Capacitors and

Inductors

Samuel Tesfay

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 Chap. 6, Capacitors and Inductors
 Introduction
 Capacitors
 Series and Parallel Capacitors
 Inductors
 Series and Parallel Inductors

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 6.1 Introduction

 Resistor: a passive element which dissipates

energy only
 Two important passive linear circuit elements:
1) Capacitor
2) Inductor

 Capacitor and inductor can store energy only and

they can neither generate nor dissipate energy.

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Michael Faraday (1971-1867)

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 6.2 Capacitors

 A capacitor consists of two conducting plates

separated by an insulator (or dielectric).
εA
C
d
   r 0
 0  8.854  10 12 (F/m)

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 εA
C
d
 Three factors affecting the value of capacitance:
1. Area: the larger the area, the greater the capacitance.
2. Spacing between the plates: the smaller the spacing,
the greater the capacitance.
3. Material permittivity: the higher the permittivity, the
greater the capacitance.

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Fig 6.4

(a) Polyester capacitor, (b) Ceramic capacitor, (c) Electrolytic capacitor

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Fig 6.5

Variable capacitors

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Fig 6.3

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Fig 6.2

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 Charge in Capacitors

 The relation between the charge in plates and the

voltage across a capacitor is given below.

q  Cv
q Linear
1F  1 C/V Nonlinear

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 Voltage Limit on a Capacitor

 Since q=Cv, the plate charge increases as the

voltage increases. The electric field intensity
between two plates increases. If the voltage across
the capacitor is so large that the field intensity is
large enough to break down the insulation of the
dielectric, the capacitor is out of work. Hence,
every practical capacitor has a maximum limit on
its operating voltage.

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I-V Relation of Capacitor

dq dv
+ i q  Cv, i  C
C
dt dt
v

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 Physical Meaning

+
dv i
iC C
dt v

-
• when v is a constant voltage, then i=0; a constant
voltage across a capacitor creates no current through
the capacitor, the capacitor in this case is the same as
an open circuit.
• If v is abruptly changed, then the current will have an
infinite value that is practically impossible. Hence, a
capacitor is impossible to have an abrupt change in
its voltage except an infinite current is applied.
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 Fig 6.7

 A capacitor is an open circuit to dc.

 The voltage on a capacitor cannot change abruptly.

Abrupt change

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 dv 1 t +
 v()  0 
i
iC v(t )   idt
dt C  v C

1 t
v(t )   idt  v(to)
C to
 v(to)  q(to) / C  -

 The charge on a capacitor is an integration of

current through the capacitor. Hence, the memory
effect counts.

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 Energy Storing in Capacitor

dv
p  vi  Cv
dt
t t dv v (t ) 1 2
w   pdt  C  v dt  C v (  ) vdv  Cv v (t )
v (  )
dt 2
1
w(t )  Cv 2 (t ) ( v ( )  0) + i
2 C
v

q 2 (t )
w(t )  -
2C

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Model of Practical Capacitor

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 Example 6.1

(a) Calculate the charge stored on a 3-pF capacitor

with 20V across it.
(b) Find the energy stored in the capacitor.

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 Example 6.1

Solution:
(a) Since q  Cv,

12
q  3  10  20  60pC

(b) The energy stored is

1 2 1 12
w  Cv   3  10  400  600pJ
2 2

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 Example 6.2

 The voltage across a 5- F capacitor is

v(t )  10 cos 6000t V

Calculate the current through it.
Solution:
 By definition, the current is
dv 6 d
iC  5  10 (10 cos 6000t )
dt dt
 5  10 6  6000 10 sin 6000t  0.3 sin 6000t A

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 Example 6.3

 Determine the voltage across a 2-F capacitor if the

current through it is
3000t
i (t )  6e mA
Assume that the initial capacitor voltage is zero.
Solution: 1 t
v  0 idt  v(0) and v(0)  0,
 Since C
1 t 3000 t 3  10 3000t t
3
v 6 0
6 e 3
dt  10  e
2  10  3000 0

3000t
 (1  e )V
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 Example 6.4

 Determine the current through a 200- F capacitor

whose voltage is shown in Fig 6.9.

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 Example 6.4

Solution:
 The voltage waveform can be described
mathematically as

 50t V 0  t 1
 100  50t V 1 t  3
v(t )  
 200  50t V 3t  4

 0 otherwise

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 Example 6.4

 Since i = C dv/dt and C = 200 F, we take the

derivative of to obtain

 50 0  t  1  10mA 0  t 1
6  50 1  t  3  10mA 1 t  3
i (t )  200  10   
50 3  t  4 10mA 3t  4
 
 0 otherwise  0 otherwise

 Thus the current waveform is shown in Fig.6.10.

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Example 6.4

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 Example 6.5

 Obtain the energy stored in each capacitor in Fig.

6.12(a) under dc condition.

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 Example 6.5

Solution:
 Under dc condition, we replace each capacitor with
an open circuit. By current division,

3
i (6mA)  2mA
3 2 4
 v1  2000 i  4 V, v 2  4000i  8 V
1 1 3
 w1  C1v1  (2  10 )(4)  16mJ
2 2

2 2
1 1 3
w2  C2 v2  (4  10 )(8)  128mJ
2 2

2 2
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Fig 6.14

Ceq  C1  C 2  C3  ....  C N

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 6.3 Series and Parallel Capacitors

i  i1  i2  i3  ...  iN
dv dv dv dv
i  C1  C2  C3  ...  C N
dt dt dt dt
 N
 dv dv
   C K   Ceq
 k 1  dt dt
Ceq  C1  C2  C3  ....  C N
 The equivalent capacitance of N parallel-connected
capacitors is the sum of the individual capacitance.

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Fig 6.15

1 1 1 1 1
    ... 
Ceq C1 C2 C3 CN

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 Series Capacitors

v(t )  v1 (t )  v2 (t )  ...  v N (t )
1 t 1 1 1 1 t
Ceq id  ( C1  C2  C3  ...  C N )id
q (t ) q (t ) q (t ) q (t )
  
Ceq C1 C2 CN

 The equivalent capacitance of series-connected

capacitors is the reciprocal of the sum of the
reciprocals of the individual capacitances.

1 1 1 C1C2
  Ceq 
Ceq C1 C2 C1  C2
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 Summary

 These results enable us to look the capacitor in this

way: 1/C has the equivalent effect as the resistance.
The equivalent capacitor of capacitors connected in
parallel or series can be obtained via this point of
view, so is the Y-△ connection and its
transformation

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 Example 6.6

 Find the equivalent capacitance seen between

terminals a and b of the circuit in Fig 6.16.

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 Example 6.6

Solution:
 20  F and 5  F capacitors are in series:
20  5
  4 F
20  5
 4  F capacitor is in parallel with the 6  F
and 20  F capacitors :
 4  6  20  30 F
 30  F capacitor is in series with
the 60  F capacitor.
30  60
Ceq  F  20 F
30  60
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 Example 6.7

 For the circuit in Fig 6.18, find the voltage across

each capacitor.

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Example 6.7

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 Example 6.7

Solution:
 Two parallel capacitors:
1
 Ceq  1 1 1 mF  10mF
 
60 30 20

 Total charge 3
q  Ceq v  10  10  30  0.3 C

 This is the charge on the 20-mF and 30-mF

capacitors, because they are in series with the 30-v
source. ( A crude way to see this is to imagine that
charge acts like current, since i = dq/dt)
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Example 6.7
 Therefore, q 0 .3
v1   3
 15 V,
C1 20  10

q 0 .3
v2   3
 10 V
C2 30  10

 Having determined v1 and v2, we now use

KVL to determine v3 by
v3  30  v1  v2  5V

 Alternatively, since the 40-mF and 20-mF

capacitors are in parallel, they have the same
voltage v3 and their combined capacitance is
q 0.3
40+20=60mF.  v3   3
 5V
60mF 60  10
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Joseph Henry (1979-1878)

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 6.4 Inductors

 An inductor is made of a coil of conducting wire

N A
2
L
l

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Fig 6.22

N 2 A
L
l
  r 0
 0  4  10 7 (H/m)

N : number of turns.
l :length.
A:cross  sectional area.
 : permeability of the core

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Fig 6.23

(a) air-core
(b) iron-core
(c) variable iron-core

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 Flux in Inductors

 The relation between the flux in inductor and the

current through the inductor is given below.

  Li
ψ Linear
1H  1 Weber/A
Nonlinear

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 Energy Storage Form

 An inductor is a passive element designed to store

energy in the magnetic field while a capacitor
stores energy in the electric field.

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 I-V Relation of Inductors

 An inductor consists of i
a coil of conducting +
wire.
d di
v L v
L
dt dt
-

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 Physical Meaning

d di
v L
dt dt

 When the current through an inductor is a

constant, then the voltage across the inductor is
zero, same as a short circuit.
 No abrupt change of the current through an
inductor is possible except an infinite voltage across
the inductor is applied.
 The inductor can be used to generate a high
voltage, for example, used as an igniting element.
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 Fig 6.25

 An inductor are like a short circuit to dc.

 The current through an inductor cannot change
instantaneously.

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 1 1 t
di  vdt i   v (t ) dt
L L  +

v
1 t
i   v(t )dt  i (to )
L

L to -

The inductor has memory.

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 Energy Stored in an Inductor

 di 
P  vi   L  i +
 dt 
t t  di  v
L
w   pdt    L idt
 dt  -
i (t ) 1 2 1 2
 L i (  ) i di  Li (t )  Li () i ()  0,
2 2

 The energy stored in an inductor 1 2

w(t )  Li (t )
2
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Model of a Practical Inductor

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 Example 6.8

 The current through a 0.1-H inductor is i(t) = 10te-5t

A. Find the voltage across the inductor and the
energy stored in it.
Solution:
di
Since v  L and L  0.1H,
dt
d
v  0.1 (10te 5t )  e 5t  t (5)e 5t  e 5t (1  5t )V
dt
The energy stored is
1 2 1
w  Li  (0.1)100t 2 e 10t  5t 2 e 10t J
2 2
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 Example 6.9

 Find the current through a 5-H inductor if the

voltage across it is
30t , t  0
2
v(t )  
 0, t0
Also find the energy stored within 0 < t < 5s.
Assume i(0)=0.
Solution:
1 t
Since i  t v(t )dt  i (t0 ) and L  5H.
L 0

1 t 2 t3
i  0 30t dt  0  6   2t 3 A
5 3
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 Example 6.9

The power p  vi  60t , and the energy stored is then

5

6
5 t 5
w   pdt  0 60t dt  60
5
 156.25 kJ
60
Alternatively, we can obtain the energy stored using
Eq.(6.13), by writing
1 2 1
w(5)  w(0)  Li (5)  Li (0)
2 2
1
 (5)(2  5 )  0  156.25 kJ
3 2

2
as obtained before.
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 Example 6.10

 Consider the circuit in

Fig 6.27(a). Under dc
conditions, find:
(a) i, vC, and iL.
(b) the energy stored in
the capacitor and
inductor.

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 Example 6.10

Solution:
(a ) Under dc condition : capacitor  open circuit
inductor  short circuit
12
i  iL   2 A, vc  5i  10 V
1 5
(b) 1 1
wc  Cvc  (1)(10 )  50J,
2
2

2 2
1 2 1
wL  Li  (2)(2 2 )  4J
2 2
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Inductors in Series

Leq  L1  L2  L3  ...  LN

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Inductors in Parallel

1 1 1 1
  
Leq L1 L2 LN

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 6.5 Series and Parallel Inductors
 Applying KVL to the loop,

v  v1  v2  v3  ...  v N
 Substituting vk = Lk di/dt results in
di di di di
v  L1  L2  L3  ...  LN
dt dt dt dt
di
 ( L1  L2  L3  ...  LN )
dt
 N
 di di
   LK   Leq
 K 1  dt dt
Leq  L1  L2  L3  ...  LN
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 Parallel Inductors

 Using KCL,
i  i1  i2  i3  ...  iN
 But 1 t
ik  t vdt  ik (t0 )
Lk o

1 t 1 t 1 t
i  t vdt  i1 (t0 )  t vdt  is (t0 )  ...  t vdt  iN (t0 )
Lk 0
L2 0 LN 0

1 1 1 t
    ...   t vdt  i1 (t0 )  i2 (t0 )  ...  iN (t0 )
 L1 L2 LN  0

N 1t N
1 t
    t vdt   ik (t0 )   vdt  i (t0 )
 k 1 Lk  Leq
0 t 0
k 1

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 The inductor in various connection has the same
effect as the resistor. Hence, the Y-Δ transformation
of inductors can be similarly derived.

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Table 6.1

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 Example 6.11

 Find the equivalent inductance of the circuit shown

in Fig. 6.31.

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 Example 6.11

 Solution:

Series : 20H, 12H, 10H

 42H
7  42
Parallel :  6H
7  42
 Leq  4  6  8  18H

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Practice Problem 6.11

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 Example 6.12

 Find the circuit in Fig. 6.33, 10t

i (t )  4(2  e )mA.
If i (0)  1 mAfind , : (a) i (0)
2 1

(b) v(t ), v1 (t ), and v2 (t ); (c) i1 (t ) and i2 (t )

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 Example 6.12

Solution:
10 t
(a ) i (t )  4(2  e )mA  i (0)  4(2  1)  4mA.
 i1 (0)  i (0)  i2 (0)  4  (1)  5mA
(b) The equivalent inductance is
Leq  2  4 || 12  2  3  5H
di 10 t 10 t
 v(t )  Leq  5(4)(1)( 10)e mV  200e mV
dt
di 10 t 10 t
v1 (t )  2  2(4)(10)e mV  80e mV
dt
10 t
 v2 (t )  v(t )  v1(t )  120e mV
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 Example 6.12

1 t
(c) i  0 v(t ) dt  i (0) 
L
1 t 120 t 10t
i1 (t )  0 v2 dt  i1 (0)   e dt  5 mA
4 4 0
10 t t
 3e  5 mA  3e 10t  3  5  8  3e 10t mA
0
1 t 120 t 10t
i2 (t )  0 v2 dt  i2 (0)   e dt  1mA
12 12 0

10 t t
 e  1mA  e 10t  1  1  e 10t mA
0
Note that i1 (t )  i2 (t )  i (t )
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