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Chapter 21: Problem 88

An RLC series circuit is connected to a \(240-\mathrm{V}\) ms power supply at a frequency of \(2.50 \mathrm{kHz}\). The elements in the circuit have the following values: \(R=12.0 \Omega, C=\) \(0.26 \mu \mathrm{F},\) and $L=15.2 \mathrm{mH},$ (a) What is the impedance of the circuit? (b) What is the rms current? (c) What is the phase angle? (d) Does the current lead or lag the voltage? (e) What are the rms voltages across each circuit element?

Short Answer

Expert verified
Answer: The impedance of the circuit is \(12.9\Omega\), the rms current is \(18.6\mathrm{A}\), the phase angle is \(-9.4^\circ\), the current leads the voltage, and the rms voltages across the resistor, capacitor, and inductor are \(223\mathrm{V}, 4.57\times10^3\mathrm{V}\), and \(4.43\times10^3\mathrm{V}\) respectively.

Step by step solution

01

Calculate angular frequency, inductive reactance, and capacitive reactance.

First, we need to find the angular frequency, inductive reactance, and capacitive reactance using the given values. Angular frequency: \(\omega = 2\pi f = 2\pi (2.50 \times 10^3 \mathrm{s^{-1}}) = 15.7 \times 10^3\mathrm{s^{-1}}\) Inductive reactance: \(X_L = \omega L = (15.7 \times 10^3\mathrm{s^{-1}}) (15.2 \times 10^{-3}\mathrm{H}) = 238\Omega\) Capacitive reactance: \(X_C = \frac{1}{\omega C} = \frac{1}{(15.7 \times 10^3\mathrm{s^{-1}}) (0.26 \times 10^{-6}\mathrm{F})} = 246\Omega\)
02

Calculate impedance.

Now, find the impedance using the resistance, inductive reactance, and capacitive reactance. \(Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{(12.0 \Omega)^2 + (238\Omega - 246\Omega)^2} = 12.9\Omega\) The impedance of the circuit is \(12.9\Omega\).
03

Calculate rms current.

Next, find the rms current using the given voltage and calculated impedance. \(I = \frac{V}{Z} = \frac{240\mathrm{V}}{12.9\Omega} = 18.6\mathrm{A}\) The rms current is \(18.6\mathrm{A}\).
04

Calculate phase angle.

Now, find the phase angle using the resistance and reactances. \(\phi = \tan^{-1}(\frac{X_L - X_C}{R}) = \tan^{-1}(\frac{238\Omega - 246\Omega}{12.0\Omega}) = -9.4^\circ\) The phase angle is \(-9.4^\circ\).
05

Determine if current leads or lags the voltage.

Since the phase angle is negative, the current leads the voltage in this circuit.
06

Calculate rms voltages across each circuit element.

To find the rms voltages across the resistor, capacitor, and inductor, use the rms current and the resistance/reactances. \(V_R = IR = (18.6\mathrm{A})(12.0\Omega) = 223\mathrm{V}\) \(V_C = IX_C = (18.6\mathrm{A})(246\Omega) = 4.57\times10^3\mathrm{V}\) \(V_L = IX_L = (18.6\mathrm{A})(238\Omega) = 4.43\times10^3\mathrm{V}\) The rms voltages across the resistor, capacitor, and inductor are \(223\mathrm{V}, 4.57\times10^3\mathrm{V}\), and \(4.43\times10^3\mathrm{V}\) respectively.

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Most popular questions from this chapter

A capacitor (capacitance \(=C\) ) is connected to an ac power supply with peak voltage \(V\) and angular frequency \(\alpha\) (a) During a quarter cycle when the capacitor goes from being uncharged to fully charged, what is the average current (in terms of \(C . V\), and \(\omega\) )? [Hint: \(\left.i_{\mathrm{av}}=\Delta Q / \Delta t\right]\) (b) What is the rms current? (c) Explain why the average and rms currents are not the same.
The voltage across an inductor and the current through the inductor are related by \(v_{\mathrm{L}}=L \Delta i / \Delta t .\) Suppose that $i(t)=I \sin \omega t .\( (a) Write an expression for \)v_{\mathrm{L}}(t) .$ [Hint: Use one of the relationships of Eq. \((20-7) .]\) (b) From your expression for \(v_{\mathrm{L}}(t),\) show that the reactance of the inductor is \(X_{\mathrm{L}}=\omega L_{\mathrm{r}}\) (c) Sketch graphs of \(i(t)\) and \(v_{\mathrm{L}}(t)\) on the same axes. What is the phase difference? Which one leads?
A generator supplies an average power of \(12 \mathrm{MW}\) through a transmission line that has a resistance of \(10.0 \Omega .\) What is the power loss in the transmission line if the rms line voltage \(8_{\text {rms }}\) is (a) \(15 \mathrm{kV}\) and (b) \(110 \mathrm{kV} ?\) What percentage of the total power supplied by the generator is lost in the transmission line in each case? (IMAGE CAN'T COPY)
An RLC series circuit is built with a variable capacitor. How does the resonant frequency of the circuit change when the area of the capacitor is increased by a factor of \(2 ?\)
A \(150-\Omega\) resistor is in series with a \(0.75-\mathrm{H}\) inductor in an ac circuit. The rms voltages across the two are the same. (a) What is the frequency? (b) Would each of the rms voltages be half of the rms voltage of the source? If not, what fraction of the source voltage are they? (In other words, \(V_{R} / \ell_{m}=V_{L} / \mathcal{E}_{m}=?\) ) (c) What is the phase angle between the source voltage and the current? Which leads? (d) What is the impedance of the circuit?
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