2.1 Ideal Memristor: Basic Properties and Signatures
29
2.1.1 Passivity
Property 2.1 An ideal memristor is passive if and only if ˆ
ϕ (q) ≥ 0, for any q, in the
charge-controlled case or ˆ
q (ϕ) ≥ 0, for any ϕ, in the flux-controlled case, namely,
if and only if its nonlinear characteristic is a monotone nondecreasing function.
Proof To verify the sufficiency of this condition, let us consider, without loss of
generality, a flux-controlled ideal memristor. Since the memductance W (ϕ) =
ˆ
q (ϕ) ≥ 0 for all ϕ, then p(t) = v(t)i(t) = W (ϕ(t))v 2 (t) ≥ 0, i.e., the memristor
always absorbs electric power and hence it is passive.
Let us now sketch the proof of necessity. Suppose the memristor has a nonmonotone characteristic ˆ
q(ϕ) and it is connected to an (ideal) voltage source v s (t)
as in Fig. 2.1. Suppose the initial flux across the memristor is ϕ(0) = 0 at the
initial instant t = 0. Apply via the voltage source a rectangular voltage pulse in
[t 0 , t 0 +ΔT ] that moves the operating point of the memristor at Q = (ϕ Q , ˆ
q(ϕ Q )) on
the characteristic curve such that ˆ
q (ϕ Q ) < 0. Hence, the memductance is negative
in a neighborhood of ϕ Q . If we then apply a sinusoidal signal v s (t) = sin(ωt), with
a sufficiently small amplitude , it can be seen that the energy w(t) =
t
0 p(τ )dτ
tends to −∞ as t → +∞. Then, the memristor is active since it is able to supply to
the voltage source an infinite amount of energy.
Example 2.1 (Active Memristor) Given a passive flux-controlled (ideal) memristor
with a characteristic q = bϕ 3 , where b > 0, we can realize an active (ideal)
memristor simply by connecting in parallel a negative conductance G < 0 obtained
for instance by means of a Negative Impedance Converter (NIC) [7, pag. 192]. It is
easily seen that such an element is equivalent to an active flux-controlled memristor
with a non-monotone characteristic q = Gϕ + bϕ 3 .
2.1.2 No Energy Storage Property
Property 2.2 A passive memristor cannot store or deliver energy.
Proof Let us consider a simple circuit given by a passive charge-controlled ideal
memristor, with an initial charge q(0) = 0, which is connected to a resistor R > 0
at t = 0. The governing equation for t ≥ 0 is
R + ˆ
ϕ
(q(t))
dq(t)
dt
= 0.
(2.6)
Since the resistor is passive, we have ϕ (q) ≥ 0, hence R + ϕ (q(t)) > 0 and
then i(t) = 0 for any t ≥ 0. It follows that q(t) = q(0) and p(t) = v(t)i(t) = 0 for
any t ≥ 0, i.e., the ideal memristor cannot deliver energy to the resistor.
29
2.1.1 Passivity
Property 2.1 An ideal memristor is passive if and only if ˆ
ϕ (q) ≥ 0, for any q, in the
charge-controlled case or ˆ
q (ϕ) ≥ 0, for any ϕ, in the flux-controlled case, namely,
if and only if its nonlinear characteristic is a monotone nondecreasing function.
Proof To verify the sufficiency of this condition, let us consider, without loss of
generality, a flux-controlled ideal memristor. Since the memductance W (ϕ) =
ˆ
q (ϕ) ≥ 0 for all ϕ, then p(t) = v(t)i(t) = W (ϕ(t))v 2 (t) ≥ 0, i.e., the memristor
always absorbs electric power and hence it is passive.
Let us now sketch the proof of necessity. Suppose the memristor has a nonmonotone characteristic ˆ
q(ϕ) and it is connected to an (ideal) voltage source v s (t)
as in Fig. 2.1. Suppose the initial flux across the memristor is ϕ(0) = 0 at the
initial instant t = 0. Apply via the voltage source a rectangular voltage pulse in
[t 0 , t 0 +ΔT ] that moves the operating point of the memristor at Q = (ϕ Q , ˆ
q(ϕ Q )) on
the characteristic curve such that ˆ
q (ϕ Q ) < 0. Hence, the memductance is negative
in a neighborhood of ϕ Q . If we then apply a sinusoidal signal v s (t) = sin(ωt), with
a sufficiently small amplitude , it can be seen that the energy w(t) =
t
0 p(τ )dτ
tends to −∞ as t → +∞. Then, the memristor is active since it is able to supply to
the voltage source an infinite amount of energy.
Example 2.1 (Active Memristor) Given a passive flux-controlled (ideal) memristor
with a characteristic q = bϕ 3 , where b > 0, we can realize an active (ideal)
memristor simply by connecting in parallel a negative conductance G < 0 obtained
for instance by means of a Negative Impedance Converter (NIC) [7, pag. 192]. It is
easily seen that such an element is equivalent to an active flux-controlled memristor
with a non-monotone characteristic q = Gϕ + bϕ 3 .
2.1.2 No Energy Storage Property
Property 2.2 A passive memristor cannot store or deliver energy.
Proof Let us consider a simple circuit given by a passive charge-controlled ideal
memristor, with an initial charge q(0) = 0, which is connected to a resistor R > 0
at t = 0. The governing equation for t ≥ 0 is
R + ˆ
ϕ
(q(t))
dq(t)
dt
= 0.
(2.6)
Since the resistor is passive, we have ϕ (q) ≥ 0, hence R + ϕ (q(t)) > 0 and
then i(t) = 0 for any t ≥ 0. It follows that q(t) = q(0) and p(t) = v(t)i(t) = 0 for
any t ≥ 0, i.e., the ideal memristor cannot deliver energy to the resistor.