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Chapter 28: Problem 31

A series \(R L C\) circuit has \(R=18 \mathrm{k} \Omega, C=14 \mu \mathrm{F},\) and \(L=0.20 \mathrm{H}\) (a) At what frequency is its impedance lowest? (b) What's the impedance at this frequency?

Short Answer

Expert verified
The impedance is lowest at the resonant frequency, which is approximately 849.83 Hz, and the impedance at this frequency is \(18k\Omega\).

Step by step solution

01

Calculate Resonant Frequency

To find the frequency at which the impedance is lowest (resonant frequency), we can use the formula:\[f=\frac{1}{2 \pi \sqrt{L C}}\]where \(L\) is the inductance, \(C\) is the capacitance. In this problem, \(L = 0.20H\) and \(C = 14\mu F = 14 \times 10^{-6} F\). Substitute the values of \(L\) and \(C\) in the formula and calculate the resonant frequency.
02

Calculate Impedance at Resonant Frequency

At resonant frequency, the reactance (X) of inductor (XL) equals the reactance of capacitor (XC), i.e, \(X_{L}=X_{C}\). The impedance (Z) of the circuit at this frequency is given by Ohm's law:\[Z=\sqrt{R^2+ (X_L - X_C)^2}\]Because \(X_{L} = X_{C}\) at resonance, the impedance (Z) equals the resistance (R) of the circuit. Therefore, the impedance Z at the resonant frequency in this problem is \(Z = R = 18k\Omega\).

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Most popular questions from this chapter

A \(0.75-\mathrm{H}\) inductor is in series with a fluorescent lamp, and the combination is across \(120-\mathrm{V}\) rms, \(60-\mathrm{Hz}\) power. If the rms inductor voltage is \(90 \mathrm{V},\) what's the rms lamp current?

Your university's FM station broadcasts at \(89.5 \mathrm{MHz}\). The \(L C\) circuit that establishes this frequency has a 47 -pF capacitor. What's the corresponding inductance?

What's meant by the statement, "A capacitor acts like a DC open circuit"?

Find the resonant frequency of an \(L C\) circuit consisting of a \(0.22-\mu \mathrm{F}\) capacitor and a 1.7 -mH inductor.

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.
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