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 inductor L and resistor R is connected to a source of alternating e.m.f. in series as shown in flg. ltnrgt The maximum voltage across R is `vec(V_(R))=vec(I_(0))R`.
As `vec(V_(R))` is in phase with current, it is represented by the vector `vec(OA)` along OX.
The maximum voltage across L is `vec(V_(L))=vec(I_(0))X_(L)`
As voltage across the inductor leads the current by `90^(@)`, it is represented by `vec(OB)` along OY, `90^(@)` ahead of `vec(I_(0))`.
The vector sum of `vec(V_(R)) and vec(V_(L))` is phasor `vec(E_(0))` is represented by `vec(OK)` making an angle `phi` with current phasor `I_(0)`.
`:. OK =sqrt(OA^(2)+OB^(2))rArr E_(0) = sqrt(V_(R)^(2)+V_(L)^(2))=sqrt(I_(0)^(2)R^(2)+I_(0)^(2)X_(L)^(2))`
`E_(0)+I_(0)sqrt(R^(2)+X_(L)^(2))`.
The impedance of the circuit is given by Z=`(E_(0))/(I_(0))=(I_(0)sqrt(R^(2)+X_(L)^(2)))/(I_(0)) rArr Z=sqrt(R^(2)+X_(L)^(2))`.