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Phase Angle Help

By — McGraw-Hill Professional
Updated on Sep 6, 2011

Introduction

Phase angle is an expression of the displacement between two waves having identical frequencies. There are various ways of defining this. Phase angles are usually expressed as values φ such that 0° ≤ φ < 360°. In radians, this range is 0 ≤ φ < 2 π . Once in awhile you will hear about phase angles specified over a range of −180° < φ ≤ + 180°. In radians, this range is − π < φ ≤ + π . Phase angle figures can be defined only for pairs of waves whose frequencies are the same. If the frequencies differ, the phase changes from moment to moment and cannot be denoted as a specific number.

Phase Coincidence

Phase coincidence means that two waves begin at exactly the same moment. They are “lined up.” This is shown in Fig. 13-13 for two waves having different amplitudes. (If the amplitudes were the same, you would see only one wave.) The phase difference in this case is 0°.

Alternating Current Phase Angle Phase Opposition

Fig. 13-13 . Two sine waves in phase coincidence.

If two sine waves are in phase coincidence, the peak amplitude of the resulting wave, which also will be a sine wave, is equal to the sum of the peak amplitudes of the two composite waves. The phase of the resultant is the same as that of the composite waves.

Phase Opposition

When two sine waves begin exactly one-half cycle, or 180°, apart, they are said to be in phase opposition . This is illustrated by the drawing of Fig. 13-14.

 

Alternating Current Phase Angle Phase Opposition

Fig. 13-14 . Two sine waves in phase opposition.

If two sine waves have the same amplitude and are in phase opposition, they cancel each other out because the instantaneous amplitudes of the two waves are equal and opposite at every moment in time.

If two sine waves have different amplitudes and are in phase opposition, the peak value of the resulting wave, which is a sine wave, is equal to the difference between the peak values of the two composite waves. The phase of the resultant is the same as the phase of the stronger of the two composite waves.

Leading Phase

Suppose that there are two sine waves, wave X and wave Y , with identical frequencies. If wave X begins a fraction of a cycle earlier than wave Y , then wave X is said to be leading wave Y in phase. For this to be true, X must begin its cycle less than 180° before Y . Figure 13-15 shows wave X leading wave Y by 90°. The difference can be anything greater than 0°, up to but not including 180°.

Alternating Current Phase Angle Leading Phase

Fig. 13-15 . Wave X leads wave Y by 90°.

Leading phase is sometimes expressed as a phase angle φ such that 0° < φ < +180°. In radians, this is 0 < φ < + π . If we say that wave X has a phase of + π /2 rad relative to wave Y , we mean that wave X leads wave Y by π /2 rad.

Lagging Phase

Suppose that wave X begins its cycle more than 180° but less than 360° ahead of wave Y In this situation, it is easier to imagine that wave X starts its cycle later than wave Y by some value between but not including 0° and 180°. Then wave X is lagging wave Y . Figure 13-16 shows wave X lagging wave Y by 90°. The difference can be anything between but not including 0° and 180°.

Alternating Current Phase Angle Vector Representations Of Phase

Fig. 13-16 . Wave X lags wave Y by 90°.

Lagging phase is sometimes expressed as a negative angle φ such that −180° < φ < 0°. In radians, this is − π < φ < 0. If we say that wave X has a phase of −45° relative to wave Y , we mean that wave X lags wave Y by 45°.

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