Question

An inductor 200 mH, capacitor `500 mu F`, resistor 10 ohm are connected in series with a 100 V, variable frequency a.c. source. Calculate (i) frequency at which power factor of the circuit is unity (ii) current amplitude at this frequency (iii) Q factor.

Answer 1

Here `L=200mH = 200xx10^(-3)=0.2H`
`C=500 mu F = 500 xx 10^(-6) = 5xx10^(-4)F`
`R=10Omega and F_(V)=100V`
(i) Power factor, `cos phi =1 ` (given)
`(R)/(Z)=1 " " Z=R`
`sqrt(R^(2)+(X_(L)-K_(C))^(2))=R`
`R^(2)+(X_(L)-X_(C))^(2)=R^(2) rArr X_(L)-X_(C)=0`
`rArr X_(L)=X_(C) rArr 2 pi v L = (1)/(2pi v C)`
`rArr 4pi^(2)v^(2)LC=1`
`:. v=(1)/(2pi sqrt(LC))=(1)/(2xx3.14sqrt(0.2xx5xx10^(-4)))=(1)/(2xx3.14xx10^(-2))=(100)/(6.28)=15.92 ~~ 16Hz`.
(ii) Current amplitude at resonance
`I_(0)=(E_(0))/(Z)=(sqrt(2 in_(rms)))/(R)=(1.414xx100)/(10)=14.14A`
(iii) Q-factor `=(1)/(R) sqrt((L)/(C))=(1)/(10)sqrt((0.2)/(5xx10^(-4)))`
`=(1)/(10)sqrt((2)/(50xx10^(-4)))=(1)/(10) sqrt((1)/(25xx10^(-4)))=(1)/(10xx5xx10^(-2))=(10^(2))/(50)=(100)/(50)=2`.