Question

If a series circuit has a capacitor of \(C=0.8 \times 10^{-6}\) farad and an inductor of \(L=0.2\) henry, find the resistance \(R\) so that the circuit is critically damped.

Short answer

Answer: The resistance required is approximately 2.53 × 10^{-3} Ω.

Step-by-step solution

Understand the Concept of Critically Damped Circuit

In a series RLC circuit, the damping ratio determines whether the circuit is underdamped, critically damped, or overdamped. A critically damped circuit is the ideal case for many applications, as it allows the circuit to return to steady-state as quickly as possible without oscillating. The damping ratio (ξ) is given by the following formula: ξ = \(\frac{R}{2} \sqrt{\frac{C}{L}}\) For a critically damped circuit, ξ = 1.

Plug in the Given Values and Solve for R

Now, we'll plug in the given values for capacitance (C) and inductance (L) into the formula for the damping ratio. 1 = \(\frac{R}{2} \sqrt{\frac{0.8 \times 10^{-6}}{0.2}}\) Let's now solve for R: R = 2 × \(\sqrt{0.2 \times 0.8 \times 10^{-6}}\) R ≈ 2 × \(\sqrt{1.6 \times 10^{-6}}\) R ≈ 2 × 1.26491 × 10^{-3} R ≈ 2.52982 × 10^{-3} Ω So, the resistance R required to make the circuit critically damped is approximately 2.53 × 10^{-3} ohms.

Key concepts

Damping Ratio

The damping ratio is a key parameter in characterizing the response of a system to disturbances. It indicates how oscillations in a system decay after a disturbance. Specifically, in an ideal case such as a critically damped system, the damping ratio is equal to 1. This ensures that the system returns to its steady state as quickly as possible without oscillating.

In terms of a series RLC circuit, the damping ratio (\(\xi\)) is calculated using the formula:

• \(\xi = \frac{R}{2} \sqrt{\frac{C}{L}}\)

where \(R\) is the resistance, \(C\) is the capacitance, and \(L\) is the inductance. When \(\xi = 1\), the system is critically damped. This means the resistance is perfectly balanced with the capacitance and inductance, providing optimal performance. Lower than this value results in underdamping, leading to oscillations, while higher values cause overdamping, slowing the system unnecessarily.

Series RLC Circuit

A series RLC circuit is a fundamental component in electronic engineering composed of three main elements: a resistor (\(R\)), an inductor (\(L\)), and a capacitor (\(C\)). These elements are connected in series, meaning their arrangement is such that the current flows through each component one after another.

This type of circuit is commonly used to analyze the temporal response of electrical systems. The behavior of a series RLC circuit depends on the relationships between its components: resistance, inductance, and capacitance. The combined effect of these components is reflected in the circuit's damping characteristics.

• When the resistance, inductance, and capacitance are perfectly balanced to achieve a critical damping condition, the circuit can rapidly settle to zero without any oscillation.
• This makes it very useful in applications where quick stabilization is necessary, such as certain radio receivers or audio filters.

Understanding a series RLC circuit requires knowing how these components interact and affect the overall electrical characteristics of the circuit.

Capacitor

A capacitor is an electronic component used to store energy in an electric field. In a series RLC circuit, it plays a crucial role in determining the system's oscillatory behavior. Capacitors are commonly used to manage power supply stability and filter out noise in circuits.

Capacitors consist of two conductive plates separated by an insulating material (dielectric). The capacitance \(C\) is measured in farads (F) and indicates the ability of a capacitor to store an electric charge. The value of the capacitance directly affects how quickly a circuit responds to changes in voltage.

• In our example, the capacitor value is given as \(C = 0.8 \times 10^{-6}\) farads, or 0.8 microfarads.
• This value plays a pivotal role in calculating the damping ratio as it determines the balance of charges in the circuit, influencing the time it takes for the circuit to return to steady state.

The efficiency of a capacitor in managing energy within an RLC circuit is crucial for the optimal performance of electronic devices.

Inductor

An inductor is a passive electrical component that stores energy in a magnetic field when electrical current flows through it. It is one of the three main elements in a series RLC circuit, alongside the resistor and capacitor.

The inductance \(L\) is measured in henrys (H) and relates to the component's ability to oppose changes in current. Inductors are crucial in circuits that require precise manipulation of current and voltage.

• In the context of the given problem, the inductor value is \(L = 0.2\) henry. This specific value influences how the circuit reacts over time to changes in electrical current.
• It affects the calculation of the damping ratio by contributing to the overall resistance to changes in the circuit's current or energy state.

Inductors are especially vital in tuning circuits and filtering applications where stable and consistent energy flow is critical.