A coil having a resistance of 10 ohms and an inductance of 0.2 H is connected in series with a 100 x 10-6 F capacitor across a 230V, 50Hz supply, Calculate: (i) The active and reactive components of the current (ii) The voltage across the coil, Draw the phasor diagram

To solve this problem, we will follow these steps:

1. Calculate the inductive reactance \( X_L \) and capacitive reactance \( X_C \).
2. Determine the total impedance \( Z \) of the series circuit.
3. Calculate the current \( I \) using Ohm's law.
4. Determine the active (real) and reactive components of the current.
5. Calculate the voltage across the coil.
6. Draw the phasor diagram.

### Step 1: Calculate Inductive and Capacitive Reactance

Given dataTo solve this problem, we will follow these steps:

1. Calculate the inductive reactance \( X_L \) and capacitive reactance \( X_C \).
2. Determine the total impedance \( Z \) of the series circuit.
3. Calculate the current \( I \) using Ohm's law.
4. Determine the active (real) and reactive components of the current.
5. Calculate the voltage across the coil.
6. Draw the phasor diagram.

### Step 1: Calculate Inductive and Capacitive Reactance

Given 
- Resistance, \( R = 10 \, \Omega \)
- Inductance, \( L = 0.2 \, \text{H} \)
- Capacitance, \( C = 100 \times 10^{-6} \, \text{F} \)
- Supply voltage, \( V = 230 \, \text{V} \)
- Supply frequency, \( f = 50 \, \text{Hz} \)

Inductive reactance \( X_L \) is given by:
\[ X_L = 2 \pi f L \]
\[ X_L = 2 \pi \times 50 \times 0.2 \]
\[ X_L = 20 \pi \]
\[ X_L \approx 62.83 \, \Omega \]

Capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{2 \pi f C} \]
\[ X_C = \frac{1}{2 \pi \times 50 \times 100 \times 10^{-6}} \]
\[ X_C = \frac{1}{2 \pi \times 0.005} \]
\[ X_C = \frac{1}{0.01 \pi} \]
\[ X_C \approx 31.83 \, \Omega \]

### Step 2: Calculate the Total Impedance

The total impedance \( Z \) of the series circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
\[ Z = \sqrt{10^2 + (62.83 - 31.83)^2} \]
\[ Z = \sqrt{100 + 31^2} \]
\[ Z = \sqrt{100 + 961} \]
\[ Z = \sqrt{1061} \]
\[ Z \approx 32.56 \, \Omega \]

### Step 3: Calculate the Current

Using Ohm's law:
\[ I = \frac{V}{Z} \]
\[ I = \frac{230}{32.56} \]
\[ I \approx 7.06 \, \text{A} \]

### Step 4: Determine Active and Reactive Components of the Current

The active component of the current \( I_R \) (in phase with voltage) is given by:
\[ I_R = I \cos \phi \]
where \( \cos \phi = \frac{R}{Z} \)
\[ \cos \phi = \frac{10}{32.56} \]
\[ \cos \phi \approx 0.307 \]
\[ I_R = 7.06 \times 0.307 \]
\[ I_R \approx 2.17 \, \text{A} \]

The reactive component of the current \( I_X \) (out of phase with voltage) is given by:
\[ I_X = I \sin \phi \]
where \( \sin \phi = \frac{X_L - X_C}{Z} \)
\[ \sin \phi = \frac{62.83 - 31.83}{32.56} \]
\[ \sin \phi = \frac{31}{32.56} \]
\[ \sin \phi \approx 0.952 \]
\[ I_X = 7.06 \times 0.952 \]
\[ I_X \approx 6.72 \, \text{A} \]

### Step 5: Calculate the Voltage Across the Coil

The voltage across the coil \( V_{coil} \) can be found using:
\[ V_{coil} = I \times \sqrt{R^2 + X_L^2} \]
\[ V_{coil} = 7.06 \times \sqrt{10^2 + 62.83^2} \]
\[ V_{coil} = 7.06 \times \sqrt{100 + 3947.29} \]
\[ V_{coil} = 7.06 \times \sqrt{4047.29} \]
\[ V_{coil} \approx 7.06 \times 63.62 \]
\[ V_{coil} \approx 449.62 \, \text{V} \]

### Step 6: Draw the Phasor Diagram

1. Draw the voltage vector \( V \) along the horizontal axis (reference axis).
2. Draw the current vector \( I \) lagging \( V \) by the phase angle \( \phi \).
3. The voltage across the resistor \( V_R \) is in phase with \( I \).
4. The voltage across the inductor \( V_L \) leads \( I \) by \( 90^\circ \).
5. The voltage across the capacitor \( V_C \) lags \( I \) by \( 90^\circ \).

The phasor diagram should show the following vectors:
- \( V \) as the reference vector.
- \( I \) lagging \( V \) by the phase angle.
- \( V_R \) in phase with \( I \).
- \( V_L \) perpendicular and leading \( I \).
- \( V_C \) perpendicular and lagging \( I \).

Would you like a detailed explanation or have any questions about the calculations? Here are some related questions to expand on this topic:

1. How is the phase angle \(\phi\) calculated in an RLC circuit?
2. What is the significance of active and reactive power in AC circuits?
3. How does the resonance frequency affect an RLC circuit?
4. What happens to the impedance of an RLC circuit at resonance?
5. How can the power factor of an RLC circuit be improved?
6. What are the practical applications of series RLC circuits?
7. How does changing the frequency affect the reactance of the inductor and capacitor?
8. What is the role of damping in RLC circuits?

**Tip:** Understanding the phase relationship between voltage and current in AC circuits is crucial for analyzing power consumption and efficiency in electrical systems.

Learn how to calculate the active and reactive components of current, as well as the voltage across a coil in a series RLC circuit. This problem involves using Ohm's law, calculating reactances, and understanding phasor diagrams. Suitable for undergraduate students and electrical engineering enthusiasts.

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