Question

Derive an expression for the impedance of an LCR circuit connected to an AC power supply. Draw the phasor diagram.

Hint:

In a series LCR circuit, the impedance depends on the resistance, inductive reactance, and capacitive reactance. The phase difference between voltage and current varies with frequency.

Answer 1

An LCR circuit consists of a resistor \( R \), an inductor \( L \), and a capacitor \( C \) connected in series to an AC power supply. The impedance \( Z \) of a series LCR circuit is defined as the opposition to the flow of alternating current, and it is a complex quantity due to the phase difference between the voltage and the current. The total impedance \( Z \) of a series LCR circuit is given by the expression:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] where: - \( R \) is the resistance,
- \( X_L = \omega L \) is the inductive reactance, and
- \( X_C = \frac{1}{\omega C} \) is the capacitive reactance.
Here: - \( \omega = 2\pi f \) is the angular frequency of the AC source,
- \( L \) is the inductance of the inductor,
- \( C \) is the capacitance of the capacitor, and
- \( f \) is the frequency of the AC supply.
Step 1: Derivation of the Impedance
In a series LCR circuit, the current \( I \) is the same through all components. The voltage across each component is given by:
- Voltage across the resistor: \( V_R = I R \),
- Voltage across the inductor: \( V_L = I X_L = I \omega L \),
- Voltage across the capacitor: \( V_C = I X_C = I \frac{1}{\omega C} \).
The total voltage \( V \) across the circuit is the vector sum of the individual voltages. Since the voltage across the resistor is in phase with the current, the voltage across the inductor leads the current by \( 90^\circ \), and the voltage across the capacitor lags the current by \( 90^\circ \), we can use the phasor diagram to represent these voltages.
Step 2: Phasor Diagram
In the phasor diagram:
- The voltage \( V_R \) is represented along the real axis (since it is in phase with the current),
- The voltage \( V_L \) is represented as a vector along the imaginary axis, pointing upwards (since it leads the current by \( 90^\circ \)),
- The voltage \( V_C \) is represented as a vector along the imaginary axis, pointing downwards (since it lags the current by \( 90^\circ \)).
The total voltage \( V \) is the vector sum of these individual voltages. Using the Pythagorean theorem, we get the magnitude of the total voltage:
\[ V = \sqrt{V_R^2 + (V_L - V_C)^2} \] Substituting the expressions for \( V_R \), \( V_L \), and \( V_C \), we get the impedance:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] This gives the magnitude of the impedance, which tells us how much the circuit resists the flow of alternating current. The phase difference \( \phi \) between the current and the voltage is given by:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \]
Step 3: Special Cases
- When \( X_L = X_C \), the impedance becomes \( Z = R \), and the circuit behaves like a purely resistive circuit.
- When \( X_L>X_C \), the circuit is inductive, and the voltage leads the current.
- When \( X_C>X_L \), the circuit is capacitive, and the voltage lags the current.
Step 4: Impedance and Power
The power \( P \) dissipated in the circuit is related to the voltage and the current as follows:
\[ P = I^2 R = \frac{V^2}{Z^2} R \] where \( V \) is the total voltage across the LCR circuit and \( I \) is the current.
Phasor Diagram
The phasor diagram for a series LCR circuit can be drawn as follows:

- \( V_R \) is the voltage across the resistor,
- \( V_L \) is the voltage across the inductor,
- \( V_C \) is the voltage across the capacitor,
- \( V \) is the total voltage across the LCR circuit.