Q Factor MCQ Quiz - Objective Question with Answer for Q Factor - Download Free PDF

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

Explanation:

Q Factor (Quality Factor)

Definition: The Q factor, or Quality Factor, is a dimensionless parameter that describes the damping or energy loss of an oscillatory system. In the context of electrical circuits, it quantifies the sharpness of the resonance of a circuit and is defined as the ratio of the energy stored in the reactive components (inductors or capacitors) to the energy dissipated per cycle in the resistive elements.

Correct Option Analysis:

The correct option is:

Option 1: Resistance / inductance of reactive element.

The Q factor is defined as:

Q = (Energy stored in the reactive element) / (Energy dissipated per cycle)

For a series RLC circuit, the Q factor can be expressed in terms of the inductance (L) and resistance (R). The formula is:

Q = ωL / R

Where:

• ω = 2πf, the angular frequency of the circuit.
• L = Inductance of the inductor.
• R = Resistance of the circuit.

This equation shows that the Q factor is directly proportional to the inductance and inversely proportional to the resistance of the circuit. A higher Q factor indicates lower energy loss and sharper resonance, which is desirable in many practical applications such as communication systems and filters.

Therefore, the Q factor is essentially the ratio of resistance (R) to the inductance (L) of the reactive element when we consider the relationship between energy dissipation and energy storage in the circuit.

Additional Information

Analysis of Other Options:

Option 2: Resistance / capacitance of reactive element.

This option is incorrect because the Q factor is not defined in terms of capacitance (C) alone. While capacitance is a reactive element, the Q factor depends on the ratio of stored energy to dissipated energy, which is more commonly expressed in terms of inductance (L) and resistance (R) in a series RLC circuit. In circuits where capacitance is the primary reactive element, the Q factor is defined using the reciprocal of capacitance (1/C), not capacitance directly.

Option 3: Resistance to reactance of reactive element.

This option is also incorrect because the Q factor is not directly defined as the ratio of resistance (R) to reactance (X). Instead, the Q factor is related to the ratio of reactance (X) to resistance (R) in a series circuit, as reactance determines the energy storage in the reactive elements. Specifically, for an inductor:

Q = X_L / R = ωL / R

For a capacitor:

Q = X_C / R = 1 / (ωCR)

Thus, the Q factor is inversely proportional to resistance and directly proportional to the reactance.

Option 4: Resistance to susceptance of reactive element.

This option is incorrect because susceptance (B) is the reciprocal of reactance (1/X), which is used in the analysis of parallel circuits. The Q factor is not defined in terms of resistance (R) to susceptance (B). The relationship between susceptance and Q factor is more relevant for parallel RLC circuits, but even in those cases, the Q factor is not expressed as the ratio of resistance to susceptance.

Conclusion:

The correct definition of the Q factor involves the ratio of resistance (R) to the inductance (L) of the reactive element, as given in Option 1. The Q factor plays a critical role in determining the performance of resonant circuits, with higher Q values indicating better performance due to lower energy losses. Understanding the Q factor and its mathematical relationship with circuit parameters is essential for designing efficient electrical systems, especially in applications requiring high selectivity and minimal energy dissipation.

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.

Concept:

For a series RLC circuit, the current versus frequency curve can be given as follows:

Bandwidth (BW) \(= \frac{{{f_0}}}{Q} = {f_2} - {f_1} = \frac{R}{L}\)

Where Q is the quality factor and f0 is the resonant frequency

\({f_1} = {f_0} - \frac{{BW}}{2}\)

\({f_2} = {f_0} + \frac{{BW}}{2}\)

Calculation:

Resonant frequency (f0) = 60 Hz

Bandwidth (BW) =  12 Hz

Quality factor (Q) = \(\frac{60}{12}\) = 5Hz

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 RLC 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

fr = frequency

ω0 = angular resonance frequency

Calculation:

Given that 

R = 10 Ω, L = 10 mH and C = 1 μF

Quality factor of inductor at resonance can be calculated as

\(Q=\frac{1}{10}\sqrt{\frac{10\times 10^{-3} }{1\times 10^{-6}}}\)

Q = 10

Concept:

The coil is a circuit with Resistance and Inductance connected in series as shown:

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:

\(Q = \frac{{{\omega _o}L}}{R}=\frac{X_L}{R}\)

The sharpness of the resonance in RLC series resonant circuit is measured by the quality factor and is explained in the figure shown below:

Observations:

• Less the Bandwidth, more the Quality factor.
• More the Bandwidth, less is the Quality factor.
• For the above figure, B2 > B1, so Q2 < Q1

Concept:

Resonant frequency for a series RLC circuit is given by:

\(\omega_0=\frac{1}{{\sqrt {LC }}}\)

Bandwidth for a series RLC circuit is given by:

\(B.W.=\frac{R}{L}\)

Also, the quality factor is given by:

\({\text{Q}} = \frac{{{\omega_o}}}{{{\text{B}}}}\)

Calculations:

Given that, R = 10 Ω, L = 25 mH, C = 100 μF

For the given RLC network, the resonant frequency will be:

\(\omega_0=\frac{1}{{\sqrt {LC} }}\)

\( = \frac{1}{{\sqrt {25 \times {{10}^{ - 3}} \times 100 \times {{10}^{ - 6}}} }} = 632.45\)

\(B = \frac{R}{L} = \frac{{10}}{{25\times 10^{-3}}} = 400\)

Qulaity factor, Q = 632.45/400 = 1.58

Concept:

The coil is a circuit with Resistance and Inductance connected in series as shown:

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:

\(Q = \frac{{{\omega _o}L}}{R}=\frac{X_L}{R}\)

The sharpness of the resonance in RLC series resonant circuit is measured by the quality factor and is explained in the figure shown below:

Observations:

• Less the Bandwidth, more the Quality factor.
• More the Bandwidth, less is the Quality factor.
• For the above figure, B2 > B1, so Q2 < Q1

Concept:

In a series resonant circuit, the resonant frequency and Bandwidth are related as under

\(Q = \frac{{{\omega _0}}}{{BW}}\)

Where,

Q: Quality factor

ω0 = resonant frequency (rad/sec)

BW = Bandwidth (rad/sec)

\(Q = \frac{{{X_L}}}{R} = \frac{{{X_C}}}{R} = \frac{1}{R}\;\sqrt {\frac{L}{C}} \)

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

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

fr = resonant frequency

For a series RLC circuit, the resonance frequency is calculated as:

\(f_r=\frac{1}{\sqrt{LC}}\)

Also, the bandwidth of a series RLC circuit is calculated as:

\(BW=\frac{R}{L}\)

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.

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

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

\(Q=\frac{{{f}_{r}}}{BW}=\frac{1}{R}\sqrt{\frac{L}{C}}\)

The sharpness of the resonance in RLC series resonant circuit is measured by the quality factor and is explained in the figure shown below:

Observations:

• Less the Bandwidth, more the Quality factor.
• More the Bandwidth, less is the Quality factor.
• For the above figure, B₂ > B₁, so Q₂ < Q₁