# Transfer function of a R+RLC low pass filter

Here's a picture of the R+RLC circuit:

I'm trying to find the frequency response of this low pass filter, my solution is below and please let me know if there is anything wrong with it.

My solution: $$\\ x=V_{in} \\ y=V_{out} \\ x-y = (i_1+i_2+i_3)R_1 \\ y=i_1R_2 \\ y=Li_2'\\ i_3 = Cy' \\ x'-y' = (i_1'+i_2'+i_3')R_1 \\ x'-y' = (\frac{y'}{R_2} +\frac{y}{L} + Cy'')R_1 \\ LR_2x' = R_1R_2y + L(R_1+R_2)y' + LR_1R_2C y'' \\ x(t) = e^{st} , y(t) = H(s)e^{st} \\ H(s) = \frac{LR_2s}{R_1R_2+L(R_1+R_2)s + LR_1R_2Cs^2}$$

$$H(s)=\frac{Z||}{Z_1+Z||}$$ $$Z|| = \left(\frac{1}{R_2}+\frac{1}{Ls} +\frac{1}{(1/Cs)}\right)^{-1} = \frac{R_2Ls}{R_2LCs^2+Ls+R_2}$$ $$H(s)=\frac{R_2Ls}{R_1R_2CLs^2 + (R_1+R_2)Ls+R_1R_2}$$
There is a fast and clean way to get there by using the fast analytical circuits techniques or FACTs. If you consider natural time constants of the circuits obtained with the input source is zeroed (replace $$\V_{in}\$$ by a short circuit), then you can determine the transfer function swiftly without writing a single line of algebra. You can rearrange the final expression to make it fit a low-entropy format, a term forged by Dr. Middlebrook. The below pictures show the steps for the time constants:
You can see that for $$\s=0\$$ the gain is 0 indicating the presence of a zero at the origin. Then you temporarily disconnect the energy-storing elements and "look" through their connecting terminals to determine the resistance $$\R\$$ driving the considered element. For $$\C_2\$$, $$\R=0\$$ and the time constant is 0. For $$\L_1\$$, the time constant involves the parallel combination of $$\R_1\$$ and $$\R_2\$$. Then you calculate the gain $$\H^1\$$ when the inductor is set in its high-frequency state (an open circuit). Finally, you combine all these time constants as in the below Mathcad sheet. Follow the guidelines from the book and rearrange the equation to unveil the peaking gain, a quality factor and a resonant frequency. This is the ultimate goal when deriving a transfer function.