5.7 Analogy Between a Nonlinear RLC Circuit and a Memristor Circuit
195
By substitution we obtain
q(t; t 0 ) = −f 1 (ϕ M 1 0 ) − f 2 (ϕ M 2 0 )
+f 1 (ϕ(t; t 0 ) + ϕ M 1 0 ) + f 2 (ϕ(t; t 0 ) + ϕ M 2 0 ).
The parallel is thus equivalent to a flux-controlled memristor with a CR
q(t; t 0 ) = ˜
f (ϕ(t; t 0 ); ϕ M 1 0 , ϕ M 2 0 )
where
˜
f (ϕ(t; t 0 ); ϕ M 1 0 , ϕ M 2 0 )
.
= ˜
f 1 (ϕ(t; t 0 ); ϕ M 1 0 ) + ˜
f 2 (ϕ(t; t 0 ); ϕ M 2 0 ).
The result is analogous to the parallel connection of two nonlinear voltagecontrolled resistors in the (v, i)-domain.
Example 5.14 (Parallel Connection of a Flux-Controlled Memristor and a Capacitor) Consider as in Fig. 5.21 the parallel connection of a flux-controlled memristor
and a capacitor. The CRs are
q M (t; t 0 ) = f (ϕ M (t; t 0 ) + ϕ M 0 ) − f (ϕ M 0 )
where ϕ M 0 = ϕ M (t 0 ) and
q C (t; t 0 ) = C
dϕ C (t; t 0 )
dt
− q C 0
where q C 0 = Cv C (t 0 ).
We have from KqL
q(t; t 0 ) = q M (t; t 0 ) + q C (t; t 0 )
while KϕL yields
ϕ(t; t 0 ) = ϕ M (t; t 0 ) = ϕ C (t; t 0 ).
By substitution we obtain the CR
q(t; t 0 ) = −f (ϕ M 0 ) − q C 0 + f (ϕ(t; t 0 ) + ϕ M 0 ) + C
dϕ(t; t 0 )
dt
.
Note that this parallel is a dynamic first-order flux-controlled two-terminal element
that is analogous to a dynamic first-order voltage-controlled two-terminal element
given by the parallel of a nonlinear resistor and a capacitor in the (v, i)-domain.