a general matching theory and its application to tunnel diode amplifiers - ieee transactions - draft...

8/16/2019 A General Matching Theory and Its Application to Tunnel Diode Amplifiers - IEEE Transactions - Draft - 1966 - Pp.16

http://slidepdf.com/reader/full/a-general-matching-theory-and-its-application-to-tunnel-diode-amplifiers- 1/13

IEEE TRANSACTIONSON CIRCUIT THEORY

VOL. CT-13, NO. 1 MARCH 1966

A General Matching Theory and Its Application

to Tunnel Diode Amplifiers

Y. T. CHAN, MEMBER, IEEE, AND E.

s. KUH,

FELLOW, IEEE

Abstract-In this paper we generalize Youla’s theory of broad-

band matching to include both passive and active l-port load imped-

ances. The design philosophy of lossless coupling networks for an

active load is different from that of a passive load; however, the

design theory is quite similar. The theory is illustrated in a detailed

example with a tunnel diode as the active load. Both series resistance

and inductance are used in the equivalent circuit representation.

New results in terms of theoretical limitation on gain-bandwidth

and a sufhcient condition for potential stability are obtained.

plete treatment. The existing sta tus of tunnel diode

I.

INTR~D~TIoN

HE CLASSICAL problem of broadband matching

was first introduced by Bode in the study of

coupling network design for vacuum tubes [l],

Fano later extended Bode’s result to arbitrary passive

load impedances and derived a set of integral constraints

on the reflection coelhcient [a]. Fano’s method depends

on, as a crucial step, the Darlington equivalent repre-

sentation of a passive load. The design procedures for

the optimum matching network was worked out, but

only for a few classes of simple impedances. Recently,

Youla deve loped an alternative theory of broadband

matching which bypassed the step of finding the Darling-

ton equivalent, and consequently, was able to handle,

more complex impedances with relative ease [3]. The

present paper is a generalization of Youla’s work to

include both passive and active load impedances.

The design philosophy for obtaining a coupling net-

work for an active load is different from that for a passive

load; however, the theory involved and the design pro-

cedure are quite similar. These concepts are brought out

in Section II along with some preliminary notions of

importance. The necessary constraints on the complex

reflection coefficient and the sufficient conditions for

the realization of the coupling network are introduced

in Section III. The presentation is maintained in a simpli-

fied form de liberately so that readers can follow the

physical significance of each step without being bogged

down by the details which are necessary for the general

theory. As a matter of fact, the succeeding four sections

are devoted to a special, but important, example using

the tunnel diode to illustrate the main ideas, while the

general matching theory is postponed until Section VIII.

For those who are specially interested in the tunnel

diode amplifier problem, this paper gives a fairly com-

Manuscript received August 13, 1965; revised November 8, 1965.

This work was supported in part by the Joint Services Electronics

Program (U. S. Army, U. S. Navy, and U. S. Air Force) under Grant

AF-AFOSR-139-65 and by the National Science Foundation under

Grant GP-2684.

The authors are with the Department of Electrical Engineering,

University of California, Berkeley, Calif.

amplifier theory is not satisfactory, since all work on

gain-bandwidth limitation and optimum design are

based on a simplified equivalent circuit representation

which is valid only at low frequencies [4]-[6]. In the

present paper the complete equivalent circuit as shown

in Fig. 1 is used. Based on the complete equivalent cir-

cuit we derived the theoretical limitation on gain-band-

width and optimum coupling networks. As a by-product

we have shown that the necessary conditions for potential

stability due to Smilen and Youla are also sufficient

[71-POI.

Fig. 1. Tunnel diode equivalent circuit.

In Section IV we derive the simple constraints on the

complex reflection coefhcient from which the driving-

point impedance of the passive coupling network which

is faced by the tunnel diode is obtained. In Section V

we introduce the integral constraints on the reflection

coefficient

and establish the theoretical limitation on

gain-bandwidth. The approximation problem is discussed

in Section VI and a proof is given to show that the

necessary conditions for potential stability are sufficient.

In Section VII we discuss again the approximation prob-

lem and give an illustrative design of a Chebyshev-type,

low-pass, tunnel diode amplifier.

II. PRELIMINARY CONCEPTS

Consider the circuit in Fig. 2, where z is a given lumped

impedance which can be either passive or active, Z is a

passive impedance to be designed. Let p(s) be the com-

plex reflection coefficient and s be the complex frequency.

Then

Z(s) - z( 7s)

p(s) = Z(s) + z(s) *

0)

Let S(s) designate the complex reflection coefficient at

the input of the coupling network normalized with re-

spect to a one-ohm resistance as shown in Fig. 2. If z is

passive and lossy, the conventional broadband matching

theory as developed by Youla can be outlined in the

following steps:

1) Obtain the constraints on p(s) imposed by the

passive load z from (1).

6



1966

CHAN AND KUH: MATCHING THEORY FOR TUNNEL DIODE AMPLIFIERS

7

2) Determine a realizable p(s) which satisfies the

above constraints. Meanwhile, the magnitude Ip(jw) 1

must approximate a constant which is kept as small

as possible over the desired band.

3) Obtain the passive impedance Z(s) from (1) and

realize the coupling network of Fig. 2.

Since the a-port network is lossless, IX(jw) 1 = Ip(jw) I.

Hence, at the input of the coupling network, a broad-

band match to the one-ohm resistance is obtained.

Fig. 2. General matching problem.

If the given z is an active impedance, the problem

can be similarly stated as above. The main differences

are the following:

1)

2)

In deriving the constants on p(s), we must consider

the stability of the over all circuit.

In determining p(j,), we must keep in mind that

the aim is to have a magnitude function which

approximates as large a constant as possible over a

given frequency band.

We named the lossless 2-port in Fig. 2 a coupling net-

work rather than a matching network because, in actuality,

a complete mismatch is desired if z is active. In the

Appendix we show that the condition Ip( = IS(

is true even if z is an active impedance. Thus by maxi-

mizing Ip(j the input is completely mismatched. A

reflection-type amplifier is obtained by means of a 3-

port coupling network such as a circulator as shown in

Fig. 3.

Fig. 3. Reflection-type amplifier.

The design procedure for either a passive or an active

load can be briefly outlined as follows:

1) Choose a desirable magnitude square function

Ip(jw) 1’ which satisfies certain necessary constraints

to be discussed later. (This is an approximation

problem. Usually either a Butterworth- or Cheby-

shev-type function may be used. Information, such,

as possible bandwidth and the level of the pass band

constant, will be studied fo r the tunnel diode ampli-

fier problem).

2) Determine p(s) from Ip(j = p(s)p(--s)l ,+,. Ad-

ditional constraints on p(s) must be imposed at

this point. These constraints will be treated in

Sections IV, V, and VIII.

3) From p(s) obtain Z(s) using (1). The constraints on

p(s) guarantee the realizability of Z(s). Synthesize

the lossless 2-port.

III. SIMPLIFIED CONSTRAINTS ON THE

REFLECTION COEFFICIENT

The crux of the whole matching problem is to de-

termine various necessary constraints on p(s), which

are imposed by z. From these necessary constraints a

set of sufficient conditions is obtained which guarantees

that Z(s) of (l), expressed in the following form, is a

positive real function

Z(s) =

44 + d-4 _ z(s).

1 - PC4

In this section we will derive these conditions in an

intuitive fashion. We further restrict our considera tion

to the special case for which all relevant poles and zeros

of various functions in (1) and (2) are simple. The general

theory is presented in Section VIII with its proof.

Type 1 constraints deal with the zeros of the function

z(s) + x( -8). The study is based on considerations of

stability and passivity. Let us digress for a moment

to introduce some definitions. Following Smilen and

Youla we say that an active, l-port with impedance

z(s) is potentially stable if there exists a passive imbedding

network with impedance Z(s), such that the zeros of

z(s) + Z(s) (see Fig. 2) are not in the closed RHP (right-

half plane) Re s 2 0. Under such a passive imbedding

the overall circuit is called absolutely stable.

The absolute stability requirement immediately im-

poses certain constraints on p(s). Equation (1) can be

rewritten in the following form:

1

z(s) + 2(--s)

- p(s) = --a

Z(s) + 44

We are particularly interested here in the restriction on

the zeros of 1 - p(s). Let us call the zeros of x(s) + 2(--s)

the transmission zeros, and let us denote transmission

zero by s,. At s, in the open RHP (Re s, > 0), since

z(s) + Z(s) cannot be zero, the function 1 - p(s) must

be zero. On the other hand, at other frequencies in the

open RHP, since Z(s) is regular, the function 1 - p(s)

cannot be zero except possibly at poles of x(s). The latter

case will be included in the type 3 constraints.

On the jw-axis, Z(jw) may have a simple pole at jmi

and it behaves as K-,(s - jw,)-’ with K-, real and

positive. Z(jw) may have a simple zero at jw, and it

behaves as K,(s -

jw,) with K1 real and positive. From

the absolute stab ility point o f view, two degenerate

cases are allowed, namely: the cases where Z(s) and Z(S)

have common poles and common zeros. At a transmission

zero jwr the function 1 - .p(s) may have a second-order

zero due to the common pole of Z(s) and z(s), the func-

tion may contain no zero because of the common zero

of Z(s) and z(s). However, the behavior of 1 - p(ju,) is



8

IEEE TRAPiSACTIONS ON CIRCUIT THEORT

MARCH

restricted by the fact that K-, and K, are real and

positive. Finally, the function 1 - p(jm) may also have

a zero which is not caused by a transmission zero but is

due to a pole of Z(jw). These restrictions in a precise

form will be stated in Section VIII.

Type 2 constraints are based on the behavior of p(s)

on the jw-axis. Let z(jw) = RL + jX, and Z(j,) =

R + jX. From (1) we have

PW =

R + jX -RL + 3,

R + jX + RL -I- jXL ’

(44

and

(R - RJ2 + (X + X,)”

“(jw)” = (R + R,)’ + (X + X,)”

(4b)

We say that an active impedance is active at a point

jw, if RL(uO) < 0, and passive at a point jw, if RL(uO) 2 0.

Thus from (4), since R 2 0, we conclude that

]p(jwO)] > 1, if z(juO) is active,

Ip(jb.hJI I 1, if z(jmO) is passive.

(5)

Type 3 constraints are introduced to p(s) from the

poles of x(s) and 2(--s) in the closed RHP. Refer to (1)

and consider the open RHP. At a pole of z(s), where

z(-s) is regular, p(s) must be zero since Z(s) is regular.

Similarly at a pole of 2(--s), where z(s) is regular, p(s)

must have a pole. On the jw-axis, since a pole of e(s) is a

pole of .2(-s), p(ju) is regular at a pole of x(s). It is also

clear from (1) that p(s) may not have poles in the closed

RHP other than the poles of x(-s), where x(s) is regular.

Type 4 constraints deal with zeros of p(s) in the closed

RHP, which are contributed by the numerator of (1).

At a point so n the open RHP, p(so) = 0 if Z(s,) = x(-so).

Since 2 is a passive impedance, Z(s,) satisfies the angle

condition for passivity [ll], [12], i.e.,

IL .GJI 5 11 sol.

(f-3

Thus, in order that p(s) have a zero in the open RHP

at so which is not a pole of Z(S), Z(S) must be passive

at so, i.e.,

IL 4---so)1 IL sol.

(7)

On the jw-axis the angle condition for passivity is equiva-

lent to the non-negativeness of the real part.

The above four types of constraints in p are used in

the next four sections with the tunnel diode as the active

load.‘It turns out that the first three types of constraints

on p(s) represent sufficient conditions for realizability

of Z(s). We will present a reasonable justification of our

claim as follows: let us consider first the real part of Z(jm) :

Re [Z(ju)] = Ev Z(S)~.,~,

=

MS) +s)1[, lp(s) 1 Ip(-s)1]i8.w

= Re [z(ju)] w.

w

(8)

Using the type 2 constraints, i.e., (5), we obtain im-

mediately Re [Z(jw)] 2 0, for all w.

Next, let us investigate the open RHP behavior of

Z(s). From (2) we see that poles of Z(s) are contributed

by zeros of 1 - p(s), poles of Z(S) and poles of 2(--s).

In the type 1 constraints, we rule out zeros of 1 - p(s)

in the open RHP except at a transmission zero, where the

zero is cancelled out by the transmission zero. In the

type 3 constraints we have restricted p(s) to be a zero

at a pole of z(s), where x(-s) is regular. Thus from (2),

Z(s) = x(-s) is regular at that frequency. Similarly,

we have restricted p(s) to have a pole at a pole of x(-s),

where z(s) is regular. Thus the poles of p(s) and 2(--s)

cancel, hence, Z(s) is also regular. Therefore, Z(s) is

analytic in the open RHP.

Finally, we must check the jw-axis poles of Z(s) and

the residues. Since we have not expressed explicit con-

ditions on the constraints ,in terms of Km,, K, etc., we

cannot give a proof at this moment. However, it is

reasonable to believe that since constraints on p(ju) are

determined from the conditions that the jw-axis poles

and zeros of Z must be simple and K-, and K, must be

real and positive, we only need to trace it backward to

complete the proof of positive reality.

We will assume, therefore, that the constraints of

types 1, 2, and 3 are necessary and sufficient for a positive

real Z(s). The complete statement is given in Section

VIII as a theorem with its proof. The purpose of this

rather crude presentation in this section is to give the

reader various concepts of the general matching theory,

which are needed for the tunnel diode problem to be

treated in the next four sections.

IV. THE TUNNEL DIODE AMPLIFIER PROBLEM

Consider the linear equ ivalent circuit of a tunnel

diode as shown in Fig. 1. Let us introduce both impedance

and frequency normalization so that C = 1 and G, = 1.

The normalized series inductance and resistance are

denoted by 1 and r, respectively. The active load im-

pedance is, therefore,

x(s) = r + Is + s+ ,

*

r<l

(94

1

2(-s) = r - Ls - --

sfl

(9b)

The even part of Z(S) is given by

$[x(s) + x(-s)] = r + & = rs2s~~ 1 r -

(10)

Let us go through the four types of constraints im-

posed on p(s) by z. For the type 1 constraints, we first

calculate the transmission zero. From (lo), the trans-

mission zeros are found to be at

S,

= hjw, = *j

4

;- 1 = *j

d

l-r

-.

r

01)

1 This is a necessary condition for potential stability.



1966

CHAN AND HUH: MATCHING THEORY FOR TUNNEL DIODE AMPLIFIERS

9

At =tj~,, Z(s) + z(s) in (3) is not zero, therefore, we

obtain

Substituting (18) and (20) in (16), we obtain the be-

havior of p(s) at ~0:

p(ztjwJ = 1.

(12)

In addition, the function 1 - p(s) has no zeros in the

open RHP.

The type 2 constraints deal with the behavior of

IP(G From (10)

Thus Re [Z&J)] is negative for 0 5 w < w,, hence, x is

active in the region 0 5 w < w,. From (5), we have

IPGJ.4l > 1,

0 5 a < 0,. (14)

In the region w 2 w,, Re [z(jw)] is non-negative, hence, x

is passive.

IP(i4I 5 1,

w 2 0,.

(15)

The type 3 constraints are dependent upon the poles

of z(s) and 2(--s) in the closed RHP. From (9) we see

that z(s) and z( -s) have a common pole of first order

at s =

a. In addition, z(s) has a first-order pole at s = 1,

where 2(--s) is regular. Let us rewrite (1) as follows:

(16)

Consider first the pole of z(s) at s, = 1. Since both Z(s,)

and 2(-s,) are regular, p(s) must have at least a simple

zero at s = s, = 1. Thus the first constraint of type 3 is

p(1) = 0.

(17)

Consider next the common pole at s, = 00. Let us ex-

pand -z(-s)/z(s) in Taylor’s series at infinity

4-4

1,-r+--&

2r

-- =

4s)

Is++&

=1-z+***’

0%

At s =

03, 2 may either contain a simple pole or be

regular. Thus at s = a,

Z(s) = K-g + K, + - - * ,

(19)

where K-, is real and positive. Taylor’s expansion of

Z(s)/z(s) can be expressed as

a4

Km,s + K, + . . .

-=

4s)

zs+r+*

A++ (-p)f+ . . . .

(20)

PM =

++1+(2 2&$+...

++1+(g-;);+ .

2r

=I-K-,+ls

1+ . . . ,

(21)

which represents the second constraint of type 3. If

K-, is nonzero, both e(s) and Z(s) have a pole at infinity,

we obtain a degenerate case. That is, the matching net-

work starts with a series inductance. Intuitively, this

additional series inductance will further degrade the

bandwidth performance. In practical design, the de-

generate matching network is often used to realize a

specific gain or bandwidth which is lower than the opti-

mum for the nondegenerate network. The constraint of

(21) becomes, for the nondegenerate case,

p(s) = 1 ++ . . . .

(22)

In addition, type 3 constraints state that p(s) has no

poles in the closed RHP.

The type 4 constraints put a restriction on the loca-

tions of zeros of p(s) in the RHP. We need to use (7) to

determine the allowable regions in the RHP of the zeros

of p(s). The angle of x(-s) is calculated from (9b) and

substituted in (7) to yield the following conditions [13]:

Let the allowable zeros of p(s) be denoted by so = u0 + jwO,

then

and

Isol2 2 @Jo + l)w,2 > w:,

(23)

UlJ <r-

I’

for wo=O,

+

for w,#O.

(24)

With the constraints of p(s) determined we can con-

sider the design. The first step is to choose a rational

magnitude square-function (p(jw)l’ which satisfies some

obvious constraints of type 2 and type 3; i.e.,

IPb.4l > 1, 0 5 0 < WV,

PW = 1,

w = w,,

IPWI 5 1,

w, <w < f=Q)

PW = 1,

w= 00.

A typical plot is shown in Fig. 4, where K2 > 1 repre-

sents the gain in the pass band and w, is the bandwidth.

A typical function is given below:

IP(%412 = 1 +

K2 - 1

_ . EWW /WC‘,

(25)



10

IEEE TRANSACTIONS ON CIRCUIT THEOR Y

MARCH

IP(

2

lll?L’

----

0

WC wr

OJ-

Fig. 4.

A

typical gain curve.

where f is to be picked such that a maximally flat or a

Chebyshev approximation is obtained in the pass band.

The approximation problem and the theoretical limita-

tion on gain-bandwidth will be discussed in the next

three sections. Assume that (25) has been chosen. The

next step is to determine p(s) from

PkdP(---s)I~=i~ = IPw12.

(26)

The factorization is not unique, we, therefore, express

p(s) in the following form:

P(S) = PO(S)PI(S) P2W,

(27)

where pa(s) is uniquely determined from (26) and contains

all the open LHP (left-half plane) poles and zeros. pi(s)

represents an all-pass function whose poles and zeros in

the RHP are specified by the constraints of type 3. In

the present problem, from (17)

pz(s) is another all-pass function of the form

Pzb) =

II 5 9 ReX;>O, i=l,2,..+,

where X,‘s are arbitrary parameters which satisfy the

constraints of type 4 as given in (23) and (24). Up to this

point p(s) has been picked from the point of view of a

desired magnitude characteristic, but in the meantime, it

satisfies constraints of type 2, type 4, and part of type 3 for

the pole of z(s) at s = 1. The remaining constraints to

be satisfied are stated below: 1 - p(s) must not have

additional poles and zeros in the closed RHP,

and

P(=tj%) = 1

(30)

Equations (30) and (31) impose limitations on the

gain K2 and the bandwidth w,. If K2 and w, are picked

to be unusually large, (30) and (31) cannot be satisfied.

Thus, in a practical design, certain guide lines are re-

quired. This information will be derived in the next

section in terms of theoretical limitation on gain-band-

width. The design details will be illustrated in Sections

VI and VII.

V.

INTEGRAL CONSTRAINTS AND

GAIN-BANDWIDTH LIMITATIONS

In this section we will use integral constraints similar

to those of Youla to derive the theoretical gain-band-

width limitations on tunnel d iodes. Consider (27) where

(32)

is a rational function which is analytic and uniformly

bounded in the closed RHP. Then, from Youla, for any

s in the open RHP [3],

In [~~(s)~2(s)l = Erj + In p2(s>

+fs,

m PO(&)4cEw

s2w2

(33)

modulo 2rj, where E is an integer equal to zero or one.

This integer is always zero unless ~~(0) is a negative-

real constant, in which case e = 1. Since pi(s) and p2(s)

are all-pass functions, their magn itudes on the

jw-axis

are unity, we can combine (32) and (33) to form

h.~ P(S) = 4 + Jn ~~(4 + In Pi

+~~“$J$b.

(34)

Consider now the following critical frequencies in the

closed RHP: the transmission zeros s, and the poles of

z(s) and 2(-s), s,. A set of constraints in terms of the

coefficients of the series expansions of those terms in (34)

with respect to these critical points can be derived. For

our present discussion of the tunnel diode amplifier, we

need to investigate the two frequencies s = a3 and

s = jw?. Note that the behavior of In p(s)

at these fre-

quencies can be obtained from (30) and (31)

hl P(S) I*+0

= F; + o(+)

h-l PC4 I*-&

= O(ls - jw,I).

(36)

To employ (34), we must extend its validity to the jw-

axis. Youla has established the following: at s = 03,

since In p(s) has a zero of order one, the first-order co-

efficients of Taylor’s expansions at infinity for all terms

in (34) must be equal. Similarly at s = jw,, since ln p(s)

has a zero of order one, the zeroth-order coefficients of

the Taylor’s expansions at jw, for all terms in (34) must

be equal. The expressions for In pi(s), ln p*(s), and

[2s/r(s’ + w”)] are given below [the forms o f pi(s) and

p2(s) are given in (28) and (29)]. At s = a

2

ln p,(s) = -; + * * * .

(374

In p2(s) = -c + + * *. .

i

2s 21

&2 + w”) = --;

+

**- *

(37c)



1966


At s = jw,,

3.000

2.667

2.333

2.000

y

1.667

4

1.333

1.000

0.667

0.333

0.000

0.

9

In pi(s) = j2 tan-’ wI

+ . . . .

(384

(38b)

2s

.2 w,

-.

7r(s2 + w”) = 3 ; w2 - w;

(38~)

Substituting (35) and (37) in (34), we obtain

-2r

-.=

1

-2 - 2

C Ai + S,- h Ip(jo>l WF

or

In \p(jw)l dw = 1 + C Xi - i*

(39)

i

Substituting (36) and (38) in (34), we obtain

0 = j2 tan-’ w,+ j Ctanm1wz~~{-12

I

or

bandwidth in terms of o, and the given parameter r/l

of the diode. The parameters L’s are to be chosen so

that the right-hand side is maximized. From the type 4

constraints given by (23) and (24), l&l2 > wz, thus the

term containing Xi in the numerator is always negative.

Clearly the maximum bandwidth is obtained by setting

all Xi zero. Therefore, the optimum bandwidth is given

by setting Xi zero in (44) and is denoted by We

%

s

m ”, (jw)[w

n- 0 w, w2

1 + 4 C tan-’ w; ~Y;,,z*

tan-’

(40)

W,

I

*

Equations (39) and (40) give all the information on gain

and bandwidth.’ Let

The solution of (45) is plotted in Fig. 5 in terms of r and

1. Interestingly, (45), which gives the theoretical limita-

tion on bandwidth, agrees with the necessary condition

for potential stability obtained by Youla and Smilen.

Their stability condition is given in terms of the normal-

ized series resistance and inductance by the following:

h-~p(j4l = H,

o<w<w,

=

0,

w > w.

(41)

represent the ideal amplifier. Then (39) and (40) become

W,H

7r

=1-i+ CXi, ’

i

and

11

r-

r = normolized series resistance

I = normalized series inductance

Fig. 5. The optimum bandwid th of tunnel diode amplifier.

1

tanh-’ (wJw,) = wr tan-1 W,

%hJ,

1

-r/l ’

(45)

r < 1,

(464

I<

1

r

1’

(46b)

1 - w, tan-’ -

(J-Q

(43)

Eliminating H from the two equations, we obtain a

formula which gives a relation for w, in terms of the

tunnel diode parameters and the Xi’s to be adjusted.

The two equations are represented by the area under

the curve wb = 0 in Fig. 5. Setting wb = 0 in (45), we

obtain the upper bound o f the condition (46b). The proof

of sufficiency will be given in the next section.

tanh-’ (0,/w?) = or tan-’ w,

+ C * tan-’ w4 Try{. 2

i

1

WC/W,

-;+ pi

’ * (44)

1

Finally we wish to discuss the gain limitation. Refer

to (42) with Xi zero, the gain-bandwidth limitation can

be expressed as

w,H (innepers) = x

(47)

The left-hand side of the equation is monotonically in-

creasing in w,/w, and is larger than unity. It gives the

It should be emphasized that by increasing 2 (using de-

generate matching for example) the optimum band-

width is reduced (see Fig. 5). However, from (47) the

1 These equations were used by Smilen and Youla in the deriva-

. ._. - -_

tion of necessary conditions fo r potential stability 1131.

gain-bandwidth measure can be increased.



12

IEEE TRANSACTIONS ON CIRCUIT THEORY

VI. THE APPROXIMATION PROBLEM AND THE

SUFFICIENCY PROOF OF POTENTIAL STABILITY

L p(ju) = n(0 - 4) + 2 tan-’ L

w’

MARCH

(56)

In this section we will present a simple maximally

flat approximation to the ideal behavior of the last sec-

where 0 and 4 denote the phase combination of a pair of

tion. Then we will demonstrate that as the number of

conjugate zeros and poles, respectively.

elements in the matching network approaches infinity,

- K’co:

the upper bound of potential stability, i.e., the curve

- w2

) 4 = tan-l d26-w - wf.

w,w - w2

We = 0 in Fig. 5, is realized.

Referring to (25), let us introduce a special Butter-

(57)

At iw,,

let & and 4, be the phase angles, the constraints

worth approximation,

I&w) I2 = [ w4 EZ;$ ; y$twqn.

The pass band behavior is monotonic and is

IP(O = K2”,

become

(4%

given by where

6 -

49 =

-1 tan-’ 1 )

WV

(5%

b(j412 = [K’(l - $) + @‘I”.

(49)

2 2

8,

= --tan-’

2/

2Kw,w,

- K w,

= -cos

-1 w,

- Kw,

- Kw,

, (59)

(50)

WV

W,

I

”

The frequency w, is a normalized cutoff frequency, how-

ever, the behavior of IpI2 at jw, deperids on the gain and

thgratio w,/w,. In the stop band

Idj412 < 1,

0, <w < 00)

(51)

IPW12 = 1,

w = w,, 03.

Thus the magnitude function satisfies the necessary

constraints. For convenience in comparing the magnitude

for different n, we further normalize the gain so that

K = k’“.

(52)

where k is a constant. Thus the dc gain is given by the

same formula

Ip(O = k”.

(53)

The factorization of (48) is straightforward. Let us

denote the LHP poles and zeros by s, and so and let us

choose p2(s) = 1 in (29). We thus have, from (27),

(8 -

P(S) = (s

som

- soy s - 1

.-.

- sJ(s - 3,)” s + 1

(54)

The locations of the poles and zeros are given by

s, = a, =t jw,

= -3d2w7wc - 0: f j$d2wrw, + WC”, (554

so = u . f jw,

= -$

2Kw,w, -

K2wz f j$d2Kw,w, +

K’w,~.

(55b)

Note that the expression for u. implies a limitation on

gain-bandwidth; i.e., Kw, < 20,. The constraints of (30)

and (31) are next introduced to determine a more precise

gain-bandwidth restriction.

Consider (30) first, which states that p(jwl) = 1. Since

in (51) we see that ]p(jw,)] = 1, we only need to calculate

the phase. From (54) and (55), we have

& = -tan-l Y~w,wc - 4

= -co,cj

-1 WI - WC

-.

WV - WC

W,

(60)

Next consider the constraint of (31). Expanding p(s)

at infinity, we have, from (54),

P(S)- =

1 + [n(s, + .$J - n(s, + So) - 21 + . . .

2r 1

= l -z+ -** *

(61)

Thus

2a, - 2a. = p l-i,

( >

or

2Kw,w, - K2w: 2w,w, - wf

-

W,

W,

or

sin 6

-sine,=-& 1-i.

( )

Equations (58) and (62) represent the two constraint

equations with two unknowns, & and 0,. Once they are

solved, w, and K can be obtained from (59) and (60) as

$2 -

-1

WI

- cash,

and

“2 = 1 - cos 0,.

(64)

Thus the quantities K and o, are expressed in terms of n

which indicates the complexity of the given network.

The solution of (58) and (62) can be expressed as

follows:



1966

CHAN AND KUH: RIATCHIXG THEORY FOR TUNNEL DIODE AMPLIFIERS

- 1 tan-’

n

b 5 0 (65a)

< 0. (65b)

As n approaches infinity,

(66)

From (63) we obtain

WC

-=I-

1-i

WV

1’

(67)

w, tan-’ -

WV

Thus, if the given tunnel diode has its parameters satis-

fying the necessary condition for potential stability, i.e.,

(46b) ; a finite w, can be obtained. In the limit,

1

1 - ?I = w, tan-’ -

1

WV

which represents the upper bound of the condition in

(46b). From (67) we see that w, = 0, which checks with

the plot in Fig. 5.

In the above treatment we have neglected to check the

remaining constraint on p(s). We have assumed in our

derivation that 1 - p(s) has no other poles and zeros in

the closed RHP. The fact that 1 - p(s) has no poles in

the closed RHP is obvious from (54). To prove that

1 - p(s) has no zeros other than those at s =

&jw,

and

s = ~0 is a little difficult. The simplest way is to con-

struct a Nyquist plot of p(jw). The magnitude behavior

is more or less known, the phase has been given by (56)

and (57). A typical plot for n = 3 is shown in Fig. 6.

It is seen that the plot goes through the critical point

p(jw) = 1 at w = fw, and m. For other values of n it is

useful to consider separately the loci for /WI < w, and

for IwI >

w,. For the former case, the locus is outside of

the unit circle; while for the latter, it is inside the unit

circle. Moreover, it can be shown from (56) and (57)

that the angle for [WI >

w, never reaches f180”. Since

p(s) has a zero at s = 1 in the closed RHP, the locus

p(jw) must have a net encirclement of the origin. Thus,

because of the phase behavior just mentioned, the en-

circlement of the origin must be contributed by the plot

for IwI < Iw,I outside the unit circle. Since the complete

locus goes through the critical point +l, at w = fw,

and 00,

the net encirclement of the point +l must be

zero, for otherwise it would encircle the origin more than

or less than one, which contradicts the information on

the zero of p(s). Thus we have shown that a lumped

network exists for the upper bound of the condition of

potential stability.

Fig. 6. A typical Nyquist plot for n = 3.

VII.

ILLUSTRATIVE EXAMPLES

Let us consider a tunnel diode with

r = 0.6 and 1 = 1.8.

The pertinent parameters are

1

0, =

d-

- - 1 = 0.815,

r

tan-’ - -

W,

= 0.888,

and

1 - ; = q.667.

First, we will use the approximating function of the

previous section as given by (48). We need to solve for

K and w, from the two constraint equations of (58) and

(62). They are

and

i (4, - e,) = tan-’ I

W,

= 0.888,

It is not difficult to see that with n =

1,

2, no solution

exists. Thus we choose n = 3 and obtain the following:

K

= 92.7, gain = 118 dB, and w, = 0.00184.

The magnitude plot is shown in Fig. 7. The synthesis

of the coupling network is straightforward since we have

already the poles and zeros of p(s) from (54) and (55).

The coupling network will contain four reactive ele-

ments. It is clear that the approximating function used

is not an efficient one in terms of bandwidth and number

of elements. For comparison let us compute the theoretical



I I I I I I1 II I

0 0.2

0.4

0.6 0.8 1.0

W--c

Fig. 7. Magnitude plot of the reflection coefficient with n = 3.

limits on gain-bandwidth. With a brick-wall type of

response the maximum bandwidth is given by (45),

We = 0.46 w, = 0.375. The maximum gain for the opti-

mum bandwidth is given by (47), gain = 48.6 dB.

Obviously a more efficient approximation can be used.

Let us refer to (25) and choose f to be a Chebyshev

polynomial of fifth order. Thus

I&412 = 1 +

K2 -

1

~2C%JJ/Wc)

033)

l + 1 - (w/wJ2

where

c,(x) = 5s - 20~~ + 16x5.

(6%

2 - wza, + w,b4 = 7 = 0.667.

(74)

Next, (70) is substituted in (68) to obtain a set of simul-

taneous equations. These equations together with (71),

(73), and (74) are solved with a computer. We obtain

We wish to determine the realizable w, and K. Using

the information on the theoretical maximum of band-

width, we know that w,/w, is considerably smaller than

0.46. The factor 1 - (w/w,)~ will clearly introduce a

distortion in the pass band. However, since w,/w, is small,

we will neglect its effect. For a 3 dB bandwidth, we set

E = 0.99763. To determine p(s), we set

WC

= 0.238,

(75)

PM =

1309s5+1922s4+1504s3+742.8s2+231.4s+37.45

1309s5+176.7s4+104.6s3+9.592s2+1.709s+0.06264

s-l

'&g-P

(76)

and

K = 599 or 55dB.

(77)

The impedance Z(s) is next obtained from (2)

s-l.-.

s+1

(70)

Z(s) =

87.66~~ + 70.28s + 22.43

872.8~~ + 699.7s2 + 293.6s + 56.27’

(78)

ao+a,s+a2s2+a,s3+a,s4+a,sb

‘(‘) = ,,+~,(~)+~2(~)2+ba(~)3+b4(;)4+b5(~)s

The constraints are now introduced. First from (30)

Next

p(jw,) =

1,

w, =

0.815,

(71)

PCS> Is-

2r 1

=l-Ts+“”

which implies that

(72)

Consider the general matching problem stated in

Section II in terms of the circuit of Fig. 2. The given

load impedance z(s) can either be active or passive but

not lossless. The main theorem is now stated and proved.

h5

a5 = ,T

-C

(73)

First, let us rewrite some of, the useful equations in-

volving Z(s) and p(s) and introduce some notation for

IEEE TRANSACTIONS ON CIRCUIT THEORY

MARCH

I249

II = 1.8 I = 0.6

I .“I

(a)

dB

60

I I I, ,

0.01

0.05 0.1

0.2

03 0.4

W-

(b)

Fig. 8. (a) A tunnel diode amplifier. (b) Frequency

response of the amplifier.

and

The circuit realization and the response is shown in

Figs. 8(a) and 8(b).

VIII. A GENERAL MATCHING THEORY



1966


15

the various series expansions which are needed for the

Condition 1 (a) : In the open RHP, the function 1 - p(s)

theorem. The useful equations are

is not zero except at a transmission zero-s, of order r or

p(s) = z&--;i;~ ’

possibly at poles of z(s). The latter case will be included

OW

in Condition 3. In the former case,

d, = c, = 1,

(87)

.

and if x(s,) is regular,

(79b)

di = 0, i = 1,2, **. ,r - 1.

z(s) ’ -

And if x(8,) is a pole of order m,

1

- PC4 =

4s) + d-s)

as> + 4s)

d--s)

1+-

=

4s) .

Ci?i=Ci, i=r+WZ,~**,r+2?YZ-l.

(90)

1+z(s)

@W

Condition l(b): On the jw-axis, the function 1 - p(s)

44

may have a first-order zero at jwO, which is neither a

z(s) + x(-s)

transmission zero nor a pole of x(s). Then

Z(S) = --

1 - P(S>

- 4s)

(814

P(S) = 1 + &(s - jd + * . a ,

where d, is real and

(81b)

Re N&b)l , o

-d, *

Zeros of z(s) + 2(--s) are called transmission zeros and

are denoted by s,. Poles #of x(s) and z( -s) in the closed

RHP are designated by a,. The Laurent series expansions

around si (s,, s, or other frequencies) for the following

functions are given by:

5(s) = a-,(s - s ,y + a-(,-,)(S - Si)-(- + * * -

+ a0 + Ul(s - si> + ’ * ’

(82)

2(-S) = b-*(S - SJ-” + b-(a-l)(S - Si)-(n-l) + * ’ a

+ bo + f&b - SJ + - - *

(83)

At a transmission zero, jtir of order r,

do = co = 1,

and if x(j,?) is regular,

CL = 0, i = 1,2, .-a ,r - 1

9

and

drVl # 0 (if degenerate).

And if x(jm,) is a pole of order m,

di=C<=O, i=l,2,a**,r+m.

-

(91)

(92)

(93)

(94

1, w-9

-z( -s)

and

~ = C-,(s - s,j-”

44

$ c-(k-l)(s - Si)-(k-l) + * * ’

di = Ci, i = r + m,

+ co Cl(SSi> *

p(s) = d-,js - si):” + d-,,-,,(s - s,)-(‘-~) + . . .

+

do

+ d,(s - si> + - - .

Z(s) = K-,(s - jwi)-' + K, + - . . ,

where K-, is real and positive.

(84)

(85)

(8’3)

Theorem

Let z(s) be a given rational impedance which may be

either active or passive but non-Foster. Then

and

r+m+l,**.,r+2m-1,

(96)

d

r+2m-1 =

C

F+P?lS--l

K-,C,+,/a-, (if degenerate),

(97)


Condition 5’: On the jw-axis, Ip( satisfies the follow-

ing :

IP(G>l ? 1,

if x(jw) is active,

i.e., Re [z(ja>] < 0,

(984

Ip( i 1,

if z(j,) is passive,

z(S) = 1 - p(s)

s) + d-4 _ z(s)

i.e., Re [x(jw)] 2 0.

(98b)

Condition 3(u): In the open RHP, p(s) is analytic

is a positive real function and Z(s) + z(s) # 0 for

except at s,, which is a pole of z(s) of qrder m and is a

Re (s) > 0 except for degenerate cases if and only if p(s)

pole of z( -s) of order n, m, n > 0. Then

satisfies the following three conditions.

di=O, i<m-n-l,

(99)



16

IEEE TRANSACTION S ON CIRCUIT THE ORY

MARCH

and

function 1 -

p(s) will then have a first-order zero, and

di = Cc, i = m - n, m - n + 1, ***

-1

an-n+,

I

if m 2 n, C,-, = 1 and

cmmn+i

I

0, i = 1, 2 * * * ) q

WQ

m - n (if m 2.

n)

2,-,-1 (if m < n)

and if m = 0, the second equation is not needed.

Condition S(b): On the jw-axis, p(s) is analytic. At

s = jw,, which is a pole of z(s) and x(-s) of order m, we

have

and

di = Cc, i = 0, 1, *a. , m - 1,

(101)

d,-I = Cm-l + K-,(1 - C ,)/u-, (if degenerate),

(102a)

and if

Co = 1 and Ci = 0, i = 1, 2, . . . , q.

di = Ci, i = 0, 1, 0.. , m + q, (102b)

d

m+sl=

C

m+a

- K-G+,Ia-m,

and if m = 1,

d

0

= Coa-, + K-1

a-, + K-1

(degenerate),

andifm = landCo = l,d, = Co = land

(103a)

d, =

a-ICI

a-, + K-1 ’


(103b)

Proof

A. Necessity

Condition 1 (a): In the open RHP, Z(s) is regular

and nonzero. Consider (SOa), due to stability, the function

1 - p(s) cannot have a zero except when z(s) + z(-s)

is zero or z(s) has poles. In the former case, 1 - p(s)

must have a zero of the same order as the transmission

zero s,. But if at s,, z(s) has a pole of order m, then z(s)

and -2(--s) must have identical first m + r - 1 terms

in their Laurent series expansions ‘with respect to s,.

Thus

-z( -s)

~ = 1 + Cr+JS - s,)r+m + **. .

44

Using (80b), we obtain

1 - PCS> I-

= [-Cr+JS - sJr+m + --*I

(104)

* 1 - E@J (s - s,)” + * * * ,

[

a-,

1

(105)

which implies (89) and (90) of Condition la.

Condition 1(b): On the jw-axis Z(s) may have a

pole at jwo which is not a pole of z(s). From (80a), the

1 - P(S)Is=i,,=

4jwo) + 4--ho)

K-,(s - jwo)-’ + . . .

= 2 Re$m+O)l (s -

jwo)’

+ . . . ,

(106)

which implies (91) and (92).

At a transmission zero, jwr of order r, if z(s) is regular,

(8Oa) requires that 1 - p(s) has a zero of the same order.

However, in the degenerate case, 1 - p(s) has a zero of

order r - 1 due to cancellation. Thus we have shown

(93) and (94).

At a transmission zero, jwr of order r, if x(s) has a pole

of order m, the situation is similar to that of (105). How-

ever, we must consider the degenerate case. If we sub-

stitute Z(s,) by K-,(s - jwI)-l in (105), we obtain

1 - P(s)]~+~. = [-C,+Js - jw,>‘+” + .. -1

*

[

1 - 5 (s - jm,)m-l + . . .

1

(107)

m

which gives

d

C

C,+mK-1

r+2m--1= 7+2n--l -

fLf8

(1’38)

Thus we have proved (95)-(97) of Condition l(b).

Condition W: This has been proved in Section III.

Condition Z(u): In the open RHP, Z(s) is regular

and nonzero. Consider that (79a), due to stability p(s),

cannot have a pole except at a pole of x(-s). At a pole of

x(s), s, of order m (also a pole of 2(--s) of order n),

-z( -s)~ = CmJS - S,)- + * - * .

4s)

(109)

Using (79b), we obtain

PC3 I *-a.

[c&sspy-” ] e (s s,)”* 1

=

1+

F (s s,)m

(110)

Thus if m = 0, we have di = 0, i < -n. If m # 0, we

have

P(S)

I *lip

[Cm-,(s spy- + * * 1

+ [““(s - spy+ - *-]}{l - 2 (s s,)”+ - - .},

(111)

which implies (99) and (100).

Condition S(b): On the jw-axis, p(s) is analytic be-

cause Z(s) + z(s) cannot have a zero due to stability.

In the degenerate case, if Z(s) and z(s) have a common

zero, Z(s) and -2(--s) also have a common zero, hence,



1966


17

they cancel out in (79a). At a pole of x(s), jwp of order m,

Substituting (93) and (95)-(97) into (81b), we obtain,

which is also a pole of x(-s) of order m, we use (111) and

for the degenerate case

substitute Z(s,) by K-,(s - jw,)-’ + . . . .

.w Is+.=

[a-,(~jw,)-m + - - a]

P(S)la-rjwp = [Co + CI(S - jw,) + -**I

e(d,+2,-1 - C,+2m-I)(~ - jw,)r+2m-1 + - - -

+

C

(8jwp)m-l

I)

-Cl+m(~ - jw,)r+m + -- -

(&+2m--l

Cr+2m-1)(~

- jwX1 + * * - .

(116)

.{ -Y?sjwJm--l

=*

1

(112)

Equation (97) guarantees that the residue is real and

positive.

which implies (101) and (102). However if m = 1, we

Finally, at the poles of z(s) and 2(--s) at jwp of order

use (110) directly to obtain (103).

m, we have restrictions given by Condition 3(b). Sub-

B. Xuficiency: To prove that Z(s) is a positive real

stituting (101) and (102) in (81b), we have

function we need to check 1) Re [Z(jw)] 2 0, 2) the

analyticity of Z(s) in the open RHP, and 3) the jw-axis

poles of Z(s) and the residues. The fact that Re [Z(jw)] 2. 0

z(s)a-,(~jw,Jmm1dm-1?Gft . ’m-

has already been proved in Section III. We thus proceed

to show the other two.

= fi (dm-1

- Cm-,)(s - jwp)-' + -- - .

(117)

In the open RHP, we saw from (8la), that poles of Z(s)

are due to zeros of 1 - p(s), poles of z(s) and poles of

Equation (102a) guarantees that the residue is real and

2(--s). Condition l(a) rules out zeros of 1 - p(s) in the positive. Similar treatment can be given for the case

open RHP except at a transmission zero of order r and

Co = 1. Form = 1,

possibly the poles of z(s). In the former case, 1 - p(s)

has a zero of the same order, hence from (81a), the zeros

Z(s) = [a-,(s - jw,)-' + - . a] 2 12 ‘J sme*e'

cancel and Z(s) is analytic at the transmission zeros.

In Condition 3(a), (99) and (100) restrict the nature of

do - Co

p(s) at a pole of z(s) of order m which is simultaneously

= a-, l--d, (s - jw,>-' + - - - .

(1 W

a pole of 2(--s) or order n. Let us consider (81b). For

Equation (103) guarantees that the residue is real and

m 2 n, we have

positive. This completes the proof of the theorem.

Z(s) I*-+*p=

[a-,(s - SJ- + - - e-j

IX. CONCLUSION

(d, - C,)(s - sJ’ + . . .

- 1 - [C,-,(s - &Jrn-- + * * -1.

(113)

In this paper we have extended Youla’s broadband

matching theory to include both passive and active load

For m < n, we have

impedances. The main theorem which gives the neces-

sary and sufficient conditions on the complex reflection

.w I I-W. =

[a-,(s - SD)- + * - *]

coefficient is stated and proved in Section VIII, while its

&n--n - C,,-J(s - sJ2’- + . - . .

special application to the tunnel diode amplifier problem

- [CL,(s - SJ- + - * *]

(114)

is treated in detail from Sections IV to VII. New results

for tunnel diode amplifiers in terms of the theoretical

In both cases the poles and zeros cancel, hence, Z(s) is

limitations on gain-bandwidth and sufficient conditions

analytic.

On the jw-axis, Condition l(b) states that 1 - p(s)

for potential stability are obtained. We acknowledge the

editor and the reviewers for their useful comments.

may have a first-order zero at jwo, which is not a trans-

mission zero and which is not a pole of z(s). Using (81a)

APPENDIX

and (go), we have

Consider the jw-axis behavior of the active impedance

z(s)I = 4jwo) + 4--jwo) z. At frequencies where Re [z(jw)] > 0, z is said to be

*‘/loo

-4s - jwo)

-

4jw0),

passive, and clearly Ip( = IS(j We only need to

=

2

Re W41 cs _ jwo)-' + . . .

consider the frequencies where Re [z(jw)] < 0, i.e., z is

-4

.

active. Let z(jw) be active at jw,, then -z( - jw.) = - Z(jw,)

is passive. Let

Equation (92), that restricts the residue, is real and

positive.

s = p1 Sl2l

At a transmission zero jw, of order r, 1 - p(s) has the

Ls21 s221

same zero of order r or r - 1, thus Z(s) is analytic at

by the scattering matrix of a lossless 2-port with respect

jwr. At a transmission zero jw, of order r, where z(s) has

to passive reference impedances 1 and -~(jw,). Thus the

a pole of order m, we have constraints given by (95)-(97).

incident and reflected waves are related by



IEEE TRANS.4CTIONS ON CIRCUIT THEO RY

In the actual situation, port-2 is terminated by z&J

which is the negative conjugate of the passive reference.

Under such a termination, b, = 0, and the input re-

flection coefficient is

But by definition,

Therefore,

Thus, for all w,

VOL. CT-13, NO. 1 hfARCH

1966

111

121

[31

[41

r51

F31

171

PI

WI

[lOI

1111

WI

[I31

REFERENCES

H. W. Bode, Network Analysis and Feedback Amplifier Design.

Princeton, N. J.: Van Nostrand, 1945.

R. M. Fano, “Theoretical limitation on the broadband matching

of arbitrary impedances,” J. Franklin Inst., vol. 249, pp. 57-83,

139-154; January, February 1950.

D. C. Youla, “A ne w theory of broadband matching,” IEEE

Trans. on Circuit Theory, vol. CT-11, pp. 30-50, March 1964.

D. C. Youla and L: I. Smilen, “Optimum negative-resistance

amplifiers,” 1960 Proc. Symp. on Active Networks and Feedback

Systems, pp. 241-318.

E. S. Kuh and J. D. Patterson, “Design theory of optimum

negative-resistance amplifiers,”

Proc. IRE,

vol. 49. RD. 1043-

1050, June 1961. - ’

. _^

E. W. Sard. “Gain-bandwidth performance of maximum flat

negative-conductance amplifiers;” 1960

Proc. Symp. on Active

Networks and Feedback Systems ,

pp. 319-344.

L. I. Smilen and D. C. Youla, “Stability criteria for tunnel

diodes,” Proc. IRE, vol. 49, pp. 1206-12 07, July 1961.

M. E. Hines, “High frequency negative resistance principles

for Esaki diode applications,”

Bell Sys. Tech. J.,

vol. 39, pp.

471-513, May 1960.

L. A. Davidson? “Optimum stability criterion for tunnel diodes

shunted by resistance and capacitance,”

Proc. IEEE (Corre-

spondence), vol. 51, p. 123 3, September 1963.

I T. Frisch, “A stability criterion for tunnel diodes,” PTOC.

I’EE,

vol. 52,.pp. 922-923, August 1964.

E. A. Guillemm,

The Mathematics of

Circuit

Analysis. New

York: Wiley, 1949.

C. A. Desoer and E. S. Kuh, “Bound on natural frequencies of

linear active networks,”

1960 Proc. Symp. on Active Networks

and Feedback Systems,

pp. 415-436.

L. I. Smilen and D. C. Youla, “On the stability of tunnel

diodes,” Microwave Res. Inst., Polytechnic Inst. of Brooklyn,

Brooklyn, N. Y., mm 49, January 30, 1962.

Two Theorems on Positive-Real Functions and Their

Application to the Synthesis of Symmetric

and Antimetric Filters

DANTE G. YOULA, FELLO\v, IEEE

Abstract-It is iirst shown that the power gain of a filter which

has been partitioned into two component parts may be expressed in

terms of a formula involving only the two impedances seen looking

to the left and the right of the common junction. By imposing the

constraints of symmetry and antimetry this formula leads quite

naturally to two global equations for positive-real (pr) functions.

Theorems 1 and 2 present necessary and suf6cient conditions for

the existence of solutions. Moreo ver, the construction of these pr

functions is made to depend on two algorithms of an extremely

simple character. The theory is fully illustrated by means of four

worked, nontrivial examples. Finally, it is pointed out that synthesis

by bisection is often wasteful of reactsnces (especially in the sym-

metric case), and a careful count of elements is presented for anti-

metric filters.

Manuscript received March 23, 1965; revised July 21, 1965.

This report is part of an Air Force research program performed under

Contract AF30 (602) 3951, with Rome Air Development Center, by

the Polytechnic Institute of Brooklyn, Brooklyn, N. Y., Secondary

Rept. PIBMRI 1252-65.

The author is with the Department of Electrophysics, Poly-

technic Institute of Brooklyn (Graduate Cen ter), Farmingdale, N. Y.

I. INTRODUCTION

HE PROBLEM of synthesizing, by physical bi-

section, the transducer power ga ins of electrically

symmetric and antimetric filters has been treated

by several authors [l]-[3]. Recently, Navot and Zeheb

[2] and Navot [4] pointed out that the method of poly-

nomial identification advocated by Guillemin [l] is valid

only for a restricted class of power gains. In [4] Navot

described algorithms (his generalized even and odd the-

orems) which are applicable in the general case.

In this paper an entirely different approach is taken.

It is first shown that the power gain of a filter which

has been partitioned into two component parts may be

expressed in terms of a formula involving only the two

impedances seen looking to the left and the right of the

common junction. By imposing the constraints of sym-

metry and antimetry, this formula leads quite naturally

IS

a general matching theory and its application to tunnel diode amplifiers - ieee transactions - draft...

Documents