I'm trying to chain a high-pass filter and a low-pass passive filter together to form a band pass filter to illustrate how you can combine filters together, but I'm having issues trying to understand it mathematically.
Consider the following schematic:
simulate this circuit – Schematic created using CircuitLab
Conceptually, I'm taking the output of a high-pass filter (with corner frequency formed by L1 and R1), putting it through a low-pass filter (with corner frequency formed by R1 and C1), and taking the output. It should work as a bandpass filter, but when I try to work it out mathematically, I get the following equation (right now, not taking into account actual values but just working it out conceptually):
Let \$Z_1 = R_2 + Z_C\$ and \$Z_2 = Z_L || Z_1\ = ((Z_L*Z_1)/(Z_L+Z_1))\$
\$V_{MID} = V_{IN} * Z_2/(Z_2+R_1)\$
\$V_{OUT} = V_{MID} * Z_C/(Z_C+R_2)\$
\$V_{OUT} = V_{IN} * (Z_2/(Z_2+R_1)) * (Z_C/(Z_C+R_2))\$
\$H(\omega) = (Z_2/(Z_2+R_1)) * (Z_C/(Z_C+R_2))\$
Expanding out transfer function yielded the following:
\$H(\omega) = (Z_LZ_C) / ((R_1+Z_L)*(R_2+Z_C) +R_1Z_L)\$
In this form, it almost mathematically works out that the output of the bandpass filter is just the transfer function of both the LPF and the HPF multiplied together, but that bottom addition term ruins it. From what I've read, I'm not accounting for the effects of load impedance on the output/inputs of the filters, which may be the cause of what I'm seeing here. If that's the case, assuming that \$R_1=R_2\$, what can I do with L and C to make this BPF work as expected?
EDIT: Sorry for not being more specific enough. What I'm looking for in my bandpass filter is to pass frequencies within a certain range, as dictated by the corner frequencies of the HPF and LPF. Relatively straightforward. If you would like some numbers, let's go with corner frequencies of 5 kHz and 10 kHz. Ideally, I imagine my BPF to pass all signals with frequencies from 5 kHz to 10 kHz, and attenuate all signals that fall outside of that range. I'm just trying to demonstrate a very simplistic concept of the bandpass filter by combining filters together.