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A Program for the Computation of the Approximate Symbolic Pole/Zero Expressions for Linear Electromechanical Systems
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ady $wte in ;witzerland, d on muiti 'CAS) 1993. mic circuits 1/112 Desi,?,ii
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florin Constantinescu, Miruna Nitcscu, Mihai Iordache, Lucia Dumitriu "Politehnica" UniversIty, Dept. of Electrical Eng" Spl. Independentei nr. 313. Bucuresti 77206, ROMANIA, tel: +401.252.47,80, fax: +401.41.12,17, e-mail:
[email protected],ro
V implementation of the symbolic LR algorithm for cigcnvalue cODl[lutation is presented. The formulation of the ~ymbolic state matrix is implcmented in C++ and MAPLE, An e~ample i:s given for illustration.
AbstracT A MAPLE
rhe Design
I. Introduction Gcneve, fme(i/'
LAD
,per2.ting EEE Truns.
01 pp. lll-
The silicoll micwlllil,'hi"jn:; provides Ilw nIlpartllnity tl.' -realize mobik slru':lUres able to play an electrical role (micro switch, vilrjable CilP'lcitorE with mOYir,g electrodes, mechanical resonators and filters, ele~rromtoCh
mentioned above. 2. C:,mpJllation ufnumericd eigerrw:hw5 corre.~pondjn5 to nomimj) Y:ll1.les of circuit r"ntI!ie,e:~
3 CJuster ident!llration ~.l. A mJximal number of poles Sm
is elim in:lted, where: -'ml«15 tl "''''1:
3.2. A maximal number oft'oles S\.j i~ eliminated, where I s~I!»isol\1,!orl 3.3. The r~mainiDg poles form the ·'cluster"". 4. Ordering ofthe state matrix so that !alll2:lanl:?:•..::::!aon" 5. LR fraalions The reduced state matrix A=[agl being given, the LR iteralions are A'=LjRJ, R]L1=A, , A I=L 2R:;
J(2L-{"A2, and so on (a sequence of I,kcompositions and mnltiplications). In the LR decomrosition ,. is a lowe;
triangular matrix with unitary diagonal and R is an upper triangular matrix.
The program goes out from the LR iteration loop after the st.ep p if lhe smallest module eingenvalue satisfies an
error condition (to be defined in the following). In." is real, the LR iteration will be continued with an (n-I) • (n 1) matrix A* obtamed from A.. by removing the last row and the las.t column Tn this case J.." = a." _ If t,,, is
complex, A* is obtained by removing tlle last two rows and cohimns. In Inis ca~e I. n and A'n_1 are (he eigenvalUl; ---
. """ Olb st::J\e
tn is
Ie - e *1 ~ Ie, *1
11
t,
II['\ I!,I
n
I;
in the case ora complex pair Sm
Sn_l
where
en
is a nnmerical eigenvalue of the 2.2 lower right
M
hand block of Ap In order to obtain simpler expressions and to shorten the computation tune, some simplifications must be made. The simplifications may be performed in the initial slate matrix A and in any intermediary stage of computation. There are two situatiom to be considered: - an em"!"y which is significantly smaller than others (with 2-3 orders of magnitude) and which may be replaced by zero (no symbolic expression); - a faetor (or many) which is significantly smaller than others in a sum representing an entry of A; in [his case this [actor{s) may be neglected. The algorithm checb all simplification possibilities. Any attempt of simplification is validated by measuring its influeoce on eigenvalue module~. To Ihis end [he numerical eigenvalue of the simp/Hied matnx flre compared with the ones ofthc initia] matrix. I[(he maximum relative error doesn't txceed C the simplifIcation is ;!(cepted
4. Implementation The cirt'utt is described by a netlist starting with lhe number of the circuit nodes and the number of the eircuit brancnes. it roiiows a sel of inpm TOWS describing Ihe circuit branehes. A circuit branch is made up ouly of one iViO-terminal element or a controlled (controlJing) port tilr the controlled sources. Each circuit branch is described by: the initial node, the final r,ode, ar,d Ihe type of eircnit element. In the case of a controlled sni..iH:e Q"e have to give also Ihe llumher of (he COlltrolling braneh. The program fol' state equation formulation is written !:n C-'+ and uses ~orne solving rotltines of MAPLE V. The program for state matrix reduction and eigcli'.'31uc OJrnputation is written in MAPLE V.
5. Example C..l'ts
A common emitter ilmpiitier whose poles have been computed in [7] i5 given in Fig 3, Ihe smal!-slgnal t::'qUivalent Cln:llit h,~ing sho·.... n in rig. 4.
6
R1~! RC
L2R2_ es an • (nJon is 'alues
en by mmlil
~ oF
VI~¢
i:
22k~ H
er.;
lower
,
I 5
:
l
Q12 p F o I
2
4
IR2
R L 47 kQ
IRE ~v,j }vee IkQ
'1'47 kQ 7
fig.]
Tile parar!'!e!~'T values are: cl =lOOnF; c2=!1IlF; c3= CBE=-30pF; c4=CBC=5pF; rlp2=(rl/h2)=3 J .97278l*rll *B8_9+2"'r121 *r8*rll +r8*r12*rl 2+2*r8*'rlS"rl 2-trI2*r rll "'ri 2+r12*ri 5*ri 2» actual error 0.138 % I "I , Es
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