Electrical – Determining the Transfer Function of a Wheatstone Bridge

Here is a 5 second answer to start. First, put a decoupling cap (0.1uF or something like that) between LM324 pins 4 and 11 (its just good practice). Just for testing connect LM324-1 to LM324-2 (unity gain), because an OpAmp is meant to be used with feedback.

Of course you know that you will need a differential amp sensing both sides of the bridge to get any advantage out of using the bridge.

Edit

Thanks for adding the schematic.

The output differential stage, as drawn is positive feedback. So, you will need to exchange pins 9 and 10 of the LM324 connection to have negative feedback.

Also, I didn't think about this much earlier, but the 4th channel would be best to be connected as a voltage follower with the (+) input connected to return. I know that each channel is supposed to be independent, but they are not quite really. The die will have parasitic structures that can affect all the channels. For example, parasitic SCRs are common in many monolithic ICs. So, it is always best to have all channels connected.

I am sure that they can be implemented with linear elements: resistors, inductors and capacitors.

I wouldn't be so sure about that. A trivial counter-example is \$H(s) = K, K > 1\$.

A less trivial counter-example is any \$H(s)\$ with RHP poles or any with a finite impulse response. RLC circuits cannot implement an arbitrary transfer function.

Transfer functions can be written as the product of 2nd order transfer functions (for even order) and as the product of 2nd order and 1st order transfer functions (for odd order).

Thus, as Andy aka answers, realizable transfer functions can be implemented as a cascade of well known 1st and 2nd order active filter circuits.

Now, if one wants to implement a realizable transfer function entirely with passive components, you cannot use the above method. In the case of active filters, there is no (or actually very little) interaction between the active filter sections due to very low output impedance of the op amp. However, with passive filter sections, there is interaction (loading effects) the make the choice of topology and components exceedingly complex.

The synthesis techniques of passive ladder filter networks for the canonical filter alignments is well known (see, for example) but, for arbitrary, realizable transfer functions, I don't know.