Three Phase Circuits

Transcription

1 Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University

2 OUTLINE Previously on ELCN102 Three Phase Circuits Balanced Star-Star Connection Balanced Star-Delta Connection Balanced Delta-Delta Connection Balanced Delta-Star Connection Unbalanced Three Phase System 2

3 Previously on ELCN102 Instantaneous Power Instantaneous power p t is the product of the instantaneous voltage v(t) across the element and the instantaneous current i(t) through it p t = v t i t v t = V m cos ωt + φ v, i t = I m cos ωt + φ i p t = V m I m cos ωt + φ v cos ωt + φ = V mi m 2 cos 2ωt + φ v + φ i + cos φ v φ i 3

4 Previously on ELCN102 Instantaneous Power p t = 1 2 V mi m cos 2ωt + φ v + φ i + cos φ v φ i 4

5 Previously on ELCN102 Average Power Average Power (P av ) is the average of p t over one period P av = 1 T න 0 T p t dt = 1 T න 0 T V m I m 2 cos 2ωt + φ v + φ i + cos φ v φ i dt = 1 2 V mi m cos φ v φ i 5

6 Previously on ELCN102 Average Power Average Power (P av ) is the average of p t over one period P av = 1 2 V mi m cos φ v φ i 6

7 Previously on ELCN102 Average Power Z = Z Z = R + jx R = Z cos Z, P av = 1 2 V mi m cos φ v φ i X = Z sin Z = I m 2 2 Z cos Z = I m 2 R 2 For impedance Z, the average power is given by P av = I m 2 R 2 R = Re Z is the resistive part of Z 7

8 Previously on ELCN102 Average Power For φ v φ i = 0 P av = 1 2 V mi m cos φ v φ i Z is pure resistive and the average power absorbed by the impedance will be maximum. For φ v φ i = ±90 o Z is pure reactive and the average power absorbed by the impedance will be zero. 8

9 Previously on ELCN102 Root Mean Square Root Mean Square (rms) or Effective value of a varying signal is defined as the value of DC signal that would produce the same power dissipation in a resistive load DC Signal Sinusoidal Signal 2 P av = I eff R 2 I eff = I m 2 2 I eff = I m 2 P av = I m 2 2 R The eff value of a sinusoidal signal = Maximum of the sinusoidal 2 9

10 Previously on ELCN102 Root Mean Square Root Mean Square (rms) or Effective value of a varying signal is defined as the value of DC signal that would produce the same power dissipation in a resistive load DC Signal 2 P av = I eff R Any periodic Signal P av = 1 T න 0 T i t v t dt I eff = 1 T න 0 T i 2 t dt = R T න 0 T i 2 t dt 10

11 Previously on ELCN102 Complex Power Complex power is important in power analysis because it contains all the information about the power absorbed by a given load. The complex power S absorbed by an AC load is the product of the voltage and the complex conjugate of the current S = 1 V I 2 V = V m φ v, I = I m φ i S = 1 2 V mi m φ v φ i 11

12 Previously on ELCN102 Maximum Average Power Transfer For maximum average power transfer, the load impedance Z L must be equal to the complex conjugate of the Thevenin/Norton impedance Z Th. For maximum power P ZL Z L = Z th = Z N 12

13 Previously on ELCN102 Maximum Average Power Transfer P ZL = V m 2 R th 2 4R th 2 P ZL = V m 2 8R th V rms = V m 2 V 2 rms = V m 2 2 Z th = R th + jx th P ZL = V 2 rms 4R th Z L = Z th = R th jx th 13

14 Previously on ELCN102 Complex Power S = V rms I rms φ v φ i = V rms I rms cos φ v φ i + jv rms I rms sin φ v φ i 2 2 = I rms R + ji rms X = P + jq P = Re S is called the Active Power and its unit is Watt. Q = Im S is called the Reactive Power and its unit is VAR. A = S = V rms I rms is the Apparent Power and its unit is VA. φ v φ i is the Power Factor Angle. 14

15 Single phase Circuits Up to now, we are dealing with a single phase AC power system which consists of An AC generator (V AC = V p φ) A load (Z L ). A pair of wires (a transmission line) to connect the generator and the load. 15

16 Polyphase Circuits Circuits or systems in which the AC sources operate at the same frequency but different phases are known as polyphase. 16

17 Polyphase Circuits Two-phase generator produces two sources having the same amplitude and frequency but out of phase with each other by 90 o. 17

18 Polyphase Circuits Three-phase generator produces three sources having the same amplitude and frequency but out of phase with each other by 120 o 18

19 Three Phase Circuits Three Phase Voltages Three phase power transmission has become the standard for power distribution, at the operating frequency of 60 Hz (in the United States) or 50 Hz in most other parts of the world. For the generation of same amount of power, the three-phase system is more economical than the single-phase. For the transmission of same amount of power, the threephase system is more economical than the single-phase. The instantaneous power in a three-phase system is constant. This results in smooth and vibration free operation of machine. A 3-phase generator can be used to feed a 1-phase load, whereas vice-versa is not possible. 19

20 Generation of Three Phase Voltages Electricity is generated by varying the magnetic flux passing through an inductor. One way to do this is by rotating a coil between two magnetic poles. 20

21 Generation of Three Phase Voltages Electricity is generated by varying the magnetic flux passing through an inductor. The other way to do this is by keeping the coil stationary (stator) and rotating a magnetic (rotor). 21

22 Generation of Three Phase Voltages By winding two coils around the stator, two waves can be generated. The two signals have the same magnitude and frequency but different phase. 22

23 Generation of Three Phase Voltages A three-phase voltage source is a generator with three separate windings distributed around the periphery of the stator. The three signals have the same magnitude and frequency but out of phase with each other by 120 o. 23

24 Generation of Three Phase Voltages Standard practice is to refer to the three phases as a, b, and c, and to use the a-phase as the reference phase. The abc sequence (+ve sequence) is produced when the rotor rotates counterclockwise. V a = V m 0 o, V b = V m 120 o, V c = V m 120 o 24

25 Generation of Three Phase Voltages Standard practice is to refer to the three phases as a, b, and c, and to use the a-phase as the reference phase. The acb sequence ( ve sequence) is produced when the rotor rotates clockwise. V a = V m 0 o, V b = V m 120 o, V c = V m 120 o 25

26 Source Connections A typical three-phase system consists of three voltage sources connected to loads by three or four wires. The first source configuration is done by connecting a, ư ሗb, and cư to one node n (called the neutral node). Star Connection 26

27 Source Connections A typical three-phase system consists of three voltage sources connected to loads by three or four wires. The second source configuration is done by connecting a to c, ư c to ሗb, and b to a. ư Delta Connection 27

28 Star Connections The phase voltages V an, V bn, and V cn are respectively between lines a, b, and c, and the neutral line n. V an = V p 0 o, V bn = V p 120 o, V cn = V p 120 o V an V p + j0 V bn V p cos 120 o + jv p sin 120 o V cn V p cos 120 o + jv p sin 120 o Total 28

29 Star Connections The phase voltages V an, V bn, and V cn are respectively between lines a, b, and c, and the neutral line n. V an = V p 0 o, V bn = V p 120 o, V cn = V p 120 o V an V bn V cn V p + j0 0.5V p + jv p sin 120 o 0.5V p jv p sin 120 o Total 0 29

30 Star Connections The phase voltages V an, V bn, and V cn are respectively between lines a, b, and c, and the neutral line n. V an = V p 0 o, V bn = V p 120 o, V cn = V p 120 o If the voltage sources have the same amplitude and frequency ω and are out of phase with each other by 120 o, the voltages are said to be balanced. 30

31 Example (1) Three Phase Circuits Determine the phase sequence of the set of voltages : v an t = 200 cos(ωt + 10 o ) v bn t = 200 cos(ωt 230 o ) v cn t = 200 cos(ωt 110 o ) 31

32 Load Connections Similar to the source, the load has two configuration Star Connection Delta Connection A balanced load is one in which the phase impedances are equal in magnitude and in phase. 32

33 Circuit Connections Because there are two configuration for the source and load connections, there are four different configuration for the whole circuit. Source Connection Load Connection System Connection Star Connection Star Connection Star-Star Connection Star Connection Delta Connection Star-Delta Connection Delta Connection Star Connection Delta-Star Connection Delta Connection Delta Connection Delta-Delta Connection 33

34 Balanced Star-Star Connection A balanced Y-Y system is a three-phase system with a balanced Y-connected source and a balanced Y-connected load. Z s is the source impedance. Z l is the line impedance. Z L is the load impedance. Z n is the neutral line impedance. Z Y = Z s + Z L + Z l 34

35 Balanced Star-Star Connection KCL at node N I n = I a + I b + I c V N Z n = V an V N Z Y + V bn V N Z Y + V cn V N Z Y 35

36 Balanced Star-Star Connection KCL at node N I n = I a + I b + I c V N 1 Z n + 3 Z Y = V an + V bn + V cn Z Y 36

37 Balanced Star-Star Connection KCL at node N I n = I a + I b + I c V an + V bn + V cn = 0 V N 1 Z n + 3 Z Y = 0 37

38 Balanced Star-Star Connection KCL at node N I n = I a + I b + I c 1 V an + V bn + V cn = 0 VV N + 3 N = 0 I= n = 0 0 Z n Z Y 38

39 Balanced Star-Star Connection For a balanced star-star connection, the neutral line can be removed without affecting the system. I a + I b + I c = 0 V N V n = 0 39

40 Balanced Star-Star Connection I a = V an Z Y = V p 0 o Z Y I b = V bn = V p 120 o = I Z Y Z a 120 o Y I c = V cn = V p 120 o = I Z a 120 o Y Z Y I a, I b, and I c are called the line currents. 40

41 Example (2) Three Phase Circuits Calculate the line currents in the three-wire Y-Y system shown. 41

42 Balanced Star-Delta Connection A balanced Y- system is a three-phase system with a balanced Y- connected source and a balanced -connected load. V an = V p 0 o, V bn = V p 120 o, V cn = V p 120 o V an, V bn, and V cn are called phase voltage 42

43 Balanced Star-Delta Connection V an = V p 0 o, V bn = V p 120 o, V cn = V p 120 o V AB = V an V bn = V p 0 o V p 120 o = V p 1 + j0 V p 0.5 j 3 2 = V p j 3 2 = 3V p 30 o V 43

44 Balanced Star-Delta Connection Similarly, V AB = 3V p 30 o = V L 30 o V V BC = V L 90 o V, V CA = V L 150 o V V AB, V BC, and V CA are called line voltage. 44

45 Balanced Star-Delta Connection V an = V p 0 o V bn = V p 120 o V cn = V p 120 o V AB = V L 30 o V V BC = V L 90 o V V CA = V L 150 o V Line voltage = 3 phase voltage Line voltage leads phase voltage by 30 o 45

46 Balanced Star-Delta Connection Phasor Diagram V AB = V an V bn V BC = V bn V cn V CA = V cn V an Line voltage = 3 phase voltage Line voltage leads phase voltage by 30 o 46

47 Balanced Star-Delta Connection Similarly, I AB = V AB Z = V L 30 o Z = I p φ p A I BC = V BC Z = V L 90 o Z = I p φ p 120 o A (= I AB 120 o ) 47

48 Balanced Star-Delta Connection Similarly, I AB = V AB Z = V L 30 o Z = I p φ p A I CA = V CA Z = V L 150 o Z = I p φ p o A = I AB 120 o 48

49 Balanced Star-Delta Connection I a = I AB I CA = I AB I AB 120 o = I AB 1 + j0 I AB j 3 2 = I AB 1.5 j 3 2 = 3I AB 30 o A 49

50 Balanced Star-Delta Connection Similarly, I a = 3I AB 30 o = I L 30 o A I b = I L 150 o A = I a 120 o I c = I L 90 o A = I a 120 o 50

51 Balanced Star-Delta Connection I a, I b, and I c are called the line currents. I a = I b = I c = I L I AB, I BC, and I CA are called the phase currents. I AB = I BC = I CA = I p 51

52 Balanced Star-Delta Connection I AB = I p 30 o A I BC = I p 90 o A I CA = I p 150 o A I a = I L 30 o A I b = I L 150 o A I c = I L 90 o A Line current = 3 phase current Line current lags phase current by 30 o 52

53 Balanced Star-Delta Connection Line voltage = 3 phase voltage. Line voltage leads phase voltage by 30 o Line current = 3 phase current. Line current lags phase current by 30 o 53

54 Example (3) Three Phase Circuits A balanced abc -sequence Y-connected source with V an = o V is connected to a -connected balanced load of Z Δ = 8 + j4 Ω per phase. Calculate the phase and line currents. 54

55 Balanced Delta-Delta Connection A balanced - system is a three-phase system with a balanced -connected source and a balanced -connected load. Line voltage = phase voltage. Line current = 3 phase current. Line current lags phase current by 30 o 55

56 Example (4) Three Phase Circuits A balanced Δ-connected load having an impedance Z Δ = 20 j15ω is connected to a Δ -connected, positive-sequence generator having V ab = o V. Calculate the phase currents of the load and the line currents. 56

57 Balanced Delta-Star Connection A balanced -Y system is a three-phase system with a balanced -connected source and a balanced Y-connected load. Line voltage = 3 phase voltage. Line voltage leads phase voltage by 30 o Line current = phase current. 57

58 Example (5) Three Phase Circuits A balanced Y-connected load with a phase resistance of 40 and a reactance of 25 is supplied by a balanced, positive sequence Δ connected source with a line voltage of 210V. Calculate the phase currents. (Hint: Use V ab as reference). 58

59 Example (6) Three Phase Circuits Determine the total average power, reactive power, and complex power at the source and at the load in the three-wire Y-Y system shown. 59

60 Unbalanced Three Phase System An unbalanced system is due to unbalanced voltage sources or an unbalanced load. Unbalanced three phase system is solved by direct application of node or mesh analysis. I a = V AN Z 1 I c = V CN Z 3 I b = V BN Z 2 I n = I a + I b + I c 60

61 Example (7) Three Phase Circuits The unbalanced Y-load shown has balanced voltages of 100 V and the acb sequence. Calculate the line currents and the neutral current if Z 1 = 15 Ω, Z 2 = 10 + j5 Ω, and Z 3 = 6 j8 Ω. 61