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# Waveforms Help

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## Introduction

If you graph the instantaneous current or voltage in an ac system as a function of time, you get a waveform. Alternating currents can manifest themselves in an infinite variety of waveforms. Here are some of the simplest ones.

## Sine Wave

In its purest form, alternating current has a sine-wave , or sinusoidal , nature. The waveform in Fig. 13-1 is a sine wave. Any ac wave that consists of a single frequency has a perfect sine-wave shape. Any perfect sine-wave current contains one, and only one, component frequency.

Fig. 13-1 . A sine wave. The period is the length of time required for one cycle to be completed.

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 other frequencies present. Imperfections are often too small to see. Utility ac in the United States has an almost perfect sine-wave shape, with a frequency of 60 Hz. However, there are slight aberrations.

## Square Wave

On an oscilloscope, a theoretically perfect square wave would look like a pair of parallel dotted lines, one having positive polarity and the other having negative polarity (Fig. 13-2a). In real life, the transitions often can be seen as vertical lines (see Fig. 13-2b).

Fig. 13-2 . (a) A theoretically perfect square wave, (b) The more common rendition.

A square wave might have equal negative and positive peaks. Then the absolute amplitude of the wave is constant at a certain voltage, current, or power level. Half the time the amplitude is + x , and the other half it is − x volts, amperes, or watts.

Some square waves are asymmetrical, with the positive and negative magnitudes differing. If the length of time for which the amplitude is positive differs from the length of time for which the amplitude is negative, the wave is not truly square but is described by the more general term rectangular wave .

## Sawtooth Waves

Some ac waves reverse their polarity at constant but not instantaneous rates. The slope of the amplitude-versus-time line indicates how fast the magnitude is changing. Such waves are called sawtooth waves because of their appearance.

In Fig. 13-3, one form of sawtooth wave is shown. The positive-going slope (rise) is extremely steep, as with a square wave, but the negative-going slope (fall or decay) is gradual. The period of the wave is the time between points at identical positions on two successive pulses.

Fig. 13-3 . A fast-rise, slow-decay sawtooth wave.

Another form of sawtooth wave is just the opposite, with a gradual positive-going slope and a vertical negative-going transition. This type of wave is sometimes called a ramp (Fig. 13-4). This waveform is used for scanning in cathode-ray-tube (CRT) television sets and oscilloscopes.

Fig. 13-4 . A slow-rise, fast-decay sawtooth wave, also called a ramp wave.

Sawtooth waves can have rise and decay slopes in an infinite number of different combinations. One example is shown in Fig. 13-5. In this case, the positive-going slope is the same as the negative-going slope. This is a triangular wave .

Fig. 13-5 . A triangular wave.

## Waveforms Practice Problem

#### Problem 1

Suppose that each horizontal division in Fig. 13-5 represents 1.0 microsecond (1.0 μs or 1.0 × 10 −6 s). What is the period of this triangular wave? What is the frequency?

Fig. 13-5 . A triangular wave.

#### Solution 1

The easiest way to look at this is to evaluate the wave from a point where it crosses the time axis going upward and then find the next point (to the right or left) where the wave crosses the time axis going upward. This is four horizontal divisions, at least within the limit of our ability to tell by looking at it. The period T is therefore 4.0 μS or 4.0 × 10 −6 s. The frequency is the reciprocal of this: f = 1/ T = 1/(4.0 × 10 −6 ) = 2.5 × 10 5 Hz.

Practice problems of these concepts can be found at: Alternating Current Practice Test

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