The Capacitor
3.33
The current through a capacitor is proportional to the
rate of change of voltage
across it. The voltage across the capacitor is proportional to the area under the
current waveform,
i.e
., the A-s product (or C)
applied through it from
t
=
-∞
where
t
=
-∞
has to be understood as the moment this capacitor came into being.
We notice that the element relationship of capacitor is similar to that of an inductor. Only that the
role of voltage and current in the relationship has been interchanged.
Inductor is an element that accumulates flux linkage (V-sec or Wb-T) and makes its
response variable,
i.e.
, current, proportional to the accumulated flux linkage. Capacitor
is an element that accumulates charge (A-s or C) and makes its response variable,
i.e
.,
voltage, proportional to the accumulated charge.
Therefore we need not enter into a detailed discussion on the implications of the element relationship
of capacitor. Such a discussion will be analogous to the line of reasoning we employed in the case of
inductor. Hence, we list the implications without detailed explanation.
i t
dv t
dt
( )
( )
=
C
⇒
• Instantaneous voltage across a capacitor can not be predicted from instantaneous
value of current through it.
• If instantaneous value of current is positive, the capacitor voltage will be increasing
at that instant, and, if it is negative the voltage will be decreasing at that instant.
• When current through a capacitor crosses
zero in the downward direction, its
voltage attains a local maximum. When the current crosses zero in the upward
direction, the capacitor voltage attains a local minimum.
• Current through a capacitor with a constant voltage across it is zero.
• Capacitor preserves the waveshape for exponential and sinusoidal inputs.
v t
C
i t dt
C
i t dt
C
i t dt
t
V
t
( )
( )
( )
( )
=
=
+
=
−∞
−∞
∫
∫
∫
−
−
1
1
1
0
0
0
V
V
C
i t dt
t
0
0
1
+
−
∫
( )
⇒
Change in capacitor voltage over [
t
1
,
t
2
],
D
v
=
(Area under capacitor current over [
t
1
,
t
2
])/
C
.
(
v
(
t
) at
t
=
t
2
) is (
v
(
t
) at
t
=
t
1
)
+
D
v
v
(
t
)
=
V
0
+
(Area under capacitor current over [0,
t
])/
C
, where
V
0
is the voltage across
the capacitor at
t
=
0 and is called initial condition of the capacitor.
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3.34
Single
Element Circuits
C
v
i t dt
i
t t
i t dt
t
t
C v
t
t
t
av
t
t
×
=
=
−
=
∫
∫
∆
∆
( )
[
,
]
( )
(
)
(
1
2
1
2
1
2
2
1
and
in
2
2
1
−
t
)
⇒
The amount of voltage change required in a capacitor decides the area-content under
current waveform to be applied to it to bring about the change. The time allowed to
bring about it decides the average current to be applied. Thus, rapid change in capacitor
voltage calls for large amplitude current through it.
C
v
i t dt
i
t t
i t dt
t
t
C v
t
t
t
av
t
t
×
=
=
−
=
∫
∫
∆
∆
( )
[
,
]
( )
(
)
(
1
2
1
2
1
2
2
1
and
in
2
2
1
−
t
)
⇒
• Voltage in a capacitor can not change instantaneously unless an impulse current is
applied or supported in the circuit.
• Unit Impulse Current will have an area-content of unity since it is a unit impulse.
Thus, unit impulse current will deposit 1 C of charge in a capacitor over [0
-
, 0
+
], i.e.,
instantaneously. Therefore, the voltage across a capacitor C changes instantaneously
by 1/C V when the circuit applies or supports a unit impulse current through it.
• Therefore, if a circuit does not apply or support impulse current, the voltage across
capacitors in that circuit will be continuous functions of time.
Capacitors absorb
rapid variations in circuit currents and tend to keep circuit voltages smooth.
• The
amplitude
of
voltage
sinusoid
in
a
capacitor
is
inversely
proportional
to
the
product
of
frequency
of
applied
current
sinusoid
and
capacitance
value.
• There can be a DC voltage across a capacitor even when the applied current
waveform is a pure alternating one. The amount of DC content depends upon the
initial condition of the capacitor and the point at which
the current waveform is
switched on to the capacitor.
• When
the
applied
current
through
a
capacitor
is
a
periodic
alternating
waveform,
the
voltage
across
the
capacitor
will
contain
an
alternating
component
with
the
same
period.
The
peak-to-peak
amplitude
of
this
alternating
component
will
be
directly
proportional
to
half-cycle
area
of current waveform and inversely proportional to capacitance value. It decreases
with increase in frequency of the current.
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The Capacitor
3.35
• Therefore, a large valued capacitor in a circuit can absorb alternating currents in the
circuit without contributing significant amount of alternating voltages to the circuit.
A large valued capacitor can hold the potential difference
across two points in a
circuit at a reasonably constant level even when large amplitude alternating currents
flow through it.
• A capacitor with zero initial voltage is a linear electrical element.
• A capacitor with non-zero initial voltage is a linear element as far as the voltage
component caused by applied current is concerned.
• The total energy delivered to a capacitor carrying a voltage V across it is (1/2)
CV
2
J
and this energy is stored in its electric field.
• Stored energy in a capacitor is also given by (1/2
C
)
Q
2
J and
QV
/2 J.
• The capacitor will be able to deliver this stored energy back to other elements in the
circuit if called upon to do so.
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