Page 136 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 127

¯
¯
This last equality holds if and only if i 1 + i 2 = i 1 + i 2 , which means that the
∗
∗
∗
current through the resistor R, that is, i = i + i , is uniquely determined.
1 2
Moreover, the variational inequality VI(F,q, ) is here equivalent to the mini-
mization problem
,
min ϒ(x) + q,x + R − ×R +
x∈R 2
so that
R 6 R 6
¯
¯
¯
¯
(i 1 + i 2 ) + q 1 i 1 + q 2 i 2 = (i 1 + i 2 ) + q 1 i 1 + q 2 i 2 .
6 6
¯
¯
We know that i 1 + i 2 = i 1 + i 2 , and thus
q 1 (i 1 − i 1 ) + q 2 (i 2 − i 2 ) = 0.
¯
¯
We get the system
A

1 1 i 1 − i 1 0
¯
= .
E 1 − uE 2 − u i 2 − i 2 0
¯
Here E 2
= E 1 , and thus det(A) = E 2 − E 1
= 0. Thus i 1 = i 1 and i 2 = i 2 .The
¯
¯
solution VI(F,q, ) is thus unique.
Example 62 (Double-diode clipper/Practical diode). Let us again consider the
circuit in Fig. 4.6 and suppose that the electrical superpotential of each diodes
D 1 and D 2 is given by (practical diode model)

ν 1 x if x ≥ 0
(∀x ∈ R) : ϕ PD (x) =
ν 2 x if x< 0,
where ν 2 < 0 <ν 1 . We suppose also that
E 2 − E 1
|ν 2 | > . (4.116)
2
We set
(∀x ∈ R) :¯ϕ PD (x) = ϕ PD (−x)

and
2
(∀x ∈ R ) : (x) =¯ϕ PD (x 1 ) + ϕ PD (x 2 ). (4.117)
The Kirchoff laws yield the system

E 1 + R(i 1 + i 2 ) − u =+V 1 ∈−∂ ¯ϕ PD (i 1 )
(4.118)
E 2 + R(i 1 + i 2 ) − u =−V 2 ∈−∂ϕ PD (i 2 ),