Phasors/Complex Numbers in AC
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Transcript Phasors/Complex Numbers in AC
Lesson 18
Phasors & Complex Numbers in
AC
Learning Objectives
Define and graph complex numbers in rectangular and polar form.
Perform addition, subtraction, multiplication and division using
complex numbers and illustrate them using graphical methods.
Define a phasor and use phasors to represent sinusoidal voltages
and currents.
Define time domain and phasor (frequency) domain
Represent a sinusoidal voltage or current as a complex number in
polar and rectangular form.
Use the phasor domain to add/subtract AC voltages and currents.
Determine when a sinusoidal waveform leads or lags another.
Graph a phasor diagram that illustrates phase relationships.
Complex numbers
A complex number is a number of the form C =
a + jb where a and b are real and j = 1
a is the real part of C and b is the imaginary
part.
Complex numbers are merely an invention
designed to allow us to talk about the quantity j.
j is used in EE to represent the imaginary
component to avoid confusion with CURRENT (i)
Geometric Representation
C = 6 + j8
(rectangular form)
C = 1053.13º
(polar form)
Conversion Between Forms
To convert between forms where
C a jb
C C
(rectangular form)
(polar form)
apply the following relations
a C cos
b C sin
C a 2 b2
1 b
tan
a
Example Problem 1
Convert (5∠60) to rectangular form.
Convert 6 + j 7 to polar form.
Convert -4 + j 4 to polar form.
Convert (5∠220) to rectangular form.
Properties of j
j 1
j ( 1)( 1) 1
2
1 1
j j
j
j
2 j
j j
Addition and Subtraction of Complex Numbers
Easiest to perform in rectangular form
Add/subtract real and imaginary parts separately
(6 j12) (7 j 2) = (6 7) j (12 2) = 13 j14
(6 j12) (7 j 2) = (6 7) j (12 2) = 1 j10
Multiplication and Division of Complex Numbers
Easiest to perform in polar form
Multiplication: multiply magnitudes and add the
angles
(670) (230) 6 2(70 30) 12100
Division: Divide the magnitudes and subtract the
angles
(670) 6
(70 30) 340
(230) 2
Example Problem 2
Given A =1 +j1 and B =2 – j3
Determine A+B and A-B.
Given A =1.4145° and B =3.61-56°
Determine A/B and A*B.
Reciprocals and Conjugates
The reciprocal of C = C , is
1
1
C C
The conjugate of C is denoted C*,
which has the same real value but
the opposite imaginary part:
C a jb C
C a jb C
Example Problem 3
And now you can try with your TI!!
(3-i4) + (10∠44)
(22000+i13)/(3∠-17)
Convert 95-12j to polar:
ANS: 10.6∠16.1
ANS: 10.2 + 2.9i
ANS: 7.3E3∠17.0
ANS: 95.8∠-7.2
Phasor Transform
To solve problems that involve sinusoids
(such as AC voltages and currents) we
use the phasor transform.
We transform sinusoids into complex
numbers in polar form, solve the problem
using complex arithmetic (as described),
and then transform the result back to a
sinusoid.
THE SINUSOIDAL WAVEFORM
Generating a sinusoidal waveform through the vertical projection of a
rotating vector.
Phasors
A phasor is a rotating vector whose projection
on the vertical axis can be used to represent a
sinusoid.
The length of the phasor is amplitude of the
sinusoid (Vm)
The angular velocity of the phasor is
Representing AC Signals with Complex
Numbers
By replacing e(t) with it’s phasor equivalent
E, we have transformed the source from
the time domain to the phasor domain.
Phasors allow us to convert from
differential equations to simple algebra.
KVL and KCL still work in phasor domain.
Using phasors to represent AC voltage and current
Looking at the sinusoid eqn, determine VPk and phase offset
.
v(t ) VPK sin(t 30 )
Using VPK, determine VRMS using the formula:
“The equivalent dc value of a sinusoidal current or voltage is
0.707 of its peak value”
V
VRMS
The phasor is then
V
PK
2
RMS
Representing AC Signals with Complex
Numbers
Phasor representations can be viewed as
a complex number in polar form.
e(t ) 2Em sin(t )
E = Erms
Example Problem 4
i1 = 20 sin (t) mA.
i2 = 10 sin (t+90˚) mA.
i3 = 30 sin (t - 90˚) mA.
Determine the equation for iT.
Phase Difference
Phase difference is angular
displacement between waveforms of
same frequency.
If angular displacement is 0° then
waveforms are in phase
If angular displacement is not 0o, they
are out of phase by amount of
displacement
Phase Difference
If v1 = 5 sin(100t) and v2 = 3 sin(100t - 30°), v1 leads v2 by
30°
Phase Difference w/ Phasors
The waveform generated by the leading
phasor leads the waveform generated by
the lagging phasor.
Formulas from Trigonometry
Sometimes signals are expressed in
cosines instead of sines.
cos(t ) sin(t 90 )
sin(t ) cos(t 90 )
cos(t 180 ) cos(t )
sin(t 180 ) sin(t )
cos(t 70 ) sin(t 160 ) sin(t 20 )
Example Problem 5
Draw the phasor diagram, determine phase
relationship, and sketch the waveform for the
following:
i = 40 sin(t + 80º) and v = -30 sin(t - 70º)