Complex Numbers,Sinusoidal Sources & Phasors

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Elements of Electrical Engineering

Dr. Ron Hayne

Images Courtesy of Allan Hambley and Prentice-Hall

Complex Numbers

Complex numbers involve the imaginary number EE’s use j instead of i because i is used for

current

A complex number Z = x+jy Has a real part x Has an imaginary part y Can be represented by a point in the complex

plane

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j 1

Basic Concepts

Pure imaginary number has real part zero Pure real number has imaginary part zero Complex numbers of the form x+jy are in

rectangular form Complex conjugate of a number in

rectangular form is obtained by changing the sign of the imaginary part ex. Complex conjugate of z3 = 3-j4 is z3

* = 3+j4

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Example A.1

Complex Arithmetic in Rectangular Form Given that z1 = 5+j5 and z2 = 3-j4, reduce the

following to rectangular form:z1+z2

z1-z2

z1z2

z1/z2

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Polar Form

Complex number z can be expressed in polar form Give length of vector that represents z

Denoted as |z|Called the magnitude of the complex number z

Give angle of vector that represents zangle between vector and positive real axisUsually represented by θ

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Polar-Rectangular Conversion

Use trigonometry and right triangles:

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z2 x 2 y 2

tan y

xx z cos y z sin

Example A.2

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Convert z3 530o to rectangular form.

Example A.3

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form.polar to510Convert 6 jz

Euler’s Identity

What do complex numbers have to do with sinusoids? Euler’s identity:

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sincos je j

Exponential Form

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number.complex a of theis This

as written becan number complex Any

sincos1

Therefore

1sincos= sincos

is of magnitude The

22

form lexponentia

j

j

j

j

AeA

A

je

je

e

Example A.4

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Express the complex number z 1060o

in exponential and rectangular forms.

Sketch the number in the complex plane.

Arithmetic Operations

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Consider two complex numbers:

z1 z11 z1 e j1 and z2 z2 2 z2 e j 2

Multiplication is easy in exponential or polar form:

z1z2 z1 z2 1 2 z1 z2 e j 1 2

Division is easy in exponential or polar form:

z1

z2

z1

z2

1 2 z1

z2

e j 1 2

Example A.5

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formpolar in + and ,/ , find

,455 and 6010Given

212121

21

zzzzzz

zz

Sinusoidal Voltage

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v t Vm cos t

Sinusoidal Signals

Same pattern of values repeat over a duration T, called the period Sinusoidal signals complete one cycle when the

angle increases by 2π radians, or ωT = 2π Frequency is number of cycles completed in one

second, or f = T-1

Units are hertz (Hz) or inverse seconds (sec-1) Angular frequency given by ω = 2πf = 2πT-1

Units are radians per second

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Sinusoidal Signals

Argument of cosine or sine is ωt+θ To evaluate cos(ωt+θ)

May have to convert degrees to radians, or vice versa

Relationship between cosine and sine

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sin z cos z 90o

Root-Mean-Square (RMS)

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Consider applying a periodic voltage v t with period T to a resistance R.

Power delivered to the resistance is given by

p t v 2 t R

The energy delivered in one period is given by

ET p t dt0

TThe average power delivered to the resistance is given by

Pavg ET

T1

Tp t dt

0

T 1T

v 2 t R

dt0

T

1

Tv 2 t dt

0

T

2

R

Root-Mean-Square (RMS)

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The root - mean - square (rms) or effective value

of the periodic voltage v t is defined as

Vrms 1

Tv 2 t dt

0

T

Therefore, Pavg Vrms2

RThe root - mean - square (rms) or effective value

of a periodic current i t is defined as

Irms 1

Ti2 t dt

0

TTherefore, Pavg Irms

2 R

RMS Value of a Sinusoid

Important Note: THIS ONLY APPLIES TO SINUSOIDS!!!

What is the peak voltage for the AC signal distributed in residential wiring in the United States?

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Consider a sinusoidal voltage given by

v t Vm cos t The RMS value for this sinusoidal voltage is given by

Vrms 1

TVm

2 cos2 t dt0

T Vm

2

Example 5.1

Suppose that a voltage given byis applied to a 50-Ω resistance. Sketch v(t) to scale versus time. Find the RMS value of the voltage. Find the average power delivered to the resistance.

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v t 100cos 100t

Example 5.1

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Exercise 5.3

Suppose that the AC line voltage powering a computer has an RMS value of 110 V and a frequency of 60 Hz, and the peak voltage is attained at t = 5 ms.

Write an expression for this AC voltage as a function of time.

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Phasors

Sinusoidal steady-state analysis Generally complicated if evaluating as time-

domain functions Facilitated if we represent voltages and currents

as vectors in the complex-number planeThese vectors are also called PHASORS

Convenient methods for adding and subtracting sinusoidal waveforms (for KCL and KVL)Standard trig. techniques too tedious

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Voltage Phasors

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For a sinusoidal voltage

v1 t V1 cos t 1 ,The phasor is defined to be

V1 V11

For a sinusoidal voltage

v2 t V2 sin t 2 ,The phasor is defined to be

V2 V2 2 90o because

sin z cos z 90o .

Current Phasors

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For a sinusoidal current

i1 t I1 cos t 1 ,The phasor is defined to be

I1 I11

For a sinusoidal current

i2 t I2 sin t 2 ,The phasor is defined to be

I2 I2 2 90o

Adding Sinusoids

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term.single a to reduce

,60sin10

and 45cos20Given

11

2

1

tvtvtv

ttv

ttv

s

Exercise 5.4

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30sin530cos10

:phasors usingby expression following theReduce

1 ttti

Phasors as Rotating Vectors

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tV

eV

tVtv

m

tjm

m

Re

Re

cos

as written becan

voltagesinusoidalA

Phase Relationships

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Consider the voltages

v1 t 3cos t 40o V1 340o

and

v2 t 4cos t 20o V2 4 20o

The angle between V1 and V2 is 60o.

Because the complex vectors rotate counterclockwise,

we say that V1 leads V2 by 60o, or V2 lags V1 by 60o.

Phase Relationships

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Exercise 5.5

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45sin10

30cos10

30cos10

:below voltagesofpair each

between iprelationsh phase theState

3

2

1

ttv

ttv

ttv

Summary

Complex Numbers Rectangular Polar Exponential

Sinusoidal Sources Period Frequency Phase Angle RMS Phasors

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