Course: EE 1103 Subject: Basic Electrical Engineering

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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