Page 146 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 137

T
so that there exists α ∈ ker{M + M } such that x − x = α. We may write
⎛ ⎞
⎛ ⎞
i 4 ⎛ ⎞
i 4 α 1
⎜ ⎟
⎜ ⎟
⎜ −V 3 ⎟ ⎜ α 2 ⎟
⎜ −V 3 ⎟
⎟, x = ⎜ ⎟ ,α = ⎜ ⎟ .
x = ⎜
⎟ ⎝ ⎠
⎜
i 1 −α 1
⎝ ⎠ i 1
⎝ ⎠
i 2 α 4
i 2
Our existence result ensures the existence of the output signal V = R(i 1 + i 4 ).
Moreover, we have
V = R(i 1 + i 4 ) = R(i 1 + α 1 + i 4 − α 1 ) = R(i 1 + i 4 ) = V,
and the uniqueness of the output signal is also guaranteed.
For both examples, we may assert that for each t ∈ R + , the rectifier output
signal V(t) is uniquely defined by
(∀t ∈ R + ) : V(t) = R(i 1 (t) + i 4 (t)), (4.126)

where for each t ∈ R + , i 1 (t) and i 4 (t) are computed as solutions of the varia-
tional inequality V I(M, ,Fu(t)). Setting
⎛ ⎞
V(t) + V s
⎜ ⎟
(∀t ∈ R + ) : q(t) = ⎝ V(t) + V s ⎠
V(t)
with V(t) defined in (4.126), we may also assert that for each t ∈ R + , the stabi-
lizer output signal V o (t) is uniquely defined by

R 2

(∀t ∈ R + ) : V o (t) = (I(t) − α I I (t)), (4.127)
K
where I(t) and I (t) are determined in solving the variational inequality

V I(
, ,q(t)) (see Fig. 4.11 and Fig. 4.12).

4.16 A COMMON EMITTER AMPLIFIER CIRCUIT
Let us consider the common emitter amplifier circuit of Fig. 4.13 involving an
ideal NPN transistor. The common emitter configuration lends itself to voltage
amplification and is the most common configuration for transistor amplifiers.
In electronics, it is useful to work with equivalent circuits under appropriate
conditions. Indeed, in the case of low frequencies the capacitors of the circuit in
Fig. 4.13 play a short-circuit role, and the circuit reduces to that of Fig. 4.14.