6.1 The Sine Wave
The sine wave is the fundamental type of alternating current (ac) and alternating voltage. It is also referred to as a sinusoidal wave, or, simply, sinusoid. The electrical service provided by the power companies is in the form of sinusoidal voltage and current. In addition, other types of repetitive wave forms are composites of many individual sine waves called harmonics.
Figure 6-1 Symbol for a sinusoidal voltage source.
Sine waves, or sinusoidals, are produced by two types of sources: rotating electrical machines (ac generators) or electronic oscillator circuits, which are used in instruments commonly known as electronic signal generators. Figure 6-1 shows the symbol used to represent either source of sinusoidal voltage.
Figure 6-2 is a graph showing the general shape of a sine wave, which can be either an alternating current or an alternating voltage. Voltage (or current) is displayed on the vertical axis and time (t) is displayed on the horizontal axis. Notice how the voltage (or current) varies with time. Starting at zero, the voltage (or current) increases to a positive maximum (peak), returns to zero, and then increases to a negative maximum (peak) before returning again to zero, thus completing one full cycle.
Figure 6-2 Graph one cycle of a sine wave.
Polarity of a Sine Wave
As you have seen, a sine wave changes polarity at its zero value; that is, it alternates between positive and negative values. When a sinusoidal voltage source (Vs) is applied to a resistive circuit, as in Figure 6-3, an alternating sinusoidal current results. When the voltage changes polarity, the current correspondingly changes direction as indicated.
During the positive alternation of the source voltage Vs, the current is in the direction shown in Figure 6-3(a). During a negative alternation of the source voltage, the current is in the opposite direction, as shown in Figure 6-3(b). The combined positive and negative alternations make up one cycle of a sine wave.
Figure 6-3 Alternating current and voltage.
Period of a Sine Wave
A sine wave varies with time (t) in a definable manner.
The time required for a given sine wave to complete one full cycle is called the period (T).
Figure 6-4(a) illustrates the period of a sine wave. Typically, a sine wave continues to repeat itself in identical cycles, as shown in Figure 6-4(b). Since all cycles of a repetitive sine wave are the same, the period is always a fixed value for a given sine wave. The period of a sine wave can be measured from a zero crossing to the next corresponding zero crossing, as indicated in Figure 6-4(a). The period can also be measured from any peak in a given cycle to the corresponding peak in the next cycle.
Figure 6-4 The period of a given sine wave is the same for each cycle.
Example 6-1
What is the period of the sine wave in Figure 6-5?
Figure 6-5
Solution
As shown in Figure 6-5, it takes twelve seconds (12s) to complete three cycles. Therefore, to complete one cycle it takes four seconds (4s), which is the period.
T=4s
Example 6-2
Show three possible ways to measure the period of the sine wave in Figure 6-6. How many cycles are shown?
Figure 6-6
Solution
1. The period can be measured from one zero crossing to the corresponding zero crossing in the next cycle (the slope must be the same at the corresponding zero crossings).
- The period can be measured from the positive peak in one cycle to the positive peak in the next cycle.
- The period can be measured from the negative peak in one cycle to the negative peak in the next cycle
These measurements are indicated in Figure 6-7, where two cycles of the sine wave are shown. Keep in mind that you obtain the same value for the period no matter which corresponding peaks or corresponding zero crossings on the waveform you use.
Figure 6-7 Measurement of the period of a sine wave.
Frequency of a Sine Wave
Frequency is the number of cycles that a sine wave completes in one second.
Figure 6-8 Illustration of frequency.
The more cycles completed in one second, the higher the frequency. Frequency (f) is measured in units of hertz. One hertz (Hz) is equivalent to one cycle per second; for example, 60 Hz is 60 cycles per second. Figure 6-8 shows two sine waves. The sine wave in part (a) completes two full cycles in one second. The one in part (b) completes four cycles in one second. Therefore, the sine wave in part (b) has twice the frequency of the one in part (a).
Relationship of Frequency and Period
The relationship between frequency and period is important. The formulas for this relationship are as follows:
Equation 6-1
Equation 6-2
Example 6-3
Which sine wave in Figure 6-9 has the higher frequency? Determine the period and the frequency of both waveforms.
Figure 6-9
Solution
The sine wave in Figure 6-9(b) has the higher frequency because it completes more cycles in 1 s than does the one in part (a).
In Figure 6-9(a), three cycles take 1 s. Therefore, one cycle takes 0.333 s (one-third second), and this is the period.
T = 0.333 s = 333 ms
The frequency is
3 Hz
In Figure 6-9(b), five cycles take 1 s. Therefore, one cycle takes 0.2 s (one-fifth second), and this is the period.
T = 0.2 s = 200 ms
The frequency is
5 Hz
Example 6-4
The period of a certain sine wave is 10 ms. What is the frequency?
Solution
Use Equation 6-1.
100 Hz
Example 6-5
The frequency of a sine wave is 60 Hz. What is the period?
Solution
Use Equation 6-2.
16.7 ms