Question #d1bd3

1 Answer
Jan 3, 2017

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)#