Page 149 - Complementarity and Variational Inequalities in Electronics

140 Complementarity and Variational Inequalities in Electronics

where

R L + R C + R 6 R C R g + R 2
K = > 0,L = > 0,N = > 0,
R 6 R L R L R 2
R C R 1 (R C + R 6 )
S = > 0,T = R C + > 0.
R 6 R 6
System (4.129) is equivalent to
A

−(R g K + N) R g K −V E
L + R 1 K −(1 + L + R 1 K) −V C
B

−R g R g S i E u g
= + .
0 −T i C 0
Since the matrix A is invertible, it follows that

−V E −1 i E −1 u g
= A B + A ,
−V C i C 0
where

1 R g (1 + L + R 1 K) −R g S(1 + L + R 1 K) + R g KT
−1
A B =
μ 0 R g (L + R 1 K) −R g (L + R 1 K)S + T(R g K + N)
with
μ 0 = R g K + N(1 + L + R 1 K).

Using
C

1 1 I
i E −α I
= ,
i C 1 − α I α N −α N 1 I
we obtain
x q

I
−V E −1 u g
= M +A ,
−V C I 0
where

1
−1 M 11 M 12
M = A BC =
μ M 21 M 22