Page 35 - Complementarity and Variational Inequalities in Electronics

The Convex Subdifferential Relation Chapter | 2 25

and simple calculations yield

⎧
⎪ V 4 if x< I R2
⎪
⎪
⎪ [V 4 ,V 2 ] if x = I R2
⎪
⎪
⎪
⎪
⎨
V 2
∂ϕ CD (x) = x if I R2 <x ≤ 0
I R2
⎪
⎪
V 1
⎪
⎪ x if
⎪ 0 <x ≤ I R1
⎪ I R1
⎪
⎪
⎩
αx − β if I R1 <x.
On the other hand, we may compute the conjugate function
⎧
⎪ +∞ if z ≤ V 4
⎪
⎪
⎪
⎪
⎪ V 2
⎪
⎪ I R2 (z − ) if V 4 <z ≤ V 2
⎪
⎪ 2
⎪
⎪
⎪
⎪
⎨ I R2 2
ϕ ∗ (z) = z if V 2 <z ≤ 0
CD 2V 2
⎪
⎪
⎪
⎪ I R1 2
⎪
⎪ z if 0 <z ≤ V 1
⎪
⎪
⎪ 2V 1
⎪
⎪
⎪ 1 2 1
⎪
⎪
⎩ − αV 1 )z + V 1 (αV 1 − I 1 ) if V 1 <z,
αz + (I R 1
2 2
and we get
⎧
⎪ ∅ if z< V 4
⎪
⎪
⎪
⎪
⎪ ]−⇔,I R2 ] if z = V 4
⎪
⎪
⎪
⎪
⎪
⎪ I R2 if V 4 <z ≤ V 2
⎪
⎪
⎨
∂ϕ ∗ (z) = I R2
CD z if V 2 <z ≤ 0
⎪
⎪ V 2
⎪
⎪
⎪
⎪
⎪ I R1
z
⎪
⎪ if 0 <z ≤ V 1
⎪
⎪
⎪ V 1
⎪
⎪
αz + (I R1 − αV 1 ) if V 1 <z.
⎩
The ampere–volt characteristic of the complete diode can then be written as
V ∈ ∂ϕ CD (i) ⇐⇒ i ∈ ∂ϕ ∗ (V ) ⇐⇒ ϕ CD (i) + ϕ ∗ (V ) = iV.
CD CD
2.3.4 Zener Diode Models
The Zener diodes are made to permit current to flow in the reverse direction
if the voltage is larger than the rated breakdown or “Zener voltage” V 2 .For
example, for a common Zener diode, V 1 0.7 V and V 2 −7 V. The Zener