Page 46 - Complementarity and Variational Inequalities in Electronics

36 Complementarity and Variational Inequalities in Electronics

The approach using variational inequalities of the second kind so as to study
electrical networks involving devices like diodes and transistors has been devel-
oped in [5] and [40]. The mathematical approach studied in [5] uses recession
tools so as to define a new class of problems that is called “semicomplementarity
problems” (see also [43]). It is first shown that the study of semicomplementar-
ity problems can be used to prove qualitative results applicable to the study of
linear variational inequalities of the second kind. By using variational inequali-
ties of the second kind the authors study diode circuits like amplitude selectors,
which are used to transmit the part of a given waveform that lies above or be-
low some given reference level, double-diode clippers that are used to limit the
input amplitude at two independent levels, sampling gates that are transmission
circuits in which the output is a reproduction of an input waveform during a
selected time interval and is zero otherwise, and other circuits involving diodes,
transistors, and operational amplifiers. Further theoretical results, applications
in electronics, and numerical simulations can be found in [7], [32], and [38].

3.3 A GENERAL CLIPPING CIRCUIT
Let us again consider the circuit of Fig. 1.2. We discuss here the case of a diode
with electrical superpotential ϕ. The Kirchhoff voltage law gives

u = U R + V + E,

where U R = Ri denotes the difference of potential across the resistor, and V ∈
∂ϕ(i) is the difference of potential across diode. Thus

E + Ri − u ∈−∂ϕ(i), (3.7)

which is equivalent to the variational inequality

(Ri + E − u)(v − i) + ϕ(v) − ϕ(i) ≥ 0,∀v ∈ R.

Moreover,

E u 1 E u 1
+ i − ∈− ∂ϕ(i) ⇐⇒ − + ∈ i + ∂ϕ(i)
R R R R R R
1 −1 u − E
⇐⇒ i = (id R + ∂ϕ) ( ).
R R
Let us now consider a driven time-dependent input t ø u(t) and define the
output-signal t ø V o (t) as
V o (t) = E + V(t) = u(t) − Ri(t).