Problem 61 - Fundamentals of Physics

Question

Figure \(31-36\) shows an ac generator connected to a "black box" through a pair of terminals. The box contains an \(R L C\) circuit, possibly even a multiloop circuit, whose elements and connections we do not know. Measurements outside the box reveal that \(\mathscr{E}(t)=(75.0 \mathrm{~V}) \sin \omega_{d} t\) and \(i(t)=(1.20 \mathrm{~A}) \sin \left(\omega_{d} t+42.0^{\circ}\right)\) (a) What is the power factor? (b) Does the current lead or lag the emf? (c) Is the circuit in the box largely inductive or largely capacitive? (d) Is the circuit in the box in resonance? (e) Must there be a capacitor in the box? (f) An inductor? (g) A resistor? (h) At what average rate is energy delivered to the box by the generator? (i) Why don't you need to know \(\omega_{d}\) to answer all these questions?

Step-by-Step Solution

Answer

(a) 0.743, (b) lead, (c) capacitive, (d) not in resonance, (e) yes, (f) yes, (g) yes, (h) 31.5 W, (i) it's unnecessary for phase and power calculations.

Step 1: Calculate the Power Factor

To calculate the power factor, use the formula: \( \cos(\phi) \), where \( \phi \) is the phase angle between the voltage and current. Given that \( \phi = 42.0^\circ \), the power factor is \( \cos(42.0^\circ) \approx 0.743 \). This value suggests that a good portion of the power is used effectively, but some is reactive.

Step 2: Determine Current Lead or Lag

The current leads or lags the emf depending on the phase angle. Since the current phase angle \( (42.0^\circ) \) is positive relative to the emf phase angle (which is zero in this case), the current leads the emf.

Step 3: Identify Circuit Type (Inductive or Capacitive)

If the current leads the emf, this implies the presence of a largely capacitive circuit as capacitors cause the current to lead the voltage.

Step 4: Check for Resonance

Resonance occurs in an RLC circuit when the inductive and capacitive reactances are equal, causing the impedance to be purely resistive and the power factor to be 1. Since the power factor is not 1, the circuit is not in resonance.

Step 5: Determine Presence of Capacitor

Given that the circuit is largely capacitive, it is likely that there is a capacitor present in the box.

Step 6: Determine Presence of Inductor

Even though the circuit is mainly capacitive, if there is no resonance, there must be an inductor as well, as the imbalance between capacitive and inductive reactance is what maintains the non-resonant condition.

Step 7: Determine Presence of Resistor

Since real power is consumed (as shown by the non-zero power factor), there must be a resistor present in the box, which is responsible for the real power consumption.

Step 8: Calculate the Average Power Delivered

The average power delivered \( P_{avg} \) can be calculated using the formula: \[ P_{avg} = V_{rms} \times I_{rms} \times \cos(\phi) \] where \( V_{rms} = \frac{75.0}{\sqrt{2}} \) and \( I_{rms} = \frac{1.20}{\sqrt{2}} \). Plugging in the values, \[ P_{avg} = \frac{75.0}{\sqrt{2}} \times \frac{1.20}{\sqrt{2}} \times 0.743 \approx 31.5 \, \text{W} \]

Step 9: Reassess the Need for Frequency \(\omega_{d}\)

The problem involves phase relationships and power calculations, all of which are independent of \(\omega_{d}\). These relationships remain consistent at any frequency, which is why knowing \(\omega_{d}\) is unnecessary.

Key Concepts

Power Factor

The power factor is an essential aspect of RLC circuit analysis. It helps determine how effectively electrical power is being converted into usable work output. In an RLC circuit, the power factor is calculated using the cosine of the phase angle between the voltage and the current, denoted as \( \cos(\phi) \). A power factor closer to 1 means that more power is being used effectively, while a value further from 1 indicates more wasted energy, known as reactive power.

• Power factor formula: \( \cos(\phi) \)
• Indicates efficiency of power usage
• Important for reducing energy waste in electrical systems

For the exercise, the phase angle \( \phi \) is provided as \( 42.0^{\circ} \), leading to a power factor of approximately 0.743. This suggests a moderate amount of energy is reactive and not contributing to the work output effectively.

Electrical Phase Angle

Understanding the concept of electrical phase angle is crucial in RLC circuit analysis. The phase angle \( \phi \) is the angle of difference in the timing of the alternating current (AC) waveform compared to the alternating voltage waveform. It indicates whether the current waveform is leading or lagging the voltage waveform.

• Positive phase angle: current leads the voltage
• Negative phase angle: current lags the voltage

In the original exercise, a positive phase angle of \( 42.0^{\circ} \) shows that the current leads the voltage. This behavior is characteristic of capacitive components in the circuit, where capacitors tend to shift the current waveform ahead of the voltage waveform.

Average Power Calculation

Calculating average power in an RLC circuit involves considering both the root mean square (RMS) values of voltage and current, as well as the power factor. The average power \( P_{avg} \) delivered to the circuit can be found with the formula:
\[ P_{avg} = V_{rms} \times I_{rms} \times \cos(\phi) \]
- \( V_{rms} \) and \( I_{rms} \) are obtained by dividing the peak values by \( \sqrt{2} \).
- The power factor \( \cos(\phi) \), calculated earlier, is crucial in determining the true power drawn by the circuit.
For the given values, \( V_{rms} = \frac{75.0}{\sqrt{2}} \) and \( I_{rms} = \frac{1.20}{\sqrt{2}} \). Substituting these into the formula gives \( P_{avg} \approx 31.5 \, \text{W} \). This quantifies the power being transferred in the circuit from the generator.

Inductive and Capacitive Reactance

Inductive and capacitive reactance are essential concepts in analyzing RLC circuits. They are part of what determines whether a circuit's behavior is predominantly inductive or capacitive.

• Inductive Reactance \( X_L \): Reactance that opposes changes in current, usually found in inductors.
• Formula: \( X_L = \omega L \) where \( \omega \) is the angular frequency and \( L \) is inductance.
• Capacitive Reactance \( X_C \): Reactance that opposes changes in voltage, usually found in capacitors.
• Formula: \( X_C = \frac{1}{\omega C} \) where \( C \) is capacitance.

An RLC circuit's behavior is determined by comparing \( X_L \) and \( X_C \). If \( X_C \) is greater, the circuit is predominantly capacitive, leading to a capacitive phase angle where current leads voltage. This aligns with the given exercise, where it is deduced that the circuit is largely capacitive since current leads voltage by \( 42.0^{\circ} \).