Page 52 - Complementarity and Variational Inequalities in Electronics

42 Complementarity and Variational Inequalities in Electronics

We have

⎛ ⎞⎛ ⎞
−1 0 0 V ze
⎝ −1 0 1 ⎠⎝ −V E ⎠
⎟
⎟⎜
⎜
0 −11 −V C
⎛ ⎞⎛ ⎞ ⎛ ⎞
−R 1 −R 1 −R 1 i z V + V s
0 0 0
⎜ ⎟⎜ ⎟ ⎜ ⎟
= ⎝ ⎠⎝ i E ⎠ + ⎝ V + V s ⎠
0 −R 2 0 i C V
and

1 I
i E 1 −α I
= ,
i C 1 − α I α N −α N 1 I

where I and I denote the currents through the diodes of the Ebers–Moll model
(see Section 2.3.7 in Chapter 2). Thus
⎛ ⎞⎛ ⎞
−1 0 0 V ze
⎝ −1 0 1 ⎠⎝ −V E ⎠
⎟⎜
⎜
⎟
0 −11 −V C
⎛ ⎞⎛ ⎞ ⎛ ⎞
−R 1 K R 1 (α N − 1) R 1 (α I − 1) i z V + V s
⎜
⎟
⎟⎜
⎟
= 1 ⎜ 0 0 0 ⎠⎝ I ⎠ + ⎝ V + V s ⎠ ,
K
⎝
0 −R 2 R 2 α I I V
where K = 1 − α I α N . Then
w
z

⎛ ⎞ ⎛ ⎞⎛ ⎞
V ze R 1 K R 1 (1 − α N ) R 1 (1 − α I ) i z
⎜ ⎟ 1 ⎜ ⎟⎜ ⎟
⎝−V E⎠ = ⎝ R 1 K R 1 (1 − α N ) + R 2 R 1 (1 − α I ) − α I R 2 ⎠⎝ I ⎠
K
−V C R 1 K R 1 (1 − α N ) R 1 (1 − α I ) I
q

⎛ ⎞
V + V s
+ ⎝ V + V s ⎠ .
⎟
⎜
V
We also have
⎧
V ze ∈−∂ R (i z ),
⎪ +
⎪
⎨
−V E ∈−∂ R (I),
+
⎪
⎪
−V C ∈−∂ +(I ).
⎩
R