Question

In a circuit \(\mathrm{L}, \mathrm{C}\) and \(\mathrm{R}\) are connected in series with an alternating voltage source of frequency \(\mathrm{f}\). The current leads the voltage by \(45^{\circ}\). The value of \(c\) is, (a) \([1 /\\{2 \pi \mathrm{f}(2 \pi \mathrm{fL}+\mathrm{R})\\}]\) (b) \([1 /\\{\pi f(2 \pi \mathrm{fL}+R)\\}]\) (c) \([1 /\\{2 \pi f(2 \pi f L-R)\\}]\) (d) \([1 /\\{\pi \mathrm{f}(2 \pi \mathrm{fL}-\mathrm{R})\\}]\)

Short answer

The value of capacitance (C) in the given circuit is \(C = [1 / \{2 \pi f(2 \pi f L-R)\}]\), which corresponds to option (c).

Step-by-step solution

Phase Angle of the Circuit

In a series RLC circuit, the phase angle (ϕ) determines how much the current leads or lags the voltage. It can be calculated using the formula: \(ϕ = arctan(\dfrac{X_L - X_C}{R})\) We know the current leads the voltage by 45°, so the phase angle is: \(ϕ = +45^{\circ}\)

Convert Degrees to Radians

To continue calculations, we need to convert the phase angle from degrees to radians: \(ϕ = +45^{\circ} \times \dfrac{\pi}{180^\circ} = \dfrac{\pi}{4}\) radians

Calculate Reactances of Inductor and Capacitor

In an RLC circuit, the reactances of the inductor (L) and capacitor (C) are given by: \(X_L = 2 \pi f L\) \(X_C = \dfrac{1}{2 \pi f C}\)

Plug the Values into Phase Angle Formula

Use the given phase angle formula and the ϕ and reactances we calculated: \(\dfrac{\pi}{4} = arctan(\dfrac{2 \pi f L - \dfrac{1}{2 \pi f C}}{R})\)

Solve for Capacitance (C)

To find the value of C, we need to solve the above equation. First, we use the tan function to get rid of arctan: \(tan(\dfrac{\pi}{4}) = \dfrac{2 \pi f L - \dfrac{1}{2 \pi f C}}{R}\) Since \(tan(\dfrac{\pi}{4}) = 1\), we have: \(1 = \dfrac{2 \pi f L - \dfrac{1}{2 \pi f C}}{R}\) Now solve for C: \(2 \pi f C(2 \pi f L - \dfrac{1}{2 \pi f C}) = R(2 \pi f C)\) \(4 \pi^2 f^2 L C^2 - 1 = 2 \pi f CR\) We want the value of C, divide by \(2 \pi f(2 \pi f L - R)\) to get: \(C = [1 / \{2 \pi f(2 \pi f L-R)\}]\) The answer is option (c).

Key concepts

Phase Angle

In a series RLC circuit, the phase angle is a crucial concept that defines the relationship between the current and voltage. The phase angle, represented by \( \phi \), tells us whether the current leads or lags behind the voltage waveform. This is vital for understanding how energy is transferred in the circuit.

The calculation of the phase angle in an RLC circuit uses the formula:

• \( \phi = \arctan\left(\dfrac{X_L - X_C}{R}\right) \)

Here, \( X_L \) and \( X_C \) are the inductive and capacitive reactances respectively, and \( R \) is the resistance.

The angle can be positive or negative indicating leading or lagging relationships. If the current leads the voltage by \( 45^{\circ} \), then it implies the system is more capacitive. This gives us essential insights into the circuit's behavior and efficiency.

Reactance

Reactance is another pivotal component of understanding RLC circuits. It's a measure of the opposition that inductors and capacitors exert against the change in current in the circuit.

Reactance is frequency-dependent and varies for inductors and capacitors:

• Inductive Reactance \( X_L = 2\pi f L \): Inductive reactance increases with frequency and inductance. It is the opposition to the change of current in inductors.
• Capacitive Reactance \( X_C = \dfrac{1}{2 \pi f C} \): Capacitive reactance decreases with frequency and capacitance. It shows how capacitors oppose changes to voltage across the terminals.

Understanding the balance between \( X_L \) and \( X_C \) helps in tuning the circuit to achieve desired performance, like minimizing power loss or optimizing energy transfer.

Alternating Current

Alternating Current (AC) is an electric current in which the flow of electric charge periodically reverses direction. It contrasts with direct current (DC) where the flow of electric charge is only in one direction.

In AC circuits,

• The voltage and current are sinusoidal, meaning they have the shape of a sine wave.
• This waveform is characterized by frequency, amplitude, and phase, all of which are essential for analyzing and designing AC circuits.

The study of AC is crucial because most appliances and power systems operate on AC due to its efficiency over long distances and ability to power high energy devices effectively.

Capacitance

Capacitance is a measure of a capacitor's ability to store charge per unit voltage. It is a fundamental property in RLC circuits that influences how they behave at different frequencies.

Key attributes include:

• The unit of capacitance is the Farad (F), which quantifies capacitance.
• It stores energy in the electric field between its plates.
• In an RLC circuit, capacitance affects the circuit's total impedance and, subsequently, the phase angle.

Understanding capacitance helps in assessing how the circuit will perform under different conditions, such as how quickly it can charge and discharge, and how it affects the AC waveform.