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9-1 A sine wave. The period is the length of time for one complete cycle.
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The frequency, denoted f, of a wave is the reciprocal of the period. That is, f = 1/T and T = 1/f .Originally, frequency was specified in cycles per second, abbreviated cps. High frequencies were sometimes given in kilocycles, megacycles, or gigacycles, representing thousands, millions, or billions of cycles per second. But nowadays, the unit is known as the hertz, abbreviated Hz. Thus, 1 Hz = 1 cps, 10 Hz = 10 cps, and so on. Higher frequencies are given in kilohertz (kHz), megahertz (MHz), or gigahertz (GHz). The relationships are: 1 kHz = 1000 Hz 1 MHz = 1000 kHz = 1,000,000 Hz 1 GHz = 1000 MHz = 1,000,000,000 Hz Sometimes an even bigger unit, the terahertz (THz), is needed. This is a trillion (1,000,000,000,000) hertz. Electrical currents generally do not attain such frequencies, although electromagnetic radiation can. 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 might also be components at odd frequencies. Some ac waves are extremely complex, consisting of hundreds, thousands, or even infinitely many different component, frequencies. In this book, most of the attention will be given to ac waves that have just one frequency.
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Sawtooth waves 167
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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 will have a perfect sine waveshape. And any perfect sine-wave current contains 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 has an almost perfect sine waveshape, with a frequency of 60 Hz.
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Earlier in this chapter, it was said that there can be an alternating current whose magnitude never changes. You might at first think this is impossible. How can polarity reverse without some change in the level The square wave is an example of this. On an oscilloscope, a perfect square wave looks like a pair of parallel, dotted 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, versus time on the horizontal scale. The transitions between negative and positive for a true square wave don t show up on the oscilloscope, because they re instantaneous. But perfection is rare. Usually, the transitions can be seen as vertical lines (Fig. 9-2B). A square wave might have equal negative and positive peaks. Then the absolute magnitude of the wave is constant, at a certain voltage, current, or power level. Half of the time it s +x, and the other half it s -x volts, amperes, or watts. Some square waves are lopsided, with the positive and negative magnitudes differing.
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Some ac waves rise and fall in straight 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. These waves provide ideal signals for control purposes. Integrated circuits can be wired so that they produce sawtooth waves having an exact desired shape.
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