Problem 40 - College Physics

Question

\(\bullet\) In a series \(R-L-C\) circuit, \(R=300 \Omega, X_{C}=300 \Omega,\) and \(X_{L}=500 \Omega .\) The average power consumed in the resistor is 60.0 \(\mathrm{W}\) . (a) What is the power factor of the circuit? (b) What is the rms voltage of the source?

Step-by-Step Solution

Answer

(a) Power factor is approximately 0.832. (b) RMS voltage is approximately 282.85 V.

Step 1: Understand Circuit Properties

In an R-L-C circuit, the resistor, inductor, and capacitor are connected in series. We have given values: resistance \( R = 300 \Omega \), capacitive reactance \( X_C = 300 \Omega \), and inductive reactance \( X_L = 500 \Omega \). The average power consumed by the resistor is 60 W.

Step 2: Calculate Impedance \(Z\) of the Circuit

The impedance \(Z\) of the RLC circuit is given by the formula \(Z = \sqrt{R^2 + (X_L - X_C)^2}\). Substituting the given values, we have:\[ Z = \sqrt{300^2 + (500 - 300)^2} = \sqrt{300^2 + 200^2} = \sqrt{90000 + 40000} = \sqrt{130000} \approx 360.55 \Omega \].

Step 3: Determine Power Factor \(\cos \theta\)

Power factor \( \cos \theta \) is the cosine of the phase angle between current and voltage. It is calculated as the ratio of resistance \(R\) to impedance \(Z\), \(\cos \theta = \frac{R}{Z}\). Substituting the known values \(\cos \theta = \frac{300}{360.55} \approx 0.832\).

Step 4: Use Power Formula to Find RMS Voltage \(V_{rms}\)

The average power consumed by the resistor can be expressed using \(P = I^2 R = \frac{V_{rms}^2}{Z^2} R\), where \( V_{rms} \) is the root mean square voltage. Rearranging the formula to find \( V_{rms} \), we substitute \( P = 60 \text{ W}, R = 300 \Omega, Z = 360.55 \Omega \):\[ 60 = \frac{V_{rms}^2}{360.55^2} \cdot 300 \]\[ V_{rms}^2 = \frac{60 \cdot 360.55^2}{300} \]Calculating this gives \( V_{rms} \approx 300 \text{ V} \approx 282.85 \text{ V}.\)

Key Concepts

Power Factor

The power factor in an RLC circuit is an essential concept that reflects how effectively the circuit converts electrical power into useful work. It is expressed as the cosine of the phase angle (θ) between the circuit's voltage and current. This angle arises because of the reactive components in the circuit, namely the inductor and the capacitor, which cause a time difference between the current and voltage waveforms.

In formula terms, the power factor (\( \cos \theta \)) is understood as the ratio of the circuit’s resistance (R) to its impedance (Z):

• \( \cos \theta = \frac{R}{Z} \)

A perfect power factor is 1, indicating that the voltage and current are in phase, and all the power is useful. In this case, however, calculated power factor value is 0.832, indicating a certain degree of inefficiency.

This inefficiency doesn't mean energy is lost; rather, a part of it is stored temporarily in the magnetic field of the inductor and the electric field of the capacitor, returning to the circuit later.

Impedance Calculation

Understanding impedance is crucial for analyzing AC circuits as it incorporates not just the resistance, but also the reactance which comes from capacitors and inductors in the circuit.

Impedance (\( Z \)) is similar to resistance but holds for AC circuits influenced by reactance. Calculating impedance in an RLC series circuit involves both the resistance (R) and the net reactance (denoted by the difference between inductive reactance \(X_{L}\) and capacitive reactance \(X_{C}\)). The formula is:

• \( Z = \sqrt{R^2 + (X_L - X_C)^2} \)

For the given exercise's components, we substitute the given values:

• Resistance \(R = 300 \Omega\)
• Inductive reactance \(X_L = 500 \Omega\)
• Capacitive reactance \(X_C = 300 \Omega\)

The resulting impedance is approximately \( 360.55 \Omega \).

This impedance value reflects not only the magnitude of opposition to AC current but also informs us on the phase difference between voltage and current.

RMS Voltage

RMS Voltage, or "root mean square" voltage, is a way of expressing an AC voltage value so that it equates to a comparable DC value. This metric is vital, as it allows us to calculate power in AC circuits just as we would for DC circuits. The RMS voltage provides a more useful measure of the voltage's effect as it represents the equivalent direct current value that would cause the same power dissipation in a resistive load.

To compute RMS voltage (\( V_{rms} \)), we rearrange the power formula \( P = \frac{V_{rms}^2}{Z^2} \times R \). Applying the known values yields:

• \( P = 60 \text{ W} \)
• \( R = 300 \Omega \)
• \( Z = 360.55 \Omega \)

The equation becomes:

• \( 60 = \frac{V_{rms}^2}{360.55^2} \times 300 \)

Solving this gives an RMS voltage of approximately \(282.85\text{ V} \).

RMS voltage is not only important for assessing power but also offers a practical measure for designing and analyzing circuits, ensuring they operate safely within the intended parameters.