Download Introduction - Electrical and Computer Engineering

Complex Numbers,
Sinusoidal Sources & Phasors
ELEC 308
Elements of Electrical Engineering
Dr. Ron Hayne
Images Courtesy of Allan Hambley and Prentice-Hall
Complex Numbers
 Complex numbers involve the imaginary
number j  1

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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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
 z1 z2
 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 z
angle between vector and positive real axis
 Usually represented by θ

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Polar-Rectangular Conversion
 Use trigonometry and right triangles:
2
z x y
2
2
y
tan  
x
x  z cos 
y  z sin  
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Example A.2
Convert z3  530 to rectangular form.
o
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Example A.3
Convert z6  10  j5 to polar form.
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Euler’s Identity
 What do complex numbers have to do with sinusoids?

Euler’s identity:
e j  cos   j sin  
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Exponential Form
The magnitude of e j is
e
j
 cos   j sin   = cos    sin    1
2
2
Therefore
e j  1  cos   j sin  
Any complex number A can be written as
A  Ae j
This is the exponentia l form of a complex number.
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Example A.4
Express the complex number z  1060
in exponential and rectangular forms.
Sketch the number in the complex plane.
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Arithmetic Operations
Consider two complex numbers :
z1  z1 1  z1 e
j 1
and z 2  z 2  2  z 2 e
j 2
Multiplication is easy in exponential or polar form
z1z 2  z1 z 2 1   2   z1 z 2 e
:
j  1  2 
Division is easy in exponential or polar form
:
z1 j  1  2 
z1 z1

1   2  
e
z2 z2
z2
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Example A.5
Given z1  1060 and z2  545 ,


find z1 z2 , z1/z2 , and z1 + z2 in polar form
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Sinusoidal Voltage
v t   Vm cost   
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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
sin z  cosz  90
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Root-Mean-Square (RMS)
Consider applying a periodic voltage
v t 
with period T to a resistance R.
Power delivered to the resistance is given by
v 2 t 
pt  
R
The energy delivered in one period is given by
ET 
 pt dt
T
0
The average power delivered to the resistance is given by
Pavg
E
1
 T 
T T

T
0
1
pt dt 
T

T
0
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v 2 t 
dt 
R
 1

 T
2

T 2
 0 v t dt 

R
17
Root-Mean-Square (RMS)
The root - mean - square (rms) or effective value
of the periodic voltage v t  is defined as
1
Vrms 
T

T
0
v 2 t dt
2
Vrms
Therefore, Pavg 
R
The root - mean - square (rms) or effective value
of a periodic current it  is defined as
1 T 2
i t dt
Irms 

0
T
2
R
Therefore, Pavg  Irms
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RMS Value of a Sinusoid
Consider a sinusoidal voltage given by
v t   Vm cost   
The RMS value for this sinusoidal voltage is given by
1
Vrms 
T
Vm
 0 V cos t   dt  2
T
2
m
2
 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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Example 5.1
 Suppose that a voltage given by v t   100cos100t 
is 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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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 timedomain functions
Facilitated if we represent voltages and currents
as vectors in the complex-number plane


These 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
For a sinusoidal voltage
v1 t   V1 cost  1,
The phasor is defined to be
V1  V11
For a sinusoidal voltage
v 2 t   V2 sin t   2 ,
The phasor is defined to be
V2  V2 2  90 o
because
sin z  cosz  90 o.
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Current Phasors
For a sinusoidal current
i1t   I1 cost  1 ,
The phasor is defined to be
I1  I11
For a sinusoidal current
i2 t   I2 sin t   2 ,
The phasor is defined to be
I2  I2 2  90
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Adding Sinusoids


Given v1 t   20 cos t  45 and



v2 t   10 sin t  60 ,

reduce vs t   v1 t   v1 t  to a single term.
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Exercise 5.4
Reduce the following expression by using phasors :
i1 t   10 cost  30   5 sin t  30 
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Phasors as Rotating Vectors
A sinusoidal voltage
can be written as
vt   Vm cost   

 Re Vm e j t  

 Re Vm t   
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Phase Relationships
Consider the voltages
v1 t   3cost  40 o V1  340 o
and
v 2 t   4 cost  20 o V2  4  20 o
The angle between V1 and V2 is 60 o.
Because the complex vectors rotate counterclockwise,
we say that V1 leads V2 by 60 o, or V2 lags V1 by 60 o.
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Phase Relationships
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Exercise 5.5
State the phase relationsh ip between
each pair of voltages below :


v t   10 cost  30 
v t   10 sin t  45 
v1 t   10 cos t  30

2

3
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Summary
 Complex Numbers



Rectangular
Polar
Exponential
 Sinusoidal Sources





Period
Frequency
Phase Angle
RMS
Phasors
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