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5 Flux-Charge Analysis Method of Memristor Circuits
In the standard (v, i)-domain, KCL states that the algebraic sum of all currents
in a cut-set vanishes for all times t (Sect. 3.1.2 in Chap. 3). This can be written
mathematically as
k
i k (t) = 0
for any cut-set and any t. In the same domain KVL states that the algebraic sum of
all voltages around a loop vanishes for all t (Chap. 3), which can be written as
k
v k (t) = 0
around any loop and for any t.
By integrating in time KCL and KVL we next establish three different forms that
can be given to Kirchhoff laws in the (ϕ, q)-domain.
1. First Form of Kirchhoff Laws in the (ϕ, q)-domain.
If we integrate KCL and KVL with respect to time in the interval (−∞, t),
we would obtain
k
q k (t) = 0
(5.3)
for any cut-set and any t and
k
ϕ k (t) = 0
(5.4)
around any loop and for any t, where
q k (t) =
t
−∞
i k (τ )dτ
is the charge and
ϕ k (t) =
t
−∞
v k (τ )dτ
is the flux of the k-th two-terminal element. In practice, −∞ is the instant
where each two-terminal element is manufactured. Clearly, q k (−∞) = 0 and
ϕ k (−∞) = 0.
Equations (5.3) and (5.4) express the principles of conservation of charge and
flux [7]: Charge and flux can neither be created nor destroyed, i.e., the quantity
of charge and flux is always conserved.