ENE 104 Electric Circuit Theory

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1 Electric Circuit Theory Lecture 11: : Dejwoot KHAWPARSUTH

2 Objectives : Ch12 Page 2 single-phase and polyphase systems Y- and Δ- connected three-phase system per-phase analysis of three-phase systems

3 ntroduction: Page 3 An example set of three voltages, each of which is 120 o out of phase with the other two. As can be seen, only one of the voltages will be zero at a particular instant.

4 Double-Subscript Notation: Page 4 (a) The definition of the voltage ab. (b) ad = ab + bc + cd = ab + cd.

5 Double-Subscript Notation: Page 5 an bn cn

6 Double-Subscript Notation: Page 6 ab an 100 nb 0 an ( 50 j86.6) bn.

7 Practice: 12.1 Page 7 Let ab = 100 0, bd = 40 80, and ca = Find (a) ad ; (b) bc ; (c) cd

8 Practice: 12.1 Page 8

9 Practice: 12.2 Page 9 Refer to the circuit of figure below and let fj = 3 A, de = 2 A, and hd = 6 A. Find (a), cd ; (b) ef ; (c) ij

10 Practice: 12.2 Page 10

11 Single-Phase Three-Wire System: Page 11

12 Single-Phase Three-Wire System: Page 12 Consider: Since Then an = nb aa Z an p And nn = Bb + Aa = Bb - aa = 0 Bb Z nb p

13 Example: Page 13 Effect of Finite Wire mpedance: Analyze the system below and determine the power delivered to each of the three loads as well as the power lost in the neutral wire and each of the two lines.

14 Example: Page 14 ENE ) 100( ) 3( ) 50( ) 100( 10) (20 0 ) 3( ) 50( j

15 Example: Page The average power drawn by each load: A. A. A. P P P20 j W. W. W.

16 Example: Page 16 The loss in each of the wires: P aa P bb P nn W. 108 W W. A. A. A.

17 Example: Page 17 The transmission efficiency, : total power delivered to total power generated % load

18 Practice: 12.3 Page 18 Modified Figure below by adding a 1.5 Ω resistance to each of the two outer lines, and a 2.5 Ω resistance to the neutral wire. Find the average power delivered to each of the three loads.

19 Practice: 12.3 Page 19

20 Practice: 12.3 Page 20

21 Three-Phase Y-Y Connection: Page 21 A balanced three-phase system: may be defined as having an bn cn and an bn cn 0 A Y-connected three-phase four-wire source.

22 Three-Phase Y-Y Connection: Page 22 Positive, or abc, phase sequence. Negative, or cba, phase sequence.

23 Three-Phase Y-Y Connection: Page 23 Line-to-Line oltage: ab ab an p 3 0 p bn 30 p 120

24 Three-Phase Y-Y Connection: Page 24 Line-to-Line oltage: ab 3 p 30 bc 90 3 p ca p

25 Three-Phase Y-Y Connection: Page 25 bb aa Z Z an p bn p an aa 120 Z p 120 cc aa 240 A balanced three-phase system, connected Y-Y and including a neutral. Therefore: Nn aa bb cc 0

26 Example: Page 26 Find the currents, voltages and the total power dissipated in the load

27 Example: Page 27 Per-phase : an bn cn ab p bc 90 3 p ca p. aa bb cc an 2 60 Z p

28 Example: Page 28 Per-phase : v AN i AN cos(120t 0) 2 2 cos(120t 60)

29 Example: Page 29 even though phase voltages and currents have zero value at specific instants in time, the instantaneous power delivered to the total load is never zero. p A ( t) v AN i AN 800 cos(120t )cos(120t 400[cos( 60) cos(240t 60) 60)] cos(240t 60)] p B ( t) cos(240t 300)] p C ( t) cos(240t 180)]

30 Example: Page 30 the instantaneous power absorbed by the total load is therefore p( t) pa( t) pb( t) pc ( t) 600W

31 Practice: 12.4 Page 31 A balanced three-phase three-wire system has a Y-connected load. Each phase contains three loads in parallel: j100 Ω, 100 Ω, and 50 + j50 Ω. Assume positive phase sequence with ab = Find (a) an ; (b) aa ; (c) the total power drawn by the load

32 Practice: 12.4 Page 32

33 Example: A balanced three-phase system with a line voltage of 300 is supplying a balanced Y- connected load with 1200W at a leading PF of 0.8. Find the line current and the per-phase load impedance. Page 33 The phase voltage is p The per-phase power is W. 3

34 Example: Page 34 1 P 2 p m m cos( ). P p eff W. 3 eff cos( ) The line current is Pp 400 L 300 p cos( ) The phase impedance is 2.89 A. Z p p L

35 Example: Page 35 The PF is 0.8 leading, so the impedance phase angle is cos 1 (0.8) 36.9 thus Z p

36 Practice: 12.5 Page 36 A balanced three-phase three-wire system has a line voltage of 500. Two balanced Y-connected loads are present. One is a capacitive load with 7 j2 Ω per phase, and the other is an inductive load with 4 + j2 Ω per phase. Find (a) the phase voltage; (b) the line current; (c) the total power drawn by the load; (d) the power factor at which the source is operating.

37 Practice: 12.5 Page 37

38 Example: A balanced 600-W lighting load is added (in parallel) to the previous example. Determine the new line current. Page per-phase circuit Z p

39 Example: The amplitude of the lighting current is determined by Page 39 P eff eff cos( ) cos 0 3 So assume And the line current is L A A. A.

40 Practice: 12.6 Page 40 Three balanced Y-connected loads are installed on a balanced three-phase four-wire system. Load 1 draws a total power of 6kW at unity PF, load 2 requires 10kA at PF=0.96 lagging, and load 3 needs 7kW at 0.85 lagging. f the phase voltage at the load is 135, if each line has a resistance of 0.1 Ω, and if the neutral has a resistance of 1 Ω, find (a) the total power drawn by the load; (b) the combined PF of the load; (c) the total power lost in the four lines; (d) the phase voltage at the source; (e) the power factor at which the source is operating.

41 Practice: 12.6 Page 41

42 Practice: 12.6 Page 42

43 The Delta ( ) Connection: Page 43 L ab bc ca p an bn cn A balanced D-connected load is present on a three-wire three-phase system. The source happens to be Y- connected. Where L 3 p and ab 30 3 p

44 The Delta ( ) Connection: The phase current Page 44 p AB BC AB CA Z Z Z ab p bc p ca BC p CA the line current: and L aa aa AB CA bb L 3 p cc

45 The Delta ( ) Connection: An example with an inductive load Page 45 p AB BC CA L aa bb cc aa AB L 3 p CA

46 -Connected: Page 46 P p p L L p cos p cos L cos 3 total power, P 3P p 3 L L cos

47 Y-connected: Page 47 P p p p L 3 p L cos cos L cos total power, P 3P p 3 L L cos

48 Practice: 12.7 Page 48 Each phase of a balanced three-phase Δ- connected load consists of a 0.2 H inductor in series with the parallel combination of a 5 μf capacitor and a 200 Ω resistance. Assume zero line resistance and a phase voltage of 200 at ω = 400 rad/s. Find (a) the phase current; (b) the line current; (c) the total power absorbed by the load

49 Practice: 12.7 Page 49

50 Example 12.5: Determine the amplitude of the line current in a three-phase system with a line voltage of 300 that supplies 1200 W to a -connected load at a lagging PF of 0.8 Page 50 P p p L L p cos p cos L cos 3 ENE 104 L.89 A., Z p

51 Example 12.6: Determine the amplitude of the line current in a three-phase system with a line voltage of 300 that supplies 1200 W to a Y-connected load at a lagging PF of 0.8 Page 51 P p p p L 3 p L cos cos L cos L ENE A., Z p Z Y Z D 3

52 Practice: 12.8 Page 52 A balanced three-phase three-wire system is terminated with two Δ-connected load in parallel. Load 1 draws 40kA at lagging PF of 0.8, while load 2 absorbs 24 kw at a leading PF of 0.9. Assume no line resistance, and let ab = Find (a) the total power drawn by the load; (b) the phase current AB1 for the lagging load; (c) AB2 ; (d) aa.

53 Practice: 12.8 Page 53

54 Power Measurement: Page 54 wattmeter

55 Power Measurement: Page 55 P 2 cos ang2 ang

56 Practice: 12.9 Page 56 Determine the wattmeter reading in Figure below, state weather or not the potential coil had to be reversed in order to obtain an upscale reading, and identify the device or devices absorbing or generating this power. The (+) terminal of the wattmeter is connected to: (a) x; (b) y; (c) z.

57 Practice: 12.9 Page 57

58 Practice: 12.9 Page 58

59 Power Measurement: The wattmeter in a Three-Phase System: Page 59

60 Power Measurement: The wattmeter in a Three-Phase System: Page 60 P A 1 T T 0 v Ax i aa dt v v v Ax Bx Cx v v v AN BN CN v v v Nx Nx Nx ENE 104 P P A P B P C 1 T T 0 ( v Ax i aa v Bx i bb v Cx i cc ) dt

61 Power Measurement: Page 61 ENE 104 The wattmeter in a Three-Phase System: dt i v i v i v T dt i i i v T dt i v i v i v T dt i v i v i v T P P P P cc CN bb BN aa T AN T cc bb aa Nx cc CN bb BN aa T AN cc Cx bb Bx aa T Ax C B A ) ( 1 ) ( 1 ) ( 1 ) ( Nx CN Cx Nx BN Bx Nx AN Ax v v v v v v v v v

62 Power Measurement: The Two-Wattmeter Method: Page 62 P P 1 2 AB CB aa cc cos cos ang ang AB CB ang ang aa cc

63 Power Measurement: Page 63 The Two-Wattmeter Method: in the case of a balance load: we can find PF P 1 L AB L cos aa cos 30 ang AB ang aa ENE 104 P 2 CB cc cos L L cos 30 ang CB ang cc

64 Power Measurement: Page 64 The Two-Wattmeter Method: in the case of a balance load: we can find PF ENE 104 P P 1 2 cos cos tan P P 2 2 P P 1 1

65 Example 12.7: Page 65 The balanced load is fed by a balanced threephase system having ab and positive phase sequence. Find the reading of each wattmeter and the total power drawn by the load. ab bc ca

66 Example: Page 66 ab ca ac aa an j15 4 j15 P 1 ac aa cos cos ang ac ang aa W

67 Example: Page 67 bc aa bb P 2 bc bb cos cos ang bc ang bb W

68 Practice: Page 68 For the circuit of Figure below, let the loads be Z A = Ω, Z B = Ω, Z C = Ω, AB = rms with (+) phase sequence, and locate point x at C. Find (a) P A ; (b) P B ; (c)p C

69 Practice: Page 69

70 Practice: Page 70

71 Example: Final 2/47 AC Circuit Power Analysis Page 71 เม อ แหล งจ ายในวงจร เป นแบบ balanced และ positive phase sequence จงหา,, และ the total complex power aa supplied by the source bb cc a A o rms + _ 10 ohm b -j10 ohm + _ B j5 ohm + _ c C

72 Example: Final 2/47 Page 72 an _ o rms a A bn cn ENE ab 3 p 30 bc ca CA BC AC Circuit Power Analysis + _ + _ b c B 10 ohm j5 ohm -j10 ohm ab AB j j10 C

73 Example: Final 2/47 AC Circuit Power Analysis Page 73 AB _ o rms a A BC _ b B 10 ohm -j10 ohm CA _ j5 ohm c C aa AB CA bb BC AB cc CA BC S total AB * AB BC * BC CA * CA

AC Circuit Power Analysis Example: Final 2/46 เม อ Wattmeters ในวงจร อ านค าได W. W 2 139,196.31 และ W.

74 AC Circuit Power Analysis Example: Final 2/46 เม อ Wattmeters ในวงจร อ านค าได W. W 2 139, และ W. เม อค า the magnitude ของ the line voltage ม ค า จงหา Z W 1 37, Page 74 tan 3 P P 2 2 P P 1 1

75 Ex: Page 75 AC Circuit Power Analysis a A o rms Wm1 - Z A = _ Z A Z B = b B + _ Z C + _ o 120 o c Wm C Z B Z C = 80 0 ให หา AB, BC, AB, BC, aa, cc และค าท อ านได ท wattmeter ท งสอง

76 Ex: Page 76 AC Circuit Power Analysis a A + _ o rms Wm1 - Z A + _ b B Z C o + _ o c Wm C Z B

77 Hw: Page 77 ENE 104 AC Circuit Power Analysis

78 Reference: Circuit Analysis and Electrical Engineering Page 78 W.H. Hayt, Jr., J.E. Kemmerly, S.M. Durbin, Engineering Circuit Analysis, Sixth Edition. Copyright 2002 McGraw-Hill. All rights reserved.

LO 1: Three Phase Circuits

Course: EEL 2043 Principles of Electric Machines Class Instructor: Dr. Haris M. Khalid Email: hkhalid@hct.ac.ae Webpage: www.harismkhalid.com LO 1: Three Phase Circuits Three Phase AC System Three phase