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. 9-3 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°.
Fig. 9-3. Two waves in phase coincidence. Graphically, they follow each other along.
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. 9-4. 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.
Fig. 9–4. Two waves in phase opposition. Graphically, they are ½ cycle apart.
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 9-5 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°.
Fig. 9–5. Wave X leads wave Y by 90°. Graphically, X appears displaced ¼ cycle to the left of (earlier than) Y.
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 9-6 shows wave X lagging wave Y by 90°.
Fig. 9–6. Wave Xlags wave Y by 90°. Graphically, X appears displaced ¼ cycle to the right of (later than) Y.
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°.
When Is A Lead Not A Lead?
If, while working out a phase problem, you find that wave X differs in phase from wave Y by some angle ø that does not fall into the range –180° < ø + 180° (-π < ø ≤ +π rad), you should reduce the phase difference, either positive or negative, to something that falls in this range. This can be done by adding or subtracting multiples of 360 (2π rad), or by adding or subtracting whole cycles until an acceptable phase difference figure is found.
Suppose, for example, you are told that wave X leads wave Y by exactly 2.75 cycles of phase. That’s 2.75 × 360°, or 990°. If you subtract three complete cycles from this, or 3 × 360° = 1080°, you end up with the fact that wave X leads wave Y by -90°. This is the same as saying that wave X lags wave Y by 90°.
Vector Representations Of Phase
If a sine wave X leads a sine wave Y by ø degrees, then the two waves can be drawn as vectors, with vector X oriented ø degrees counterclockwise from vector Y. The waves, when expressed as vectors, are denoted in non-italicized boldface. If wave X lags Y by ø degrees, then X is oriented ø degrees clockwise from Y. If two waves are in phase, their vectors overlap (line up). If they are in phase opposition, they point in exactly opposite directions.
The drawings of Fig. 9-7 show four phase relationships between waves X and Y. Wave X always has twice the amplitude of wave Y, so that vector X is always twice as long as vector Y. At A, wave X is in phase with wave Y. At B, wave X leads wave Y by 90° (π/2 rad). At C, waves X and Y are in phase opposition. In drawing D, wave X lags wave Y by 90° (π/2 rad).
Fig. 9–7. Vector representations of phase difference. At A, wave X is in phase with wave Y . At B, X leads Y by 90°. At C, X and Y are in phase opposition. At D, X lags Y by 90°.
In all cases, with the passage of time, the vectors rotate counterclockwise at the rate of one complete circle per wave cycle. Mathematically, a sine wave is a vector that goes around and around, just like the ball goes around and around your head when you put it on a string and whirl it. The sine wave is a representation of circular motion because the sine function is a circular function.
Phase Angle Practice Problems
Practice 1
Suppose there are three waves, called X, Y, and Z. Imagine that wave X leads wave Y by 0.5000 rad, while wave Y leads wave Z by precisely 1/8 cycle. By how many degrees does wave X lead or lag wave Z?
Solution 1
To solve this, convert all phase angle measures to degrees. One radian is approximately equal to 57.296°. Therefore, 0.5000 rad = 57.296° × 0.5000 = 28.65° (to four significant figures). One-eighth of a cycle is equal to 45.00° (that is 360°/8.000). The phase angles therefore add up, so wave X leads wave Y by 28.65° + 45.00°, or 73.65°.
Practice 2
Suppose there are three waves X, Y, and Z. Imagine that wave X leads wave Y by 0.5000 rad; wave Y lags wave Z by precisely 1/8 cycle. By how many degrees does wave X lead or lag wave Z?
Solution 2
The difference in phase between X and Y in this scenario is the same as that in the previous problem, namely 28.65°. The difference between Y and Z is also the same, but in the opposite sense. Wave Y lags wave Z by 45.00°. This is the same as saying that wave Y leads wave Z by -45.00°. Thus, wave X leads wave by 28.65° + (-45.00°), which is equivalent to 28.65° - 45.00° or = 16.35°. It is better in this case to say that wave X lags wave Z by 16.35°, or that wave Z leads wave X by 16.35°.
Practice problems for these concepts can be found at: Waves and Phase Practice Test
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From Trigonometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.