The Sine Wave

The terms defined in the previous section can be applied to any type of periodic waveform, whether smooth or discontinuous. The sinusoidal waveform is of particular importance, however, since it lends itself readily to the mathematics and the physical phenomena associated with electric circuits. Consider the power of the following statement: In other words, if the voltage across (or current through) a resistor, coil, or capacitor is sinusoidal in nature, the resulting current (or voltage, respectively) for each will also have sinusoidal characteristics, as shown in [Fig. 1]. If a square wave or a triangular wave were applied, such would not be the case. The unit of measurement for the horizontal axis of [Fig. 2] is the degree. A second unit of measurement frequently used is the radian (rad). It is defined by a quadrant of a circle such as in [Fig. 3] where the distance subtended on the circumference equals the radius of the circle.

Arc Length Formula

In general, if the length of the arc is $s$ units and radius is $r$ units as shown in the [Fig. 3.1], then the ratio of $s$ to $r$ is and if we rotate the radius around the circle for $360^\circ$, then the arc length will be the circumference of the circle, which is comparing eq.(ii) and eq.(iii), we can say Therefore, there are $2\pi$ rad around a $360^\circ$ circle, as shown in [Fig. 4], and with A number of electrical formulas contain a multiplier of $\pi$. For this reason, it is sometimes preferable to measure angles in radians rather than in degrees. $\pi$ has been determined to an extended number of places primarily in an attempt to see if a repetitive sequence of numbers appears. It does not. A sampling of the effort appears below: Although the approximation $\pi = 3.14$ is often applied, all the calculations in this text will use the $\pi$ function as provided on all scientific calculators. For $180^\circ$ and $360^\circ$, the two units of measurement are related as shown in [Fig. 4]. The conversion equations between the two are the following: Applying these equations, we find
$90^\circ:$ $30^\circ:$ ${\pi \over 3} rad: $ Using the radian as the unit of measurement for the abscissa, we would obtain a sine wave, as shown in [Fig. 5]. It is of particular interest that the sinusoidal waveform can be derived from the length of the vertical projection of a radius vector rotating in a uniform circular motion about a fixed point. Starting as shown in [Fig. 6(a)] and plotting the amplitude (above and below zero) on the coordinates drawn to the right [Figs. 6(b)] through [(i)], we will trace a complete sinusoidal waveform after the radius vector has completed a $360^\circ$ rotation about the center. The velocity with which the radius vector rotates about the center, called the angular velocity, can be determined from the following equation: [$$ \bbox[10px,border:1px solid grey]{\text{Angular Velocity} = {\text{distance (degrees or radians)} \over \text{time}} } \tag{4}$$] Substituting into Eq. (4) and assigning the Greek letter omega ($\omega$) to the angular velocity, we have and Since $\omega$ is typically provided in radians per second, the angle $\alpha$ obtained using Eq. (4) is usually in radians. In [Fig. 6], the time required to complete one revolution is equal to the period (T) of the sinusoidal waveform of [Fig. 6(i)]. The radians subtended in this time interval are $2\pi$. Substituting, we have In words, this equation states that the smaller the period of the sinusoidal waveform of [Fig. 6(i)], or the smaller the time interval before one complete cycle is generated, the greater must be the angular velocity of the rotating radius vector. Certainly this statement agrees with what we have learned thus far. We can now go one step further and apply the fact that the frequency of the generated waveform is inversely related to the period of the waveform; that is, $f =1/T$. Thus, This equation states that the higher the frequency of the generated sinusoidal waveform, the higher must be the angular velocity. Equations (7) and (8) are verified somewhat by [Fig. 7], where for the same radius vector, $\omega = 100 \,rad/s$ and $500 \,rad/s$.