Books in black and white
 Main menu Share a book About us Home
 Books Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics

# 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 Previous << 1 .. 67 68 69 70 71 72 < 73 > 74 75 76 77 78 79 .. 85 >> Next Consider the following type of non-linearity which contains memory
oo CO P(q) r t 1nr
–£—è(t) = E (¬£,)*,,...,–ø—è(—á) –ü / —Ö.]—è–≥{—Ç)6.—Ç\ (9.94)
n,=0 ¬ªn,)= –û 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! ‚Ä¢ ‚Ä¢ ' ! 01,nj ;....; ¬ÆP(q). 1 ¬ª‚Ä¢‚Ä¢‚Ä¢. ^P(9),Tlp(,))
('W‚Äô)n, ....—Ç–≥–∏/.–ª , .
(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¬ª–æ(1) (9.96)
—Ç—á. .... –∑ .‚Ä¢.¬£ . j _ i:____*.____________ ‚Ä¢ - T>:__/-> –æ / 1, ‚Äôt ...L ‚Äû ‚Äû 1‚Äû .J ‚Äû
in*? moumeu –ø–∏–∏-1—à–µ–∞,–≥ svsiemis btluwit m i y.oi uj wucic —à—Å uncdi <auu nun-
linear parts of the overall network have been separated. Using the algorithm in
–´–Ø = Ai,i(/)*i(-/) + ¬£1–õ(/).51(–Ø (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
–ü
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), ((?; . (–£1 i.....tf;.,- 1. is given by
- {;‚ñÝ 1";;;; is.¬ª)
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 ‚ñÝ 0‚Äôi,2) ‚Äî ^ (*iM^i,i + –§—Ö–∞)
(9.105)
where (it )–≥.(—Ñ\1\ + —Ñ\,–≥) can be derived from
= ‚Äî 2 (Gi)n(‚Äôi‚Äôi,!, ‚Äô/‚Äô1,2) –ù\(–§1,1)
The third order transfer function can be determined in a similar way as , >!'‚ñÝ , ,h. -I - - . x 1/,. ‚Äû x ‚Äû1
J\ V- X ,1 ) ri.J ' '*‚Ä¢' 1 ,J ! ‚Äî /‚ñÝ> ‚Äô‚ñÝ -I, i i V- 1 I V J ,–û–£
6 '
where (1)3(0111 4- ^i 2 + Pi,:;) can be derived from (i'i)3(wi,i + —Ñ 1,2 4- ‚Äô/‚Äô1,3)
= ~ 2 |(Gi)2(i/‚Äôl,b tii,2 + t/'l,3) (xl )l('01,1) (i'l)2(V‚Äôl,2 + ^1,3)
+ (G'i)2('i‚Äôi,2,¬ª/'l,l + 01,–∑) (‚ñÝ'Cl)l(V‚Äôl,2) (-'^1 )–≥( 01,1 + V^l ,3)
+ (GiMl&i,3,01,1 + 01,2) (–≥ 1 )l(01,3) (-^ 1)2('01,1 + '01,2) j X ll\ ( 01,1 + 01,2 4- V'1,3 )
‚Äî 6 (Cl )–∑( 01,1 - 01,2, 01.3) (/l )l! 01,1) (jl)l (01,2) (*l)l(01,3)
x ff, (‚Ä¢!*,,. + 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 —è,(01–ª) —è,(–π.1>2) –ù—Ö(—Ñ1–õ + 0!,2) (9.107)
–æ
/^(^,,i) /–ª(0i,2) tf 1 (ii‚Äôi,3)
X Hl( '/‚Äô1.1 + t‚Äôl.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  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, (–∑) Original network; (b) modified network where the non-Unear subsystems –õ–ì0 from the iinc<ir system (note tnMt the ori^mcii –ø–µ–≥–ª—É–æ–≥–∫ is per‚Äôiirbird ov the
resistance Kv to avoid non-umque system matrices and vectors).
–ì fl !/Al
vc(t) = uc( ic) = > c‚Äû I I ic(r)'H (9.111)
‚Äû.-1 LJ-oo 1
4 It should be noted that there are two errors in [>] regarding this example, in Equation (j.lyi the factor ‚Äî ~ should be ‚Äîand in Equation (.3.20) 'he factor ‚Äî f should be -t-7.
248
9. Multi-port Volterra, transfer functions
where cn, is a real constant for all —â and
iz(t) = —á(–≥'.) = –£ 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(–§\,¬ª,) is determined as Previous << 1 .. 67 68 69 70 71 72 < 73 > 74 75 76 77 78 79 .. 85 >> Next 