IEEE Circuits and Systems Magazine - Q2 2018 - 19

These 5 values of x can also be obtained by translating
the POP in Fig. 2 upward by V volts if V > 0, or downward by - V volts if V < 0, and extracting the value of
x where the translated POP intersects the x -axis. In
this paper, we choose V = - 7 V, and pick the leftmost
equilibrium point Q 0 with x = X 0 = 2. This equilibrium
point can be identified as the DC operating point located
at V = - 7 V on the negative-slope portion of the DC V-I
curve (see inset) in Fig. 4.
To derive the small-signal equivalent circuit of
the 4-lobe Chua corsage memristor at an equilibrium
point Q on the DC V-I curve let us defined x = X + dx,
v = V + dv, and i = I + di. Taylor series expansion of the
current i = G (x) v about the equilibrium point x (Q) = X
and v (Q) = V is given by
i = I + di = a 00 (Q) + a 11 (Q)dx + a 12 (Q)dv + h.o.t

2G (x)
2x
a 12 (Q) = G (x) 2v
2v

+

δv

Q

Q

= G (X )

= X2

(a)
δi
+

Lx

-35.7 mH
Ry

Rx

(10b)

and I = a 00 (Q) = G (x) v Q = G (X ) V. Neglecting the higher order terms (h.o.t) by assuming dx % 1 and dv % 1
we can rewrite (9) as

-
(b)

(11)

where
b 11 (Q) = 2 g (x, v) = - 1,
2x
Q
2
b 12 (Q) =
g (x, v) = 1,
2v
Q

10
5
-8.7

-4.35

0

4.35

8.7

V

-5

(13a)

(c)

(13b)

Figure 5. (a). deriving the small-signal model of a nonlinear
memristive device. (b). small-signal equivalent circuit of the
4-lobe chua corsage memristor at V = - 7 V. Here a 11 (Q)
and a 12 (Q) from (10a) and (10b) are calculated with x = X 0 = 2.
(c). Pole-zero diagram of Y (s, V) of the 4-lobe chua corsage
memristor over the range - 8.7 V # V # 8.7 V.

and g (X, V ) = 0 because (X, V ) is a point on the DC V-I
curve. Neglecting the h.o.t gives the linearized form of
(12) as follow:
d (dx)
= b 11 (Q)dx + b 12 (Q)dv.
dt

p, z

15

Similarly, expanding g (x, v) in (8) about the equilibrium
point (x (Q), v (Q)) = (X, V ), we obtain
g (X + dx, V + dv) = g (X,V ) + b 11 (Q)dx + b 12 (Q)dv + h.o.t
(12)

250 mΩ

-35.7 mΩ

20

di = a 11 (Q)dx + a 12 (Q)dv.

G (x )

v = V + δv

-

(10a)
Q

-

7V

δv

= 2XV

+

(9)

where
a 11 (Q) = v

i = I + δi

(14)

By following the technique given in [10], the Laplace
transformation of (11) and (14) provides,
it(s) = a 11 (Q) xt (s) + a 12 (Q)vt (s),

(15)

s xt (s) = b 11 (Q) xt (s) + b 12 (Q)vt (s).

(16)

The small-signal admittance function Y(s, Q) about the
equilibrium point Q is obtained by computing (15) and
(16) as follow [3]:
it (s)
a 11 (Q)b 12 (Q)
Y (s, Q) _ t
=
+ a 12 (Q).
v(s)
s - b 11 (Q)

(17)

and

sEcOnd quartEr 2018

The small-signal parameters of 4-lobe Chua corsage
memristor can be extracted [3] from:
IEEE cIrcuIts and systEms magazInE

19



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