Problem 36 - College Physics

Question

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 0.832. (b) RMS voltage is 161.19 V.

Step 1: Calculate the Impedance

First, calculate the net impedance of the circuit. The impedance in a series \(R-L-C\) circuit is given by:\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]Here, \(R = 300 \Omega\), \(X_C = 300 \Omega\), and \(X_L = 500 \Omega\). Therefore,\[Z = \sqrt{300^2 + (500 - 300)^2} = \sqrt{300^2 + 200^2}\]\[Z = \sqrt{90000 + 40000} = \sqrt{130000}\]\[Z = 360.56 \Omega\]

Step 2: Determine the Power Factor

The power factor (pf) is given by the formula:\[\text{pf} = \frac{R}{Z}\]Using the calculated impedance \(Z = 360.56 \Omega\) and resistor \(R = 300 \Omega\),\[\text{pf} = \frac{300}{360.56} = 0.832\]

Step 3: Calculate Current using Average Power

The average power consumed by the resistor is \(P = 60.0 \text{ W}\). This can also be expressed as\[P = I^2 R\]Rearranging to solve for \(I_{rms}\),\[I = \sqrt{\frac{P}{R}} = \sqrt{\frac{60.0}{300}} = \sqrt{0.2} = 0.447 \text{ A}\]

Step 4: Calculate the RMS Voltage of the Source

The root mean square (rms) voltage \(V_{rms}\) is given by\[V = IZ\]where \(I = 0.447 \text{ A}\) and \(Z = 360.56 \Omega\). Thus,\[V_{rms} = 0.447 \times 360.56 = 161.19 \text{ V}\]

Key Concepts

Power Factor

The power factor is an important concept in electrical circuits, especially in AC circuits. It measures how effectively electrical power is being converted into useful work output. A power factor closer to 1 indicates that the circuit's power consumption is highly efficient. In an RLC circuit, the power factor is influenced by the phase difference between the voltage and current due to reactive components like inductors and capacitors.
To calculate the power factor, use the formula:

• Power factor (pf) = \(\frac{R}{Z}\)

Here, \(R\) is the resistance, and \(Z\) is the impedance of the circuit. For our circuit, the power factor is calculated by dividing 300 \(\Omega\) (resistance) by 360.56 \(\Omega\) (impedance), resulting in a power factor of 0.832. This indicates that a significant portion of the supplied power is being used effectively, but there is still some reactive power present.

Impedance Calculation

Impedance is a key concept in AC circuits, combining resistance and reactance. Unlike resistance, which only opposes the flow of direct current (DC), impedance also accounts for the effects of the circuit's inductance and capacitance on alternating current (AC).
In an RLC circuit, impedance (\(Z\)) is calculated with the formula:

• \(Z = \sqrt{R^2 + (X_L - X_C)^2}\)

Where:

• \(R\) is resistance,
• \(X_L\) is inductive reactance,
• \(X_C\) is capacitive reactance.

In our given exercise, impedance calculations use a resistance of 300 \(\Omega\), an inductive reactance of 500 \(\Omega\), and a capacitive reactance of 300 \(\Omega\). Performing these calculations results in a total impedance of approximately 360.56 \(\Omega\). This impedance value is crucial for determining other parameters like the power factor and voltage.

Average Power

Average power in an AC circuit is the real power consumed and is especially of interest in resistive components. It is calculated using the formula \(P = I^2 R\), where \(P\) represents power, \(I\) is the rms current, and \(R\) is the resistance.
In the example problem, it is given that the resistor consumes an average power of 60 watts. By rearranging the power formula as \(I = \sqrt{\frac{P}{R}}\), you can solve for the rms current. This is calculated to be approximately 0.447 amperes.
Knowing the average power helps in understanding the total energy being utilized in the circuit over time, thus helping in efficient circuit design and load management.

RMS Voltage

Root Mean Square (RMS) voltage is a concept used to determine the effective voltage in an AC circuit. It provides the same power deliverance as a DC circuit would. Calculating the RMS voltage involves considering the current flowing through the circuit and the total impedance.
For the given RLC circuit, RMS voltage is computed with the formula:

• \(V_{rms} = I \times Z\)

Where:

• \(I\) is the rms current,
• \(Z\) is the impedance.

Plugging in the previously calculated current (0.447 A) and impedance (360.56 \(\Omega\)), the RMS voltage is given as 161.19 volts. Understanding RMS voltage is fundamental in determining the critical voltage level required by AC devices for optimal performance.