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4.3 Phasors: multiphase power

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This module is part of the collection, A First Course in Electrical and Computer Engineering . The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

The electrical service to your home is a two-phase service.This means that two 110 volt, 60 Hz lines, plus neutral, terminate in the panel. The linesare π radians (180) out of phase, so we can write them as this in circuits and power courses.

x 1 ( t ) = 110 c o s [ 2 π ( 60 ) t + φ ] = Re { 110 e j [ 2 π ( 60 ) t + φ ] } = Re { X 1 e j 2 π ( 60 ) t } X 1 = 110 e j φ x 2 ( t ) = 110 c o s [ 2 π ( 60 ) t + φ + π ] = Re { 110 e j [ 2 π ( 60 ) t + φ + π ] } = Re { X 2 e j 2 π ( 60 ) t } X 2 = 110 e j ( φ + π ) .

These two voltages are illustrated as the phasors X 1 and X 2 in [link] .

Figure one is a cartesian graph with a line passing through the origin from the third to first quadrant with a shallow positive slope. The line on the left side is labeled X_2, and on the right side is labeled X_1. The angle from the positive horizontal axis to the portion of the line in the first quadrant is measured and labeled Φ. The angle from the positive horizontal axis to the portion of the line in the third quadrant (angle measured in a counter-clockwise direction) is labeled Φ+π. Above the line in the first quadrant is the number 110. Figure one is a cartesian graph with a line passing through the origin from the third to first quadrant with a shallow positive slope. The line on the left side is labeled X_2, and on the right side is labeled X_1. The angle from the positive horizontal axis to the portion of the line in the first quadrant is measured and labeled Φ. The angle from the positive horizontal axis to the portion of the line in the third quadrant (angle measured in a counter-clockwise direction) is labeled Φ+π. Above the line in the first quadrant is the number 110.
Phasors in Two-Phase Power

You may use x 1 ( t ) to drive your clock radio or your toaster and the difference between x 1 ( t ) and x 2 ( t ) to drive your range or dryer:

x 1 ( t ) - x 2 ( t ) = 220 cos [ 2 π ( 60 ) t + φ ] .

The phasor representation of this difference is

X 1 - X 2 = 220 e j φ .

The breakers in a breaker box span the x 1 - t o -neutral bus for 110 volts and the x 1 -to- x 2 buses for 220 volts.

Figure two is a cartesian graph with three arrows pointing in various directions. The first is labeled X_1 and is drawn along the horizontal axis to the right. The second is labeled X_2 and is drawn with a sharp negative slope up into the second quadrant. The third is labeled X_3 and is drawn with sharp positive slope into the fourth quadrant. Figure two is a cartesian graph with three arrows pointing in various directions. The first is labeled X_1 and is drawn along the horizontal axis to the right. The second is labeled X_2 and is drawn with a sharp negative slope up into the second quadrant. The third is labeled X_3 and is drawn with sharp positive slope into the fourth quadrant.
Three-Phase Power

Constant Power. Two- and three-phase power generalizes in an obvious way to N-phase power. In such a scheme, the N signals x n ( n = 0 , 1 , ... , N - 1 ) are

x n ( t ) = A cos ( ω t + 2 π N n ) = Re [ A e j 2 π n / N e j ω t ] ↔ X n = A e j 2 π n / N .

The phasors X n are A e j 2 π ( n / N ) . The sum of all N signals is zero:

∑ n = 0 N - 1 x n ( t ) = Re { A ∑ n = 0 N - 1 e j 2 π n / N e j ω t } = Re { A 1 - e j 2 π 1 - e j 2 π / N e j ω t } = 0 .

But what about the sum of the instantaneous powers? Define the instantaneous power of the n t h signal to be

p n ( t ) = x n 2 ( t ) = A 2 cos 2 ( ω t + 2 π N n ) = A 2 2 + A 2 2 cos ( 2 ω t + 2 2 π N n ) = A 2 2 + Re { A 2 2 e j ( 2 π / N ) 2 n e j 2 ω t } .

The sum of all instantaneous powers is (see [link] )

P = ∑ n = 0 N - 1 p n ( t ) = N A 2 2 ,

and this is independent of time!
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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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