# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Consider the following type of non-linearity which contains memory

oo CO P(q) r t 1nr

Уя(t) = E (£,)*,,...,пя(ч) П / х.]яг{т)6.т\ (9.94)

n,=0 »n,)= О r = 1 L/-co J

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

(*^9 )"1 (01,1! • • ' ! 01,nj ;....; ®P(q). 1 »•••. ^P(9),Tlp(,))

('W’)n, ....тги/.л , .

(9.9o)

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

9.6 Examples

The theory presented in the previous sections will be used to derive the non-linear Volterra transfer functions in four examples. Maple V source code for a program to determine algebraic expressions for transfer functions and for the example are included i appendix E.

9.6.1 Example 1

Here the non-linear Volterra transfer function between input Si and response v0 in Figure 9.3(a) will be determined. The time domain relation for the non-linear conductance is given by

i0(t) = i0(v0) = giV0(t) + !}2»о(1) (9.96)

тч. .... з .•.£ . j _ i:____*.____________ • - T>:__/-> о / 1, ’t ...L „ „ 1„ .J „

in*? moumeu пии-1шеа,г svsiemis btluwit m i y.oi uj wucic шс uncdi <auu nun-

linear parts of the overall network have been separated. Using the algorithm in

ЫЯ = Ai,i(/)*i(-/) + £1Л(/).51(Я (9.97)

where X\(/) is the controlling voltage across and y\(f) is the controlled current through the non-linear conductance. Thus it can be shown from Figure 9.3(b) that

9.6. Examples

245

(a)

,Sln

П

j/iui)

Figure 9.3: Example 1: Non-linear network with one signal input, port and one non-linear element, (a) Original network; (b) modified network where the non-linear subsystem is separated from the linear system.

^i,i(/) = -(oi + J'2-fC) (0.9S)

jBi.1 (/) = 1 (9.99)

Furtiiennore

v(f) = ai(f) Xlif) + 611; / i Si[f) (9.100)

Thus as v(f) = xi(/) then

ai(f) = 1 (9.101)

b\[f) = 0 (9.102)

As seen from Equation (9.96) and Type I in section 9.5 the non linear trnnsfer function ielating Xj if) and y\ (f), ((?; . (У1 i.....tf;.,- 1. is given by

- {;■ 1";;;; is.»)

Using Equations (9.38) and (9.53) the first order transfer function is given by

246

9. Multi-port Volterra transfer Iunctions

The second order non-linear frequency domain Volterra transfer function is deter-

'0i,i ■ 0’i,2) — ^ (*iM^i,i + Фха)

(9.105)

where (it )г.(ф\1\ + ф\,г) can be derived from

= — 2 (Gi)n(’i’i,!, ’/’1,2) Н\(Ф1,1)

The third order transfer function can be determined in a similar way as , >!'■ , ,h. -I - - . x 1/,. „ x „1

J\ V- X ,1 ) ri.J ' '*•' 1 ,J ! — /■> ’■ -I, i i V- 1 I V J ,ОУ

6 '

where (1)3(0111 4- ^i 2 + Pi,:;) can be derived from (i'i)3(wi,i + ф 1,2 4- ’/’1,3)

= ~ 2 |(Gi)2(i/’l,b tii,2 + t/'l,3) (xl )l('01,1) (i'l)2(V’l,2 + ^1,3)

+ (G'i)2('i’i,2,»/'l,l + 01,з) (■'Cl)l(V’l,2) (-'^1 )г( 01,1 + V^l ,3)

+ (GiMl&i,3,01,1 + 01,2) (г 1 )l(01,3) (-^ 1)2('01,1 + '01,2) j X ll\ ( 01,1 + 01,2 4- V'1,3 )

— 6 (Cl )з( 01,1 - 01,2, 01.3) (/l )l! 01,1) (jl)l (01,2) (*l)l(01,3)

x ff, (•!*,,. + i/>,,2 + i',i3) (9.109)

Thus using Equation ^9 103) the third order frequency domain Volterra transfer function is given as

X-ffl(01,2) #l(01.1 + 01,2)

(9.106)

Using Equation (9.103) gives

#2(01.1, 01,2)

<72 я,(01л) я,(й.1>2) Нх(ф1Л + 0!,2) (9.107)

о

/^(^,,i) /л(0i,2) tf 1 (ii’i,3)

X Hl( '/’1.1 + t’l.v 4- ,3 )

(9.110)

9.6. Examples

247

These results for the transfer functions are the same as those obtained by Bussgans. Ehrman and Graham [3] using the traditional probing method.1

A somewhat similar example to the one in Figure 9.3 has been given by Maas [10, pp. 179-186] which includes a Type 5 non-linearity given in section 9.5. Using the method in this chapter, results are obtained which are in agreement with Maas.

9.6.2 Example 2

Here the non-linear Volterra transfer function between input ^ and response u,.(ic) + v, in Figure 9.4(a) will be determined. The overall network contains two single-port non-linear elements where the time domain relations are given as

Figure 9.4: Example 2: Xon-hnear network with one signal input, port, rmd two non-linonr elements, (з) Original network; (b) modified network where the non-Unear subsystems ЛГ0 from the iinc<ir system (note tnMt the ori^mcii пеглуогк is per’iirbird ov the

resistance Kv to avoid non-umque system matrices and vectors).

Г fl !/Al

vc(t) = uc( ic) = > c„ I I ic(r)'H (9.111)

„.-1 LJ-oo 1

4 It should be noted that there are two errors in [>] regarding this example, in Equation (j.lyi the factor — ~ should be —and in Equation (.3.20) 'he factor — f should be -t-7.

248

9. Multi-port Volterra, transfer functions

where cn, is a real constant for all щ and

iz(t) = ч(г'.) = У I

+Sn, «?*(*) (9.112)

where l,u and gni are real constants for all ni. A linear resistor, Rp, is introduced in tlie modified network in Figure 9.4(b). This is necessary because the system matrices and vectors otherwise will be non-unique (infinity elements in the matrices and vectors). Each Volterra transfer function from Figure 9.4(b) is denoted with a prime. Thus the wanted Volterra transfer function Hmi(Ф\,»,) is determined as

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