IEEE Circuits and Systems Magazine - Q2 2018 - 44
F ^v^ah (t), v^a + 1h (t), v^a + 2h (t) .., i^ bh (t ), i^ b + 1h (t), i^ b + 2h (t), ... h = 0
(9)
Eq. (9) can be considered a model of a two-terminal
dynamical system composed of individual building
blocks, including the prospective HOEs. In the limiting
case, when the equation describes a single HOE, Eq. (9)
becomes its constitutive relation.
LM
CM
RM
-
C
L
+
R
+
Vaudio
-
Assume that the elements fulfill the homothety
fingerprint. It holds for arbitrary integers j, k that
v n(a +j )(t ) = n a + j + 1 v (a + j )(nt ) and i n( b + k)(t) = n b + k + 1 i ( b + k) (nt).
For the accelerated set of signals, Eq. (9) can be therefore
written in the form
F (n a + 1 v^ah (nt), n a + 2 v^a + 1h (nt), n a + 3 v^a + 2h (nt), ...,
n b + 1 i^bh (nt), n b + 2 i^b + 1h (nt), n b + 3 i^b + 2h (nt), ...) = 0 (10)
It is obvious that if the nonlinear equation (9) is to
hold also for the second case (10), then the function
F ( ) must depend only on the variables which are multiplied by the number n with the exponent equal to zero.
Then Eq. (9) must be in the form F (v (-1), i (-1) ) = 0, where
v (-1) = {, i (-1) = q. However, this is the constitutive relation of the ideal memristor.
On the assumption that Eq. (9) is linear, it holds true
also for the voltage and current pair with the same order
c of the derivative or integral, thus (v (c), i (c) ), because
F (n c + 1 v (c) (nt ), n c + 1 i (c) (nt )) = F (v (c) (nt ), i (c) (nt )). The corresponding HOE (c, c) is located in the storeyed structure
in the "horizontal position". If the constitutive relation is
linear, then all such elements behave as a linear resistor.
Note that more general exceptions also follow from
(9) and (10) than those resulting from the analysis
of the definition (2) of memristive systems. The homothety fingerprint can also be fulfilled for elements
whose Eq. (9) depends not only on the charge and
flux but also on the signs of the remaining variables
v (a j), i ( b k), a j ! -1, b k ! -1:
(a)
F ^{, q, sign ^v^a j hh, .., sign ^i^ bkhhh = 0
M e m
r
i
s
to
r
(b)
Figure 7. One-port containing six fundamental elements
from Chua's table driven by audiosignal. the constitutive relations of rCL elements are: resistor i = 10 5 (v 4 + 1) v; capacitor: q = 10 -6 (v 4 + 1) v; inductor: { = 10 -3 (i 5 + 1) i; memelements: memristor model [22], R on = 100 X, R off = 16 kX,
R init = 1 kX, D = 10 n, p = 1, n v = 2e-12, Joglekar window function; memcapacitor model [29], C min = 10 nF,
C max = 10 nF, C init = 16 nF, k = 5e9, p = 1, J o g l e k a r
window function; meminductor model [30], L min = 1 nH,
L max = 100 mH, L init = 80 mH, k = 3e3, p = 10, Joglekar
window function. the audio signal is the digitized word
"memristor" pronounced by Leon Chua during his memorable lecture in Berkeley [11].
44
IEEE CIrCuItS aND SyStEmS magazINE
(11)
The systems (11) can be regarded as pseudomemristors whose constitutive relation is discontinuously
modified via the polarity of the above circuit variables.
Notwithstanding that these elements are outside the classical definitions of memristors and memristive systems,
their behavior is governed by the homothety theorem.
Fig. 5 illustrates the characteristics of two-state voltagecontrolled memristor with piece-wise-linear (PWL) constitutive relation, where the flux threshold, initiating the
change of the memristor state, depends on the sign of
the derivative of voltage. The simulation results in Fig. 6
clearly show that in spite of being rather complicated, the
shape of the pinched hysteresis loops is preserved when
the voltage and frequency increase simultaneously.
An interesting case may arise if the flux swings within the
values { p and { n in the characteristics in Fig. 5. The element then toggles between the states Ron and R off due to
the changing signs of the derivative of voltage, and therefore it behaves as a linear time-varying resistor controlled
SECOND quartEr 2018
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