Page 48 - Complementarity and Variational Inequalities in Electronics

38 Complementarity and Variational Inequalities in Electronics

FIGURE 3.3 Rectifier circuit.

FIGURE 3.4 Stabilizer circuit.

We denote by V i the voltage of diode D i of electrical superpotential ϕ i (1 ≤
i ≤ 4), V the voltage of the capacitor, and use the other notation indicated on
Fig. 3.3. The Kirchhoff laws yield the system

⎧
V dV
⎪ i 1 + i 4 = + C 1 ,
R dt
⎪
⎪
⎪
⎪
⎪ −V 4 = V + V 3 ,
⎪
⎨
i 3 = i 4 + i 1 − i 2 ,
⎪
⎪
⎪
⎪ −V 1 = V + V 3 − u,
⎪
⎪
⎪
⎩
−V 2 =−V 3 + u.
We also have
⎧
(i 1 ),
⎪ −V 1 ∈−∂ϕ D 1
⎪
⎪
⎪
⎨ (i 2 ),
−V 2 ∈−∂ϕ D 2
(i 3 ),
⎪
⎪ −V 3 ∈−∂ϕ D 3
⎪
⎪
⎩
(i 4 ).
−V 4 ∈−∂ϕ D 4
Moreover,
(i 3 ) ⇔ i 3 ∈ ∂ϕ ∗ (V 3 ).
V 3 ∈ ∂ϕ D 3
D 3