Class 12 - Unit # 20 : AC Circuits -Solved Numericals

1. Series RC Circuit (Total Impedance)

A resistor (R) of 20 ohms is connected in series with a capacitor (C) of 10 μF in an AC circuit with a frequency of 50 Hz. Calculate the total impedance?

Data:

• Resistance (R) = 20 Ω
• Capacitance (C) = 10 μF = 10 × 10⁻⁶ F
• Frequency (f) = 50 Hz
• Total impedance (Z) = ?

Solution:

The total impedance of this circuit is given by: Z = √(R² + Xc²) ----(i)

The capacitive reactance of the circuit is given by: Xc = 1 / (2πfC)

Xc = 1 / (2 × 3.1416 × 50 × 10 × 10⁻⁶) ≈ 318.31 Ω

Putting values in eq(i), we get: Z = √(20² + 318.31²)

Z = √(400 + 101321.26) ≈ 318.94 Ω

Result: The total impedance of the circuit is 318.94 Ω.

2. Inductive Reactance Calculation

For an inductor with an inductance (L) of 0.5 H and a frequency of 100 Hz, calculate the inductive reactance?

Data:

• Inductance (L) = 0.5 H
• Frequency (f) = 100 Hz
• Inductive reactance (X_L) = ?

Solution:

The inductive reactance of the circuit is given by: X_L = 2πfL

X_L = 2 × 3.1416 × 100 × 0.5

X_L ≈ 314.16 Ω

Result: The inductive reactance of the circuit is 314.16 Ω.

3. Series RL Circuit (Total Impedance)

In an RL circuit, the resistance (R) is 30 ohms, and the inductance (L) is 0.2 H. Calculate the total impedance at a frequency of 60 Hz?

Data:

• Resistance (R) = 30 Ω
• Inductance (L) = 0.2 H
• Frequency (f) = 60 Hz
• Total impedance (Z) = ?

Solution:

The total impedance of this circuit is given by: Z = √(R² + X_L²) ----(i)

The inductive reactance of the circuit is given by: X_L = 2πfL

X_L = 2 × 3.1416 × 60 × 0.2 ≈ 75.40 Ω

Putting values in eq(i), we get: Z = √(30² + 75.40²)

Z = √(900 + 5685.16) ≈ 81.15 Ω

Result: The total impedance at a frequency of 60 Hz is 81.15 Ω.

4. RC Circuit (Capacitive Reactance)

In an RC circuit, the resistance (R) is 50 ohms, and the capacitance (C) is 20 μF. Calculate the capacitive reactance?

Data:

• Resistance (R) = 50 Ω
• Capacitance (C) = 20 μF = 20 × 10⁻⁶ F
• Frequency (f) = 50 Hz (Utility Standard Value)
• Capacitive reactance (Xc) = ?

Solution:

The capacitive reactance of the circuit is given by: Xc = 1 / (2πfC)

Xc = 1 / (2 × 3.1416 × 50 × 20 × 10⁻⁶)

Xc ≈ 159.15 Ω

Result: The capacitive reactance is equal to 159.15 Ω.

5. RLC Circuit Impedance

An AC circuit has a resistance of 40 ohms, an inductive reactance of 30 ohms, and a capacitive reactance of 20 ohms. Calculate the total impedance?

Data:

• Resistance (R) = 40 Ω
• Inductive Reactance (X_L) = 30 Ω
• Capacitive Reactance (Xc) = 20 Ω
• Total impedance (Z) = ?

Solution:

The total impedance of this circuit is given by: Z = √(R² + (X_L - Xc)²)

Z = √(40² + (30 - 20)²)

Z = √(40² + 10²)

Z = √(1600 + 100)

Z = √1700 ≈ 41.23 Ω

Note: The Impedance Triangle consists of a base of 40 Ω, a vertical net reactance leg of 10 Ω, and a hypotenuse (Z) of 41.23 Ω.

Result: The total impedance of this circuit is 41.23 Ω.

6. Series RL Circuit (Phase Angle & Impedance)

In a series RL circuit, the resistance (R) is 25 ohms, and the inductance (L) is 0.1 H. Calculate the phase angle and impedance at a frequency of 80 Hz?

Data:

• Resistance (R) = 25 Ω
• Inductance (L) = 0.1 H
• Frequency (f) = 80 Hz
• Total impedance (Z) = ?
• Phase Angle (θ) = ?

Solution:

The total impedance of this circuit is given by: Z = √(R² + X_L²) ---(i)

The inductive reactance of the circuit is given by: X_L = 2πfL

X_L = 2 × 3.1416 × 80 × 0.1 ≈ 50.27 Ω

Putting values in eq(i), we get: Z = √(25² + 50.27²)

Z = √(625 + 2527.07) ≈ 56.14 Ω

Phase angle is given by: tan(θ) = X_L / R

θ = tan⁻¹(50.27 / 25)

θ = tan⁻¹(2.0108) ≈ 63.56°

Result: The phase angle is equal to 63.56° and impedance of the circuit is 56.14 Ω.

7. Parallel RC Circuit (Total Current)

In a parallel RC circuit, the resistance (R) is 60 ohms, and the capacitance (C) is 30 μF. Calculate the total current flowing through the circuit at a frequency of 120 Hz under a 60V supply?

Data:

• Voltage (V) = 60 V
• Resistance (R) = 60 Ω
• Capacitance (C) = 30 μF = 30 × 10⁻⁶ F
• Frequency (f) = 120 Hz
• Total Current (I_T) = ?

Solution:

According to Ohm’s Law: I = V / X

Current in Resistor: I_R = V / R = 60 / 60 = 1 A

Capacitive Reactance: Xc = 1 / (2πfC)

Xc = 1 / (2 × 3.1416 × 120 × 30 × 10⁻⁶) ≈ 44.21 Ω

Current in Capacitor: I_C = V / Xc = 60 / 44.21 ≈ 1.36 A

Total current is given by: I_T = √(I_R² + I_C²)

I_T = √(1² + 1.36²) ≈ 1.69 A

Result: The total current flowing through the circuit is 1.69 A.

8. RLC Circuit (Resonant Frequency)

In an RLC circuit, the resistance (R) is 50 ohms, the inductance (L) is 0.1 H, and the capacitance (C) is 50 μF. Calculate the resonance frequency?

Data:

• Resistance (R) = 50 Ω
• Inductance (L) = 0.1 H
• Capacitance (C) = 50 μF = 50 × 10⁻⁶ F
• Resonant Frequency (fr) = ?

Solution:

The resonant frequency is given by: fr = 1 / (2π√(LC))

fr = 1 / (2 × 3.1416 × √(0.1 × 50 × 10⁻⁶))

fr = 1 / (6.2832 × √(5 × 10⁻⁶))

fr = 1 / (6.2832 × 0.002236) ≈ 71.18 Hz

Result: The resonant frequency of this circuit is 71.18 Hz.