Derive an expression for the impedance of an a.c. circuit consisting of an inductor and a resistor.

Explanation

`A.C.` circuit containing inductor and resistor: Let an iductor `L` and resistor `R` is connected to a source of alternating `e.m.f` in series as shown in fig (i).
The maximum voltage across `R` is
`vecV_(R)=vecI_(0)R`
As `vecV_(R)` in phase with current it is represented by the vector `vec(OA)` along `OX`.
The maximum voltage across `L` is `vecV_(L)=vecI_(0)X`
As voltage across the inductor leads the current by `90^(@)`, it is represented by `vec(CB)` along `(OY)` , `90^(@)` ahead at `I_(0)` , The vector sum of `vecV_(R)` and `vecV_(L)` is phasor `vecE_(0)` is represented by `vec(OK)` making an angle `phi` with current phasor
`I_(0)therefore OK=sqrt(OA^(2)+OB^(2))`
`E_(0)=sqrt(V_(R)^(2)+V_(L)^(2))`
`Z=E_(0)/I_(0)=(I_(0)sqrt(R^(2)+X_(L)^(2)))/I_(0)`
`rArr Z=sqrt(R^(2)+X_(L)^(2))`