5.4 Kirchhoff Laws for Memristor Circuits
177
= − (q M (t) − q M (t 0 )) .
(5.13)
From (5.13) we can draw two basic results.
(i) First, the nonlinear dynamical behavior for t ≥ t 0 of the M–C circuit is
described in the (ϕ, q)-domain by the following initial value problem (IVP) for a
first-order SE (derived from (5.13) and (5.11))
dϕ M (t)
dt
= −
f (ϕ M (t))
C
+
f (ϕ M 0 )
C
+ v C 0
ϕ M (t 0 ) = ϕ M 0
(5.14)
where the state variable is ϕ M (t) and the initial condition is the known quantity
ϕ M 0 . It is also worth remarking that the initial conditions v C 0 and ϕ M 0 for the state
variable in the (v, i)-domain appear as constant inputs in the r.h.s. of (5.14).
We can summarize these observations as follows. The IVP (5.10) and (5.11) for
a second-order SE in the (v, i)-domain can be reduced to the IVP (5.14) for a firstorder SE in the (ϕ, q)-domain where the r.h.s. depends on the initial conditions v C 0
and ϕ M 0 at t 0 for the state variables in the (v, i)-domain.
This reduction is crucial in analyzing nonlinear dynamics and bifurcations in
the M–C circuit. We will pursue such an analysis in detail later in the book in
Example 5.15. Dual considerations hold in a circuit composed of one chargecontrolled memristor connected to an inductor. In such a case the state variable in
the (ϕ, q)-domain is the memristor charge q M (t) and the initial condition is q M (t 0 ).
(ii) Second, Eq. (5.13)—which yields (5.14)—is obtained from the integration of
KCL over [t 0 , t], where t > t 0 , and it can be formulated as “the sum of the capacitor
incremental charge
q C (t; t 0 ) = q C (t) − q C (t 0 ) = C(v C (t) − v C (t 0 ))
and the memristor incremental charge
q M (t; t 0 ) = q M (t) − q M (t 0 ) = f (ϕ M (t)) − f (ϕ M (t 0 ))
is zero” for any t ≥ t 0 .
It turns out that the fundamental step in reducing by one the order of SEs in
the IVP (5.10) and (5.11) is the integration of the KCL in (t 0 , t), stating that “the
algebraic sum of the incremental charge in any cut-set is zero,” exactly as in the
third form of Kirchhoff law in the (ϕ, q)-domain given in (5.7). With reference to
the M–C circuit in Fig. 5.5a, we can indeed rewrite (5.13) as follows for t ≥ t 0
q C (t; t 0 ) + q M (t; t 0 ) = 0.
(5.15)
Remark 5.1 Since ϕ M (t; t 0 ) = ϕ C (t; t 0 ), (5.14) can also be rewritten as an IVP for
the first-order SE in the state variable ϕ C (t; t 0 )
177
= − (q M (t) − q M (t 0 )) .
(5.13)
From (5.13) we can draw two basic results.
(i) First, the nonlinear dynamical behavior for t ≥ t 0 of the M–C circuit is
described in the (ϕ, q)-domain by the following initial value problem (IVP) for a
first-order SE (derived from (5.13) and (5.11))
dϕ M (t)
dt
= −
f (ϕ M (t))
C
+
f (ϕ M 0 )
C
+ v C 0
ϕ M (t 0 ) = ϕ M 0
(5.14)
where the state variable is ϕ M (t) and the initial condition is the known quantity
ϕ M 0 . It is also worth remarking that the initial conditions v C 0 and ϕ M 0 for the state
variable in the (v, i)-domain appear as constant inputs in the r.h.s. of (5.14).
We can summarize these observations as follows. The IVP (5.10) and (5.11) for
a second-order SE in the (v, i)-domain can be reduced to the IVP (5.14) for a firstorder SE in the (ϕ, q)-domain where the r.h.s. depends on the initial conditions v C 0
and ϕ M 0 at t 0 for the state variables in the (v, i)-domain.
This reduction is crucial in analyzing nonlinear dynamics and bifurcations in
the M–C circuit. We will pursue such an analysis in detail later in the book in
Example 5.15. Dual considerations hold in a circuit composed of one chargecontrolled memristor connected to an inductor. In such a case the state variable in
the (ϕ, q)-domain is the memristor charge q M (t) and the initial condition is q M (t 0 ).
(ii) Second, Eq. (5.13)—which yields (5.14)—is obtained from the integration of
KCL over [t 0 , t], where t > t 0 , and it can be formulated as “the sum of the capacitor
incremental charge
q C (t; t 0 ) = q C (t) − q C (t 0 ) = C(v C (t) − v C (t 0 ))
and the memristor incremental charge
q M (t; t 0 ) = q M (t) − q M (t 0 ) = f (ϕ M (t)) − f (ϕ M (t 0 ))
is zero” for any t ≥ t 0 .
It turns out that the fundamental step in reducing by one the order of SEs in
the IVP (5.10) and (5.11) is the integration of the KCL in (t 0 , t), stating that “the
algebraic sum of the incremental charge in any cut-set is zero,” exactly as in the
third form of Kirchhoff law in the (ϕ, q)-domain given in (5.7). With reference to
the M–C circuit in Fig. 5.5a, we can indeed rewrite (5.13) as follows for t ≥ t 0
q C (t; t 0 ) + q M (t; t 0 ) = 0.
(5.15)
Remark 5.1 Since ϕ M (t; t 0 ) = ϕ C (t; t 0 ), (5.14) can also be rewritten as an IVP for
the first-order SE in the state variable ϕ C (t; t 0 )