1.3 Higher-Order Circuit Elements
17
where
W (ϕ(t)) = ˆ
q
(ϕ(t)) = ˆ
q
t
−∞
v(τ )dτ
is the small-signal memductance at ϕ(t).
Remark 1.1 In the framework of Linear Circuit Theory the terms Resistor, Inductor,
and Capacitor are usually used interchangeably without any ambiguity with the
respective terms resistance, inductance, and capacitance. In nonlinear network
theory, however, this usage becomes ambiguous and only the name of the element
(Resistor, Inductor, and Capacitor) should be used.
Example 1.9 (More Realistic Josephson Junction Circuit Model) The classical
circuit model of a Josephson junction is made up of the parallel connection of
a linear capacitor C, a linear resistor R, and a nonlinear inductor L with the
CR in (1.9). A more rigorous quantum mechanical analysis of the Josephson
junction dynamics reveals the presence of an additional small current component
due to interference among quasi-particle pairs [6]. This heretofore neglected current
component is given approximately by
i = G cos(k 0 ϕ)v
(1.12)
where G and k 0 are device constants. One of the reasons why this component had
been ignored in the past is due to the lack of a familiar circuit element for its
representation. Observe, however, that if we define a flux-controlled memristor with
CR
q = G 0 sin(k 0 ϕ)
(1.13)
where G 0 = G/k 0 , then differentiating both sides of (1.13) would give us
exactly (1.12). In other words, the neglected current component defined in (1.12)
is nothing more than that flowing into a memristor, which can be simply added in
parallel to the classical model to obtain the more realistic Josephson junction circuit
model shown in Fig. 1.8. It is instructive to note that the Josephson junction provides
us with the simplest nonlinear circuit model of a device made from all four basic
circuit elements introduced so far.
1.3 Higher-Order Circuit Elements
The axiomatic approach used in Definition 1.1 can be generalized to introduce an
infinite variety of higher-order circuit elements as follows.
17
where
W (ϕ(t)) = ˆ
q
(ϕ(t)) = ˆ
q
t
−∞
v(τ )dτ
is the small-signal memductance at ϕ(t).
Remark 1.1 In the framework of Linear Circuit Theory the terms Resistor, Inductor,
and Capacitor are usually used interchangeably without any ambiguity with the
respective terms resistance, inductance, and capacitance. In nonlinear network
theory, however, this usage becomes ambiguous and only the name of the element
(Resistor, Inductor, and Capacitor) should be used.
Example 1.9 (More Realistic Josephson Junction Circuit Model) The classical
circuit model of a Josephson junction is made up of the parallel connection of
a linear capacitor C, a linear resistor R, and a nonlinear inductor L with the
CR in (1.9). A more rigorous quantum mechanical analysis of the Josephson
junction dynamics reveals the presence of an additional small current component
due to interference among quasi-particle pairs [6]. This heretofore neglected current
component is given approximately by
i = G cos(k 0 ϕ)v
(1.12)
where G and k 0 are device constants. One of the reasons why this component had
been ignored in the past is due to the lack of a familiar circuit element for its
representation. Observe, however, that if we define a flux-controlled memristor with
CR
q = G 0 sin(k 0 ϕ)
(1.13)
where G 0 = G/k 0 , then differentiating both sides of (1.13) would give us
exactly (1.12). In other words, the neglected current component defined in (1.12)
is nothing more than that flowing into a memristor, which can be simply added in
parallel to the classical model to obtain the more realistic Josephson junction circuit
model shown in Fig. 1.8. It is instructive to note that the Josephson junction provides
us with the simplest nonlinear circuit model of a device made from all four basic
circuit elements introduced so far.
1.3 Higher-Order Circuit Elements
The axiomatic approach used in Definition 1.1 can be generalized to introduce an
infinite variety of higher-order circuit elements as follows.