The Series RLC Circuit MCQ Quiz - Objective Question with Answer for The Series RLC Circuit - Download Free PDF

Concept:

In an RLC circuit at resonance, the current is at its maximum when the driving frequency is equal to the resonance frequency ω₀ = 1 / √(LC). The quality factor Q is a measure of the sharpness of the resonance and is defined as:

Q = ω₀ L / R

Where, ω₀ is the resonance angular frequency, L is the inductance, and, R is the resistance.

Calculation:

The quality factor is given by:

Q = ω₀ L / R

The quality factor Q is ω₀ L / R. Thus, option 2 is correct.

Concept:

In an AC circuit containing an inductor (L) and a capacitor (C) in parallel, the currents through these components are out of phase by 180°. This means that the total current drawn from the source is the phasor difference between the individual currents through the inductor and capacitor.

Inductor Current (I_L): The current through an inductor lags the voltage by 90°.

Capacitor Current (I_C): The current through a capacitor leads the voltage by 90°.

Phase Relationship: Since they are 180° out of phase, their phasor sum is given by:

Calculation:

Given:

Current through the inductor, I_L = 0.9 A

Current through the capacitor, I_C = 0.6 A

Since the currents are in opposite phase, the total current drawn from the AC source is:

⇒ I_total = I_L - I_C

⇒ I_total = 0.9 A - 0.6 A

⇒ I_total = 0.3 A

∴ The total current drawn from the source is 0.3 A.

Calculation: 

Quality factor is given by, Q = \(\frac{1}{\mathrm{R}} \sqrt{\frac{\mathrm{L}}{\mathrm{C}}}\)

Bandwidth of LCR circuit, ω = \(\frac{R}{L}\)

Now, \(\frac{\mathrm{Q}}{\omega}=\frac{\mathrm{L}}{\mathrm{R}^2} \sqrt{\frac{\mathrm{L}}{\mathrm{C}}}\)

= \(\frac{3}{100} \sqrt{\frac{3}{27 \times 10^{-6}}}\) = 10

∴ the Correct answer is 10

CONCEPT:
RLC CIRCUIT:

•  An RLC circuit is an electrical circuit consisting of an 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 

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

• Quality factor is

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

Where, XL & XC = Impedance of inductor and capacitor, L, R & C = Inductance, resistance, and capacitance, f = frequency and, ω0 = angular resonance frequency

EXPLANATION:

From the above explanation, we can see that, 

• The quality factor of the RLC combination can be expressed as: 

\(\Rightarrow Q=\frac{{{\omega }_{0}}L}{R}\)

• Hence option 1 is correct.

Concept:

If E₀ and I₀ are the peak voltage and current respectively then

E = E₀ sin ωt and I = I₀ sin (ωt - ϕ)

Rms current 

I = \({E_0\over Z}\)

where Z = \(\sqrt{R^2+X^2_L} \)

Calculation:

Given: 

E = 12 V; ω = 400 rad/s; R = 17.32 Ω and L = 0.025 H

Inductive Resistance X_L = ωL = 10Ω 

Impedance Z = \(\sqrt{R^2+X^2_L} \)  = 19.9Ω 

Rms Current I = \({E\over Z}\)

= \({12 \over 19.9}\)

= 0.60 A

Phase Angle ϕ = tan^-1\({Z \over R}\)

= 30° 

The correct answer is option (1).

The correct answer is option 4) i.e. ω2LC

CONCEPT:

• LCR circuit: An electrical circuit constituting an inductor (L), capacitor (C), and resistor (R) connected in series or parallel is called an LCR circuit.

For a given inductance (L) and capacitance (C): 

XL = Lω and \(X_C = \frac{1}{Cω} \)

Where XL is the inductive reactance, XC  is the capacitive reactance, R is the resistance, and ω is the angular frequency.

EXPLANATION:

Inductive reactance, X_L = Lω

Capacitive reactance, \(X_C = \frac{1}{Cω} \)

Ratio = \(\frac{X_L}{X_C} = \frac{L\omega}{1/C\omega} = \omega^2LC\)

CONCEPT:

RLC CIRCUIT:

•  An RLC circuit is an electrical circuit consisting of an 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 

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

• Quality factor is

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

Where, XL & XC = Impedance of inductor and capacitor, L, R & C = Inductance, resistance, and capacitance, f = frequency and, ω0 = angular resonance frequency

CALCULATION:

Given that: R = 10 Ω, C = 30 μF = 30 × 10^-6 F, and L = 27mH = 27 × 10-3 H

From the above discussion,

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

\(\Rightarrow Q = \frac{1}{10}\sqrt{\frac{27×10^{-3}}{30×× 10^{-6}}}\)

\(\Rightarrow Q= \frac{3×10}{10} = 3\)

• Hence option (1) is correct.

CONCEPT:
RLC CIRCUIT:

•  An RLC circuit is an electrical circuit consisting of an inductor (L), Capacitor (C), Resistor (R) it can be connected either parallel or series.
• When the LCR circuit is set to resonate (X_L = X_C), the resonant frequency is expressed as 

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

• Quality factor is

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

Where,

X_L & X_C = Impedance of inductor and capacitor 

L, R & C = Inductance, resistance, and capacitance 

f = frequency

 ω₀ = angular resonance frequency

EXPLANATION:

From the above explanation, we can see that, 

The quality factor of the RLC combination can be expressed as: 

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

Hence option 2 is correct among all

CONCEPT:

• The ac circuit containing the capacitor, resistor, and the inductor is called an LCR circuit.
• For a series LCR circuit, the total potential difference of the circuit is given by:

\(V = \sqrt {{V_R^2} + {{\left( {{V_L} - {V_C}} \right)}^2}} \)

Where VR = potential difference across R, VL =  potential difference across L and VC =  potential difference across C

• For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)
Where R = resistance, XL =induvtive reactance and XC = capacitive reactive

CALCULATION:

Given - V1 = V2 = 300V and V = 220V

• For a series LCR circuit, the total potential difference of the circuit is given by:

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

\(⇒ 220 = \sqrt {{V_3^2} + {{\left( {{300} - {300}} \right)}^2}} =\sqrt {V^2_3}=V_3\)

⇒ V₃ = 220V

• The reading on the ammeter will be

\(⇒ I=\frac{V_3}{R}=\frac{220}{100}=2.2A\)

• The reading on the voltmeter V₃ will be 

⇒ V₃ = IR

⇒ V3 = 2.2 × 100 = 220 V

CONCEPT:

• The ac circuit containing the capacitor, resistor, and inductor is called an LCR circuit.
• For a series LCR circuit, the total potential difference of the circuit is given by:

\(⇒ V = \sqrt {{V_R^2} + {{\left( {{V_L} - {V_C}} \right)}^2}} \)

Where VR = potential difference across R, VL =  potential difference across L and VC =  potential difference across C

• For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(⇒ Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)
Where R = resistance, XL =induvtive reactance and XC = capacitive reactive

• When the LCR circuit is set to resonance, the resonant frequency is

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

EXPLANATION:

• For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(⇒ Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

• Resonance is that condition when Inductive reactance is equal to capacitive reactance i.e. XL = XC.

\(⇒ Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_L}} \right)}^2}} =R\)

• So Impedance will be minimum i.e. equal to R

• Hence at resonance, the impedance is purely resistive and it is minimum.
• Current in the circuit is given by

⇒ I = V/Z

• As impedance is minimum the current is maximum.

• From the above curve, we conclude that the frequency applied to a series LRC circuit is increased, the current of the circuit; first increases then decreases after reaching a maximum value.

CONCEPT:

Series RLC circuit:

• The ac circuit containing the capacitor, resistor, and the inductor is called an LCR circuit.
• For a series LCR circuit, the total potential difference of the circuit is given by:

\(V = \sqrt {{V_R^2} + {{\left( {{V_L} - {V_C}} \right)}^2}} \)    

Where VR = potential difference across R, VL =  potential difference across L and VC =  potential difference across C

• For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)     
Where R = resistance, XL =induvtive reactance and XC = capacitive reactive

• The resonant frequency of a series LCR circuit is given by

\(⇒ ν =\frac{1}{2π\sqrt{LC}}\)   

where L = inductance and C = capacitance

• The phase difference between the current and the voltage in the series RLC circuit is given as,

\(\Rightarrow tan\phi=\frac{X_L-X_C}{R}=\frac{V_L-V_C}{V_R}\)

EXPLANATION:

• The phase difference between the current and the voltage in the series RLC circuit is given as,

\(\Rightarrow tan\phi=\frac{X_L-X_C}{R}=\frac{V_L-V_C}{V_R}\)     -----(1)

• The phasor diagram of the series RLC circuit is given as,

• From the phasor diagram, it is clear that if the current is leading the voltage, then the phase difference should be negative.
• From equation 1 it is clear that if X_C is greater than X_L in the circuit then the phase difference will be negative.
• So we can say that in the series RLC circuit if the current is leading the voltage, then XC is greater than XL. Hence, option 1 is correct.

The correct answer is option 2) i.e. tan-1 (2π).

CONCEPT:

• LCR circuit: An electrical circuit constituting an inductor (L), capacitor (C), and resistor (R) connected in series or parallel is called an LCR circuit.

• For an inductor (L), if we consider the current (I) to be the reference axis, then voltage leads by 90°. For the capacitor (C), the voltage lags by 90°. This is represented by the phasor diagram. ​
  • Phasor diagrams are representations of the voltage (V) - current (I) relationship in AC circuits.
  • A phasor is a vector that rotates about the origin.

• The angle between the phasor and current is called the phase angle and is denoted by θ.

From the phasor diagram, \(tan θ = \frac{V_L - V_C}{V_R}\)      ----(1)

Using Ohm's law, V = IR

VL = I XL, VR = I R, VC = I Xc      ----(2)

Where XL is the inductive reactance, XC  is the capacitive reactance, and R is the resistance

Substituting (2) in (1), we get,

\(tan θ = \frac{X_L - X_C}{R}\)

For a given inductance (L) and capacitance (C):

 XL = ωL and X_C = \( \frac{1}{ω C} \)

Where ω is the angular frequency given by:

ω = 2 πf and f is the frequency in Hz.

CALCULATION:

Given that:

V = 220 V

f = 50 Hz ⇒ ω = 2πf = 2π × 50 = 100π rad/s

R = 20 Ω 

L = 0.4 H ⇒ X_L = ωL = 100π × 0.4 = 40 Ω 

In the given circuit, capacitor is absent.

\(tan θ = \frac{X_L - X_C}{R} = \frac{X_L }{R} = \frac{40π}{20} =2\pi\)

Therefore, θ = tan^-1 (2π) 

Since the capacitor is absent, we may assume C = 0. Hence, we can calculate XC = \( \frac{1}{ω C} = \frac{1}{\omega \times 0} = ∞\)

Substituting X_C = ∞ in  \(tan θ = \frac{X_L - X_C}{R}\) we get,

tanθ = ∞ ⇒ θ = tan^-1(∞) 

This is wrong as calculating X_C has to be done only when there is a capacitor present in the circuit. When there is no capacitor connected in the circuit, terms related to capacitance is safely neglected from the formulas.

Concept:

• In the LCR series circuit,  the resistor of resistance R, capacitor of capacitance C, and inductor of inductance L are connected in series across the A.C voltage.

• Impedance, \(Z=\sqrt{R^2+(X_L-X_C)^2} \)
• At resonance, XL = XC, Z = R, the circuit is purely resistive in nature.
• Phase difference,  \(\phi =tan^{-1}(\frac{X_L-X_C}{R})\)
• The power dissipated in the circuit, P = VrmsIrms cosϕ 
• The power factor is given as, \(cos\phi=\frac{R}{Z}=\frac{R}{\sqrt{R^2+(X_L-X_C)^2}}\)
• The current flowing in the circuit, \(i=\frac{V_{rms}}{Z}=\frac{V_{rms}}{\sqrt{R^2+(X_L-X_C)^2}}\)

Calculation:

Given,

The resistance, R = 30Ω 

The inductive reactant, XL = 40 

The capacitive reactant,  XC = 80 Ω

The AC voltage, V_rms = 200 volt

The angular frequency, ω = 50Hz

The power dissipated in the circuit, P = V_rmsI_rms cosϕ  . . . . . . . . . . .(1)

The current flowing in the circuit, \(i=\frac{V_{rms}}{Z}=\frac{V_{rms}}{\sqrt{R^2+(X_L-X_C)^2}}\)

\(i_{rms}=\frac{200}{\sqrt{(30)^2+(40-80)^2}}\)

i_rms = 4 A

The power factor is given as, \(cos\phi=\frac{R}{Z}=\frac{R}{\sqrt{R^2+(X_L-X_C)^2}}\)

\(cos\phi=\frac{30}{\sqrt{30^2+(80-40)^2}}\)

\(cos\phi=\frac{3}{5}\)

From equation (1), \(P=200\times 4\times \frac{3}{5} = 480 ~W\)

The correct answer is option 2) i.e. 90^∘ .

CONCEPT:

• LCR circuit: An electrical circuit constituting an inductor (L), capacitor (C), and resistor (R) connected in series or parallel is called an LCR circuit.

• For an inductor (L), if we consider the current (I) to be the reference axis, then voltage leads by 90°. For the capacitor (C), the voltage lags by 90°. This is represented by the phasor diagram. ​
  • Phasor diagrams are representations of the voltage(V) - current(I) relationship in AC circuits.
  • A phasor is a vector that rotates about the origin.

• The angle between the phasor and current is called the phase angle and is denoted by θ.

From the phasor diagram, \(tan θ = \frac{V_L - V_C}{V_R}\)      ----(1)

Using Ohm's law, V = IR

VL = I XL, VR = I R, VC = I Xc      ----(2)

Where XL is the inductive reactance, XC  is the capacitive reactance, and R is the resistance

Substituting (2) in (1), we get,

\(tan θ = \frac{X_L - X_C}{R}\)

For a given inductance (L) and capacitance (C):

 \(X_L = ω L \) and \(X_C = \frac{1}{ω C} \)

Where ω is the angular frequency given by:

ω = 2 πf and f is the frequency in Hz.

CALCULATION:

Given that:

R = 0 Ω, XL = 8 Ω and XC = 6 Ω 

\(tan θ = \frac{X_L - X_C}{R}\)

Phase difference, θ = \(tan^{-1}(\frac{X_L - X_C}{R}) = tan^{-1}(\frac{8 - 6}{0}) = tan^{-1}(\infty ) =\) 90∘

CONCEPT:

• Resistor: It is an electrical component that has two terminals and it is used for either limiting or regulating the flow of electric current in electrical circuits.
• Inductor: Inductors are coil-like structures that are used in electrical circuits. The coil is an insulated wire with a central core. An inductor is used to store energy in the form of magnetic energy when ac electricity is applied to the circuit. One of the main properties of an inductor is that it opposes the change in the amount of current flowing through it.
• Capacitor: A capacitor is a device that is used for the temporary storage of energy in circuits and can be made to release it when required.
• Reactance: It is basically the inertia against the motion of the electrons in an electrical circuit.
• Reactance is of two types:
  1. Capacitive reactance
  2. Inductive reactance

​\(⇒ X_{C}=\frac{1}{2\pi fC}\)

⇒ XL = 2πfL

Where f = frequency of ac current, C = capacitance of the capacitor, and L = self-inductance of the coil

• Impedance: It is a combination of resistance and reactance. It is essentially everything that obstructs the flow of electrons within an electrical circuit.

EXPLANATION:

• The impedance of the RC circuit is given as,

\(⇒ Z=\sqrt{R^{2}+X_{C}^{2}}\)      -----(1)

\(⇒ X_{C}=\frac{1}{2\pi fC}\)     -----(2)

• From equation 2 it is clear that if the frequency of the AC current increases, the value of X_C decreases.
• So by equation 1, it is clear that when the vale of X_C decreases, the impedance of the RC circuit also decreases.

The impedance of the RL circuit is given as,

\(⇒ Z=\sqrt{R^{2}+X_{L}^{2}}\)      -----(3)

⇒ XL = 2πfL     -----(4)

• From equation 4 it is clear that if the frequency of the AC current increases, the value of X_L also increases.
• So by equation 3, it is clear that when the value of XL increases, the impedance of the RL circuit also increases.

The impedance of the circuit containing only resistance is given as,

⇒ Z = R

Where R = resistance in ohm

• We know that the value of resistance does not depend on the frequency of the AC current, so the impedance of the circuit containing only resistance is independent of the frequency of the AC current. Hence, option 2 is correct.