5.4 Kirchhoff Laws for Memristor Circuits
175
For t > 0 the C-R circuit is thus described in the (ϕ, q)-domain by the first-order
SE
d
dt
ϕ C (t; 0) = −
ϕ C (t; 0)
RC
+ v C 0
in the state variable ϕ C (t; 0). Note that the initial condition vanishes by definition
of incremental flux, i.e.,
ϕ C (0; 0) = 0.
From the theory of linear differential equations the solution is given by
ϕ C (t; 0) = RCv C 0 (1 − e
−
t
RC )
for t ≥ 0. By differentiation this yields the well-known transient (discharge of C on
R)
v C (t) =
d
dt
ϕ C (t; 0) = v C 0 e
−
t
RC
for t ≥ 0.
5.4 Kirchhoff Laws for Memristor Circuits
For circuits without memristors like those considered in the previous examples, as it
is intuitively reasonable, the analysis in the (ϕ, q)-domain may result more complex
than that in the (v, i)-domain. However, when there are memristors, it is found that it
is more convenient to analyze the circuit in the (ϕ, q)-domain. Some advantages of
the analysis in the (ϕ, q)-domain are illustrated via an elementary memristor circuit
in the next example. The remaining part of the chapter then develops in a systematic
way the flux-charge analysis approach for a broad class of memristor circuits.
Example 5.7 (Memristor-Capacitor Circuit) Consider for t ≥ t 0 , where t 0 is a
finite initial instant, a simple circuit composed by one flux-controlled memristor M
with CR q M = f (ϕ M ) 2 andϕ M (t) for t < t 0 as well (i.e., S 1 and S 3 are closed
while S 2 is open). The two-terminal circuit elements L a and L b can be any linear
networks made of resistors, capacitors, inductors, voltage, and current sources and
2 For convenience, from now on in the book we will use the simplified notation q M = f (ϕ M ),
instead of the notation q M = ˆ
q M (ϕ M ) introduced in Chap. 1, to denote a flux-controlled memristor.
Analogously, we will use ϕ M = h(q M ), instead of ϕ M = ˆ
ϕ M (q M ), to denote a charge-controlled
memristor.
175
For t > 0 the C-R circuit is thus described in the (ϕ, q)-domain by the first-order
SE
d
dt
ϕ C (t; 0) = −
ϕ C (t; 0)
RC
+ v C 0
in the state variable ϕ C (t; 0). Note that the initial condition vanishes by definition
of incremental flux, i.e.,
ϕ C (0; 0) = 0.
From the theory of linear differential equations the solution is given by
ϕ C (t; 0) = RCv C 0 (1 − e
−
t
RC )
for t ≥ 0. By differentiation this yields the well-known transient (discharge of C on
R)
v C (t) =
d
dt
ϕ C (t; 0) = v C 0 e
−
t
RC
for t ≥ 0.
5.4 Kirchhoff Laws for Memristor Circuits
For circuits without memristors like those considered in the previous examples, as it
is intuitively reasonable, the analysis in the (ϕ, q)-domain may result more complex
than that in the (v, i)-domain. However, when there are memristors, it is found that it
is more convenient to analyze the circuit in the (ϕ, q)-domain. Some advantages of
the analysis in the (ϕ, q)-domain are illustrated via an elementary memristor circuit
in the next example. The remaining part of the chapter then develops in a systematic
way the flux-charge analysis approach for a broad class of memristor circuits.
Example 5.7 (Memristor-Capacitor Circuit) Consider for t ≥ t 0 , where t 0 is a
finite initial instant, a simple circuit composed by one flux-controlled memristor M
with CR q M = f (ϕ M ) 2 andϕ M (t) for t < t 0 as well (i.e., S 1 and S 3 are closed
while S 2 is open). The two-terminal circuit elements L a and L b can be any linear
networks made of resistors, capacitors, inductors, voltage, and current sources and
2 For convenience, from now on in the book we will use the simplified notation q M = f (ϕ M ),
instead of the notation q M = ˆ
q M (ϕ M ) introduced in Chap. 1, to denote a flux-controlled memristor.
Analogously, we will use ϕ M = h(q M ), instead of ϕ M = ˆ
ϕ M (q M ), to denote a charge-controlled
memristor.