Page 31 - Complementarity and Variational Inequalities in Electronics

The Convex Subdifferential Relation Chapter | 2 21

where
0 0
−
β (i ) = lim β (z)
z→i,z<i
and
0
0
β (i ) = lim β (z).
+
z→i,z>i
Any ampere–volt characteristic that can be described by a maximal mono-
tone graph can thus also be formulated as a convex subdifferential relation
V ∈ ∂ϕ(i)

for some ϕ ∈ 0 (R;R ∪{+∞}). Recall also that

∗
∗
V ∈ ∂ϕ(i) ⇐⇒ i ∈ ∂ϕ (V ) ⇐⇒ ϕ(i) + ϕ (V ) = iV.
The function ϕ is called the electrical superpotential (determined up to an
additive constant) of the device. Roughly speaking, the electrical superpoten-
tial ϕ appears as a “primitive” of F in the sense that the “derivative” (in the
generalized sense determined by the convex subdifferential) of ϕ recovers the
set-valued function F.
The notion of superpotential has been introduced by Moreau [67] for convex
but generally nondifferentiable energy functionals so as to manage nonlinear
phenomena like unilateral contact and Coulomb friction. This approach has led
to a major generalization of the concept of superpotential by Panagiotopou-
los [71] so as to recover the case of nonconvex energy functionals. The approach
of Moreau and that of Panagiotopoulos are now well established and often used
for the treatment of various problems in elasticity, plasticity, fluid mechanics,
and robotics (see e.g. [44], [46], [68], [71] [72]). More recently, the superpo-
tential approach of Moreau and Panagiotopoulos has been used to develop a
suitable method for the formulation and mathematical analysis of circuits in-
volving devices like diodes, diacs, and thyristors in [6]. The case of circuits
with transistors has been studied in [40], and a mathematical general theory
applicable to a large class of electrical networks has been developed in [5].

2.3.1 Ideal Diode Model
Let us come back again in this section to the ideal diode model. Fig. 1.1 illus-
trates the ampere–volt characteristic of this kind of diode. We have previously
seen that the ideal diode is described by the complementarity relation

0 ≤−V ⊥ i ≥ 0,