ct lecture 2

7/31/2019 CT Lecture 2

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Complex Numbers and Phasors

Chapter Objectives:

Understand the concepts of sinusoids and phasors. Apply phasors to circuit elements.

Introduce the concepts of impedance and admittance.

Learn about impedance combinations. Apply what is learnt to phase-shifters and AC

bridges.


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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)

x is the REAL part.

y is the IMAGINARY part.

ris the MAGNITUDE.

is the ANGLE.

A second way of representing the complex number is by specifying the

MAGNITUDE and rand the ANGLE in POLAR form.

z = x+ jy= zPOLAR FORM

=r

The third way of representing the complex number is the EXPONENTIAL form.

z = x+

EXPONENT

jy= z

IAL FORM

= jre


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Complex Numbers

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

2 2 -1

j

2 2 -1

j

j

z = x+ jy j= -1

cos y sin

z=

=tan

z= e

RECTANGULAR FORM

POLAR FORM

EXPO

=tan

z = x + jy=

NE

= e

e =cos +j

NTIAL FORM

x r r

r

yr x y

x

r

yr x y

x

r r

j

j

sin

cos Re

Euler's Identity

Real part

Imaginary p

e

sin Im e art


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Complex Number Conversions

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

z = (cos sin )jx jy r re r j

2 2 1 Rectangular to Polar

Polar to Rectangu

z = (cos sin )

, tan

cos , sin lar

jx jy r re r j

yr x y

x

x r y r


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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 11 2

2 2

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

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

z z = r r +

ADDITION:

SUBTRACTION:

MULTIPLICATION:

DIVISION:

RECIPROCAL:

SQUARE ROOT:

z r= -

z r

1 1

= -z r

z=

COMPLEX CO

r2

NJUGATE: z

rj

x jy re


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Phasors

(Time Domain Re pr.) (Phasor Domain Re presentation)

( ) Re{ } (Converting Phasor back to time)

( ) cos( )m m

j t

v t V t V

v t e

V

V

A 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 Eulers identity.

Given a sinusoidv(t)=Vmcos(t+).

j

j

j

e =cos jsin

co

Euler's Identity

Real part

Imaginary pa

s Re e

s rtin Im e

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

PHA S .OR REP

j t j t j t

m m

j

m m

j

mv t V t V e e

V

e e

V

V

e

V

V


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Phasors

Given the sinusoids i(t)=Imcos(t+I) and v(t)=Vmcos(t+ V) we can obtain the

phasor forms as:


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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 ourproceeding study. If a voltage or current expression is inthe form of a sine, it will be changed to a cosine bysubtracting from the phase.

Example

Transform the following sinusoids to phasors:

i = 6cos(50t40o) A

v =4sin(30t + 50o) V

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


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Phasor as Rotating Vectors

( )

( ) cos( )

( ) Re

( ) Re ( )

Rotating Phasor

m

j t

m

m

v t V t

v t V e

v t V j t


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Phasor Diagrams

The SINOR

Rotates on a circle of radius Vm at an angular velocity of in the counterclockwise

direction

j te

V

Ph D


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Phasor Diagrams

cos( )

sin(

Time

) 90

cos

Domain Representation Phasor Domain Re

( )

sin( ) 0

p.

9

m m

m m

m m

m m

V t V

V t V

I t I

I t I

Ph


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Time Domain Versus Phasor Domain


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Differentiation and Integration in Phasor Domain

(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

V

V

V

Differentiating a sinusoid is equivalent to multiplying its corresponding phasor by j.

( ) cos( ) Re

( )sin( ) cos( 90 )

= Re

j t

m

m m

j t

v t V t e

dv t

d

V t V t dt

ev

j Jdt

V V

V

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


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20cos(5 30 ) At 15

2 F1

H10

S l i AC Ci it


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We can derive the differential equations for the followingcircuit in order to solve for vo(t) in phase domain Vo.

20

02

5 40020 sin(4 15 )

3 3

ood v dv v tdt dt

However, the derivation may sometimes be very tedious.

Is there any quicker and more systematic methods to do it?

Instead of first deriving the differential equation and then

transforming it into phasor to solve for Vo, we can transform all the

RLC componentsinto phasor first, then apply the KCL laws and other

theorems to set up a phasor equation involving Vo directly.

Solving AC Circuits

ct lecture 2

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