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Find the value of phase lag/lead between...

Find the value of phase lag/lead between the current and voltage in the given CLR circuit, fig. Without making any other change, find the value of additional capacitor, such that when joined switably to `C = 2 mu F`, would make the power factor of this circuit unity.

Text Solution

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(a) ` V=V_(0) sin (1000 t + phi)`
` V=V_(0) sin ( omegat+ phi) `
`omega=1000`
` X_(L) = omegaL`
` X_(L) = 1000 xx 100 xx10^(-3) =10^(2) = 1000 Omega`
`X_(C) = 1/(omegaC) = 1/(1000 xx 2xx 10^(-6)) =1/2 xx 10^(3) = 500 Omega`
Phase difference
` tan phi= ( X_(L) -X_(C))/R`
` tan phi = (X_(C)-X_(L))/R ,X_(C) gt X_(L)`
` tan phi = (500 -100)/400 = 400/400 =1`
` tan phi =1`
` tan phi tan 45^(@)`
`phi = 45`
Now ` sqrt(R^(2) +(X_(L) -X_(C))^(2))`
` = sqrt(400^(@)(100-500)^(2)) = sqrt(160000 + 160000) = sqrt(320000) = 565.68`
power factor = ` cos phi R/Z= 400/565.68 = 0.707`
(b) ` X_(L)=X_(C) . " Let new , capatior " = C_(1)`
` omegaL= 1/(omegaC)`
`C= = 1/(omega^(2)L) =1/(2pif)^(2) X_(L)) = 1/(4pi^(2)f^(2)L)`
` 2xx 1-^(-6) +C_(1) = 1/( 4xx (3.14)^(2) xx (50)^(2) xx 100 xx 10^(-3)) = 1/( 4xx 9.8596 xx 2500 xx 10^(-1))`
` 1/(4xx 9.8596 xx250) = 1/(9859.6)`
` C_(1) =1/(9859.6) -2 xx 10^(-6) = 1/(9859.6) -2/10^(6) = 0.000101424-0.000002= 0.00009924=99.42 muF`
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