COMPUTER-AIDED SENSITIVITY ANALYSIS OF TRANSISTOR MICROWAVE OSCILLATORS

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Nebil Tanzi and Thomas T.Y. Wong Department of Electrical and Computer Engineering

lllinoie Institute of Technology Chicago, Illinois 60616

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Abstract

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A methodology to perform sensitivity analysis on a given oscillator circuit prior to its construction is presented. The method uses a computer program developed to obtain the solution of the oscillation equations, and a commercially available curve-fitting routine. Results of the sensitivity analysis performed on a 2GHs oscillator circuit are presented.

I. INTRODUCTION

Over the years much effort has been expended to formulate an accurate and repeatable oscillator design methodology that allows one to predict the performance of the oscillator with a high level of accuracy. Although good results have been obtained with the developed methods, the sensitivity of the resultant circuits to the changes in the active device’s 8 -parameters has not been sufficiently investigated. Part of the difficulty in performing such a sensitivity analysis s t e m from the nonlinear nature of the oscillation equations. For instances where the oscillation equations are linear and closed form solutions for the embedding elements exist, it is found that the closed form expressions are not in terms of the s-parameters of the active device [ll.

A direct method of determining the sensitivity of the oscillator circuit is to fabricate multiple embedding circuits, and place another active device of the same type in each one and compare the performances of the circuits. Although this method can give the designer an idea about how tolerant the circuit is to changes in the s-parameters of the active device, it would prove to be quite time consuming and cumbersome.

In this work we have implemented a methodology which allows one to perform sensitivity analysis on a given oscillator circuit prior to its construction. The method uses a computer program developed to obtain the solution of the oscillation equations, and a commercially available curve-fitting routine. The sensitivity analysis is initiated by varying one s -parameter over a specified range, while retaining all others constant, and solving the oscillation equations to obtain the embedding elements [l]. Once completed, the results of these computations are used to characterbe thevariation of each of the embedding elements with the 8-parameters of the active device. The computational results are plotted and an equation 0-7803-1254-6/93$03.00 (0 1993 IEEE

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describing the dependence of each of the embedding elements on the s -parameters of the transistor are obtained with the use of the curve-fitting software. Classical sensitivity analysis methods are applied to these equations to obtain a measure of the fractional change in the embedding elements with respect to each of the s -parameters of the transistor. Since in a real circuit more than one of the s-parameters of the active device may deviate from its typical value, multi-parameter sensitivity analysis may be used to obtain an overall measure of the fractional change in the embedding elements of the oscillator circuit if the second and higher order sensitivities are sufficiently small [2,3].

11. THE OSCILLATOR CIRCUIT

The developed sensitivity analysis methodology has been applied to a previously designed oecillator circuit at 2 GHs (1,4,5]. The lumped element prototype and the ~.

microstrip implementation of the circuit are shown Figure 1.

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n

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Figure 1. Lumped circuit prototype and microstrip layout of the series resonant oscillator circuit.

The nominal values of the device s -parameters are given in Equation (1).

sll = I sI1 I Le 11 = 0 . 3 9 ~ 1 6 6 0

sal = I s21 I = 3 . 5 4 ~ 6 0 0

s12 = I s12 I Le 12 = 0.08L58 0

s22 = I sZ2 I L e , = 0 . 4 6 ~ - 37 0

( 1)

In Figure 1 the three impedances that comprise the embedding circuit of the oscillator are as follows: RI and XI represent the general impedance 21, and X 2 and X3 represent the general impedances 2 2 and 2 3 respectively. In the microstrip implementation of the circuit, X 2 is realized with the transmission line section TL1, X3 is realised with transmission line sections TL2, and the transmission line sections TL3, TL4 and TL5 transform the 50 Cl load to the general impedance 21 seen at the base of the device.

111. SENSITIVITY ANALYSIS OF THE CIRCUIT

The sensitivity analysis of the circuit is initiated by varying each of the s-parameters of the transistor by f 20% about the nominal value one at a time, and solving the oscillation equations to obtain the embedding elements. The computationalresults are used to obtain a graph of each of the embedding element versus the 8 -parameters of the transistor. These graphs are shown in Figures 2-5. It is seen from these graphs that certain parameters of the transistor, namely, 811, I S21 I , and I S22 I have significant second or higher order effects on the embedding elements, while the remaining parameters can characterize the embedding elements by polynomials with small contributions from second and higher order terms. For smallvariations near the nominal values of I SI1 I , I S I 2 I , 6'12,821 and On, the curves for each of the embedding elements can be approximated by a straight line and therefore the f i t -order sensitivity would accurately predict the fractional change in the embedding elements.

In the next stage, commercially available curve-fitting software is used to obtain a polynomial describing the dependence of the embedding elements on the s -parameters. The polynomial will enable one to asses the contribution of second and higher order terms. The first-order sensitivity of each of the embedding elements to each of the s-parameters is computed using the well-known sensitivity expression [2,3]

The results of these computations are given in Equations (3)-(15).

SI XI s,, I = - 1.781, se,: X = - 0.295. (15)

The computed first-order sensitivities show that the circuit is sensitive to changes in the $-parameters. In particular, the load resistance, R I , and the feedback element, XJ, are more sensitive than X1 and X2 respectively to changes in the 8 -parameters of the transistor.

The fractional change of the embedding elements to simultaneous changes in all the 8-parameters can be computed, by multi-parameter sensitivity analysis using the expression in Equation (16), if the second and higher order sensitivities are sufficiently small [2,3].

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It is expected, based on the curves shown in Figures 2-5, that the expression in Equation (16) would accurately predict the fractional changes in values of the embedding elements to small simultaneous variations in I SI1 1 , I $12 I , 812, 821 and Om about their nominal values. The embedding elements are computed from the expression in Equation (16) and also from the oscillation equations for comparison purposes. The results of these computations are given in Tables 1 and 2.

Table 1. +5% Change in I SII I , I S12 , 812, 821 and 822

Embedding Sensitivity Oscillation Element Analysis Equation

R l R SI s,, I = - 1.950, so,: = - 0.414, (5)

76.260 R1 PI 77.845 Xi [Cl] -190.917 -182.452 x2 [fll 175.182 172.553 X3 [fl] -115.568 -111.5094

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Table 2. - 5% Change in I S11 I, I S12 1 , 012, 021 and On

Embedding Sensitivity Oscillation Element Analysis Equation

R1 In1 77.845 76.260

Xi [n] -190.917 -182.452 x2 In1 175.182 172.553 X3 [nl -115.568 -111.5094

It is seen from Tables 1 and 2 that the multi-parameter sensitivity analysis accurately predicted the values embedding elements for small changes in I S11 I , I S12 I ,012, 021 and 022 about their nominal values. Inclusion of ell, I S2l 1 , I S22 1 , in the multi-parameter sensitivity

computation, results in large errors due to the fact that these parameters have significant second and higher order contributions. The results of the multi-parameter sensitivity computations with these parameters included are shown in Tables 3-6. Tables 3 and 4 show the results with I S22 I included, and Tables 5 and 6 show the results when all the s -parameters are included in the computations.

Table 3. + 5% Deviation in I S11 1 , 1 S12 I, 012, Ozl, IS221 a n d e n

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Embedding Sensitivity Oscillation Element Analysis Equation

R1 IQ1 66.168 59.152 Xi [n] -183.465 -153.736 x2 IQ1 169.546 159.506 X3 [n] -101.976 -87.889

Embedding Sensitivity Oscillation Element Analysis Equation

RI PI 173.380 168.046

x2 In1 221.694 215.8519 Xi [n] -301.088 -286.032

X3 [a] -203.311 -195.015

Table 5. + 5% Deviation in all the S-Parameters

Embedding Sensitivity Oscillation Element Analysis Equation

R1 In1 38.802 10.202 Xi [n] -199.864 -47.239 x2 PI 170.393 123.9458 x3 In1 -92.738 -24.849

Table 6. - 5% Deviation in ad the S-Parameters

Embedding Sensitivity Oscillation Element Analysis Ecluation

200.747 180.3814 R1 PI x1 [n] -284.688 -236.460

x2 PI 220.847 203.033 X3 [n] -212.549 -183.212

It is seen that including I S n l in the computations introduces small but significant errors, while including 011 and 15’21 I in the computations increases the amount of error due to significant second and higher order contributions from these parameters.

IV. CONCLUSIONS

A methodology to perform sensitivity analpia of transistor microwave oscillators and the computational results are presented. It is seen that the embedding elements are sensitive to variations in the magnitudes of the s -parameters, in particular to I Sl2 I and I S21 I, while being relatively less sensitive to variations in the phase angles of the device’s s -parameters. Furthermore, multi-parameter sensitivity analysis is performed on the circuit. For smalldeviations about the nominalvalues of the s -parameters, it is seen that when only I S11 1 , I S12 I , 812, Ozl and 0n are included that the results are accurate, while when all the s -parameters are included the amount of error increases due to significant second and higher order contributionsfrom 011, I S21 I and I Sn I.

REFERENCES

[l] Tanzi, N., “Microwave Oscillator Design and Circuit Implementation Using Computer-Aided Design Tools,” Ph.D. Dissertation, Il1;nois Institute of Technology,Chicago, Illinois, May 1992.

[2] Chen, W-K., “Passive and Active Passive Filters Theory and Implementations,” John Wiley and Sons Inc., New York, NY, 1986.

(31 Brutton, L.T., “RC-Active Circuits: Theory and Design ”Prentice-Hall, Englewood Cliffs, NJ, 1980.

[4] Tanzi, N., and Wong, T.T.Y., “A Comprehensive Approach to High-Frequency Oscillator Design and Implementation Using CAD Tools,” in 1992 IEEE AP/URSI/NEM/ Joint Symposia URSI Digest, p.192, Chicago, July 1992.

151 Tanzi, N. and Wong, T.T.Y., “Series and Parallel Resonant Microwave Oscillator Design and Implementation Using Computer- Aided Design Routines,” in Proceedings of the 35th Midwe~t Symposium on Circuits and Systems, pp.tbd Washington D.C., August 1992.

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811 (Degrees) Figure 2. Variation of the Embedding Elements of the

Series Resonant Oscillator Circuit with S11

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Figure 3. Variation of the Embedding Elements of the Series Resonant Oscillator Circuit with Si2

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Series Resonant Oscillator Circuit with 5'21

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