Page 152 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 143

⎧
I i − I 1 + I 2 = 0,
⎪
⎪
⎨
R i I i + R 1 I 1 − U i + V D = 0,
⎪
⎪
−γR i I i + R 1 I 1 + (R 2 + R 0 )I 2 = 0
⎩
with currents I i ,I 1 ,I 2 and voltages V D ,U i as defined in Fig. 4.16. Therefore,
A ϒ B D

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 −1 1 I i 0 0
⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟
R i R 1 0 ⎠⎝ I 1 ⎠ − ⎝ −1 ⎠ V D + ⎝ −1 ⎠ U i = 0,
⎝
−γR i R 1 (R 2 + R o ) I 2 0 0
and
V D ∈ ∂ϕ ZD (I i ), (4.133)
where ϕ ZD denotes the electrical superpotential of the Zener diode, that is,

V 1 x if x ≥ 0
(∀x ∈ R) : ϕ ZD (x) =
V 2 x if x< 0
with V 2 < 0 <V 1 . Setting

C ⎛ ⎞
I i

⎟
⎜
y = 100 ⎝ I 1 ⎠ ,
I 2
we may write relation (4.133) equivalently as

V ∈ ∂ϕ ZD (y),

and we may consider problem NRM(A,B,C,D,U i ,ϕ ZD ).
We check that the assumptions of Proposition 21 are satisfied. Indeed, As-
sumption (H1) is satisfied since ϕ ZD is convex and continuous on R. Assump-
tion (H2) holds with
⎛ ⎞
0 −1 0
0 0 −1 ⎠ .
⎜ ⎟
P = ⎝
−1 0 0
As a consequence of Proposition 21, problem NRM(A,B,C,D,U i ,ϕ ZD )
can be studied via the problem VI(−PA,−PDU i , ), where
3
(∀x ∈ R ) : (x) = ϕ ZD (Cx) = ϕ ZD (x 1 ).