296
7 Pulse Programming of Memristor Circuits
7.6 Programming Memristor Circuits with Pulses and
Time-Varying Inputs
Theorems 7.3 and 7.4 permit to investigate the dynamic properties of autonomous
and non-autonomous memristor circuits N = N D
N R by studying how the
independent sources drive solutions of N through different manifolds in the (n M +
n C + n L )-dimensional state space in the (v, i)-domain. In practical applications, the
following three main cases can be considered:
• autonomous memristor circuits N with no external sources, i.e., e(t) = a(t) = 0
for any t ≥ t 0 ;
• non-autonomous memristor circuits N subject to external pulses with finite time
duration, i.e., voltage e(t) and current a(t) sources vary over a finite time interval
[t 0 , t 1 ] with 0 < t 1 − t 0 = Δ < ∞. The external sources are zero for any t ≥ t 1 .
This class of pulses 7 defines ϕ e (t; t 0 ) and q a (t; t 0 ) as sources with “constant
momentum” in the (ϕ, q)-domain for t ≥ t 1 (following the nomenclature
introduced in [15]) and it is widely used in the experimental characterization
of memristor devices and the programming of memristor-based neuromorphic
systems;
• non-autonomous memristor circuits N with voltage sources e(t) and/or current
sources a(t) varying over the infinite time interval [t 0 , +∞), i.e., N has sources
with “time-varying momentum” in the (ϕ, q)-domain. Typical examples used in
several applications are:
– the class of sinusoidal voltage e(t) and/or current a(t) generators (usually
considered in memristor-based filters), giving sources ϕ e (t; t 0 ) and q a (t; t 0 )
with sinusoidal momentum in the (ϕ, q)-domain;
– the class of constant voltage e(t) and/or current a(t), providing sources
ϕ e (t; t 0 ) and q a (t; t 0 ) with linear time-varying momentum in the (ϕ, q)domain.
7.6.1 Memristor Circuits with No External Sources
Let us consider memristor circuits N = N D
N R satisfying (A1)–(A4) with no
voltage and/or current sources, that is N is an autonomous memristor circuit with
e(t) = a(t) = 0 for any t ≥ t 0 . It turns out that ϕ e (t; t 0 ) = 0 and q a (t; t 0 ) = 0
in the (ϕ, q)-domain as well. Hence, ˙
u x (t; t 0 ) = 0 and ˙
u y (t; t 0 ) = 0 and (7.33)
becomes
7 For simplicity, impulsive voltage and current sources instantaneously applied at t 0 can be
considered by including their effects in the ICs.
7 Pulse Programming of Memristor Circuits
7.6 Programming Memristor Circuits with Pulses and
Time-Varying Inputs
Theorems 7.3 and 7.4 permit to investigate the dynamic properties of autonomous
and non-autonomous memristor circuits N = N D
N R by studying how the
independent sources drive solutions of N through different manifolds in the (n M +
n C + n L )-dimensional state space in the (v, i)-domain. In practical applications, the
following three main cases can be considered:
• autonomous memristor circuits N with no external sources, i.e., e(t) = a(t) = 0
for any t ≥ t 0 ;
• non-autonomous memristor circuits N subject to external pulses with finite time
duration, i.e., voltage e(t) and current a(t) sources vary over a finite time interval
[t 0 , t 1 ] with 0 < t 1 − t 0 = Δ < ∞. The external sources are zero for any t ≥ t 1 .
This class of pulses 7 defines ϕ e (t; t 0 ) and q a (t; t 0 ) as sources with “constant
momentum” in the (ϕ, q)-domain for t ≥ t 1 (following the nomenclature
introduced in [15]) and it is widely used in the experimental characterization
of memristor devices and the programming of memristor-based neuromorphic
systems;
• non-autonomous memristor circuits N with voltage sources e(t) and/or current
sources a(t) varying over the infinite time interval [t 0 , +∞), i.e., N has sources
with “time-varying momentum” in the (ϕ, q)-domain. Typical examples used in
several applications are:
– the class of sinusoidal voltage e(t) and/or current a(t) generators (usually
considered in memristor-based filters), giving sources ϕ e (t; t 0 ) and q a (t; t 0 )
with sinusoidal momentum in the (ϕ, q)-domain;
– the class of constant voltage e(t) and/or current a(t), providing sources
ϕ e (t; t 0 ) and q a (t; t 0 ) with linear time-varying momentum in the (ϕ, q)domain.
7.6.1 Memristor Circuits with No External Sources
Let us consider memristor circuits N = N D
N R satisfying (A1)–(A4) with no
voltage and/or current sources, that is N is an autonomous memristor circuit with
e(t) = a(t) = 0 for any t ≥ t 0 . It turns out that ϕ e (t; t 0 ) = 0 and q a (t; t 0 ) = 0
in the (ϕ, q)-domain as well. Hence, ˙
u x (t; t 0 ) = 0 and ˙
u y (t; t 0 ) = 0 and (7.33)
becomes
7 For simplicity, impulsive voltage and current sources instantaneously applied at t 0 can be
considered by including their effects in the ICs.