Characteristics of Sinusoids

3.2 Characteristics of Sinusoids

Consider the sine waveform shown in Figure 3.2, where ft () may represent either a voltage or a current function, and let ft () = A sin t where is the amplitude of this function. A sinusoid (sine A or cosine function) can be constructed graphically from the unit circle, which is a circle with radius of one unit, that is, A = 1 as shown, or any other unit. Thus, if we let the phasor (rotating vector) travel around the unit circle with an angular velocity ω , the cos ωt and sin t ω functions are gen- erated from the projections of the phasor on the horizontal and vertical axis respectively. We

observe that when the phasor has completed a cycle (one revolution), it has traveled 2 π radians or 360 ° degrees, and then repeats itself to form another cycle.

ft ()

Sine Wavef orm

of rotation

Figure 3.2. Generation of a sinusoid by rotation of a phasor

At the completion of one cycle, t = T (one period), and since is the angular velocity, com- ω monly known as angular or radian frequency, then

T = 2 ------ π π

ωT = 2 or

The term frequency in Hertz, denoted as Hz , is used to express the number of cycles per second. Thus, if it takes one second to complete one cycle (one revolution around the unit circle), we say

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Characteristics of Sinusoids

that the frequency is 1 Hz or one cycle per second. The frequency is denoted by the letter and in terms of the period and (3.1) we have f T

f = --- 1 or

ω = 2 π f (3.2)

The frequency is often referred to as the cyclic frequency to distinguish it from the radian fre- f quency ω .

Since the cosine and sine functions are usually known in terms of degrees or radians, it is conve- nient to plot sinusoids versus ωt (radians) rather that time . For example, t vt () = V max cos ωt ,

and it () = I max sin t ω are plotted as shown in Figure 3.3.

Figure 3.3. Plot of the cosine and sine functions

By comparing the sinusoidal waveforms of Figure 3.3, we see that the cosine function will be the same as the sine function if the latter is shifted to the left by π2 ⁄ radians, or 90 ° . Thus, we say that the cosine function leads (is ahead of) the sine function by π2 ⁄ radians or 90 ° . Likewise, if we shift the cosine function to the right by π2 ⁄ radians or 90 ° , we obtain the sine waveform; in this case, we say that the sine function lags (is behind) the cosine function by π2 ⁄ radians or 90 ° .

Another common expression is that the cosine and sine functions are out-of-phase by 90 ° , or there is

a phase angle of 90 ° between the cosine and sine functions. It is possible, of course, that two sinusoids are out-of-phase by a phase angle other than 90 ° . Figure 3.4 shows three sinusoids which are out- of-phase. If the phase angle between them is 0 ° degrees, the two sinusoids are said to be in-phase.

We must remember that when we say that one sinusoid leads or lags another sinusoid, these are of the same frequency. Obviously, two sinusoids of different frequencies can never be in phase.

Numerical Analysis Using MATLAB ® and Excel ® , Third Edition

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