2.1 Ideal Memristor: Basic Properties and Signatures
33
or charge). To see this, consider a flux-controlled memristor subject to a sinusoidal
voltage v(t) = A sin(ωt) for t ≥ 0. We may write
ϕ(t) = ϕ(0) +
t
0
A sin(ωτ )dτ = ϕ(0) +
A
ω
(1 − cos(ωt)) ϕ(0)
(2.10)
for any t, as ω → +∞.
It can also be seen that, for the same sinusoidal input, the shape of the pinched
hysteresis loop is strongly dependent upon the initial condition.
Example 2.3 (Influence of Initial Conditions on Pinched Hysteresis Loops) Consider a flux-controlled memristor with a piecewise linear characteristic
q = ˆ
q(ϕ) = 0.01ϕ + 0.04(|ϕ + 0.25| − |ϕ − 0.25|)
as in Fig. 2.3a that is subject to the voltage v(t) = 1.2 sin(t). For two different initial
conditions ϕ(0) = −0.25 and ϕ(0) = −0.3 we observe two quite different pinched
hysteresis loops as shown in Fig. 2.3b, c, respectively.
Example 2.4 (Experiment to Characterize an Ideal Memristor) Consider a fluxcontrolled memristor with CR q = sin(ϕ) and apply three different voltage
signals v(t) = sin(ωt) with ω = 3.3, ω = 6.6 and ω = 9.9, respectively. The
corresponding waveforms of v(t), ϕ(t) =
t
−∞ v(τ )dτ = ϕ(0) +
t
0 v(τ )dτ ,
q(t) = sin ϕ(t) and i(t) = dq(t)/dt, are plotted in Fig. 2.4, assuming the initial
flux ϕ(0) = 0.
Suppose we did not know the collection of signals depicted in Fig. 2.4 is derived,
or measured, from an ideal memristor characterized by such sinusoidal CR, and we
are asked to develop a circuit model which would reproduce with sufficient accuracy
the waveforms in Fig. 2.4.
To this end we may depict such curves in the planes v–i, ϕ–i, v–q, and ϕ–q,
respectively, as shown in Fig. 2.5a–d. Since the input is periodic, each curve is also
periodic. However, it is seen that in the planes v–i, ϕ–i, and v–q the obtained curves
are closed loops, hence the two-terminal element cannot be modeled by a nonlinear
resistor, inductor, or capacitor (cf. Chap. 1). Only in the plane ϕ–q we obtain
an open-ended loci which is traced back and forth during the application of the
sinusoidal signal. We conclude that, based on these experiments, the considered twoterminal element may be reasonably described by a memristor. Other experiments
with different signals would of course be needed to confirm this finding.
Example 2.5 (Constitutive Relation of an Ideal Memristor) Let us consider a simple
circuit with a voltage source v s and a linear resistor R connected to an unknown
device D (see Fig. 2.6). The voltage source v s in series with the resistance R can
model a real voltage source supplying the device D. The circuit is subject to different
waveforms v s (t) with zero mean value (i.e., no dc component), namely sinusoids
v s (t) = A sin(ωt), a combination of sinusoids and triangular waveforms.
33
or charge). To see this, consider a flux-controlled memristor subject to a sinusoidal
voltage v(t) = A sin(ωt) for t ≥ 0. We may write
ϕ(t) = ϕ(0) +
t
0
A sin(ωτ )dτ = ϕ(0) +
A
ω
(1 − cos(ωt)) ϕ(0)
(2.10)
for any t, as ω → +∞.
It can also be seen that, for the same sinusoidal input, the shape of the pinched
hysteresis loop is strongly dependent upon the initial condition.
Example 2.3 (Influence of Initial Conditions on Pinched Hysteresis Loops) Consider a flux-controlled memristor with a piecewise linear characteristic
q = ˆ
q(ϕ) = 0.01ϕ + 0.04(|ϕ + 0.25| − |ϕ − 0.25|)
as in Fig. 2.3a that is subject to the voltage v(t) = 1.2 sin(t). For two different initial
conditions ϕ(0) = −0.25 and ϕ(0) = −0.3 we observe two quite different pinched
hysteresis loops as shown in Fig. 2.3b, c, respectively.
Example 2.4 (Experiment to Characterize an Ideal Memristor) Consider a fluxcontrolled memristor with CR q = sin(ϕ) and apply three different voltage
signals v(t) = sin(ωt) with ω = 3.3, ω = 6.6 and ω = 9.9, respectively. The
corresponding waveforms of v(t), ϕ(t) =
t
−∞ v(τ )dτ = ϕ(0) +
t
0 v(τ )dτ ,
q(t) = sin ϕ(t) and i(t) = dq(t)/dt, are plotted in Fig. 2.4, assuming the initial
flux ϕ(0) = 0.
Suppose we did not know the collection of signals depicted in Fig. 2.4 is derived,
or measured, from an ideal memristor characterized by such sinusoidal CR, and we
are asked to develop a circuit model which would reproduce with sufficient accuracy
the waveforms in Fig. 2.4.
To this end we may depict such curves in the planes v–i, ϕ–i, v–q, and ϕ–q,
respectively, as shown in Fig. 2.5a–d. Since the input is periodic, each curve is also
periodic. However, it is seen that in the planes v–i, ϕ–i, and v–q the obtained curves
are closed loops, hence the two-terminal element cannot be modeled by a nonlinear
resistor, inductor, or capacitor (cf. Chap. 1). Only in the plane ϕ–q we obtain
an open-ended loci which is traced back and forth during the application of the
sinusoidal signal. We conclude that, based on these experiments, the considered twoterminal element may be reasonably described by a memristor. Other experiments
with different signals would of course be needed to confirm this finding.
Example 2.5 (Constitutive Relation of an Ideal Memristor) Let us consider a simple
circuit with a voltage source v s and a linear resistor R connected to an unknown
device D (see Fig. 2.6). The voltage source v s in series with the resistance R can
model a real voltage source supplying the device D. The circuit is subject to different
waveforms v s (t) with zero mean value (i.e., no dc component), namely sinusoids
v s (t) = A sin(ωt), a combination of sinusoids and triangular waveforms.