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Course: EE 1103
Subject: Basic Electrical Engineering
Topic: Phasor Algebra
Edited and Presented by-
Dr. Mohiuddin Ahmad Dept. of EEE, KUET, Khulna-9203,
Bangladesh
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Contents of the Topic
Contents of Lecture
Sinusoid
Phasors
Phasor operations
References
1. Alternating current circuits – Kechner & Corcoran
2. Fundamentals of Electric Circuit – Alexander & Sadiku
Sinusoids
Sinusoids
A sinusoid is a signal that has the form of the sine or
cosine function.
Consider the sinusoidal voltage
Vm = amplitude of the sinusoid
ω = the angular frequency in radians/s
ωt = the argument of the sinusoid
tVv m sin
Sinusoids
Sketch of Vmsinωt
Sinusoid repeats itself every T seconds; thus, T is
called the period of the sinusoid.
(a) as a function of ωt
(b) as a function of t.
2T
Sinusoids
The fact that v(t) repeats itself every T seconds is
shown by replacing t by in t +T
Hence
that is, v has the same value at t+T as it does at t and is
said to be periodic.
In general,
A periodic function is one that satisfies f(t ) = f(t + nT),
for all t and for all integers n.
)()( Ttvtv
Phasors
A phasor is a complex number that represents the
amplitude and phase of a sinusoid.
Phasors provide a simple means of analyzing linear
circuits excited by sinusoidal sources; solutions of
such circuits would be intractable otherwise.
Before we completely define phasors and apply them
to circuit analysis, we need to be thoroughly familiar
with complex numbers.
A complex number z can be written in rectangular
form as
x is the real part of z;
y is the imaginary part of z.
jyxz 1j
Phasors
The complex number z can also be written in polar or
exponential form as
where r is the magnitude of z, and ϕ is the phase of z.
z can be represented in three ways:
jrerz
jyxz
rz
jrez
Rectangular form
Polar form
Exponential Form
Phasors
Relation between polar and rectangular form
Given x and y, we can get r and ϕ as
On the other hand, if we know r and ϕ we can obtain x and y as
Thus, z may be written as
22 yxr x
y1tan
cosrx sinry
)sin(cos jrrjyxz
Phasor Operations
Phasor operations
Addition of phasors
Subtraction of phasors
Multiplication of vectors
Division of complex quantities or vectors
Raising a vector to a given power
Extracting the roots of a vector
Logarithm of a vector
Phasor Operations
Addition and subtraction of complex numbers are
better performed in rectangular form; multiplication
and division are better done in polar form
Given the complex numbers
rjyxz
11111 rjyxz
22222 rjyxz
Cartesian or
Rectangular
Coordinate
system
Polar
Coordinate
system
Phasor Operations
Phasor Operations
Phasor Operations
Operator j
Euler’s identity
Real and Imaginary part of ejϕ
where Re and Im stand for the real part of and the
imaginary part of
jj
1
sincos je j
)Im(sin
)Re(cos
j
j
e
e
Phasor Operations
Given sinusoid
Thus,
Where,
V is thus the phasor representation of the sinusoid
jtj
m
tj
m
m
eeVtv
eVtv
tVtv
Re)(
Re)(
)cos()(
)(
tjetv VRe)(
m
j
m VeVV
Phasor Operations
Here, v(t)
Time-domain representation
Phasor-domain representation
)cos()( tVtv m
m
j
m VeVV
Phasor Operations
Sinusoid-phasor transformation
Phasor Operations
Difference between v(t) and V
1. v(t) is the instantaneous or time domain representation, while is V is the frequency or phasor domain representation.
2. v(t) is time dependent, while V is not. (This fact is
often forgotten by students.)
3. v(t) is always real with no complex term, while V is
generally complex.
Example – 9.3(a)
Evaluate the complex number
Solutions:
Example – 9.3(b)
(b) Problem:
Using polar-rectangular transformation, addition,
multiplication, and division,
Practice Problem – 9.3
Evaluate the following complex numbers:
Example – 9.4
Transform these sinusoids to phasors
Solutions
Practice Problem – 9.4
Express these sinusoids as phasors:
Example – 9.5
Find the sinusoids represented by these phasors:
Solutions
Practice Problem – 9.5
Find the sinusoids corresponding to these phasors:
Example – 9.6
Find the sum
Solution
Here is an important use of phasors—for summing
sinusoids of the same frequency. Current i1(t) is in the
standard form. Its phasor is
We need to express i2(t) in cosine form. The rule for
converting sine to cosine is to subtract 900. Hence,
Example – 9.6 Cont’d
Solution
and its phasor is
Transforming this to the time domain, we get
Practice Problem – 9.6
Find the sum
Example – 9.7
Example – 9.7
Solution: We transform each term in the equation from time
domain to phasor domain.
Practice Problem – 9.7
Find the sum
Example
Find the sum
Solution:
689.3610 0 jA
2.531206 0 jA
Example
Graphical illustration or Vector addition for the
example
Problem – Self Practice
Example – Subtraction
Subtract vector B from vector A
Solution:
Subtract vector A from vector B
06030A )160sin160(cos21 00 jB
Example - Multiplication
Multiply C=AB where
0402A01003B
00000 140)10040(10040 663.2 jjjj eeee ABC
000
00
14061004032
1003402
ABC
Example - Multiplication
It can be shown that
Also,
Again, if
Vector product
BAAB
CBAABC ABC
ACBBCAABC
aja A bjb B
Example - Multiplication
Magnitude of the resulting vector
Phase angle of F
Example - Division
Division example
Example - Division
Again, if
Similar way
aja A bjb B
Example - Division
Again, if
Reduced to polar form
3.1710 jA 5.233.4 jB
Raising Vector to a Given Power
A vector of phasor may be raised to a given power n,
where n is an integer
If
Then
Similarly,
AA A
A
nn nA A
BA
nnnn nnBA BA
Raising Vector to a Given Power
A vector of phasor may be raised to a given power n,
where n is an integer
If
Then
866.05.01201 0 ja
02 2401a003 013601 a
004 12014801 a
Extracting the Roots of a Vector
A vector of phasor may be raised to a given power n,
where n is an integer
If
Then
Cartesian form of above equation is,
AA A
n
qA Ann 2A
n
qj
n
qA AAnn 2
sin2
cosA
where, q=0, 1, 2, … ,n – 1
Extracting the Roots - Example
Find the square roots of A if
Solution: Transformed into polar form
445.808.3 jA
Extracting the Roots - Example
Find the square roots of A if
Vector diagram
445.808.3 jA
Vector
A=9∟700
and its two
roots
Extracting the Roots - Problem
Problem:
Logarithm of a Vector
The logarithm of a vector A is the inverse of the exponential of
A.
In other words, the logarithm of the vector A=Aejθ to the base e
is defined as the power to which e must be raised to equal Aejθ
By definition:
Here, θ is the phase angle of the vector A, must be considered
in radians !!!
jAejA
eAAe
eee
j
ee
j
ee
logloglog
loglogloglog A
jAeA A
Logarithm of a Vector
Example
jAeA A
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