Lecture 02 COMPLEX NUMBERS and PHASORS OBJECTIVES A

Lecture 02 COMPLEX NUMBERS and PHASORS

OBJECTIVES A. Use a phasor to represent a sine wave. B. Illustrate phase relationships of waveforms using phasors. C. Explain what is meant by a complex number. D. Write complex numbers in rectangular or polar form, and convert between the two. E. Perform addition, subtraction, multiplication and division using complex numbers. F. Convert between the phasor form and the time domain form of a sinusoid. G. Explain lead and lag relationships with phasors and sinusoids.

Ex. For the sinusoid given below, find: a) The b) The c) The d) The amplitude phase angle period, and 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

Ex. For the sinusoid given below, calculate: a) The amplitude (Vm) b) The phase angle ( ) c) Angular frequency ( ) d) The period (T), and e) The frequency (f)

1. INTRODUCTION TO PHASORS PHASOR: ◦ a vector quantity with: Magnitude (Z): the length of vector. Angle ( ) : measured from (0 o) 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. ( =2 f) � 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,

Ex. 1. i(t) = 3 A sin (2 ft+30 o) 2. v(t) = 4 V sin ( -60 o) 3. p(t) = 1 A +5 A sin ( t-150 o) 3 A<30 o 4 V<-60 o 5 A<-150 o 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. 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: ◦ Write the phasors for A and B, if wave A is the reference wave. t = 5 ms per division

Example: 1. What is the instantaneous voltage at t = 3 s, if: Vp = 10 V, f = 50 k. Hz, =0 o (t measured from the “+” going zero crossing) 2. 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 90 o using the “j” operator. � Express in: ◦ Rectangular coordinates (Re, Im) ◦ Polar (A< ) coordinates - like phasors

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

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

POLAR to RECTANGULAR �V P = Z< to V R = A + j B j Y-Axis -Re Z B X-Axis Re X-Axis A Y-Axis -j

MATH OPERATIONS � ADDITION/ form SUBTRACTION - use Rectangular Øadd real parts to each other, add imaginary parts to each other; Øsubtract real parts from each other, subtract imaginary parts from each other � ex: Ø(4+j 5) + (4 -j 6) = 8 -j 1 Ø(4+j 5) - (4 -j 6) = 0+j 11 = j 11 � 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 to Polar The final answer is; Conjugate: + to -