582 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 6, JUNE 1987

Unconditional Stability of a Three-PortNetwork Characterized with S-Parameters

JOHN F. BOEHM, MEMBER, IEEE, AND WILLIAM G. ALBRIGHT

AMiwct —An analytical solution is presented which establishes nine

conditions necessary for determining the unconditional stability of anetwork described with three-port S-parameters. In contrast to the uncon-

ditional stability conditions of a two-port network the unconditionalstability conditions of a three-port network are dependent on both thethree-port S-parameters and the port terminations. These criteria form the

basis for three-port amplifier design, and are used to anafyze measured

three-port S-parameter data of a silicon BJT at 2.4 GHz.

1. lNTRODUCTION

T HE IDEA OF using three-port S-parameters in the

design of high-frequency transistor circuits is not new

[1], but only recently have three-port S-parameters been

exploited in such designs, particularly in microwave oscil-

lator design [2], [3]. The use of three-port S-parameters in

the design of microwave amplifiers has not materialized

due to the lack of an analytical criteria for predicting the

unconditional stability of such networks.

A typical example of this lack of analytical criteria is

found in series-feedback amplifier design. Most series-

feedback amplifier designs rely on empirical knowledge for

determining the third-port termination which results in the

unconditional stability between the input and output ports.

Once unconditionally stable, the input and output ports

may be simultaneously conjugate matched. One third of

the three-port unconditional stability conditions provide

this termination information. It is true that analysis em-

ploying parameter system conversions can be used in this

example, but the use of a three-port S-parameter approach

is superior to these methods since parameter conversion

can multiply measurement errors up to five times [4]. The

significance of measurement error in a design is evident

above 12 GHz, where obtaining accurate S-parameters of

an active device is difficult.

The use of a three-port unconditional stability criterion

is not limited to a feedback-type amplifier, which is actu-

ally operated in a two-port sense. It is plausible that truly

three-port circuits (circuits which have more than one

output port), such as active power-splitters or active circu-

lators [5], can be designed from a three-port approach.

Marmscript received December 13, 1985; revised December 18, 1986.This work is a portion of thesis research conducted at the University ofIllinois in 1985.

J. F. Boehm is with the Watkins-Johnson Comrmrw. Pafo Alto. CA94304.

. .

W. G. Albright is with the Department of Electrical and ComputerEngineering, University of Illinois, Urbana, IL 61801.

IEEE Log Number 8714118.

Both of these circuits would require a three-port uncondi-

tional stability criterion if the port terminations providing

the appropriate conjugate match conditions are to be

easily determined. Such circuits will be of major impor-

tance in silicon and GaAs MMIC’S.

II. THE APPROACH TO UNCONDITIONAL STABILITY

OF A THREE-PORT NETWORK

In the past, two works have dealt with the problem of

unconditional stability of a three-port network from a

negative-resistance approach. An early work employing

three-port Z-parameters [6] established unconditional sta-

bility conditions for a two-port network with series feed-

back. The analysis is based on the envelope of a function

[7], which, though mathematically correct, requires a

numerical solution. A second work [8] employs three-port

S-parameters to establish an unconditional stability crite-

rion. In this work, stability is approached by mapping the

&’~-plane and the S’~~-plane onto the I_’q-plane. Unfor-

tunately, two ports must be terminated before any useful

information is found, and in doing so, a degree of freedom

is lost. In addition, this criterion is based on the two-port

unconditional stability “conditions” ISIII <1, ISZ2I <1, K

>1, which Woods [9] has shown to be insufficient.

Unlike these previous works, the present analysis con-

siders stability between two of the ports with a fixed

termination on the third port. This approach is advanta-

geous, since it becomes clear that the location of the

stability circles, and hence unconditional stability, is de-

pendent on the third-port termination. One also notices

that the three-port unconditional stability conditions di-

rectly reduce to the form of the two-port unconditional

stability conditions when the three-port matrix is degener-

ated to a two-port matrix.

III. DERIVATION OF THE THREE-PORT

UNCONDITIONAL STABILITY CONDITIONS

Unconditional stability conditions are found by solving

two converse problems. First, the S:-plane is mapped

onto the I“-plane for a fixed r~. Imposing the requirement

that the IS,J’I >1 region cannot intersect with the lrjl <1

region leads to the derivation of the stability factors.

Second, the rj-plane is mapped onto the S~-plane. After

imposing the condition that the Irj I <1 region cannot

intersect with the IS,~ I >1 region, a second condition is

found. For each pair of ports considered for stability, only

0018-9480/87/0600-0582$01.00 01987 IEEE

BOEHM AND ALBRiGHT: UNCONDITIONAL STABILITY OF THREE-PORT NETWORK 583

S: rl r2 “,,S.2~

Fig. 1. Configuration of the three-port network characterized withS-parameters.

three conditions (instead of four conditions) arise due to

the symmetry of the stability factors.

A. The Stability Factors

In order to find the image of the S#plane in the

rl-plane, consider the configuration of the three-port net-

work shown in Fig. 1. In a 50-0 system, it is seen that

where

Al, = S#33 – S3#23

A2Z= SIIS33 – S31S13

where

CJ(r3) = [(SJ – A$3SZ2)+(S2ZA’; – A;2)~$

+ (AllA& – S;s33)r3

+ ir312(s33A3z – AIIA? )] (11)

E = [– IA31211’312+ISI112+ IA221211’312–IA3312

-2 Re[S11A;21’$] +2 Re[A:A33A:]] (12)

.qr3) = (A11A,22– s3A)JY

+ (A3 + S33A33 – S1lA1l – S22A22)r3

+(s~~szz–A33). (13)

Unconditional stability between ports 1 and 2 will be

guaranteed when

Rearranging (14) and taking the magnitude squared yields

llJ(r3)l-lCJ(r3)112>E2. (15)

Squaring (15) and regrouping the terms, one sees

21J(r3)lp; (r3)l<lc$(r3) 12+lJ(r3)12-E2. (16)

A33 = s11s22 – S21S12 Squaring (16) yields the expression

and As is the determin~t of the three-port S-parameter 41 J(r3) 121c; (r3) 12s (\C:(r3) 1’ + lJ(r3) 12)2+ E’

matrix.

Considering 173 fixed, solving for 171, and substituting -2 E’([c:(r3)12 +lJ(r3)12). (17)

Sjj = (Z2 – 1): (Z2 + 1), it is found that

(I- s22- s,sr, + WV% + (M’, - s22+ sqqr,-1) (2)“= (s,l-A22r3 +A3r3-A33).Zz +( AJ’3-s,,+AJ’--A33)

which is a linear fractional transformation between rl and ,

22. It can be shown that the center and radius of this ‘sing ‘he ‘dentity ’10]

transformation are given by lC;(1’3)12 =lJ(173)12-t EF (18)

BC*+AD*T= (3)

where

DC*+(DC*)*F= [1– 1S2212+ ls33121r312–lAl112ir3~2

AD – BC

‘= DC*+(DC*)*(4) –2Re[s331’3] +2 Re[S~A11r3]]

and substituting (18) into (17), one sees that

where

A = (1 – S22 – S33r3 + A11r3)(5) 41J(r3)12(lJ(r3)12+EF) < (21 J(r3)112+EF)2

B = (A111’3 – S22 + S33r3 – 1) (6) + E4–2E2(21J(I’3)12 + EF). (19)

C=[S1l– A2ZI’3+A31’3-A33) (7) After factoring out E2, (19) reduc$s to

D=(A221’3- S11+A31’3-A33). (8) 0< {–41J(1’3)12 +( E2-2EF+F’)}E2. (20)

Substituting (5)–(8) into (3) and (4) leads to Further simplification of (20) shows that

c:(r3) (F-E )>21J(I’3)11. (21)

T=E

(9)Equation (21) implies that

R=lJ(r3)l

lE1(lo) (22)

584 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 6, JUNE 1987

for unconditional stability between ports 1 and 2. A plot Equations (29) and (30) describe the second set of condi-

of K3( r~ ) >1, one of the three three-port stability factors, tions necessary to guarantee unconditional stability of a

in the 17~-plane will show which r~ terminations provide three-port S-parameter network.

unconditional stability between ports 1 and 2.

Generalizing the port designations, the three-port stabil-

ity factors are given by

~ ~rk) _ [(1- l~t12-l~,j12+lA~~12)+lr~12(l~~~12-1A,,12-lAj,12+IA,12)3-

21(AliAJj – &A3)r~ +(A3 +Sid,u – S,,A,, – ‘jjA~~)rk

‘2Re[Skkrk] +2 Re[Sz:A,jrk] +2Re [S’:AIZr,] –2Re[A~Akkr:]]

‘(sizs’j - ‘kk)/

B. The Converse Problem

In order to establish sufficient conditions for stability,

the image of the rl-plane must be found in the S;j’-plane.

In a manner similar to that employed in the derivation of

the stability factors, it can be shown that the center and

radius of this image are

T= cl(rJ

F21(24)

~=lJ(r3)l

1F211(25)

where

cl(r~) - [(s22 – sfiAsJ)+(A;zA33 – s22sfi)r;

Fzl = [1– IS1J2+ lS33121r~12–IA221211’312

–2Re[sq3r3] +2 Re[S;Az2r3]].

In order to guarantee stability in this view,

lTl+@/<1

or, eqtiivalently,

ITI <1- IRI. (26)

Since IT I could equal zero, it must be guaranteed that

IRI <1. Consequently, IS;;I e 1 for all lrll <1 when

F21>l.l(r3)l. (27)

Similarly solving for where the r2-plane maps into the

S{;-plane, one sees that 1$’~1<1 for all lr21 <1 when

F12>l.l(r3)l.

Generalizing (27) and (28), it is found that

Zj>lJ(rk)l

qt>lJ(rk)l

where

~j=l– Is’jl’ + Iskkl’lrkl’– @ji121rk12

–2Re[Sk#k] +2Re [Sj;Aiirk]

~, =1– Isi,l’ + ls~~l’lr~l’– IAjjl’lrkl’

–2Re[Skkr~] +2Re [S,:AJjrk]

J(rk) = (ALZAjj–Sk~Aj)I’/

+ (As+ SkkAkk – S,, A,, – Sj,Ajj)rk

+ (Slisjj ‘Akk).

(28)

(29)

(30)

(23)

IV. SUMMARY OF THE CONDITIONS

Stability between two of the ports has been investigated.

However, all three driving-point S-parameters must be

considered for three-port stability. It has been shown that

for 173values which result in K3(r3) >1, all passive loads

on port 2 will result in stability at port 1. It is not

necessarily true that all r2 terminations result in stability

between ports 3 and 1. In order to find the subset of rz

terminations which result in stability between ports 3 and

1, K3(I’2) is found and plotted as a function of fixed I’2. If

r2 and I’s are chosen from the respective set where K3(r3)

>1 and K3(172) >1, then for those terminations lS{~l <1.

Similar reasoning will show that to guarantee IS;;I <1, it is

required that both K3 ( rl) >1 and KS (~~) >1 for some

pair of rl and r2 terminations. Consequently, to guaran-tee that IS;’ 1<1 for i =1,2,3, simultaneously, 171, r2, and

r3 must belong to the subset of terminations where K3(rl)

>1, K3(r2) >1, and K3(rq) >1, respectively.Table I summarizes the possible combinations of i, j,

and k for (23), (29), and (30). The nine equations repre-

sented in Table I establish the conditions necessary for

three-port unconditional stability. These equations state

which passive terminations result in passive driving-point

S-parameters for all ports. Explicitly, the requirements for

three-port unconditional stability are

i) ~3(rJ >1 F23 > l@’l)t ’32> lJ(rl)l (31a)

ii) ~3(r2) >1 F13 > wr2)l F31>v(r2)l (3W

iii) K~(r’3) >1 F1’> lJ(rj)l F’1> IJ(L)I (31c)

for some set of terminations rl, r2, I’3. If any one of these

equations is violated, then the network is conditionally

stable in the three-port sense for this set of terminations.

V. ANALYSIS OF MEASURED THREE-PORT

S-PARAMETERS

The three-port S-parameters of an NE46734 transistor

biased at VCB =10 V and Ic = 50 mA were measured withan automatic network analyzer at 2.4 GHz. An error-cor-

recting scheme [10] using the indefinite property for this

three-terminal device (with base port 1, emitter port 2, and

collector port 3) was used to obtain the following S-

BOEHM AND ALBRIGHT : UNCONDITIONAL STABILITY OF THREE-PORT NETWORK 585

TABLE ICOMBINATIONSOF i, j, AND k FOR EQUATIONS (23), (29),

AND (30)

I

i j k

2 3 1

3 2 1

1 3 2,

3 1 2

1 2 3

2 1 3

TABLE IIPREDICTEDCOMMON-COLLECTORS-PARAMETERSFROMTHE

THREE-PORTMATRIX AND MSASURSDCOMMON-COLLECTOR S-PARAMETERS

~

I %2 I 0.773 &j.J& I 0.790 ~ I 0.017 3.51.I

Fig. 2. Stability between ports 2 and 3 viewed as a function of rl(Crosshatched area represents values of 1’1 where K3 (rl) >1, F23 >

l~(rl)l, fi, > I.qrl)l.)

parameters:

[030@X o’~’k O.’”k 1

10.659/ – 74.01° 0.466/55.53°

1

0.526/10.90° .

1.052/14.53° 0.06i/ – 60.24° 0.346/ – 102.28°

In order to check the validity of these error-corrected

S-parameters, the common-collector two-port S-parame-

Fig. 3. Stability between ports 3 and 1 viewed as a function of 17z.(Crosshatched area represents values of ra where K3 (r2) >1, F31 >

lqr,)l,F1, > I.qr,y.)

Fig. 4. Stability between ports 1 and 2 viewed as a function of rq.(Crosshatched area represents values of r~ where K3 (r3 ) >1, F12 >l~(r,)l, fil > I.r(r,)l.)

ters were measured and compared with the corresponding

two-port S-parameters found from terminating the collec-

tor port with a short circuit (l_’3= 0.998/176 .940). The

corresponding data of Table II show that good agreement

exists between the measured and calculated two-port S-

pararneters.

I’-plane plots of rl, I’z, and r~ are shown in Figs. 2, 3,

and 4, respectively. These plots show which terminations

satisfy conditions i), ii), and iii) respectively, in Section IV.

586 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHN?QUSS, VOL. MT”C-35, NO. 6, JUNE 1987

Any such set of rl, I’z, 173which satisfies i), ii), and iii) will

guarantee that the three-port is unconditionally stable.

In addition, if one wishes to determine two-port series-

feedback configurations, Figs. 2–4 give vital termination

information. For example, a capacitive terr&ation in the

emitter (port 2) results in unconditional stability between

ports 3 and 1 (see Fig. 3). In Fig. 2, one sees that feedback

bver a large area in the base lead results in unconditional

stability between ports 2 and 3. The designer may choose a

particular termination in these areas after considering gainor noise requirements.

VI. CONCLUSIONS

An analytical solution has been presented which estab-

lishes conditions for the unconditional stability of a net-

work described with three-port S-parameters. It has been

shown that nine conditions, dependent O! the termina-

tions, must be satisfied to guarantee that lS;l <1 for

i =1,2,3. Measured data have been analyzed and plotted,

thus providing the designer with a grap~cal r~presentation

of the type of termination (capacitive, inductive, etc.)

necessary to provide three-port unconditional stability.

These plots also provide stability information for all possi-

ble two-port configurations.

ACKNOWLEDGMENT

The authors would like to thank Dr. J. D. Dyson for his

support of this work and his invaluable suggestions and

comments. The authors would also like to thank Dr. J.

Galli and D. E. Peck of the Watkins-Johnson Company

for the use of facilities and for guidance in microwave

devi~e characterization.

[1]

[2]

[3]

[4]

REFERENCES

G. E. Bodway, “Circuit design and characterization of transistorsby means of three-port scattering parsugeters,” Microwave J., vol.11, no. 5, May 1968.A. P. S. Khanna, “Parallel feedback fetdro design using three-portS-parameters,” in Proc. IEEE A4TT-S (SW Francisco), 1984, pp.181-183.A. P. S. I@rma, “Efficient low noise three-port X-band FEToscillator using two dielectric resonat~rs,” in Proc. IEEE MTT-S(Dallas), 1982, pp. 277-279.D. Woods, “Review of methods for the characterization of activedevices in terms of two-port and three-port pqameters,” in IEEColloquim Dig. Q.ondon), vol. 13, RF Measurements on Solid StateDeuices, 1971, pp. 1-8.

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[10]

S. Tanaka, N. Shumonura, and K. Ohtake, “Active circulators—Therealization of circulators using transistors,” Proc. IEEE, vol. 53,

pp. 260–267, Mar. 1965.W. H. Ku, “ Stabifity of linear active nonreciporical N-ports; J.Franklin Inst., vol. 276, pp. 207-224, Sept. 1963.H. Kleinwatcher, ‘:.Zur Widerstandstransformation Linear 2N-Polej” Arch. Elek. (,lbertragrorg, vol. 10, pp. 26-28, Jan. “1956.M. F. Abulela, “Studies of some aspects of linear amplifier designin terms of measurable two-port Wd three-port scattering parame-ters: Ph.D. dissertation, Manchester Univ Manchester, England,1972.D. Woods, “Reappraisal of the unconditional stability criteria foractive 2-Port networks’ in terms of S-parameters,” IEEE Trans.Circuits Syst., vol. CAS-23, pp. 73-81, Feb. 1976.J. F. Boehm, “Microwave oscillator design using two-port andthree-port scattering parameters,” Master’s thesis, Univ. Illinois,Urbana, Oct. 1985.

John F. Boehm (S82—M85) was born inChicago, IL, on February 22, 1961. He receivedthe B.S. and M.S. degrees in electncaf engineer-ing in 1983 and 1985, respectively, from theUniversity of Illinois, Urbana.

In 1985, he joined the Subsystems R&D De-partment of the Watkins-Johns~n Company, PaloAlto, CA, where he is currently involved in mul-tiport device characterization and GaAs MMICamplifier development. His research interests in-clude multiDort device characterization and cir-.

cuit design, low-noise amplifier design, and opticaf communications.Mr. Boehm is a member of Eta Kappa Nu.

William G. Albnght is Professor of ElectricalEngineering Emeritus at the University of 11-linois, Urbana.