32
2 Fundamental Properties of Mem-Elements
Example 2.2 (Hysteresis Loops of a Passive Ideal Memristor) Consider a passive
ideal flux-controlled memristor with characteristic q = ϕ +
1
3 ϕ 3 , suppose it is
subject to a voltage v(t) = sin(ωt) and the initial condition is ϕ(0) = 0. Simple
calculations permit to derive that:
ϕ(t) =
1
ω
(1 − cos(ωt))
(2.7)
q(t) =
1
ω
[1 − cos(ωt)] +
1
3ω 3 [1 − cos(ωt)]
3
(2.8)
i(t) =
1 +
1
ω 2 [1 − cos(ωt)]
2
sin(ωt).
(2.9)
Figure 2.2 reports the pinched hysteresis loops observed in the (v, i)-plane when
ω = 1, ω = 3, and ω = 9, respectively. Note that the hysteresis loops occur because
maxima and minima of the sinusoidal input current take place at a different instant
with respect to the corresponding voltage. Clearly, hysteresis loops are pinched since
i = 0 if and only if v = 0.
As seen in Example 2.2, hysteresis loops depend upon the frequency of the
applied sinusoidal signals and their area shrink as the frequency increases. When
the frequency tends to infinity, then a pinched hysteresis loop tends to a straight line,
i.e., the ideal memristor behaves as a linear resistor whose resistance is independent
of the amplitude of the input signal but depends on the initial condition (for the flux
Fig. 2.2 Pinched hysteresis
loops of a passive
flux-controlled memristor in
response to voltage sinusoidal
inputs with the same
amplitude but different
angular frequencies