18
1 Device Modeling and Circuit Elements
C
R
L
M
i
−
+
v
i 1
i 2
i 3
i 4
Fig. 1.8 More realistic circuit model of a Josephson junction consisting of a parallel connection
of four basic circuit elements, namely, a linear capacitor C, a linear resistor R, a nonlinear inductor
L with CR i 3 = I 0 sin(kϕ 3 ), and a nonlinear memristor with CR q 4 = G 0 sin(k 0 ϕ 4 ) which
incorporates the small, hitherto neglected current component due to interference among quasiparticle pairs. Note that the Josephson junction is represented with the symbol of a nonlinear
flux-controlled inductor and not with that introduced in Fig. 1.6
Definition 1.2 ((α, β)-Element) A two-terminal circuit element is called an
(α, β)-element if and only if it is defined by an algebraic CR involving only
the signal pair v (α) and i (β) given in (1.2) and (1.3), where α and β are integers.
The electric symbol of an (α, β)-element is reported in Fig. 1.9, while the same
figure also depicts a typical curve representing its characteristic in the v (α) − i (β)
plane (in the case of a i (β) -controlled element).
Such an infinite family of circuit elements is defined not only for the sake of
generality. Rather, these circuit elements are essential for developing a rigorous and
comprehensive mathematical theory of nonlinear circuits in the following sense: for
any integer k, if one excludes all elements with |α| > k and |β| > k, then it is
possible to construct hypothetical circuits whose dynamic behavior is pathological
since solutions are defined up until some finite time instants but cannot be prolonged
in time thereafter due to the presence of “singularities” called impasse points
[5, 7, 8]. Examples of circuits with impasse points shall be discussed in Chaps. 4
and 6.
It is unlikely, however, that (α, β)-elements with |α| > 2 and |β| > 2 will be
needed in modeling most real-world devices. 4 In the next section the higher-order
elements named memcapacitor and meminductor are introduced by exploiting the
electrical variables defined for α = −2 and β = −2, i.e., v (−2) and i (−2) . The reader
is referred to Chap. 2 for further properties of these elements.
Finally, it is important to remark that every (α, β)-element can be synthesized
via the procedure illustrated in [3, 5] using a family of linear active two ports called
mutators. They can also be emulated via various off-the-shelf digital components,
4 It can be proved that any (α, β)-element with |α| > 2 and |β| > 2 is active in the sense that it can
be built only with active components, such as transistors and operational amplifiers, which require
a power supply.
1 Device Modeling and Circuit Elements
C
R
L
M
i
−
+
v
i 1
i 2
i 3
i 4
Fig. 1.8 More realistic circuit model of a Josephson junction consisting of a parallel connection
of four basic circuit elements, namely, a linear capacitor C, a linear resistor R, a nonlinear inductor
L with CR i 3 = I 0 sin(kϕ 3 ), and a nonlinear memristor with CR q 4 = G 0 sin(k 0 ϕ 4 ) which
incorporates the small, hitherto neglected current component due to interference among quasiparticle pairs. Note that the Josephson junction is represented with the symbol of a nonlinear
flux-controlled inductor and not with that introduced in Fig. 1.6
Definition 1.2 ((α, β)-Element) A two-terminal circuit element is called an
(α, β)-element if and only if it is defined by an algebraic CR involving only
the signal pair v (α) and i (β) given in (1.2) and (1.3), where α and β are integers.
The electric symbol of an (α, β)-element is reported in Fig. 1.9, while the same
figure also depicts a typical curve representing its characteristic in the v (α) − i (β)
plane (in the case of a i (β) -controlled element).
Such an infinite family of circuit elements is defined not only for the sake of
generality. Rather, these circuit elements are essential for developing a rigorous and
comprehensive mathematical theory of nonlinear circuits in the following sense: for
any integer k, if one excludes all elements with |α| > k and |β| > k, then it is
possible to construct hypothetical circuits whose dynamic behavior is pathological
since solutions are defined up until some finite time instants but cannot be prolonged
in time thereafter due to the presence of “singularities” called impasse points
[5, 7, 8]. Examples of circuits with impasse points shall be discussed in Chaps. 4
and 6.
It is unlikely, however, that (α, β)-elements with |α| > 2 and |β| > 2 will be
needed in modeling most real-world devices. 4 In the next section the higher-order
elements named memcapacitor and meminductor are introduced by exploiting the
electrical variables defined for α = −2 and β = −2, i.e., v (−2) and i (−2) . The reader
is referred to Chap. 2 for further properties of these elements.
Finally, it is important to remark that every (α, β)-element can be synthesized
via the procedure illustrated in [3, 5] using a family of linear active two ports called
mutators. They can also be emulated via various off-the-shelf digital components,
4 It can be proved that any (α, β)-element with |α| > 2 and |β| > 2 is active in the sense that it can
be built only with active components, such as transistors and operational amplifiers, which require
a power supply.