Question

An \(L\)-\(R\)-\(C\) series circuit has \(L = 0.600\) H and \(C = 3.00 \, \mu \mathrm{F}\). (a) Calculate the angular frequency of oscillation for the circuit when \(R =\) 0. (b) What value of \(R\) gives critical damping? (c) What is the oscillation frequency \(\omega'\) when \(R\) has half of the value that produces critical damping?

Short answer

(a) \( \omega \approx 2363.6 \) rad/s (b) \( R_c \approx 894.43 \) ohms (c) \( \omega' \approx 2047.9 \) rad/s

Step-by-step solution

Identify Given Values and Formulas

We have an L-R-C series circuit with \( L = 0.600 \) H and \( C = 3.00 \, \mu \mathrm{F} = 3.00 \times 10^{-6} \) F. We need to use the following formulas for angular frequency of oscillation \( \omega \), critical damping resistance \( R_c \), and the new frequency \( \omega' \) when \( R = R_c/2 \).1. Angular frequency \( \omega = \frac{1}{\sqrt{LC}} \).2. Critical damping resistance \( R_c = 2 \sqrt{\frac{L}{C}} \).3. Damped frequency formula \( \omega' = \sqrt{\omega_0^2 - (\frac{R}{2L})^2} \), where \( \omega_0 = \frac{1}{\sqrt{LC}} \).

Key concepts

Angular Frequency

In the context of an LRC circuit, angular frequency is an essential concept that defines the oscillation speed of the circuit without any resistance. It is represented by the symbol \( \omega \). In an LRC circuit with an inductor (\(L\)) and a capacitor (\(C\)), the angular frequency \( \omega \) is calculated using the formula:

• \( \omega = \frac{1}{\sqrt{LC}} \)

This formula shows that the angular frequency is inversely proportional to the square root of the product of the inductance and capacitance.
Therefore, by increasing either the inductance or capacitance or both, the angular frequency decreases. In simple terms, you can imagine angular frequency as how quickly the circuit can oscillate in the absence of resistance, illustrating the natural frequency of the system.

Critical Damping

Critical damping in an LRC circuit is achieved when the resistance is just enough to prevent oscillations. It allows the circuit to return to equilibrium as quickly as possible without oscillating. In the context of an LRC circuit, the critical damping resistance \( R_c \) can be found using:

• \( R_c = 2 \sqrt{\frac{L}{C}} \)

This formula implies that \( R_c \) is influenced by both the inductance \(L\) and capacitance \(C\).
Critically damped systems are particularly significant in applications where we want systems to stabilize quickly without overshooting, such as in car suspensions or electrical circuits where fast response times are necessary.

Damped Oscillation

Damped oscillation in an LRC circuit occurs when the circuit includes resistance, which causes the amplitude of oscillations to decrease over time. This damping effect links the resistance \( R \) with the circuit's oscillatory behavior. The frequency of the damped oscillation \( \omega' \) is given by:

• \( \omega' = \sqrt{\omega_0^2 - \left(\frac{R}{2L}\right)^2} \)

where \( \omega_0 \) is the undamped angular frequency.
As you increase the resistance, the frequency of oscillation \( \omega' \) decreases, indicating stronger damping. In practice, damped oscillations are important for systems that require gradual energy disipation, such as in tuning circuits or shock absorbers.

Series Circuit

A series circuit is a configuration where components are connected one after another, forming a single pathway for the current to flow through. In the case of an LRC circuit, the inductor (\(L\)), resistor (\(R\)), and capacitor (\(C\)) are connected in a series one after another.
This type of circuit allows the same current to flow through each component, which affects the overall behavior of the circuit. Some characteristics of series circuits include:

• Voltage is divided among the components.
• Current remains constant throughout the circuit.
• Series circuits are often simpler to analyze but also more vulnerable to failure if any component fails.

Understanding series circuits is crucial in electronics because they form the basis of many electrical devices, providing basic insights into how energy is distributed in a circuit.