Page 138 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 129

FIGURE 4.8 Double-diode clipper: Practical diode.

4.14.2 Clipping Circuit/Ideal Diode and Nonlinear Resistor

Let us again consider the clipping circuit of Fig. 1.2 involving a load resistance
R> 0, an input-signal source u and corresponding instantaneous current i,an
ideal diode as a shunt element, and a supply voltage E. We suppose here that the
resistor is nonlinear with ampere–volt characteristics described by the relation
5
U R = Ri ,

where V R is the difference of potential across the resistor. The Kirchoff voltage
law gives

u = U R + V + E.
Thus
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0 ≤ i ⊥−V ≥ 0 ⇔ 0 ≤ i ⊥ E + Ri − u ≥ 0.
Our problem is equivalent to the variational inequality VI(F,q, ), that is,

i ∈ R : F(i) + q,v − i + (v) − (i) ≥ 0,∀v ∈ R, (4.121)

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with F(i) = Ri , (i) = R + (i), and q = E − u.Wehave
6
(∀i ∈ R + ) : F(i)i = Ri .
Therefore, for some σ> 0 large enough, we get

2
(∀i ∈ R + ,|i|≥ σ) : F(i)i ≥ R|i| .
n
We may thus use Theorem 9 to ensure that for all q ∈ R , the variational in-
equality VI(F,q, ) has at least one solution. If i 1 and i 2 are two solutions of