Expressions of Phase Difference

The phase difference, also called the phase angle, between two waves can have meaning only when those two waves have identical frequencies. If the frequencies differ, even by just a little bit, the relative phase constantly changes, and it’s impossible to specify a value for it. In the following discussions of phase angle, let’s assume that the two waves always have identical frequencies.

Phase Coincidence

Phase coincidence means that two waves begin at exactly the same moment. They are “lined up.” This is shown in following figure for two waves having different amplitudes. The phase difference in this case is 0°. You could say it’s some whole-number multiple of 360°, too—but engineers and technicians rarely speak of any phase angle of less than 0° or more than 360°. If two sine waves are in phase coincidence, and if neither wave has dc superimposed, then the resultant is a sine wave with positive or negative peak amplitudes equal to the sum of the positive and negative peak amplitudes of the composite waves. The phase of the resultant is the same as that of the composite waves.

Two sine waves in phase coincidence.

Phase Opposition

When two sine waves begin exactly 1⁄ 2 cycle, or 180°, apart, they are said to be in phase opposition This is illustrated by the drawing of following figure. In this situation, engineers sometimes say that the waves are out of phase, although this expression is a little nebulous because it could be taken to mean some phase difference other than 180°.

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

If two sine waves are in phase opposition, and if neither wave has dc superimposed, then the resultant is a sine wave with positive or negative peak amplitudes equal to the difference between the positive and negative peak amplitudes of the composite waves. The phase of the resultant is the same as the phase of the stronger of the two composite waves.

Any sine wave without superimposed dc has the unique property that, if its phase is shifted by 180°, the resultant wave is the same as turning the original wave upside down. Not all waveforms have this property. Perfect square waves do, but some rectangular and sawtooth waves don’t, and irregular waveforms almost never do.

Intermediate Phase Differences

Two sine waves can differ in phase by any amount from 0° (phase coincidence), through 90° ( phase quadrature, meaning a difference a quarter of a cycle), 180° (phase opposition), 270° (phase quadrature again), to 360° (phase coincidence again).

Imagine two sine waves, called wave X and wave Y, with identical frequency. 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. Following Figure shows wave X leading wave Y by 90°.

Note that if wave X (the dashed line in following figure) is leading wave Y (the solid line), then wave X is displaced to the left of wave Y. In a time-domain graph or display, displacement to the left represents earlier moments in time, and displacement to the right represents later moments in time.

Wave X leads wave Y by 90° of phase (1⁄4 of a cycle)

Lagging Phase

Suppose that some sine 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 0° and 180°. Then wave X is not leading, but lagging, wave Y. Following Figure shows wave X lagging wave Y by 90°.

Wave X lags wave Y by 90° of phase (1⁄4 of a cycle)