [(sg-yongp‘lju (4-10)

Being the negative of a conductance matrix,,é(0) can be

conveniently represented as

an “Cu ‘9“ '9“

15(0) - '61: Gas ’92s ’6“ ’
‘Gia 'Gaa GI! “Gas
_'G1‘ ’93. ’Gae Ghe‘

 

 

where the G11 > 0 for all 1,].

63

 

 

Also, let
"c, o o o ‘
0 ca 0 o
g I
o 0 Ca 0 ,
L0 0 o c,L
then
'0 o o 01
0 0 03C; 0
‘5! 0 'k/C;Cs 0 0 .
_o o o o,

 

 

It will be convenient to use the notation p -=/C;C;, then

(4-10) becomes__

 

 

T
[361*Ch1 ”Gas 'Gis 'Gie .
'Ghs sC.+G,g -G,.+kp 'Gae
213(3) '
'Gia ’Gss‘kp 39a+caa 'Gse
_'Gie ’Gse 'Gae 366+G1¢i
11

 

 

(4-11)
As might be expected the evaluation of {4-11) leads to
unwieldy functions. They can be simplified quite meaningfully,
however, by allowing k to become large and discarding all terms
which remain small in the process. The denominator in this

case can be approximated by
D e a s s
(s) - C1C‘C.C‘[s + (El}+§2?+§29+§1’)s + k s
C; C, C, C.

+ gaggle”: + (fixerciokj

C1 Ca 3134

64
With 61‘ small (it will be seen later to be desirable that
6;. - 0), it is a reasonably good approximation to write D(s)
in the factored form

0(3) - c CaCBC‘(8+G:1)(8+C:‘)[(S+Ggg+gég)a + k3]

2c. 2ca (4'12)

which gives a good idea of the pole locations when k is large
enough.

The numerators for large k are:

N,,(s) c,,c,c,[sa +(ca+c,.+c,, a” + 183 + 186‘,

Cscsce Ce

N,.(s) C;C.C,[sa + (91i?§ZE?§!2?32 + kas + k?§;;]

Ci Ca Cs 01

Nu“) " C,C,G;{sa + (EEIL’G"+G‘__’G“)B
Ca Ca 0,0“ 6391‘

+ k3 +._E_ (GisGas‘GiaG..)]
G14

 

“4.1 (3) " Cacscura +(Gga+G”+GuG“+G“G“)s
Cs 0. 0361‘ Cacao

+ ka" (c,,c.,-c,.c,,)]
G“

The main diagonal numerators will approximately factor into

A
N11(8) - Cscscs(8+§2_4_)[(8+§s_s_+§9;)a + 18’]
2c. 2c.
A (4-13)
mm - C:C:C.(s+<_:u) (81.1%)“ + k’]
01 2c, 20.

65
By combining (4~12) and (4-13) the main diagonal impedances are

found to be approximated by

211(8) 3 ___l;___ (4-14)
611+8C1
2,,(8) = 1 (4-15)

The off-diagonal numerators approach a common expression
as k increases, which in factored form is

c,,c,c, (smaamgficl ”Gas-+6136“? + k3] .

Comparing this with (4-12) the complex zeros are seen to
be uncomfortably close to the complex zeros of the denominator
‘3(s). However, if Cl. - Cg, a 0, the off-diagonal numerators

become exactly

/\
“14(9) ‘ (GisGseCa +’GisGsecs)3 +'GisGsecaa
+ GisGuGaa 4' “(918334 " G139“) a and

[1341(8) " (31:92.3: + GisGsGCs)3 + GiaGuGss
+ stcsecss ' kp(31sGae ’ GhsGbe) .

The most useful form of the transfer impedance expressions

are therefore given by

214(3) "fii4(3)
D(s)

(4-16)

Z“ (8) '3 fi (8)
3(8) (4-17)

66
A numerical example is considered in Chapter V which indicates

the degree of approximation involved in equations (4-14) through

(4-17).

4. A LINEAR RECIPROCAL PASSIVE TIME-VARYING REALIZATION OF THE

FOURTH-ORDER EXAMPLE

A sufficient condition for a real symmetric nxn matrix to
be the conductance matrix of an n-port resistive network is
that the matrix be dominant. A scheme for realizing such a
matrix [16] can be described in terms of Figure 4-1. Connected
across each port is a conductance Eii given by

n
811 " 911 " Z) lGimIt m 7‘ 1 . (4-18)
m-l

Dominance will insure that none of the 311 are negative.
There are four conductances connected between each pair

of ports. To describe those between ports 1 and 1, let the

ports be defined by the terminals 11’ and jj' respectively.

Then the four conductances are
811 ‘ 81’3’ ‘ Icijl ‘ Gij (4-19)
and 81’] - 311: B lcijl + Gij (4-20)

For a time—invariant network either (4-19) or (4-20) will be
zero depending on whether the particular off-diagonal term of
the conductance matrix is positive or negative. For the time-
varying network described by (4-7), which has off-diagonal terms

which alternate in sign, (4-19) will be zero every other half-

67

 

 

 

 

1w

 

 

 

 

“3111*”

n n
Gm: |Gim| G11": lGJml
m-l mFl'
mi‘i «6‘1

[Gijl-IGU
NVWr
IGij'-Gij

Fig. 4-1. Scheme for realizing any dominant
matrix as a short-circuit conductance matrix.

68

cycle during which periods (4-20) will be non-zero, and vice-

versa. Also from equations (4-7a) and (4-18)

81103) " Cu " [Cu + flgglcoukttg) |W9u|c°3(kt'z_l) I]

l
9
.

g“ (t) - - [6“ + HG“, Icos(kt+1ZT) IWGM, Icos(kt-E) I]

333 (t) = G; - _C_9_G”sin2kt - [63,lc932ktl

Cs

+ [2033 Icos(kt+r_1) IWG“ [cos (kt-+12) I]
4

833(t) a G33 ’ 6:69.81n2kt "[Gb‘ICO82kt| +
Ca

[26, . Icos (kt -4) “[26“, [cos (kt-E) I]

These can be simplified somewhat by setting

G1‘ ' Ga. a 0.
Then g“ (t) - G“ - f2[G;. Icos(kt+'r_'r) I+Gu lcos (kt-11) I]
4 4
8“ (t) G“ ' WC“ |C08 (kt‘l'fl) |+GO6 |COS (kt'fi) l]
4 4
822 03 Gas " fl(Gl ewes) l3“ (kt?|

833 (t) g Gas ' fl(c1a+ca4) |C08(kt-TZT)|

This realization is shown in Figure 4-2, the values
indicated corresponding to the numerical example in Chapter V.
Figure 4-3 is an active network realization of the fixed network

to which Figure 4-2 is equivalent. Variations of this example

69

811.(t) - 3-|1.531n2c|-|1.5coszt|

322.(t) = 4-|I.Ssin2tI-ll.5(cosZt+sin2t)I

s33.(t) - 4-ll.5cosZt|-Il.5(cosZt-sin2t)I

s44.(t) - 9-Il.5(cosZt+sin2t)l-ll.5(cosZt-sin2t)I

8120(t) ' 8112(t) = 1.5(|81n2t|-81n2t)

812(t) - 31.2.(c) l.5(|sin2t|+sin2t)

l.5(|cos2tI-cosZt)

813|(t) ' 81I3(t)

sl3<t) - 31.3.(c) l.S(|cosZt|+cosZt)

324.(c) - 32.4(c) = l.5[lcosZt+sin2tI-(cosZt+sin2t)]

824(t) ' 32.4,(t) l.5[|cosZt+sin2t|+(cosZt+sin2t)]

1.5[IcosZt-sin2tl-(cosZt-sinZt)]

834c(t) ' 83:4(t)

g34(t) - 33.4.(c) l.SEIcosZt-sinZt|+(cosZt-sin2t)]

Fig. 4-2. Realization of the fourth-order time-varying
equivalent network example.

71

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 1.5
1 2 3 4
0—1 r-———o
1.5
4V
__ ’ __ J-
01 F .5 2 .5 1 2 6 01
I: 4v,
.7 0 —o

 

 

Fig. 4-3. Time-invariant network equivalent
to the fourth-order example.

72

can be generated from equations (4-4) with h(t) - 1. In
particular, conditions other than (4-7) can be applied to
these equations, allowing for example the setting to zero of
G1, and 634 without uncoupling the network. This results in

a transfer impedance of the form

241(5) '_53_ ,
D(s)

Alternatively, letting 61, = G,‘ = 0 yields

z¢1(3) ‘ 83%
D(S)

 

This indicates that there is a fair degree of control over the
transfer function properties of.§(s).

A final remark should be made aboutԤ(s). Note that there
is no choice of initial conductances that will prevent the
complex pair of zeros of 211(8) from almost coinciding with the
complex pole pair. This is unfortunate in that it precludes
the possibility of synthesizing transfer functions with two-
terminal third-order networks having complex driving point

impedance poles.

3. A SIXTH-ORDER GC TIME-VARYING NETWORK

The success with the fourth-order example leads to the
consideration of a sixth-order time-varying network as a
means of realizing a transfer function having two pairs of
complex conjugate poles. The first and sixth rows and columns

of the §,mmtrix will play the same role as the first and fourth

73
of the fourth-order case. For the present, therefore, we will

work with the central 4x4 principal minor, which will be

h

designated g’. The 5 matrix will be constant and the (E1
":1

terms are symbolized by:
P‘ LT“ . ‘1',/.’£a_1 . r"/“_u.
Mas . H“ “as
. - Free . u we. : v - __«.
"as “a “as °

Note that pa =- q, qv =- r, sv - u and pu =- r.

 

 

Thus
r 0 Pks s qks a rk; 31
'kg I 0 81(3‘ “R's
P
2' °
’ ’52:. “Lu 0 Vku
q 8
'33.; $.29. ’11:. 0
b r u v _

The characteristic equation for g' is
x‘ + (k2, + k3, + k2. + k3, + k2. + 13.))?
+ kgakas + kgekga 4' kgakge 4’ zhskssksel‘es

' stsksskeskes " stsksskuku " 0 (4'21)

For a periodic 9 matrix the eigenvalues of g’ must be
purely imaginary and have magnitudes whose ratios are rational.
Writing equation (4-21) as

x‘+x‘x’+Q‘-o,

74

where Q ‘ (kgak‘s ' kggkgs + ka‘kag), then

x3=-_1g‘: -Q .
2 4.

Since periodicity requires |A1|a - m:|x.|3, set
n

('1; W2?" 'Qa) (4-22)

-5? + - Q2 a
2

:47;

”J a”

Equation (4-22) reduces to

K? - m?+n?Q - 0.
mn

Now substitute for K and Q and group the terms as

k:. +'k:s ' Eiifli. ksskss +'kgs + kgs +‘EE12:.ksskss
mn mn

+ kg. + k2; ‘ m?+n? k.‘kg. ' 0.
mn

or

(Elm-mx‘eku-M) + (.Ekaa+koa)%kag+kaa>
“felt“ “kas)(§ke4'kss) - o.

This is satisfied by choosing one condition out of each of the

following pairs:

1‘“ " Elks: kss "' ‘Ekse kss " 2k“
n n n

Rea ' Eksa ksa ' 'Ekss kss ' 23st
m m m

A typical selection yields

Q n k2 + 3 + k2 , a d
5K as Rae as) n

75

K3 B (1 + £203: + k2. 4' 1‘25)

n

Let kg, +k2‘ +16. llrka, andm-w;
n

then the characteristic equation becomes
x‘ + (1+w3)ka}.a + wak‘ =- 0,

so that the eigenvalues are
= _Jk and = :jwk .

The subscripts can be reduced without introducing any ambiguity

with the substitutions

ks: ' ks: kse ' kc: and kit ' ks .

In this particular case the simplest method for finding the
fundamental matrix of g is to form a modal matrix. A fairly

compact one is

’-q(\_c,k.+1kk.) «km-11g) 151;. £117

ke‘H‘s kfiks k k
g = .
1%) If???) 155 "133‘
1 1 'L’fi 11“.:
k k
L o o 1 1 _

 

 

Now take the inverse of‘g;

76

 

 

'-k!k.+jk}3 35”th 5".4-1‘: o“
2qu 281:” 21183

'kgks'Jklg kah'ikfl 133%” o
qu2 28k2 21;:3

N" -
1.151 '11s .15. l ’
Zrk 2uk 2vk 2
'15s. 11‘; £1 1.
2rk 2uk ka 2
and form

Q! " Eefiflfl, where

 

 

Pejkt o o o 7
o e'Jkt o o
e/~\t -
0 0 ejwkt 0 °
0 O o e-jwkt

Since the 9 matrix is too large to present in an array the

entries are listed:

Wu " c9w " 1

¢ba ' kg+k2 coskt + kgcoswkt
k3 i;-

cpgg - pair. (coskt-coswkt) + pfisinkt
k3 k

q)“ - qukd-coskt-i-coswkt) + qhsinkt
k“ k

i
I

r_k_._s inkt
k

77

¢§3 - pk.k5(ccskt-coswkt) - pkasinkt
k? k

 

$53 ' k3+k§coskt + Eicoswkt
k3 13

¢bs ' kaks(coskt-coswkt) + kgsinkt

Ska sk

cp" - -uk. sinwkt
k

mg; - kaEg(-coskt+coswkt) - Eisinkt
qka qk

¢ga - kak‘(coskt-coswkt) - Egsinkt
3k sk

¢k4 - Eg+kgcoskt + Egcoswkt
ks R?

q“. - vklsinwkt
k

Cpgg - 2118 1W1“:
rk

%. - 13-8 1Wkt
uk

¢5. - -§!sinwkt
vk

$5. I (:08th

All other entries in the 6x6 array are zero.

Expanding g(é(O)-y)gr is, to say the least, tedious.
Furthermore, the result would cover several pages. A Fortran
listing is given in Appendix B for a program*which expands the

product, checks the resulting g(t) for dominance and evaluates

78
the coefficients of the Fourier series representation of each
entry. With w-2, and all the ki-l, the highest harmonic of any~
of the expansions is the fourth. For other rational values of w
the harmonic content is much higher. These results are tabulated
in Chapter V.
Corresponding to (4-11) of the fourth-order example we can

evaluate the transfer impedance for the sixth-order network from

 

 

Faci+911 “Gin 'Gis
“G13 8Cg+Gga “638+VE;E;kI
211(3) '
‘Gia -G..q/C;C;k. sca+Gss
'Gis 'Gss‘/6;5;ks -G.¢~/C§5;ko
'61 s ”Gas 'V cscs *5 'Gsa'W (330th
_ 'Gis 'Gas “Gas
L.
_1
‘Gia ‘91s '51s1

~Ga.+¢CEC;k. 'Gss+VCaCs”ks ’Gss
"Gas‘W Cacaks 4» 'V Cscsfl‘s 'Gas

 

 

3°4+Gsa 'Gss+/C:63Wks “Gas
’Gssfi/UICsts 30s+Gss ’Gla
-G‘. ’Gs. 8C.+G'.d 11 (4 23)

The size of this matrix greatly discourages an analytical
study of the pole and zero locations in the general case. Also,
in a manner similar to the fourth-order network, if the numerator
has complex roots they tend to cancel the complex poles. It is

therefore desirable to eliminate from the numerator all the terms

79
involving a2 and higher. Since the network is tightly coupled
through the dependent generators (see Figure 4-4) it is possible
to let all the cij-o except one to each of the nodes 1 and 6.
This will greatly increase the possibilities of removing the

powers of s from the numerator. Thus, in place of (4-23) write

 

 

 

 

rhcz+911 0 -G1.
0 sc.+c., c,c,k,
0 c.c,k, sc,+c,,
211(3) -
'Gia fi/E;Esks q/C;C;k.
0 fi/E;5sts lfiifilwk.
_ o o o
o o o ‘ "fl
mic. mac. 0
/E;E;ks 'VEgEsts 0
9c¢+Gs6 /E:Es‘ks "Gas
c,c,wk, sons... 0
’Gas 0 8Qs+ng
“ {4-24)
The numerator of 2.1(3) is seen from (4-24) to be given by
_ “Galena flTcm mwkfl
Nu(s) -9139“ mks £70.19 mm .
__‘/C;Es‘ks C‘C.wk, 9°s+Gas_

 

 

Examining the above determinant will reveal that there is
only one term involving 83. To ensure a real zero, this term can

be removed by choosing k5 = 0. Then

.3383 noouounuxuo on”. 3 oceans—.60 xuosuoc “Hanoi—«-939 .«uo .wam

 

 

8O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r o
:5 «E .5 «3 use I
H n H S + 9 NH e + g N. HM H q i. e HH 1
J_H :6 g
e m

 

 

 

 

 

81

Naa(s> ‘ Giacwsksks C,C‘[C.C,(w3-1)s + Wacaacs ' GssCa] (4'25)
The zero can be located at the origin by letting

Gas 3 V8392

Ca Ca

When k5=0, and k, = k. - kA/Z, the denominator can be
roughly approximated in factored form for sufficiently large
k by the expression

’fi(s) - K1(s+G_1_1_) (992;) (s + g; + (3!; + 91.92 + 13]

,[(s +c_,_,_+ 9.2+ 9.292 +w’k’]
4C3 “Cs 20.

(4-26)

The degree of approximation involved can be evaluated from the

numerical examples considered in Chapter V.

CHAPTER.V

NUMERICAL EXAMPLES AND RESULTS

1. INTRODUCTION

While sufficient conditions for the equivalence of a time-
varying and a fixed network have been proved to exist in Chapter II,
showing that two such networks do indeed have the same response to
a given input has a certain asthetic appeal. Such illustrations
are therefore included for both the fourth and sixth-order
GC networks discussed in Chapter IV. It has been necessary to do
this numerically due to the difficulty in solving time-varying
differential equations.

A particular fourthworder example has been worked from con-
ception to completion to show that time-varying equivalent
networks with useful properties do exist, at least in theory.

In addition, theԤ(t) matrix has been found for a particular
sixth-order example since the Operation was too cumbersome to
carry out in general form in Chapter IV. The programs for each

of these cases are described and listed in the appendices.

2. SYNTHESIS OF A FOURTH-ORDER.TIME-VARIINC CC NETWORK

To illustrate that a transfer function can be synthesized
to a good approximation within the context of the 211(s) functions
derived in Chapter IV we use the following example.

Let the transfer function to be synthesized be

2‘1 (3) "' (5'1)

 

s
(s+30)(s+90)[(s+2)5 + 2‘]

82

83
Referring to the numerator Ng1(s) of (4-17) it can be seen
that in order to have a zero at the origin with k positive,
either 61, or 63‘ must be zero. Letting c,,=o, the zero will be

at the origin if

k ' 922334

Gem

Turning now to the denominator, it can be concluded directly

from (4‘12) that

.99
9.;
II
(JD
0
Q
.09
0
l
\O
O
U
5"
I
N
m
D
0..

we

Let c“ - 6,, and c; 0,. Then

Ca

Values compatible with the above relationships are now
selected such as to insure dominance of the éfit) matrix. The
following choices are satisfactory, (all values are in mhos

or farads):

611 - 39 G“ 3 9, C1 3 C‘ - 0.1

84

The state-variable equations for the time-varying GC network

corresponding to this choice of values and k=2 are, from

equations (4-4)

 

 

 

 

 

 

3210:)“ rM30 153 150 o .. _ x, (t) l
x.(t) .755 «2 0 .75(C+S) . x,(t)
i3(t) . .75C 0 -2 .7S(C-S) x,(t)
_x,(t1 _ 0 15(C+S) 15(C-S) -90 d _ x‘(t).
—10is(tf
0
+ ’ (5-2)
0
_ 0 -

 

 

where S = sin2t and C a cosZt.
These equations and the state equations of the equivalent
fixed network have been solved numerically for x.(t) with the

source current

18(t) = cosl.2t.

The Fortran IV program listing and the computed responses
are included in Appendix A. The slight differences observed are
less than the error specification of the integrating subroutine.

The time-varying network realization of equations (5-2) has
been introduced in Figure 4-2 in Chapter IV. Likewise, the
equivalent fixed network is shown in Figure 4-3. The poles and
zeros of this network have been computed using an analysis
program (see Appendix A). Of particular interest is the transfer

impedance, found to be

85

241(9) 3 (5'3)

 

8
(s+30.4)(s+90.2)[(s+1.6§3’ + 1.9f'T

This compares quite favorably with the nominal values in
equation (5-1), the greatest deviation being in the real part
of the complex pole pair. This is not unexpected since k=2 is
a relatively low value and (5-1) represents an approximation (in
the denominator) good only for sufficiently large k.

The magnitude of 241(jw) is plotted in Figure 5-1 on a log
frequency abcissa for k a 2, 8, and 32. The capacitances Ca and
C, have been adjusted in each case to maintain a zero at the
origin. The deterioration of the symmetry as k increases is
due to the relative decrease in the magnitude of the real poles.

This could be corrected by decreasing the values of C1 and Ca.

3. SYNTHESIS OF A SIXTH-ORDER.TIHE-VARYING CC NETWORK

As a numerical example of the application of the synthesis
equations for the sixth-order network developed in Chapter IV,
consider the problem

2.1(3) II K8 o
(3+70)‘[(s+4)a + 32][(s+10)a + 128]

 

Referring to equations (4-25) and (4-26), the above

expression for 2.1(8) is seen to result if:

cz‘cla-laGIIEGVV‘7

“3-688'6‘4’43
6.. - 16, (to locate the zero at the origin)

c. - c, - c, - c, - 1

86

 

 

 

0015 I _________ _ 31
I
I
l
I
I
I
.01 ___ _ _
k=2 |
I
l
Izsxl ;
I
I
I
.005 g ,_ _ m _ _I
I
I
I
I
I E
I I
' I
I .
.1 l 10

log f

Fig. 5-1. Frequency response of the fourth-order example.

87

kfi-oaksghg4a

k - 4/2 and w a 2 .
The computed transfer impedance for the above values is
shown in List B-l of Appendix B to be

291(8) 8 K8
(3+7o . 7) (3+7o .5) [(s+3. 7)3+31 .6][(s+9. 7)‘+114.8]

 

The equivalence of the time-varying and invariant sixth-
order example networks has been verified with a program very
similar to that used to solve the fourth-order differential
equations. Due to the large number of terms involved in the

expansion of
T
~Mt) - 93mm

this operation was programmed as shown in List B-2 of

Appendix B. In the particular example illustrated none of the
initial A11 were made zero and the kij all were equated to unity.
Calculations were made for the rational multiplier w-2,3,4,5.
and ;, Since'éfit) is known to be periodic the Fourier series
coefficients for each Aij(t> were evaluated. Values for the
'typical entries A.g(t), A,.(t), A,.(t) and Ag.(t) are given

for each value of w in Table 5-1. Two observations deserve
attention. The number of harmonics for all integral values of w
appears to be at most four. For non-integer values of w there
is an unexpectedly rich harmonic content even for the relatively
simple ratio %, It has not been determined whether this is due

to the imprecision of the routine for evaluating the Fourier

coefficients.

88

 

 

 

 

 

 

a A
mooo. Haoo. mace. maoo. mace. muoo. mace.“ mmoo. mac. «sec. «moo. «\n
o o o o o name. o o o ooeo. o n
. o o o o o o ammo., o .o some. o e
w o o o o o o o m name. o some. o m
m o M o o o o o o w o ammo. some. o N
_ H 0N4
_I I I
i once. « mace. mace. Hwoo._ muoo. «Noe. mNo. flee. Name. mo. «do. «\m
. so. . o oo. o sumo. o some. o o o o n
m o _ o o o o sumo. o some. o o o e
w o o o o oo. o sumo. o ammo. o o n
J o o o o o o me. name. o ammo. o N
. mm
M a
nmoo. emoo. ooo. Boo. mmoo. moao. «sac. nno. man. sees. no. N\«
keno. o o o «no. o ”me. o ans. 0 ooao. n
o o Reno. o o nmo. o one. and. o mono. e
_ o o o o News. o «me. o H. o ooao. n
_ o o o o o o Reno. ammo. and. fine. mono. N nu
_ _ . a
mooo. mooo. Naoo. “ace. euoo. «co. nmoo. Hues. nee. some. me.“ ~\«
memo. o o o oooo. e ”on. o nae. o m~.a n
o 0 meme. o 0 some. o Hoe. an”. o m~.H e
o o o o gene. a sons. 0 Ham. o nu.a m
o o o o o o name. see. ans. Hoe. n~.H- N
«Na
VI III I . I
soon has one one new new one can ecu one .o.o a. mmmmm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

xauuma Auvfl_uovuoInux«m man no moauuoo Hosanhu you

munowoamwooo moauom uoauaomII.HIn ugnda

 

89

The time-varying differential equations written in terms
of the Fourier coefficients are solved in List B-3 of Appendix B.
This particularly involved example was chosen in an effort to
provide a convincing test of the sufficiency of the constraints.
The solutions of the time-varying and equivalent invariant
equations are again listed together so they can be easily
compared. The discrepancies between the two solutions are
somewhat larger than in the fourth-order example. This can be

attributed to the inexactness of the Fourier series representation

of ~g(t) .

4. CONCLUSION

The existence of time-varying networks with selected
invariant network functions has been demonstrated. Sufficient
conditions for a class of such networks have been given as
constraints on the time-variation of the network components
in the form of linear time-varying differential equations.

There is considerable motivation for establishing conditions
which are, to some degree, necessary. Particularly one would
like to be able to answer the question, is there not some simpler
equivalent network? A great deal of unsuccessful effort was
directed toward attempting to show constraint equations (2-4)
and (2-5) to be necessary for the validity of

33(2.t) - some») (2-19)

oz °

On the other hand, a counter-example could not be found.

90

A counterwexample to the necessity of the constraints
derived for continuously equivalent invariant networks in
Chapter I has appeared in the literature [17]. The example
does not fall within the class K, however, since the network
contains a cutset of inductors. Thus, it is fair to say that
the question of necessity remains unanswered.

The constraint equations have been shown to have solutions
which yield realizable (albeit none too practical) solutions of
linear, passive time-varying networks which are equivalent to
invariant networks. Further, we have seen that such networks
can be used to realize complex poles using only resistances and
capacitances. There has been no attempt to construct physical
examples of such networks beyond simulating their existence on

the digital computer.

91

List of References

Howitt, N. "Group Theory and Electric Circuits," Physics

Review 37:1583-1595. 1941.

Guillemin, E. A. "Synthesis of Passive Networks,"

J6hn.Wiley & Sons,_lnc,, New York, N. Y., 141-157, 1957.

Schoeffler, J. D. "Continuously Equivalent Networks and
their Application," Case Institute of Technolo , Rep. No. S,

O.N.R. Contract Nonr-114l (10), Dec., 1962.

Schoeffler, J. D. "The Synthesis of Minimum Sensitivity
Networks," I.E.E.E. Transactions on Circuit Theory CT-ll,

No. 2, June, 1964.

Leeds, J. V., and Urgon, G. I. "Simplified multiparameter

Sensitivity Calculation and Continuously Equivalent Networks,"

I.E.E,E. Transactions on Circuit Theogy CT-l4, No. 2,
June, 1967.

Calahsn, D. A. "Equivalence of Time-Varying Systems,"

groc. I.F.E.E. 52, No. 11, November, 1964.

Franks, L. E., and Sandberg, I. W. "An Alternative Approach
to the Realization of Network Transfer Functions: the N-

Path Filter," Bell System Technical Journal 39, No. 5,
September, 1960.

Saraga, W. "A New Class of Time-Varying Filters," Electronic

Letters 3, No. 4, April, 1967.

10.

11.

12.

13.

14.

15.

92
Anderson, B. D. 0., and Newcomb, R. W. "On Reciprocity

and Time-Variable Networks," Proc. I.E.E.E. 53, No. 10,

October, 1965.

Anderson, B. D. 0., and Newcomb, Re W. "A Capacitor-
Transformer Gyrator Realization," Proc. I.E.E2§. 53,

No. 10, October, 1965.

Desoer, C. A., and Wong, K. K. "Constant Resistance
One-Ports which Include Non-Linear Time-Varying ;

Elements," I.E,E.E. Transactions on Circuit Theogy
CT-13, No. 4, December, 1966.

 

Silverman, L. M., and Meadows, H. E. "Controllability
and Unilateral Time-Varying Networks," I.E.E.E.

Transactions on Circuit Theogy CT-lZ, No. 3,
September, 1965.

Sen, J. K. "On the Equivalence and Stability of
Linear Time-Varying Networks," Ph.D. Thesis,

University of Illinois, 1965.

Bellman, R. "Introductions to Matrix Analysis,"

McCraw-Hill Inc., New York, N. Y., 175, 1960.

DeRusso, P. H., Roy, R. J., and Close, C. N. "State

Variables for Engineers," John Wiley and Sons, Inc.,
New York, No Yo. 364-366, 19650

 

16.

17.

18.

19.

93

Weinberg, L. "Network.Ana1ysis and Synthesis," McGraw-

Hill Inc., New York, N. Y., 366-68, 1962.

Newcomb, R. W. "The Non-Completeness of Continuously
Equivalent Networks," I.E.E.E. Traggactions on Circuit

Theory 01-13, No. 2, June, 1966.

Frame, J. S. Textbook to be published.

Pottle, C. "Comprehensive Active Network.Ana1ysis by
Digital Computer - A State-Space Approach," Proceedings
Third Annual Allerton Conference on Circuit and System

Theo , 659-668, October, 1965.

 

u ¢- v
o

 

 

APPENDIX A

This section contains examples of the computer print-out
associated with the computations for the fourth—order network.

The program and output of the numerical solutions of the
differential equations for both the time-varying and fixed
networks shown in Figures 4-2 and 4-3 reapectively are given
in List A-l. The differential equation solver DEQ is a
standard library subroutine and is therefore not included.

It should be noted that the two networks are solved
separately, the solution of the time-varying network being
completed before the program branches into the routine for
the invariant network. The outputs are stored, then listed
side by side at regular intervals so they can be readily
compared.

The poles, zeros and frequency response of the equivalent
fixed network were computed with the aid of an analysis program.
This particular program uses the state variable analysis approach
developed by C. Pottle [19] as its core. The frequency response
feature and the facility for stepping selected components was
added by D. A. Calahsn. Just the data input and the computed
output are shown in List A-2. The node numbers identifying the
components correspond to the numbering in Figure 4-3.

Perhaps it is worth remarking that the unrealistic
resistance and capacitance values have been chosen to accommodate
the analysis program which requires that the input data be
normalized. There would be no difficulty in translating the

example into something with a more useable impedance level.

94

50

200

LIST A-l

Comparison of the Responses of the Equivalent
Time-Varying and Fixed Fourth-Order Networks

DIMENSION V(60),DV(60),VV(1000),VF(1000),TIME(lOOO)
READ(S,2) AK,W,CYCLES
FORMAT(3F10.5)
WRITE(6,SO)AK,W,CYCLES
F0RMAT(///.3F20-5.///)
PD=6.2831853/AK

T IF-PD*CYCLES

NV=4

JUMPa-l

T=0.

TLIM=.OOOOl

TMAX-.02*PD

ERROR=.001

DT=.000001

INC=O

DO 200 I-l,4

V( 1) =0 .

CALL DEQ(DV,T,TLDH,V,ERROR,NV,DT,TMAX,JUMP)
IF (JUMP)4,5 ,6
FA-15.*SIN(AK*T)

FB= . 75*SIN(AK*T)

FC= . 75*cos (AKfl?)

FD=FB+FC

FEaFc-FB

FF=20.*FD

FG=20.*FE
DV(1)=-30.*V(l)+FA*V(2)+20.*FC*V(3)+10.*COS(l.2*T)
DV(2)=FB*V(1)-2.*V(2)+FD*V(4)
DV(3)=FC*V(1)-2.*V(3)+FE*V(4)
DV(4)=FF*V(2)+FG*V(3)-90.*V(4)
GO TO 3

INC=INC+1

VV(INC)=V(1)

TIME(INC) =1:

TLIM=T+TMAX

IF(T-TIF)3,3,4

DO 8 1:1,4

V(I)=0.

INC-=0

NV=4

JUMP-~1

T=0.

TLIMP.00001

TMAX=.02*PD

DT=.000001

INC=O

ERROR=.001

95.

96

LIST A-l (continued)

7 CALL DEQ(DV,T,TLIM,V,ERROR,NV,DT,THAX,JUMP)
IF(JUMP)12,10,11
10 DV(l)--30.*V(l)+15.*V(3)+10.*COS(1.2*T)
DV(2)--2.*V(2)-AK*V(3)+.75*V(4)
DV(3)c.7S*V(1)+AK*V(2)-2.*V(3)+.75*V(4)
nv<4)=15.*V(2)+15.*V(3)-90.*V(4)
co TO 7
11 INC-INC+1
VF(INC)=V(1)
TLIM=T+TMAX
IF(T-TIF)7,7,12
12 WRITE(6,13)
13 FORMAT(SX,'TIME-VARYING FIXED TIME',/)
NTHBINC
WRITE(6,14)(VV(I),VF(I),TIME(I),I-1,NTH)
14 (FORMAT(5X,1F11,8,SX,1F11.8,4X,1F8.S)
IF(AK—100000.)15,15,16
16 STOP
END
TIME-VARYINC FIXED TIME
0.00009998 0.00009998 0.00001
0.28425354 0.28425360 0.06284
0.33069247 0.33069265 0.12567
0.33963645 0.33963645 0.18851
0.34070933 0.34070927 0.25134
0.33832592 0.33832598 0.31417
0.33317292 0.33317280 0.37700
0.32540971 0.32540953 0.43983
0.31513971 0.31513941 0.50266
0.30247587 0.30247545 0.56550
0.28754705 0.28754652 0.62833
0.27049714 0.27049649 0.69116
0.25148237 0.25148153 0.75399
0.23066908 0.23066789 0.81682
0.20823079 0.20822990 0.87966
0.18434858 0.18434793 0.94249
0.15920693 0.15920675 1.00532
0.13299251 0.13299209 1.06815
0.10589516 0.10589457 1.13098
0.07810563 0.07810563 1.19381
0.04981338 - 0.04981143 1.25664
0.02120080 0.02119996 1.31947
-0.00754198 -0.00754335 1.38230
-0.03623562 -0.03623771 1.44514
-0.06470263 -0.06470609 1.50797
-0.09276915 -0.09277099 1.57080
-O.12026739 -0.12026864 1.63363
-O.14703560 -0.14703858 1.69646

 

v_i_¢h"

 

97

LIST A-l (continued)

TINE-VAR!ING FIXED TIME
-0.17291749 -0.17291921 1.75929
-0.19776529 -0.19776642 1.82212
-0.22143573 -0.22143704 1.88495
-0.24379802 -0.24379772 1.94778
-0.26472384 -0.26472467 2.01061
-0.28409904 -0.28410023 2.07344
-0.30181921 -0.30181795 2.13628
-0.31778318 -0.31778193 2.19911
-0.33190680 -0.33190596 2.26194
-0.34411353 -0.34411210 2.32477
-0.35433823 -0.35433596 2.38760
-0.36252624 -0.36252362 2.45043
-0.36863470 -0.36863196 2.51326
-0.37263304 -0.37262994 2.57609
-0.37450147 -0.37449771 2.63892
-O.37423217 -0.37422788 2.70175
-0.37182933 -0.37182450 2.76458
-0.36730844 -0.36730385 2.82742
-0.36069793 -0.36069256 2.89025
-0.35203624 -0.35203075 2.95308
-O.34137452 -0.34l36844 3.01591
-0.32877326 -0.32876778 3.07874
-0.3l430703 -0.31430048 3.14157
-0.29805732 -0.29805118 3.20440
-0.28011686 -0.28011078 3.26723
-0.26058817 -0.26058125 3.33006
-0.23958266 -0.23957688 3.39289
-0.21721900 -0.21721458 3.45572
-0.19362497 -0.19361830 3.51856
-0.16893321 -O.16892904 3.58139
-0.14328700 -0.14328194 3.64422
-0.11682689 -0.11682367 3.70705
-0.08970910 ~0.08970475 3.76988
-0.06208248 -0.06207822 3.83271
-0.03410530 ~0.03410143 3.89554
-0.00593642 -0.00593160 3.95837
0.02226517 0.02226852 4.02120
0.05033797 0.05034043 4.08403
0.07812375 0.07812536 4.14686
0.10546458 0.10546726 4.20970
0.13220567 0.13220769 4.27253
0.15819496 0.15819770 4.33536
0.18328500 0.18328649 4.39819
0.20733410 0.20733446 4.46102
0.23020369 0.23020434 4.52385
0.25176585 0.25176579 4.58668
0.27189749 0.27189785 4.64951
0.29048419 0.29048365 4.71234

 

LIST A-l (continued)

TIME-VARYING

0.30742049
0.32261032
0.33596742
0.34741533
0.35688978
0.36433643
0.36971331
0.37298989
0.37414736
0.37317890
0.37009019
0.36489874
0.35763371
0.34833759
0.33706236
0.32387114

98

FIXED

0.30741954
0.32260895
0.33596569
0.34741312
0.35688680
0.36433321
0.36970961
0.37298566
0.37414247
0.37317371
0.37008476
0.36489320
0.35762846
0.34833163
0.33705503
0.32386535

TIME

4.77517
4.83801
4.90084
4.96367
5.02650
5.08933
5.15216
5.21499
5.27782
5.34065
5.40348
5.46631
5.52915
5.59198
5.65481
5.71764

 

'. mb’n—‘u

LIST A-2

Transfer Function and Frequency Response of the Fourth-
Order Network as Computed by 8 Linear Analysis Program [19]

W POLES AND ZEROS OF A FOURTH ORDER PROBLEM, K-l,2,4,8,16,32
THE GRAPH OF THIS NETWORK Is DESCRIBED BY THE FOLLOWING BRANCHES

BRANCH NODE OUT CON- ELEMENT CNTRL
LABEL NOS. PUT TROL VALUE BRANCH
110 1 0 0.0

C10 1 0 V 1.0000E-01

C40 4 0 V 1.0000E-01

R10 1 0 6.6667E-01

R13 1 3 6.6667E-01

C20 2 0 2.00008 00

C30 3 0 2.0000E 00

R20 2 0 4.00008-01

120 0 2 -4.0000E 00 C30
I30 0 3 4.0000E 00 C20
R24 2 4 6.6667E-01

R34 3- 4 6.6667E-01

R30 3 0 1.0000E 00

R40 4 0 1.6667E-01

TRANSFER FUNCTION NUMERATOR POLYNOMIALS

OUTPUT VARIABLE - V
SOURCE VARIABLE -

POLYNOMIAL
COEFFICIENTS

-1.0000003E 01
-9.4000293E 02
-3.4550010E O3
-6.7498828E 03

OUTPUT VARIABLE - V
SOURCE VARIABLE -

POLYNOMIAL
COEFFICIENTS

-1.1ZSOOZOE 02
4.8828125E-03

C10
110

DEGREE
IN S

OHNW

C40
110

DEGREE
IN S

1
O

99

ZERO POSITIONS
REAL PART IMAGINARY PART

-9.0254364E 01 0.0

'1.8725891E 00 1.99300868 00
-1.8725891E 00 -1.9930086E 00

ZERO POSITIONS
REAL PART IMAGINARY PART

4.3402688E-05 0.0

LIST A-2 (continued)

100

TRANSFER FUNCTION DENOMINATOR POLYNOMIAL

POLYNOMIAL
COEFFICIENTS

1.0000000E 00
1.2400031E 02
3.1542598E 03
1.0004984E 04
1.8352438E 04

FREQ(CPS)

0.0100000
0.0158489
0.0251188
0.0398106
0.0630955
0.0999992
0.1584881
0.2511864
0.3981037
0.6309500
0.9999883
1.5848722
2.5118542
3.9810095
6.3094730
9.9998312
15.8486633
25.1183624
39.8099213
63.0943909
99.9978943
158.4855042
251.1826019
398.0969238
630.9414062
999.9714355

DEGREE

IN S

C>hlnauac~

FREQUENCY RESPONSE

REAL PART

-9.025543ZE 01
-3.0391739E 01
-l.676579SE 00
-1.676579SE 00

OHMS

0.0003852
0.0006106
0.0009680
0.0015355
0.0024384
0.0038806
0.0061796
0.0095666
0.0121691
0.0100245
0.0064596
0.0039238
0.0022808
0.0012220
0.0005767
0.0002333
0.0000807
0.0000244
0.0000067
0.0000018
0.0000004
0.0000001
0.0000000
0.0000000
0.0000000
0.0000000

POLE POSITIONS
IMAGINARY PART

1.9696894E 00
-1.9696894E 00

PHASE(DEG)

'91.924
'93.087
'94.921
“97.833
-102.493
-110.082
-122.873
-145.538
-183.331
-224.122
-252.938
85.417
64.954
42.771
18.617
5.970
-28.910
-48.118
-62.463
-72.306
-78.750
-82.880
-85.502
-87.160
-88.208
-88.869

 

IT "

APPENDIX B

Two of the programs in this section, those in Lists
B-1 and B-3 are essentially those described in Appendix A
adapted to the sixth-order problem. There is, in addition,
the program in List B-2 which evaluates g(t) in Fourier
series form by expanding g(t)g(0)gr(t). In the process it
also checks g(t) for dominance for all t. All of the

Operations are straightforward and the subroutines standard.

101

l "

 

IT

LIST B-l

Fortran Program and Example Output for the Computation of the
Time-Varying Differential Equations for the Sixth-Order Example

DIMENSIONQ(6,6,52),Y(60),CT(20),ST(20)
DIMENSIONA(50,50),B(6,6),ROOTR(50),ROOTI(50),C(6,6),D(6,6)
DIMENSIONES (6,6) ,E(6,6) ,CS(6,6) ,P(6,6) ,SUB(6)
READ(592)((B(IOJ)9J-196)OI-196)
2 FORMAT(6F10.5)
17 READ(5,3)R,W
3 FORMAT(2F10.5)
IF(R°100.)5,5,18
5 ‘WRITE(6,4)R,W
4 FORHAT(///,1X,2F10.5)
RTHBSQRT(3.)
DO 8 I'l,6
D0 8 J31,6
8 D(I,J)=0.
W-W/RTH
D(2,3)¢W
D(2 ,4) =11
D(2,5)=R*W
D(3,2)=-W
D(3,4)-W
D(3,5)'-R*W
D(4,2)=-W
D(4,3)=-W
D(4,5)=R*W
D(5,2)'-R*W
D(5,3)=R*W
D(5,4)=-R*W
WiW*RTH
D0 9 131,6
D0 9 J81,6
E(I,J)-B(I,J)-D(I,J)
9 A(I,J)-E(I,J)
CALL'HPRINT(A,6,2,4HA “3,50)
CALL HESSEN(A,6)
CALL QREIG(A,6,ROOTR,ROOTI,1)
DT=6.2831853/(W*50.)
T'0.-DT
K-0
6 T=T+DT
KPK+1
Cl-COS(W*T)
CZBCOS (RW’VI')
SI‘RTH*SIN(W*T)
SZHRTH*SIN(R*W*T)
DO 7 I‘I,6
D0 7 J31,6
7 C(I,J)=O.

102

103

LIST B-l (continued)

10

11

12

15

14

31

21

22

23

24

C(1,l)-1.
C(2,2)=(2.*C1+C2)/3.
C(2,3)-(C1-C2+Sl)/3.
C(2,4)=(-C1+C2+Sl)/3.
C(2,5)-82/3.
C(3,2)=(C1-02-81)/3.
C(3,3)-(2.*Cl+C2)/3.
C(3,4)-(Cl-C2+Sl)/3.
C(3,5)-(-SZ)/3.
C(4,2)-(-C1+Cz-SI)/3.
C(4,3)-(C1-CZ-Sl)/3.
C(4,4)-(2.*C1+C2)/3.
C(4,5)-S2/3.
C(5,2)=(-s2)/3.
C(5,3)-s2/3.
C(5.4)-(~32)/3.
C(5,5)=Cz

C(6,6)-1.

D0 10 I-l,6

DO 10 J-l,6
CS(I,J)-C(I,J)
ES(I,J)-E(I,J)

CALL HAMULT(CS,ES,P,6,6,6,6)
DO 11 I-1,6

DO 11 141,6
CS(J,I)-C(I,J)

CALL MANULT(P,CS,ES,6,6,6,6)
DO 12 I-1,6

D0 12 J=1,6
P(I,J)-ES(I,J)+D(I,J)
Q(I.J.K)=P(I.J)

DO 14 I-l,6
SUB(I)-2.*ABS(P(I,I))
DO 15 J-1,6
SUB(I)-SUB(I)-ABS(P(I,J))
CONTINUE
IF(T-6.2/W)6,31,31
D0 20 I-1,6

D0 20 J=1,6

DO 21 x-1,50
Y(K)=Q(I.J.K)

DC-O.

D0 22 R=1,50
DC-DC+Y(K)

DC--DC/50.
WRITE(6,23)I,J
F0RMAT(///////.1X,212.//)
WRITE(6,24)DC
FORMAT(5X,'THE DC TERM IS', IF10.5)
DO 25 v-1,16

CT(N)-O.

 

104
LIST B-l (continued)

ST(N)-0.
DO 26 x:1,50
EN=N
AKeK-l
ARG-ENsAR*6.28318/50.
CT (N) =CT (N) +Y (11) *COS (ARC)
26 ST(N)-ST(N)+Y(K)*SIN(ARC)
CT(N)-CT(N)/25.
25 ST(N)-ST(N)/25.
WRITE(6,27)
27 FORMAT(1X,//)
WRITE (6 , 28)
28 FORMAT(1X,'THE FIRST THROUGH SIXTEENTH COS HARNONICS ARE')
WRITE(6,29)(CT(N),N-1,16)
29 FORMAT(/.1X,16F7.4,///)
‘WRITE(6,30)
3o FORMAT(1X,'THE FIRST THROUGH SIXTEENTH SIN HARNONICS ARE')
WRITE(6,29)(ST(N),N-1,16)
20 CONTINUE
GO TO 17
18 STOP
END

2 3
THE DC TERM IS 0.02039
THE FIRST THROUGH SIXTEENTH COS HARMONICS ARE
0.0101 0.1185 0.0143-0.0072-0.0051-0.0042-0.0037-0.0035
-0.0033-0.0032-0.0031-0.0030-0.0030-0.0030-0.0029-0.0029
THE FIRST THROUGH SIXTEENTH SIN HARMONICS ARE

-0.0404 0.0888-0.0319-0.0122-0.0090-0.0071-0.0059-0.0050
-0.0043-0.0037-0.0033-0.0029-0.0025-0.0022-0.0020-0.0017

 

UN

LIST B-2

Comparison of the Responses of the Equivalent
Time-Varying and Fixed Sixth-Order Networks

DIMENSIONV(60),DV(60),VV(1000),VF(1000),TINE(1000)
READ(5,1)R,W,SOURCE,CYCLES
FORMAT(4F10.5)
PD-6.2831853/w
TIF-PD*CYCLES

Nv=6

JUHP=~1

T80.

TLIN=.OOOO1
TMAX-.01*PD
ERROR-.0001
DT-.000001

INC-0

DA--.9

DB-.03333333
EA-.0577
DD-.01666666
EB-.0289
DE-.06666666
EC-.O962

DF-.05

ED-.1347
DG=~1.2333333
EE-.0096
DH-.13333333
EF-.0385

DI-.1

DJ-.0833333

EG-.0770

DR--1 . 2

EH-.O481

EI-.0192
EJ-.08666666

D0 2 I-1,6

V(I)-0.

CALL DEO(Dv,T,TLIH,V,ERROR,NV,DT,THAX,JUHP)
IF (JUMP) 4 , 5 , 6

CA=COS (W*T)
CB=COS(2.*W*T)
CC-COS(3.*W*T)
CD-COS(4.*W*T)
SA-SIN(W*T)

SB=SIN (2 . *W’Vl‘)
SC-SIN(3.*W*T)
SD-SIN(4.*W*T)
TP-DF*CB-EB*SB
TA-DB*CA+DD*CB+EA*SA+EB*SB
TB-DE*CA-DD*CB-EB*SB

105

 

106

LIST B-2 (continued)

10

TC-DB*CA+DD*CB-EA*SA+EB*SB

TD-DF*CB-EB*SB
TE=DB*(CB-CA-CC-CD)+EC*SA+ED*SB-FA*SC+EE*SD+DG
TF--DF*CA+DI*CB-DD*CC+DB*CD-EE*SA+EC*SB+EB*SC-EE*SD-DD
TG-DB*(-CA-CC-CD)+DH*CB+EF*(SA-SB)+EE*SD+DD
TH-DG*CA-DF*CC+DD*CD-EB*SA+EB*SA+EB*SC+EA*SD
TI-DE*CA+DH*CB+DE*CC-DB*CD-FG*SA-EF*SB+EE*SD+DG
TJ-DD*CA+DB*CB-DD*CC+DB*CD+EH*SA-ED*SB-EB*SC-EE*SD-DD
TKPDE*CA-DD*CD+EA*(SA+SC-SD)
TL--DI*CA-DI*CB-DB*CC-DB*CD-EI*SA-EC*SB+EA*SC+EE*SD+DG
The-DD*CA+DF*CC+DD*CD+EJ*SA+EB*SC+EA*SD
TN=DI*CD-EB*SD+DK

DV(1) -- .9*v (l)+TA*V (2)+TB*V(3)+TC*V (4) +TD*‘V (5)+SOURCE*SIN

1(.65*T)

Dv (2) -TA*V (1)+TE*V (2)+TF*V (3) +TG*V (4) +TH*V (5) +TA*V (6)
Dv (3) -TB*V (l)+TF*V (2) +TI*V(3)+TJ*V (4)-+wa (5)+TB*V (6)
DV (4) -TC*V (1) +TG*V (2)+TJ*V (3) +TL*‘V (4) +TM*V (S)+TC*V (6)
Dv (5)-TD*V(1)+TH*V(2)+TK*V(3)+TM*V (4)+TN*V (5)+'rF*v (6)
Dv (6)-TA*V(2)+TB*V(3)+TC*V (4)+TP*V (5) -1 . 1W (6)

GO TO 3

INC-1NC+1

VV(INC)-V(6)

TIME(INC)=T

TLIHeT+THAx

IF(T-TIF)3,3,4

D0 8 1:1,6

V(I)=0.

ZA=28.8175232

23-28.9l75232

20:57.6850464

ZD-57.7850464

INC-0

NV-6

JUMPa-l

T20.

TLIM-.00001

TMAX-.01*PD

ERROR-.0001

DT-.000001

INC-0

CALL DEQ(DV,T,TLD4,V,ERROR,NV,DT,TMAX,JUMP)
IF(JUMP)12,10,11
DV(1)--.9*V(1)+.05*V(2)+.05*V(3)+.05*V(4)+.05*V(5)+SOURCE

1*SINW*T)

DV(2)-.05*V(1)-1.3*V(2)-ZA*V(3)-ZA*V(4)-ZC*V(5)+305*V(6)
DV(3)-.05*V(l)+ZB*V(2)-l.*V(3)-ZA*V(4)+ZD*V(5)+305*V(6)
DV(4)-.05*V(l)+ZB*V(2)+ZB*V(3)-1.5*V(4)-ZC*V(5)+.05*V(6)
DV(5)-.05*V(1)+ZD*V(2)-ZC*V(3)+ZD*V(4)-1.1*V(5)+u05*V(6)
DV(6)=.05*V(2)+.05*V(3)+.05*V(4)+.05*V(5)-1.1*V(6)

GO TO 7

107

LIST B-2 (continued)

11 INc-INC+1
VF(INC)-V(6)

TLIMdT+THAX
IF(T-TIF)7,7,12

12 WRITE(6,13)

13 FORMAT(sx,'TIHE-VARTINC FIXED TIHE',/)
NTHaINC
‘WRITE(6,14)(VV(I),VF(I),TINE(I),I-1,NTH)

14 FORHAT(SX,1F11.8,5X,1Fll.8,4X,1F8.5)
IF(SOURCE-100000.)15,15,16

16 STOP

TIME-VARYING FIXED

0.00000000 0.00000000
0.00000000 0.00000000
0.00000001 0.00000001
0.00000003 0.00000003
0.00000009 0.00000009
0.00000021 0.00000021
0.00000043 0.00000043
0.00000080 0.00000080
0.00000135 0.00000135
0.00000215 0.00000215
0.00000326 0.00000326
0.00000473 0.00000473
0.00000665 0.00000665
0.00000908 0.00000907
0.00001210 0.00001209
0.00001579 0.00001577
0.00002023 0.00002021
0.00002550 0.00002547
0.00003169 0.00003164
0.00003887 0.00003880
0.00004712 0.00004702
0.00005652 0.00005640
0.00006714 0.00006699
0.00007905 0.00007888
0.00009232 0.00009212
0.00010700 0.00010679
0.00012316 0.00012294
0.00014084 0.00014062
0.00016010 0.00015988
0.00018097 0.00018076
0.00020349 0.00020330
0.00022769 0.00022752
0.00025360 0.00025346
0.00028122 0.00028111
0.00031058 0.00031050
0.00034168 0.00034162

TIME

0.00001
0.00127
0.00252
0.00378
0.00504
0.00629
0.00755
0.00881
0.01006
0.01132
0.01258
0.01383
0.01509
0.01635
0.01760
0.01886
0.02012
0.02137
0.02263
0.02389
0.02514
0.02640
0.02766
0.02891
0.03017
0.03143
0.03268
0.03394
0.03520
0.03645
0.03771
0.03897
0.04022
0.04148
0.04274
0.04399

108

LIST B~2 (continued)

0.00255177

0.00256183

TIME-VARYING FIXED TIME
0.00037452 0.00037446 0.04525
0.00040908 0.00040902 0.04651
0.00044536 0.00044528 0.04776
0.00048332 0.00048321 0.04902
0.00052295 0.00052278 0.05028
0.00056420 0.00056395 0.05153
0.00060703 0.00060668 0.05279
0.00065138 0.00065091 0.05405
0.00069720 0.00069660 0.05530
0.00074443 0.00074368 0.05656
0.00079299 0.00079208 0.05782
0.00084281 0.00084174 0.05907
0.00089380 0.00089257 0.06033
0.00094588 0.00094451 0.06159
0.00099897 0.00099745 0.06284
0.00105297 0.00105132 0.06410
0.00110777 0.00110601 0.06535
0.00116329 0.00116145 0.06661
0.00121941 0.00121751 0.06787
0.00127603 0.00127411 0.06912
0.00133304 0.00133112 0.07038
0.00139032 0.00138845 0.07164
0.00144776 0.00144597 0.07289
0.00150525 0.00150356 0.07415
0.00156266 0.00156112 0.07541
0.00161986 0.00161850 0.07666
0.00167673 0.00167560 0.07792
0.00173314 0.00173226 0.07918
0.00178896 0.00178838 0.08043
0.00184406 0.00184380 0.08169
0.00189830 0.00189840 0.08295
0.00195155 0.00195203 0.08420
0.00200367 0.00200455 0.08546
0.00205452 0.00205583 0.08672
0.00210396 0.00210571 0.08797
0.00215183 0.00215406 0.08923
0.00219800 0.00220072 0.09049
0.00224232 0.00224555 0.09174
0.00228463 0.00228841 0.09300
0.00232480 0.00232915 0.09426
0.00236267 0.00236762 0.09551
0.00239809 0.00240368 0.09677
0.00243093 0.00243720 0.09803
0.00246104 0.00246802 0.09928
0.00248831 0.00249603 0.10054
0.00251259 0.00252108 0.10180
0.00253378 0.00254306 0.10305

0.10431

.‘M’bm

 

109

LIST B-Z (continued)

TIME-VARYING FIXED TIME
0.00256648 0.00257729 0.10557
0.00257782 0.00258933 0.10682
0.00258572 0.00259784 0.10808
0.00259012 0.00260275 0.10934
0.00259098 0.00260395 0.11059
0.00258825 0.00260138 0.11185
0.00258192 0.00259498 0.11311
0.00257195 0.00258468 0.11436
0.00255834 0.00257045 0.11562
0.00254105 0.00255226 0.11687
0.00252010 0.00253009 0.11813
0.00249546 0.00250392 0.11939
0.00246714 0.00247378 0.12064
0.00243512 0.00243967 0.12190
0.00239940 0.00240164 0.12316

LIST B-3

Transfer Function and Frequency Reaponse of the Sixth-
Order Network as Computed by 8 Linear Analysis Program [19]

***** POLES AND ZEROS OF A SIXTH ORDER PROBLEM DEC 8 1967
THE GRAPH OF THIS NETWORK IS DESCRIBED BY THE FOLLOWING BRANCHES

BRANCH NODE OUT CON- ELEMENT CNTRL
LABEL NOS. PUT TROL VALUE BRANCH
110 1 0 0.0

010 1 0 V 1.00008-01

R10 1 0 2.0000E-01

R13 1 3 5.0000E-01

C20 2 0 1.0000E 00

R20 2 0 2.5000E-01

C30 3 0 1.0000E 00

R30 3 0 5.0000E-01

C40 4 0 1.0000E 00

R40 4 0 5.0000E-01

050 5 0 1.0000E 00

R50 5 0 6.2500E-02

C60 6 0 V 1.0000E-01

R60 6 0 2.0000E-01

IA02 0 2 V 4.0000E 00 C30
1802 0 2 V 4.0000E 00 C40
IA03 0 3 V -4.0000E 00 C20
1803 0 3 V -8.0000E 00 050
IAO4 0 4 V -4.0000E 00 020
1804 0 4 V 8.0000E 00 C50
IAOS 0 5 V 8.0000E 00 C30
1805 0 5 V -8.0000E 00 C40
R46 4 6 5.0000E-01

TRANSFER FUNCTION NUMERATOR POLYNOMIALS

OUTPUT VARIABLE - V C10

SOURCE VARIABLE - IIO
POLYNOMIAL DEGREE ZERO POSITIONS
COEFFICIENTS IN 8 REAL PART IMAGINARY PART

-l.0000003E 01
-9.8000024E 02
-2.3200004E 04
-2.9535944E 05
-1.7497700E 06
-6.1433210E 06

-7.0588074E 01 0.0

-9.8598022E 00 9.46351532 00
-9.85980228 00 -9.4635153E 00
-3.8460369E 00 5.6395559E 00
-3.8460369E 00 -5.6385559E 00

OHkam

OUTPUT VARIABLE - V C60
SOURCE VARIABLE - 110

110

LIST B-3 (continued)

POLYNOMIAL

COEFFICIENTS

-1.9200004E
0.0

04

111

DEGREE
IN S

l
0

ZERO POSITIONS
REAL PART IMAGINARY PART

-0.0 0.0

TRANSFER FUNCTION DENOMINATOR POLYNOMIAL

POLYNOMIAL

COEFFICIENTS

1.0000000E
1.6800003E
9.1400000E
1.8817594E
2.1679330E
1.2236806E
4.0958416E

FREQ(CPS)

0.0010000
0.0015849
0.0025119
0.0039811
0.0063095
0.0099999
0.0158488
0.0251186
0.0398104
0.0630950
0.0999988
0.1584873
0.2511855
0.3981010
0.6309474
0.9999833
1.5848665
2.5118361
3.9809923
6.3094406
9.9997921
15.8485527
25.1182556
39.8097076
63.0941467

00
02
03
05
06
07
07

DEGREE
IN S

OHNwDMO‘

OHMS

0.0000029
0.0000047
0.0000074
0.0000117
0.0000186
0.0000295
0.0000467
0.0000740
0.0001173
0.0001861
0.0002955
0.0004706
0.0007544
0.0012253
0.0019996
0.0027378
0.0019614
0.0007856
0.0002135
0.0000470
0.0000087
0.0000013
0.0000002
0.0000000
0.0000000

POLE POSITIONS
REAL PART IMAGINARY PART

-7.0676743E 01
-7.0499939E 01
-9.70962SZE 00 .33116728 00
-9.70962528 00 -9.331167ZE 00
-3.70205128 00 5.623414OE 00
-3.70205128 00 -5.62341408 00

.0
.0

\OOO

PHASE(DEG)

-90.108
-90.171
-90.270
-90.428
-90.679
-91.076
-91.705
-92.702
-94.284
-96.794
“100.788
-107.178
-117.544
-134.926
-166.106
-223.701
60.831
-7.834
-63.775
-108.777
~148.596
-183.830
-212.049
-232.332
-245.930