Chapter 31: Problem 45
In an \(L-R-C\) series circuit, \(R\) = 300 \(\Omega\), X\(_C\) = 300 \(\Omega\), and X\(_L\) = 500 \(\Omega\). The average electrical power consumed in the resistor is 60.0 W. (a) What is the power factor of the circuit? (b) What is the rms voltage of the source?
Short Answer
Expert verified
(a) 0.832, (b) 161.1 V.
Step by step solution
01
Calculating the Impedance
To find the impedance, we need to know both the resistance and the reactance. The total reactance \( X \) in the circuit is \( X = X_L - X_C = 500 \Omega - 300 \Omega = 200 \Omega \). The total impedance \( Z \) is given by \( Z = \sqrt{R^2 + X^2} \). So, substitute the known values: \[ Z = \sqrt{300^2 + 200^2} = \sqrt{90000 + 40000} = \sqrt{130000} \approx 360.55 \, \Omega \].
02
Calculating the Power Factor
The power factor (PF) is the ratio of the resistance to the total impedance, given by \( \text{PF} = \frac{R}{Z} \). Plug in the values for \( R \) and \( Z \): \[ \text{PF} = \frac{300}{360.55} \approx 0.832 \].
03
Calculating the RMS Current
The average power \( P \) consumed by the resistor is given by \( P = I^2 R \). Rearrange to find the RMS current \( I \): \[ I = \sqrt{\frac{P}{R}} = \sqrt{\frac{60.0}{300}} = \sqrt{0.2} \approx 0.447 \, \text{A} \].
04
Calculating the RMS Voltage
The RMS voltage \( V \) of the source is given by \( V = I Z \). Use the current \( I \) and impedance \( Z \) we calculated previously: \[ V = 0.447 \times 360.55 \approx 161.1 \, \text{V} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Factor
The power factor in an L-R-C circuit is a measure of how effectively the electrical power is being converted into useful work. It is defined as the ratio of the real power (the power actually used in the circuit) to the apparent power (the total power flowing through the circuit). In simpler terms, the power factor indicates how well the current is aligned with the voltage in the circuit.
For our circuit, the power factor (PF) is calculated using the formula \( \text{PF} = \frac{R}{Z} \), where \( R \) is the resistance and \( Z \) is the total impedance.
For our circuit, the power factor (PF) is calculated using the formula \( \text{PF} = \frac{R}{Z} \), where \( R \) is the resistance and \( Z \) is the total impedance.
- Resistance \( R = 300 \Omega \)
- Total impedance \( Z \approx 360.55 \Omega \)
Impedance Calculation
Impedance is a crucial concept when analyzing AC circuits. It combines resistance with reactance, which is the opposition to a change in current. Total impedance \( Z \) in an L-R-C circuit can be found with the formula \( Z = \sqrt{R^2 + X^2} \), where \( R \) is resistance and \( X \) is the total reactance derived from inductive and capacitive reactance.
To find reactance:
To find reactance:
- Inductive reactance \( X_L = 500 \Omega \)
- Capacitive reactance \( X_C = 300 \Omega \)
- \( Z = \sqrt{300^2 + 200^2} = \sqrt{130000} \approx 360.55 \Omega \)
Average Power Consumption
In an L-R-C series circuit, understanding how much power is being consumed is essential. The average power consumed is the real power used in the circuit, which can be calculated using the formula \( P = I^2 R \). In this formula, \( P \) represents the power, \( I \) is the RMS current, and \( R \) is the resistance.
Given:
Given:
- Average power \( P = 60 \text{ W} \)
- Resistance \( R = 300 \Omega \)
- \( I = \sqrt{\frac{P}{R}} = \sqrt{\frac{60.0}{300}} \approx 0.447 \text{ A} \)
RMS Voltage
The RMS (Root Mean Square) voltage is vital to understand because it reflects the effective value of the voltage in an AC circuit. It corresponds to the amount of DC voltage that would produce the same power in a resistor as the AC voltage.
In the problem, the RMS voltage \( V \) can be calculated using the impedance \( Z \) and the RMS current \( I \) with the formula \( V = I Z \).
In the problem, the RMS voltage \( V \) can be calculated using the impedance \( Z \) and the RMS current \( I \) with the formula \( V = I Z \).
- RMS Current \( I \approx 0.447 \text{ A} \)
- Impedance \( Z \approx 360.55 \Omega \)
- \( V = 0.447 \times 360.55 \approx 161.1 \text{ V} \)