Problem 69 - Fundamentals of Physics

Question

A generator of frequency \(3000 \mathrm{~Hz}\) drives a series \(R L C\) circuit with an emf amplitude of \(120 \mathrm{~V}\). The resistance is \(40.0 \Omega\), the capacitance is \(1.60 \mu \mathrm{F},\) and the inductance is \(850 \mu \mathrm{H} .\) What are (a) the phase constant in radians and (b) the current amplitude? (c) Is the circuit capacitive, inductive, or in resonance?

Step-by-Step Solution

Answer

(a) Phase constant is approximately -0.403 radians. (b) Current amplitude is about 2.78 A. (c) The circuit is capacitive.

Step 1: Determine Angular Frequency

First, calculate the angular frequency \(\omega\) using the equation \(\omega = 2\pi f\), where \(f\) is the frequency. Given \(f = 3000 \mathrm{~Hz}\), we have:\[\omega = 2\pi \times 3000 = 6000\pi \mathrm{~rad/s}.\]

Step 2: Calculate Impedances

Find the inductive reactance \(X_L\) and capacitive reactance \(X_C\):- Inductive reactance \(X_L = \omega L\), where \(L = 850 \mu H = 850 \times 10^{-6} H\): \[X_L = 6000\pi \times 850 \times 10^{-6} \approx 16.02 \Omega.\]- Capacitive reactance \(X_C = \frac{1}{\omega C}\), where \(C = 1.60 \mu F = 1.60 \times 10^{-6} F\): \[X_C = \frac{1}{6000\pi \times 1.60 \times 10^{-6}} \approx 33.24 \Omega.\]

Step 3: Calculate Phase Constant

The phase constant \(\phi\) is given by:\[\tan \phi = \frac{X_L - X_C}{R},\]where \(R = 40 \Omega\). Thus:\[\tan \phi = \frac{16.02 - 33.24}{40} = -0.43.\]So, \(\phi = \arctan(-0.43) \approx -0.403\) radians.

Step 4: Calculate Current Amplitude

The current amplitude \(I\) is calculated using:\[I = \frac{\text{emf amplitude}}{Z},\]where \(Z = \sqrt{R^2 + (X_L - X_C)^2}\) is the impedance:\[Z = \sqrt{40^2 + (16.02 - 33.24)^2} = \sqrt{40^2 + (-17.22)^2} \approx 43.24 \Omega.\]Thus:\[I = \frac{120}{43.24} \approx 2.78 \mathrm{~A}.\]

Step 5: Determine Circuit Type

The circuit characteristic is determined by comparing \(X_L\) and \(X_C\):- If \(X_L > X_C\), it’s inductive.- If \(X_L < X_C\), it’s capacitive.- If \(X_L = X_C\), it’s in resonance.Since \(X_L = 16.02 \Omega < X_C = 33.24 \Omega\), the circuit is capacitive.

Key Concepts

Understanding Phase Constant

The phase constant, often symbolized as \( \phi \), is a crucial parameter in the analysis of RLC circuits. It represents the phase difference between the voltage across the circuit and the current flowing through it. In an RLC circuit, this phase difference arises due to the reactive components, namely the inductor and the capacitor, which cause the current to either lead or lag the voltage.
To calculate the phase constant, we use the formula \( \tan \phi = \frac{X_L - X_C}{R} \), where \( X_L \) is the inductive reactance, \( X_C \) is the capacitive reactance, and \( R \) is the resistance. The sign of \( \phi \) indicates whether the circuit is predominantly inductive or capacitive.

• If \( \phi > 0 \), the voltage leads the current, and the circuit is inductive.
• If \( \phi < 0 \), the current leads the voltage, and the circuit is capacitive.
• If \( \phi = 0 \), the circuit is in resonance, and voltage and current are in phase with each other.

In our example, we calculated \( \phi \) as approximately \(-0.403\) radians, indicating a capacitive circuit where the current leads the voltage.

Exploring Reactances

Reactance is a measure of how much a circuit resists the flow of alternating current due to its inductive and capacitive components. Unlike resistance, which dissipates energy, reactance stores and releases energy.

• Inductive Reactance \( (X_L) \): This occurs due to the presence of inductors which oppose changes in current. It is frequency dependent and calculated as \( X_L = \omega L \), where \( \omega = 2\pi f \) is the angular frequency, and \( L \) is the inductance.
• Capacitive Reactance \( (X_C) \): This occurs due to capacitors which oppose changes in voltage. It is also frequency dependent and calculated as \( X_C = \frac{1}{\omega C} \), where \( C \) is the capacitance.

In our circuit, we found \( X_L \) to be approximately \( 16.02 \Omega \) and \( X_C \) to be about \( 33.24 \Omega \). This difference indicates whether the circuit behaves more like an inductor or a capacitor at a given frequency. Since \( X_C > X_L \), the circuit is primarily capacitive.

Impedance and its Impact

Impedance, represented by \( Z \), is a comprehensive measurement of opposition a circuit presents to the flow of AC, encompassing both resistance and reactance. Impedance is expressed as a complex quantity but often simplified to its magnitude for analysis.
The impedance in RLC circuits is calculated using the formula \( Z = \sqrt{R^2 + (X_L - X_C)^2} \). This formula combines resistance and the difference between inductive and capacitive reactances, providing insight into the overall difficulty the current faces as it flows through the circuit.
A higher impedance means that the circuit opposes the current more, reducing the current amplitude for a given voltage. In our example, the impedance was calculated to be approximately \( 43.24 \Omega \), and this played a vital role in determining the current amplitude of around \( 2.78 \text{ A} \).
Impedance not only informs us about the "ease" of current flow but also helps determine the phase relationships between voltage and current, crucial for predicting circuit behavior.