1. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Lecture 16 Phasor Circuits, AC Power, Thevenin

2. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Kirchhoff’s Laws in Phasor Form We can apply KVL directly to phasors. The sum of the phasor voltages equals zero for any closed path. The sum of the phasor currents entering a node must equal the sum of the phasor currents leaving.

3. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Circuit Analysis Using Phasors and Impedances 1. Replace the time descriptions of the voltage and current sources with the corresponding phasors. (All of the sources must have the same frequency.)

4. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. 2. Replace inductances by their complex impedances Z L = jωL. Replace capacitances by their complex impedances Z C = 1/(jωC). Resistances have impedances equal to their resistances. 3. Analyze the circuit using any of the techniques studied earlier in Chapter 2, performing the calculations with complex arithmetic.

5. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.3 Find i(t):

6. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.3 Find the phasor voltage across each element:

7. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.4 Find the voltage v c (t) in steady state:

8. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.4

9. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.4 Find the phasor current through each element:

10. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.5 Use the node voltage technique to find v 1 (t):

11. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.5

12. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.9 Find i(t):

13. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.9 Construct the a phasor diagram showing all three voltages and the current:

14. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Find the phasor voltage and phasor current through each element: Exercise 5.10

15. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.10 Find the phasor voltage and current through each element:

16. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.11 Solve for the mesh currents: 100j -200j

17. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.11

18. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power in AC Circuits For  >0 the load is called “inductive” since Z=j  L for an inductor For  <0 the load is “capacitive” since Z=-j/  C for a capacitor

19. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power in AC Circuits Current, Voltage and Power for a Resistive Load

20. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power in AC Circuits Current, Voltage and Power for an Inductive Load “Reactive” power using cos(x)sin(x) = (1/2)sin(2x)

21. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power in AC Circuits For a pure inductance half of the time the power is positive and power flows to the inductor where it is stored as energy in the magnetic field Half of the time the power is negative and power flows from the inductor to the source Even though the average power is zero for a pure inductor, current flows between the source and the inductor and power is dissipated in the lines connecting the source to the inductor Reactive Power

22. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power in AC Circuits Current, Voltage and Power for a Capacitive Load Reactive power for a capacitor opposite the sign for an inductor

23. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power in AC Circuits Inductor Capacitor If a load contains both inductance and capacitance with reactive powers of equal magnitude, the reactive powers cancel.

24. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power for a General Load For a general RLC circuit in which the phase can be any value between -90  to +90  PF is the “Power Factor”

25. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power for a General Load Power angle: If the phase angle for the voltage is not zero, we define the power angle  :

26. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. AC Power Calculations Average Power: Reactive Power: Apparent Power:

27. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Power Triangles Average power Reactive power Average power Apparent power

28. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Load Impedance in the Complex Plane

29. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Additional Power Relations Average power Reactive power

30. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. AC Power Calculations Compute the power and reactive power from the source in this circuit:

31. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. AC Power Calculations

32. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. AC Power Calculations The reactive power delivered to the inductor and capacitor:

33. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. AC Power Calculations The power delivered to the resistor:

34. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Using Power Triangles Find the power, reactive power and power factor for the source. Also find the phasor current I: Load A: 10kVA apparent power, PF=0.5 leading (capacitive) Load B: 5kW power, PF=0.7 lagging (inductive)

35. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Using Power Triangles Load A: 10kVA apparent power, PF=0.5 leading (capacitive) Reactive power Q A and the power angle  A are negative Load B: 5kW power, PF=0.7 lagging (inductive) Reactive power Q B and the power angle  B are positive

36. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Using Power Triangles

37. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Using Power Triangles

38. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Using Power Triangles

39. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Using Power Triangles

40. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.12 Voltage source: Delivers 5 kW to a load with a power factor of 100 percent. Find the reactive power and the phasor current:

41. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.12 If the PF=20% lagging, cos(  ) = 0.20   = arcos(0.20) = 78.46   Q = 24.49kVAR P = 5kW V rms I rms

42. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Thevenin Equivalent Circuits The Thevenin equivalent for an ac circuit consists of a phasor voltage source V t in series with a complex impedance Z t

43. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. The Thévenin voltage is equal to the open-circuit phasor voltage of the original circuit. We can find the Thévenin impedance by zeroing the independent sources and determining the impedance looking into the circuit terminals. Thevenin Equivalent Circuits

44. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. The Thévenin impedance equals the open-circuit voltage divided by the short-circuit current. Thevenin Equivalent Circuits

45. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Norton Equivalent Circuit The Norton equivalent for an ac circuit consists of a phasor current source I n in parallel with a complex impedance Z t

46. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.9

47. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.9 Circuit with the voltage and current source zeroed to find the Thevenin impedance

48. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.9 Circuit with the output shorted to find the short circuit current.

49. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Example 5.9

50. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. The Thevenin equivalent of a two-terminal circuit delivering power to a load impedance. Maximum Average Power Transfer

51. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. If the load can take on any complex value, maximum power transfer is attained for a load impedance equal to the complex conjugate of the Thévenin impedance. If the load is required to be a pure resistance, maximum power transfer is attained for a load resistance equal to the magnitude of the Thévenin impedance. Maximum Average Power Transfer

52. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Maximum Average Power Transfer Find the load for the maximum power transfer if the load can have any complex value and if the load must be a pure resistance:

53. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.14 Find the Thevenin impedance, the Thevenin voltage and the Norton current. Zero the source Z eff

54. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.14

55. ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc. Exercise 5.14