# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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During calculation reference from the former table to the latter is made.

9.4.3 Program

The program named volfun consists of three table-generating procedures, an initialization procedure which defines arrays and reads tables into memory and a procedure to calculate the multi-port frequency domain Volterra transfer function. Once the tables are calculated only the latter two are used. The current version of the program volfun allows determination of up to tenth order transfer functions and non-linear subsystems with up to two input ports.

The program can be used in three different modes:

• ‘algf’ ~ A full algebraic evaluation where the multi-port frequency domain Volterra transfer function is given by a single expression as a function of the elements in the linear network and Volterra transfer functions of the non-linear subsystems in the overall network. This is usually used for visual inspection of relatively low order Volterra transfer functions.

• ‘algr’ - A recursive algebraic evaluation where a given oth order contribution is given by lower order contributions of the controlling ж-vectors. The result from this mode can be directly translated into FORTRAN 77 code using an internal Maple V procedure. This mode is used for relatively high orders and where a high speed FORTRAN 77 version is to be determined.

• ‘num’ - A numerical evaluation of the Volterra transfer function. As this version does not use hardware floating-point arithmetic this mode is usually used for numerical evaluation of relatively low order Volterra transfer functions. For higher order numerical evaluation the ‘algr' mode is used to produce FORTRAN 77 code

The user must specify: (i) the number A of signal input ports, (ii) the number Q of controlled variables, (iii) the number R of controlling variables. (:v) index lists j for ail q t {1,2,. .., Ii) to specify which variables control each о I the nonlinear subsystems, (v) procedures to calculate A~L(f), B(f), a(f) and b(f), (vi) procedures to calculate the multi-port frequency domain Volterra transfer functions

(GVbi,...,np(<,)(') lor all q £ {1,2......Q} of the non-linear subsystems, and (vii)

mode of operation which may be either of the types ‘algf'. ‘algr’ or 'num’.

242

9. Multi-port Volterra. transfer functions

9.5 Some types of non-linear subsystems

This section gives some examples of types of non-linear subsystems. The time

domain relation between inputs {ij j(t).......•cj,p(3)(^)} where jq,r £ {!,'2...., Д}

for r fc {1,2,.... p(q)} and output yq(t) for subsystem q fe {I, 2,. .., Q} is given, and the corresponding multi-port Volterra transfer function from Equation (9.45), (G,)n,..nPW(9\,l, • • ■ .....• ■. ,Sp(?),npU)), is presented without derivation. In the following (/C?)„,.nP(t;) is a real constant related to the trans-

fer oi signals tor the multi-port попТшеаг system ш Figure 9.2.

9.5.1 Type 1

A common type of non-linearity is given by the time domain input-output relation

CO CO P(?)

. y,w = .пр,„ П ^.до (9.86)

711=0 n^7)=0 Г—1

It can be shown that the corresponding multi-port frequency domain Volterra transfer function is given by

,...,rcP(ql (^1,1 > ■ * • > BUni ,.; ^Pf7)a : - - - • ) — (^(/)ni,...,nF(^

(9.87)

This type of non-linearity (note from Equation (9.86) that it does not have memory) can be used to represent e.g. a non-linear resistance, conductance, transresistance, transconductance, and current and voltage ga,in transfer.

9.5.2 Type 2

Consider the following type of non-linearity which contains memory:

I rpM

-[П ft)

(9.88)

yjt) = T (A),......

гц=0 np(q^=0

It can be shown that the corresponding multi-port frequency domain \ nlt.prra. trans-fer function is given by

(G,)

1/m nP(q 1

»1,1.

•: tt

P(q), 1> ' ' • ! &P(q),np(q) )

Р{ч)

(9.89)

9.5. Some types of non-linear subsystems

243

This type of non-linearity which contains memory can be used to represent e.g. a non-linear current-voltage capacitance.

9.5.3 Type 3

Consider the following type of non-linearity which contains memory:

оо CO .( P(q)

Уч(*-) = 51 ••• 51 / П (9-90)

n1=0 npM= 0 J 00 r = l

It can be shown that the corresponding multi-port frequency domain Volterra transfer function is given by

(G’g)„b...,Ilp(rt(^i,i,--------;................; &p(q),\-• • •, 0p(,).„

_ (^(? )ni ,....np(

j-2z XIpl-1 c'rp

(9.91')

This type of non-linearity which contains memory can be used to represent e.g. a non-linear voltage-current capacitance.

9.5.4 Type 4

Consider the following type of non-linearity which contains memory:

P(.i)

d

(9.92)

It can be shown that the corresponding multi-port frequency domain Volterra trans-fer function is siven bv

1^1

..0

'Р(П) П П °r.P

r=l p =1

This type of non-linearity which contains memory can be used to represent e.g. a non-iinear current-voltage inductance.

244

9. Multi-port Volterra. transfer functions

9.5.5 Type 5

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