Question

Question #d1bd3

Answer 1

Cross-multiply, equate real and imaginary parts, then eliminate `R_1`

Explanation:

Cross-multiply:
`(R_1+j omega L)(R_4-j/(omega C))=R_2R_3`
Multiply out left-hand side and group real and imaginary:

`R_1R_4+(cancel(omega) L)/(cancel(omega) C) + j(omega L R_4 - R_1/(omega C))=R_3R_2+j0`
Equate real parts: `R_1R_4+L/C=R_3R_2` hence `L=CR_3R_2-CR_1R_4`
Equate imaginary parts: `omega LR_4=R_1/(omega C)` hence `R_1=omega^2LCR_4`
Substitute for `R_1`:
`L=CR_3R_2-omega^2C^2LR_4^2`
Collect `L` on left-hand side:
`L(omega^2C^2R_4^2+1)=CR_2R_3`
Hence:
`L=(CR_2R_3)/(omega^2C^2R_4^2+1)`