Problem 35 - Physics

Question

In a series RCL circuit the dissipated power drops by a factor of two when the frequency of the generator is changed from the resonant frequency to a nonresonant frequency. The peak voltage is held constant while this change is made. Determine the power factor of the circuit at the nonresonant frequency.

Step-by-Step Solution

Answer

The power factor at the nonresonant frequency is 0.5.

Step 1: Understand the Power Expression

The power dissipated in an RCL circuit at a nonresonant frequency can be expressed as \( P = \frac{V_m^2}{2Z} \cdot \cos \phi \), where \( V_m \) is the peak voltage, \( Z \) is the impedance, and \( \phi \) is the phase angle. At resonance, \( \cos \phi = 1 \) because the circuit is purely resistive, thus the power is \( P_0 = \frac{V_m^2}{2R} \).

Step 2: Relate Power at Resonance and Nonresonance

Let the dissipated power at resonance be \( P_0 \) and at nonresonance be \( P_{nr} \). Given \( P_{nr} = \frac{1}{2}P_0 \), we have \( \frac{1}{2} \cdot \frac{V_m^2}{2R} = \frac{V_m^2}{2Z} \cdot \cos \phi \).

Step 3: Solve for Power Factor

By equating the expressions for \( P_{nr} \) and substituting \( P_0 \), we get \( \frac{1}{2} = \frac{R}{Z} \cdot \cos \phi \). Since \( Z = \sqrt{R^2 + (X_L - X_C)^2} \) and \( X_L \) and \( X_C \) represent the inductive and capacitive reactances, solve \( \cos \phi = \frac{1}{2} \cdot \frac{Z}{R} \).

Step 4: Apply Power Factor Definition

Since the power factor \( \cos \phi \) is desired, noting that \( \frac{R}{Z} = \frac{1}{2} \cdot \cos \phi \), rearrange to find \( \cos \phi = 0.5 \cdot \frac{Z}{R} = 0.5 \). Thus, \( \cos \phi = 0.5 \).

Key Concepts

Resonant Frequency in RCL Circuits

In an RCL circuit, resonant frequency is a key concept that affects how the circuit behaves and performs. It is the frequency at which the reactive components (inductance and capacitance) cancel each other out, meaning the impedance is at its minimum and is purely resistive. This frequency can be found using the formula:

• Resonant frequency, \( f_0 = \frac{1}{2\pi\sqrt{LC}} \), where \( L \) is inductance and \( C \) is capacitance.

At resonance, energy oscillates between the inductor and capacitor with no loss, maximizing the power transfer to the resistive load. This is important because at this frequency, the circuit's efficiency in power use is maximized. When the circuit operates at resonant frequency, the overall impedance is simply the resistance, \( R \), which results in a power factor of 1. This implies that all the power supplied is used effectively by the circuit components.

Understanding Power Factor

The power factor is a crucial measure in RCL circuits when analyzing the efficiency of power usage. It is defined as the cosine of the phase angle (\( \cos \phi \)) between the voltage and current in the circuit. At resonance, the power factor is 1, meaning that voltage and current are in phase, and all the power is being effectively utilized:

• Power factor formula: \( \text{Power factor} = \cos \phi \).

A power factor less than 1 indicates that some power is not being effectively converted into useful work, often due to reactance in the circuit. This scenario occurs at nonresonant frequencies where the circuit contains both inductive and capacitive reactance, potentially resulting in either a lagging or leading power factor. Nonresonant operation adds additional reactance, increasing total impedance and reducing the portion of power used effectively.

Exploring Dissipated Power in Circuits

Dissipated power in an RCL circuit represents the actual power consumed by the resistive component of the circuit. This can be a critical factor, as it's the portion of the power supply that does useful work (e.g., generating heat or running a motor). The formula to calculate the dissipated power at any frequency is:

• \( P = \frac{V_m^2}{2Z} \cdot \cos \phi \).

Here, \( V_m \) is the peak voltage, \( Z \) is impedance, and \( \phi \) is the phase angle. At the resonant frequency, the impedance \( Z \) equals the resistance \( R \), maximizing power dissipation. If the frequency changes away from resonance, the dissipated power drops due to increased impedance from net reactance not being zero. In fact, the exercise specifically illustrates how the power drop is constricted when moving off resonance.

Impedance and Its Role

Impedance is a fundamental concept in RCL circuits, which expands upon the idea of resistance by also considering capacitive and inductive reactances. It is the total opposition a circuit offers to the flow of alternating current (AC) at a given frequency. The impedance \( Z \) in a series RCL circuit is calculated as:

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

Here, \( X_L \) and \( X_C \) are the inductive and capacitive reactances, respectively. At the resonant frequency, \( X_L \) exactly cancels out \( X_C \), causing impedance to equal the resistance \( R \). Outside of this frequency, the imbalance between \( X_L \) and \( X_C \) increases total impedance. This increased impedance affects both the power factor and the amount of power dissipated by the circuit. Understanding and calculating impedance is essential for designing circuits that function efficiently at specific frequencies.