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Phase Angle Help (page 2)

By — McGraw-Hill Professional
Updated on Aug 30, 2011

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).

Waves and Phase Phase Angle Vector Representations Of Phase

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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