# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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2In case a non-linear subsystem contains a first order (linear) contribution this must be included in the linear network and not in the non linear subsystem.

9.3. Theory

221

ling variables. At the signal input ports independent voltage or current generators {З] , sk{/)} are applied. The complementary currents or voltages of the (voltage or current respectively) signal generators , .sa'(/)} are not of interest

in this analysis. At the designated output port the response v(f) is either the oDen circuited voltage or the short circuited current. Thus the complementary current or voltage of the (voltage or current respectively) response v(f) is zero.

Figure 9.1: Non-lmear network separated into a purely linear network and purely nonlinear subsystems.

Each non-linear subsystem in the overall non-linear network is described by the general non-linear multi-port type shown in Figure 9.2. The variables

{x;,.i(/)»”->*i,.n,)(/)} where jq,T 6 {1-2........R} for q e {1,2,....Q} and r 6

{1,2,..., -P(i?)} are the controlling voltage or current variables and yq(f) is the cor-responding controlled voltage or current output variable for non-linear subsystem q £ {l.2,...,Q}. The subscripts for the controlling variables are collected in the

vectors {j,,.. .,jQ} where j.. = ..,jbP(,)]T 6 2£t7'*' for q G [1,2-----,Q}.

Each controlling input port is Either short: or open circuited and hence the port, volt age or current respectively is zero. The complementary current or voltage oi tlie controlled (voHa.geor current respectively) output variable yq{ f) where q С {!-• 2.Q)

is not of interest in the analysis. The general type of non-linear multi-port subsystem shown in Figure 9.2 can be used for one-port non-linear elements as weil as multi-port non-linear subsystems. One-port non-linear elements are represented

22 2

9. Multi-port Volterra. transfer functions

from Figure 9.2 by appropriate use of feedback between the output and the input port.

Figure 9.2: The used type of general non-linear multi-port subsystem. At the controlling signal input pores, {«Sj, v(/)s Я( >(/)} '^re the controlling variables where

jqx 6 {1,2, . . ., /2}, r £ {1,2_and q £ {1,2,.. Q}. At the output port, yq (/)

is the controlled output variable for non-linear subsystem 7 £ {1,2,...,Q}. P(q) is the number of controlling variables for non-linear subsystem q 6 {1,2,...,Q}. This multi-port non-linear subsystem can also be used to represent non-iinear one-port elements, e.g. non-linear resistors and capacitors, by appropriate feedback between the output and input ports.

The problem to be solved in the following is to determine the multi-port frequency domain Volterra transfer function relating the input signals {si to the output response v(f').

9.3.1 Relations for the linear system

By appropriate arrangement of the network variables and usmg the above properties for the non-linear subsystem?, the linear system can, alter soma considerations, be described by the matrix equation

Г / r\ 1 y\ j } Л\Л ' x(f) + " В\J) ~ s(f) (9.12)

I v{f) J . aT(f) . L ьТи) \

where

9.3. Theory

223

and

y(J) = ЫЛ- ■ -,г/с(/)]т

г- Г n

^(/)

я(Л

Г-r. (' П

l~ L\j

-^1,1 (/) ЛЯл{1)

Di.iif)

bqAI)

mT

[fll (/).

-■*<?,«(Л

вы I)

Bq.kU) J

CQxi

у 1

а(Л = [si(/)7-----------6 CK*'

6 С

QxR

ew>

с fflxi

• • • “Л-, j /j c-

(9.13)

(9.14) (9.151

(9.16)

(9.17

(9.18)

KS)

W/).....W/)f

(9.19)

As seen from Equations (9.16)—(9.19) the sizes of the system matrices and vectors depend on the number of controlling variables, the number of controlled variables and the number of signal input ports. Thus the sizes of A(f), B(f), a(f) and b(J) do not depend on the number of linear elements in the overall network. However, it is obvious that the linear system matrices and vectors get more complicated the more linear elements there are in the network. That is, the number of multiplications and additions required to determine A(f), B(f), a[f) and b(f) increases with the number of linear network elements. The system matrices and vectors of the linear network A(f), B(f), a(f) and b(f) may be determined using standard techniques [9]. However, when the system vectors and matrices are determined, {wj(/),. . ., i/ol /)) should be interpreted as the port voltages or currents and not as generators since

{«!(/)■

still be viewed as generators since these are indenendent variables.

{/)} are dependent variables — the variables

SKU -

9.3.2 Non-linear response

An expression for the non-linear response v(f) is determined with the signals at the К signal input ports as given by Equation (9.8) and h = mj, for all к £ {1,2,..., A }. Thus the output response vif) is given bv

224

9. Multi-port Volterra transfer functions

CO £ •X) £

mi = Э m =0

Hmi fc ( Vl

r A' rnk

exp ;££

L i (=i

• £ • £

1 AM 1 * Я',тд" — 1

■>........i "’Л',1АМ >•••> Фк,1к,,

К mk .

From Equation (9.12) it is seen that another expression for v(f) is ’-'(/) = + bT(f) s(f)

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