ELE 225

Electric Circuits and Devices

Department of Electrical Engineering

College of Engineering
American University of Sharjah

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 ELE 225
Electric Circuits and Devices

Chap 3.
Capacitance and Inductance

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 Learning Objectives
 Perform current and voltage calculations for
capacitors and inductors.
 Combine capacitors or inductors in series and
parallel.
 Calculate stored energies in capacitors and
inductors.

Capacitance and Inductance | Learning Objectives 3

Energy Storage Devices
 Resistors convert electrical energy into heat.
 Capacitors and inductors are energy-storage
elements.
 Store energy and later return it to the circuit.
 They don’t generate energy  Passive elements.
 Capacitance: energy stored in electric fields.

 Inductance: energy stored in magnetic fields.

Capacitance and Inductance | Energy Storage Devices 4

Capacitors
Capacitance
 A typical capacitor is made of two sheets of conductors
separated by a layer of insulating material.
 Capacitance value dependent on dimensions and material.
 The insulating material is called dielectric.
 This could be air, polyester, …

 Unit of capacitance is Farad (F).

 Typical values for capacitors are in μF.
 Capacitor circuit symbol and passive sign convention:
i (t ) C

+
Capacitance and Inductance | Capacitance
v(t ) − 6
Some Typical Capacitors

Capacitance and Inductance | Some Typical Capacitors 7

How does a capacitor work?
 As current flows through a capacitor, charges of opposite signs
collect on the respective plates.
 Electrical energy is stored in material between the plates.

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | How does a capacitor work? 8
 Parallel-Plate Capacitor
 An example of a simple capacitor is a parallel-plate capacitor.
 It consists of two conductive plates separated by a dielectric
layer.

Dielectric constant of material

ε = ε 0ε r εA Surface area of plates (W × L)

Permittivity of Air
C=
ε 0 = 8.854 ×10 F/m −12 d Separation between plates

ε r relative dielectric constant

Source: Hambley’s Book Supplementary Material
Capacitance and Inductance | Parallel-Plate Capacitor 9
Capacitor: Voltage and Current Relationships
 The capacitance C of a device is defined as charge Q per voltage
V.
 One Farad (F) is the capacitance of a device that can store one
Coulomb of charge per one Volt.
 Thus:

 Recall, current definition:

 Further:

 So, if voltage is constant i = 0.

 A capacitor in steady state acts an open circuit.

Source: Hambley’s Book Supplementary Material
Capacitance and Inductance | Voltage and Current Relationships 10
Capacitor as Energy Storage Device
 Remember to calculate the instantaneous power of a device

 Recall capacitor voltage-current-charge relationships

 Next to calculate the energy stored in the capacitor

 Assuming t0 = 0 and w(0) = 0,

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Energy Stored 11
Example 1
 The voltage through a 1 μF is shown in figure below. Plot q(t)
and i(t).

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Example 1 12
Example 2
 The current through a 0.1 μF is given by i(t) = 0.5 sin (104 t).
Plot i(t), q(t), and v(t). Assume 𝑞𝑞 𝑡𝑡 = 0 = 0.

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Example 2 13
Example 2
 The current through a 0.1 μF is given by i(t) = 0.5 sin (104 t).
Plot i(t), q(t), and v(t). Assume 𝑞𝑞 𝑡𝑡 = 0 = 0.

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Example 2 14
 Example 3
 The voltage through a 10 μF is shown in figure below. Find and
plot the current, power delivered, and energy stored for time
between 0 and 5 s.

10 𝜇𝜇𝜇𝜇

 1000t 0 < t < 1  10 ×10 −3 A 0 < t < 1

 dv(t ) 
v(t ) =  1000 1 < t < 3
500(5 − t ) 3 < t < 5
i(t ) = C =  0A 1 < t < 3
 dt − 5 ×10 −3 A 3 < 3 < 5

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 Example 3 (cont’d)
 1000t 0 < t < 1

v(t ) =  1000 1 < t < 3
500(5 − t ) 3 < t < 5


 10 ×10 −3 A 0 < t < 1

dv(t ) 
i(t ) = C =  0A 1 < t < 3
dt − 5 ×10 −3 A 3 < 3 < 5


p (t ) = v(t )i (t )

1
w(t ) = Cv 2 (t )
2
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Capacitances in Parallel and Series
Parallel Capacitors Series Capacitors

 To get a high voltage than the source, charge capacitors in

parallel then connect them in series.
Source: Hambley’s Book Supplementary Material
Capacitance and Inductance | Capacitors in Parallel and Series 17
Example 4
 Find Ceq

Source: Irwin’s Book Supplementary Material, 10th edition. Example 6.12, p. 267.
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Example 4
 Find Ceq 6µ F

3µ F
3
C eq = µ F
2
C eq → 4µ F

12 µ F Source: Irwin’s Book Supplementary Material, 10th edition. Example 6.12, p. 267.
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Example 5
 Find Ceq
 All capacitors are 4 µF

Source: Irwin’s Book Supplementary Material, 10th edition.

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Example 5
 Find Ceq
 All capacitors are 4 µF
8µ F

8µ F

4µ F

8
+8=
32
C eq
32
µF 8µ F
3 3 12
8µ F

Source: Irwin’s Book Supplementary Material, 10th edition.

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Inductors
Inductance
 A typical inductor is made of wire wound around a core.
 Inductance value dependent on dimensions and core material.
 The core is usually a magnetic material
 This could be steel, ferrite, …

 Unit of inductance is Henry (H).

 Typical values for inductors are in mH.
 Inductor circuit symbol and passive sign convention:

+ v(t ) −

i (t ) L
Source: Hambley’s Book Supplementary Material
Capacitance and Inductance | Inductance 23
Some Typical Inductors

Capacitance and Inductance | Some Typical Inductors 24

How does an inductance work?
 As voltage across the inductor changes, magnetic flux is
induced and magnetic energy is concentrated inside the core.
 Direction of magnetic field determined using right-hand rule.
 Typical inductor forms:

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | How does an inductor work? 25
 Simple Inductor
 An example of a simple inductor, is a wire wound around a
cylindrical core.

µA
Cross-sectional area of core
A
L= N 2 Number of turns
l Core length N l
Permeability of the core
µ = µ0 µr
Permeability of air
µ 0 = 4π ×10 −7 H/m
µ r relative permeability of the core

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Simple Inductor 26
Inductor: Voltage and Current Relationships
 The inductance L of a device is defined as total magnetic flux φ
per current I.
 Voltage across an inductor is equal to rate of change of
magnetic flux.
 One Henry (H) is the inductance of a device that can store one
Weber of magnetic flux per one ampère of current.
 Thus:

 Further
 So, if current is constant v = 0.

 A inductor in steady state acts a short circuit.

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Voltage and Current Relationships 27
Inductor as Energy Storage Device
 To calculate the instantaneous power of a device

 Recall inductor voltage-current relationship

 Next to calculate the energy stored in the inductor

 Assuming t0 = 0 and w(0) = 0,

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Energy Storage 28
Example 6
 The current i(t) is applied to a 5 H inductance. Find and plot the
voltage, power delivered and energy stored for time between 0 and 5 s.

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Capacitance and Inductance | Example 4 Source: Hambley’s Book Supplementary Material
Example 6
 The current i(t) is applied to a 5 H inductance. Find and plot the
voltage, power delivered and energy stored for time between 0 and 5 s.

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Capacitance and Inductance | Example 4 Source: Hambley’s Book Supplementary Material
Example 7
 Consider the circuit shown in the figure. In this circuit, we have a switch
that closes at t = 0, connecting a 10 V source to a 2-H inductance. Find
the current as a function of time.
t
1
i (t ) = ∫ v(t )dt + i (t0 )
L to
t
1
= ∫ 10dt = 5t t > 0
2 to

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Capacitance and Inductance | Example 5 Source: Hambley’s Book Supplementary Material
Inductors in Series and Parallel
Series Inductors

Parallel Inductors

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Inductors in Series and Parallel 32
Example 8
 Find the equivalent inductance for each of the circuits shown
in the figure.

Source: Hambley’s Book Supplementary Material

Capacitance and Inductance | Example 6 33
Example 9
 Find the equivalent inductance for each of the circuits shown
in the figure.
ALL INDUCTORS ARE 4mH

Source: Irwin’s Book Supplementary Material, 10th edition.

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LEARNING EXTENSION

ALL INDUCTORS ARE 4mH

CONNECT COMPONENTS BETWEEN NODES

a 4mH
WHEN IN
DOUBT… 4mH
REDRAW!
Leq d c
2mH
2mH
2mH
IDENTIFY ALL NODES
PLACE NODES IN CHOSEN LOCATIONS
b

Leq = (6mH || 4mH ) + 2mH = 4.4mH

Source: Irwin’s Book Supplementary Material, 10th edition.

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Example 8
 Find the equivalent inductance for each of the circuits shown
in the figure.
ALL INDUCTORS ARE 6mH

Source: Irwin’s Book Supplementary Material, 10th edition. E6.14, p. 269.

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LEARNING EXTENSION
ALL INDUCTORS ARE 6mH a

2mH
6 || 6 || 6
Leq b
6mH
6mH

6mH c

48 24
Leq = 6 + [(6 + 2) || 6] = 6 + = 6 mH
14 7

NODES CAN HAVE COMPLICATED SHAPES. 66

Leq = mH
KEEP IN MIND DIFFERENCE BETWEEN 7
PHYSICAL LAYOUT AND ELECTRICAL
CONNECTIONS

Source: Irwin’s Book Supplementary Material, 10th edition. E6.14, p. 269.

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 Summary
Energy Storage Devices

Resistor Capacitor Inductor

i (t ) i (t ) C + v(t ) −

+ v(t ) − + v(t ) − i (t ) L

Unit: Ohm (Ω) Farad (F) Henry (H)

At Steady State: Normal Open-circuit Short-circuit

Capacitance and Inductance | Summary 38

Example 9
 FIND THE TOTAL ENERGY STORED IN THE CIRCUIT

Source: Irwin’s Book Supplementary Material, 10th edition. Example 6.5, p. 255.
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 LEARNING EXAMPLE FIND THE TOTAL ENERGY STORED IN THE CIRCUIT

In steady state inductors act as

short circuits and capacitors act
as open circuits

1 1 2
WC = CVC W L = LI L
2

2 2

81
⇒ VA − [V ]
5

I L1 + 3 A = I L 2 ⇒ I L1 = −1.2 A V = 6 V = 10.8V
C2
6+ 3 A

VC 1 = 9 − 6 I L1 ⇒ VC 1 = 16.2V VA
I L2 = = 1.8 A
9

Source: Irwin’s Book Supplementary Material, 10th edition. Example 6.5, p. 255.
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 ELE 225
Electric Circuits and Devices
End of Lecture 6. Capacitance and Inductance
Some materials/figures in this presentation are adapted/taken from (with permission):
Basic Engineering Circuit Analysis, 10th Edition. By Irwan, J.D, & Nelms, R.M.,
Electrical Engineering, Principles and Applications, 6th Edition. By Hambley, A.R.,
Lecture notes from Dr. Mohamed Hassan, Dr. Amer Zakaria, Dr. Ming, Dr. Mostafa Shaaban
and Other online resources.

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