Question

An A.C. generator generating an emf of ε = 300 sin (100 πt) V is connected to a series combination of 16μ F capacitor, 1 H inductor and 100 Ω resistor.

Calculate :

1) An impedance of the circuit at the given frequency.

2) Resonant frequency `f_0`

3)Power factor at the resonant frequency `f_0`.

Answer 1

`E = 300sin(100pit)`

C =  capacitor = `16muF`

L = Inductar = 1H

`E = 300sin(100pit)`

From `E = 300 sin (100pit)`

`omega = 100 pi`

`2pif = 100pi`

`:. f = 50 Hz`

`:. X_C =1/(2pifC) = 1/(2pixx50xx16xx10^(-6))= 1/(5026.5 xx 10^(-6)) = 1.98 xx 10^(-4) xx 10^6`

`:. X_C = 198 Ω`

`X_L = 2pifl = 2pi xx 50 xx 1 = 3.14.1 Ω`

1) Impedence of the circuit (Z

`Z =  sqrt(R^2 + (X_L -X_C)^2) `

`= [(100)^2 + (314 - 198 )^2]^("1/2")`

= 153.15 Ω

2) Resonant frequency `(f_0)`

`f_0 = 1/(2pisqrt(LC))`

`= 1/(2pisqrt(1xx16xx10^(-6)))`

= 39.84 Hz

3) at Resonance `X_C = X_L`

`:. Z =R`

`cos phi = R/Z`

`cos phi = R/R`

`cos phi = 1`

`"power factor" (cos phi) = 1`