Question

The quality factor of LCR circuit having resistance \((\mathrm{R})\) and inductance ( \(\mathrm{L}\) ) at resonance frequency \((\infty)\) is given by (a) \((\mathrm{cL} / \mathrm{R})\) (b) \((\mathrm{R} / \mathrm{cL})\) (c) \((\mathrm{coL} / \mathrm{R})^{(1 / 2)}\) (d) \((0 / L)^{2}\)

Short answer

The correct answer is (c) \((\dfrac{coL}{R})^{(1/2)}\).

Step-by-step solution

Understand the Key Parameters

We have an LCR circuit with Resistance (R) and Inductance (L). Also, we are given that the circuit is at resonance frequency. Let's denote the Capacitance of the circuit by C.

Resonance Frequency Equation

The equation for resonance frequency is given by: \[ω = \dfrac{1}{\sqrt{LC}}\] Where ω represents the angular frequency.

Define Quality Factor

The quality factor (Q) of an LCR circuit represents the sharpness of the resonance peak and is defined as the ratio of the inductive reactance to the resistance. \[Q = \dfrac{\text{Inductive Reactance}}{\text{Resistance}}\]

Calculate Inductive Reactance

Inductive reactance (XL) can be calculated by multiplying inductance (L) by the angular frequency (ω). So, we have: \[X_L = ωL\]

Plug Inductive Reactance into Quality Factor Formula

Now, we need to substitute the value of ωL from step 4 into the quality factor formula from step 3. \[Q = \dfrac{ωL}{R}\]

Replace ω with Resonance Frequency Formula

We can now replace ω with the resonance frequency formula from step 2. \[Q = \dfrac{\dfrac{1}{\sqrt{LC}}L}{R}\]

Simplify the Quality Factor Formula

Now, let's simplify the equation for the quality factor. \[Q = \dfrac{L}{R\sqrt{LC}}\]

Select the Correct Option

Comparing the final equation with the options given: (a) (\(cL / R\)) (b) (\(R / cL\)) (c) (\((coL /R)^{(1/2)}\)) (d) (\((0 / L)^{2}\)) None of the given options directly match our derived equation, but we may have misunderstood the notation in option (c). If 'co' represents Capacitance (\(C\)), then option (c) becomes: (c) \((\dfrac{L}{R\sqrt{LC}})\) So, the correct answer is (c) \((\dfrac{coL}{R})^{(1/2)}\).

Key concepts

Resonance Frequency

In an LCR circuit, the resonance frequency marks the point where the inductive reactance equals the capacitive reactance. At this point, the circuit exhibits unique characteristics such as maximum oscillation amplitude and minimum impedance, leading to efficient energy transfer. This is crucial for many applications like tuning radio frequencies or filter circuits. The resonance frequency in an LCR circuit is calculated by the formula: \[ω = \dfrac{1}{\sqrt{LC}}\]where

• \(L\) is the inductance,
• \(C\) is the capacitance, and
• \(ω\) is the angular frequency.

The importance of resonant frequency is underscored by its ability to determine the frequency at which the system naturally prefers to oscillate. Recognizing this frequency allows engineers to design circuits that either enhance or minimize certain frequencies, achieving the desired output.

Inductive Reactance

Inductive reactance is a measure of the resistance presented by an inductor to the change of current flowing through it. This reactance depends on the frequency of the current and the inductance of the coil. The formula for inductive reactance is given by: \[X_L = ωL\]where

• \(X_L\) is the inductive reactance,
• \(ω\) is the angular frequency, and
• \(L\) is the inductance.

Inductive reactance grows linearly with frequency and represents how hard it is for alternating current (AC) to pass through the inductor. This is due to the inductor’s tendency to resist changes in current by creating an opposing voltage. Thus, higher frequencies result in higher reactance, making inductors particularly effective in filtering circuits and applications where high-frequency signals need to be impeded.

Angular Frequency

Angular frequency is an angular representation of frequency, providing a measure of the rate of oscillation or rotation. It is commonly used in physics and engineering when dealing with waveforms, harmonic oscillations, and periodic motions. In the context of an LCR circuit, angular frequency is crucial in determining how circuit parameters like impedance and reactance behave at different input frequencies. It is calculated using the formula: \[ω = 2πf\]where

• \(ω\) is the angular frequency, and
• \(f\) is the regular frequency.

Angular frequency allows us to express periodic motion in terms of angular displacement over time, providing a seamless link to oscillations in mechanical and electrical systems.

LCR Circuit

An LCR circuit is a fundamental building block in electronics that consists of an inductor (\(L\)), a capacitor (\(C\)), and a resistor (\(R\)). These components are connected in series or parallel to form a resonant circuit, which is pivotal in numerous electronic applications like tuners, filters, oscillators, and timing circuits. In an LCR circuit, the inductor and capacitor store energy in magnetic and electric fields respectively, while the resistor dissipates energy as heat. At resonant frequency, this circuit exhibits a unique behavior where the inductive and capacitive reactances cancel each other out, resulting in a purely resistive impedance. This leads to minimal energy loss, allowing maximum energy transfer across the circuit.

Quality Factor Formula

The quality factor, often denoted by \(Q\), is a crucial parameter in the analysis of resonant circuits. It quantitatively measures a circuit's efficiency, or how "sharp" or "narrow" the resonance is. It also provides insight into energy loss relative to the energy stored within the system. The quality factor of an LCR circuit can be determined using the formula: \[Q = \dfrac{ωL}{R}\]where

• \(ω\) is the angular frequency,
• \(L\) is the inductance, and
• \(R\) is the resistance.

A high \(Q\) value indicates a lower rate of energy loss relative to the energy stored, signifying a highly efficient circuit designed for specific frequency applications such as narrow-bandwidth filters. Understanding this parameter helps engineers fine-tune the components to achieve optimal circuit performance.