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Chapter 28: Problem 54

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?

Short Answer

Expert verified
To determine if the capacitor's rating is likely to be exceeded, the peak capacitor voltage should be calculated across a range of frequencies. From the resulting plot, the regions that should be avoided can be identified as those where the peak capacitor voltage crosses the capacitor's rated voltage.

Step by step solution

01

Understanding RLC Circuits

In an RLC circuit with an AC supply, the current and voltage are not in phase due to the reactive components (inductor and capacitor). The combined effect of these three components in the circuit is called impedance (Z). It is given by the formula: \(Z = \sqrt{R^2 + (X_L - X_C)^2}\), where \(X_L = 2\pi f L\) (inductive reactance) and \(X_C = 1 / (2\pi f C)\) (capacitive reactance).
02

Calculating Immitance in the Circuit

For given values of R, L and C, immitance, \(X = X_L - X_C\), can be calculated for each frequency points. To find the voltages across each component recall that \(V = I X\), where I is the current in the circuit. The peak value of the current can be calculated for given peak source voltage, \(V_{peak}\), and impedance Z, from \(I_{peak} = V_{peak} / Z\). The voltage across capacitor then can be calculated for each frequency point from \(V_C = I X_C\).
03

Plotting the Peak Capcitor Voltage as a Function of Frequency

Plot the calculated peak capacitor voltage, \(V_C\), against the frequency. The plot will show how the peak capacitor voltage varies with frequency.
04

Analyzing the Danger Zone

From the plot identify the ranges of frequency for which the capacitor voltage, \(V_C\), exceeds the rated capacitor voltage, \(V_rated\). Highlight those danger zones to avoid when operating the circuit.

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Most popular questions from this chapter

For safety, medical equipment connected to patients is often powered by an isolation transformer, whose primary is connected to 120 -V AC power and whose secondary delivers \(120-\mathrm{V}\) power. What's the turns ratio of such a transformer?

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?

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

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.

If you measure the rms voltages across the resistor, capacitor, and inductor in a series \(R L C\) circuit, will they add to the rms generator voltage?
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