Problem 21 - College Physics

Question

\(\bullet\) In an \(R-L-C\) series circuit, \(R=150 \Omega, L=0.750 \mathrm{H},\) and \(C=0.0180 \mu \mathrm{F}\) . The source has voltage amplitude \(V=150 \mathrm{V}\) and a frequency equal to the resonance frequency of the circuit. (a) What is the power factor? (b) What is the average power delivered by the source? (c) The capacitor is replaced by one with \(C=0.0360 \mu \mathrm{F}\) and the source frequency is adjusted to the new resonance value. Then what is the average power delivered by the source?

Step-by-Step Solution

Answer

(a) Power factor is 1. (b) Average power is 75 W. (c) Average power remains 75 W.

Step 1: Determine the Resonant Frequency

The resonant frequency in an RLC circuit is given by the formula \(f_0 = \frac{1}{2\pi\sqrt{LC}}\). For the original configuration, substituting the values \(L = 0.750\, \text{H}\) and \(C = 0.0180 \times 10^{-6}\, \text{F}\), we calculate: \[ f_0 = \frac{1}{2\pi\sqrt{0.750 \times 0.0180 \times 10^{-6}}} \approx 13.77 \text{ kHz} \]

Step 2: Calculate the Power Factor at Resonance

At resonant frequency, the impedance is purely resistive, which means the power factor is 1. This is because the inductive reactance \(X_L\) cancels out the capacitive reactance \(X_C\), making the potential difference across the inductor-capacitor pair zero at resonance.

Step 3: Calculate the Average Power at Resonance

The average power \(P_{avg}\) delivered by the source at resonance is given by the formula:\[ P_{avg} = \frac{V_{rms}^2}{R} \] where \(V_{rms} = \frac{V}{\sqrt{2}}\). So, plugging in the numbers:\[ V_{rms} = \frac{150}{\sqrt{2}} \approx 106.07 \, \text{V} \]\[ P_{avg} = \frac{(106.07)^2}{150} \approx 75.0 \, \text{W} \]

Step 4: Determine New Resonant Frequency

The new resonance frequency \(f_0'\) can be found by replacing the capacitance in the original resonant frequency formula with the new capacitance \(C = 0.0360 \times 10^{-6} \text{ F}\). So:\[ f_0' = \frac{1}{2\pi\sqrt{0.750 \times 0.0360 \times 10^{-6}}} \approx 9.74 \text{ kHz} \]

Step 5: Calculate the Average Power with New Capacitor

At the new resonance frequency, the impedance is again purely resistive. Thus, the power factor remains 1, and the average power formula applies as before:\[ P_{avg}' = \frac{V_{rms}^2}{R} = 75.0 \, \text{W} \]The average power remains the same as the resistance and \(V_{rms}\) are unchanged.

Key Concepts

Resonant Frequency in RLC Circuits

In electrical circuits featuring resistors, inductors, and capacitors (RLC circuits), the resonant frequency is an important concept. It refers to the frequency at which the circuit naturally oscillates. At this frequency, the reactance of the inductor and the capacitor are equal and opposite, effectively canceling each other out.
This means the circuit behaves like a pure resistor with no imaginary component to the impedance, which simplifies analysis. The formula to calculate the resonant frequency, often denoted as \(f_0\), is given by:

• \(f_0 = \frac{1}{2\pi\sqrt{LC}}\)

Here, \(L\) is the inductance in henries, and \(C\) is the capacitance in farads. To find the resonant frequency, simply plug in the values of \(L\) and \(C\) into the formula. This tells you at what frequency the circuit's impedance is minimized, making it behave most efficiently.

Understanding Power Factor at Resonance

The power factor is a key indicator of efficiency in AC circuits. It’s defined as the cosine of the phase angle between the voltage and the current. In simpler terms, it tells us how much of the power is being effectively used. A power factor of 1 means all the power is being used effectively, while a value less than 1 indicates inefficiency due to phase differences caused by reactance.
In an RLC circuit at resonant frequency, something special happens—the power factor becomes 1. Why? Because the inductance and capacitance reactances cancel each other out, leaving only the resistance to affect the phase angle. This means that the total impedance equals the resistance without any reactive components, maximizing energy transfer. When designing circuits or troubleshooting, a power factor of 1 is a sign that your circuit is operating ideally.

Calculating Average Power in RLC Circuits

The average power in an AC circuit, especially one operating at resonant frequency, is an important measure of effectiveness. It tells us how much power is being transferred from the source to the load.
The formula for calculating the average power, \(P_{avg}\), in such a circuit at resonance is given by:

• \(P_{avg} = \frac{V_{rms}^2}{R}\)

Where \(V_{rms}\) is the root mean square (RMS) voltage, which is calculated as \(V_{rms} = \frac{V}{\sqrt{2}}\) for a peak voltage \(V\), and \(R\) is the resistance.
The implication here is that at resonance, the total power is efficiently used due to the absence of reactive losses, characterized by a power factor of 1. For example, in our exercise, the average power stays consistent before and after the capacitor change, as \(V_{rms}\) and \(R\) are unchanged, showing that power delivery efficiency is maintained at resonance.