180
5 Flux-Charge Analysis Method of Memristor Circuits
where A ∈ R (n−1)×b is the reduced incidence matrix and i ∈ R b is the vector of
two-terminal element currents. Moreover, KVL yields b − n + 1 fundamental loop
equations (Chap. 3)
Bv(t) = 0
where B ∈ R (b−n+1)×b is the fundamental loop matrix and v ∈ R b is the vector of
two-terminal element voltages. By integrating between t 0 and t ≥ t 0 we obtain
Aq(t) = Aq(t 0 )
(5.17)
Bϕ(t) = Bϕ(t 0 )
(5.18)
for any t ≥ t 0 , where q(t) (resp., ϕ(t)) is the vector of charges (resp., fluxes) of the
b two-terminal elements.
These equations are expressed in the (ϕ, q)-domain, however they involve all the
initial conditions q(t 0 ), ϕ(t 0 ) of two-terminal circuit elements. As already discussed
in Example 5.6, the initial conditions q C k (t 0 ) = C k v C k (t 0 ) and ϕ L k (t 0 ) = L k i L k (t 0 )
can be obtained by the measurement at the instant t 0 of voltages v C k (t 0 ) across
capacitors and currents i L k (t 0 ) through inductors by means of a voltmeter or
an ammeter. Instead, initial conditions as q L k (t 0 ) =
t 0
−∞ i L k (t)dt, ϕ C k (t 0 ) =
t 0
−∞ v C k (t)dt and q R k (t 0 ) =
t 0
−∞ i R k (t)dt cannot be obtained via measurements
at t 0 . Rather, to evaluate them we would need to know the past circuit history for
t < t 0 (see for example Fig. 5.5b), an information that is usually unavailable or
difficult to obtain in practice. In addition, it may happen that q a (t 0 ) (ϕ e (t 0 )) are not
finite for some current (voltage) ideal sources. 3
Kirchhoff Laws can be made independent of initial conditions by using the
incremental fluxes and charges of two-terminal elements defined as
ϕ k (t; t 0 ) ˙
= ϕ k (t) − ϕ k (t 0 ) =
t
t 0
v k (τ )d τ
(5.19)
q k (t; t 0 ) ˙
= q k (t) − q k (t 0 ) =
t
t 0
i k (τ )d τ
(5.20)
for any t ≥ t 0 , where v k (t) is the voltage across and i k (t) is current through any
two-terminal element in LM, respectively. Indeed, using the vector of incremental
charges q(t; t 0 ), the Kirchhoff Charge Law (KqL) takes the simpler form
Aq(t; t 0 ) = 0
(5.21)
3 It is readily derived that if a(t) = A (e(t) = E) for all t ∈ (−∞, t 0 ], with A = 0 (E = 0)
constant, then q a (t 0 ) =
t 0
−∞ A dτ (ϕ e (t 0 ) =
t 0
−∞ E dτ) are infinite.
5 Flux-Charge Analysis Method of Memristor Circuits
where A ∈ R (n−1)×b is the reduced incidence matrix and i ∈ R b is the vector of
two-terminal element currents. Moreover, KVL yields b − n + 1 fundamental loop
equations (Chap. 3)
Bv(t) = 0
where B ∈ R (b−n+1)×b is the fundamental loop matrix and v ∈ R b is the vector of
two-terminal element voltages. By integrating between t 0 and t ≥ t 0 we obtain
Aq(t) = Aq(t 0 )
(5.17)
Bϕ(t) = Bϕ(t 0 )
(5.18)
for any t ≥ t 0 , where q(t) (resp., ϕ(t)) is the vector of charges (resp., fluxes) of the
b two-terminal elements.
These equations are expressed in the (ϕ, q)-domain, however they involve all the
initial conditions q(t 0 ), ϕ(t 0 ) of two-terminal circuit elements. As already discussed
in Example 5.6, the initial conditions q C k (t 0 ) = C k v C k (t 0 ) and ϕ L k (t 0 ) = L k i L k (t 0 )
can be obtained by the measurement at the instant t 0 of voltages v C k (t 0 ) across
capacitors and currents i L k (t 0 ) through inductors by means of a voltmeter or
an ammeter. Instead, initial conditions as q L k (t 0 ) =
t 0
−∞ i L k (t)dt, ϕ C k (t 0 ) =
t 0
−∞ v C k (t)dt and q R k (t 0 ) =
t 0
−∞ i R k (t)dt cannot be obtained via measurements
at t 0 . Rather, to evaluate them we would need to know the past circuit history for
t < t 0 (see for example Fig. 5.5b), an information that is usually unavailable or
difficult to obtain in practice. In addition, it may happen that q a (t 0 ) (ϕ e (t 0 )) are not
finite for some current (voltage) ideal sources. 3
Kirchhoff Laws can be made independent of initial conditions by using the
incremental fluxes and charges of two-terminal elements defined as
ϕ k (t; t 0 ) ˙
= ϕ k (t) − ϕ k (t 0 ) =
t
t 0
v k (τ )d τ
(5.19)
q k (t; t 0 ) ˙
= q k (t) − q k (t 0 ) =
t
t 0
i k (τ )d τ
(5.20)
for any t ≥ t 0 , where v k (t) is the voltage across and i k (t) is current through any
two-terminal element in LM, respectively. Indeed, using the vector of incremental
charges q(t; t 0 ), the Kirchhoff Charge Law (KqL) takes the simpler form
Aq(t; t 0 ) = 0
(5.21)
3 It is readily derived that if a(t) = A (e(t) = E) for all t ∈ (−∞, t 0 ], with A = 0 (E = 0)
constant, then q a (t 0 ) =
t 0
−∞ A dτ (ϕ e (t 0 ) =
t 0
−∞ E dτ) are infinite.