I want to find the pole associated with this circuit design
KCL for the node is
$$
\begin{align}
i_{in} ={}&amp; i_1 + i_2\\
={}&amp; \frac{v_o}{R + 1/sC_1} + \frac{v_o}{1/sC_2}\\
={}&amp; v_o (\frac{1}{R + 1/sC_1} + \frac{1}{1/sC_2})\\
={}&amp; v_o \left(\frac{R + 1/sC_1 + 1/sC_2}{(R + 1/sC_1)(1/sC_2)}\right)\\
\end{align}
$$
Re arrange this equation to find \$\frac{v_o}{i_{in}}\$. The last line of the above equation set gives a clue to why the pole location is dependent on series or parallel combination of the elements.
Even though the circuit contains two energy storing elements, there is only one pole. To see why, think of the case where the input is fixed as 0 (figure below). Then we can see that the three elements \$R,\ C_1,\ C_2\$ are in series and there is effectively only one capacitance which is the series combination of \$C_1\$ and \$C_2\$. Note that this type of simplification may not be applicable if this was part of a bigger circuit and some other currents were entering or exiting this circuit.



simulate this circuit &ndash; Schematic created using CircuitLab