Q Factor MCQ Quiz in বাংলা - Objective Question with Answer for Q Factor - বিনামূল্যে ডাউনলোড করুন [PDF]

Concept:

In a series RLC circuit, at resonance

resonant frequency 

Quality factor, 

Bandwidth, 

Calculation:

Given that, Quality factor = 11.18

 rad/sec.

C = 1 μF.

We can redraw the above circuit as follows

Now the circuit is series RLC circuit.

Quality factor in a series RLC circuit is

For critically damped system, ξ = 1

⇒ R = 60 Ω 

For series circuit,

R || 160 = 80 Ω

⇒ R = 160 Ω

Concept:
RLC series circuit:

 An RLC circuit is an electrical circuit consisting of Inductor (L), Capacitor (C), Resistor (R) it can be connected either parallel or series.

When the LCR circuit is set to resonate (XL = XC), the resonant frequency is expressed as 

Quality factor:

The quality factor Q is defined as the ratio of the resonant frequency to the bandwidth.

Mathematically, for a coil, the quality factor is given by:

Where,

XL & XC = Impedance of inductor and capacitor respectively

L, R & C = Inductance, resistance, and capacitance respectively

f_r = frequency

ω0 = angular resonance frequency

Calculation:

Given that 

f_r = 5000/2π hz

Impedance at resonance (Z) = resistance (R)= 56 Ω

ω₀ = 2π f_r = 5000 rad/sec

∴ 

L = 0.28 H

Given that

L = 50 × 10^-3 H

C = 5 × 10^-6 F

Resonant frequency,

Given that, current = 10 mA

At resonance,

⇒ R = 5 KΩ

Quality factor,

Concept

In a resonant series circuit, the quality factor (Q) is a measure of how underdamped the system is and how sharp the resonance is.

It is given by:

where, ω_o = Resonance frequency

BW = Bandwidth

Calculation

Given, QF = 100

ω_o = 1 MHz

BW = 1kHz

CONCEPT:

The Quality factor: Quality factor of resonance is a dimensionless parameter that describes how underdamped an oscillator or resonator is.

Mathamaticaly, Q factor =  

Where, L, C and R are the inductance, capacitance and resistance respectively.

EXPLANATION:

We know,

⇒ Q =  

⇒ Q =

⇒Q=50

Calculation:
The formula for the quality factor (Q) of an LC circuit is given by:

Q = (1 / R) × √(L / C)

Where:

• R = resistance = 100 Ω
• L = inductance = 80 mH = 80 × 10^-3 H
• C = capacitance = 2 μF = 2 × 10^-6 F

Substituting the values into the formula:

Q = (1 / 100) × √((80 × 10^-3) / (2 × 10^-6))

Q = (1 / 100) × √(40 × 10³)

Q = (1 / 100) × 200

Q = 2

The quality factor of the circuit is 2.

Explanation:

Series RLC Circuit Excited by DC Source

Definition: A series RLC circuit is an electrical circuit consisting of a resistor (R), inductor (L), and capacitor (C) connected in series. When this circuit is excited by a DC source, the behavior of the circuit is determined by the transient response, as there is no steady-state AC behavior due to the DC nature of the input.

Working Principle: Upon the application of a DC voltage, the capacitor initially acts as a short circuit, and the inductor as an open circuit. Over time, the capacitor charges, and the inductor allows current to flow. The transient response of the circuit involves oscillations that decay over time, and these oscillations are characterized by the damping ratio (ξ).

The damping ratio (ξ) is a dimensionless measure describing how oscillations in a system decay after a disturbance. It is defined as:

ξ = (R/L) / (2×√(1/LC))

Here’s the detailed step-by-step derivation of the correct option:

Step 1: Write the Standard Form of the Damping Ratio

The damping ratio for an RLC circuit is given by:

ξ = (R/2L) / (1/√(LC))

Step 2: Simplify the Expression

To simplify the expression, we can rewrite it as:

ξ = (R / 2L) × √(LC)

Therefore, the damping ratio ξ is:

ξ = (R/L) / (2×√(1/LC))

Conclusion:

The correct expression for the damping ratio in a series RLC circuit excited by a DC source is:

(R/L) / (2×√(1/LC))

This corresponds to Option 2, making it the correct choice.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: (2/√LC)/(L/R)

This option incorrectly places the inductor (L) and resistor (R) terms in the numerator and denominator, respectively. The correct relationship should involve the resistor in the numerator and inductor in the denominator as seen in the correct expression.

Option 3: (2/√LC)/(R/L)

This option is incorrect as it misplaces the inductor (L) and resistor (R) terms in the numerator and denominator. Additionally, the correct form should involve (R/L) in the numerator and (2×√(1/LC)) in the denominator.

Option 4: (L/R)/(2/√LC)

This option is incorrect because it incorrectly places the inductor (L) and resistor (R) terms in the numerator and denominator. The expression should involve (R/L) in the numerator and (2×√(1/LC)) in the denominator.

Conclusion:

Understanding the correct formulation of the damping ratio is crucial in analyzing the transient response of an RLC circuit. The correct option, as derived, is Option 2, which accurately represents the relationship between the circuit components and the damping behavior of the system.

Explanation:

Q Factor of a Coil

Definition: The Q factor, or quality factor, of a coil is a dimensionless parameter that describes the efficiency or quality of the coil in terms of its ability to store energy versus dissipating it. It is commonly used in the analysis of resonant circuits and is a measure of the sharpness of the resonance in the circuit.

Formula: The Q factor for a coil is defined as:

Q = X_L / R

Where:

• Q: Quality factor of the coil.
• X_L: Inductive reactance of the coil.
• R: Resistance of the coil.