Bode plot of circuit with unknown values
Boy are you up for a treat...

If you are only searching for the right configuration, you can probably get away with roughly estimating the poles and zero's in the circuit. If you calculate correctly, your first circuit leads to:

$$TF(s) = \frac{R_2C_1s}{1+(R_1+R_2)C_1s + L_1C_1s^2}$$

For the second circuit:

$$TF(s) = \frac{R_2C_1s(R_1+L_1s)}{R_1 + (C_1R_1R_2+L_1)s + (R_1+R_2)C_1L_1s^2}$$

The second circuit has one extra zero. If this zero is absent, the first circuit is the one you want.

If you additionally want to find the coefficients of the transfer function, you'll have to work a bit more. It is technically not hard if you have numerical software available (eg. Matlab), but getting there is much harder.

Given a rational transfer function:

$$\hat H(j\omega)=\frac{\sum_{i=0}^mb_i(j\omega)^i}{\sum_{k=0}^na_k(j\omega)^k} = \frac{B(j\omega)}{A(j\omega)}$$

Where we choose \$a_0 = 1\$ (normalization). We can compare our model with the measured data (possibly noisy) to find the error we are making. We take a few frequency points \$\omega_i\$, and measure the frequency response \$H_i\$. The error our model makes can be estimated using the expression:

$$e_i = \lVert B(j\omega_i) - A(j\omega_i)H_i \rVert^2$$

$$e_{tot} = \sum_i e_i = \sum_i \lVert B(j\omega_i) - A(j\omega_i)H_i \rVert^2$$

Our job is then to minimize this error by choosing \$a_n\$ and \$b_m\$ carefully. This can be calculated by computing the partial derivatives to \$a_n\$ and \$b_m\$, which have to equal 0.

So for your case:


Construct the matrix \$H\$ with all your measurements:


$$H = \left[ \begin{matrix} H_1 \\ H_2 \\ \vdots \\ H_N \end{matrix} \right]$$


Construct the matrix \$U\$, which can compute estimates by multiplying with the unknown parameters \$\theta\$:


$$U = \left[ \begin{matrix}
-j\omega_1 H_1 &amp; -(j\omega_1)^2 H_1 &amp; 1 &amp; j\omega_1 &amp; (j\omega_1)^2 \\
-j\omega_2 H_2 &amp; -(j\omega_2)^2 H_2 &amp; 1 &amp; j\omega_2 &amp; (j\omega_2)^2 \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots \\
-j\omega_N H_N &amp; -(j\omega_N)^2 H_N &amp; 1 &amp; j\omega_N &amp; (j\omega_N)^2
\end{matrix} \right]
$$


Compute the best estimate under least-squared error: \$\hat \theta\$.


$$\hat \theta = (\Re\{U^*\cdot U\})^{-1}\cdot \Re\{U^*\cdot H\}$$

Where

$$\hat \theta = \left[ \begin{matrix} a_1 \\ a_2 \\ b_0 \\ b_1 \\ b_2 \end{matrix}\right]$$

(\$\Re\$ means real part, \$\ ^*\$ means conjugated)

which you can plug in the transfer function:

$$H(s) = \frac{b_0 + b_1s + b_2s^2}{1 + a_1s + b_2s^2}$$

If you are interested in how you get there: check here

These coefficients can then be used to find your component values.

Measuring the frequency response

Sweeping the frequency can work, but is not always a good idea as you always have to wait for transients to die out. Another way is by cramming in all frequencies at the same time, in which case you only have to wait only once for all transients to go away.

If you choose a random-phase multi-sine excitation signal, you can measure the input and output at the same sample rate and simply divide the output spectrum by the input spectrum. Please take care to avoid aliasing, eg. by an AA filter though.