1. COMPLEX NUMBERS and PHASORS
Lecture 02
COMPLEX NUMBERS and PHASORS

2. 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.

3. For the sinusoid given below, find:
Ex.
For the sinusoid given below, find:
The amplitude
The phase angle
The period, and
The frequency

4. Solution Compare with the general sinusoid equation: Thus, we get:
The amplitude is Vm= 12 V
The phase angle is,  = 10
The angular frequency is,  = 50 rad/s
The period is, T = 2/ = s
The frequency is, f = 1/T = Hz

5. For the sinusoid given below, calculate:
Ex.
For the sinusoid given below, calculate:
The amplitude (Vm)
The phase angle ()
Angular frequency ()
The period (T), and
The frequency (f)

6. 1.INTRODUCTION TO PHASORS
a vector quantity with:
Magnitude (Z): the length of vector.
Angle () : measured from (0o) horizontal.
Written form:

7. 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.
( w =2pf)
One revolution of the phasor ,through 360°,
= 1 cycle of a sinusoid.

8. 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.

9. USE OF PHASORS Phasors are used to compare phase differences
The magnitude of the phasor is the Amplitude (peak)
The angle measurement used is the PHASE ANGLE, 

10. DC offsets are NOT represented.
Ex.
i(t) = 3A sin (2pft+30o) A<30o
v(t) = 4V sin (q-60o) V<-60o
p(t) = 1A +5A sin (wt-150o) 5A<-150o
DC offsets are NOT represented.
Frequency and time are NOT represented unless the phasor’s w is specified.

11. GRAPHING PHASORS Note: A leads B B leads C C lags A etc
Positive phase angles are drawn counterclockwise from the axis;
Negative phase angles are drawn clockwise from the axis.
Note:
A leads B
B leads C
C lags A
etc

12. 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.

13. Example: t = 5ms per division
Write the phasors for A and B, if wave A is the reference wave.
t = 5ms per division

14. Example:
What is the instantaneous voltage at t = 3 s, if: Vp = 10V, f = 50 kHz, =0o
(t measured from the “+” going zero crossing)
What is your phasor?

15. Solution 1. General sine wave equation:
Substitute all the values given,
At t=3μs,
2. The sine wave equation obtained:
In phasor form,

16. 2. COMPLEX NUMBER SYSTEM
COMPLEX PLANE:

17. 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

18. COORDINATE SYSTEMS RECTANGULAR: POLAR:
addition of the real and imaginary parts:
V R = A + j B
POLAR:
contains a magnitude and an angle:
V P = Z<
like a phasor!
Y-Axis
X-Axis B A q Z j -j Re
-Re

19. CONVERTING BETWEEN FORMS
Rectangular to Polar:
V R = A + j B to V P = Z<
Y-Axis
X-Axis B A q Z j -j Re
-Re

20. POLAR to RECTANGULAR V P = Z< to V R = A + j B j Z B q Re -Re A -j
Y-Axis
X-Axis B A q Z j -j Re
-Re

21. MATH OPERATIONS ADDITION/ SUBTRACTION - use Rectangular form ex:
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

22. MULTIPLICATION/ DIVISION - use Polar form
Multiplication: multiply magnitudes, add angles;
Division: divide magnitudes, subtract angles

23. Examples:
Evaluate these complex numbers:

24. Solution (a)
Polar to Rectangular conversion:
Adding them up gives:

25. Rectangular to Polar conversion:
Taking square root of this;

26. Solution (b) Polar to Rectangular Rectangular to Polar
Conjugate: + to -
The final answer is;