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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 cos50t 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 4t 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. ( =2f) 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 (2ft+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 40V B 2.557.6V 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; 4302 20 4.250 (20) 830 • Division: divide magnitudes, subtract angles 450 4 50 (20) 270 2 20 2 Ex. • Evaluate these complex numbers: a) 4050 20 30 1/2 10 30 3 j 4 b) 2 j 43 j5