Problem

A coil having a resistance of $70 \, \Omega$ and an inductance of $31.8 \, \mathrm{mH}$ is connected to a $230 \, \mathrm{V}$, $50 \, \mathrm{Hz}$ supply. Calculate:

1. The circuit current,
2. The phase angle,
3. The power factor,
4. The power consumed.

Data

• Inductance: $L = 31.8 \, \mathrm{mH} = 31.8 \times 10^{-3} \, \mathrm{H}$
• Resistance: $R = 7 \, \Omega$
• AC Frequency: $f = 50 \, \mathrm{Hz}$
• RMS Voltage: $V_{\text{rms}} = 230 \, \mathrm{V}$

Prerequisite Concepts

1. Inductive Reactance:

$$X_L = 2 \pi f L$$

2. Impedance in an RL Circuit:

$$Z = \sqrt{R^2 + X_L^2}$$

3. RMS Current:

$$I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}$$

4. Phase Angle:

$$\theta = \tan^{-1} \left( \frac{X_L}{R} \right)$$

5. Power Factor:

$$\cos \theta$$

6. Power Consumed:

$$P = I_{\text{rms}} V_{\text{rms}} \cos \theta$$

Solution

Step 1: Calculate Inductive Reactance

$$X_L = 2 \pi f L$$ $$X_L = 2 \times 3.14 \times 50 \, \mathrm{Hz} \times 31.8 \times 10^{-3} \, \mathrm{H}$$ $$X_L = 9.985 \, \Omega$$

Step 2: Calculate Impedance

$$Z = \sqrt{R^2 + X_L^2}$$ $$Z = \sqrt{(7 \, \Omega)^2 + (9.985 \, \Omega)^2}$$ $$Z = \sqrt{49 + 99.7} = \sqrt{148.7}$$ $$Z = 12.194 \, \Omega$$

Step 3: Calculate RMS Current

$$I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}$$ $$I_{\text{rms}} = \frac{230 \, \mathrm{V}}{12.194 \, \Omega}$$ $$I_{\text{rms}} = 18.86 \, \mathrm{A}$$

Step 4: Calculate Phase Angle

$$\theta = \tan^{-1} \left( \frac{X_L}{R} \right)$$ $$\theta = \tan^{-1} \left( \frac{9.985 \, \Omega}{7 \, \Omega} \right)$$ $$\theta = 54.95^\circ \approx 55^\circ$$

Step 5: Calculate Power Factor

$$\cos \theta = \cos 55^\circ = 0.573$$

Step 6: Calculate Power Consumed

$$P = I_{\text{rms}} V_{\text{rms}} \cos \theta$$ $$P = 18.86 \, \mathrm{A} \times 230 \, \mathrm{V} \times 0.573$$ $$P = 2484.24 \, \mathrm{W}$$

Answer

1. Circuit Current: $I_{\text{rms}} = 18.86 \, \mathrm{A}$
2. Phase Angle: $\theta = 55^\circ$
3. Power Factor: $\cos \theta = 0.573$
4. Power Consumed: $P = 2484.24 \, \mathrm{W}$

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