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Chapter 28: Problem 15

An industrial electric motor runs at \(208 \mathrm{V}\) rms and \(400 \mathrm{Hz}\). What are (a) the peak voltage and (b) the angular frequency?

Short Answer

Expert verified
The peak voltage of the motor is \(208V \times \sqrt{2} \approx 294.4V\) and the angular frequency is \(2\pi \times 400 Hz = 800\pi rad/s\).

Step by step solution

01

Calculate the Peak Voltage

The peak voltage can be calculated from the root mean square (rms) voltage using the following formula: \[V_{peak} = V_{rms} \times \sqrt{2}\] where \(V_{rms}\) is the rms voltage. Therefore, plug the given rms voltage of 208 V into the formula to find the peak voltage.
02

Calculate the Angular Frequency

Angular frequency can be calculated by using the following formula: \[\omega = 2\pi f\] where \(f\) is the frequency. Given that the frequency of the motor is 400 Hz, plug this value into the formula to find the angular frequency.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

RMS Voltage
RMS voltage stands for root mean square voltage, a critical concept in alternating current (AC) circuits. Unlike direct current (DC) where the voltage remains constant, AC voltages fluctuate over time in a sinusoidal wave. The RMS voltage is a type of average voltage that represents the equivalent DC voltage which would deliver the same amount of power to a load as the AC voltage does over a complete cycle.

Peak Voltage Calculation
The peak voltage of an AC circuit is the maximum voltage reached by the wave during its cycle and is higher than the RMS voltage. To calculate the peak voltage from the given RMS voltage, a simple formula is used: \[V_{\text{peak}} = V_{\text{rms}} \times \sqrt{2}\]. Let's apply this to our industrial motor example where the motor runs at 208 volts RMS. The peak voltage, therefore, can be found by multiplying 208 volts by \(\sqrt{2}\), approximately 1.414. This gives us a peak voltage value that is useful in assessing the maximum electrical stress on the components in the circuit.
Angular Frequency
Angular frequency is a measure of how quickly the AC voltage cycle repeats itself, expressed in radians per second. It's related to the frequency of the cycle, which is typically measured in hertz (Hz). The formula to calculate angular frequency from the regular frequency is \[\omega = 2\pi f\]. In the motor example, with a frequency of 400 Hz, the angular frequency would be calculated by multiplying 400 by 2\(\pi\), which equals roughly 2512 radians per second. Understanding angular frequency is crucial when analyzing the behavior of AC circuits and in the design of electronic equipment, such as the industrial electric motor in our example.

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Most popular questions from this chapter

A triangle wave swings linearly between voltages \(-V_{\mathrm{p}}\) and \(+V_{\mathrm{p}}\) Show that the rms voltage of a triangle wave is \(V_{\mathrm{p}} / \sqrt{3}\)

A power supply like that of Fig. 28.23 is supposed to deliver 22-V DC at a maximum current of 150 mA. The transformer's peak output voltage can charge the capacitor to a full \(22 \mathrm{V},\) and the primary is supplied with \(60-\mathrm{Hz}\) AC. What capacitance will ensure that the output voltage stays within \(3 \%\) of the rated \(22 \mathrm{V} ?\)

The applied voltage in a series \(R L C\) circuit lags the current. Is the frequency above or below resonance?

An LC circuit with \(C=18\) mF undergoes oscillations with period \(2.4 \mathrm{s}\). Find the inductance.

The voltage across two components in series is zero. Is it possible that the voltages across the individual components aren't zero? Give an example.
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