Problem 46 - College Physics

Question

A series \(\mathrm{RLC}\) circuit with a resistance of \(400 \Omega\) has capacitive and inductive reactances of \(300 \Omega\) and \(500 \Omega\) respectively. (a) What is the power factor of the circuit? (b) If the circuit operates at \(60 \mathrm{~Hz},\) what additional capacitance should be connected to the original capacitance to give a power factor of unity, and how should the capacitors be connected?

Step-by-Step Solution

Answer

(a) The power factor is approximately 0.895. (b) Add 2.65 \( \mu F \) in parallel to reach unity power factor.

Step 1: Calculate the Impedance of the Circuit

The total impedance (\( Z \)) of an \( RLC \) circuit is given by the formula:\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]where \( R \) is the resistance, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance.Substitute the given values: \( R = 400 \Omega \), \( X_L = 500 \Omega \), \( X_C = 300 \Omega \).\[Z = \sqrt{400^2 + (500 - 300)^2} = \sqrt{400^2 + 200^2}\]Calculate: \[Z = \sqrt{160,000 + 40,000} = \sqrt{200,000} = 447.21 \Omega\]

Step 2: Determine the Current Power Factor

The power factor (PF) is calculated using the formula:\[\text{PF} = \frac{R}{Z}\]With \( R = 400 \Omega \) and \( Z = 447.21 \Omega \), substitute to find the power factor:\[\text{PF} = \frac{400}{447.21} \approx 0.895\]

Step 3: Calculate the Required Capacitive Reactance for Unity Power Factor

For unity power factor, the inductive and capacitive reactances must be equal \( X_L = X_C \). Thus, the capacitive reactance should be:\[X_C = X_L = 500 \Omega\]Currently, \( X_C = 300 \Omega \), therefore additional reactance needed is:\[\Delta X_C = 500 - 300 = 200 \Omega\]

Step 4: Determine the Value of the Additional Capacitance

The capacitive reactance is related to capacitance by the formula:\[X_C = \frac{1}{2\pi f C}\]Rearranging for the capacitance needed to add the additional reactance \( \Delta X_C = 200 \Omega \):\[\Delta C = \frac{1}{2 \pi f \Delta X_C} = \frac{1}{2 \pi \times 60 \times 200}\]Calculating:\[\Delta C \approx \frac{1}{75472} \,F \approx 2.65 \text{ \(\mu F\)}\]

Step 5: Determine How Capacitors Should Be Connected

To achieve the total required capacitive reactance for unity power factor, the original and additional capacitances should be connected in parallel. This is because reactances in parallel result in a lower total reactance, thereby increasing total capacitance.

Key Concepts

Power Factor

The power factor in an RLC circuit is a measure of how effectively the circuit converts electrical power into usable work. It is the ratio of the circuit's real power, which performs work, to its apparent power, which is the total power flowing through the circuit.
The formula for power factor (PF) is given by:

• \[\text{PF} = \frac{R}{Z}\]

where \( R \) is the resistance and \( Z \) is the impedance. In simple terms, a power factor near 1 means the circuit is efficient, whereas a value near 0 indicates inefficiency.
In our given RLC circuit with a resistance of \( 400 \Omega \), the calculated impedance is \( 447.21 \Omega \). Substituting these values into the power factor formula, we arrive at a power factor of approximately 0.895.
This value signifies that the circuit converts about 89.5% of the electrical power into useful work, with the remaining 10.5% being reactive power, which does not perform any real work but affects the circuit's energy flow.

Impedance

Impedance is a critical factor in RLC circuits and measures the total opposition that a circuit offers to the flow of alternating current (AC). It combines both resistance and reactance into one quantity and is measured in ohms (\( \Omega \)).
The formula to calculate the impedance \( Z \) in an RLC circuit is:

• \[Z = \sqrt{R^2 + (X_L - X_C)^2}\]

Here, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance. It is important to note the directionality in which reactance affects impedance. If \( X_L \) is greater than \( X_C \), like in our exercise, the circuit presents more inductive behavior.
In our specific circuit, with \( R = 400 \Omega \), \( X_L = 500 \Omega \), and \( X_C = 300 \Omega \), the impedance is calculated to be \( 447.21 \Omega \).
This impedance value is a vector sum where both the resistive and the reactive components are considered in the context of AC circuits.

Capacitive Reactance

Capacitive reactance is a resistance offered by a capacitor in an AC circuit. It is inversely proportional to the frequency of the AC signal and the capacitance, described by the formula:

• \[X_C = \frac{1}{2\pi f C}\]

where \( f \) is the frequency and \( C \) is the capacitance. Capacitive reactance decreases with higher frequency and larger capacitance, which means capacitors allow more AC to flow at high frequencies.
In the given RLC circuit, the initial capacitive reactance is \( 300 \Omega \). To achieve a unity power factor, which means perfect energy efficiency, the capacitive reactance needs to equal the inductive reactance. Hence, an additional reactance of \( 200 \Omega \) is required.
This can be achieved by adding an extra capacitance, calculated by rearranging the capacitive reactance formula. The solution indicates that adding \( 2.65 \mu F \) of capacitance in parallel to the existing capacitor aligns the reactance with inductive reactance, thus achieving the desired power factor of 1.

Inductive Reactance

Inductive reactance is the reactance in a circuit due to inductors, similar to capacitive reactance but works in the opposite manner. It is dependent on the frequency of the applied AC signal and is given by:

• \[X_L = 2\pi f L\]

where \( L \) is the inductance. Unlike capacitive reactance, inductive reactance increases with frequency, meaning that inductors impede higher frequency currents more than lower frequencies.
In the RLC circuit under review, the inductive reactance is \( 500 \Omega \). This high reactance value contributes to the circuit's energy storage in its magnetic field, influencing the overall impedance undeniably more than the capacitive reactance.
For achieving efficient energy usage (power factor of 1), the circuit's capacitive and inductive reactances must balance each other out. This alignment reduces unnecessary reactive power circulation, ensuring the circuit operates optimally.