Page 131 - Complementarity and Variational Inequalities in Electronics

122 Complementarity and Variational Inequalities in Electronics

FIGURE 4.6 Double-diode clipper.

4.14.1 A Double-Diode Clipper
Let us here discuss a double-diode clipper to illustrate the results established in
Proposition 21 and Corollary 8.
Example 60 (Double-diode clipper/Ideal diode). Let us consider the circuit in
Fig. 4.6 involving a load resistance R> 0, two ideal diodes, an input-signal
source u, and two supply voltages E 1 and E 2 . We assume that E 1 <E 2 .We
denote by i the current through the resistor R, and we set i = i 1 + i 2 , where i 1
denotes the current through diode D 1 , and i 2 is the current through diode D 2 .
The Kirchoff voltage laws yield the system

E 1 + R(i 1 + i 2 ) − u =+V 1 ,
(4.103)
E 2 + R(i 1 + i 2 ) − u =−V 2 ,
where
(i 1 ) (4.104)
V 1 ∈ ∂ R + (−i 1 ) =−∂ R −
is the difference of potential across diode D 1 , and

(i 2 ) (4.105)
V 2 ∈ ∂ R +
is the difference of potential across diode D 2 . Setting

R R E 1 − u i 1
M = ,q = ,I = , ≡ R − ×R + , (4.106)
R R E 2 − u i 2
we see that system (4.103) is equivalent to the variational inequality
VI(M,q, ), that is,
2
2
I ∈ R : MI + q,v − I + (v) − (I) ≥ 0,∀v ∈ R . (4.107)