Electrical Resonance MCQ Quiz - Objective Question with Answer for Electrical Resonance - Download Free PDF

Explanation:

Resonance in a Series Circuit

Definition: Resonance in an electrical series circuit occurs when the inductive reactance (XL) is equal to the capacitive reactance (XC), resulting in their effects canceling each other out. This leads to a purely resistive circuit where the impedance is minimized and is equal to the resistance of the circuit. At resonance, the circuit oscillates at its natural frequency, which is determined by the inductance (L) and capacitance (C) in the circuit.

Correct Option Analysis:

The correct option is:

Option 4: The phase angle of the circuit is 90°.

This option is NOT correct because, at resonance in a series circuit, the phase angle between the voltage and current is 0°, not 90°. At resonance, the inductive and capacitive reactances cancel each other out, and the circuit behaves as a purely resistive circuit. In a purely resistive circuit, the voltage and current are in phase, meaning the phase angle is 0°. A phase angle of 90° would indicate a completely reactive circuit (either purely inductive or purely capacitive), which is not the case at resonance in a series circuit.

Important Information: At resonance, the following conditions are observed:

• Inductive reactance (XL) = Capacitive reactance (XC).
• The impedance of the circuit is equal to the resistance (Z = R).
• The circuit behaves as a purely resistive circuit, with no reactive component.
• The current is at its maximum value, as the impedance is minimized.
• The voltage across the inductor and capacitor is equal in magnitude but opposite in phase, effectively canceling each other out.

Additional Information

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

Option 1: The voltage across the capacitor and inductor is equal and opposite.

This statement is correct. At resonance, the voltage across the inductor and capacitor is equal in magnitude but opposite in phase. This means that the inductive and capacitive voltages cancel each other out, leaving only the resistive voltage in the circuit.

Option 2: The circuit behaves as a purely resistive circuit.

This statement is also correct. At resonance, the inductive and capacitive reactances cancel out, leaving only the resistive component. The impedance of the circuit is purely resistive, and the current and voltage are in phase.

Option 3: The net reactance of the circuit is zero.

This statement is correct as well. At resonance, the inductive reactance (XL) is equal to the capacitive reactance (XC), resulting in a net reactance of zero. This is why the circuit behaves as a purely resistive circuit at resonance.

Option 5: [This option is not provided in the question, so no analysis is required for it.]

Conclusion:

Understanding resonance in a series circuit is crucial for analyzing the behavior of electrical circuits at specific frequencies. At resonance, the circuit achieves unique characteristics such as maximum current flow, zero net reactance, and purely resistive behavior. While most statements provided in the options align with the principles of resonance, the incorrect statement is Option 4, as the phase angle of a resonant series circuit is 0° and not 90°.

Explanation:

Resonant Frequency in a Parallel RLC Circuit

Definition: Resonant frequency (f₀) in a parallel RLC circuit is the frequency at which the inductive reactance and capacitive reactance become equal in magnitude, causing the circuit to exhibit purely resistive behavior. At this frequency, the impedance of the circuit is maximized, and the circuit resonates.

Formula: The resonant frequency (f₀) for a parallel RLC circuit is given by the following equation:

f₀ = 1 / (2 × π × √(L × C))

Where:

• L: Inductance of the inductor (in henries, H)
• C: Capacitance of the capacitor (in farads, F)

Working Principle: Resonance occurs in a parallel RLC circuit when the energy alternates between the inductor and capacitor, creating oscillations at the resonant frequency. At this point, the inductive reactance (X_L) and capacitive reactance (X_C) cancel each other out, leading to the circuit behaving as though only the resistance is present.

Impact of Decreasing Capacitance (C):

From the formula for resonant frequency (f₀):

f₀ = 1 / (2 × π × √(L × C))

It is evident that the resonant frequency is inversely proportional to the square root of the product of inductance (L) and capacitance (C). Specifically:

• If C decreases, the denominator (√(L × C)) becomes smaller.
• This causes the value of f₀ to increase because the resonant frequency is inversely proportional to the square root of C.

Therefore, when the capacitance (C) in a parallel RLC circuit decreases, the resonant frequency (f₀) increases.

Correct Option Analysis:

The correct option is:

Option 2: Increases

This option correctly identifies the effect of decreasing capacitance on the resonant frequency in a parallel RLC circuit. As explained above, a reduction in capacitance leads to an increase in the resonant frequency due to their inverse relationship in the resonant frequency formula.

Additional Information

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

Option 1: Decreases

This option is incorrect because decreasing capacitance does not cause the resonant frequency to decrease. As shown in the formula, resonant frequency is inversely proportional to the square root of capacitance (C); hence, a reduction in C increases the value of f₀.

Option 3: Remains constant

This option is incorrect because the resonant frequency depends on the values of both inductance (L) and capacitance (C). Changes in either L or C will affect the resonant frequency. If capacitance decreases, f₀ will increase, as explained earlier.

Option 4: Becomes zero

This option is incorrect because the resonant frequency cannot become zero simply by decreasing capacitance. Even with very small values of C, the resonant frequency remains finite and increases. Resonant frequency becoming zero would imply the absence of oscillation, which is not the case here.

Conclusion:

Understanding the relationship between capacitance and resonant frequency is crucial for analyzing parallel RLC circuits. The resonant frequency is determined by the values of inductance and capacitance. When capacitance decreases, the resonant frequency increases due to their inverse proportionality. This principle is fundamental in electronics and telecommunications, where resonant circuits are used for signal filtering and frequency selection.

Explanation:

Power Factor of Series R-L Circuit

Definition: In an R-L series circuit, the power factor is a measure of how effectively electrical power is converted into useful work. It represents the cosine of the angle (\(\theta\)) between the voltage and current vectors in the circuit. The power factor is crucial in AC circuits as it determines the efficiency of power usage.

The power factor of a series R-L circuit is given by:

\(\rm \cos \theta = \frac{R}{Z}\)

Where:

• \(R\): Resistance of the circuit
• \(Z\): Impedance of the circuit

Impedance (\(Z\)): In an R-L series circuit, the impedance is the vector sum of the resistance (\(R\)) and the reactive inductance (\(X_L\)). It is given by:

\(\rm Z = \sqrt{R^2 + X_L^2}\)

Here:

• \(X_L\): Reactive inductance, which is calculated as \(X_L = \omega L\), where \(\omega\) is the angular frequency and \(L\) is the inductance.

Detailed Explanation:

In an R-L series circuit, the total impedance (\(Z\)) is a combination of the resistive and inductive components. The resistance (\(R\)) represents the real part of the impedance and is associated with energy dissipation as heat. The inductive reactance (\(X_L\)) represents the imaginary part of the impedance and is associated with energy storage in the magnetic field of the inductor.

The phase angle (\(\theta\)) between the voltage and current in the circuit is determined by the relationship between the resistance (\(R\)) and the reactive inductance (\(X_L\)):

\(\rm \tan \theta = \frac{X_L}{R}\)

From this, the cosine of the phase angle (\(\cos \theta\)) is calculated as:

\(\rm \cos \theta = \frac{R}{Z}\)

This formula indicates that the power factor depends on the ratio of the resistance to the total impedance of the circuit. A higher resistance relative to the impedance results in a better power factor, whereas a higher reactive inductance results in a lower power factor.

Correct Option Analysis:

The correct option is:

Option 3: \(\rm \cos \theta = \frac{R}{Z}\)

This option correctly defines the power factor of a series R-L circuit. The power factor is the ratio of the resistance (\(R\)) to the total impedance (\(Z\)) of the circuit. It is a measure of how efficiently the electrical power is being converted into useful work.

Additional Information

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

Option 1: \(\rm \cos \theta = \frac{X_L}{Z}\)

This option is incorrect. The ratio of the reactive inductance (\(X_L\)) to the impedance (\(Z\)) does not represent the power factor. Instead, it represents the sine of the phase angle (\(\sin \theta\)), which is associated with the reactive component of the circuit.

Option 2: \(\rm \cos \theta = \frac{Z}{R}\)

This option is incorrect. The ratio of impedance (\(Z\)) to resistance (\(R\)) does not define the power factor. Instead, it is inversely related to the power factor and does not have a direct physical interpretation in this context.

Option 4: \(\rm \cos \theta = \frac{R}{X_L}\)

This option is incorrect. The ratio of resistance (\(R\)) to reactive inductance (\(X_L\)) is related to the tangent of the phase angle (\(\tan \theta\)), not the cosine of the phase angle (\(\cos \theta\)). It is not a measure of the power factor.

Conclusion:

In summary, the power factor of a series R-L circuit is given by \(\rm \cos \theta = \frac{R}{Z}\), which accurately represents the ratio of the resistance to the total impedance. This formula is essential for understanding the efficiency of power usage in AC circuits. Correctly identifying the power factor formula is crucial for analyzing and optimizing the performance of R-L circuits, especially in applications where power efficiency is critical.

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.

We have RMS value of source

\({{\rm{V}}_{{\rm{RMS}}}} = \frac{{{{\rm{V}}_{\rm{m}}}}}{{\sqrt 2 }} = 150\)

Then, \({{\rm{V}}_{\rm{L}}} = \sqrt {{\rm{V}}_{{\rm{RMS}}}^2 - {\rm{V}}_{\rm{R}}^2} = \sqrt {{{150}^2} - {{120}^2}}\)

\(\Rightarrow {{\rm{V}}_{\rm{L}}} = 90{\rm{V}}\)

Now,  \({\rm{I}} = \frac{{{{\rm{V}}_{\rm{R}}}}}{{\rm{R}}} = \frac{{120}}{{1{\rm{k}}}} = 120 {\rm{mA}}\)

Again \({{\rm{V}}_{\rm{L}}} = {\rm{I}} \times{\rm{\omega L}}\)

\(\begin{array}{l} \Rightarrow 90 = 0.12 \times 500 \times {\rm{L}}\\ \Rightarrow {\rm{L}} = \frac{{90}}{{0.12 \times 500}} = 1.5{\rm{H}} \end{array}\)

The quality factor is defined as the ratio of the maximum energy stored to maximum energy dissipated in a cycle

\(Q = 2\pi \frac{{Maximum\;energy\;stored}}{{Total\;energy\;lost\;per\;period}}\)

In a series RLC, Quality factor \(Q = \frac{{\omega L}}{R} = \frac{1}{{\omega RC}} = \frac{{{X_L}}}{R} = \frac{{{X_C}}}{R}\)

In a parallel RLC, \(Q = \frac{R}{{{\rm{\omega L}}}} = \omega RC = \frac{R}{{{X_L}}} = \frac{R}{{{X_C}}}\)

It is defined as, resistance to the reactance of reactive element.

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

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

For parallel RLC Circuit \({{\rm{Q}}_1} = {\rm{R}}\sqrt {\frac{{\rm{C}}}{{\rm{L}}}}\)

For Series RLC Circuit \({{\rm{Q}}_2} = \frac{1}{{\rm{R}}}\sqrt {\frac{{\rm{L}}}{{\rm{C}}}}\)

Concept:

The quality factor is defined as the ratio of the maximum energy stored to maximum energy dissipated in a cycle

\(Q = 2\pi \frac{{Maximum\;energy\;stored}}{{Total\;energy\;lost\;per\;period}}\)

The quality factor in a series RLC circuit is given by:

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

At resonance, the circuit acts as pure resistive and the impedance equals the resistance.

Calculation:

Resonance frequency (f) = 5000 / 2π Hz

Resonant frequency (ω) = 2πf = 5000 rad/sec

Impedance at resonance (Z) = R = 56 Ω

Quality factor = 25

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

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

⇒ L = 0.28 H

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

\( \Rightarrow 25 = \frac{1}{{56}}\sqrt {\frac{{0.28}}{C}} \)

⇒ C = 0.143 μF

Parallel Resonance:

The input admittance is given by:

\(Y_{in} = {1 \over R}+j(\omega C-{1\over \omega L})\)

\(Y_{in} = G+j(B_C-B_L)\)

where, Y_in = Admittance

G = Conductance

B_C = Capacitive susceptance

B_L = Inductive susceptance

At resonance, BC = BL 

Hence, Yin = G and is minimum.

Additional Information

Concept:

In series LCR circuits, Resonance is a condition in which the inductive reactance and capacitive reactance are equal and lie opposite in phase, so cancel out each other. Only, resistance is left as impedance.

The frequency at which the series LCR circuit goes into resonance is:

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

At resonance, the impedance of the series LCR circuit is minimum and hence the current is maximum.

A series LCR circuit is as shown:

The phasor diagram is as shown below:

At resonance, X_L = X_C, i.e. the voltage across the inductor and capacitor are equal and opposite. The resultant phasor diagram at resonant frequency will, therefore, be:

The impedance - frequency curve is shown below:

In parallel resonant circuit, inductive reactance (XL) is equal to capacitive reactance (XC). The circuit becomes pure resistive.

At resonance XL = XC.

At parallel resonance, impedance becomes maximum.

At parallel resonance line current Ir = IL cos ϕ

\(\frac{V}{{{Z_r}}} = \frac{V}{{{Z_1}}}X\frac{R}{{{Z_1}}}or\frac{1}{{{Z_r}}} = \frac{R}{{Z_L^2}}\;\)

\(\frac{1}{{{Z_r}}} = \frac{{\frac{R}{L}}}{C} = \frac{{CR}}{L}\left( {as\;Z_L^2 = \frac{L}{C}} \right)\)

Therefore, the circuit impedance will be given as

Parallel RLC circuit:

The characteristic equation is given as,

\({s^2} + \frac{1}{{RC}}s + \frac{1}{{LC}} = 0\)

The bandwidth of a parallel RLC network is given as:

\({\rm{BW}} = \frac{{{{\rm{\omega}}_{\rm{r}}}}}{{\rm{Q}}}=\frac{1}{RC}\)

Observations:

• The bandwidth is inversely proportional to the resistance, i.e. as the resistance increases, the bandwidth decreases.
• The bandwidth is independent of inductance. Hence the bandwidth of the circuit remains the same if L is increased.
• At resonance, the imaginary part of the net impedance becomes zero, which makes the input impedance a real quantity.
• Also, at resonance, the input impedance for a parallel resonant circuit attains a maximum value. This is opposite to a series resonance where at resonance, the impedance attains a minimum value.

Concept:

RLC series circuit:

For a series RLC circuit, the net impedance is given by:

Z = R + j (XL - XC)

XL = Inductive Reactance given by:

XL = ωL

XC = Capacitive Reactance given by:

XL = 1/ωC

ω = 2 π f

ω = angular frequency

f = linear frequency

The magnitude of the impedance is given by:

\(|Z|=\sqrt{R^2+(X_L-X_C)^2}\)

\(V = \sqrt {V_1^2 + {{\left( {{V_2} - {V_3}} \right)}^2}} \)

Where, V₁ = voltage across the resistor

V₂ = voltage across the inductor

V3 = voltage across the capacitor

V = resultant voltage

The current flowing across the series RLC circuit will be:

\(I=\frac{V}{|Z|}\)

At resonance, XL = XC, resulting in the net impedance to be minimum. This eventually results in the flow of the maximum current of:

∴\(I=\frac{V}{R}\)

Calculation:

V1 = 100 V, V2 = 50 V, V3 = 50 V 

V2 = V3 (As data provided in the question)

So, the circuit is in series resonance.

V = V1 = 100 V

The source voltage is 100 V

Concept:

The graph between impedance Z and the frequency of the parallel RLC circuit:

Here,

f1 is the lower cutoff frequency

f2 is the upper cutoff frequency

fr is the resonant frequency

BW is the bandwidth

Formula:

BW = f2 – f1

\({f_1} = {f_r} - \left( {\frac{{BW}}{2}} \right)\)

\({f_2} = {f_r} + \left( {\frac{{BW}}{2}} \right)\)

Calculation:

Given

Resonant frequency fr = 20 kHz

Bandwidth = 2 kHz

The upper cutoff frequency is given as:

\({f_2} = {f_r} + \left( {\frac{{BW}}{2}} \right)\)

\({f_2} = {20kHz} + \left( {\frac{{2kHz}}{2}} \right)\)

f₂ = 21 kHz

The selectivity/sharpness of the resonance amplifier is measured by the quality factor and is explained in the figure shown below:

Selectivity of a resonant circuit is defined as the ratio of resonant frequency f_r to the half power bandwidth.

Selectivity = resonance frequency/3-dB Bandwidth
\( = \frac{{{f_r}}}{{{f_2} - {f_1}}}\)

Important Points

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

\(Q=\frac{{ω}_{r}}{BW}\)

ωr = Resonant frequency

B.W. = Bandwith of the amplifier