Introduction to Phase Angle
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 ø such that 0° ≤ ø < 360°. In radians, that range is 0 ≤ ø < 2π. Once in a while, you will hear about phase angles specified over a range of –180° < ø + 180°. In radians, that range is –π < ø ≤ + π. Phase angle, also called phase difference, can be defined only for pairs of waves whose frequencies are the same.
Phase Coincidence
Two waves are in phase coincidence if and only if they have the same frequency and each cycle begins at exactly the same instant in time. Graphically, waves in phase coincidence appear “lined up.” This is shown in Fig. 93 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°.
If two sine waves are in phase coincidence, the peak amplitude of the resultant wave, which is also 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 have the same frequency and they begin exactly half a cycle, or 180°, apart, they are said to be in phase opposition. This is illustrated in Fig. 94. 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 resultant 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 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 95 shows wave X leading wave Y by 90°. When one wave leads another, the phase difference can be anything greater than 0° but less than 180°.
Leading phase is sometimes expressed as a positive phase angle ø such that 0° < ø < + 180°. In radians, this is 0 < ø < + 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 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 96 shows wave X lagging wave Y by 90°.
Lagging phase is sometimes expressed as a negative angle ø such that –180° < ø 0°. In radians, this is stated as –π < ø < 0. If we say that wave X has a phase of –90° relative to wave Y, we mean that wave X lags wave Y by 90°.

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