Page 38 - Complementarity and Variational Inequalities in Electronics

28 Complementarity and Variational Inequalities in Electronics

Moreover,
⎧
⎪ (V 1 −V 3 )
⎪ x + V 3 if x< 0
I 1
⎪
⎨
∂ϕ Z (x) = [V 3 ,V 4 ] if x = 0
⎪
⎪
⎩ (V 2 −V 4 )
⎪
x + V 4 if x> 0
I 2
and
⎧
I 1
⎪ (z − V 3 ) if z< V 3
⎪
⎪ V 1 −V 3
⎨
∗
∂ϕ (z) = 0 if V 3 ≤ z ≤ V 4
Z
⎪
⎪
I 2
(z − V 4 ) if V 4 <z.
⎪
⎩
V 2 −V 4
The ampere–volt characteristic of the concrete Zener diode can thus be written
as
∗
∗
V ∈ ∂ϕ Z (i) ⇐⇒ i ∈ ∂ϕ (V ) ⇐⇒ ϕ Z (i) + ϕ (V ) = iV.
Z Z
2.3.5 Empirical Diode Model
An empirical model used in electronics to describe the ampere–volt characteris-
tic of a diode is
V T i
V(i) = ln( + 1)(i > −I S ),
η I S
where I S is the saturating reverse current (10 −15 ≤ I S ≤ 10 −12 A), V T is the
thermodynamic voltage (25 mV), and η is the emission coefficient (1 ≤ η ≤ 2).
This model is usually considered in the engineering literature when a rigor-
ous mathematical analysis taking care of the domain of V is not required. It is
however possible to proceed to a suitable mathematical treatment as before in
defining the set-valued function V : R ⇒ R by
⎧
⎨ V T ln( i + 1) if i> −I S
η
V(i) = I S
∅ if i ≤−I S .
⎩
Then setting
⎧
V T i V T I S
⎪ ln( + 1)(i + I S ) − i if i> −I S
η η
⎪ I S
⎪
⎨
ϕ ED (i) = V T I S 2
η if i =−I S
⎪
⎪
⎪
⎩
+∞ if i< −I S ,
we see that
(∀i ∈ R) : V(i) = ∂ϕ ED (i).