1. EE301 Phasors, Complex Numbers, And Impedance

2. Learning Objectives Define a phasor and use phasors to represent sinusoidal voltages and currents Determine when a sinusoidal waveform leads or lags another Graph a phasor diagram that illustrates phase relationships Define and graph complex numbers in rectangular and polar form Perform addition, subtraction, multiplication and division using complex numbers and illustrate them using graphical methods Represent a sinusoidal voltage or current as a complex number in polar and rectangular form

3. Learning Objectives Define time domain and phasor (frequency) domain Use the phasor domain to add/subtract AC voltages and currents For purely resistive, inductive and capacitive elements define the voltage and current phase differences Define inductive reactance Understand the variation of inductive reactance as a function of frequency Define capacitive reactance

4. Learning Objectives Understand the variation of capacitive reactance as a function of frequency Define impedance Graph impedances of purely resistive, inductive and capacitive elements as a function of phase

5. Complex numbers A complex number is a number of the form C = a + jb where a and b are real and j = a is the real part of C and b is the imaginary part. Complex numbers are merely an invention designed to allow us to talk about the quantity j. j is used in EE to represent the imaginary component to avoid confusion with CURRENT (i)

6. Geometric Representation C = 6 + j8 (rectangular form) C = 10  53.13º (polar form)

7. Conversion Between Forms To convert between forms where apply the following relations

8. Example Problem 1 Convert (5 ∠ 60) to rectangular form. Convert 6 + j 7 to polar form. Convert -4 + j 4 to polar form. Convert (5 ∠ 220) to rectangular form.

9. Properties of j

10. Addition and Subtraction of Complex Numbers Easiest to perform in rectangular form Add/subtract real and imaginary parts separately

11. Multiplication and Division of Complex Numbers Easiest to perform in polar form Multiplication: multiply magnitudes and add the angles Division: Divide the magnitudes and subtract the angles

12. Example Problem 2 Given A =1 +j1 and B =2 – j3 Determine A+B and A-B. Given A =1.41  45° and B =3.61  -56° Determine A/B and A*B.

13. Reciprocals and Conjugates The reciprocal of C = C , is The conjugate of C is denoted C *, which has the same real value but the opposite imaginary part:

14. Example Problem 3 And now you can try with your TI!! (3-i4) + (10 ∠ 44) ANS: 10.6 ∠ 16.1 ANS: 10.2 + 2.9i (22000+i13)/(3 ∠ -17) ANS: 7.3E3 ∠ 17.0 Convert 95-12j to polar: ANS: 95.8 ∠ -7.2

15. Phasor Transform To solve problems that involve sinusoids (such as AC voltages and currents) we use the phasor transform. We transform sinusoids into complex numbers in polar form, solve the problem using complex arithmetic (as described), and then transform the result back to a sinusoid.

16. Phasors A phasor is a rotating vector whose projection on the vertical axis can be used to represent a sinusoid. The length of the phasor is amplitude of the sinusoid ( V m ) The angular velocity of the phasor is 

17. Representing AC Signals with Complex Numbers By replacing e(t) with it’s phasor equivalent E, we have transformed the source from the time domain to the phasor domain. Phasors allow us to convert from differential equations to simple algebra. KVL and KCL still work in phasor domain.

18. Using phasors to represent AC voltage and current Looking at the sinusoid eqn, determine V Pk and phase offset. Using V PK, determine V RMS using the formula: The phasor is then

19. Representing AC Signals with Complex Numbers Phasor representations can be viewed as a complex number in polar form. E = E rms 

20. Example Problem 4 i 1 = 20 sin (  t) mA. i 2 = 10 sin (  t+90˚) mA. i 3 = 30 sin (  t - 90˚) mA. Determine the equation for i T.

21. Phase Difference Phase difference is angular displacement between waveforms of same frequency. If angular displacement is 0° then waveforms are in phase If angular displacement is not 0 o, they are out of phase by amount of displacement

22. Phase Difference If v 1 = 5 sin(100t) and v 2 = 3 sin(100t - 30°), v 1 leads v 2 by 30°

23. Phase Difference w/ Phasors The waveform generated by the leading phasor leads the waveform generated by the lagging phasor.

24. Formulas from Trigonometry Sometimes signals are expressed in cosines instead of sines.

25. Example Problem 5 Draw the phasor diagram, determine phase relationship, and sketch the waveform for the following: i = 40 sin(  t + 80º) and v = -30 sin(  t - 70º)

26. R, L and C circuits with Sinusoidal Excitation R, L, C have very different voltage-current relationships Sinusoidal (ac) sources are a special case Review

27. The Impedance Concept Impedance ( Z ) is the opposition that a circuit element presents to current in the phasor domain. It is defined Ohm’s law for ac circuits

28. Impedance Impedance is a complex quantity that can be made up of resistance (real part) and reactance (imaginary part). Unit of impedance is ohms (  ).  R X Z

29. Resistance and Sinusoidal AC For a purely resistive circuit, current and voltage are in phase.

30. Resistors For resistors, voltage and current are in phase.

31. Example Problem 1 Two resistors R 1 =10 kΩ and R 2 =12.5 kΩ are in series. If i(t) = 14.7 sin (ωt + 39˚) mA a) Compute V R1 and V R2 b) Compute V T =V R1 + V R2 c) Calculate Z T d) Compare V T to the results of V T =IZ T

32. Inductance and Sinusoidal AC Voltage-Current relationship for an inductor It should be noted that for a purely inductive circuit voltage leads current by 90º.

33. Inductive Impedance Impedance can be written as a complex number (in rectangular or polar form): Since an ideal inductor has no real resistive component, this means the reactance of an inductor is the pure imaginary part:

34. Inductance and Sinusoidal AC Voltage leads current by 90˚

35. Inductance For inductors, voltage leads current by 90º.

36. Since X L =  L = 2  fL, inductive reactance is directly proportional to frequency. Extreme case f = 0 Hz (DC): inductor looks like a short circuit! Variation with Frequency

37. Impedance and AC Circuits Solution technique 1. Transform time domain currents and voltages into phasors 2. Calculate impedances for circuit elements 3. Perform all calculations using complex math 4. Transform resulting phasors back to time domain (if reqd)

38. Example Problem 2 For the inductive circuit: v L = 40 sin (ωt + 30˚) V f = 26.53 kHz L = 2 mH Determine V L and I L Graph v L and i L

39. Example Problem 2 solution v L = 40 sin (ωt + 30˚) V i L = 120 sin (ωt - 60˚) mA vLvL iLiL Notice 90°phase difference!

40. Example Problem 3 For the inductive circuit: v L = 40 sin (ωt + Ө) V i L = 250 sin (ωt + 40˚) μA f = 500 kHz What is L and Ө?

41. Capacitance and Sinusoidal AC Current-voltage relationship for an capacitor It should be noted that, for a purely capacitive circuit current leads voltage by 90º.

42. Capacitive Impedance Impedance can be written as a complex number (in rectangular or polar form): Since a capacitor has no real resistive component, this means the reactance of a capacitor is the pure imaginary part:

43. Capacitance and Sinusoidal AC

44. Capacitance For capacitors, voltage lags current by 90º.

45. Variation with Frequency Since, capacitive reactance is inversely proportional to frequency. Extreme case f = 0 Hz (DC): capacitor looks like an open circuit!

46. Example Problem 4 For the capacitive circuit: v C = 3.6 sin (ωt-50°) V f = 12 kHz C=1.29 uF Determine V C and I C

47. Example Problem 5 For the capacitive circuit: v C = 362 sin (ωt - 33˚) V i C = 94 sin (ωt + 57˚) mA C = 2.2 μF Determine the frequency

48. ELI the ICE man E leads I I leads E When voltage is applied to an inductor, it resists the change of current. The current builds up more slowly, lagging in time and phase. Since the voltage on a capacitor is directly proportional to the charge on it, the current must lead the voltage in time and phase to conduct charge to the capacitor plate and raise the voltage Voltage Inductance Current Voltage Capacitance Current

49. Frequency dependency Inductors Capacitors