ECE 1311 Ch9

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ECE 1311

Chapter 9 Sinusoids and Phasors

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Outlines

Sinusoids

Phasors

Phasor relationships for circuit elements

Impedance and admittance

Kirchoffslaws in the frequency domain

Impedance and combinations

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Introduction

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Thus far our analysis only concentrates on dc circuits.

Now we begin the analysis in which the source is time

varying (ac circuits).

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Sinusoids

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A sinusoid is a signal that has the form of the sine andcosine function

Vm= the amplitude of the sinusoid

= the angular frequency of the sinusoid

t= the argument of the sinusoid

tVtv m sin)( Sinusoidal voltage

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Sinusoids

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Period of the sinusoid (T):

Cyclic frequency (f):

And:

2T Measured in second

2

1

Tf

f 2

Measured in herts (Hz)

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General Expression - Sinusoid

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Expressed insine

form:

)sin()( tVtv m

t Argument of the sinusoid phase

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Example 1

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Given the sinusoid, calculate its amplitude, phase,

,period and frequency.

)604sin(5 t

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Sinusoids

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If two sinusoids are given:

v2starts first in time.

v2leads v1by .

0.

v2and v1are out of

phase.

)sin()(2 tVtv m)sin()(1 tVtv m

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Sinusoids

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A sinusoid can be expressed in either sine or cosineform.

When comparing two sinusoids, it is easier if both are in

sine or both in cosine forms with positive amplitudes.

This can be achieved by using the Trigonometric

identities:

BABABA

BABABA

sinsincoscos)cos(

sincoscossin)sin(

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Trigonometric Identities

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It is easy to show using the trigonometric identities that:

tt

tt

tt

tt

sin)90cos(

cos)90sin(

cos)180cos(

sin)180sin(

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Graphical Approach for Sinusoids

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Alternative approach to trigonometric identities.

Eliminates memorization.

Do not confuse the sine and cosine axes with the axes for complex numberstobe discuss later in this chapter.

.

)90sin(cos tt )180sin(sin tt

Angle:

clockwise+ve counterclockwise-ve


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Example 2

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Calculate the phase angle between v1and v2.

State which sinusoid is leading.

)50cos(101 tv )10sin(122 tv

Note: when comparing two sinusoids, express them in the same form.

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Graphical Approach for Sinusoids

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The approach can be used to add two sinusoids of the samefrequency when one is in sine form and the other in cosineform.

For example:

Where:

.

22 BAC

A

B1tan

)cos(sincos tCtBtA


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Example 3

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Add the two sinusoids

Hence:

Answer:

tt sin4cos3

543 22 C

1.5334tan 1

)13.53cos(5 t


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Phasors

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Sinusoids are easily expressed in terms of phasors.

A phasor is a complex number that represents the

amplitudeand phaseof a sinusoid,

Phasors are written in boldface.

Before completely define and apply phasors to circuit

analysis, we have to be familiar with complex numbers.

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Complex Number

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A complex number can be written in rectangular form as:

Where:

x is the real part ofz.

yis the imaginary part ofz.

Also can be written in:

Polar form

Exponential form

jyxz

11

2

jandj

rzjrez

Graphical representation of a complex

numberz.


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Complex Number Conversions

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Rectangular form to polar form conversion:

Polar form to rectangular form conversion:

Hence:

rztojyxz

22 yxr

x

y1tan

jyxztorz

sincos ryandrx

sincos jrrjyxz

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Example 4

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Convert the following complex numbers into (a) polarform (b) rectangular form.

ob

ja

3010.

)25.(

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Operations on Complex Numbers

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Example 5

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Evaluate the following complex numbers.

5301043403510.

*605)41)(25(.

jj

jb

jja

oo

o

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Time Domain to Phasor Domain

Transformation (SinusoidPhasor)

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)cos()( tVtv m

mVV

Time domain

representation of a sinusoid

Phasor domainrepresentation of a sinusoid

Note: A sinusoid should be expressed in a cosine form before it can be

transformed into a phasor form.

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A Phasor Diagram

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If two sinusoids are given: and

Then, the phasor diagram is shown below:

mVV mII


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Time Domain to Phasor Domain

Transformation (SinusoidPhasor)

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Example 6

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Express these sinusoids as phasors.

)1010sin(4.

)402cos(7.

o

o

tib

tva

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Example 7

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Find the sinusoids corresponding to these phasors.

)125(.

3010.

jjIb

Va o

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Example 8

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If v1= -10sin(t-30o) V and v

2= 20cos(t+45o) V, find

v=v1+v2.

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Phasors Differentiation and Integration

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Example 9

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Find the voltage v(t)in a circuit described by theintegrodifferential equation using phasor approach.

)305cos(501052 0 tvdtvdt

dv

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Phasor Relationships for Circuit Elements

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Example 10

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If voltage v=10cos(100t+30)is applied to a 50F capacitor,

calculate the current through the capacitor.

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Impedance (Z)

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It is the ratio of the phasor voltage Vto the phasor

current I.

It is measured in ohms ().

Where:

R = Real Z (resistance)

X = Imaginary Z (reactance)

jXRIVZ

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Admittance (Y)

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It is the reciprocal of impedance.

It is measured in siemens (S).

V

I

ZY 1

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Impedance and Admittance - summary

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Equivalent Circuits at Dc and High

Frequencies

34The Mc-Graw-Hill Companies

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Example 11

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Determine v(t)and i(t)for the following circuit.


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Impedance Combinations

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KCL and KVL both hold true in the frequency (phasor)

domain.

Series Configurations:

ImpedanceZeq=Z1+Z2

Admittance

Voltage division rule:

21

111

YYYeq

VZZ

ZVV

ZZ

ZV

21

22

21

11


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Parallel Configurations:

Impedance

AdmittanceYeq=Y1+Y2

Current division rule:

21

111

ZZZeq

IZZ

ZI

IZZ

ZI

21

12

21

21


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Example 12

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Determine the input impedance of the circuit at=10rad/s.


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Example 13

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Determine v0.


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Delta to Y transformation:

Y to delta transformation:

cba

ba

cba

ca

cba

cb

ZZZ

ZZZ

ZZZ

ZZZ

ZZZ

ZZZ

3

21

3

133221

2

133221

1

133221

Z

ZZZZZZZ

Z

ZZZZZZZ

ZZZZZZZZ

cb

a


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Example 14

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Determine I.

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ECE 1311 Ch9