# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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0.6. Examples

253

9.6.4 Example 4

Here the non-linear Volterra transfer function between input si and response vr in Figure 9.6(a) will be determined. In Figure 9.6, R\ = 4 Q, ~ 2 Q. and С ~ 1/2 F

Hi{xi)

Ri

(b) | j +

31 с ) lie ! gi\\zi =

j <ii ^ “

' лл 1 x‘

О Уг(х2)

Figure 9.6: Example 4: Non-iinear bridge network with one signal input port and two nonlinear elements, (a) Original network; (b) modified network where the non-linear subsystems aiO separated from trie нсси. *i^:urvijLK.

The overall network contains two single-port non-linear elements where the time domain relations are given as

irCt) — M iv' =

Ф) = L'H‘l) = (9.142)

HI ----

n:=l

where a\ = 1 H, a2 = 1/3 HA. gx = 1/2 S and ~ 1/5 SV. From Figure 9.6(b) the system matrices and vectors can be determined as

254

0. Multi-port Volterra transfer functions

3(.s2 + 2) —3s

(9.144)

(9.143)

where D(s) = 2sz + 3« + 4 and $ = j2~f has been used as frequency variable, and

a(s) =

Ms) = [0]

(9.145)

(9.146)

\T

The Volterra transfer functions for the two non-linear elements are determined from Equations (9.141)—(9.142) as

The results obtained are in agreement with Chua and Ng [6, Section 6.2] who have determined the Volterra transfer functions up to second order.

9.7 Conclusion

A method has been presented to determine the frequency domain Volterra transfer functions of non-linear multi-port, networks containing non-linear multi-port subsystems. The method is based on a generalization of the probing method to allow arbitrary frequencies. The Volterra transfer functions are described as functions of network specific vectors of the linear part of the overall network and as functions of variables which control the non-linear multi-port subsystems. A method has been derived to determine these controlling variables in гесшь'е form.

The method has been implemented in a symbolic programming language ( WapIc V- Re/ease ■'!:. which makes it possible to determine algebraic expressi. >iis lor : !:■ Volterra transfer functions. This implementation can directly translate the resulting Volterra transfer functions in recursive algebraic form into FORTRAN 77 code.

Four examples have been presented. The results obtained are in agreement with existing literature in the cases where comparison has been possible.

(9.147)

and

for щ = 2 otherwise

9.8. References

255

9.8 References

[1] Steer, М. B., Chang, C.-R. к Rhyne, G. W.: “Computer-aided analysis of nonlinear microwave circuits using frequency-domain nonlinear analysis techniques: the state of the art’, Int. J. Microwave and Millimeter-wave Computer-aided Engine.e.rma. vn! [ no. 2, pp. 181-200, 1991.

[2] Gilmore. R. J. к Steer, М. B.: "Nonlinear circuit analysis using the method of harmonic balance — a review of the art. II. Advanced concepts", Int. I. Microwave and Millimeter-wave Computer-aided Engineering, vol. 1. no. 2, pp. 159-180, 1991.

[3] Cussgang, Л. J., Ehrman, L. к Graham, J. W.: “Analysis of nonlinear systems with multiple inputs'’, Proc. IEEE, vol. 62, no. 8, pp. 1088-1119, 1974.

[4| Maas, S. A.: “A general-purpose computer program for the Volterra-series analysis of nonlinear microwave circuits”, IEEE Microwave Theory and Techniques Symposium Digest, pp. 311-314. 1988.

[5] Maas, S. A.: “C/NL linear and nonlinear microwave circuit analysis and optimization” , Artech House, 1990.

Г6] Chua. L. O. к Ng. C.-Y.. “Frequency domain analysis of nonlinear systems: formulation of transfer functions”, IEE J. Electronic Circuits and Systems, vol. 3. no. 6, pp. 257-269, 1979.

[7] Chua, h. О. к Ng, C.-Y.: "Frequency domain analysis of nonlinear systems: general theory’’. IEE •/. Electronic Circuits and Systems, vol. 3, no. 4, pp. 165-185, 1979.

[S] Lighthill, M. .1.: '‘Introduction to Fourier analysis and generalized functions’", Cambridge University Press, 1958-

p! Chua, L. О. к Lin, P.-L.: '‘'Computer-aided analysis of electronic circuits — algorithms and computational techniques", Prentice Hall, New Jersey, 1975.

Г10] Maas, S. A.: '“Nonlinear microwave circuits'’. Artech House, 1988.

[11] Minaaian, R. A.: “Intermodulation distortion analysis of MESFET amplifiers using the Volterra series representation'’, IEEE Traris. Microwave Theory and Techniques, vol. 25, no. 1, рл. 1-8, 1980.

А

Mathematical concepts

This appendix discusses the mathematical representation of noise. First there is a short introduction to stochastic processes which is the basis for the mathematical treatment of noise. Next the representation of noise in the frequency domain using a Fourier series expansion is discussed in detail. Finally, there is an investigation of signal energies and powers of random variables.

A.l Stochastic processes

Noise in electrical networks and systems can be considered as random fluctuations in time of. for example, voltage, current or charge. Statistical methods are used to describe the influence of such random noise signals in a qualitative way. These are, for example, to analyze the mean value (which is usually assumed identical to zero), variance, autocorrelation and spectral density of the noise signals.

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