I'm having difficulties finding complex impedance of the following circuit:
What I did first is find the impedance of \$C\$ and \$R\$ in parallel, which is:
$$\underline{Z}_{CR}=\frac{R}{1+j\omega CR}$$
Then, I added two inductor impedances:
$$\underline{Z}_{CRL}=2j\omega L+ \frac{R}{1+j\omega CR}$$
and finally calculated the parallel of that with capacitor impedance \$\frac{1}{j\omega C}\$ which yields:
$$\underline{Z}_{in}=\frac{\omega CR+2 \omega L(1+(\omega CR)^2)-j(1+2\omega CR^2)}{\omega CR + j(2\omega ^2 LC(1+(\omega CR)^2) -2(\omega CR)^2 -1)}$$
However, I'm not getting the correct result:
$$\underline{Z}_{in_{correct}}=\frac{R(1-2LC \omega ^2) + j2L\omega}{1-2LC\omega ^2 +j2RC\omega (1-LC\omega ^2)}$$
Best Answer
\$\small R//C\$: $$\small \frac{R/sC}{R+1/sC}=\frac{R}{1+sCR}$$
Add series L's: $$\small \frac{R}{1+sCR}+2sL=\frac{R+2sL+2s^2RLC}{1+sCR} $$
Parallel C: $$\large \frac{\frac{R+2sL+2s^2RLC}{sC(1+SCR)}}{\frac{1}{sC}+\frac{R+2sL+2s^2RLC}{1+sCR}}$$
Clear denominators:
$$\small \frac{{R+2sL+2s^2RLC}}{1+sRC+sRC+2s^2LC+2s^3RLC^2} $$
Simplify: $$\small \frac{{R+2sL+2s^2RLC}}{1+2sRC+2s^2LC+2s^3RLC^2} $$
\$\small s \rightarrow j\omega\$:
$$\small \frac{{R(1-2\omega^2LC)+j2\omega L}}{(1-2\omega^2LC)+j2\omega RC(1-\omega^2LC)} $$