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1 Device Modeling and Circuit Elements
1.2.3 Inductor
An inductor is defined by the CR f L (ϕ, i) = 0, which corresponds to a curve,
a.k.a., inductor characteristic, in the ϕ–i (or i-ϕ) plane. The inductor is linear if and
only if f L is linear. In such case the characteristic is a straight line passing through
the origin and the inductor satisfies ϕ = Li, where L is a constant parameter named
inductance. The description of an inductor in terms of voltage v and current i results
to be v = L
di
dt , which is the classical CR of a linear inductor. If the CR of the
inductor is not a linear function, then the inductor is said to be nonlinear.
The inductor is current-controlled if it is possible to explicitly write ϕ = ˆ
ϕ(i),
i.e., the flux is a (single-valued) function of the current. Similarly, it is said to be
flux-controlled if it is possible to explicitly write i = ˆ
i(ϕ), i.e., the current is a
(single-valued) function of the flux.
By considering a current-controlled inductor, the slope L(i P ) = ˆ
ϕ (i P ) of the
characteristic at an operating point P = (i P , ˆ
ϕ(i P )) is named small-signal or
differential inductance of the inductor at P . Note that, in terms of current and
voltage, a nonlinear inductor exhibits a CR in differential form
v = L(i)
di
dt
where L(i) = ˆ
ϕ (i) is the small-signal inductance .
The inductor is said to be locally passive at P if we have L(i P ) ≥ 0. Otherwise,
if L(i P ) < 0, it is said to be locally active at P .
Example 1.8 (Josephson Junction) The Josephson junction is made up of two
superconductors separated by an insulating layer. Superconductors physics permits
to derive the algebraic equation describing the device in terms of the current and the
flux
i(t) = ˆ
i(ϕ(t)) = I 0 sin (kϕ(t))
(1.9)
where I 0 is a device parameter and k = 4π(e/ h) (e = electron charge and h =
Planck’s constant). The Josephson junction is a flux-controlled (but not currentcontrolled) inductor with a characteristic as shown in Fig. 1.6 and a reciprocal smallsignal inductance (in H −1 )
Γ (ϕ) = ˆ
i
(ϕ) = kI 0 cos(kϕ).
It is important to observe that the physical variable ϕ is proportional to the difference
between two quantum mechanical phases, and is not related to the familiar magnetic
flux in a winding coil known from electromagnetic field theory.
1 Device Modeling and Circuit Elements
1.2.3 Inductor
An inductor is defined by the CR f L (ϕ, i) = 0, which corresponds to a curve,
a.k.a., inductor characteristic, in the ϕ–i (or i-ϕ) plane. The inductor is linear if and
only if f L is linear. In such case the characteristic is a straight line passing through
the origin and the inductor satisfies ϕ = Li, where L is a constant parameter named
inductance. The description of an inductor in terms of voltage v and current i results
to be v = L
di
dt , which is the classical CR of a linear inductor. If the CR of the
inductor is not a linear function, then the inductor is said to be nonlinear.
The inductor is current-controlled if it is possible to explicitly write ϕ = ˆ
ϕ(i),
i.e., the flux is a (single-valued) function of the current. Similarly, it is said to be
flux-controlled if it is possible to explicitly write i = ˆ
i(ϕ), i.e., the current is a
(single-valued) function of the flux.
By considering a current-controlled inductor, the slope L(i P ) = ˆ
ϕ (i P ) of the
characteristic at an operating point P = (i P , ˆ
ϕ(i P )) is named small-signal or
differential inductance of the inductor at P . Note that, in terms of current and
voltage, a nonlinear inductor exhibits a CR in differential form
v = L(i)
di
dt
where L(i) = ˆ
ϕ (i) is the small-signal inductance .
The inductor is said to be locally passive at P if we have L(i P ) ≥ 0. Otherwise,
if L(i P ) < 0, it is said to be locally active at P .
Example 1.8 (Josephson Junction) The Josephson junction is made up of two
superconductors separated by an insulating layer. Superconductors physics permits
to derive the algebraic equation describing the device in terms of the current and the
flux
i(t) = ˆ
i(ϕ(t)) = I 0 sin (kϕ(t))
(1.9)
where I 0 is a device parameter and k = 4π(e/ h) (e = electron charge and h =
Planck’s constant). The Josephson junction is a flux-controlled (but not currentcontrolled) inductor with a characteristic as shown in Fig. 1.6 and a reciprocal smallsignal inductance (in H −1 )
Γ (ϕ) = ˆ
i
(ϕ) = kI 0 cos(kϕ).
It is important to observe that the physical variable ϕ is proportional to the difference
between two quantum mechanical phases, and is not related to the familiar magnetic
flux in a winding coil known from electromagnetic field theory.