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Chapter 28: Problem 77

You're concerned about a circuit that will be used in a remote communications installation. The series RLC circuit with \(R=5.5 \Omega, L=180 \mathrm{mH},\) and \(C=0.12 \mu \mathrm{F}\) is connected across a sine-wave generator. The inductor can safely handle 1.5 A of current. The peak generator output when it's tuned to resonance will be \(8.0 \mathrm{V} .\) Will the inductor current stay within a safe limit?

Short Answer

Expert verified
Yes, the inductor current will stay within the safe limit of 1.5A as the calculated current at resonance is approximately 1.45A.

Step by step solution

01

Calculate the Resonant Frequency

The resonant frequency is given by the formula \[f = \frac{1}{2\pi\sqrt{LC}}\], where L is inductance and C is capacitance. Substituting the given values, \[ f = \frac{1}{2\pi\sqrt{(180 \times 10^{-3}) \times (0.12 \times 10^{-6})}} = 992.83 Hz\]
02

Calculate the Impedance at Resonance

At resonance, impedance Z is equal to the resistance R in an RLC series circuit, so Z = R = 5.5 Ω.
03

Calculate the Current

The current in the circuit at resonance can be calculated using the formula \[I = \frac{V}{Z}\], where V is peak voltage and Z is impedance. Hence, \[ I = \frac{8}{5.5} = 1.45A\].
04

Check against the Limit

Upon the comparison, it is seen that the inductor's current of 1.45A is less than its maximum safe current of 1.5A. Therefore, the inductor current will stay within safe limits when the generator is at its resonance peak output.

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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?

A series \(R L C\) circuit with \(R=47 \Omega, L=250 \mathrm{mH},\) and \(C=\) \(4.0 \mu \mathrm{F}\) is connected across a sine-wave generator whose peak output voltage is independent of frequency. Find the frequency range over which the peak current will exceed half its value at resonance.

A series \(R L C\) circuit with \(R=1.3 \Omega, L=27 \mathrm{mH},\) and \(C=\) \(0.33 \mu \mathrm{F}\) is connected across a sine-wave generator. If the capacitor's peak voltage rating is \(600 \mathrm{V},\) what's the maximum safe value for the generator's peak output voltage when it's tuned to resonance?

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?

A series \(R L C\) circuit has resistance \(100 \Omega\) and impedance \(300 \Omega\) (a) What's the power factor? (b) If the rms current is 200 mA, what's the power dissipation?
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