Page 12 - Complementarity and Variational Inequalities in Electronics

2 Complementarity and Variational Inequalities in Electronics

or
n
n
R U ⊥ V ∈ R .
+ +
Example 1. We have
⎞ ⎛ ⎞
⎛
0 2
0 ≤ ⎝ 1 ⎠ ⊥ ⎝ 0 ⎠ ≥ 0.
⎜
⎟
⎟
⎜
4 0
It is easy to check that the complementarity relation is equivalent to the fol-
lowing set of relations (∀ i ∈{1,2,...,n}):
⎧
⎪ U i ≥ 0
⎪
⎪
⎪
⎨ V i ≥ 0
⎪ U i > 0 =⇒ V i = 0
⎪
⎪
⎪
⎩
V i > 0 =⇒ U i = 0,
which is also equivalent to the equation
min{U,V }= 0.
Indeed, we have
n

U,V = U i V i ,
i=1
and for U,V ≥ 0, the equation U,V = 0 is equivalent to the system
⎧
⎪ U 1 V 1 = 0
⎪
⎪
⎪
⎪
⎨ U 2 V 2 = 0
.
⎪ .
⎪ .
⎪
⎪
⎪
U n V n = 0.
⎩
1.2 THE COMPLEMENTARITY RELATION IN ELECTRONICS
The diode is a device that constitutes a rectifier that permits the easy flow of
charges in one direction but restrains the flow in the opposite direction. Diodes
are used in power electronics applications like rectifier circuits, switching in-
verter and converter circuits. Fig. 1.1 illustrates the ampere–volt characteristic
of an ideal diode. This kind of diode is a simple switch. Denoting by i the diode
current and by V the voltage across the diode, if V< 0, then i = 0 and the diode
is blocking, whereas if i> 0, then V = 0 and the diode is conducting.