The A.C. voltage and current in an L-C-R A.C. series circuit are given by the following expression `V = 200 sqrt(2) cos ( 3000t - 55^(@) V`, `I = 10 sqrt(2) cos ( 3000 t - 10^(@) ) A`. Calculate the impedance and the resistance of the above circuit.

Explanation

`delta_(1) = 10^(@), delta_(2) = 55^(@)`
`:. delta = delta_(2) - delta_(1) = 55^(@) - 10^(@) = 45^(@)`
`V_(m) = 200 sqrt(2) V, I_(m) = 10sqrt(2) A`
here, `delta= 45^(@)`
`:. tan delta = tan 45^(@)`
`:. tan delta =1 `
Now for L-C-R- series circuit,
`:. tan delta = ( omega l - (1)/( omega C ))/(R )`
`:. 1= ( omega L - ( 1)/( omega C ))/( R )`
`R = omega L - ( 1)/( omega C )` ....(i )
and `| Z | = sqrt( R^(2) + ( omega L - ( 1)/( omega C ))^(2)) `
`= sqrt( R^(2) + R^(2))` [ `:. ` From result (i) ]
`| Z | = sqrt( 2).R` ....(ii)
and `|Z | ( V_(m))/( I_(m))`
`:. | Z | ( 200 sqrt(2))/( 10sqrt(2))`
`:. | Z | = 20 Omega` ....(iii)
From equation (ii) and (iii)
`:. 20 = sqrt( 2) xx R `
`:. 10 sqrt(2) = R `
`:. 10 xx 1.414 = R `
`:. R = 14.14 Omega`