# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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(9.113)

From Figure 9.4(b) the system matrices and vectors can be determined as

Mf) = B(f) =

-[j2~fR?^cx) -f2'/

j2-f j'Zirf -U2Tf9i+h)

Rp

0

(9.114)

(9.115)

and

a(f) = [-Rp, 0]T b(f) = [йв]т

(9.116)

(9.117)

The frequency domain Volterra transfer functions for the two non-linear elements are determined from Types 1 and 5 in section 9.5 as

(Gl)ni ■ ■ ■ I 01.ni)

0'2тг)п‘ ^i,i •

(9.118)

and

9n\

• • • n/li(

an"; Equation (9.3S) the first order controlling variables arn determined as

9.119)

Хт(Фц) =

(]2-ipi,igi + li){j2TTibhlRp + ci) - 4к2ф^1

h 1T 9i + -rz—:— , 1 J^’i’i.i

(9.120)

9.6. Examples

'249

Then by use of Equation (9.53) the first order Volterra transfer function is determined as

J 2-c-i.i ' j2*il>i,igi + h

(9.121)

(9.122)

Then, using Equations (9.68) and (0.78) the second order controlling variables are determined as

ж2('г1,1 + vA;i)

2 (Gi)-)(’ii.i-'/'i,j) U'iM'i'i.i) (ri)i( 0i,2)

- (<^2)2( (i-'l. 11 v’l.s) (X2)l(^i,l) (-'2 )l( U1.2)

(9.123)

Thus by use of Equations (9.41) and (9.120) the second order frequency domain Volterra transfer function is determined as

-^2(^1,ъ ^1.2)

^1 \ lira I - (-ЛР, 0! x2(u'i,i +• t-'i 2) tD—»oo V2 ’ /

1^4,2 \ -i'1 ^’l Л Ф1,

- 92

j2~{

2)

j’2« t'1,101 4- U j2t^i 2J1 + h

(9.124)

(9.125)

Higher order Volterra transfer functions can be determined in the same way. The results obtained are in agreement with Chua and Ng [6j who have determined the Volterra transfer functions up to second order.

9.6.3 Example 3

Here the non-linear Volterra transfer functions between inputs {st,«2,«3} and response vjs in Figure 9.5(a) will be determined. The overall network contains two one-port non-linear elements and one non-linear transconductance with the following lime domain relations:

\'jl4 = lj'"3/ = 2_j l3’7m ~J~t v,'j 1 (9.Ii6)

71-1=1

3

id\t) — ' :1 ' 1 — У ^ 9*n,ni r:; 1 '■ * (9.12 f J

•ii=l

250

9. Multi-port Volterra. transfer functions

fa1)

-II-

*0

\io{vi$)

с Ju(vg) Г г -] ] С is — с > !'‘L

а

У2(Х1,*2)

~^y~~

, ^ 9m, 1*^1 j |$o,l Cds — =r X2 I 4^ ъут,

figure 0.5: Example ‘6: Non-hnear network with three non-linear elements and three signal

input pOitS. (a) Original network.', ^b) mOuintu lictwovk where the ViOli-liiii-d.L: b’uubv bteiiis

are separated from the linear system.

9.6. Examples

251

and

з

*'Д0 = ‘o(y,!s) = 53 So.n, i-S(t) Г9Л28)

n, =1

where cg,ni, and уЛП) are real constants for щ G {1.2,3}. The modified

overall network is shown in Figure 9.5(b) where the non-linear conductance and the non-linear transconductance are represented at the same controlled generator, У?{-съ :t,2). Thus the overall network has two controlling variables and two controlled variables. It should be noted that the controlling variable x\ controls both the non-linear capacitor and the non-linear transconductance. From Figure 9.5(b) the system matrices and vectors can be determined as

A(f)

B(J)

YAf) RtYs(f) 0 0 0 -i

(9.129)

(9.130)

/here

Ys(f)

Y,(f)

Yo(f)

1

Ri + Za(f)

YS{ f } T ji-jCg l

Яо, 1 + YL(f) + ji-xfCi,

(9.131)

(9.132)

(9.133)

“(/) = l0’ !]Г b(f) = [0, 0. 0]T

(9.134)

(9.135)

From Equations (9.126), (9.127) and (9.128) and by use of Types I and 2 in section 9.5 the Volterra transfer functions for the two non-linear subsystems are derived

•252

9. Multi-port Volterra transfer functions

С 9m,n, for 711 6 {2,3} Л n2 - 0 \ 'Jo,n2 ior »J - 0 A ";t {2,3} (5.137)

( 0 otherwise

In this example there are К = 3 signal input ports. This means that there are three first order, six second order and ten third order Volterra transfer functions as seen from Table 9.3. In the following only a few of these are shown as the equations

are quite lengthy. Using the algorithm in section 9.3 the following Volterra transfer

functions are obtained. The first order frequency domain Volterra transfer function from port 1 is given by

tfi.o,o№,i) = (9'Ш)

-1 H Vl,l) Jot <r-l,l )

The second order frequency domain Volterra transfer function from port 1 is given bv

-^2,0,o( 01,1) 01,2) =

_____________________УВ(Ф1Л)УАА,2)

i i( 01,1 + 01,2/ '’0(01,1 + Ф 1,2) l'i(01,l) ^'г(. 7/-’l ,2.) Уо (01,1 ) ^з(01,2) x|j'2t(01,i + 01,2) c3,2 Jm.l У о ( 01,1) Ув(ф1,г)

- go,2 д^л к:(01л + 01,2)

+ 9m,2 Vi(01 д Г 01,2) io(01.l) ^o(f/’l,2)| (9.139)

and the second order multi-port frequency domain Volterra transfer function from input ports 1 and 3 is given by

и ( 1 j 2 gm 1 g0 2 У$(0i л)

#i,o,i(^i.b03,ij = ГГТ---------——-■ . , ■ , ---7 (9.140)

io(^'l,l T b’3,1 ) Yi[pi,i) 1 a (1.15'j(03.1J

The results for the Volterra transfer functions from port ! from this example are in agreement with Minasian [11] for orders up to three which is the highest order considered by him. It should be noted that Minasian has- represented, the non-linear capacitor and transconuuctam.e as two separate one-port non-linear generators and not by one two-port generator as m this work. Using one two-port non-linear element instead of two one-port non-linear elements, the number of non-linear elements in the modified network is reduced by 1 which reduces the sizes of the system matrices A(f) and B( f ).

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