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transformation capabilities is invaluable in the design of balanced mi-crowave circuits such as mixers, push–pull amplifiers, and frequency doublers.
The authors would like to thank M. Blewett and D. Granger for their technical support in fabricating the circuits.
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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
Index Terms—Poles, -parameters, transistors, zeros.
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 , 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.
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: firstname.lastname@example.org).
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  with some necessary circuit element modifications as fol-lows: C0 gs=1 + gCgs mRS (1) R0 i= Ri+ RS (2) C0 ds=1 + gCds mRS (3) R0 ds= Rds1 (1 + gmRS) (4) C0 gd= Cgd (5) g0 m=1 + ggm m1 RS (6)
wheregm = gmo1 exp(0j!) and gmois 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= ZZin0 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) + s3C0 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, usuallyR0ican be ne-glected, as well as the product ofR0iCds0 . 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  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 ofS11, 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= ZZin0 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= VVO2 1 1 2 = 02AM 1 0 sCg0gd0 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 ofVO2=V11 !Z5= gm0 =Cgd0 .
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= ZZout0 ZO2 out+ ZO2 = Rds0 + Rd0 ZO2 R0 ds+ Rd+ ZO2 1 N2(s)D(s) =RRds00 + 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(R0ds+ Rd0 ZO2)=(R0ds+ Rd+ ZO2) is the reflection
coefficient thatV2“sees” at dc frequency.
As forS12, the physical meaning ofS12is 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 gdZO1D(s)1 = 2 1 Rds0 R0 ds+ Rd+ ZO2 1 sC0 gdZO1 1 + s! P 1 1 + s!P 2 (13)
where(R0ds1 sCgd0 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  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
EXPRESSIONS OFPOLES ANDZEROS OFS ANDS BYDOMINANTPOLE
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!p2given in Table I has the physical meaning of the inverse of the time constant of the par-allel combination ofZout0 ,R0ds, 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 ofS22,!Z3, and!Z4can be obtained easily by replacingZO2 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
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 ofjS11j 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 ofS11. The two zeros ofS22fall 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 ofS22observed 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.
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 ofS22 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.
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