Alternating Current Help (page 2)
Introduction to Alternating Current
Trigonometry is important in electricity and electronics, particularly in the analysis of alternating current (AC) and waves.
In electrical applications, direct current (DC) has a polarity, or direction, that stays the same over a long period of time. Although the intensity of the current might vary from moment to moment, the electrons always flow in the same direction through the circuit. In alternating current (AC), the polarity reverses repeatedly and periodically. The electrons move back and forth; the current ebbs and flows.
Period and Frequency
In a periodic AC wave, the kind discussed in this chapter, the mathematical function of the amplitude (the level of current, voltage, power, magnetic-field intensity, or some other variable quantity) versus time repeats precisely and indefinitely. The period is the length of time between one repetition of the pattern, or one wave cycle, and the next. This is illustrated in Fig. 9-1 for a simple AC wave.
The period of an alternating wave can be as short as a minuscule fraction of a second, or as long as thousands of centuries. Some electromagnetic fields have periods measured in quadrillionths of a second or smaller. The charged particles held captive by the magnetic field of the sun reverse their direction over periods measured in years, and large galaxies may have magnetic fields that reverse their polarity every few million years. The period of an AC wave, when expressed in seconds, is symbolized T .
The frequency of an AC wave, denoted f , is the reciprocal of the period. That is, f = 1 /T , and T = 1 /f . Prior to the 1970s, frequency was expressed in cycles per second , abbreviated cps. High frequencies were expressed in kilocycles, megacycles , or gigacycles , representing thousands, millions, or billions of cycles per second. Nowadays, the standard unit of frequency is known as the hertz , abbreviated Hz. Thus, 1 Hz = 1 cps, 10 Hz = 10 cps, and so on.
Higher frequencies are expressed in kilohertz (kHz), megahertz (MHz), gigahertz (GHz), and terahertz (THz). The relationships are:
1 kHz = 1000 Hz = 10 3 Hz
1 MHz = 1000 kHz = 10 6 Hz
1 GHz = 1000 MHz = 10 9 Hz
1 THz = 1000 GHz = 10 12 Hz
The Sine Wave
In its purest form, alternating current has a sine-wave, or sinusoidal, nature. The waveform in Fig. 9-1 is a sine wave. Any AC wave that concentrates all of its energy at a single frequency has a perfect sine-wave shape. Conversely, any perfect sine-wave electrical signal contains one, and only one, component frequency.
In practice, a wave can be so close to a sine wave that it looks exactly like the sine function on an oscilloscope, when in reality there are traces of signals at other frequencies present. The imperfections in a signal are often too small to see using an oscilloscope, although there are other instruments that can detect and measure them. Utility AC in the United States has an almost perfect sine-wave shape, with a frequency of 60 Hz.
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 positive-going. 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. 9-2.
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 end-to-end 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.
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 positive-going.
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
What is the angular frequency of household AC in radians per second? Assume the frequency of utility AC is 60.0 Hz.
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.
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.
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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