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Theory of Linear and nonlinear circuits - Engberg J.

Engberg J. Theory of Linear and nonlinear circuits - Wiley & sons , 1995. - 154 p.
ISBN 0-47-94825
Download (direct link): noisetheoryoflinearandnonlinear1995.pdf
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9.3.4 Controlled variables
From Equation (9.12) the controlled variable vector y(f) is given by
y(f) = Mf)xU) ~ B(})s(f) (9,12)
Thus
R К
У,(Л = Y1 A4.r(f) ЫЛ - E BqA(f) Mf) (9.43)
Г=1 k = i
for a given q g {1,2,...,Q}. Substituting Equations (9.8) with Ik = mk and (9.-33) into Equation (9.43) gi^es
D .,c
■= YlYm A»(i!<T:i) ‘А'^Ч) exp!j <pL q]
r=i и— 1 q
f\ "lk
4- \ Bq l I ?/'■ l- /) pxn[i / I /*( / - i) {9-14)
k= 1 /=1
The response yq(f) from the purely non-linear subsystem q 6 {1,2..................Q} can also
be described by a multi-port Voiterra series as
228
9. Multi-port Volterra transfer functions
JjU )
'-*J 00 y»CO ^CO
E ••• E j ■■■]
n1=0 np(,,=0 J~x
-CWIHI)
(G,?)n,t...,npu)(^i,b • ■ ■, 0i,ni;.....; вр(я),1,- - •, вр(я),пРМ)
Р(я) Hr , Р(я) Пг ч Р(я) лг
П П *i,.r(*r,„) «(/-ЕЕМ П П ^.P (9.45)
Г = 1 р=1 Г = 1 р=1 г=1 р=г I
where
?РШ > “■0+
(9.46)
and (G,)n]......np(,,(') is the partly symmetrical multi-port frequency domain Volter-
ra transfer function between inputs {xjq ,(/),..., xj p(,,(/)} and output jf,(/) for non-linear subsystem 9 G {1, 2,..-,Q}-3 In case the original (Gq)rli................np{i)(-) trans-
fer function is not partly symmetrical, it must be made partly symmetrical using for example Equation (9.5). The reason for requiring a partly symmetrical (Gij)ni,...,np(,)(') transfer function is that it significantly reduces the amount of computations that must be carried out to determine the overall Hmi.....................mA-(') partly
symmetrical transfer function. When (G7)nil...,np(7)(-) is partly symmetrical then НШ1,...,шк(') automatically becomes partly symmetrical when the present method is used. Note that yq(f) where q £ {1, 2...., 0) contains only non-linear contributions. Insertion of Equation (9.33) into (9.45) gives
3In section 9.5 some common types of non-linear time domain relations between input-
s ...^ pi,) (0} and output i/q(t) are given as well as the corresponding multi-port
\0Q)n,,transfei functions.
9-3. Theory
229
yq(f)
со
= E E E E- E E-
71! 2=0 ПP(,)=0 0],i=l ] 01|Я1=1 glirtl
E E E E
°р{я)л^ Яр(„),i °P(i),
P(7) 71.
A.,=o(IH) П П 'VP(il<? JO
r=I p— 1
\G,)ni....nP(„ (i>Tqun ■ • •. Фтяi,nj; • ■ •
- •; ^тчр(Ч),ъ ■ ■ ■■.'Ф' чр(,),Лр(г))
^(?) ПГ
П П ^)ог.г{фТ Чг,Р)
Г = 1 Р=1
*г т , Р?) П.
ехр
j^EE^.p < /-f ЕЕ«г,
г = \ р = 1
where
*7г,р ~ [<7г,р, 1Д-. ■ • ■ 1 (?г,р,1,771] J.......I ?г,р, А',17 1 ?г,р,А\тк|
and the summation over qrp is defined symbolically as
Or,p °ЛР Jr,p Or p
E = E E ........................................ E - E
Qr,p ?r,p,1.1—0 7»".p,A',l “0 3r,p, к ,-n г,- —0
where r £ {1,2,.... P(q)j and p £ {1. 2... ., п.Л.
9,3,5 Controlling variables
(9.47)
7limi|x1 ~o+
(9.48)
As seen from Equations (9.38) and (9.41) it is necessary to determine Х||тц( This is done using the fact that Equations (9.44) and (9.47) must be identical:
230
9. Multi-port Volterra transfer functions
CO CO oo
E- E E E- E E- («-so)
n 1=0 7iP(t()=o o.(1 = i q,д oi,„]=i 7, „j
- E E ■■■ E E
"W-1 9p«,),i йР(-1)^р(,)-! 9/ч«).пл,(
P(7) r.r
^.oo(llnll) П П V,(IK„II)
г — 1 p=l
(Gq)nb...,npU) ,'0тчг,,м; ■ • •
• • •; i>T4PM,u ■■■, Фтчрь),пр(Ч))
Р(я) nr
П П (хлЛгА-ФгЯг,Р)
Г— 1 p= 1
г ^(?) ПГ , Pi?) tv
exPUTEEJ *(/- *TEI>,,
L „^.1 ^_i J \ ,— I „—I
R ‘to
= E E E M!WI) Aq.T{^Tq) {xr)0{i>Tq) exp[j<pT<j] 6{f -■ фТq)
Г— i o=i g
+ ЕЕ ВЧ,к(Фк, i) ехР[? Фк.’.} #(/ - V'fc.i) (9.51)
^ = l /=1
The controlling variables can be determined by using the fact that the coefficient of uxp[j ||</>||] <5(/ — Ill’ll) on both sides of Equation (9.51) must be identical.
To determine the first order controlling variables сс\{фк,1) for given Ar G {1,2,...,
A'} and I £ {1, 2,..., rrik} Equation (9.51) is used to yield
R
E V(*w) (*rM*w) - - (9.52)
Г = 1
where q £ {1,2?.. .,Q}. Now ail the first order controlling variables x\(iby.,i) can be determined as
X[(ibk,i) = -A~*( <£•*,/) *( Фш) (9.5-3)
bkli!>ki) - I'i?.,. (м. .)...Вок(Фи)? C’Qxi (9.51)
Thus the vector is the fcth column vector of the matrix. The
solution vector in Equation (9.53), xi( ta-,/), can be determined provided the matrix A{Vkj) is unique and non-singular. If Inis is not the case, the original network can be perturbed bv some linear elements to obtain a unique and non-singular
9.3. Theory
231
A(ibkti) matrix. (Practically, the matrix A(i-kj) is not determined since it is actually only А~1(фк.1) that is used. This matrix may be determined by setting s(f) = [0, 0,. ■., 0]T {0}Лх1 and then using y{ f) as excitation signals instead of x(f). This technique is used in Example 2 in Section 9.6. If Q Ф R then A(J) is non-
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