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DC CIRCUITS:
CHAPTER 4
Capacitors and Inductors
• Introduction
• Capacitors: terminal behavior in terms of
current, voltage, power and energy
• Series and parallel capacitors
• Inductors: terminal behavior in terms of
current, voltage, power and energy
• Series and parallel inductors
Introduction
• Two more linear, ideal basic passive circuit
elements.
• Energy storage elements stored in both magnetic
and electric fields.
• They found continual applications in more
practical circuits such as filters, integrators,
differentiators, circuit breakers and automobile
ignition circuit.
• Circuit analysis techniques and theorems applied
to purely resistive circuits are equally applicable
to circuits with inductors and capacitors.
Capacitors
• Electrical component that consists of two
conductors separated by an insulator or dielectric
material.
• Its behavior based on phenomenon associated
with electric fields, which the source is voltage.
• A time-varying electric fields produce a current
flow in the space occupied by the fields.
• Capacitance is the circuit parameter which
relates the displacement current to the voltage.
A capacitor with an applied voltage
Plates – aluminum foil
Dielectric – air/ceramic/paper/mica
Circuit symbols for
capacitors
(a) Fixed capacitor
(b) Variable capacitor
Circuit parameters
• The amount of charge stored, q = CV. (1)
• C is capacitance in Farad, ratio of the
charge on one plate to the voltage
difference between the plates. But it does
not depend on q or V but capacitor’s physical
dimensions i.e.,
C
A
d
 = permeability of dielectric in
Wb/Am
A = surface area of plates in m2
d = distance btw the plates m
Current – voltage
relationship of a capacitor
• To obtain the I-V characteristic of a
capacitor, we differentiate both sides of
dV
eq.(1) . We obtain,
(2)
iC
dt
• Integrating both sides of eq.(2) we obtain,
1
V 
C
1

C

t


t
to
i dt
i dt  V (t o )
(3)
Instantaneous power and
energy for capacitors
• The instantaneous power delivered to a
capacitor is,
dV
p  Vi  CV
dt
(4)
• The energy stored in the capacitor,
t
dV
1
w   p dt  C  V
dt  C  V dV  CV 2



dt
2
t
t
t
t  
• At V(-∞) = 0 (cap. uncharged at t = -∞,
hence
2
1
w  CV 2
2
or w 
q
2C
(6)
(5)
Important properties of
a capacitor
• A capacitor is an open circuit to dc.
- When the voltage across capacitor is not changing
with time (constant), current thru it is zero.
• The voltage on a capacitor cannot change
abruptly.
- The voltage across capacitor must be continuous.
Conversely, the current thru it can change
instantaneously.
Practice problem 6.1
• What is the voltage across a 3-F
capacitor if the charge on one plate
is 0.12 mC? How much energy is
stored? (Ans: 40V, 2.4mJ)
Practice problem 6.2
• If a 10-F capacitor is connected to
a voltage source with
v(t) = 50sin2000t V
• Calculate the current through it.
(Ans: cos2000t A)
Practice problem 6.3
• The current through a 100-F
capacitor is i(t) = 50sin120t mA.
Calculate the voltage across it at t =
1 ms and t = 5 ms Take v(0) = 0. (Ans:
93.137V, 1.736V)
Practice problem 6.4
• An initially uncharged
1-mF capacitor has
the current shown in
Figure 6.11 across it.
Calculate the voltage
across it at t = 2 ms
and t = 5 ms. (Ans:
100mV, 400mV)
Practice problem 6.5
• Under dc conditions, find the energy
stored in the capacitors in Fig. 6.13.
(Ans: 405J, 90 J)
Series/parallel
capacitances
• Series-parallel combination is powerful
tool for circuit simplification.
• A group of capacitors can be combined
to become a single equivalent
capacitance using series-parallel rules.
Parallel capacitances
• The equivalent capacitance of N parallelconnected capacitors is the sum of the
individual capacitances.
C eq  C1  C 2  C 3    C N
Parallel Nconnected
capacitors
(7)
Equivalent
circuit
Series capacitances
• The equivalent capacitance of series-connected
capacitors is the reciprocal of the sum of the
reciprocals of the individual capacitances.

1
1
1
Ceq  C1  C2  C3    C N
Series Nconnected
capacitors

1 1
(8)
Equivalent
circuit
Practice problem 6.6
• Find the equivalent capacitance seen at the
terminals of the circuit in Fig. 6.17. (Ans: 40F)
50 F
60 F
Ceq
70 F
20 F
120 F
Practice problem 6.7
• Find the voltage across each of the capacitors
in Fig. 6.20 (Ans: 30V, 30V, 10V, 20V)
+ V1 -
+ V3 20 F
+ V4 -
60 F
+ V2 -
60 V
40 F
30 F
Inductors
• Electrical component that opposes any change in
electrical current.
• Composed of a coil or wire wound around a nonmagnetic core/magnetic core.
• Its behavior based on phenomenon associated
with magnetic fields, which the source is current.
• A time-varying magnetic fields induce voltage in
any conductor linked by the fields.
• Inductance is the circuit parameter which relates
the induced voltage to the current.
Typical form of an
inductor
Circuit symbols for
inductors
Air-core
iron-core
Variable
iron-core
Current – voltage
relationship of an inductor
• The voltage across an
inductor,
• L is the constant
proportionality called
inductance measured
in Henry.
1
• To obtain current
integrate eq. (7), i 
di
V L
dt
L
1

L
t

(9)
V (t ) dt
t
 V (t ) dt  i (t
to
o
) (10)
Instantaneous power and
energy fir inductors
• The instantaneous power delivered to a capacitor
is,
 di 
p  Vi   L
 i (11)
 dt 
• The energy stored in the capacitor,
t
di
1 2
w   p dt   L i dt  L  i di  Li

  dt

2
t
t
t
t  
(12)
• At V(-∞) = 0 (ind. uncharged at t = -∞, hence
1
w  Li 2
2
(13)
Important properties of
an inductor
• An inductor acts like a short circuit to dc.
- When the current thru inductor is not changing
with time (constant), voltage across it is zero.
• The current thru an inductor cannot change
instantaneously.
- An important property is its opposition to the
change in current flowing thru it. However the
voltage across it can change abruptly.
Practice problem 6.8
• If the current through a 1-mH inductor
is i(t) = 20cos100t mA, find the terminal
voltage and the energy stored.
(Ans: -2sin100t mV, 0.2cos2100t J)
Practice problem 6.9
• The terminal voltage of a 2-H inductor is
V = 10(1 – t) V. find the current flowing
thru it at t=4s and the energy stored in
it within 0 < t < 4s. Assume i(0)=2A.
(Ans: -18 A, 320 J)
Practice problem 6.10
• Determine VC, iL and the energy stored in
the capacitor and inductor in the circuit
below under dc conditions.
(Ans: 3V, 3A, 1.125J)
0.25 H
iL
3
1
+ VC -
4A
F
Series inductances
• The equivalent inductance of N seriesconnected inductors is the sum of the
individual inductances.
Leq  L1  L 2  L 3    LN
Series Nconnected
inductors
(14)
Equivalent
circuit
Parallel inductances
• The equivalent inductance of series-connected
inductors is the reciprocal of the sum of the
reciprocals of the individual inductances.

1
1
1
1
Leq  L  L2  L3   LN
Parallel N-connected
inductors

1 1
(15)
Equivalent
circuit
Practice problem 6.11
• Calculate the equivalent inductance for
the inductive ladder network in Figure
below. (Ans: 25 mH)
20 mH
Leq
50 mH
100 mH
40 mH
40 mH
30 mH
20 mH
Practice problem 6.12
• In the circuit of Figure below, given i1(t)=0.6e2t. If i(0)=1.4A,
find (a)i2(0); (b) i2(t) and i(t); (c) V1(t), V2(t) and
V (t).
i
1
i
3H
+ V1 -
+
-
i2
6H
+ V2 -
V
8H