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The Variational Inequality Problem Chapter | 3 37


















                           FIGURE 3.1 Clipping circuit 1: Practical diode as shunt element, V 1 = 0.1, V 2 =−90, E = 1.

















                           FIGURE 3.2 Rectifier–Stabilizer circuit.


                           The time-dependent current t  ø  i(t) is given by
                                                   1   −1  u(t) − E
                                       i(t) = (id R +  ∂ϕ)  (     )
                                                  R          R
                                                      1     u(t) − E  2  1
                                          = argmin x∈R { |x − (    )| +   ϕ(x)}.       (3.8)
                                                      2        R        R

                           Numerical simulations depicting input and output signals are given in Fig. 3.1.

                           3.4 A RECTIFIER–STABILIZER CIRCUIT

                           In this section, we illustrate our mathematical modeling approach with a
                           rectifier–stabilizer circuit (Fig. 3.2).
                              The rectifier–stabilizer circuit involves four diodes D 1 , D 2 , D 3 , and D 4 ,
                           a Zener diode D z , an NPN transistor T , two resistors R 1 and R 2 , and two ca-
                           pacitors C 1 and C 2 . This circuit is supplied by the signal input u. We first follow
                           a classical compartmental approach to split it into two blocks: the “rectifier” cir-
                           cuit depicted in Fig. 3.3 and the “stabilizer” one presented in Fig. 3.4.
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