# Basic Components and Electric Circuits

Chapter 12 Polyphase Circuits 1 Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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 McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2 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 McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3 KVL equations can be written correctly without

reference to a circuit diagram: Vad Vab Vbd Vad Vab Vbc Vcd Vab Vba Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 4 Vab Van Vnb

Van Vbn This is an example three-phase source with a neutral. 173.2 30 o Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 5

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 McGraw-Hill Companies, Inc. Permission required for reproduction or display. 6

The typical North American household is provided single-phase 3-wire power, where V1110 V and Vab=2V1. Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 7 When the A and B loads are balanced, the neutral wire from n to N carries no

current: Van Vnb I aA I Bb Zp Zp I nN I Bb I aA I Bb I Aa 0 Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

8 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: P50 = 206 W P100 =117 W P20+j10 = 1763 W PaA = 126 W PbB = 108 W

PnN =3 W Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 9 A balanced three phase source has Van Vbn Vcn and

Van Vbn Vcn 0 Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10 The abc (positive) and cba (negative) sequences:

Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11 The line voltages have amplitude VL 3Vp Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

12 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 McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13

Find the total power delivered to the loads. Answer: P=600 W Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14 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 = 60ej36.9 Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 15 A delta-connected load is also commonly used (note the absence of the neutral wire).

Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 16 An example phasor diagram for an inductive phase impedance. Note that IL 3Ip

Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 17 The wattmeter is a four-terminal device that measures power delivered to the network if connected as shown: the wattmeter Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18 The sum of the powers measured by the wattmeters is the total power delivered. The common node x is arbitrary. Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for

reproduction or display. 19 One wattmeter can be eliminated if the point x is moved to a line (as shown, B). Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for

reproduction or display. 20

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