Page 129 - Complementarity and Variational Inequalities in Electronics

120 Complementarity and Variational Inequalities in Electronics

FIGURE 4.5 Four-diode bridge.

4.14 FOUR-DIODE BRIDGE FULL-WAVE RECTIFIER

Let us consider the circuit in Fig. 4.5 involving four diodes D 1 , D 2 , D 3 , and D 4 ,
∈ (R;R ∪{+∞})
a resistor R> 0, and a capacitor C> 0. We denote by ϕ D i
(1 ≤ i ≤ 4) the electrical superpotential of diode D i . The input-signal source is
denoted by U.
Let us denote by V i the voltage of diode D i (1 ≤ i ≤ 4),by x the voltage
of the capacitor, and use the other notation indicated in Fig. 4.5. The Kirchhoff
laws yield the system

⎧
x dx
⎪
⎪ + C
⎪ i 1 + i 4 =
⎪
⎪ R dt
⎪
⎪
⎪
⎪
−V 4 = x + V 3
⎨
i 3 = i 4 + i 1 − i 2
⎪
⎪
⎪
⎪
⎪ −V 1 = x + V 3 − U
⎪
⎪
⎪
⎪
−V 2 =−V 3 + U.
⎩
We have
⎧
(i 4 ),
⎪ −V 4 ∈−∂ϕ D 4
⎪
⎨
(i 1 ),
−V 1 ∈−∂ϕ D 1
⎪
⎪
(i 2 ).
⎩
−V 2 ∈−∂ϕ D 2
Moreover,
(i 3 ) ⇔ i 3 ∈ ∂ϕ ∗ (V 3 ).
V 3 ∈ ∂ϕ D 3
D 3
We set
(x) = ϕ ∗ (−x)
(∀x ∈ R) : θ D 3
D 3