Chapter 12 Polyphase Circuits 1 Copyright 2013 The

Chapter 12 Polyphase Circuits 1 Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.

Polyphase / Three-phase Sources Polyphase (in practice, three-phase) voltage sources are important for transmission over the grid, and for large industrial loads. These three sources are 120° out of phase with each other. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 2

Double-Subscript Voltage The notation Vab indicates the voltage between point a (+) and point b (−). Indicating the + and − terminals is redundant if double-subscript voltages are used. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 3

KVL with Double-Subscripts KVL equations can be written correctly without reference to a circuit diagram: Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 4

Example: Voltage Calculations This is an example three-phase source with a neutral. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 5

Currents and Double-Subscripts The double subscript notation Iab means the current from a to b by the direct path When the direct path is not obvious (e. g. between c and d) we need to use a different notation. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 6

Household Power: not Polyphase The typical North American household is provided single-phase 3 -wire power, where V 1≅110 V and Vab=2 V 1. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 7

Current in Neutral Wire When the A and B loads are balanced, the neutral wire from n to N carries no current: Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 8

Example: Effect of Line Resistance 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. Answer: P 50 = 206 W P 100 =117 W P 20+j 10 = 1763 W Pa. A = 126 W Pb. B = 108 W Pn. N =3 W Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 9

Three-Phase Y-Y Connection A balanced three phase source has and Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 10

Phase Sequences The abc (positive) and cba (negative) sequences: Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 11

Line-to-Line Voltages The line voltages have amplitude Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 12

Balanced Y-Y: Zero Neutral Current For a balanced source and balanced loads, the neutral wire current INn=0 and so the neutral wire could have any impedance, including ∞. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 13

Example: Y-Y Power Find the total power delivered to the loads. Answer: P=600 W Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 14

Example: Per-phase Analysis A balanced three-phase system with a line voltage of 300 V is supplying a balanced Yconnected load with 1200 W at a leading PF of 0. 8. Find the line current and the per-phase load impedance. Answer: IL = 2. 89 A, Zp = 60 ej 36. 9◦ Ω Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 15

The Delta Connection A delta-connected load is also commonly used (note the absence of the neutral wire). Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 16

Line and Phase Currents for a Delta Load An example phasor diagram for an inductive phase impedance. Note that Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 17

Power Measurements: The Wattmeter The wattmeter is a four-terminal device that measures power delivered to the network if connected as shown: the wattmeter Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 18

Measuring Three-Phase Power The sum of the powers measured by the wattmeters is the total power delivered. The common node x is arbitrary. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 19

The Two-Wattmeter Method One wattmeter can be eliminated if the point x is moved to a line (as shown, B). Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 20