Transfer Function of 2 Loop RLC Circuit

Question

^{simulate this circuit – Schematic created using CircuitLab}

For a problem I am tasked with finding the transfer function \$G(s) = \frac{V_o(s)}{V_i(s)}\$. I am having trouble defining the 2nd loop in my node voltage analysis since there is no circuit element between the 2 nodes.

These are the current equations that I am working with \$V_1(s)[\frac{1}{sL+R_1}+\frac{1}{sc_1}] = \frac{V_i(s)}{sL+R_1}\$ and then \$\frac{V_o(t)}{\frac{1}{sc_2}+R_2}=0\$

Answer 1

Well, when we have the circuit:

^{simulate this circuit – Schematic created using CircuitLab}

We know that the transfer function is given by:

$$\underline{\mathcal{H}}\left(\text{j}\omega\right):=\frac{\underline{\text{V}}_{\space\text{o}}\left(\text{j}\omega\right)}{\underline{\text{V}}_{\space\text{i}}\left(\text{j}\omega\right)}=\frac{\underline{\text{Z}}_{\space2}}{\underline{\text{Z}}_{\space1}+\underline{\text{Z}}_{\space2}}\tag1$$

And in your case it is not hard to see that:

$$\underline{\text{Z}}_{\space1}=\text{R}_1+\text{j}\omega\text{L}\tag2$$

And:

\begin{equation} \begin{split} \underline{\text{Z}}_{\space2}&=\frac{1}{\text{j}\omega\text{C}_1}\space\text{||}\space\left(\text{R}_2+\frac{1}{\text{j}\omega\text{C}_2}\right)\\ \\ &=\frac{\frac{1}{\text{j}\omega\text{C}_1}\left(\text{R}_2+\frac{1}{\text{j}\omega\text{C}_2}\right)}{\frac{1}{\text{j}\omega\text{C}_1}+\text{R}_2+\frac{1}{\text{j}\omega\text{C}_2}}\\ \\ &=\frac{\frac{\text{j}\omega\text{C}_1}{\text{j}\omega\text{C}_1}\left(\text{R}_2+\frac{1}{\text{j}\omega\text{C}_2}\right)}{\frac{\text{j}\omega\text{C}_1}{\text{j}\omega\text{C}_1}+\text{j}\omega\text{C}_1\text{R}_2+\frac{\text{j}\omega\text{C}_1}{\text{j}\omega\text{C}_2}}\\ \\ &=\frac{\text{R}_2+\frac{1}{\text{j}\omega\text{C}_2}}{1+\text{j}\omega\text{C}_1\text{R}_2+\frac{\text{C}_1}{\text{C}_2}}\\ \\ &=\frac{\text{j}\omega\text{C}_2}{\text{j}\omega\text{C}_2}\cdot\frac{\text{R}_2+\frac{1}{\text{j}\omega\text{C}_2}}{1+\text{j}\omega\text{C}_1\text{R}_2+\frac{\text{C}_1}{\text{C}_2}}\\ \\ &=\frac{\text{j}\omega\text{C}_2\text{R}_2+\frac{\text{j}\omega\text{C}_2}{\text{j}\omega\text{C}_2}}{\text{j}\omega\text{C}_2\cdot1+\text{j}\omega\text{C}_2\text{j}\omega\text{C}_1\text{R}_2+\frac{\text{j}\omega\text{C}_2\text{C}_1}{\text{C}_2}}\\ \\ &=\frac{\text{j}\omega\text{C}_2\text{R}_2+1}{\text{j}\omega\text{C}_2-\omega^2\text{C}_1\text{C}_2\text{R}_2+\text{j}\omega\text{C}_1}\\ \\ &=\frac{1+\omega\text{C}_2\text{R}_2\text{j}}{\left(\text{C}_1+\text{C}_2\right)\omega\text{j}-\omega^2\text{C}_1\text{C}_2\text{R}_2} \end{split}\tag3 \end{equation}