Degrees and Radians of Phase
Degrees Of Phase
One method of specifying fractions of an AC cycle is to divide it into 360 equal increments called degrees of phase, symbolized ° or deg (but it’s okay to write out the whole word “degrees”). The value 0° is assigned to the point in the cycle where the signal level is zero and positivegoing. The same point on the next cycle is given the value 360°. Halfway through the cycle is 180°; a quarter cycle is 90°; 1/8 cycle is 45°. This is illustrated in Fig. 92.
Radians Of Phase
The other method of specifying fractions of an AC cycle is to divide it into 2π, or approximately 6.2832, equal parts. This is the number of radii of a circle that can be laid endtoend around the circumference. One radian of phase, symbolized rad (although you can write out the whole word “radian”), is equal to about 57.29583°. Physicists use the radian more often than the degree when talking about fractional parts of an AC cycle.
Sometimes, the frequency of an AC wave is measured in radians per second (rad/s), rather than in hertz (cycles per second). Because there are 2π radians in a complete cycle of 360°, the angular frequency of a wave, in radians per second, is equal to 2π times the frequency in hertz. Angular frequency is symbolized by the lowercase, italicized Greek letter omega (ω). Angular frequency can also be expressed in degrees per second (deg/s or °/s). The angular frequency of a wave in degrees per second is equal to 360 times the frequency in hertz, or 57.29583 times the angular frequency in radians per second.
Instantaneous Amplitude
In a sine wave, the amplitude varies with time, over the course of one complete cycle, according to the sine of the number of degrees or radians measured from the start of the wave cycle, or the point on the wave where the amplitude is zero and positivegoing.
If the maximum amplitude, also called the peak amplitude, that a wave X attains is x _{pk} units (volts, amperes, or whatever), then the instantaneous amplitude, denoted x _{i} , at any instant of time is:
x _{i} = x _{pk} sin ø
where ø is the number of degrees or radians between the start of the cycle and the specified instant in time.
Alternating Current Practice Problems
Practice 1
What is the angular frequency of household AC in radians per second? Assume the frequency of utility AC is 60.0 Hz.
Solution 1
Multiply the frequency in hertz by 2π. If this value is taken as 6.2832, then the angular frequency is:
ω = 6.2832 × 60.0 = 376.992 rad/s
This should be rounded off to 377 rad/s, because our input data is given only to three significant figures.
Practice 2
A certain wave has an angular frequency of 3.8865 × 10 ^{5} rad/s. What is the frequency in kilohertz? Express the answer to three significant figures.
Solution 2
To solve this, first find the frequency in hertz. This requires that the angular frequency, in radians per second, be divided by 2π, which is approximately 6.2832. The frequency f _{Hz} is therefore:
f _{Hz} = (3.8865 × 10 ^{5} )/6.2832
= 6.1855 × 10 ^{4} Hz
To obtain the frequency in kilohertz, divide by 10 ^{3} , and then round off to three significant figures:
f _{kHz} = .1855 × 10 ^{4} /10 ^{3}
= 61.855 kHz
= 61.9 kHz
Practice problems for these concepts can be found at: Waves and Phases Practice Test
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