User:AbyxDev/sandbox

User:abyxdev

$$ \begin{align} z_1 &= R + R_1 + iwL \\ z_2 &= R_2 \\ z_3 &= R_3 - \frac{i}{wC_3} \\ z_4 &= -\frac{i}{wC_4} \\ \end{align} $$

$$ \begin{align} \text{Substitute each term into }\frac{z_1}{z_2} &= \frac{z_3}{z_4} \\ \frac{R + R_1 + iwL}{R_2} &= \frac{R_3 - \frac{i}{wC_3}}{-\frac{i}{wC_4}} \\ (R + R_1 + iwL)\left (-\frac{i}{wC_4}\right ) &= R_2(R_3 - \frac{i}{wC_3}) \\ -\frac{R}{wC_4}i - \frac{R_1}{wC_4}i - i^2 \frac{\cancel{w}L}{\cancel{w}C_4} &= R_2R_3 - \frac{R_2}{wC_3}i \\ -\frac{R + R_1}{wC_4}i + \frac{L}{C_4} &= R_2R_3 - \frac{R_2}{wC_3}i \\ \text{Real part:}& \\ \frac{L}{C_4} &= R_2R_3 \\ L &= R_2R_3C_4 \\ \text{Imaginary part:}& \\ -\frac{R + R_1}{\cancel{w}C_4} &= -\frac{R_2}{\cancel{w}C_3} \\ (R + R_1)(C_3) &= R_2C_4 \\ RC_3 + R_1C_3 &= R_2C_4 \\ RC_3 &= R_2C_4 - R_1C_3 \\ R &= \frac{R_2C_4 - R_1C_3}{C_3} \\ &= \frac{R_2C_4}{C_3} - R_1 \\ \end{align} $$