Question

What do you mean by impedance of LCR series circuit? Derive an expression for it by using phasor diagram.

Hint:

Impedance is the total opposition to current in an AC circuit, and in an LCR circuit, it depends on the resistance, inductive reactance, and capacitive reactance.

Answer 1

Step 1: Definition of impedance in an LCR series circuit.
Impedance \( Z \) is the total opposition to the flow of alternating current (AC) in a circuit containing resistors (R), inductors (L), and capacitors (C). It is a complex quantity that takes both the resistance and reactance into account and is measured in ohms. The impedance \( Z \) in an LCR series circuit is the combined effect of the resistance \( R \), inductive reactance \( X_L \), and capacitive reactance \( X_C \).
Step 2: Phasor diagram for an LCR series circuit.
In an LCR series circuit, the voltage across the resistor, inductor, and capacitor are all out of phase with each other: - The voltage across the resistor \( V_R \) is in phase with the current. - The voltage across the inductor \( V_L \) leads the current by \( 90^\circ \). - The voltage across the capacitor \( V_C \) lags the current by \( 90^\circ \).
Using a phasor diagram, we represent these voltages as vectors in the complex plane. The total voltage across the LCR circuit \( V \) is the vector sum of \( V_R \), \( V_L \), and \( V_C \).
The total impedance \( Z \) is the magnitude of the total voltage divided by the current \( I \): \[ Z = \dfrac{V}{I} \]
Step 3: Expression for impedance.
The total impedance \( Z \) can be calculated using the Pythagorean theorem: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] where: - \( R \) is the resistance, - \( X_L = \omega L \) is the inductive reactance, - \( X_C = \dfrac{1}{\omega C} \) is the capacitive reactance, - \( \omega = 2 \pi f \) is the angular frequency of the AC source.
Thus, the impedance in an LCR series circuit is a combination of the resistance and the net reactance (the difference between inductive and capacitive reactance). Step 4: Conclusion.
The impedance \( Z \) of an LCR series circuit is given by: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]