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: a)The amplitude b)The phase angle c)The period, and d)The frequency

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

5. PHASORS

6. INTRODUCTION TO PHASORS PHASOR: –a vector quantity with: Magnitude (Z): the length of vector. Angle (  ) : measured from (0 o ) 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. (  =2  f)

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

10. Thus, the vertical distance from the end of a rotating phasor represents the instantaneous value of a sine wave at any time, t. INSTANTANEOUS VALUES

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. 1.i(t) = 3A sin (2  ft+30 o ) 3A<30 o 2.v(t) = 4V sin (  -60 o ) 4V<-60 o 3.p(t) = 1A +5A sin (  t-150 o ) 5A<-150 o DC offsets are NOT represented. Frequency and time are NOT represented unless the phasor’s  is specified. DC offsets are NOT represented. Frequency and time are NOT represented unless the phasor’s  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. 1.What is the instantaneous voltage at t = 3  s, if: Vp = 10V, f = 50 kHz,  =0 o (t measured from the “+” going zero crossing) 2.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 90 o using the “j” operator. Express in: – Rectangular coordinates (Re, Im) – Polar (A<  ) coordinates - like phasors

21. 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! Y-Axis X-Axis Y-Axis B A  Z j -j Re -Re

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

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

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: