1. EKT103 ELECTRICAL ENGINEERING
Chapter 1
Three-Phase System

2. COURSE OUTCOME (CO)
CO1: Ability to define and explain the concept of single-phase and three-phase system.

3. Revision
A sinusoid is a signal that has the form of the sine or cosine function.
A general expression for the sinusoid,
where
Vm = the amplitude of the sinusoid
ω = the angular frequency in radians/s
Ф = the phase

4. Revision
A periodic function is one that satisfies v(t) = v(t + nT), for all t and for all integers n.
Only two sinusoidal values with the same frequency can be compared by their amplitude and phase difference.
If phase difference is zero, they are in phase; if phase difference is not zero, they are out of phase.

5. Revision
Example 1
Given a sinusoid, , calculate its amplitude, phase, angular frequency, period, and frequency.

6. Revision
Example 1
Given a sinusoid, , calculate its amplitude, phase, angular frequency, period, and frequency.
Solution:
Amplitude = 5, phase = –60o, angular frequency = 4p rad/s, Period = 0.5 s, frequency = 2 Hz.

7. Revision Find the phase angle between and , does i1 lead or lag i2?
Example 2
Find the phase angle between and , does i1 lead or lag i2?

8. Revision Find the phase angle between and , does i1 lead or lag i2?
Example 2
Find the phase angle between and , does i1 lead or lag i2?
Solution:
Since sin(ωt+90o) = cos ωt
therefore, i1 leads i2 155o.

9. Revision Impedance transformation

10. Single-Phase Circuit
A single phase circuit consists of a generator connected through a pair of wires
to a load
Three wired system
same magnitude
same phase
Two wire system

11. Second source with 90° out
Two-Phase Circuit A a
Three wired system
Second source with 90° out
of phase
Three wired system
same magnitude
different phase

12. What is a Three-Phase Circuit?
It is a system produced by a generator consisting of three sources having the same amplitude and frequency but out of phase with each other by 120°.
Three sources with 120° out
of phase
Four wired system

13. Balance Three-Phase Voltages
A three-phase generator consists of a rotating magnet (rotor) surrounded by a stationary winding (stator).
A three-phase generator
The generated voltages

14. Balance Three-Phase Voltages
Two possible configurations:
Three-phase voltage sources: (a) Y-connected ; (b) Δ-connected

15. Balance Three-Phase Voltages
Phase sequences
a) abc or positive sequence b) acb or negative sequence

16. Balance 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

17. Balance Three-Phase Voltages
abc sequence or positive sequence:
is the effective or rms value
acb sequence or negative sequence:

18. Balance Three-Phase Voltages
Example 1
Determine the phase sequence of the set of voltages.

19. Balance Three-Phase Voltages
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.

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

21. Balance 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:

22. Balance Three-Phase Connection
Four possible connections
Y-Y connection (Y-connected source with a Y-connected load)
Y-Δ connection (Y-connected source with a Δ-connected load)
Δ-Δ connection
Δ-Y connection

23. Balance Y-Y Connection
A balanced Y-Y system is a three-phase system with a balanced y-connected source and a balanced y-connected load.

24. Balance Y-Y Connection
Source impedance
Line impedance
Load impedance
Total impedance per phase
Since all impedance are in series, Thus

25. Balance Y-Y Connection
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26. Balance Y-Y Connection
Applying KVL to each phase:

27. Balance Y-Y Connection
Line to line voltages or line voltages:
Magnitude of line voltages:
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28. Balance Y-Y Connection
Example 2
Calculate the line currents in the three-wire Y-Y system shown below:

29. Balance Y-Y Connection
Example 2
Calculate the line currents in the three-wire Y-Y system shown below:

30. Balance Y-Δ Connection
A balanced Y-Δ system is a three-phase system with a balanced y-connected source and a balanced Δ-connected load.

31. Balance Y-Δ Connection
A single phase equivalent circuit
ale29559_12016.jpg

32. Balance Y-Δ Connection
A single phase equivalent circuit
Line voltages:
ale29559_12016.jpg

33. Balance Y-Δ Connection
A single-phase equivalent circuit of a balanced Y- circuit
Line currents:
Phase currents:
ale29559_12015.jpg

34. Balance Y-Δ Connection
A single-phase equivalent circuit of a balanced Y- circuit
Magnitude line currents:
ale29559_12015.jpg

35. Balance Y-Δ Connection
Example 3
A balanced abc-sequence Y-connected source with ( ) is connected to a Δ-connected load (8+j4) per phase. Calculate the phase and line currents.
Solution
Using single-phase analysis,
Other line currents are obtained using the abc phase sequence

36. Balance Δ-Δ Connection
A balanced Δ-Δ system is a three-phase system with a balanced Δ -connected source and a balanced Δ -connected load.

37. Balance Δ-Δ Connection
Phase currents:
Line voltages:
Line currents:
Magnitude line currents:
Total impedance:

38. Balance Δ-Δ Connection
Example 4
A balanced Δ-connected load having an impedance 20-j15  is connected to a Δ-connected positive-sequence generator having ( ). Calculate the phase currents of the load and the line currents.
Ans:
The phase currents
The line currents

39. Balance Δ-Y Connection
A balanced Δ-Y system is a three-phase system with a balanced y-connected source and a balanced y-connected load.

40. Balance Δ-Y Connection
Applying KVL to loop aANBba:
From:
Line currents:

41. Balance Δ-Y Connection
Replace Δ connected source to equivalent Y connected source.
Phase voltages:

42. Balance Δ-Y Connection
A single phase equivalent circuit

43. Balance Δ-Y Connection
Example 5
A balanced Y-connected load with a phase impedance 40+j25  is supplied by a balanced, positive-sequence Δ-connected source with a line voltage of 210V. Calculate the phase currents. Use Vab as reference.
Answer
The phase currents

44. Power in a Balanced System
Comparing the power loss in (a) a single-phase system, and (b) a three-phase system
If same power loss is tolerated in both system, three-phase system use only 75% of materials of a single-phase system

45. Power in a Balanced System
For Y connected load, the phase voltage:

46. Power in a Balanced System
Phase current lag phase voltage by θ. If
The phase current:

47. Power in a Balanced System
Total instantaneous power:
Average power per phase:
Reactive power per phase:
Apparent power per phase:
Complex power per phase:

48. Power in a Balanced System
Total average power:
Total reactive power:
Total complex power:

49. 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

50. 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.

51. 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.
The apparent power is
The real power is
The power factor is

52. 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.

53. 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.
Answer :

54. Exercise 7 For the Y-Y circuit in Exercise 2, calculate the
complex power at the source and at the load.