Question

An inductor, capacitor, and resister are connected in series to an a.c. source V = V sin ωt.

1. Draw a circuit diagram of L.C.R. series circuit with applied a.c. voltage. 

2. Find an expression for impedance of L.C.R. series circuit using phasor diagram.

3. What is impedance of L.C.R. series circuit at resonance?

Answer 1

For impedance of LCR circuit.

From the right angled triangle OAE

Final voltage, V = \(\sqrt{v^2_n+(v_L-v_c)^2}\)

V = \(\sqrt{V^2_R+(IX_L-IX_c)^2}\)

V= \( \sqrt{V^2_R+\Big[ILω-{\frac{I}{cω}}}\Big]^2\)

V = \(\sqrt{I^2R^2+I^2\Big[Lω-{\frac{I}{cω}}}\Big]^2\)

V = \(I\sqrt{R^2+\Big[Lω-{\frac{I}{cω}}}\Big]^2\)

\(\frac{V}{I}=\sqrt{R^2+\Big[Lω-{\frac{I}{cω}}}\Big]^2\)

Z = \(\sqrt{R^2+\Big[Lω-{\frac{I}{cω}}}\Big]^2\)

Where Z is called impedance of LCR circuit

3. Impedance, Z=R