406 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 49, NO. 2, FEBRUARY 2001

transformation capabilities is invaluable in the design of balanced mi-crowave circuits such as mixers, push–pull amplifiers, and frequency doublers.

ACKNOWLEDGMENT

The authors would like to thank M. Blewett and D. Granger for their technical support in fabricating the circuits.

REFERENCES

*[1] T. Chen et al., “Broadband monolithic passive baluns and monolithic*
*double balanced mixer,” IEEE Trans. Microwave Theory Tech., vol. 39,*
pp. 1980–1986, Dec. 1991.

[2] P. C. Hsu, C. Nguyen, and M. Kintis, “Uniplanar broad-band push–pull
*FET amplifiers,” IEEE Trans. Microwave Theory Tech., vol. 45, pp.*
2150–2152, Dec. 1997.

[3] S. A. Maas and Y. Ryu, “A broadband, planar, monolithic resistive
*frequency doubler,” in IEEE Int. Microwave Symp. Dig., 1994, pp.*
443–446.

[4] A. M. Pavio and A. Kikel, “A monolithic or hybrid broadband
*compen-sated balun,” in IEEE Int. Microwave Symp. Dig., 1990, pp. 483–486.*
[5] W. R. Brinlee, A. M. Pavio, and K. R. Varian, “A novel planar

*double-balanced 6–18 GHz MMIC mixer,” in IEEE Microwave*

*Millimeter-Wave Monolithic Circuit Symp. Dig., 1994, pp. 139–142.*

*[6] M. C. Tsai, “A new compact wide-band balun,” in IEEE Microwave and*

*Millimeter Wave Monolithic Circuit Symp. Dig., 1993, pp. 123–125.*

[7] K. Nishikawa, I. Toyoda, and T. Tokumitsu, “Compact and broad-band
*three-dimensional MMIC balun,” IEEE Trans. Microwave Theory Tech.,*
vol. 47, pp. 96–98, Jan. 1999.

*[8] Y. J. Yoon et al., “Design and characterization of multilayer spiral *
*trans-mission-line baluns,” IEEE Trans. Microwave Theory Tech., vol. 47, pp.*
1841–1847, Sept. 1999.

*[9] N. Marchand, “Transmission line conversion transformers,” Electronics,*
vol. 17, no. 12, pp. 142–145, Dec. 1944.

[10] C. M. Tsai and K. C. Gupta, “A generalized model for coupled lines and
*its applications to two-layer planar circuits,” IEEE Trans. Microwave*

*Theory Tech., vol. 40, pp. 2190–2099, Dec. 1992.*

[11] R. Schwindt and C. Nguyen, “Computer-aided analysis and design of
*a planar multilayer Marchand balun,” IEEE Trans. Microwave Theory*

*Tech., vol. 42, pp. 1429–1434, July 1994.*

[12] R. H. Jansen, J. Jotzo, and R. Engels, “Improved compactions of a planar
multilayer MMIC/MCM baluns using lumped element compensation,”
*in IEEE Int. Microwave Symp. Dig., 1997, pp. 227–280.*

[13] T. Wang and K. Wu, “Size-reduction and band-broadening design
tech-nique of uniplanar hybrid coupler techtech-nique of uniplanar hybrid ring
*coupler using phase inverter for M(H)MIC’s,” IEEE Trans. Microwave*

*Theory Tech., vol. 47, pp. 198–206, Feb. 1999.*

*[14] S. A. Maas, Microwave Mixers, 2nd ed.* Norwood, MA: Artech House,
1992.

**A Novel Interpretation of Transistor** **-Parameters by**
**Poles and Zeros for RF IC Circuit Design**

Shey-Shi Lu, Chin-Chun Meng, To-Wei Chen, and Hsiao-Chin Chen

**Abstract—In this paper, we have developed an interpretation of ****tran-sistor** **-parameters by poles and zeros. The results from our proposed**
**method agreed well with experimental data from GaAs FETs and Si**
**MOSFET’s. The concept of source-series feedback was employed to**
**analyze a transistor circuit set up for the measurement of the**
**-param-eters. Our method can describe the frequency responses of all transistor**
**-parameters very easily and the calculated** **-parameters are scalable**
**with device sizes. It was also found that the long-puzzled kink phenomenon**
**of** **observed in a Smith chart can be explained by the poles and zeros**

**of** **.**

**Index Terms—Poles,****-parameters, transistors, zeros.**

I. INTRODUCTION

Wireless circuit design has recently become an important field all over the world. However, as far as RF circuit design is concerned, mi-crowave circuit designers are talking aboutS-parameters, while analog circuit designers are thinking in terms of poles and zeros. Obviously, there is a gap between the thought processes of microwave circuit en-gineers and analog circuit enen-gineers. For a long time,S-parameters have been understood in terms ofY - or Z-parameters. These Y - or Z-parameters, though very useful in calculating S-parameters, cannot give insight into the behaviors or physical meanings ofS-parameters. For example, it is difficult forY - or Z-parameters to describe the fre-quency responses ofS-parameters directly or to explain the kink be-havior ofS22observed in a Smith chart [1], such as the one shown in Fig. 1. In this paper, we present an interpretation ofS-parameters from a poles’ and zeros’ point-of-view. By doing this, we can predict the frequency responses ofS-parameters very easily and explain the kink behavior ofS22 in Smith charts. Our calculated values of transistor S-parameters showed excellent agreement with the experimental data from 0.25-m-gate Si MOSFETs and sub-micrometer gate GaAs FETs with different gate width.

II. THEORY

First, consider the circuit shown in Fig. 2(a), where an FET is con-nected for the measurement of itsS-parameters. S11andS21can be measured by settingV2 = 0 and V1 6= 0, while S22andS12can be measured by settingV1= 0 and V26= 0. ZO1at input port andZO2

at output port are both equal to 50, but are intentionally labeled dif-ferently. The reason for this will become clear later on. In general, the circuit in Fig. 2(a) is not easy to handle. However, the problem will be much easier to solve if this circuit is viewed as a dual feedback circuit, in whichRs is the local series–series feedback element andCgdthe local shunt–shunt feedback element. In order to simplify circuit anal-ysis, we temporarily neglect the inductors and transform the circuit of Fig. 2(a) into that of Fig. 2(b) by using local series–series feedback

Manuscript received December 9, 1999. This work was supported under Grant 89-E-FA06-2-4, under Grant NSC89-2219-E-002-044, and under Grant NSC88-2219-E-005-003.

S.-S. Lu, T.-W. Chen, and H.-C. Chen are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: sslu@cc.ee.ntu.edu.tw).

C.-C. Meng is with the Department of Electrical Engineering, Chung-Hsing University, Taichung, Taiwan 40227, R.O.C.

Publisher Item Identifier S 0018-9480(01)01089-4. 0018–9480/01$10.00 © 2001 IEEE

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 49, NO. 2, FEBRUARY 2001 407

Fig. 1. Comparison of the experimental and calculatedS-parameters of a sub-micrometer gate GaAs FET with gate width of 4 mm on a Smith chart. —: experimental data. —— : calculated values by our theory. Note that the kink phenomenon ofS is indicated by an arrow.

theory [2] with some necessary circuit element modifications as
fol-lows:
C0
gs=_{1 + g}Cgs
mRS (1)
R0
i= Ri+ RS (2)
C0
ds=_{1 + g}Cds
mRS (3)
R0
ds= Rds1 (1 + gmRS) (4)
C0
gd= Cgd (5)
g0
m=_{1 + g}gm
m1 RS (6)

whereg_{m} = g_{mo}1 exp(0j!) and g_{mo}is the dc transconductance
and all the other symbols have their usual meanings.

SetV2 = 0 and V1 6= 0 for the discussions of S11 andS21. If the input impedance seen to the right-hand side ofZO1in Fig. 2(b) is denoted byZin, thenS11is given by

S11= Z_{Z}in0 ZO1

in+ ZO1: (7)

By definition, the poles ofS11are the roots ofL(s) = Zin+ZO1= 0.

Zincan be calculated andL(s) = 0 can be proven to be equivalent to

equationD(s) = 0 given by
D(s) = 1 + s C0
gs ZO1+ Rg+ R0i + Cds0 (ZO2+ Rd)
+ C0
gd g0m(ZO1+ Rg)(ZO2+ Rd)
+(ZO1+ Rg) + (ZO2+ Rd)) + s2
1 Cgd0 Cgs0 + Cgd0 Cds0 + Cgs0 Cds0 (ZO1+ Rg)
1 (ZO2+ Rd) + Cgs0 Cgd0 + Cgs0 Cds0 R0i
1(ZO2+ Rd) + Cgs0 Cgd0 R0i(ZO1+ Rg)
+ s3_{C}0
gsCgd0 Cds0 R0i(ZO1+ Rg)(ZO2+ Rd) = 0: (8)

The expression ofD(s) in (8) may be complicated at a first glance
because it involves a cubic equation. However, usuallyR0_{i}can be
ne-glected, as well as the product ofR0_{i}C_{ds}0 . Thus, (8) is reduced to a

Fig. 2. Setup for the measurement of transistorS-parameters. (a) Complete circuit. (b) Simplified circuit with the local series–series feedback element (R ) absorbed.

quadratic equation as follows, by which two poles!_{p1} and!_{p2} can
be solved easily:
D(s) = 1 + s C0
gs(ZO1+ Rg) + Cds0 (ZO2+ Rd)
+ C0
gd(gm0 (ZO1+ Rg)(ZO2+ Rd)
+(ZO1+ Rg) + (ZO2+ Rd))
+ s2 Cgd0 Cgs0 + Cgd0 Cds0 + Cgs0 Cds0
1(ZO1+ Rg)(ZO2+ Rd) = 0: (9)
For the discussions that follows, for convenience, we will call (8)
the three-pole approximation and (9) the two-pole approximation. In
solving (9), the method of dominant pole approximation [3] can be
utilized if a dominant pole indeed exists, i.e., the lowest frequency
pole is at least two octaves lower than the other pole. As an illustrative
example, the solutions (poles) of (9) found by the dominant pole
approximation are listed in Table I.

The zeros ofS11are the roots ofZ(s) = Zin0 ZO1 = 0. This

zero equation can be viewed as the transformation of the pole equation
L(s) = Zin + ZO1 = 0 with ZO1 inL(s) is replaced by 0ZO1.
Equivalently, the zero equation ofS_{11}, which we call N1(s) = 0,
can be obtained easily by replacingZO1 inD(s) = 0 with 0ZO1.
At dc frequency,S11= 1 because Cgs0 andCgd0 are open circuits and,

therefore,S11can be written as follows for all frequencies:
S11= Z_{Z}in0 ZO1
in+ ZO1
= 1 1 N1(s)_{D(s)}
= 1 1
1 + s_{!}
Z1 1 + s!Z2
1 + s_{!}
P 1 1 + s!P 2
(10)

408 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 49, NO. 2, FEBRUARY 2001

The physical meaning of S21 is twice that of the voltage gain VO2=V1. Since the poles of all the fourS-parameters are the same, S21can be given easily by the inspection of Fig. 2(b) as follows:

S21= V_{V}O2
1 1 2
= 02AM
1 0 sC_{g}_{0}gd0
m
1 + sC0
gsR0i
D(s)
= 02AM
1 0 s_{!}
Z5
1 + s_{!}
P 1 1 + s!P 2
(11)

whereAM= g0mZO2(Rds0 k(Rd+ZO2))=(ZO2+Rd) is the midband

gain ofV_{O}2=V_{1}1 !_{Z5}= g_{m}0 =C_{gd}0 .

We now turn our attention intoS22andS12by settingV1= 0 and

V2 6= 0. If the output impedance seen to the left-hand side of ZO2is denoted byZout, thenS22is given by

S22= Z_{Z}out0 ZO2
out+ ZO2
= Rds0 + Rd0 ZO2
R0
ds+ Rd+ ZO2 1 N2(s)D(s)
=R_{R}ds0_{0} + Rd0 ZO2
ds+ Rd+ ZO2 1
1 + s_{!}
Z3 1 + s!Z4
1 + s_{!}
P 1 1 + s!P 2
(12)

whereN2(s) is the zero equation of S22and is obtained by replacing
ZO2 inD(s) = 0 with 0ZO2.!Z3and!Z4are the zeros ofS22.
The factor(R0_{ds}+ Rd0 ZO2)=(R0ds+ Rd+ ZO2) is the reflection

coefficient thatV2“sees” at dc frequency.

As forS_{12}, the physical meaning ofS_{12}is twice that of the voltage
gainVO1=V2.S12is given by the inspection of Fig. 2(b) as follows:

S12= 2 1 R
0
ds
R0
ds+ Rd+ ZO2 1 sC
0
gdZO1_{D(s)}1
= 2 1 Rds0
R0
ds+ Rd+ ZO2 1
sC0
gdZO1
1 + s_{!}
P 1 1 + s!P 2
(13)

where(R0_{ds}1 sC_{gd}0 1 ZO1)=(R0ds+ Rd+ ZO2) is the gain of VO1=V2

at low frequencies.

III. EXPERIMENTALRESULTS ANDDISCUSSIONS

(1)–(13) have been applied to Fujitsu GaAs FETs with different gate width (0.5, 1, 2, and 4 mm). The effects of inductors in the circuit of Fig. 2(a) were included by replacingRg,Rs, andRdwithRg+j!Lg, Rs+ j!Ls, andRd+ j!Ldin related formulas, respectively. The S-parameters of the transistor with gate width of 4 mm were plotted in a Smith chart, as shown in Fig. 1. Excellent agreement between cal-culated values and experimental data [4] can be seen clearly in Fig. 1. The method has been applied to Si MOSFETs with good agreement as well.

The frequency responses of the fourS-parameters of the Fujitsu GaAs FETs with gate width of 2 mm are shown in Fig. 3(a) and (b), respectively. The characteristics of the devices with gate width of 0.5 and 1 mm are similar to those of the device with gate width of 2 mm

TABLE I

EXPRESSIONS OFPOLES ANDZEROS OFS ANDS BYDOMINANTPOLE

(ZERO) APPROXIMATION

and, therefore, for clarity, are not shown in Fig. 3(a) and (b). The
S-pa-rameters of the device with gate width of 4 mm also shows similar
frequency responses, except for itsS22, whose frequency response is,
therefore, included in Fig. 3(b), as well as for comparison. The
loca-tions of poles and zeros have to be known in order to discuss the
charac-teristics of the frequency responses of theseS-parameters. In general, it
involves solving the cubic equation of (8) to find the three poles, which
is, of course, difficult. However, as we mentioned previously, if the
third-order term is negligible, then (8) becomes quadratic equation (9),
which can be solved easily. As can be seen clearly in Fig. 3(a) and (b),
the two-pole approximation based on (9) indeed gives very satisfactory
results. Dominant pole (zero) approximation can be used to obtain the
expressions of the poles and zeros. The results are given in Table I. It
can be shown that the expression of the second pole!_{p2}given in Table I
has the physical meaning of the inverse of the time constant of the
par-allel combination ofZ_{out}0 ,R0_{ds}, andRd+ZO2. A significant advantage
of our theory is that, once the expressions of the two poles are known,
the expressions of the zeros ofS11andS22are readily obtained. The
zeros ofS11,!Z1, and!Z2can be obtained easily by replacingZO1

with0ZO1in the expressions of!p1and!p2, respectively. The zeros
ofS_{22},!_{Z3}, and!_{Z4}can be obtained easily by replacingZ_{O2} with
0ZO2in the expressions of!p1and!p2, respectively.

From (11) and the two-pole approximation, the frequency response ofS21has two poles and one zero. This zero is usually much larger than the two poles, which results in the shape of the Bode plot ofjS21j

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 49, NO. 2, FEBRUARY 2001 409

(a)

(b)

Fig. 3. Frequency responses of the magnitude of the fourS-parameters of a Fujitsu sub-micrometer gate GaAs FET with gate width of 2 mm. (a)S and S . (b) S and S . —: experimental data. —— : three-pole approximation. 4

—— : two-pole approximation. The frequency responses of the other size FETs show similar characteristics, except for the one with 4-mm gate width, whose S behaves differently.

shown in Fig. 3(a). From Fig. 3(a), the first pole!p1has the physical meaning of 3-dB bandwidth of the voltage gainS21.

From (10) and the two-pole approximation,S11has two poles and
two zeros. The shape of the frequency response ofjS_{11}j is dependent

on the locations of its two zeros with respect to its two poles. From Table I, it is easy to see that the two zeros fall between the two poles and, therefore, a dip will occur in a Bode plot ofjS11j, as can be seen

in Fig. 3(b).

According to (12) and the two-pole approximation, S22 has two
poles and two zeros just like the case ofS_{11}. The two zeros ofS_{22}fall
between its two poles in the cases of 0.5, 1, and 2 mm. Hence, a dip is
observed in the Bode plot ofjS22j, as shown in Fig. 3(b), which is one

reason that causes the kink phenomenon ofS22observed in a Smith chart. The kink effect inS22for a smaller device (0.5 mm) or inS11

for all sizes of devices are obscured by pole–zero cancellation. When the device size is increased to 4 mm,!Z3ofS22becomes smaller than !P 1 and, thus,jS22j looks like a sloped step, as shown in Fig. 3(b).

The alternative appearance of poles and zeros is another reason for the
kink phenomenon ofS_{22}observed in a Smith chart (see Fig. 1).

According to (13) and the two-pole approximation, S12 has two poles and one zero. This zero occurs at zero frequency and explains the bell-shape frequency response ofjS12j in Fig. 3(a) well.

IV. CONCLUSIONS

In summary, theS-parameters of transistors have been interpreted in terms of poles and zeros. All the fourS-parameters have the same two poles. It is found that the two zeros ofS11always fall between its two poles and, hence, a dip is observed in the frequency response ofjS11j.

The locations of the two zeros ofS22with respect to its two poles are
dependent on the device size. For smaller transistors,S22behaves
sim-ilarly toS11. For larger transistors, one zero ofS22becomes smaller
than its first pole and, therefore, the shape of a sloped step is observed,
which is one reason for the kink phenomenon ofS_{22} observed in a
Smith chart.S12has a zero at zero frequency and this explains why
the frequency response ofjS12j looks like a bell shape. Our proposed

method thus provides a certain insight into the behavior of the S-pa-rameters and, hence, may be helpful for RF integrated circuit (RFIC) designs.

REFERENCES

[1] B. Bayrajtariglu, N. Camilleri, and S. A. Lambert, “Microwave
perfor-mance of n-p-n and p-n-p AlGaAs/GaAs heterojunction bipolar
*transis-tors,” IEEE Trans. Microwave Theory Tech., vol. 36, pp. 1869–1873,*
Dec. 1988.

*[2] P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated*

*Circuits.* New York: Wiley, 1993, pp. 579–584.

*[3] A. S. Sedra and K. C. Smith, Microelectronic Circuits, 4th ed.* Oxford,
U.K.: Oxford Univ. Press, 1993, pp. 595–601.

*[4] Y. Aoki and Y. Hirano, “High-power GaAs FETs,” in High Power GaAs*

*FET Amplifiers, J. L. B. Walker, Ed.* Norwood, MA: Artech House,
1993, p. 81.