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Chap 1. An Introduction to
Electrical Engineering
Jean de Dieu IYAKAREMYE
October, 201511/07/15 1
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
Assignments and CAT
20/10/2015 Homework 1,
22/10/2015 , Assignment
11/07/15 3
What is electrical engineering?
The study of ELECTRICITY along with its
numerous applications
11/07/15 4
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
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
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
A brief history
1958 – integrated circuit
developed by Jack Kilby
1968 – first microprocessor is
developed
Integrated circuits
Microprocessor11/07/15 8
So where is the field now?
11/07/15 9
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
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
Basic concepts
✴ Electricity
✴ Charge
✴ Current
✴ Voltage
✴ Power and Energy
11/07/15 12
Electricity
Physical phenomenon arising from the
existence and interactions of electric charge
11/07/15 13
Charge
Where can we observe/experience/use charge?
11/07/15 14
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
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
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
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
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
Electrical Circuits
11/07/15 20
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
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
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
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
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
Circuit schematic example
11/07/15 26
Circuit elements
11/07/15 27
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
Resistors
11/07/15 29
Resistors
11/07/15 30
Ohm’s Law
(remember, R is in Ω
and ρ is in Ω-m)
11/07/15 31
Resistance and temperature
11/07/15 32
Capacitors
11/07/15 33
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
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
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
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
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
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
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
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
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
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
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
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
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
Dependent Sources
A dependent or controlled source depends upon a different
voltage or current in the circuit
11/07/15 47
• 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
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.
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
2.1 Series and Parallel Circuits
Key Question:
How do series and parallel circuits work?
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.
Electrical engineering, students notes
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)
Electrical engineering, students notes
2.1 Total resistance in a series
circuit
• Light bulbs, resistors, motors, and heaters usually have
much greater resistance than wires and batteries.
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?
Electrical engineering, students notes
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.
Electrical engineering, students notes
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.
Electrical engineering, students notes
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.
Electrical engineering, students notes
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.
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.
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.
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.
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.
Electrical engineering, students notes
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.
2.2 Analysis of Circuits
Key Question:
How do we analyze
network circuits?
*Students read Section 20.2
AFTER Investigation 20.2
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.
2.2 Three circuit laws
2.2 Voltage divider circuit
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)
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
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.
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?
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.
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?
2.3 Electric Power, AC, and DC
Electricity
Key Question:
How much does
electricity cost and
what do you pay for?
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.
20.3 Reviewing terms
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.
2.3 Power
P = VI Current (amps)
Voltage (volts)
Power (watts)
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.
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.
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?
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.
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.
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.
Electrical engineering, students notes
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.
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%
Application: Wiring in Homes and
Buildings
Application: Wiring in Homes and
Buildings
MAGNETISM &
ELECTROMAGNETISM
Lecture 3
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.
Permanent Magnets and Magnetic
Fields
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
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.
Electrical engineering, students notes
• The magnitude is given by the formula
Electrical engineering, students notes
Electrical engineering, students notes
MAGNETISM AND CURRENT
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
Magnetic Field of a Wire Loop
• N = number of turns. I = current in the wire. R
is the radius of the wire
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
Electrical engineering, students notes
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.
Magnetic Field of a Solenoid
• where n = number of turns per unit length and
I is the current in the loop.
Electrical engineering, students notes
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.
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
• 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.
Electrical engineering, students notes
Electrical engineering, students notes
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.
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.
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:
Electrical engineering, students notes
Electrical engineering, students notes
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
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
ELECTROMAGNETISM
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?
Magnetic Flux
• Definition: What is Magnetic Flux
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
ELECTROMAGNETISM
APPLICATIONS:
•Electric generator
•Electric motor
•Transformers
Electrical machines
Transformers
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
Electrical engineering, students notes
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
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
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
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.
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.
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
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).
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
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
Parallel RLC Circuit
Discussions and exercises on parallel RLC
Circuits during the lecture will be conducted
Power in RLC circuits both in // and
Series
• Discussed in class
• Exercises
• Assignments

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Electrical engineering, students notes

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

Hinweis der Redaktion

  1. AC = alternating current, not to be confused with air conditioner
  2. 1) You are asked to calculate current. 2) You are given the voltage and resistances. 3) Use Ohm’s law, I = V÷R, and add the resistance in series. 4) Solve: Resistance = R1 + R2 + R3 = 1Ω + 1Ω + 1Ω = 3Ω Current, I = (1.5 V) ÷ (3Ω)=0.5 A
  3. 1) You are asked for the current. 2) You are given the voltage and resistance. 3) Use Ohm’s law: I = V ÷ R. 4) For the 3Ω bulb: I = (3 V) ÷ (3 Ω) = 1 A. For the 0.5 Ω bulb: I = (3 V) ÷ (0.5 Ω) = 6 A. The battery must supply the current for both bulbs, which adds up to 7 amps.
  4. 1) You are asked for the resistance. 2) You are given the circuit diagram and resistances. 3) Use the rule for parallel resistances. 4) Solve: 1/R total = 1/2 Ω + 1/4 Ω = 2/4 Ω +1/4 Ω = 3/4 Ω R= 4/3 Ω = 1.33 Ω
  5. The total resistance of this circuit is 10 ohms (9Ω + 1Ω). The total current is 1 amp from Ohm’s law (10V ÷ 10Ω). What is the voltage at point (A)? We can use Kirchhoff’s voltage law to find the answer. Assume the current flows according to the light blue loop. The battery starts at +10V. Resistor R1 drops the voltage by 9V. This voltage drop is calculated from Ohm’s law and the resistance, V = 1A × 9Ω. Resistor R2 drops the voltage by 1V (1A × 1Ω). Around the whole loop the sum of the voltage gains and drops is zero (+10 - 9 - 1 = 0).
  6. 1) You are asked to calculate the resistance. 2) You are told it is a series circuit and given the voltage, total current, and one resistance. 3) Use Ohm’s law, R = V ÷ I, and add the resistance in series. 4) Solve: Total resistance = 6V ÷ 1A = 6Ω. SInce the bulb is 1Ω, the additional resistor must be 5Ω to get a total 6Ω of resistance.
  7. 1) You are asked to calculate the currents. 2) You are given the circuit diagram, voltages, and resistances. 3) Use Ohm’s law, R = V ÷ I, and the series and parallel resistance formulas. 4) First, reduce the circuit by combining the two parallel resistances. 1/R total = 1/3 Ω + 1/3 Ω = 2/3 Ω R= 3/2 Ω = 1.5 Ω 5) Calculate the total resistance of 4.5Ω by adding up the remaining series resistances. Calculate the total current using Ohm’s law: I = 3V ÷ 4.5Ω = 0.67A. The two bulbs in parallel have the same resistance, so they divide the current equally; each one gets 0.33 amps. The single bulb in series gets the full current of 0.67 amps, but the other two bulbs get only 0.33 amps each. That means the bulbs in parallel will be much dimmer since they only get half the current.
  8. 1) You are asked to find the power used by the light bulb. 2) You are given the voltage of the battery and the bulb’s resistance. 3) Use Ohm’s law, I = V/R, to calculate the current; then use the power equation, P=VI, to calculate the power. 4) Solve: I = 1.5V ÷ 1.5Ω = 1A P = 1.5V × 1A = 1.5 W; the bulb uses 1.5 watts of electric power.
  9. 1) You are asked to find the cost of using the coffee maker. 2) You are given the power in watts and the time. 3) Use the power formula P = VI and the fact that 1 kWh = 1kW x 1h. 4) Solve: Find the number of kilowatts of power that the coffee maker uses. 1,050 W × 1 kW/1,000 W = 1.05 kW Find the kilowatt-hours used by the coffee maker each month. 1.05 kW × 1 hr/day x 30 days/month = 31.5 kWh per month. Find the cost of using the coffee maker. 31.5 kWh/month × $0.14/kWh = $4.41 per month.