Phase Angle Help (page 2)
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 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°.
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.
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.
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.
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°.
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.
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°.
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°.
Vector Representations Of Phase
If a sine wave X is leading a sine wave Y by x degrees, then the two waves can be drawn as vectors, with vector X oriented x degrees counterclockwise from vector Y . If wave X lags Y by y degrees, then X is oriented y 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.
Figure 13-17 shows four phase relationships between waves X and Y . Wave X always has twice the amplitude of wave Y , so vector X is always twice as long as vector Y . In part a , wave X is in phase with wave Y . In part b , wave X leads wave Y by 90°. In part c , waves X and Y are 180° opposite in phase. In part d , wave X lags wave Y by 90°.
Fig. 13-17 . Vector representations of phase, (a) waves X and Y are in phase; (b) wave X leads wave Y by 90 degrees; (c) waves X and Y are in phase opposition; (d) wave X lags wave Y by 90 degrees.
In all cases, 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.
In a sine wave, the vector magnitude stays the same at all times. If the waveform is not sinusoidal, the vector magnitude is greater in some directions than in others. As you can guess, there exist an infinite number of variations on this theme, and some of them can get complicated.
Phase Angle Practice Problem
Suppose that there are three waves, called X, Y , and Z Wave X leads wave Y by 0.5000 rad; wave Y leads wave Z by precisely one-eighth cycle. By how many degrees does wave X lead or lag wave Z?
To solve this, let’s 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 Z by 28.65° + 45.00°, or 73.65°.
Suppose that there are three waves X, Y , and Z Wave X leads wave Y by 0.5000 rad; wave Y lags wave Z by precisely one-eighth cycle. By how many degrees does wave X lead or lag wave Z?
The difference in phase between X and Y in this problem is the same as that in the preceding 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 Z 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°.
As you can see, phase relationships can get confusing. It’s the same sort of thing that happens when you talk about negative numbers. Which number is larger than which? It depends on point of view. If it helps you to draw pictures of waves when thinking about phase, then by all means go ahead.
Practice problems of these concepts can be found at: Alternating Current Practice Test
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