42
2 Fundamental Properties of Mem-Elements
Pinched hysteresis loops of ideal memristors present further mathematical
properties when sinusoidal/periodic inputs are considered. A detailed study can be
found in [9]. The next example illustrates the odd symmetry property with respect
to the origin.
Example 2.7 (Pinched Hysteresis Loops with Odd Symmetry with Respect to the
Origin) It can be seen that, as long as the input is sinusoidal, the pinched hysteresis
loop displayed by an ideal memristor is odd symmetric about the origin. To verify
this, consider a flux-controlled ideal memristor q = ˆ
q(ϕ) and suppose it is subject
to v(t) = sin(ωt). We have for t ≥ 0
i(t) = ˆ
q
(ϕ(t))v(t) = ˆ
q
(ϕ(t)) sin(ωt).
Since
ϕ(t) = ϕ(0) +
t
0
sin(ωτ )dτ = ϕ(0) +
1
ω
[1 − cos(ωt)]
then ϕ(t) is even and hence i(t) is odd.
2.2 Ideal Memristors and Non-volatile Memories
Before discussing how ideal memristors can be used as a nonvolatile memory
capable to store information in a “physical state variable,” it is worth to mention
that many electronic memories made of passive two-terminal circuit elements are
typically volatile. In other words, a continuous power supply (and hence power
consumption) is needed to keep on an electronic circuit in its state and then to realize
a nonvolatile memory. This principle is illustrated by the following simple example.
Example 2.8 (Binary Memory with Tunnel Diode) Consider the simple circuit in
Fig. 2.16a with an Esaki (tunnel) diode having a non-monotone current-voltage CR
i = ˆ
i(v). When power is on, i.e., E = 0, the intersection between the load line
E = Ri + v and the diode characteristic i = ˆ
i(v) shows that the circuit has three
different solutions (i.e., operating points). From a real implementation of the circuit
it can be observed that two of these solutions (Q 1 and Q 2 ) are asymptotically stable
(Fig. 2.16b), while one solution (Q 0 ) is unstable (see also Example 4.4 in Chap. 4
for a discussion on stability of these solutions). As such the circuit can implement
a binary memory in the states Q 1 and Q 2 . However, when power is turned off (i.e.,
E = 0), there is a unique solution where voltage and current vanish and the circuit
“forgets Q 1 and Q 2 .” Then, the circuit acts as a volatile memory (Fig. 2.16c) and
the power source E is essential to retain the two states Q 1 and Q 2 .
Now, let us discuss the ideal memristor as a nonvolatile memory and how to
operate it.
2 Fundamental Properties of Mem-Elements
Pinched hysteresis loops of ideal memristors present further mathematical
properties when sinusoidal/periodic inputs are considered. A detailed study can be
found in [9]. The next example illustrates the odd symmetry property with respect
to the origin.
Example 2.7 (Pinched Hysteresis Loops with Odd Symmetry with Respect to the
Origin) It can be seen that, as long as the input is sinusoidal, the pinched hysteresis
loop displayed by an ideal memristor is odd symmetric about the origin. To verify
this, consider a flux-controlled ideal memristor q = ˆ
q(ϕ) and suppose it is subject
to v(t) = sin(ωt). We have for t ≥ 0
i(t) = ˆ
q
(ϕ(t))v(t) = ˆ
q
(ϕ(t)) sin(ωt).
Since
ϕ(t) = ϕ(0) +
t
0
sin(ωτ )dτ = ϕ(0) +
1
ω
[1 − cos(ωt)]
then ϕ(t) is even and hence i(t) is odd.
2.2 Ideal Memristors and Non-volatile Memories
Before discussing how ideal memristors can be used as a nonvolatile memory
capable to store information in a “physical state variable,” it is worth to mention
that many electronic memories made of passive two-terminal circuit elements are
typically volatile. In other words, a continuous power supply (and hence power
consumption) is needed to keep on an electronic circuit in its state and then to realize
a nonvolatile memory. This principle is illustrated by the following simple example.
Example 2.8 (Binary Memory with Tunnel Diode) Consider the simple circuit in
Fig. 2.16a with an Esaki (tunnel) diode having a non-monotone current-voltage CR
i = ˆ
i(v). When power is on, i.e., E = 0, the intersection between the load line
E = Ri + v and the diode characteristic i = ˆ
i(v) shows that the circuit has three
different solutions (i.e., operating points). From a real implementation of the circuit
it can be observed that two of these solutions (Q 1 and Q 2 ) are asymptotically stable
(Fig. 2.16b), while one solution (Q 0 ) is unstable (see also Example 4.4 in Chap. 4
for a discussion on stability of these solutions). As such the circuit can implement
a binary memory in the states Q 1 and Q 2 . However, when power is turned off (i.e.,
E = 0), there is a unique solution where voltage and current vanish and the circuit
“forgets Q 1 and Q 2 .” Then, the circuit acts as a volatile memory (Fig. 2.16c) and
the power source E is essential to retain the two states Q 1 and Q 2 .
Now, let us discuss the ideal memristor as a nonvolatile memory and how to
operate it.