IEEE Circuits and Systems Magazine - Q2 2018 - 20
Y (s, Q) =
1
+ 1 ,
s Lx + Rx R y
(18a)
where
Lx =
1
,
a 11 (Q)b 12 (Q)
(18b)
Rx =
- b 11 (Q)
,
a 11 (Q)b 12 (Q)
(18c)
Ry =
1
.
a 12 (Q)
(18d)
The small-signal equivalent circuit computed at
V = - 7 V is shown in Fig. 5(b). Observe from Fig. 5(b)
that both L x and R x are negative, whereas R y is positive.
The pole s = p and the zero s = z of the small-signal
admittance Y (s, V ) of 4-lobe Chua corsage memristor
shown in Fig. 5(c) can be obtained as follow [3]:
b 1 (V ) s + b 0 (V )
K^s - z h
Y (s, V ) =
=
a 1 (V ) s + a 0 (V )
^s - ph
(19)
where
K=
b 1 (V )
= a 12 (Q),
a 1 (V )
(20a)
p =-
a 0 (V )
= b 11 (Q),
a 1 (V )
z =-
b 0 (V )
[a 11 (Q) b 12 (Q) - a 12 (Q) b 11 (Q)]
=. (20c)
b 1 (V )
a 12 (Q)
(20b)
The pole p depicted in Fig. 5(c) is constant whereas
the zero z changes with the voltage, V. However, a pair
of complex-conjugate poles on the imaginary axis are
needed to design an oscillating circuit.
Re Y (iω, V ), Im Y (iω, V )
V = -7 V
20
Im Y (iω ∗) = 9.799 S
Re Y (-iω ∗) = 0 S 10
ω
-20
-10
ω ∗ = -2.4495 rad/s
0
-10
10
20
ω ∗ = 2.4495 rad/s
Re Y (iω ∗) = 0 S
Im Y (-iω ∗) = -9.799 S
-20
Re Y (0) = -24 S
-30
Figure 6. Frequency response of the 4-lobe chua corsage
memristor at V = - 7 V over the range -20 rad/s # ~ # 20 rad/s.
20
IEEE cIrcuIts and systEms magazInE
Since the small-signal admittance Y (s, V ) at v = - 7 V
has only a real pole p = - 1 in Fig. 5(c), it is necessary to
add at least one energy storage element in series with the
4-lobe Chua corsage memristor in order to make the circuit oscillate [3] at some frequency ~ 2 0. To determine
the type and the value of the energy-storage element, we
must derive the frequency response of the admittance
function of the 4-lobe Chua corsage memristor. The frequency response Y (i~, V ) of the 4-lobe Chua corsage
memristor is obtained by substituting s = i~ in (19):
Y (i~, V ) =
a 0 (V )b 0 (V ) + ~ 2 a 1 (V )b 1 (V )
a 02 (V ) + ~ 2 a 12 (V )
~ ^ a 0 ( V ) b 1 ( V ) - a 1 ( V ) b 0 (V ) h
+i
a 02 (V ) + ~ 2 a 12 (V )
(21)
where
Re Y ^i~, V h =
a 0 (V ) b 0 (V ) + ~ 2 a 1 (V ) b 1 (V )
,
a 02 (V ) + ~ 2 a 12 (V )
(22a)
I m Y ^i~,V h =
~ ^ a 0 (V ) b 1 (V ) - a 1 (V ) b 0 (V ) h
.
a 02 (V ) + ~ 2 a 12 (V )
(22b)
T h e f r e q u e n c y r e s p o n s e o f Re Y (i~, V ) a n d
I m Y (i~, V ) at V = - 7 V are shown in Fig. 6 over the
range - 20 rad/s # ~ # 20 rad/s. Fig. 6 shows Re Y (i~*) =
0 at ~* = ! 2.4495 rad/s whereas the I m Y (i~*) = ! 9.799
for a given input voltage V = - 7 V. Similar to [3], the 4-lobe
Chua corsage memristor needs to add a positive inductance
with value L = L * H and the positive value of I m Y (i~*)
is used to determine the value of the inductance L* via the
following Chua equation specified by Chua at [3]:
L* =
1
~* I m Y (i~*)
= 41.67 mH.
(23)
It is predicted that the externally added inductance
L* satisfies the prime condition of the oscillator circuit by compensating the imaginary part of admittance
Y (i~, V ) as well as by making the total impedance of the
composite circuit to zero at operating point V = - 7 V.
The admittance at ~ = 0 is given by Y (0) = (- 24 + i $ 0)
Siemens, which is equal to the admittance of the 1-port
in Fig. 5(b) with the inductor replaced by a short circuit.
Moreover, Y (0) = - 24 S is equal to the slope at V = - 7 V
(calculated from the coordinates at V = - 7.045 V and
V = - 6.955 V (as shown in Fig. 4 right inset)).
III. 4-lobe Chua Corsage Memristor Oscillator
The 4-lobe Chua corsage memristor oscillator circuit is
shown in Fig. 7(a) where the 1-st order 4-lobe Chua corsage memristor is connected in series with an external
inductor L* and a battery, V = - 7 V. Fig. 7(a) shows that
the input current, i = i L = i M , whereas the input voltage
v = v L + v M and v M = i M /G (x). The state - dependent
sEcOnd quartEr 2018
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