AC Voltage Applied to a Series LCR Circuit MCQ Quiz - Objective Question with Answer for AC Voltage Applied to a Series LCR Circuit - Download Free PDF

Calculation:

The potential difference (PD) across the inductor is given by:

VL = L × (di/dt)

Substitute the given values:

VL = 5 × (-1.0) = -5 V

The voltage across the resistor is:

VR = i × R = 2 × 10 = 20 V

The total voltage across the circuit is the sum of the voltage across the resistor and the inductor, and the applied voltage:

Vab = VE + VR + VL

Substituting the values:

Vab = 20 + 20 + (-5) = 35 V

Calculation:

The formula for the cosine of the phase angle (φ) is:

cos(φ) = R / Z = R / √(R² + (X_C - X_L)²)

⇒ cos(φ) = 80 / √(80² + 60²)

⇒ cos(φ) = 80 / √(6400 + 3600)

⇒ cos(φ) = 8 / 10

Final Answer: x = 8 / 10

Calculation:

The amplitude will be maximum when there will be resonance condition

The value of resonance frequency is f = 1 / (2π√(L C))

⇒ 2000 Hz = 1 / (2π√(L × 62.5 × 10^–9))

⇒ L = 1 / (4π² × 2000² × 62.5 × 10^–9) = 0.1 H = 100 mH

Calculation:

Given Expressions for  (P), (Q), (R), and (S):

(P): Given voltage: v_z = 10 sin(400t)   (volts)

Impedance: Z = R   ⇒   Purely resistive

(Q): Z = jωL,   ω = 400,   L = x   (inductive)

Voltage: V = 6 sin(400t - 53°)   V

(R): Z = jωL + R = 50 + j10,   ⇒   |Z| = √(50² + 10²) = √2600

Voltage: V = 6 sin(400t + 53°)   V

(Phase shift is positive, indicating a capacitive nature.)

(S): Z = jωL + R = 60 + j60,   ⇒   |Z| = √(60² + 60²) = 60√2

Voltage: V = 300 sin(400t) / 60√2 = 5 sin(400t)

Solution:  

−V^″ + L(dI/dt) + IR = 0     (V^″ is source voltage of B₁)

dI/dt + (R/L) · I = V^″/L

Integrating factor  = e^Rt/L,

⇒ I · e^Rt/L = ∫ (V^″/L) · e^Rt/L dt + c

⇒ I = V^″/(LR) + c · e^−Rt/L

⇒ V = V^″ + c · e^−Rt/L

At t = 0, V = 0 → 0 = V^″ + c → c = −V^″

V = 25 · [1 − e^−Rt/L]     (From graph: At t → ∞, V = 25)

At t = 1, V = 16:

16 = 25 · [1 − e^−R/L]

⇒ L = R

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:

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

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

• Inductive reactance,

⇒ XL = Lω  

• Capacitive reactance

​\(\Rightarrow X_c=\frac{1}{C\omega}\)

• Resonance will take place when XL = X_C.

​⇒ XL = X_C

\(\Rightarrow L\omega =\frac{1}{C\omega}\)

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

EXPLANATION :

• Resonance of an LCR circuit is the frequency at which capacitive reactance is equal to the inductive reactance, X_c = X_L, then the value of impedance is given by

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

\(\Rightarrow Z = \sqrt{R^{2}+0}\)

\(\Rightarrow Z = R\)

• At resonance, the value of impedance is purely resistive. Hence, option 1 is the answer

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

CALCULATION:

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

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

• Inductive reactance,

⇒ XL = Lω  

• Capacitive reactance

​\(\Rightarrow X_c=\frac{1}{Cω}\)

• Resonance will take place when XL = XC.

​⇒ XL = XC

\(\Rightarrow Lω =\frac{1}{Cω}\)

\(\Rightarrow ω =\frac{1}{\sqrt{LC}}\)

As we know, ω = 2πf

Where f = frequency

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

Given:

\(R=30\Omega,X_L=200\Omega,X_C=160\Omega,V=220V\)

Concept:

Impedance is the opposition to the alternating current presented by the combined effect of resistance and reactance in a circuit.

Impedance is given by the formula,

• \(Z=\sqrt{R^2+(X_L-X_C)^2}\)

Where \(R\) is Resistance,

 \(X_L\) is inductive reactance,

\(X_C\) is capacitive reactance

Explanation:

• \(Z=\sqrt{R^2+(X_L-X_C)^2}\)
• \(Z=\sqrt{30^2+(200-160)^2}=\sqrt{30^2+40^2}=\sqrt{2500}=50\Omega\)
• \(Z=\sqrt{30^2+40^2}=\sqrt{2500}=50\Omega\)

Current, I\(=\frac{V}{Z}=\frac{220}{50}=\frac{22}{5}A\)

Potential drop across the resistor is \(V_d=IR=\frac{22}{5}\times30=132V\)

The correct answer is Option-1-132V.

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

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

CALCULATION:

• From the above, it is clear that the resonant frequency is given by

\(⇒ f = {\frac{1}{2\pi\sqrt{LC}}}\)

• From the above equation, it is clear that the resonant frequency is independent of resistance, so if the value of R is decreased by 20 Ω  then there will be no change in the value of resonant frequency. Therefore the correct answer is option 4.   

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

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

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

​Reactance: 

• It is basically the inertia against the motion of the electrons in an electrical circuit.

• There are two types of reactance:

  1. Capacitive reactance (X_C) (Ohms is the unit)

  2. Inductive reactance (X_L) (Ohms is the unit)​

​CALCULATION:

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

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

• For resonating condition,

\(\Rightarrow X_{L}=X_{C}\)

\(\Rightarrow Z= R\)

• Hence the impedance is minimum for the resonating condition.
• Hence, option 2 is correct.

CONCEPT:

LCR CIRCUIT:

•  An LCR Circuit is an electrical circuit consisting of Inductor (L), Capacitor (C), Resistor (R) it can be connected either parallel or series.
• 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 =inductive reactance and XC = capacitive reactive

• At resonance X_c = X_L,  the frequency of oscillation is given by

\(⇒ \nu = \frac{1}{2\pi\sqrt{LC}}\)

Where L = Inductance, C = Capacitance

EXPLANATION :

• The power factor of LCR is given by

⇒ P  = VI cosϕ  ----(1)

\(⇒ Cosϕ = \frac{R}{Z}\)

at resonance R = Z

\(⇒ Cosϕ = \frac{Z}{Z} = 1\)

Substituting the above  value in equation 1

⇒ P = VI

• From the above equation, it is clear that the power factor is not zero. So, option 4 is wrong
• Hence, option 4 is the answer.

CONCEPT:

• Power Factor: The power factor of series-connected L and R circuit in AC voltage is

cos ϕ = R/Z

where R is resistance and Z is overall impedance.​

EXPLANATION:

In an AC circuit, the power factor is given by:

cos ϕ = R/Z

So the correct answer is option 2.

Additional Information

• Voltage in AC: In an AC voltage source, the voltage of the source keeps changing with time and is defined as

V = V0 sin ωt

where V is the voltage at any time t, V0 is the max value of voltage, and ω is the angular frequency.

• With AC source capacitance, In a series combination of a resistor (R), an inductance (L)

Inductance Reactance is defined as:

 \(X_L = ω L=2\pi fL\)

Where ω is the angular frequency, ƒ is the frequency in Hertz, C is the AC capacitance in Farads, and XL is the Inductive Reactance in Ohms,.

Overall Impedance (Z) in a series combination of a resistor (R), an inductance (L):

\(Z = \sqrt{R^2 + {ω L}^2}\)

where R is resistance and ωL is Inductance reactance.

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

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

\(⇒ \nu =\frac{1}{2\pi\sqrt{LC}}\)

If ϕ is the phase difference between the current and voltage in L-C-R circuit, then,

\(⇒ tanϕ=\frac{X_L-X_C}{R}=\frac{V_L-V_C}{V_R}\)

​Reactance: 

• It is basically the inertia against the motion of the electrons in an electrical circuit.

• There are two types of reactance:

  1. Capacitive reactance (XC) (Ohms is the unit)

  2. Inductive reactance (XL) (Ohms is the unit)​

​CALCULATION:

We know that for a series LCR circuit, the resonating condition is given by:

⇒ X_L = X_C = X     -----(1)

If ϕ is the phase difference between the current and voltage in L-C-R circuit, then,

\(⇒ tanϕ=\frac{X_L-X_C}{R}\)     -----(2)

By equation 1 and equation 2,

\(⇒ tanϕ=\frac{X_L-X_C}{R}\)

\(⇒ tanϕ=\frac{X-X}{R}\)

⇒ tanϕ = 0

⇒ ϕ = 0

• Hence, option 1 is correct.

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

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

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

CALCULATION:

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

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

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

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

• So Impedance will be minimum i.e. equal to R
• And hence the LCR circuit will behave like a normal circuit with resistance.
• This means that in an LCR circuit at resonance the current and voltage are in phase