44
2 Fundamental Properties of Mem-Elements
Then:
An ideal flux-controlled (resp., charge-controlled) memristor has a continuum
of nonvolatile memory states and corresponding memductance (resp., memristance) values, i.e., an ideal memristor is a nonvolatile memory memristor.
Actually, if the power is switched off at an instant T , i.e., if we let v(t) = 0, for
any t ≥ T , then the final state value ϕ(T ) is a stable EP and then the corresponding
memductance value ˆ
q (ϕ(T )) is nonvolatile because it is kept by the memristor for
all t > T .
By definition, ϕ(T ) depends just on the (arbitrary) waveform of v(t) for all t ≤
T , thus ϕ(T ) can be changed with continuity by choosing suitable inputs v(t) over
the interval (−∞, T ]. Typically, input pulses with specified amplitude and duration 4
are used to tune ϕ(T ) and then the nonvolatile memory state.
The next example discusses a simple experiment to show the nonvolatility
property of an ideal memristor.
Example 2.9 (Ideal Memristor as Nonvolatile Memory Memristor) Consider a
circuit given by a voltage source v s and a passive ideal flux-controlled memristor
q = ˆ
q(ϕ) as shown in Fig. 2.17a. Suppose the initial flux is ϕ(t 0 ) = 0, hence the
initial charge q(t 0 ) = Q(t 0 ) = 0, as shown in Fig. 2.17b. Then, apply a voltage
pulse v s (t) = E, t ∈ [t 0 , t 0 + ΔT ] and v s (t) = 0, t ≥ t 0 + ΔT (see Fig. 2.17c).
Although v s (t) = 0 (hence i(t) = 0), for all t ≥ t 0 + ΔT , i.e., the power supply
turns off, the memristor is in a steady state with
ϕ(t) = ϕ(t 0 + ΔT ) =
t 0 +ΔT
t 0
v s (t)dt = EΔT = ϕ Q
(2.13)
for all t ≥ t 0 + ΔT (see Fig. 2.17d). Thus the ideal memristor stores its final
memductance value at t 0 + ΔT , i.e., ˆ
q (ϕ(t 0 + ΔT )) = ˆ
q (ϕ Q ), in a permanent
way by means of the steady state (flux) ϕ Q = EΔT (see Fig. 2.17b). The ideal
memristor thus acts as a nonvolatile memory.
2.2.2 Ideal Memristor as Continuum Memory Memristor
A flux-controlled memristor can be used as a nonvolatile analog memory when it
can store a continuum of different memductance values in correspondence with the
stable EPs of the state (flux), 5 as illustrated next.
4 Amplitude and duration of pulses depend on technological processes and materials used to
fabricate memristor devices.
5 Similar considerations hold for charge-controlled memristors.
2 Fundamental Properties of Mem-Elements
Then:
An ideal flux-controlled (resp., charge-controlled) memristor has a continuum
of nonvolatile memory states and corresponding memductance (resp., memristance) values, i.e., an ideal memristor is a nonvolatile memory memristor.
Actually, if the power is switched off at an instant T , i.e., if we let v(t) = 0, for
any t ≥ T , then the final state value ϕ(T ) is a stable EP and then the corresponding
memductance value ˆ
q (ϕ(T )) is nonvolatile because it is kept by the memristor for
all t > T .
By definition, ϕ(T ) depends just on the (arbitrary) waveform of v(t) for all t ≤
T , thus ϕ(T ) can be changed with continuity by choosing suitable inputs v(t) over
the interval (−∞, T ]. Typically, input pulses with specified amplitude and duration 4
are used to tune ϕ(T ) and then the nonvolatile memory state.
The next example discusses a simple experiment to show the nonvolatility
property of an ideal memristor.
Example 2.9 (Ideal Memristor as Nonvolatile Memory Memristor) Consider a
circuit given by a voltage source v s and a passive ideal flux-controlled memristor
q = ˆ
q(ϕ) as shown in Fig. 2.17a. Suppose the initial flux is ϕ(t 0 ) = 0, hence the
initial charge q(t 0 ) = Q(t 0 ) = 0, as shown in Fig. 2.17b. Then, apply a voltage
pulse v s (t) = E, t ∈ [t 0 , t 0 + ΔT ] and v s (t) = 0, t ≥ t 0 + ΔT (see Fig. 2.17c).
Although v s (t) = 0 (hence i(t) = 0), for all t ≥ t 0 + ΔT , i.e., the power supply
turns off, the memristor is in a steady state with
ϕ(t) = ϕ(t 0 + ΔT ) =
t 0 +ΔT
t 0
v s (t)dt = EΔT = ϕ Q
(2.13)
for all t ≥ t 0 + ΔT (see Fig. 2.17d). Thus the ideal memristor stores its final
memductance value at t 0 + ΔT , i.e., ˆ
q (ϕ(t 0 + ΔT )) = ˆ
q (ϕ Q ), in a permanent
way by means of the steady state (flux) ϕ Q = EΔT (see Fig. 2.17b). The ideal
memristor thus acts as a nonvolatile memory.
2.2.2 Ideal Memristor as Continuum Memory Memristor
A flux-controlled memristor can be used as a nonvolatile analog memory when it
can store a continuum of different memductance values in correspondence with the
stable EPs of the state (flux), 5 as illustrated next.
4 Amplitude and duration of pulses depend on technological processes and materials used to
fabricate memristor devices.
5 Similar considerations hold for charge-controlled memristors.