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Chapter 28: Problem 14

Much of Europe uses AC power at \(230 \mathrm{V}\) rms and \(50 \mathrm{Hz}\). Express this AC voltage in the form of Equation \(28.3,\) taking \(\phi_{V}=0\)

Short Answer

Expert verified
Therefore, the AC voltage can be expressed in the form \(V=230V \times \sqrt{2} \cos(2\pi \times 50 Hz \times t)\).

Step by step solution

01

Convert RMS voltage to peak voltage

To change the RMS value of a sinusoidal wave to its peak value, the RMS value is multiplied by \(\sqrt{2}\). Therefore, the peak voltage \(V_{m}\) of the AC signal can be calculated as \(V_{m}=230V \times \sqrt{2}\).\n
02

Calculate Angular Frequency

The angular frequency \(\omega\) is calculated as \(2\pi\) times the frequency in Hz. Therefore, \(\omega\) is calculated as \(\omega=2\pi \times 50 Hz\).\n
03

Substitute into the AC Voltage Equation

With the calculated peak voltage and angular frequency, the AC Voltage can now be written in the form of the equation \(V=V_{m}\cos (\omega t+\varphi)\), substituting \(\phi_{V}=0\) . Thus, the equation will look like \(V=230V \times \sqrt{2} \cos(2\pi \times 50 Hz \times t)\).

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Most popular questions from this chapter

You're asked to experiment with a series \(R L C\) circuit consisting of a \(10-\Omega\) resistor, \(50-\mathrm{mH}\) inductor, and \(1.5-\mu \mathrm{F}\) capacitor rated at \(1200 \mathrm{V} .\) You're to apply a sinusoidal AC voltage peaking at \(100 \mathrm{V} .\) But you're worried there might be a chance you'll exceed the capacitor's rated voltage. Your lab partner claims this can't happen, since the capacitor rating is 12 times the peak voltage of the AC source. Who's right? To find out, plot the peak capacitor voltage as a function of frequency. Is there a frequency range you should avoid?

An electric water heater draws \(20 \mathrm{A}\) rms at \(240 \mathrm{V}\) rms and is purely resistive. An AC motor has the same current and voltage, but inductance causes the voltage to lead the current by \(20^{\circ} .\) Find the power consumption in each device.

The same AC voltage appears across a capacitor and a resistor, and the same rms current flows in each. Is the power dissipation the same in each?

The FM radio band covers the frequency range \(88-108\) MHz. If the variable capacitor in an FM receiver ranges from \(10.9 \mathrm{pF}\) to \(16.4 \mathrm{pF},\) what inductor should be used to make an \(L C\) circuit whose resonant frequency spans the FM band?

For \(R L C\) circuits in which the resistance isn't too high, the \(Q\) factor may be defined as the ratio of the resonant frequency to the difference between the two frequencies where the power dissipated in the circuit is half the power dissipated at resonance. Using suitable approximations, show that this definition leads to \(Q=\omega_{0} L / R,\) with \(\omega_{0}\) the resonant frequency.
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