Derive an expression for the impedance of a series LCR, circuit, when an AC voltage is applied to it.

Explanation

Consider a voltage source `V=V_m sin omegat` connected to a series LCR with inductance L , capacitance C and resistance R.
If q is the charge on the capacitor and .T. is the current at any time .t. then from Kirchhoff.s rule we have ,
`V_L+V_R+V_C=0`

From the diagram `I=I_m sin (omegat+phi)` where `phi`=phase difference between voltage across the source and the current in the circuit.
`vecV=vecV_L+vecV_C+vecV_R`
`vecV_C` and `vecV_L` are along same line but in opposite direction.
`therefore` the magnitude `|V_(Cm)-V_(Lm)|`
`therefore V_m^2 = V_(Rm)^2 +[V_(Cm)-V_(Lm)]^2`
`V_m^2=(I_m R)^2 + (I_m X_C-I_m X_L)^2`
`V_m^2 =i_m^2 [R^2 + (X_C - X_L)^2]`
`Z=V_m/I_m=sqrt(R^2+(X_C-X_L)^2)`
`Z=sqrt(R^2+(X_C+X_L)^2)`
`I_m=V_m/sqrt(R^2+(X_C-X_L)^2)` or `I_m=V_m/sqrt(R^2+(X_L -X_C)^2)`