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 1317 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. 1317 . 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
Problem 1
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 oneeighth cycle. By how many degrees does wave X lead or lag wave Z?
Solution 1
To solve this, let’s convert all phaseangle 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). Oneeighth 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°.
Problem 2
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 oneeighth 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 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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