The series and parallel circuits shown in figure have the same impedance and the same power factor. If `R= 3Omega and X= 4Omega`, find the values of `R_(1) and X_(1)`. Also, find the impedance and power factor.

Explanation

Impedance, `Z=sqrt(R^(2)+X^(2))=5Omega`
Power factor, `cosphi=R/Z = 3/5rArr phi=cos^(-1)(0,6)`
`rArr tanphi=X/R = 4/3`

Now, we have, `(I_(R))_(L) = V_(0)/R_(1), (I_(X1))_(0) = V_(0)/X_(1)`

`I_(0) = (sqrt((I_(R_(1)))^(2)+(I_(X_(1)))_(0)^(2))=sqrt((V_(0)/R_(1))^(2)+(V_(0)/X_(1))^(2))`
`rArr V_(0)/Z = V_(0)sqrt(1/R_(1)^(2)+1/X_(1)^(2)) rArr Z=1/sqrt(1/R_(1)^(2)+1/X_(1)^(2))=5`
`tanphi^(')=(I_(X_(L)))_(0)/((I_(R_(1)))_(0))=(V_(0)//X_(1))/(V_(0)//R_(1))=R_(1)/X_(1)=tanphi = 4/3`
Putting `X_(1)= 3/4R_(1)` in Eq. (i)
`rArr 1/R^(2) + 1/(3/4R_(1))^(2)=1/5^(2)`
`rArr 1/R_(1)^(2)(1+16/9) = 1/25 = 1/R^(2)=9/(25)^(2)`
`rArr R_(1) = 25/3Omega`
`rArr X_(1)=3/4R_(1)=25/4Omega`