The document provides an overview of an introduction to electrical engineering, including a brief history of the field from Gilbert discovering electricity in 1600 to the development of integrated circuits and microprocessors in the late 20th century. It also outlines some of the core concepts and principles of electrical engineering like charge, current, voltage, power, resistors, capacitors, inductors, and circuit analysis. Assignments and expectations for the course are presented as well.

1. Chap 1. An Introduction to
Electrical Engineering
Jean de Dieu IYAKAREMYE
October, 201511/07/15 1

2. Overview
-Brief history, disciplines, curriculum
-Review of electrical principles
Lecturer: Jean de Dieu IYAKAREMYE, Msc
Book: Electricity for Agriculture
(Available in the Library)
11/07/15 2

3. Assignments and CAT
20/10/2015 Homework 1,
22/10/2015 , Assignment
11/07/15 3

4. What is electrical engineering?
The study of ELECTRICITY along with its
numerous applications
11/07/15 4

5. A brief history
In 1600, William Gilbert called
the property of attracting
particles after being rubbed
“electricus”.
De Magnete was a treatise of
electricity and magnetism,
noting a long list of elements
that could be electrified.
A versorium
Gilbert invented the
versorium, a device that
detected statically-charged
bodies
William Gilbert, arguably the first electrical engineer
11/07/15 5

6. A brief history
1800 – voltaic pile developed by Alessandro
Volta, a precursor to the battery
1831 – Michael Faraday discovers
electromagnetic induction
1873 – Electricity and Magnetism
published by James Maxwell, describing
a theory for electromagnetism
Voltaic pile
Circuits containing inductors
Maxwell’s equations11/07/15 6

7. A brief history
1888 – Heinrich Hertz transmits and
receives radio signals
1941 – Konrad Zuse introduces the first
ever programmable computer
1947 – invention of transistor
Spark-gap transmitter
Z3 computer
Transistor
11/07/15 7

8. A brief history
1958 – integrated circuit
developed by Jack Kilby
1968 – first microprocessor is
developed
Integrated circuits
Microprocessor11/07/15 8

9. So where is the field now?
11/07/15 9

10. Fields of study
Power:
Creation, storage, and distribution of electricity
Control:
Design of dynamic systems and controllers for the
systems
Electronics/Microelectronics:
Design of integrated circuits, microprocessors, etc.
Signal Processing: Analysis of signals
11/07/15 10

11. Telecommunications:
Design of transmission systems (voice, data)
Computer:
Design and development of computer systems
Instrumentation:
Design of sensors and data acquisition equipment
Fields of study
11/07/15 11

12. Basic concepts
✴ Electricity
✴ Charge
✴ Current
✴ Voltage
✴ Power and Energy
11/07/15 12

13. Electricity
Physical phenomenon arising from the
existence and interactions of electric charge
11/07/15 13

14. Charge
Where can we observe/experience/use charge?
11/07/15 14

15. Charge
Characteristic property of subatomic
particles responsible for electric phenomena
1.602×10−19
C−1.602×10−19
C
- +Electron Proton
The unit of quantity of electric charge is coloumb (C)
1 coulomb = 6.25 × 1018
e
e = elementary charge = charge of proton11/07/15 15

16. Charge
“Charged” particles exhibit forces
Opposite charges attract one another
Like charges repel each other
- -
+-
Charge is the source of one of the fundamental forces in nature (others?)11/07/15 16

17. Coulomb’s Law
q1 q2
r (meters)
(Newtons)
F1,2 is the electrostatic force exerted on charge 1 due
to the presence of charge 2
ke is the Coulomb constant ke = 8.987 x 109
N*m2
*C-2
11/07/15 17

18. Electric current
Describes charge in motion, the flow of charge
This phenomenon can result from moving electrons in a
conductive material or moving ions in charged solutions
11/07/15 18

19. Electric current
An ampere (A) is the number of electrons having a total
charge of 1 C moving through a given cross section in 1 s.
As defined, current flows in direction of positive charge flow
n=Q/e , where n=number of electrons, e= -1.6x10exp-19
11/07/15 19

20. Electrical Circuits
11/07/15 20

21. Electric circuit
An electric circuit is an interconnection of electrical elements
linked together in a closed path so that electric current may
flow continuously
Circuit diagrams are the standard for electrical engineers
11/07/15 21

22. Rate of flow of charge form node a to node b
Rate of flow of charge form node b to node a
(i = current)
A direct current (dc) is a current of constant magnitude
An alternating current (ac) is a current of varying
magnitude and direction
11/07/15 22

23. Voltage
Driving “force” of electrical current between two points
Vab
Vba
Voltage at terminal a with respect to terminal b
Voltage at terminal b with respect to terminal a
Vab = -Vba
Note: In a circuit, voltage is often defined relative to
“ground”11/07/15 23

24. Voltage
The voltage across an element is the work (energy) required to move a
unit of positive charge from the “-” terminal to the “+” terminal
A volt is the potential difference (voltage) between
two points when 1 joule of energy is used to move 1
coulomb of charge from one point to the other
11/07/15 24

25. Power
The rate at which energy is converted or work is performed
A watt results when 1 joule of energy is converted or used in 1 second
11/07/15 25

26. Circuit schematic example
11/07/15 26

27. Circuit elements
11/07/15 27

28. Resistors
Resistivity (ρ) is the ability of a
material to resist current flow. The
units of resistivity are Ohm-meters
(Ω-m)
Resistance (R) is the physical
property of an element that
impedes the flow of current . The
units of resistance are Ohms (Ω)
1.68×10−8
Ω·m
Example:
Resistivity of copper
Resistivity of glass 1010
to 1014
Ω·m11/07/15 28

29. Resistors
11/07/15 29

30. Resistors
11/07/15 30

31. Ohm’s Law
(remember, R is in Ω
and ρ is in Ω-m)
11/07/15 31

32. Resistance and temperature
11/07/15 32

33. Capacitors
11/07/15 33

34. Capacitors
A capacitor consists of a pair of
conductors separated by a
dielectric (insulator).
(ε indicates how penetrable a subtance is to an
electric field)
Electric charge is stored in the
plates – a capacitor can become
“charged”
When a voltage exists across the conductors,
it provides the energy to move the charge
from the positive plate to the other plate.11/07/15 34

35. Capacitors
Capacitance (C) is the ability of a material to store charge in the
form of separated charge or an electric field. It is the ratio of
charge stored to voltage difference between two plates.
Capacitance is measured in Farads (F)
11/07/15 35

36. Capacitors
The capacitor plate attached to the negative
terminal accepts electrons from the battery.
The capacitor plate attached to the positive
terminal accepts protons from the battery.
What happens when the light bulb is
initially connected in the circuit?
What happens if you replace the battery
with a piece of wire?
11/07/15 36

37. Energy storage
Work must be done by an external influence (e.g. a battery) to
separate charge between the plates in a capacitor. The charge is
stored in the capacitor until the external influence is removed and
the separated charge is given a path to travel and dissipate.
Work exerted to charge a capacitor is given by the equation:
11/07/15 37

38. Inductors
An inductor is a two terminal element
consisting of a winding of N turns capable
of storing energy in the form of a magnetic
field
Inductance (L) is a measure of the ability of
a device to store energy in the form of a
magnetic field. It is measured in Henries (H)
11/07/15 38

39. Inductors
μ0 = permeability of free space = 4π × 10−7
H/m
K = Nagaoka coefficient
N = number of turns
A = area of cross-section of the coil in m2
l = length of coil in m
Inductance in a cylindrical coil
11/07/15 39

40. Inductors
The magnetic field from an inductor can generate an induced
voltage, which can be used to drive current
While building the magnetic field, the inductor resists current flow
11/07/15 40

41. Inductors
What happens to the light bulb when the switch is closed?
What happens to the light bulb when the switch is then opened?
11/07/15 41

42. Energy storage
The work required to establish current through the
coil, and therefore the magnetic field, is given by
Inductors can store energy in the form of a magnetic
field when a current is passed through them.
11/07/15 42

43. Transformers and alternators
Inductors are located in both transformers and alternators,
allowing voltage conversion and current generation, respectively
Transformer converts from
one voltage to another Alternator produces AC current
11/07/15 43

44. Electrical sources
An electrical source is a voltage
or current generator capable of
supplying energy to a circuit
Examples:
-AA batteries
-12-Volt car battery
-Wall plug
11/07/15 44

45. Ideal voltage source
An ideal voltage source is a circuit element where the voltage
across the source is independent of the current through it.
Recall Ohm’s Law: V=IR
The internal resistance of an ideal voltage source is zero.
If the current through an ideal voltage source is
completely determined by the external circuit, it
is considered an independent voltage source
11/07/15 45

46. Ideal current source
An ideal current source is a circuit element where the current
through the source is independent of the voltage across it.
Recall Ohm’s Law: I = V/R
The internal resistance of an ideal current source is infinite.
If the voltage across an ideal current source is
completely determined by the external circuit, it
is considered an independent current source
11/07/15 46

47. Dependent Sources
A dependent or controlled source depends upon a different
voltage or current in the circuit
11/07/15 47

48. • 2.1 Series and Parallel Circuits
• 2.2 Analysis of Circuits
• 2.3 Electric Power, AC, and DC Electricity
Chapter 2. Electric Circuits and Power

49. Chapter 2. Objectives
1. Recognize and sketch examples of series and parallel circuits.
2. Describe a short circuit and why a short circuit may be a
dangerous hazard.
3. Calculate the current in a series or parallel circuit containing
up to three resistances.
4. Calculate the total resistance of a circuit by combining series
or parallel resistances.
5. Describe the differences between AC and DC electricity.
6. Calculate the power used in an AC or DC circuit from the
current and voltage.

50. Chapter 2. Vocabulary Terms
 series circuit
 parallel circuit
 short circuit
 network circuit
 circuit analysis
 power
 Kirchhoff’s voltage law
 voltage drop direct
 current (DC)
 alternating current
(AC)
 kilowatt
 Kirchhoff’s current law
 horsepower
 power factor
 circuit breaker
 watt
 kilowatt-hour

51. 2.1 Series and Parallel Circuits
Key Question:
How do series and parallel circuits work?

52. 2.1 Series and Parallel Circuits
• In series circuits, current can only take one
path.
• The amount of current is the same at all
points in a series circuit.

54. 2.1 Adding resistances in series
• Each resistance in a series
circuit adds to the total
resistance of the circuit.
Rtotal = R1 + R2 + R3...
Total
resistance
(ohms) Individual resistances (Ω)

56. 2.1 Total resistance in a series
circuit
• Light bulbs, resistors, motors, and heaters usually have
much greater resistance than wires and batteries.

57. 2.1 Calculate current
• How much current flows in a circuit with a 1.5-volt battery
and three 1 ohm resistances (bulbs) in series?

59. 2.1 Voltage in a series circuit
• Each separate resistance creates a
voltage drop as the current passes
through.
• As current flows along a series
circuit, each type of resistor
transforms some of the electrical
energy into another form of energy
• Ohm’s law is used to calculate the
voltage drop across each resistor.

61. 2.1 Series and Parallel Circuits
• In parallel circuits the current can take more than one path.
• Because there are multiple branches, the current is not the
same at all points in a parallel circuit.

63. 2.1 Series and Parallel Circuits
• Sometimes these paths are called branches.
• The current through a branch is also called the branch current.
• When analyzing a parallel circuit, remember that the current
always has to go somewhere.
• The total current in the circuit is the sum of the currents in all
the branches.
• At every branch point the current flowing out must equal the
current flowing in.
• This rule is known as Kirchhoff’s current law.

65. 2.1 Voltage and current
in a parallel circuit• In a parallel circuit the voltage is the same
across each branch because each branch has a
low resistance path back to the battery.
• The amount of current in each branch in a
parallel circuit is not necessarily the same.
• The resistance in each branch determines the
current in that branch.

66. 2.1 Advantages of parallel circuits
Parallel circuits have two big advantages over
series circuits:
1. Each device in the circuit sees the full
battery voltage.
2. Each device in the circuit may be turned off
independently without stopping the current
flowing to other devices in the circuit.

67. 2.1 Short circuit
• A short circuit is a parallel path in a circuit with zero or very
low resistance.
• Short circuits can be made accidentally by connecting a wire
between two other wires at different voltages.
• Short circuits are dangerous because they can draw huge
amounts of current.

68. 2.1 Calculate current
• Two bulbs with different resistances are connected in
parallel to batteries with a total voltage of 3 volts.
• Calculate the total current supplied by the battery.

69. 2.1 Resistance in parallel circuits
• Adding resistance in parallel provides another
path for current, and more current flows.
• When more current flows for the same
voltage, the total resistance of the circuit
decreases.
• This happens because every new path in a
parallel circuit allows more current to flow for
the same voltage.

71. 2.1 Adding resistance in parallel
circuits
• A circuit contains a 2 ohm resistor and a 4 ohm
resistor in parallel.
• Calculate the total resistance of the circuit.

72. 2.2 Analysis of Circuits
Key Question:
How do we analyze
network circuits?
*Students read Section 20.2
AFTER Investigation 20.2

73. 2.2 Analysis of Circuits
• All circuits work by manipulating currents and
voltages.
• The process of circuit analysis means figuring
out what the currents and voltages in a circuit
are, and also how they are affected by each
other.
• Three basic laws are the foundation of circuit
analysis.

74. 2.2 Three circuit laws

75. 2.2 Voltage divider circuit

76. 2.2 Voltage divider
• A circuit divides any supplied voltage by a ratio
of the resistors.
Output
voltage
(volts)
resistor ratio
(Ω)
V0 = R2 Vi
R1 + R2
Input
voltage
(volts)

77. 2.2 Solving circuit problems
1. Identify what the problem is asking you to
find. Assign variables to the unknown
quantities.
2. Make a large clear diagram of the circuit. Label
all of the known resistances, currents, and
voltages. Use the variables you defined to
label the unknowns.
3. You may need to combine resistances to find
the total circuit resistance. Use multiple steps

78. 2.2 Solving circuit problems
4. If you know the total resistance and current,
use Ohm’s law as V = IR to calculate voltages
or voltage drops. If you know the resistance
and voltage, use Ohm’s law as I = V ÷ R to
calculate the current.
5. An unknown resistance can be found using
Ohm’s law as R = V ÷ I, if you know the
current and the voltage drop through the
resistor.

79. 2.2 Solving circuit problems
• A bulb with a resistance of 1Ω is to be
used in a circuit with a 6-volt battery.
• The bulb requires 1 amp of current.
• If the bulb were connected directly to the
battery, it would draw 6 amps and burn
out instantly.
• To limit the current, a resistor is added in
series with the bulb.
• What size resistor is needed to make the
current 1 amp?

80. 2.2 Network circuits
• In many circuits, resistors are connected both
in series and in parallel.
• Such a circuit is called a network circuit.
• There is no single formula for adding resistors
in a network circuit.
• For very complex circuits, electrical engineers
use computer programs that can rapidly solve
equations for the circuit using Kirchhoff’s laws.

81. 2.2 Calculate using network circuits
• Three bulbs, each with a resistance
of 3Ω, are combined in the circuit
in the diagram
• Three volts are applied to the
circuit.
• Calculate the current in each of the
bulbs.
• From your calculations, do you
think all three bulbs will be equally
bright?

82. 2.3 Electric Power, AC, and DC
Electricity
Key Question:
How much does
electricity cost and
what do you pay for?

83. 2.3 Electric Power, AC, and DC
Electricity
• The watt (W) is a unit of
power.
• Power is the rate at which
energy moves or is used.
• Since energy is measured in
joules, power is measured in
joules per second.

84. 20.3 Reviewing terms

85. 2.3 Power in electric circuits
• One watt is a pretty small amount of power.
• In everyday use, larger units are more
convenient to use.
• A kilowatt (kW) is equal to 1,000 watts.
• The other common unit of power often seen
on electric motors is the horsepower.
• One horsepower is 746 watts.

86. 2.3 Power
P = VI Current (amps)
Voltage (volts)
Power (watts)

87. 2.3 Calculate power
• A light bulb with a
resistance of 1.5Ω is
connected to a 1.5-volt
battery in the circuit shown
at right.
• Calculate the power used
by the light bulb.

88. 2.3 Paying for electricity
• Electric companies charge for the
number of kilowatt-hours used
during a set period of time, often a
month.
• One kilowatt-hour (kWh) means that
a kilowatt of power has been used
for one hour.
• Since power multiplied by time is
energy, a kilowatt-hour is a unit of
energy.
• One kilowatt-hour is 3.6 x 106
joules.

89. 2.3 Calculate power
• Your electric company charges 14 cents per kilowatt-hour.
Your coffee maker has a power rating of 1,050 watts.
• How much does it cost to use the coffee maker one hour
per day for a month?

90. 2.3 Alternating and direct current
• The current from a battery is
always in the same direction.
• One end of the battery is
positive and the other end is
negative.
• The direction of current flows
from positive to negative.
• This is called direct current, or
DC.

91. 2.3 Alternating and direct current
• If voltage alternates, so does
current.
• When the voltage is positive, the
current in the circuit is
clockwise.
• When the voltage is negative the
current is the opposite direction.
• This type of current is called
alternating current, or AC.

92. 2.3 Alternating and direct current
• AC current is used for almost all high-power
applications because it is easier to generate
and to transmit over long distances.
• The 220 volt AC (VAC) electricity used in
homes and businesses
• AC electricity is usually identified by the
average voltage, (220 VAC) not the peak
voltage.

94. 2.3 Power in AC circuits
• For a circuit containing a
motor, the power calculation is
a little different from that for a
simple resistance like a light
bulb.
• Because motors store energy
and act like generators, the
current and voltage are not in
phase with each other.
• The current is always a little
behind the voltage.

95. 20.3 Power for AC circuits
• Electrical engineers use a power factor (pf) to calculate power
for AC circuits with motors
P = VI x pf
Avg. current (amps)Avg. voltage
(volts)
Power (watts)
power factor
0-100%

96. Application: Wiring in Homes and
Buildings

97. Application: Wiring in Homes and
Buildings

98. MAGNETISM &
ELECTROMAGNETISM
Lecture 3

99. Magnets and Magnetic Fields
• This is sometimes a complicated subject, because although
we use it every day in every motor, we don't really come in
contact with magnetism.
• Also, the way the force of magnetism acts is unlike any we
have come in contact with yet in our study of physical
phenomena. It is a property that always requires three
dimensions to describe.
• Because magnetism involves three dimensions, we often have
to draw vectors into the plane of the paper or out of the
plane of the paper. We represent vectors like that as arrows.
But all we see is either the tip of the arrow , if the field is
coming out of the page, or the tail of the arrow, , if the field
is going into the page.

100. Permanent Magnets and Magnetic
Fields

104. Magnetic Field of a Wire (Hans
Christian Oersted – 1820)
• The field goes in circles around the wire. The
direction is given by the right hand rule. Thumb is
in direction of current.
• The fingers curl in the direction of the magnetic
field. What is its magnitude? A drawing of the
field lines shows they go in circles around the
wire and are denser near the wire.
• Where the field lines are denser, the magnetic
field is stronger.
• It is stronger near the wire.

106. • The magnitude is given by the formula

109. MAGNETISM AND CURRENT

110. Magnetic Field of a Wire Loop
• What about a loop of wire? Each section of
the wire gives a magnetic field with the
direction determined by the right hand rule.
• The magnetic field inside the wire is in a
different direction from the magnetic field
outside the wire. At the center of the loop

111. Magnetic Field of a Wire Loop
• N = number of turns. I = current in the wire. R
is the radius of the wire

112. Ampere's Law
• There is a fundamental principle which allows
us to calculate the magnetic field from any
wire carrying a current.
• André Marie Ampere determined that if we
take any closed path around a current
carrying wire and looked only at the vector
component of the magnetic field parallel to
that closed path, you would find

114. Magnetic Field of a Solenoid
• If I now take many turns of wire, and pack
them tightly, I get a solenoid. Inside the
solenoid, the magnetic field is approximately
constant and outside the solenoid the
magnetic field is approximately zero. See
Figure below. The direction of the field is
given by the RHR and the magnitude can be
determined using Ampere's law.

115. Magnetic Field of a Solenoid
• where n = number of turns per unit length and
I is the current in the loop.

117. Magnetic Fields Exert a Force on a
Moving Charge
Characteristics of the Force
•A magnetic field can create a force on an object. However, for the
object to feel a force, and the magnetic field to affect the object, three
things must be true
1. The object must have an electric charge.
2. The charged object must be moving.
3. The velocity of the moving charged object must have a component
that is perpendicular to the direction of the magnetic field.
•A slightly different "Right Hand Rule" (RHR) is used to determine the
direction of the force on a charged particle from a magnetic field.
Index finger points in direction of the particle's velocity, the middle
finger points in the direction of the magnetic field, and the thumb
points in the direction of the force on a positive charge - the direction
a positively charged particle will accelerate.

118. Definition of Magnetic Field
• Remember that there was a relationship
between the electric field and the force that a
charged particle felt. That relationship was
qE=F. Similarly, there is a relationship
between a magnetic field and the force a
charged particle feels. It is

119. • Remember that force is a vector, and there is
a direction to this force. The direction is given
by the RHR as explained above. If we look at
this equation we can see four different ways
to increase the force a particle feels from a
magnetic field? Increase B. Increase q.
Increase v. Maximize sin theta. The sign of q
must be used.

122. Magnetic Fields Exert a Force on a
Current in a Wire
• What is a current in a wire? Moving charges.
What does a magnetic field do to moving
charges? Exerts a force. What does a magnetic
field do to a wire. Exerts a force.

123. Magnetic Fields Exert a Force on a
Current in a Wire
• A stereo speaker works by using this principle.
If current increases the force increases. The
direction is given by RHR.

124. Magnetic Fields Exert a Torque on a
Current in a Coil
• Now put a current carrying loop of wire in an
already present magnetic field.
• There is a force on the loop due to the current
in the wires and the force on all the wires
creates a net torque.
• The torque is given by force multiplied by the
length of the lever arm:

127. Force Between Two Wires
• Okay, what do we know about currents in
wires.
• They produce a magnetic field.
• They feel a force from magnetic fields

131. ELECTROMAGNETISM

132. Introduction
• The main emphasis of this chapter can be
summed up in one sentence. A Changing
Magnetic Flux induces an emf (electromotive
force, or voltage difference). This statement
immediately brings up three questions that we
need to answer?
• What is a magnetic flux?
• How do you change it?
• What are the consequences of the induced emf
and what is its polarity?

133. Magnetic Flux
• Definition: What is Magnetic Flux

141. ELECTROMAGNETISM
APPLICATIONS:
•Electric generator
•Electric motor
•Transformers
Electrical machines

142. Transformers

147. AC Circuits lecture 4
Objectives:
Understanding AC circuits and their use in
electric network
RLC series and parallel circuits
Power factor
Economic power factor improvement

148. Phasors
A phasor is a vector whose magnitude is the maximum value of a quantity (eg V or I) and
which rotates counterclockwise in a 2-d plane with angular velocity ω. Recall uniform
circular motion:
The projections of r (on the
vertical y axis) execute
sinusoidal oscillation.
⇒
⇒
⇒ i
L
tL
m= −
ε
ω
ωcos
i C tC m= ω ε ωcos
i
R
tR
m=
ε
ωsin
x r t= cosω
y r t= sinω
V Ri tR R m= = ε ωsin• R: V in phase with i
V
Q
C
tC m= = ε ωsin• C: V lags i by 90°
V L
di
dt
tL
L
m= = ε ωsin• L: V leads i by 90°
ω
x
y y

149. Phasors for
L,C,R i
ωt
ω
V
R
i
ωt
ωV
L
i
ωt
ω
V
C
⇓
Suppose:
tii m ωsin=
V Ri tR m= sinω
V
C
i tC m= −
1
ω
ωcos
V Li tL m= ω ωcos
t
i
V
R
0
V
C
0
i
i
0
V
L

150. Phasors: LCR
From these equations, we can draw the phasor diagram
to the right.
• Assume:
⇒
φ
ω
φ
φ
im
R
im
ωL
im
ω C
εm
LC
∼
ε
R
• Given: ε ε ω= m tsin
i i tm= −sin( )ω φ ⇒
Q
i
tm= − −
ω
ω φcos( )
di
dt
i tm= −ω ω φcos( )
V Ri Ri tR m= = −sin( )ω φ
V L
di
dt
Li tL m= = −ω ω φcos( )
V
Q
C C
i tC m= = − −
1
ω
ω φcos( )
This picture corresponds to a snapshot at t=0. The
projections of these phasors along the vertical axis
are the actual values of the voltages at the given
time.

151. Phasors: LCR
• The phasor diagram has been relabeled in terms of the reactances
defined from:
φ
ω
φ
φ
im
R
εm
imXC
imXL
LC
∼
ε
R
X
C
C ≡
1
ω
X LL ≡ ω
The unknowns (im,φ) can now be solved for graphically
since the vector sum of the voltages
VL + VC + VR must sum to the driving emf ε.

152. Phasors:LCR
⇒
im
R
εm
φ
im(XL-XC)
⇓
φ
φ
φ
im
R
εm
imXC
imXL
X
C
C ≡
1
ω
X LL ≡ ω
( )Z R X XL C≡ + −2 2
tanφ =
−X X
R
L C
( )( )εm m L Ci R X X2 2 2 2
= + −
( )
i
R X X Z
m
m
L C
m=
+ −
=
ε ε
2 2

153. Phasors:Tips
• This phasor diagram was drawn as a snapshot of
time t=0 with the voltages being given as the
projections along the y-axis.
φ
φ
φ
im
R
εm
imXC
imXL
y
x
φ
imR
imXL
imXC
εm
“Full Phasor Diagram”
From this diagram, we can also create a triangle
which allows us to calculate the impedance Z:
X XL C−
φ
Z
R
“ Impedance Triangle”
• Sometimes, in working problems, it is easier to
draw the diagram at a time when the current is
along the x-axis (when i=0).

154. Phasors:LCRWe have found the general solution for the driven LCR circuit:
X LL ≡ ω
X
C
C ≡
1
ω
( )Z R X XL C≡ + −2 2
R
XL
XC
φ
Ζ
tanφ =
−X X
R
L C
i
Z
m
m=
ε
 
ε = i Zm
the loop
eqn
XL - XC
i i tm= −sin( )ω φ
φ
imR
imXL
imXC
εm ω

155. Lagging & Leading
The phase φ between the current and the driving emf depends on the relative
magnitudes of the inductive and capacitive reactances.
φ
R
XL
XC
Ζ
tanφ =
−X X
R
L Ci
Z
m
m=
ε X LL ≡ ω
X
C
C ≡
1
ω
XL > XC
φ > 0
current
LAGS
applied voltage
φ
R
XL
XC
Ζ
XL < XC
φ < 0
current
LEADS
applied voltage
XL = XC
φ = 0
current
IN PHASE
applied voltage
R
XL
XC
Ζ

156. Parallel RLC Circuit
Discussions and exercises on parallel RLC
Circuits during the lecture will be conducted

157. Power in RLC circuits both in // and
Series
• Discussed in class
• Exercises
• Assignments