Q Factor MCQ Quiz in தமிழ் - Objective Question with Answer for Q Factor - இலவச PDF ஐப் பதிவிறக்கவும்

We can redraw the above circuit as follows

Now the circuit is series RLC circuit.

Quality factor in a series RLC circuit is

\(Q = \frac{1}{R}\sqrt {\frac{L}{C} = } \frac{1}{{\left( {\frac{{30R}}{{30 + R}}} \right)}}\;\sqrt {\frac{{1 \times {{10}^{ - 3}}}}{{0.01 \times {{10}^{ - 3}}}}} \)

\(Q = \frac{{30 + R}}{{3R}}\)

\(\xi = \frac{1}{{2Q}} = \frac{1}{{2\left( {\frac{{30 + R}}{{3R}}} \right)}} = \frac{{3R}}{{60 + 2R}}\)

For critically damped system, ξ = 1

\(\Rightarrow \frac{{3R}}{{60 + 2R}} = 1\)

⇒ R = 60 Ω 

For series circuit,

\({R_0} = \sqrt {\frac{{4L}}{C}} = \sqrt {\frac{{4 \times 16 \times {{10}^{ - 3}}}}{{10 \times {{10}^{ - 6}}}}} = 80{\rm{\Omega }}\) 

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 

 \(f = \frac{1}{{2π }}\sqrt {\frac{1}{{LC}}}\)

Quality factor:

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

\(Q=\frac{{{f}_{r}}}{BW}\)

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

 \(Q=\frac{{{\omega }_{0}}L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}}\)

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

∴ \(Q=\frac{{{\omega }_{0}}L}{R}\)

\(L=\frac{{}25\times 56}{5000}\)

L = 0.28 H

Given that

L = 50 × 10^-3 H

C = 5 × 10^-6 F

Resonant frequency, \({\omega _0} = \frac{1}{{\sqrt {LC} }}\)

\({\omega _0} = \frac{1}{{\sqrt {50 \times {{10}^{ - 3}} \times 5 \times {{10}^{ - 6}}} }} = 2000rad/sec\;\;\)

Given that, current = 10 mA

At resonance, \({I_0} = \frac{V}{R}\)

\(\Rightarrow 10 \times {10^{ - 3}} = \frac{{50}}{R}\)

⇒ R = 5 KΩ

Quality factor, \(Q = \frac{{{\omega _0}L}}{R} = \frac{{2000 \times 50 \times {{10}^{ - 3}}}}{{5 \times {{10}^3}}} = 0.02\)

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:

\(QF={ω_o\over BW}\)

where, ω_o = Resonance frequency

BW = Bandwidth

Calculation

Given, QF = 100

ω_o = 1 MHz

\(100={10^6\over BW}\)

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 =  \( \frac{1}{R} \sqrt{ \frac{L}{C}}\)

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

EXPLANATION:

We know,

⇒ Q =  \( \frac{1}{R} \sqrt{ \frac{L}{C}}\)

⇒ Q =\( \frac{1}{2} \sqrt{ \frac{1\times10^{-3}}{0.1\times10^{-6}}}\)

⇒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.

Explanation:

Quality Factor (Q) of a Series Resonant Circuit

Definition: The quality factor (Q) of a series resonant circuit is a parameter that measures the sharpness or selectivity of the resonance in the circuit. It is defined as the ratio of the reactive power stored in the inductor or capacitor to the average power dissipated in the resistor at resonance. This parameter is significant in determining the efficiency and performance of resonant circuits used in applications such as filters, oscillators, and communication systems.

Correct Option Explanation:

The correct option is:

Option 4: The ratio of the reactive power of either the inductor or the capacitor to the average power of the resistor at resonance.

This definition accurately captures the essence of the quality factor (Q). In a series resonant circuit, resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC), resulting in a purely resistive impedance. At this condition, the circuit achieves maximum current flow, and the reactive power in the inductor or capacitor is at its peak. The quality factor (Q) is then calculated using the formula:

Q = (Reactive Power of Inductor or Capacitor) / (Average Power of Resistor)

Let us break this down:

• Reactive Power: Reactive power is the energy alternately stored and released by the inductor or capacitor during each cycle of AC oscillation. It is given by:
• Reactive Power of Inductor: \( Q_L = I^2 × X_L \)
• Reactive Power of Capacitor: \( Q_C = I^2 × X_C \)
• Average Power: Average power is the energy dissipated in the resistor during each cycle. It is given by:
• \( P_R = I^2 × R \)

At resonance, the inductive reactance (XL) equals the capacitive reactance (XC), and the current (I) in the circuit is maximized. Therefore, the quality factor (Q) can be expressed as:

\( Q = \frac{X_L}{R} \) or \( Q = \frac{X_C}{R} \)

This ratio indicates the sharpness of the resonance. Higher values of Q signify narrower bandwidth and greater selectivity, making the circuit more effective in applications requiring precise frequency selection.

Analysis of Other Options:

To further understand why the other options are incorrect, let us evaluate them:

Option 1: The sum of the reactive power of either the inductor or the capacitor and the average power of the resistor at resonance.

This option is incorrect because the quality factor (Q) is defined as a ratio, not a sum. The reactive power and average power represent entirely different types of energy in the circuit, and their sum does not provide any meaningful measure of resonance sharpness or selectivity.

Option 2: The ratio of the average power of the resistor to the reactive power of either the inductor or the capacitor at resonance.

This option is incorrect because it reverses the definition of the quality factor (Q). The correct definition involves the reactive power being divided by the average power, not the other way around. Such a reversal would yield a value that does not correspond to the intended characteristic of resonance sharpness.

Option 3: The product of the reactive power of either the inductor or the capacitor and the average power of the resistor at resonance.

This option is incorrect because the quality factor (Q) is not defined as a product. Multiplying the reactive power and average power would result in a value that does not have any physical relevance to the concept of resonance or selectivity.

Option 5: No additional information provided in the context of the problem.

This option is invalid as no specific details or explanation are given. It does not align with the standard definition or formula for the quality factor (Q).

Conclusion:

The quality factor (Q) is a crucial parameter in resonant circuits, serving as a measure of their sharpness and efficiency at resonance. It is defined as the ratio of the reactive power in the inductor or capacitor to the average power dissipated in the resistor. Among the given options, only Option 4 correctly describes this relationship, making it the correct answer. Understanding the Q factor is essential for designing and analyzing resonant circuits in various electrical and electronic applications.