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COMPLEX NUMBERS and PHASORS

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.

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COMPLEX NUMBERS and PHASORS

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

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

  5. PHASORS

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

  7. Ex: A< 

  8. 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)

  9. PHASORS & SINE WAVES • One revolution of the phasor ,through 360°, = 1 cycle of a sinusoid.

  10. 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.

  11. 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,

  12. 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.

  13. GRAPHING PHASORS • Positive phase angles are drawn counterclockwise from the axis; • Negative phase angles are drawn clockwise from the axis.

  14. GRAPHING PHASORS Note: A leads B B leads C C lags A etc

  15. 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.

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

  17. Ex. • 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?

  18. COMPLEX NUMBERS

  19. COMPLEX NUMBER SYSTEM • COMPLEX PLANE:

  20. 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

  21. j Z Y-Axis B q Re -Re X-Axis X-Axis A Y-Axis -j COORDINATE SYSTEMS • RECTANGULAR: • 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!

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

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

  24. 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

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

  26. Ex. • Evaluate these complex numbers:

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