LCR Circuit Questions

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Very Short Answers Questions [1 Mark Questions]

Ques. What is an LCR circuit?

Ans. An LCR circuit is an electrical circuit that consists of three main components: an inductor (L), a resistor (R), and a capacitor (C) connected in series or parallel configurations. It is also known as an acceptor circuit, tuned circuit, or filter circuit.

Ques. What is the role of a resistor in the LCR circuit?

Ans. The resistor in an LCR circuit limits the flow of current flowing through it, provides damping, and dissipates electrical energy.

Ques. What is the role of a capacitor in the LCR circuit?

Ans. The role of a capacitor in an LCR circuit is to store electrical energy and introduces phase shifts in the circuit. It also helps in controlling the voltage or power that is supplied to the LCR circuit.

Ques. What is the role of an inductor in the LCR circuit?

Ans. The role of the inductor in an LCR circuit is to store energy in a magnetic field and opposes the changes to the flow of electric current.

Ques. At resonance, the series LCR circuit is equivalent to

1. Purely resistive
2. Purely Capacitive
3. Purely Inductive
4. None of the above

Ans. The correct option is a. Purely resistive

Explanation: The series LCR circuit behaves as a purely resistive circuit at resonance.

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Short Answers Questions [2 Marks Questions]

Ques. Explain electrical resonance in the LCR circuit.

Ans. Electrical resonance is said to take place in a series LCR circuit when the circuit allows maximum current for a given frequency of the source of alternating supply for which capacitive reactance becomes equal to the inductive reactive.

Ques. What is the full form of L, C, and R in the LCR circuit?

Ans. In an LCR circuit

• The term L represents an inductor
• The term C represents a capacitor
• The term R represents a resistor

Ques. Define the Quality factor of the series LCR circuit.

Ans. The quality factor, also known as the Q factor of the LCR circuit is defined as the ratio of the voltage developed across the inductor or voltage developed across the capacitor at the resonance to the voltage applied (i.e. voltage across the resistor) to the circuit.

Ques. What is the role of an inductor in the LCR circuit?

Ans. The role of an inductor in the LCR circuit is to oppose the change to the flow of current. The presence of the inductor in the LCR circuit affects the frequency response of the circuit. The capacitor and resistor together with the inductor determine the resonant frequency of the circuit.

Also Read:

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Long Answers Questions [3 Marks Questions]

Ques. A series LCR circuit is connected across an ac source E = 10 sin[100∏t - ∏/6]. Current from the supply is I = 2 sin[100∏t + ∏/12], what is the average power dissipated?

Ans. The phase difference between voltage and current is given by

∏ = ∏/12 - (- ∏/6) = ∏/4

Power factor = cos ∏ = cos (∏/4) = 1/√2

Average power dissipated, P_avg = V_mI_m cosⲫ / 2

⇒ P_avg = (10 x 2 x 1/√2 ) / 2 = 5√2 watt

Ques. Is there any difference between the LCR circuit and the RLC circuit?

Ans. In an LCR circuit, L represents the inductor, C represents the capacitor, and R represents the resistor. Rearranging the LCR to RLC makes no difference in the behavior of the circuit. Therefore there is no difference between the LCR circuit and the RLC circuit.

Ques. What are the uses of LCR circuits?

Ans. The uses of LCR circuits are

• The resonant behavior of LCR circuits is used to make low-pass, high-pass, band-pass, and band-stop filters.
• LCR circuits can produce stable oscillations used in applications such as signal generators, clock generators, and radio frequency (RF) circuits.
• LCR circuits at resonant frequencies are used in devices like electric guitars, resonant transformers, and magnetic resonance imaging (MRI) machines.

Ques. In a series LCR circuit, obtain an expression for the resonant frequency.

Ans. The phenomenon of resonance occurs at a particular frequency called resonant frequency. At this frequency, the amplitude of the oscillations becomes very large.

If the inductive reactance X_L becomes equal to the capacitive reactance X_C at frequency fo, then the impedance of the LCR circuit will be minimum. And hence, the current will be maximum.

The condition for electrical resonance is, 

X_L = X_C

But inductive reactance X_L = ω₀L and

capacitive reactance X_C = 1/ω₀C

Where, ω₀ is angular frequency.

⇒ ω₀L = 1/ω₀C

⇒ ω₀² = 1/LC

⇒ ω₀ = 1/√(LC)

Also, ω₀ = 2πf₀

Where f₀ is resonant frequency.

⇒ 2πf₀ = 1/√(LC)

⇒ f₀ = 1/√2π(LC)

Above expression is the formula for resonant frequency in a series LCR circuit.

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Very Long Answers Questions [5 Marks Questions]

Ques. Draw the phasor diagram of the LCR circuit and derive an expression for the impedance of an LCR circuit connected to an AC power supply.

Ans. Consider an LCR series circuit. Let the current flow through the circuit be I.

The potential difference across the inductor, V_L = IX_L ….(i)

Where X_L is the inductive reactance of the inductor.

This potential difference leads the current I by an angle π/2

The potential difference across the capacitor, V_C = IX_C ….(ii)

Where X_C is the capacitive reactance of the capacitor.

This potential difference lags behind the current I by an angle π/2.

The potential difference across the resistor, V_R = IR ….(iii)

Where R is the resistance of the resistor.

This potential difference is in phase with the current I.

The phasor diagram of the LCR circuit is shown below.

Since V_L and V_C are in opposite directions so its resultant vector will be V_L - V_C (Here V_L > V_C).

Now, the resultant of V_L - V_C and V_R is represented by the OA line. Therefore

OA = \(\sqrt{V_R^2+(V_L-V_C)^2}\)

⇒ V = \(\sqrt{V_R^2+(V_L-V_C)^2}\)

Using equations (i), (ii), and (iii), we get

V = \(\sqrt{I^2R^2 + (IX_L – IX_C)^2}\) = \(\sqrt{IR^2 + (X_L – X_C)^2}\)

⇒ \(\frac{V}{I}\) = \(\sqrt{R^2 + (X_L – X_C)^2}\)

But V/I = Z, the impedance of the circuit defined as the effective opposition of the LCR circuit to alternating current. Therefore

Z = \(\sqrt{R^2 + (X_L – X_C)^2}\)

Ques. Write the expression for the current in a series LCR circuit.

Ans. According to Ohm’s law, the current flowing any circuit is given by

I = Potential difference (V) / Resistance of the circuit (R)

Let V be the potential difference across the end of the LCR circuit and the resistance of the LCR circuit is known as the impedance (Z), the current flowing through the circuit is given by

I = V/Z

But impedance of the LCR circuit is given by

Z = \(\sqrt{R^2 + (X_L – X_C)^2}\)

Where

• R is the resistance
• X_L = ω_L, is the inductive reactance of the inductor of inductance L.
• X_C = 1/ω_C, is the capacitive reactance of the capacitor of capacitance C.
• ω = angular frequency of the supply voltage.

Therefore, the expression of the current through the LCR circuit is given by

I = \(\frac{V}{Z} = \frac{V}{\sqrt{R^2 + (X_L – X_C)^2}}\)

I = \(\frac{V}{\sqrt{R^2 + ( \omega L – \frac{1}{\omega C)^2}}}\)

Ques. Find the potential difference across the resistance, capacitance, and inductance in a series LCR circuit where L = 2.0 H, C = 1 µF, and R = 1000 Ω. The applied emf has the form of V = 100√2 sin(1000t).

Ans. The impedance of a series LCR circuit is given by

Z = \(\sqrt{R^2 + (X_L – X_C)^2}\) = \(\sqrt{R^2 + ( \omega L – \frac{1}{\omega C)^2}}\)

Given

• L = 2.0 H
• C = 1 µF = 10^-6 F
• R = 1000 Ω

Also from the equation V = 100√2 sin(1000t), we have

• angular frequency, ω = 1000 rad/s
• Peak voltage, V₀ = 100√2 V

Substituting the above value, we get

Z = \(\sqrt{1000^2+ (1000 \times 2- \frac{1}{1000 \times 10^{-6}})^2}\) = 1000 x \(\sqrt{2}\) Ω

As, I_rms = V_rms/Z = V₀/√2Z = (100√2) / (√2 x 1000√2) = 0.0707 A

Now, the potential difference across the inductor, V_L = I_rmsX_L = I_rmsωL

⇒ V_L = 0.0707 x 1000 x 2 = 141.4 V

Potential difference across the capacitor, V_C = I_rmsX_C = I_rms/ωC

⇒ V_C = 0.0707/ (1000 x 10^-6) = 70.7 V

Potential difference across the resistor, V_R = I_rmsR

⇒ V_R = 0.0707 x 1000 = 70.7 V

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