1. CHAPTER 4: THREE-PHASE CIRCUIT
Balanced Three-Phase Voltages
Balance Connections: Wye-Wye, Wye-Delta, Delta-Delta, Delta-Wye
Power in a Balanced System

2. a) two-wire type b) three-wire type
Introduction
Single phase systems
a) two-wire type b) three-wire type

3. Introduction
Three Phase System

4. Balanced Three Phase Voltages
Three-phase voltage sources
a) wye-connected source b) delta-connected source

5. Balanced Three Phase Voltages
Phase sequences
a) abc or positive sequence b) acb or negative sequence

6. Balanced Three Phase Voltages
If the voltage source have the same amplitude and frequency ω and are out of phase with each other by 120o, the voltage are said to be balanced.
Balanced phase voltages are equal in magnitude and out of phase with each other by 120o

7. Balanced Three Phase Voltages
abc sequence or positive sequence:
is the effective or rms value
acb sequence or negative sequence:

8. Balanced Three Phase Voltages
Two possible three-phase load configurations:
a) a wye-connected load b) a delta-connected load

9. Balanced Three Phase Voltages
A balanced load is one in which the phase impedances are equal in magnitude and in phase.
For a balanced wye connected load:
For a balanced delta connected load:

10. Example 1 Determine the phase sequence of the set of voltages
van = 200 cos(ωt + 10◦)
vbn = 200 cos(ωt − 230◦), vcn = 200 cos(ωt − 110◦)
Solution:
The voltages can be expressed in phasor form as
We notice that Van leads Vcn by 120◦ and Vcn in turn leads Vbn by 120◦.
Hence, we have an acb sequence.

11. Exercise 1 Given that , find Van and Vcn, assuming a
positive (abc) sequence.
Answer:

12. Balanced Wye-Wye Connection
A balanced Y-Y system is a three phase system with a balanced Y connected source and balanced Y connected load.

13. Balanced Wye-Wye Connection
Source impedance
Line impedance
Load impedance
Total impedance per phase

14. Balanced Wye-Wye Connection
Line to line voltages or line voltages:
Magnitude of line voltages:

15. Balanced Wye-Wye Connection
Phasor diagram illustrating the relationship between line voltages and phase voltages.

16. Balanced Wye-Wye Connection
Balanced Y-Y connection

17. Balanced Wye-Wye Connection
Applying KVL to each phase:

18. Balanced Wye-Wye Connection
A single phase equivalent circuit

19. Example 2
Calculate the line currents in the three wire Y-Y system of figure below.
Reference : Alexander, Sadiku Chapter 12 - page 511

20. Exercise 2
A Y-connected balanced three-phase generator with an impedance of 0.4+j0.3 Ω per phase is connected to a Y-connected balanced load with an impedance of j19 Ω per phase. The line joining the generator and the load has an impedance of j0.7 Ω per phase.
Assuming a positive sequence for the source voltages and that
Find: (a) the line voltages
(b) the line currents
Reference : Alexander, Sadiku Chapter 12 – PP 12.2

21. Balanced Wye-Delta Connection
A balanced Y- Δ system consists of balanced Y connected source feeding a balanced Δ connected load.

22. Balanced Wye-Delta Connection
Line voltages:
Phase currents:
Line currents:

23. Balanced Wye-Delta Connection
A single-phase equivalent circuit of a balanced Y- circuit

24. Balanced Wye-Delta Connection
From phasor diagram
Magnitude line currents:

25. Balanced Wye-Delta Connection
A single phase equivalent circuit

26. Example 3 A balanced abc sequence Y-connected source with
Is connected to a Δ-connected balanced load
(8+j4)Ω per phase. Calculate the phase and line
currents.
Reference : Alexander, Sadiku Chapter 12 - page 513

27. Exercise 3 One line voltage of a balanced Y-connected source is
If the source is connected to a Δ -connected load
of , find the phase and line currents.
Assume the abc sequence.
Reference : Alexander, Sadiku Chapter 12 – PP 12.3

28. Balanced Delta-Delta Connection
A balanced Δ - Δ system is one in which both balanced source and balanced load are Δ connected.

29. Balanced Delta-Delta Connection
Phase currents:
Line voltages:
Line currents:
Magnitude line currents:
Total impedance:

30. Example 4 A balanced Δ connected load having an impedance
20-j15 Ω is connected to a Δ connected, positive sequence
generator having
Calculate the phase currents of the load and the line
currents.
Reference : Alexander, Sadiku Chapter 12 - page 515

31. Exercise 4 A positive-sequence, balanced -connected source
supplies a balanced Δ-connected load. If the
impedance per phase of the load is 18+j12 Ω
and , find IAB and VAB.
Reference : Alexander, Sadiku Chapter 12 – PP 12.4

32. Balanced Delta-Wye Connection
A balanced Δ -Y system consists of balanced Δ connected source feeding a balanced Y connected load.

33. Balanced Delta-Wye Connection
Applying KVL to loop aANBba:
From:
Line currents:

34. Balanced Delta-Wye Connection
Replace Δ connected source to equivalent Y connected source.
Phase voltages:

35. Balanced Delta-Wye Connection
A single phase equivalent circuit

36. Example 5 A balanced Y connected load with a phase resistance
of 40 Ω and a reactance of 25 Ω is supplies by a
balanced, positive sequence Δ connected source with
a line voltage of 210 V. Calculate the phase currents.
Use Vab as reference.
Reference : Alexander, Sadiku Chapter 12 - page 519

37. Exercise 5 In a balanced -Y circuit, and ZY = (12 + j15) Ω.
Calculate the line currents.
Reference : Alexander, Sadiku Chapter 12 – PP 12.5

38. POWER IN A BALANCED SYSTEM
For Y connected load, the phase voltage:

39. POWER IN A BALANCED SYSTEM
Phase current lag phase voltage by θ. If
The phase current:

40. POWER IN A BALANCED SYSTEM
Total instantaneous power:
Average power per phase:
Reactive power per phase:
Apparent power per phase:
Complex power per phase:

41. POWER IN A BALANCED SYSTEM
Total average power:
Total reactive power:
Total complex power:

42. POWER IN A BALANCED SYSTEM
Power loss in two wires:
Power loss in three wires:
PL : power absorbed by the load
IL : magnitude of line current
VL : line voltage
R : line resistance

43. Example 6 A three-phase motor can be regarded as a balanced
Y-load. A three-phase motor draws 5.6 kW when the
line voltage is 220 V and the line current is 18.2 A.
Determine the power factor of the motor.
Reference : Alexander, Sadiku Chapter 12 - page 523

44. Exercise 6 Calculate the line current required for a 30-kW
three-phase motor having a power factor of 0.85
lagging if it is connected to a balanced source
with a line voltage of 440 V.
Reference : Alexander, Sadiku Chapter 12 – PP 12.7

45. Exercise 7 For the Y-Y circuit in Exercise 2, calculate the
complex power at the source and at the load.
Reference : Alexander, Sadiku Chapter 12 – PP 12.6