1.2 Four Basic Two-Terminal Circuit Elements
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1.2.2 Capacitor
A capacitor is defined by the CR f C (q, v) = 0, which corresponds to a curve, a.k.a.,
capacitor characteristic, in the q–v (or v–q) plane. The capacitor is linear if and
only if f C is linear. In such case the characteristic is a straight line passing through
the origin and the capacitor satisfies q = Cv, where C is a constant parameter
named capacitance. The description of a capacitor in terms of voltage v and current
i results to be i = C
dv
dt , which is the classical CR of a linear capacitor. If the CR of
the capacitor is not a linear function, then the capacitor is said to be nonlinear.
The capacitor is voltage-controlled if it is possible to explicitly write q = ˆ
q(v),
i.e., the charge is a (single-valued) function of the voltage. Similarly, it is said to be
charge-controlled if it is possible to write v = ˆ
v(q), i.e., the voltage is a (singlevalued) function of the charge.
By considering a voltage-controlled capacitor, the slope C(v P ) = ˆ
q (v P ) of
the characteristic at an operating point P = (v P , ˆ
q(v P )) is named small-signal
or differential capacitance of the capacitor at P . Note that, in terms of current and
voltage, a nonlinear capacitor exhibits a CR in differential form
i = C(v)
dv
dt
where C(v) is the small-signal capacitance at v.
The capacitor is said to be locally passive at P if we have C(v P ) ≥ 0. Otherwise,
if C(v P ) < 0, it is said to be locally active at P .
Example 1.7 (Varactor Diode) Varactor diodes are widely used in communication
systems. The physical structure is a pn-junction diode designed to exploit the
capacitive phenomena associated with the depletion layer. The physical approach
permits to derive that the charge accumulated on the top layer is equal to
q(t) = −K
V 0 − v(t) = ˆ
q(v(t)), ∀v < V 0
where V 0 ∈ (0.2 V , 0.9 V ) is the contact potential and K is a constant parameter
related to the physical and geometrical parameters of the semiconductor structure.
The varactor diode results to be a voltage-controlled capacitor for v < V 0 with a
small-signal capacitance
C(v) = ˆ
q
(v) =
1
2
K
1
√
V 0 − v
, ∀v < V 0 .
For all values v > V 0 the q–v characteristic is not defined and the varactor diode
behaves like a nonlinear resistor.
13
1.2.2 Capacitor
A capacitor is defined by the CR f C (q, v) = 0, which corresponds to a curve, a.k.a.,
capacitor characteristic, in the q–v (or v–q) plane. The capacitor is linear if and
only if f C is linear. In such case the characteristic is a straight line passing through
the origin and the capacitor satisfies q = Cv, where C is a constant parameter
named capacitance. The description of a capacitor in terms of voltage v and current
i results to be i = C
dv
dt , which is the classical CR of a linear capacitor. If the CR of
the capacitor is not a linear function, then the capacitor is said to be nonlinear.
The capacitor is voltage-controlled if it is possible to explicitly write q = ˆ
q(v),
i.e., the charge is a (single-valued) function of the voltage. Similarly, it is said to be
charge-controlled if it is possible to write v = ˆ
v(q), i.e., the voltage is a (singlevalued) function of the charge.
By considering a voltage-controlled capacitor, the slope C(v P ) = ˆ
q (v P ) of
the characteristic at an operating point P = (v P , ˆ
q(v P )) is named small-signal
or differential capacitance of the capacitor at P . Note that, in terms of current and
voltage, a nonlinear capacitor exhibits a CR in differential form
i = C(v)
dv
dt
where C(v) is the small-signal capacitance at v.
The capacitor is said to be locally passive at P if we have C(v P ) ≥ 0. Otherwise,
if C(v P ) < 0, it is said to be locally active at P .
Example 1.7 (Varactor Diode) Varactor diodes are widely used in communication
systems. The physical structure is a pn-junction diode designed to exploit the
capacitive phenomena associated with the depletion layer. The physical approach
permits to derive that the charge accumulated on the top layer is equal to
q(t) = −K
V 0 − v(t) = ˆ
q(v(t)), ∀v < V 0
where V 0 ∈ (0.2 V , 0.9 V ) is the contact potential and K is a constant parameter
related to the physical and geometrical parameters of the semiconductor structure.
The varactor diode results to be a voltage-controlled capacitor for v < V 0 with a
small-signal capacitance
C(v) = ˆ
q
(v) =
1
2
K
1
√
V 0 − v
, ∀v < V 0 .
For all values v > V 0 the q–v characteristic is not defined and the varactor diode
behaves like a nonlinear resistor.