The A.C. voltage and the current in a circuit are given by the following expressions. `V=110sqrt2 cos (2000t + 25^@)V` and `I = 10sqrt2` cos (2000t - `20^@`)A. Calculate impedance and resistance of the circuit.

Options

A. 7.679 `Omega`
B. 7.579 `Omega`
C. 7.779 `Omega`
D. 7.879 `Omega`

Answer

C

Explanation

Here `delta_1=-25^@` and `delta_2=20^@`
`V_m=110sqrt2V`
`I_m=10sqrt2A`
`omega`=2000 rad/s
`delta=delta_2-delta_1=20^@-(-25^@)=45^@`
Fro L-C-R series circuit
`tan delta=(omegaL-1/(omegaC))/R`
`tan45^@ = (omegaL-1/(omegaC))/R`
`therefore R=omegaL-1/(omegaC)`....(1) `[because tan45^@=1]`
Now, `|Z| = sqrt(R^2+(omegaL-1/(omegaC))^2)`
`=sqrt(R^2+R^2)` [From (1)]
`therefore |Z|=sqrt2R` ...(2)
But `|Z|=V_m/I_m=(110sqrt2)/(10sqrt2)=11Omega`
`therefore` From equation (2),
`|Z|=sqrt2R`
`11=sqrt2R`
`therefore R=11/sqrt2`
`=11/1.414`
`therefore R=7.779 Omega`