COMPLEX NUMBERS and PHASORS Lecture 02 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.

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

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/ = 0.1257 s The frequency is, f = 1/T = 7.958 Hz

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)

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

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.

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.

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, 

DC offsets are NOT represented. Ex. i(t) = 3A sin (2pft+30o) 3A<30o v(t) = 4V sin (q-60o) 4V<-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.

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

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.

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

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?

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

2. COMPLEX NUMBER SYSTEM COMPLEX PLANE:

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: 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

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

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

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

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

Examples: Evaluate these complex numbers:

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

Rectangular to Polar conversion: Taking square root of this;

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