Unit 2
Inductance and Capacitance
Inductors and capacitors are devices which store energy:
– Inductors store energy in magnetic fields – eg., fields within a coil. According
to Faraday’s law, a time-varying magnetic field arising from time varying currents
within the material of the inductor induces a voltage which, according to Lenz’s law,
opposes the change in the current which caused it. Inductors may be used to delay
and shape the form of an alternating current in a circuit. Sometimes they are used in
electrical filters to block alternating currents of particular frequencies from entering
parts of a circuit, and when used in this fashion they are often referred to as chokes.
– Capacitors store energy in electric fields – eg., fields between conducting plates
separated by a dielectric (i.e. an insulator).
Both of these circuit elements are passive – they store energy (and release stored
energy) but they cannot generate energy.
2.1
Inductance
We have seen that resistance links current through and voltage across a resistor.
Analogously, inductance L in henrys (H) links voltage v induced across an inductor
by a magnetic field to the time, t, rate of change of current i in the inductor. Taking
the reference direction of the current as shown below (i.e., using the passive sign
convention),
the voltage drop across the inductor is given by
v=L
1
di
.
dt
(2.1)
Here v is in volts, i in amperes and t in seconds. Equation (2.1) reveals two very
important properties of inductors:
• The current i through an inductor cannot change instantaneously, for that would
mean that di/dt → ∞ and v would be infinite.
• Furthermore, if the current does not vary with time – i.e. the current is constant
(as in dc current) – then di/dt = 0 and v = 0 which corresponds to a short
circuit.
It should be noted that while the current cannot change instantaneously, the voltage
across an inductor can do so. Also, ideal inductors have no resistance and, while we
will often use such inductors in calculations, it must be remembered that in the lab it
may be necessary to account for both the inductance and resistance associated with
an inductor.
Example: Consider a sinusoidal current given by
i(t) = i0 sin ωt
where i0 is the amplitude of the current while ω is the radian frequency in radians/second (recall that the radian frequency is related to the frequency f in hertz
(Hz) as ω = 2πf ; recall hertz=1/second). We wish to find the voltage across an
inductance L through which this current flows. Then we will sketch the waveforms
of i and v.
Clearly,
v=
and it is obvious that for sinusoidal/cosinusoidal currents and voltages that the voltage
leads the current by 90◦ as depicted below:
Equation (2.1) may be used to obtain an expression for the current i (which of
course is understood to be a function of time, that is, i ≡ i(t)) as follows :
v=L
di
=⇒
dt
2
Z
i(t)
i(t0 )
1
dη =
L
Z
t
v(x)dx
t0
where, in the required integrations, we have used x and η as ‘dummy’ variables for
time and current, respectively, and t0 and t as the beginning and end of a particular
time interval. Then,
i(t) =
and, if the initial time is given as t0 = 0, then
i(t) =
.
(2.2)
Power and Energy in an Inductor
Using equation (2.1) above,
di
p = vi = L
i.
dt
(2.3)
Furthermore, symbolizing energy as w,
p=
dw
dt
from which, if we assume a zero reference level for the energy (no current in the
inductor initially) integration gives
Z w
du =
0
That is, the energy stored in an inductor carrying a current i is given by
1
w = Li2
2
where it is understood that both w and i are functions of time.
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(2.4)
Example: (Similar to Assessment Problem 6.1, page 182 of text.) Find (a) the
voltage across the inductor for t > 0; (b) the power in mW at the inductor terminals
when t = 200 ms; (c) is the inductor delivering or absorbing power at t = 200 ms?;
(d) the energy stored in the inductor at t = 200 ms; (e) the maximum energy stored
in the inductor and the time at which it occurs. The inductor current is given by
iL = 50te−10t A ; t ≥ 0.
+
ig
• i
L
2 mH
v
•
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2.2
Combinations of Inductors
Inductors in Series
For a series combination of inductors as depicted below
i
L1
•
+ v1 -
L2
•
L3
+ v2 -
•
+ v3 -
•....•
Ln
+ vn -
•
the equivalent inductance Leq is given by
Leq =
n
X
Li .
(2.5)
i=1
This is easily deduced as follows:
The current i is the same through all the inductors in series, but the overall or
equivalent voltage drop across the combination, which we shall call v, is the sum of
the individual voltages. Therefore, using these facts and equation (2.1),
v = v1 + v2 + . . . + vn = L1
Thus,
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di
di
di
+ L2 + . . . + Ln
dt
dt
dt
Inductors in Parallel
For a parallel combination of inductors as depicted below
i
•
+
v
•
•
L1
•
i1
L2
•
•
i2
....
L3
•
•
i3
in
Ln
....
•
•
the equivalent inductance Leq is given by
n
X 1
1
=
Leq
Li
i=1
. . . (5)
(2.6)
This is easily deduced as follows:
The voltage v is the same across all the inductors in parallel, but the overall or
equivalent current through the combination, which we shall call i, is the sum of the
individual currents. Therefore, using these facts and equation (2.1),
v = L1
di2
din
di1
; v = L2
; . . . v = Ln
dt
dt
dt
Thus,
Example: Problem 6.20, page 207 of text.
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2.3
Capacitance
A capacitor is a device which “stores” charge. The capacitor symbol and passive sign
convention for the voltage across it and the current “through” it are illustrated below:
The capacitance C of a capacitor measured in farads (F) is by definition the ratio of
the charge Q stored on either of its “plates” to the voltage v across it. That is,
C=
Q
v
(2.7)
or
Q = Cv .
If we consider a differential voltage dv across the capacitor due to a differential charge
dq on its plates we may write
dq = Cdv .
Then, recalling that dq = idt, so that equation (2.7) may be written as
i=C
dv
dt
(2.8)
Equation (2.8) reveals two very important features of capacitors:
• The voltage across the capacitor cannot change instantaneously for that would
mean that dv/dt → ∞ and i would be infinite.
• Furthermore, if there is no change in the voltage across the capacitor, then
dv/dt = 0 and the current i = 0. This means that capacitors block dc
voltages.
In terms of voltage (taking the initial time to be 0), (2.8) may be written as
Z
1 t
v(t) =
idx + v(0)
(2.9)
C 0
Of course, the capacitance of the device is taken to be a constant (as was done for
the resistor and inductor).
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Power and Energy in a Capacitor
Using (2.8), the power is given by
p = vi = Cv
dv
.
dt
(2.10)
From equation (2.10) and recalling that power is the rate of change of energy (i.e.
p = dw/dt)
1
w = Cv 2
2
(2.11)
where the reference for zero energy is taken to correspond to zero voltage.
Example: See Assessment Problems 6.2 and 6.3, page 186 of the text. It is given
that for a 0.6 µF capacitor shown below the voltage at the terminals is
0,
t<0
v(t) =
−15,000t
40e
sin 30, 000t V, t ≥ 0
Find (a) i(0); (b) the power delivered to the capacitor at t = π/80 ms; (c) the energy
stored in the capacitor for the t of part (b); (d) the new v(t) if the current is 0 for
t < 0 and 3 cos 50, 000t A for t ≥ 0; (e) the maximum power delivered to the capacitor
for the case in (d); and the maximum energy stored in the capacitor for the case in
(d).
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2.4
Combination of Capacitors
Capacitors in Series
Consider the following series combination of capacitors across which at a given instant
of time the shown voltages exist:
v1
v2
i
•
+ -
C1
•+
•
+ -
v3
+ -
•
....•
C3
C2
vn
+ -
•
Cn
-•
v
Using equation (2.9) above and noting from KVL that v = v1 + v2 + . . . + vn we have
Thus, for a series combination of capacitors,
n
X 1
1
=
.
Ceq
C
i
i=1
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(2.12)
Capacitors in Parallel
It is similarly easy to show on using KCL and noting, for capacitors in parallel, the
voltage across each element is the same, that the combination depicted below
i
•
+
v
•
•
C1
•
i1
C2
•
•
i2
i3
in
Cn
C3
•
.... •
.... •
•
gives an equivalent capacitance according to (DO THIS)
Ceq =
n
X
Ci .
(2.13)
i=1
Summary Observation: While inductors in series/parallel add in a manner similar
to resistors in series/parallel, capacitors in series/parallel add in a form similar to
resistors in parallel/series. Be careful!
Example: Problem 6.26, page 208 of text.
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