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

Explanation

Consider a series LCR circuit connected to an AC source.
AC voltage applied is `v=v_(m) sin omega t`
Let the current through the series be, `i=i_(m) sin(omega t + O/)`
where `O/` is phase difference between voltage and current.

Let `vec I` phasor respresenting current and `vec V_(R) , vec V_(L) , vec V_(C) and vec V` be the phasors respresenting voltage across R,L,C and source respectively and we assume that `vec V_(C) gt vec V_(L)`

The length of the phasors `vec V_(R) vec V_(C) and vec V_(L)` are
`V_(Rm) = i_(m) R`
`V_(cm) = i_(m) X_(c) and `
`V_(Lm) = i_(m) X_(L)`
From Kirchhoff.s loop rule `V_(L) + V_(R) + V_(C) = V`
and the phasor relation is respresented as,
`vecV_(L) + vecV_(R) + vecV_(C) = vecV`
Expression for Impedance:
Applying Phythagorean Theorem to the right angle triangle,
`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)]`
`V_(m) =i_(m) sqrt (R^(2) + (X_(c) - X_(L))^(2))`
`i_(m) =(V_(m))/ sqrt (R^(2) + (X_(c) - X_(L))^(2))`
`i_(m) =(V_(m))/Z`
Where Z is called impedance of the coil
`Z = sqrt (R^(2) + (X_(c) - X_(L))^(2))`