Page 137 - Complementarity and Variational Inequalities in Electronics

128 Complementarity and Variational Inequalities in Electronics

which is equivalent to the variational inequality VI(M,q, ), that is,

2 2
I ∈ R : MI + q,v − I + (v) − (I) ≥ 0,∀v ∈ R , (4.119)
with M and q as in (4.106) and as in (4.117).Here

2
D( ) ∞ = R , ∞ ≡ , N(M, ) = ker{M},
and thus

2
D( ) ∞ ∩ ker{M}∩ N(M, ) = ker{M}={v ∈ R : v 2 =−v 1 }.
Let v ∈ ker{M}, v
= 0. Then

q,v + ∞ (v) = v 2 (E 2 − E 1 ) + ϕ PD (−v 1 ) + ϕ PD (v 2 )
= v 2 (E 2 − E 1 ) + 2ϕ PD (v 2 ).

Therefore, if v 2 > 0, then

q,v + ∞ (v) = v 2 (E 2 − E 1 ) + 2ν 1 v 2 > 0,

whereas if v 2 < 0, then

q,v + ∞ (v) =−v 2 (2|ν 2 |− (E 2 − E 1 )) > 0.

We may then apply Corollary 8, which ensures that system (4.119) has at least
one solution.
∗ ∗ T
∗
If I = (i i ) is a solution of system (4.119), then from the first relation
1 2
∗
∗
∗
in part (c) of Corollary 8 we deduce that the current i = i + i through the
1 2
resistor R is uniquely determined.
So, for a driven time-dependent input t à u(t), the time-dependent current
∗
t à i (t) through the resistor R is given by
∗
∗
∗
i (t) = i (t) + i (t), (4.120)
2
1
∗
∗
where I = (i (t),i (t)) is a solution of VI(M,q(t), ) with
∗
1 1

E 1 − u(t)
q(t) = ,
E 2 − u(t)
and the output-signal V o (see Fig. 4.8) can then be determined by the formula
∗
V o (t) = u(t) − Ri (t).