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Chapter 28: Problem 30

Find the impedance at \(10 \mathrm{kHz}\) of a circuit consisting of a \(1.5-\mathrm{k} \Omega\) resistor, \(5.0-\mu \mathrm{F}\) capacitor, and \(50-\mathrm{mH}\) inductor in series.

Short Answer

Expert verified
The impedance of the given RLC series circuit at \(10 kHz\) is approximately \(3155.08 \Omega\).

Step by step solution

01

Calculate the Inductive and Capacitive Reactance

First, calculate the inductive reactance \(X_L\) and the capacitive reactance \(X_C\). Remember to convert the given values into suitable units before putting in the values in their respective formulas.For \(X_L\):\(X_L = 2\pi fL = 2\pi(10^4 Hz)(50 \times 10^{-3} H) = 3141.59 \Omega\)For \(X_C\):\(X_C = 1/(2\pi fC) = 1/{2\pi(10^4 Hz)(5 \times 10^{-6} F)} = 3.1831 \Omega\)
02

Calculate the impedance

Now, you can substitute the values of \(R\), \(X_L\), and \(X_C\) in the impedance formula:\(Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{(1500 \Omega)^2 + (3141.59 \Omega - 3.1831 \Omega)^2} = 3155.08 \Omega\)
03

Round off and present the answer

As a final step round off the calculated impedance to a suitable number of decimal places keeping in view the significant figures. Here, rounding off up to two decimal places would be sufficient. Therefore, the impedance \(Z \approx 3155.08 \Omega\).

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Most popular questions from this chapter

A 2.0 - \(\mu\) F capacitor has \(1.0-\mathrm{k} \Omega\) reactance. (a) What's the frequency of the applied voltage? (b) What inductance would give the same reactance at this frequency? (c) How would the reactances compare if the frequency were doubled?

Resonance is a phenomenon which occurs in RLC circuit, during which impedance due to capacitor and inductor become equal and peak current is achieved in the circuit. At resonance the impedance of the circuit is lowest and equal to the resistance in the circuit The value of peak current / is given as follows: \(I=\frac{V}{r}\) Here, \(V\) is voltage in RLC circuit and \(r\) is resistance in the circuit.

An LC circuit includes a \(0.025-\mu \mathrm{F}\) capacitor and a \(340-\mu \mathrm{H}\) inductor. (a) If the peak capacitor voltage is \(190 \mathrm{V},\) what's the peak inductor current? (b) How long after the voltage peak does the current peak occur?

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.

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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