An inductor and a bulb are connected in series to an AC source of 220V, 50Hz. 7C A current of 11A flows in the circuit and phase angle between voltage and current is `(pi)/(4)` radians. Calculate the impedance and inductance of the circuit

Explanation

Here ,`V=220 V , f=50 Hz I=11 A , phi = pi //4 .Z =? L=?`
`I_(rms) = 0.7071 I_0 `
` I_0=(I_(rms))/(0.707 ) = (11)/(0.707 ) = 15.55 A`
`V_(rms)=0.707 V_0`
`V_0 =(V_(rms))/( 0.707 )= (220 )/( 0.707 ) = 311.17 V `
indedence `Z=(V_0)/(I_C) = (311.17 )/( 15.55) = 20.01 Omega `
`cos phi = (R )/(Z)`
` R=Z cos phi = 20.01 cos ((pi)/(4))`
` R= 20.01 xx (1)/(sqrt(2)) = (20.01)/(1.414 )= 14.151 Omega `
` Z= sqrt(R^2 + ( omega L)^2)`
` Z^2 = R^2 + (2 pi fL)^2 " " ( :. omega = 2 pi f )`
`(Z^2 -R^2)/( 4 pi ^2 f^2) = L^2 implies L^2 =((20.01)^2 - (14.151)^2)/( 4 xx (3.14 )^2 xx (50)^2)`
`L^2 = ( 400.40 - 200.25 ) /( 98596 )`
`= (200.15 )/( 98596 )= 0.00202 H`
`L= 0.0449 H = 44.9 mH .`