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144 Alternating-Current Basics
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9-1 A sine wave. The period
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is the length of time it takes for one cycle to be completed.
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1 kHz = 1000 Hz 1 MHz = 1000 kHz = 1,000,000 Hz = 106 Hz 1 GHz = 1000 MHz = 1,000,000,000 Hz = 109 Hz Sometimes an even bigger unit, the terahertz (THz), is used to specify ac frequency. This is a trillion (1,000,000,000,000, or 1012) hertz. Electrical currents generally do not attain such frequencies, although some forms of electromagnetic radiation do. The frequency of an ac wave, denoted f, in hertz is the reciprocal of the period in seconds. Mathematically, these two equations express the relationship: f = 1/T and T = 1/f
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Some ac waves have only one frequency. These waves are called pure. But often, there are components at multiples of the main, or fundamental, frequency. There can also be components at odd frequencies. Some ac waves have hundreds, thousands, or even infinitely many different component frequencies.
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The Sine Wave
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Sometimes, alternating current has a sine-wave, or sinusoidal, nature. This means that the direction of the current reverses at regular intervals, and that the current-versus-time curve is shaped like the trigonometric sine function. The waveform in Fig. 9-1 is a sine wave. Any ac wave that consists of a single frequency has a perfectly sinusoidal shape. Any perfect si-
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Square Waves 145
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nusoidal ac source has only one component frequency. In practice, a wave might 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 other frequencies present. Imperfections are often too small to see. But pure, single-frequency ac not only looks perfect, but actually is a perfect replication of the trigonometric sine function. The current at the wall outlets in your house is an almost perfect ac sine wave with a frequency of 60 Hz.
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Square Waves
Earlier in this chapter, it was said that there can be an ac wave whose instantaneous amplitude remains constant, even though the polarity reverses. Does this seem counterintuitive Think some more! A square wave is such a wave. On an oscilloscope, a square wave looks like a pair of parallel, dashed lines, one with positive polarity and the other with negative polarity (Fig. 9-2A). The oscilloscope shows a graph of voltage on the vertical scale and time on the horizontal scale. The transitions between negative and positive for a theoretically perfect square wave would not show up on the oscilloscope, because they would be instantaneous. But in practice, the transitions can often be seen as vertical lines (Fig. 9-2B). True square waves have equal negative and positive peaks. Thus, the absolute amplitude of the wave is constant. Half of the time it s +x, and the other half of the time it s x (where x can be expressed in volts, amperes, or watts). Some squared-off waves are lopsided; the negative and positive amplitudes are not the same. Still others remain at positive polarity longer than they remain at negative polarity (or vice versa). These are examples of asymmetrical square waves, more properly called rectangular waves.
9-2 At A, a perfect square wave; the transitions are instantaneous and therefore
do not show up on the graph. At B, the more common rendition of a square wave, showing the transitions as vertical lines.
146 Alternating-Current Basics
Sawtooth Waves
Some ac waves rise and/or fall in straight, sloping lines as seen on an oscilloscope screen. The slope of the line indicates how fast the magnitude is changing. Such waves are called sawtooth waves because of their appearance. Sawtooth waves are generated by certain electronic test devices. They can also be generated by electronic sound synthesizers.
Fast Rise, Slow Decay Figure 9-3 shows a sawtooth wave in which the positive-going slope (called the rise) is extremely steep, as with a square wave, but the negative-going slope (called the decay) is not so steep. The period of the wave is the time between points at identical positions on two successive pulses. Slow Rise, Fast Decay Another form of sawtooth wave is just the opposite, with a defined, finite rise and an instantaneous decay. This type of wave is often called a ramp because it looks like an incline going upward (Fig. 9-4). This waveshape is useful for scanning in television sets and oscilloscopes. It tells the electron beam to move, or trace, at constant speed from left to right across the screen during the rise. Then it retraces, or brings the electron beam back, instantaneously during the decay so the beam can trace across the screen again. Variable Rise and Decay Sawtooth waves can have rise and decay slopes in an infinite number of different combinations. One common example is shown in Fig. 9-5. In this case, the rise and the decay are both finite and equal. This is known as a triangular wave.