Question

Derive an expression for the impedance of a series LCR circuit connected to an AC power supply.

Answer 1

Given:

Resistance R
Inductance L
Capacitance C

It is connected to an alternating voltage source: V(t) = V_0​sinωt

To derive an expression for Impedance (Z) of the circuit.

Voltage drop across each component

Voltage across resistor: VR = IR
Voltage across inductor: VL = IXL = IωL
Voltage across capacitor: VC = IX_C = `I/(ωC)`

Net voltage across circuit:

`V=sqrt((V_R)^2 + (V_L-V_C)^2) = I sqrt(R^2+(X_L-X_C)^2)`

Ohm's Law for AC circuits:

V = IZ

`Z=V/I = sqrt(R^2+(X_L-X_C)^2)`

`Z = sqrt(R^2+(ωL - 1/(ωC))^2)`