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Question Number 51250 by Tawa1 last updated on 25/Dec/18

Given that   z_1  = R_1  + R + jωL ;   z_2  = R_2  ;  z_3  = (1/(jωC_3 ))  and  z_4  = R_4  + (1/(jωC_4 ))  and also that   z_1 z_3   =  z_2 z_4  ,   express   R and L in terms of the real constants  R_1 , R_2 , R_4 , C_3  and C_4     Answer:      R = ((R_2 C_3  − R_1 C_4 )/C_4 ) ,        L = R_2 R_4 C_3

$$\mathrm{Given}\:\mathrm{that}\:\:\:\mathrm{z}_{\mathrm{1}} \:=\:\mathrm{R}_{\mathrm{1}} \:+\:\mathrm{R}\:+\:\mathrm{j}\omega\mathrm{L}\:;\:\:\:\mathrm{z}_{\mathrm{2}} \:=\:\mathrm{R}_{\mathrm{2}} \:;\:\:\mathrm{z}_{\mathrm{3}} \:=\:\frac{\mathrm{1}}{\mathrm{j}\omega\mathrm{C}_{\mathrm{3}} } \\ $$$$\mathrm{and}\:\:\mathrm{z}_{\mathrm{4}} \:=\:\mathrm{R}_{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{j}\omega\mathrm{C}_{\mathrm{4}} }\:\:\mathrm{and}\:\mathrm{also}\:\mathrm{that}\:\:\:\mathrm{z}_{\mathrm{1}} \mathrm{z}_{\mathrm{3}} \:\:=\:\:\mathrm{z}_{\mathrm{2}} \mathrm{z}_{\mathrm{4}} \:,\:\:\:\mathrm{express}\: \\ $$$$\mathrm{R}\:\mathrm{and}\:\mathrm{L}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{real}\:\mathrm{constants}\:\:\mathrm{R}_{\mathrm{1}} ,\:\mathrm{R}_{\mathrm{2}} ,\:\mathrm{R}_{\mathrm{4}} ,\:\mathrm{C}_{\mathrm{3}} \:\mathrm{and}\:\mathrm{C}_{\mathrm{4}} \\ $$$$ \\ $$$$\mathrm{Answer}:\:\:\:\:\:\:\mathrm{R}\:=\:\frac{\mathrm{R}_{\mathrm{2}} \mathrm{C}_{\mathrm{3}} \:−\:\mathrm{R}_{\mathrm{1}} \mathrm{C}_{\mathrm{4}} }{\mathrm{C}_{\mathrm{4}} }\:,\:\:\:\:\:\:\:\:\mathrm{L}\:=\:\mathrm{R}_{\mathrm{2}} \mathrm{R}_{\mathrm{4}} \mathrm{C}_{\mathrm{3}} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Dec/18

z_1 z_3 =z_2 z_4   (R_1 +R+jwL)(1/(jwC_3 ))=R_2 (R_4 +(1/(jwC_4 )))  comparing real and imaginary part  ((R_1 +R)/(jwC_3 ))=(R_2 /(jwC_4 ))  R_(1 ) C_4 +RC_4 =R_2 C_3   R=((R_2 C_3 −R_1 C_4 )/C_4 )  (L/C_3 )=R_2 R_4     [L=C_3 R_2 R_4

$${z}_{\mathrm{1}} {z}_{\mathrm{3}} ={z}_{\mathrm{2}} {z}_{\mathrm{4}} \\ $$$$\left({R}_{\mathrm{1}} +{R}+{jwL}\right)\frac{\mathrm{1}}{{jwC}_{\mathrm{3}} }={R}_{\mathrm{2}} \left({R}_{\mathrm{4}} +\frac{\mathrm{1}}{{jwC}_{\mathrm{4}} }\right) \\ $$$${comparing}\:{real}\:{and}\:{imaginary}\:{part} \\ $$$$\frac{{R}_{\mathrm{1}} +{R}}{{jwC}_{\mathrm{3}} }=\frac{{R}_{\mathrm{2}} }{{jwC}_{\mathrm{4}} } \\ $$$${R}_{\mathrm{1}\:} {C}_{\mathrm{4}} +{RC}_{\mathrm{4}} ={R}_{\mathrm{2}} {C}_{\mathrm{3}} \\ $$$${R}=\frac{{R}_{\mathrm{2}} {C}_{\mathrm{3}} −{R}_{\mathrm{1}} {C}_{\mathrm{4}} }{{C}_{\mathrm{4}} } \\ $$$$\frac{{L}}{{C}_{\mathrm{3}} }={R}_{\mathrm{2}} {R}_{\mathrm{4}} \:\:\:\:\left[{L}={C}_{\mathrm{3}} {R}_{\mathrm{2}} {R}_{\mathrm{4}} \right. \\ $$$$ \\ $$

Commented by Tawa1 last updated on 25/Dec/18

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 26/Dec/18

thank you...

$${thank}\:{you}... \\ $$

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