1. Balanced Three-Phase Circuits
CHAPTER 11
Balanced
Three-Phase Circuits

2. CHAPTER CONTENTS 11.1 Balanced Three-Phase Voltages
11.2 Three-Phase Voltage Sources
11.3 Analysis of the Wye-Wye Circuit
11.4 Analysis of the Wye-Delta Circuit
11.5 Power Calculations in Balanced Three-Phase Circuits
11.6 Measuring Average Power in Three-Phase Circuits

3. CHAPTER OBJECTIVES
Know how to analyze a balanced, three-phase wye-wye connected circuit.
Know how to analyze a balanced, three-phase wye-delta connected circuit.
Be able to calculate power (average, reactive, and complex) in any three-phase circuit.

4. 11.1 Balanced Three-Phase Voltages
Three sinusoidal voltages
Identical amplitudes and frequencies
Out of phase 120° with each other by exactly
As the a-phase voltage, the b-phase voltage, and the c-phase voltage.
abc (or positive) phase sequence
acb (or negative) phase sequence

5. Figure 11.1 A basic three-phase circuit.

6. abc (or positive) phase sequence
Figure Phasor diagrams of a balanced set of three-phase voltages. (a) The abc (positive) sequence.

7. acb (or negative) phase sequence
Figure Phasor diagrams of a balanced set of three-phase voltages. (b) The acb (negative) sequence.

8. 11.2 Three-Phase Voltage Sources
Figure A sketch of a three-phase voltage source.

9. n in Fig. 11.4(a), is called the neutral terminal of the source.
There are two ways of interconnecting the separate phase windings to form a three-phase source: in either a wye (Y) or a delta (Δ)
n in Fig. 11.4(a), is called the neutral terminal of the source.
Figure The two basic connections of an ideal three-phase source. (a) A Y-connected source. (b) A ∆-connected source.

10. Three-phase sources and loads can be either Y-connected or Δ-connected

11. Three-phase source with winding impedance
Figure A model of a three-phase source with winding impedance: (a) a Y-connected source; and (b) a ∆-connected source.

12. 11.3 Analysis of the Wye-Wye Circuit
Figure A three-phase Y-Y system.

13. Conditions for a balanced three-phase circuit
The voltage sources form a set of balanced three-phase voltages. In Fig. 11.6, this means that and are a set of balanced three-phase voltages.
The impedance of each phase of the voltage source is the same. In Fig. 11.6, this means that Zga = Zgb = Zgc.
The impedance of each line (or phase) conductor is the same. In Fig. 11.6, this means that Z1a = Z1b = Z1c.
The impedance of each phase of the load is the same. In Fig. 11.6, this means that ZA = ZB = ZC.

14. where a balanced three phase circuit,

15. When the system is balanced, the three line currents are

16. Single-phase equivalent circuit
The current in the a-phase conductor line is simply the voltage generated in the a-phase winding of the generator divided by the total impedance in the a-phase of the circuit.
Figure A single-phase equivalent circuit.

17. Line-to-line and line-to-neutral voltages
Figure Line-to-line and line-to-neutral voltages.

18. Relationship between line-to-line and line-to-neutral voltages
The magnitude of the line-to-line voltage is times the magnitude of the line-to-neutral voltage.
The line-to-line voltages form a balanced three-phase set of voltages.
The set of line-to-line voltages leads the set of line-to-neutral voltages by 30°.

19. Phasor diagrams
Figure Phasor diagrams showing the relationship between line-to-line and line-to-neutral voltages in a balanced system. (a) The abc sequence. (b) The acb sequence.

20. Line voltage v.s. Phase voltage
Line voltage refers to the voltage across any pair of lines; phase voltage refers to the voltage across a single phase.
Line current refers to the current in a single line; phase current refers to current in a single phase.
In a Δ connection, line voltage and phase voltage are identical.
In a Y connection, line current and phase current are identical.

21. Example 11.1
A balanced three-phase Y-connected generator with positive sequence has an impedance of j0.5Ω/f and an internal voltage of 120 V/f The generator feeds a balanced three-phase Y-connected load having an impedance of 39 + j28Ω/f. The impedance of the line connecting the generator to the load is j1.5Ω/f. The a-phase internal voltage of the generator is specified as the reference phasor.
a) Construct the a-phase equivalent circuit of the system.
b) Calculate the three line currents IaA, IbB, and IcC.

22. Example 11.1
c) Calculate the three phase voltages at the load, VAN, VBN, and VCN. d) Calculate the line voltages VAB, VBC, and VCA at the terminals of the load. e) Calculate the phase voltages at the terminals of the generator, Ian, Ibn, and Icn. f) Calculate the line voltages Iab, Ibc, and Ica and at the terminals of the generator. g) Repeat (a)–(f) for a negative phase sequence.

23. Example 11.1
Figure The single-phase equivalent circuit for Example 11.1.

24. Example 11.1

25. Example 11.1

26. Example 11.1

27. Example 11.1

28. Example 11.1

29. 11.4 Analysis of the Wye-Delta Circuit
When the load is balanced, the impedance of each leg of the wye is one third the impedance of each leg of the delta.
Relationship between three-phase delta-connected and wye-connected impedance

30. After the D load has been replaced by its Y equivalent, the a-phase can be modeled by the single phase equivalent circuit shown in Fig
Figure A single-phase equivalent circuit.

31. When a load (or source) is connected in a delta, the current in each leg of the delta is the phase current, and the voltage across each leg is the phase voltage.
Figure A circuit used to establish the relationship between line currents and phase currents in a balanced ∆ load.

32. To demonstrate the relationship between the phase currents and line currents, we assume a positive phase sequence and let If represent the magnitude of the phase current

34. Figure Phasor diagrams showing the relationship between line currents and phase currents in a ∆-connected load. (a) The positive sequence. (b) The negative sequence.

35. Example 11.2
The Y-connected source in Example 11.1 feeds a -connected load through a distribution line having an impedance of j0.9Ω/f. The load impedance is j85.5Ω/f .Use the a-phase internal voltage of the generator as the reference.
a) Construct a single-phase equivalent circuit of the three-phase system.
b) Calculate the line currents IaA, IbB, and IcC.
c) Calculate the phase voltages at the load terminals.
d) Calculate the phase currents of the load.
e) Calculate the line voltages at the source terminals.

36. Example 11.2
Figure The single-phase equivalent circuit for Example 11.2.

37. Example 11.2

38. Example 11.2

39. Example 11.2

40. Example 11.2

42. 11.5 Power Calculations in Balanced Three-Phase Circuits
Average Power in a Balanced Wye Load
Figure A balanced Y load used to introduce average power calculations in three-phase circuits.

43. Average Power in a Balanced Wye Load

44. Total real power in a balanced three-phase load

45. Complex Power in a Balanced Wye Load
Total reactive power in a balanced three-phase load
Total complex power in a balanced threephase load

46. Power Calculations in a Balanced Delta Load
Figure A ∆-connected load used to discuss power calculations.

48. The total power delivered to a balanced Δ-connected load

49. Instantaneous Power in Three-Phase Circuits
The total instantaneous power is the sum of the instantaneous phase powers, which reduces to 1.5VmIm cosθf; that is,
Note this result is consistent with Eq since and

50. Example 11.3
a) Calculate the average power per phase delivered to the Y-connected load of Example b) Calculate the total average power delivered to the load. c) Calculate the total average power lost in the line. d) Calculate the total average power lost in the generator. e) Calculate the total number of magnetizing vars absorbed by the load. f) Calculate the total complex power delivered by the source.

51. Example 11.3

52. Example 11.3

53. Example 11.3

54. Example 11.4
a) Calculate the total complex power delivered to the D-connected load of Example b) What percentage of the average power at the sending end of the line is delivered to the load?

55. Example 11.4

56. Example 11.5
A balanced three-phase load requires 480 kW at a lagging power factor of 0.8. The load is fed from a line having an impedance of j0.025Ω/f. The line voltage at the terminals of the load is 600V.
a) Construct a single-phase equivalent circuit of the system.
b) Calculate the magnitude of the line current.
c) Calculate the magnitude of the line voltage at the sending end of the line.
d) Calculate the power factor at the sending end of the line.

57. Example 11.5
Figure The single-phase equivalent circuit for Example 11.5.

58. Example 11.5

59. Example 11.5

60. Example 11.5

61. Example 11.5

63. 11.6 Measuring Average Power in Three-Phase Circuits
Electrodynamometer wattmeter: current coil, potential coil.
The wattmeter deflects upscale when
(1) the polarity-marked terminal of the current coil is toward the source, and
(2) the polarity-marked terminal of the potential coil is connected to the same line in which the current coil has been inserted.
Figure The key features of the electrodynamometer wattmeter.

64. The Two-Wattmeter Method
To measure the total power at the terminals of the box, we need to know n – 1currents and voltages.
Figure A general circuit whose power is supplied by n conductors.

65. Figure A circuit used to analyze the two-wattmeter method of measuring average power delivered to a balanced load.

66. For a positive phase sequence,
To find the total power, we add W1 and W2 thus

67. Readings of the two wattmeters
A closer look at Eqs and reveals the following about the readings of the two wattmeters:
If the power factor is greater than 0.5, both wattmeters read positive.
If the power factor equals 0.5, one wattmeter reads zero.
If the power factor is less than 0.5, one wattmeter reads negative.
Reversing the phase sequence will interchange the readings on the two wattmeters.

68. Example 11.6
Calculate the reading of each wattmeter in the circuit in Fig if the phase voltage at the load is 120 V and (a) (b) (c) and (d) (e) Verify for (a)–(d) that the sum of the wattmeter readings equals the total power delivered to the load

69. Example 11.6

70. Example 11.6

71. Example 11.6

72. Example 11.6

73. Summary
When analyzing balanced three-phase circuits, the first step is to transform any connections into Y connections, so that the overall circuit is of the Y-Y configuration.
A single-phase equivalent circuit is used to calculate the line current and the phase voltage in one phase of the Y-Y structure. The a-phase is normally chosen for this purpose.

74. Summary
Once we know the line current and phase voltage in the a-phase equivalent circuit, we can take analytical shortcuts to find any current or voltage in a balanced threephase circuit, based on the following facts:
The b- and c-phase currents and voltages are identical to the a-phase current and voltage except for a 120° shift in phase. In a positive-sequence circuit, the b-phase quantity lags the a-phase quantity by 120°, and the c-phase quantity leads the a-phase quantity by 120°. For a negative sequence circuit, phases b and c are interchanged with respect to phase a.

75. Summary
The set of line voltages is out of phase with the set of phase voltages by ±30°. The plus or minus sign corresponds to positive and negative sequence, respectively.
In a Y-Y circuit the magnitude of a line voltage is times the magnitude of a phase voltage.
The set of line currents is out of phase with the set of phase currents in D-connected sources and loads by The minus or plus sign corresponds to positive and negative sequence, respectively.
The magnitude of a line current is times the magnitude of a phase current in a D-connected source or load.

76. Summary
The techniques for calculating per-phase average power, reactive power, and complex power are identical to those introduced in Chapter 10.
The total real, reactive, and complex power can be determined either by multiplying the corresponding per phase quantity by 3 or by using the expressions based on line current and line voltage, as given by Eqs , 11.38, and

77. Summary
The total instantaneous power in a balanced three-phase circuit is constant and equals 1.5 times the average power per phase.
A wattmeter measures the average power delivered to a load by using a current coil connected in series with the load and a potential coil connected in parallel with the load.
The total average power in a balanced three-phase circuit can be measured by summing the readings of two wattmeters connected in two different phases of the circuit.