Three-Phase System 1 by Dr Rosemizi Abd Rahim Click here to watch the three phase animation video

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

3 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 Revision

4 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 = –60 o, angular frequency = 4 rad/s, Period = 0.5 s, frequency = 2 Hz.

7 Revision Example 2 Find the phase angle between and, does i 1 lead or lag i 2 ?

8 Revision Example 2 Find the phase angle between and, does i 1 lead or lag i 2 ? Solution : Since sin(ωt+90 o ) = cos ωt therefore, i 1 leads i o.

9 Revision Impedance transformation

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

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

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°. 12 What is a Three-Phase Circuit? Three sources with 120° out of phase Four wired system

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

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

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

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

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

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

Solution: The voltages can be expressed in phasor form as We notice that V an leads V cn by 120° and V cn in turn leads V bn by 120°. Hence, we have an acb sequence. 19 Balance Three-Phase Voltages

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

21 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: Balance Three-Phase Voltages

Four possible connections 1.Y-Y connection (Y-connected source with a Y- connected load) 2.Y-Δ connection (Y-connected source with a Δ- connected load) 3.Δ-Δ connection 4.Δ-Y connection 22 Balance Three-Phase Connection

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

Balance Y-Y Connection

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

Line to line voltages or line voltages: Magnitude of line voltages:

Example 2 Calculate the line currents in the three-wire Y-Y system shown below: 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

30 Balance Connection 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.

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

Balance Connection Balance Y-Δ Connection A single phase equivalent circuit Line voltages:

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

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

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 35 Balance Connection Balance Y-Δ Connection

36 Balance Connection 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 Line voltages:Line currents: Magnitude line currents:Total impedance: Phase currents: Balance Connection 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 38 Balance Connection Balance Δ-Δ Connection

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

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

42 A single phase equivalent circuit Balance Connection 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 V ab as reference. Answer The phase currents 43 Balance Connection Balance Δ-Y Connection

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

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

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

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

49 Power loss in two wires: Power loss in three wires: P L : power absorbed by the load I L : magnitude of line current V L : line voltage R : line resistance Power in a Balanced System

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.