Chapter 5.6, Problem 42E

Electric Generator The armature in an electric generator is rotating at the rate of 100 revolutions per second (rps). If the maximum voltage produced is 310 V, find an equation that describes this variation in voltage. What is the RMS voltage? (See Example 6 and the margin note adjacent to it.)

To determine

The equation that describes the variation of voltage and the RMS voltage.

Answer to Problem 42E

The equation that describes the variation of voltage is $E\left(t\right)=310\cos 200\pi t$ and the root mean square voltage is $219\text{V}$.

Explanation of Solution

Given:

The armature is rotating at the rate of $100$ revolutions per second. The maximum voltage produced is $310\text{V}$.

Formula used:

Voltage generated at time t is given by,

$E\left(t\right)=E_0\cos \omega t$ (1)

Where,

$E_0$ is the maximum voltage, $\frac{\omega }{2\pi }$ is the number of revolutions per second.

Calculation:

Formula to calculate $\omega$ is,

$\text{Revolutions}\text{per}\text{second}=\frac{\omega }{2\pi }$

Substitute 100 for revoluations in the above formula.

$\begin{array}{c}100=\frac{\omega }{2\pi }\\ \omega =100\times 2\pi \\ \omega =200\pi \end{array}$

The maximum voltage is $310\text{V}$.

Substitute $310$ for $E_0$ and $200\pi$ for $\omega$, in equation (1).

$E\left(t\right)=310\cos 200\pi t$

The root mean square voltage is $\frac{1}{\sqrt{2}}$ times the maximum voltage.

$\text{RMS}\text{voltage}=\frac{\text{Maximum}\text{voltage}}{\sqrt{2}}$

Substitute $310\text{V}$ for maximum voltage in the above formula.

$\begin{array}{c}\text{RMS}\text{voltage}=\frac{310}{\sqrt{2}}\\ =\frac{310}{1.414}\\ \approx 219\text{V}\end{array}$

Therefore, the equation that describes the variation of voltage is $E\left(t\right)=310\cos 200\pi t$ and the root mean square voltage is $219\text{V}$.