IEEE Circuits and Systems Magazine - Q2 2018 - 42

Let i, v, and x in Eqs. (2) for the memristive system
denote respectively the original excitation i (t), the response to this excitation v (t), and the corresponding
waveform of the state vector x (t). If the system exhibits
the homothety fingerprint, then these equations also
hold for the n-times accelerated and n-times amplified current excitation i n (t) = n $ i (nt), for the n-times
accelerated and n-times amplified voltage response
o n (t) = n $ v (nt), and for the corresponding vector x n (t)
of yet unspecified waveform. Substituting the above
conditions in the port equation (2) and arranging yield
v ^nt h = R M ^x n (t), ni ^nt hh i ^nt h

(3)

It follows from (3) that, if the homothety fingerprint
is fulfilled, then the waveform of the memristance R M
will be only n-times accelerated in comparison with the
original excitation, or
R M ^x n (t), ni ^nt hh = R M ^x (nt), i ^nt hh

(4)

The condition (4) must hold for all possible waveforms of current and state vector. This is possible if
x n (t) = x (nt) and, concurrently, if the function R M does
not depend on the value of current. The latter condition
can be fulfilled in two ways: 1) The memristance will not
depend on the current at all. 2) The memristance will be
modulated by the current, but regardless of the current
level; in other words, only the polarity of the current will
apply. Two special types of the port equation (2) correspond to these two cases:
o = R M (x) i

(5a)

q, RM

dv
/d
q,

q, d
v

RM, dv /dt > 0

/dt >

0

t<

0

Roff

RM, dv /dt < 0
Ron

0

ϕp

ϕn

ϕ

Figure 5. Demonstration of the characteristics of a memristor whose constitutive relation depends on the sign of the derivative of voltage. the PWL flux-charge constitutive relation
(solid curve) and the memristance vs. flux (dashed curve).
the flux threshold depends on the sign of the derivative of
voltage: { p for dv/dt > 0, { n for dv/dt < 0.

42

IEEE CIrCuItS aND SyStEmS magazINE

o = R M ^ x, sign ^ i hh i

(5b)

where sign ( ) is the sign function.
Since x n (t) = x (nt) holds for both cases, it signifies
that the state vector is only n-times accelerated without
modifying the gain, and its derivative with respect to
time will therefore be n-times higher than for the original excitation. This implies a condition for the function
f in the state equation (2), which must be fulfilled for all
current and state waveforms:
f ^x (nt), ni ^nt hh = nf ^x (nt), i ^nt hh

(6)

This is possible for two special types of the state
equation
xo = f (x) i

(7a)

xo = f (x, sign (i )) i

(7b)

The combination of port equation (5a) and state
equation (7a) represents the ideal generic memristor
(i.e. with a scalar state x ) or ideal memristor for x = q
or generic memristor with a special right side of the
state equation.
Equations (5b) and (7b) are pathological cases when
the governing rules for the memristance R M and the
state x depend on the direction of the flowing current.
Such cases belong to exceptions to the rule that the
homothety fingerprint only holds for ideal memristors,
ideal generic memristors, and a class of generic memristors with the state equation (7a).
The combination of equations (5a) and (7b) leads to
a special extended memristor, which can be regarded as
the generic memristor with abruptly commutated state
depending on the direction of current. The memristor
with nonlinear dopant drift modeled by the Biolek window [22] is a typical representative of such elements. If
port Eq. (5b) holds, then the memristance of the system
is switched when the current changes its direction.
However, the above specification of various memristive systems, which surely comply with the homothety
fingerprint, does not exclude the possible existence of
elements outside the group of memristive systems (2)
which are also governed by this fingerprint. Let us seek
them within the group of quite general one-ports defined
by nonlinear port and state equations, representing a
generalization of the model (4) of memristive systems:
y = g (x, u), xo = f (x, u)

(8)

Here x is the state vector; u and y are the input and
output (I/O) variables as arbitrary derivatives or integrals of port voltages or currents, thus v (a) or i ( b), a
SECOND quartEr 2018



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