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Chapter 1Complex Numbers and Phasors

Chapter Objectives:

Understand the concepts of sinusoids and phasors.

Apply phasors to circuit elements.

Introduce the concepts of impedance and admittance.

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Introduce the concepts of impedance and admittance.

Learn about impedance combinations.

Apply what is learnt to phase-shifters and AC bridges.

Complex Numbers

A complex number may be written in RECTANGULAR FORM as:

( )RECTANGULAR FORM

z = x+ jy

j= -1, x=Re z , y=Im(z)A second way of representing the complex number is by specifying the MAGNITUDE and r and the ANGLE θ in POLAR form.

z = x+ jy= z

POLAR FORM

=rθ θ∠ ∠

• x is the REAL part.

• y is the IMAGINARY part.

• r is the MAGNITUDE.

• φ is the ANGLE.

z = x+ jy= z =rθ θ∠ ∠ The third way of representing the complex number is the EXPONENTIAL form.

z = x+

EXPONENT

jy= z

IAL FORM

= jre θθ∠

Complex Numbers

A complex number may be written in RECTANGULAR FORM as: forms.

2 2 -1

z = x+ jy j= -1

cos y sin

z=

=tan

RECTANGULAR FORM

POLAR FORM

x r r

r

yr x y

x

θ θ

φ

θ

= =

∠

= +

j

2 2 -1

j

j

z= e EXPO

=tan

z = x + jy=

NE

= e

e =cos +j

NTIAL FORMr

yr x y

x

r r

φ

φ

φ

θ

φ

φ

= +

∠

j

j

sin

cos Re

Euler's Identity

Real part

Imaginary p

e

sin Im e art

φ

φ

φφ

φ

=

=

Complex Number Conversions

We need to convert COMPLEX numbers from one form to the other form.

z = (cos sin ) jx jy r re r jφφ φ φ= + = ∠ = +

2 2 1 Rectangula

z = (cos sin )

, ta r to Polar

Pola

n

cos , sin r to Rectangular

jx jy r re r j

yr x y

xx r y r

φφ φ φ

φ

φ φ

−

= + = ∠ = +

= + =

= =

Mathematical Operations of Complex Numbers

Mathematical operations on complex numbers may require conversions from one form to other form.

1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 1 2

1 1

z + z =(x + x )+j(y + y )

z - z =(x -x )+j(y - y )

z z = r r +

ADDITION:

SUBTRACTION:

MULTIPLICATION:

DIVISION:

z r = -

φ φ

φ φ

∠

∠1 11 2

2 2

DIVISION:

RECIPROCAL:

SQUARE ROOT:

z r = -z r

1 1 = - z r

z=

COMPLEX CO

r 2

NJUGATE:z

φ φ

φ

φ

∗

∠

∠

∠

= jx jy r re φφ −− = ∠ − =

PhasorsA phasor is a complex number that represents the amplitude and phase of a sinusoid.

Phasor is the mathematical equivalent of a sinusoid with time variable dropped.

Phasor representation is based on Euler’s identity.

j

j

j

e =cos jsin

co

Euler's Identity

Real part

Imaginary pa

s Re e

s rtin Im e

φ

φ

φ

φ φφ

φ

± ±

=

=

(TimeDomain Repr.) (Phasor Domain Representation)

( ) Re (Converting Phasor back to time)

( ) cos( )m m

j t

v t V t V

v t e ω

ω φ φ= + ⇔ =

=

∠V

V

Given a sinusoidv(t)=Vmcos(ωt+φ).

Imaginary pas rtin Im eφ =

( )( ) cos( ) Re( ) Re( ) Re( )

PHA S .OR REP

j t j t j tm m

jm m

jmv t V t V e e

V

e e

V

V

e

ω φ ω ω

φ

φω φφ

+= + = = =

= = ∠ =

V

V

Phasors

Given the sinusoids i(t)=Imcos(ωt+φI) and v(t)=Vmcos(ωt+ φV) we can obtain the phasor forms as:

Phasors Amplitude and phase difference are two principal

concerns in the study of voltage and current sinusoids.

Phasor will be defined from the cosine function in all our proceeding study. If a voltage or current expression is in the form of a sine, it will be changed to a cosine by subtracting from the phase.

• Example

• Transform the following sinusoids to phasors:

– i = 6cos(50t – 40o) A

– v = –4sin(30t + 50o) V

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Solution:

a. I A

b. Since –sin(A) = cos(A+90o);

v(t) = 4cos (30t+50o+90o) = 4cos(30t+140o) V

Transform to phasor => V V

°−∠= 406

°∠= 1404

Phasors

• Example 5:

• Transform the sinusoids corresponding to

phasors:

a)

b)

V 3010 °∠−=VA j12) j(5 −=I

Solution:

a) v(t) = 10cos(ωt + 210o) V

b) Since

i(t) = 13cos(ωt + 22.62o) A

°∠=∠+=+= − 22.62 13 )12

5( tan 512 j512 122I

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Phasor Diagrams

cos( )

sin(

Time

) 90

cos

Domain Representation Phasor Domain Re

( )

sin( ) 0

p.

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m m

m m

m m

m m

V t V

V t V

I t I

I t I

ω φ φω φ φω θ θω θ θ

+ ∠+ ∠ − °+ ∠+ ∠ − °

Time Domain Versus Phasor Domain

Differentiation and Integration in Phasor Domain Differentiating a sinusoid is equivalent to multiplying its corresponding phasor by jω.

( ) cos( ) Re

( )sin( ) cos( 90 )

= Re

j tm

m m

j t

v t V t e

dv t

d

V t V tdt

ev

j Jdt

ω

ω

ω θ

ω ω θ ω ω

ω ω

θ

= + =

= − + = − + + °

⇔ V V

V

Integrating a sinusoid is equivalent to dividing its corresponding phasor by jω.

(Time Domain) (Phasor Domain)

( ) cos( )

( ) sin( ) 90

V

m m

m m

v t V t V

v t V t V

dvJ

dt

vdtJ

ω φ φω φ φ

ω

ω

= + ⇔ = ∠= + ⇔ = ∠ − °

⇔

⇔∫

V

V

V

Adding Phasors Graphically

Adding sinusoids of the same frequency is equivalent to adding their corresponding phasors.

V=V1+V2

20cos(5 30 ) At − ° 1

5Ω 2 F

1H

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