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10 Extended Memristor Devices
10.2 Simple Circuit with C and an Extended Memristor
The aim of this section is to show that if we use a piecewise linear approximation of
the nonlinear resistor, it is possible to exploit the Flux-Charge Analysis Method
(FCAM) in Chap. 5 for investigating the nonlinear dynamics in circuits with
extended memristors modeling nonvolatile switching memory devices.
Let us consider an extended memristor D ext made of:
(a) a (passive) nonlinear resistor approximated by a piecewise linear function
F R (v) = G 1 v for v ≥ 0 while F R (v) = G 2 v for v < 0, where G 1 , G 2 ≥ 0 and
G 1 = G 2 ;
(b) a locally active flux-controlled memristor described by q M = f (ϕ M ) =
−aϕ M + bϕ 3
M with a, b > 0, hence W (ϕ M ) = −a + 3bϕ 2
M .
Then, consider a simple circuit obtained by connecting for t ≥ t 0 an ideal passive
capacitor C in parallel to the extended memristor (denoted by eM) as defined above
in (a) and (b).
The SEs of the eM–C circuit for t ≥ t 0 are immediately obtained as the system
of two first-order ODEs
C
dv C
dt
= −W (ϕ M )v C − F R (v C )
(10.12)
dϕ M
dt
= v C
(10.13)
for any t ≥ t 0 . The state variables in the (v, i)-domain are v C and ϕ M and the
initial conditions v C (t 0 ) = v 0 , ϕ M (t 0 ) = ϕ 0 . Since f and F R are locally Lipschitz,
given any initial condition (v 0 , ϕ 0 ) T ∈ R 2 , there exists a unique solution (v C , ϕ M )
of (10.12)–(10.13), that is defined on a maximal interval of existence [t 0 , t 0 + T ),
where T > 0 is possibly +∞ [10].
It can be easily checked that the following properties hold.
P 1 : there exists a continuum of EPs {( ¯
v, ¯
ϕ) ∈ R 2 : ¯
v = 0}
P 2 : if the initial condition v 0 = 0, then (v C (t), ϕ M (t)) = (0, ϕ 0 ) for any t ≥ t 0 .
Properties P 1 and P 2 imply that the whole state-space (v C , ϕ M ) T ∈ R 2 in the
(v, i)-domain can be split into two regions R + and R − separated by the line of EPs,
i.e., R + = {(v C , ϕ M ) ∈ R 2 : v C > 0} whereas R − = {(v C , ϕ M ) T ∈ R 2 : v C < 0}.
Hence, for any initial condition in R + , that is v 0 > 0, the dynamic route analysis
of (10.12) and (10.13) and the uniqueness of the solution imply that v C (t) > 0
for all t ≥ t 0 . The r.h.s. of (10.12) is W + (ϕ M ) = −(a + G 1 ) + 3bϕ 2
M for any
t ∈ [t 0 , t 0 + T ). It follows that the eM–C circuit is equivalent to a capacitor in
parallel to an ideal memristor defined by f + (ϕ M ) = −(a + G 1 )ϕ M + bϕ 3
M and
thus FCAM can be used in order to analyze the dynamic behavior in R + . Similar
considerations hold in R − and permit to reduce the analysis of the eM–C circuit to
10 Extended Memristor Devices
10.2 Simple Circuit with C and an Extended Memristor
The aim of this section is to show that if we use a piecewise linear approximation of
the nonlinear resistor, it is possible to exploit the Flux-Charge Analysis Method
(FCAM) in Chap. 5 for investigating the nonlinear dynamics in circuits with
extended memristors modeling nonvolatile switching memory devices.
Let us consider an extended memristor D ext made of:
(a) a (passive) nonlinear resistor approximated by a piecewise linear function
F R (v) = G 1 v for v ≥ 0 while F R (v) = G 2 v for v < 0, where G 1 , G 2 ≥ 0 and
G 1 = G 2 ;
(b) a locally active flux-controlled memristor described by q M = f (ϕ M ) =
−aϕ M + bϕ 3
M with a, b > 0, hence W (ϕ M ) = −a + 3bϕ 2
M .
Then, consider a simple circuit obtained by connecting for t ≥ t 0 an ideal passive
capacitor C in parallel to the extended memristor (denoted by eM) as defined above
in (a) and (b).
The SEs of the eM–C circuit for t ≥ t 0 are immediately obtained as the system
of two first-order ODEs
C
dv C
dt
= −W (ϕ M )v C − F R (v C )
(10.12)
dϕ M
dt
= v C
(10.13)
for any t ≥ t 0 . The state variables in the (v, i)-domain are v C and ϕ M and the
initial conditions v C (t 0 ) = v 0 , ϕ M (t 0 ) = ϕ 0 . Since f and F R are locally Lipschitz,
given any initial condition (v 0 , ϕ 0 ) T ∈ R 2 , there exists a unique solution (v C , ϕ M )
of (10.12)–(10.13), that is defined on a maximal interval of existence [t 0 , t 0 + T ),
where T > 0 is possibly +∞ [10].
It can be easily checked that the following properties hold.
P 1 : there exists a continuum of EPs {( ¯
v, ¯
ϕ) ∈ R 2 : ¯
v = 0}
P 2 : if the initial condition v 0 = 0, then (v C (t), ϕ M (t)) = (0, ϕ 0 ) for any t ≥ t 0 .
Properties P 1 and P 2 imply that the whole state-space (v C , ϕ M ) T ∈ R 2 in the
(v, i)-domain can be split into two regions R + and R − separated by the line of EPs,
i.e., R + = {(v C , ϕ M ) ∈ R 2 : v C > 0} whereas R − = {(v C , ϕ M ) T ∈ R 2 : v C < 0}.
Hence, for any initial condition in R + , that is v 0 > 0, the dynamic route analysis
of (10.12) and (10.13) and the uniqueness of the solution imply that v C (t) > 0
for all t ≥ t 0 . The r.h.s. of (10.12) is W + (ϕ M ) = −(a + G 1 ) + 3bϕ 2
M for any
t ∈ [t 0 , t 0 + T ). It follows that the eM–C circuit is equivalent to a capacitor in
parallel to an ideal memristor defined by f + (ϕ M ) = −(a + G 1 )ϕ M + bϕ 3
M and
thus FCAM can be used in order to analyze the dynamic behavior in R + . Similar
considerations hold in R − and permit to reduce the analysis of the eM–C circuit to