Series Resonance MCQ Quiz - Objective Question with Answer for Series Resonance - Download Free PDF
Last updated on Jun 26, 2025
Latest Series Resonance MCQ Objective Questions
Series Resonance Question 1:
Which of the following statements is NOT correct regarding resonance in a series circuit?
Answer (Detailed Solution Below)
Series Resonance Question 1 Detailed Solution
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°.
Series Resonance Question 2:
If R is the resistance of an R-L series circuit and reactive inductance is XL , then the power factor of a series R-L circuit is given by _______.
Answer (Detailed Solution Below)
Series Resonance Question 2 Detailed Solution
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.
Series Resonance Question 3:
Q factor of a coil is given by
Answer (Detailed Solution Below)
Series Resonance Question 3 Detailed Solution
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 = XL / R
Where:
- Q: Quality factor of the coil.
- XL: Inductive reactance of the coil.
- R: Resistance of the coil.
Series Resonance Question 4:
The quality factor Q of a series resonant circuit is defined as:
Answer (Detailed Solution Below)
Series Resonance Question 4 Detailed Solution
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.
Series Resonance Question 5:
If the capacitance in a series RLC circuit is increased, the Q-factor will _________.
Answer (Detailed Solution Below)
Series Resonance Question 5 Detailed Solution
Quality factor
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:
From the above expression, it is observed that the quality factor is inversely proportional to capacitance.
If the capacitance in a series RLC circuit is increased, the Q-factor will decrease.
Top Series Resonance MCQ Objective Questions
Which of the following phasor diagram represents the series LCR circuit at resonance?
Answer (Detailed Solution Below)
Series Resonance Question 6 Detailed Solution
Download Solution PDFConcept:
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:
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, XL = XC, 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 the circuit shown below, readings of the voltmeter are V1 = 100 V, V2 = 50 V, V3 = 50 V. What will be the source voltage?
Answer (Detailed Solution Below)
Series Resonance Question 7 Detailed Solution
Download Solution PDFConcept:
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:
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:
At resonance, XL = XC, resulting in the net impedance to be minimum. This eventually results in the flow of the maximum current of:
∴
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
For the series RLC circuit, the Current- Frequency Curve is shown below. which of the following statement is incorrect
Answer (Detailed Solution Below)
Series Resonance Question 8 Detailed Solution
Download Solution PDFConcept:
1. Consider the information Shown in the figure below
2. For the series RLC circuit ωr is geometrical mean of ωL and ωU or
3. At upper and lower cutoff frequencies current through the circuit is 70.7% of I0 because at these frequency Z = √ 2 R
4. At resonant frequency, the circuit behaves as purely resistive circuit and hence voltage and current are in same phase. Also power factor become unity.
Analysis:
1. At point A and B current through the circuit is 70.7% of I0 because at these frequency Z = √ 2 R
As we know power factor = cos ϕ =
ϕ = 45° hence option 1 is correct.
2.The curve is symmetrical about resonance frequency
So ωU - ωr = ωr - ωL
hence option 2 is incorrect.
3. At resonant frequency, the circuit behaves as purely resistive circuit and hence voltage and current are in same phase. Also power factor become unity.
4. for ω > ωr XL > XC so circuit behave as inductive circuit.
for ω r XL C so circuit behave as Capacitive circuit.
A RLC series circuit has a resistance of 1 kΩ. Its half power frequencies are 10 kHz and 90 kHz. Quality factor of the circuit is:
Answer (Detailed Solution Below)
Series Resonance Question 9 Detailed Solution
Download Solution PDFConcept:
In a series resonant circuit,
Bandwidth (BW)
Where fr is the resonant frequency
Q is the quality factor
fH is the higher half-power frequency
fL is the lower half-power frequency
Resonant frequency (fr) is defined as:
Calculation:
Given that,
Higher half-power frequency (fH) = 90 kHz
Lower half-power frequency (fL) = 10 kHz
Bandwidth (BW) = fH – fL = 90 – 10 = 80 kHz
Resonant frequency (fr) will be:
Quality Factor Q = fr / BW = 30 / 80 = 0.375
How is the transient current in a loss-free R-L-C circuit?
Answer (Detailed Solution Below)
Series Resonance Question 10 Detailed Solution
Download Solution PDFThe correct answer is option 4.
Concept:
The RLC circuit means the series or parallel connection of the resistance(R), inductor(L), and capacitor(C).
A series circuit is given below.
Since it is mentioned that the circuit is a loss-free circuit the current will behave similarly to that of an LC series circuit i.e. an undamped sinusoidal waveform will be observed.
In a loss-free circuit, the energy keeps shifting between L and C.
Important PointsIf the case would have been of loss-making RLC series circuit then the current would have a damped sinusoidal equation.
In a RLC circuit Inductance is 20 mH and capacitance is 200 micro Farad. Find the resonance frequency of the circuit.
Answer (Detailed Solution Below)
Series Resonance Question 11 Detailed Solution
Download Solution PDFRLC 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
fr = frequency
ω0 = angular resonance frequency
Calculation:
Given, L = 20 mH = 20 × 10-3 H
C = 200 μF = 200 × 10-6 F
In a series AC circuit, XL = 2350 ohms, C = 0.005 μF, and R = 500Ω. What is the impedance at resonance?
Answer (Detailed Solution Below)
Series Resonance Question 12 Detailed Solution
Download Solution PDFFor 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
The magnitude of the impedance is given by:
At resonance, XL = XC,
|Z| = R
Hence, the impedance at resonance is |Z| = R = 0.5 kΩ
Note:
The current flowing across the series RLC circuit will be:
∴
Hence impedance is minimum and current will be maximum at resonance.
In the series RLC circuit, the current (I) Vs frequency (f) graph for the series resonance circuit is shown below. :-
From above we can conclude:-
- For resonance to occur in any circuit it must have at least one inductor and one capacitor.
- Resonance is the result of oscillations in a circuit as stored energy is passed from the inductor to the capacitor.
- Resonance occurs when XL = XC and the imaginary part of the transfer function is zero.
- At resonance, the impedance of the circuit is equal to the resistance value as Z = R.
- At low frequencies the series circuit is capacitive as XC > XL, this gives the circuit a leading power factor.
- At high frequencies the series circuit is inductive as XL > XC, this gives the circuit a lagging power factor.
- The high value of current at resonance produces very high values of voltage across the inductor and capacitor.
- Because impedance is minimum and current is maximum, series resonance circuits are also called Acceptor Circuits.
To double the resonant frequency of an LC circuit with a fixed value of L, the capacitance must be
Answer (Detailed Solution Below)
Series Resonance Question 13 Detailed Solution
Download Solution PDFConcept:
An LC circuit is represented as:
There is a characteristic frequency at which the circuit will oscillate, called the resonant frequency given by:
Application:
For a fixed value of L, the resonant frequency is proportional to:
To double the resonant frequency, C must be reduced to 1/4 (quarter), i.e. for C' = C/4, f0 becomes:
f'0 = 2 f0
A series RLC circuit resonates at 1.5 kHz and consumes 50 W from a 50 V AC source operating at the resonant frequency. If the bandwidth is 0.75 kHz, then what are the values of the circuit elements R and L?
Answer (Detailed Solution Below)
Series Resonance Question 14 Detailed Solution
Download Solution PDFConcept:
At resonance, only the resistance will remain (as both the inductive and capacitive reactance will be equal), because of which all the power will be delivered to the resistance only.
So, power will be:
The bandwidth of a resonant circuit is the difference between the upper cut-off frequency and lower cut-off frequency (with the cut-off frequency being the frequency at which the power is half of the maximum power).
For a series RLC circuit, the bandwidth is calculated as:
Where Q = quality factor and is defined as:
From both these equations, the Bandwidth can be calculated as:
Calculation:
Given resonant frequency (f0) = 1.5 kHz
Bandwidth (BW) = 0.75 kHz
Voltage (V) = 50 V
Power (P) = 50 W
B.W. in radians = R/L
B.W. in Hertz =
At half power points of a resonance curve, the current is
Answer (Detailed Solution Below)
Series Resonance Question 15 Detailed Solution
Download Solution PDFAt half-power points of a resonance curve, the current is 0.707 times the maximum current.
Important points regarding half-power points are:
- The current is Imax/√2.
- Impedance is √2 R or √2 Zmin.
- The circuit phase angle → ±45° or π/4 rad