Download Introduction II

COMPLEX NUMBERS and
PHASORS
OBJECTIVES




Use a phasor to represent a sine wave.
Illustrate phase relationships of waveforms using phasors.
Explain what is meant by a complex number.
Write complex numbers in rectangular or polar form, and
convert between the two.
 Perform addition, subtraction, multiplication and division
using complex numbers.
 Convert between the phasor form and the time domain form
of a sinusoid.
 Explain lead and lag relationships with phasors and
sinusoids.
Ex.
• For the sinusoid given below, find:
v(t )  12 cos50t  10
a) The amplitude
b) The phase angle
c) The period, and
d) The frequency
Ex.
• For the sinusoid given below, calculate:
i(t )  5 sin 4t  60
a) The amplitude (Vm)
b) The phase angle ()
c) Angular frequency ()
d) The period (T), and
e) The frequency (f)
PHASORS
INTRODUCTION TO PHASORS
• PHASOR:
– a vector quantity with:
• Magnitude (Z): the length of vector.
• Angle () : measured from (0o)
horizontal.
• Written form:
Z
Ex: A< 
90
A

180
0
270
PHASORS & SINE WAVES
• If we were to rotate a phasor and plot the
vertical component, it would graph a sine
wave.
• The frequency of the sine wave is
proportional to the angular velocity at
which the phasor is rotated.
(  =2f)
PHASORS & SINE WAVES
• One revolution of the phasor ,through 360°,
= 1 cycle of a sinusoid.
Z
90
Z

180
270
0
d

dt
t
INSTANTANEOUS VALUES
• Thus, the vertical distance from the end of a rotating
phasor represents the instantaneous value of a sine
wave at any time, t.
vinst  Z sin(t   )
90
Z
Z

180
270
Vinst
0
t
USE OF PHASORS in EE
• Phasors are used to compare phase
differences
• The magnitude of the phasor is the
Amplitude (peak)
• The angle measurement used is the PHASE
ANGLE, 
Ex.
1. i(t) = 3A sin (2ft+30o)
3A<30o
2. v(t) = 4V sin (-60o)
4V<-60o
3. p(t) = 1A +5A sin (t-150o)
5A<-150o
DC offsets are NOT represented.
Frequency and time are NOT
represented unless the phasor’s  is
specified.
GRAPHING PHASORS
• Positive phase angles are drawn
counterclockwise from the axis;
• Negative phase angles are drawn
clockwise from the axis.
GRAPHING PHASORS
90
A
3A
30
0
180
5A -150
4V
C
B
270
-60
Note:
A leads B
B leads C
C lags A
etc
PHASOR DIAGRAM
• Represents one or more sine waves (of the
same frequency) and the relationship between
them.
• The arrows A and B rotate together. A leads B
or B lags A.
90
A

0

180
B
270
Ex:
– Write the phasors for A and B, if wave A is the reference
wave.
4V
t = 5ms per division
B
A
-4 V
  57.6
A  40V
B  2.557.6V
Ex.
1. What is the instantaneous voltage at t = 3 s,
if: Vp = 10V, f = 50 kHz, =0o
(t measured from the “+” going zero crossing)
2. What is your phasor?
COMPLEX NUMBERS
COMPLEX NUMBER SYSTEM
• COMPLEX PLANE:
j
90
-Re
180
Re
X-Axis
X-Axis
270
-j
0
FORMS of COMPLEX NUMBERS
• Complex numbers contain real and imaginary (“j”)
components.
– imaginary component is a real number that has been
rotated by 90o using the “j” operator.
• Express in:
– Rectangular coordinates (Re, Im)
– Polar (A<) coordinates - like phasors
COORDINATE SYSTEMS
– RECTANGULAR:
j
Y-Axis
– addition of the real and
imaginary parts:
– VR = A + j B
Z
B

– POLAR:
-Re
X-Axis
Re
X-Axis
A
Y-Axis
– contains a magnitude and
an angle:
– V P = Z<
– like a phasor!
-j
CONVERTING BETWEEN FORMS
• Rectangular to Polar:
j
V R = A + j B to V P = Z<
Y-Axis
Z  A B
2
2
-Re
Z
B

X-Axis
Re
X-Axis
A
Y-Axis
1  B 
  tan  
 A
-j
POLAR to RECTANGULAR
• V P = Z< to V R = A + j B
j
Y-Axis
A  Z cos
Z
B

-Re
X-Axis
Re
X-Axis
A
Y-Axis
B  Z sin 
-j
MATH OPERATIONS
• ADDITION/ SUBTRACTION - use Rectangular
form
 add real parts to each other, add imaginary parts to
each other;
 subtract real parts from each other, subtract imaginary
parts from each other
• ex:
 (4+j5) + (4-j6) = 8-j1
 (4+j5) - (4-j6) = 0+j11 = j11
• OR use calculator to add/subtract phasors directly
• MULTIPLICATION/ DIVISION - use Polar form
• Multiplication: multiply magnitudes, add angles;
4302  20  4.250  (20)  830
• Division: divide magnitudes, subtract angles
450
4
 50  (20)   270
2  20 2
Ex.
• Evaluate these complex numbers:
a) 4050  20  30
1/2
10  30  3  j 4
b)
2  j 43  j5 