Background
I'm working on a circuit involving an AC-excited sensor bridge, and am trying to figure out ways to detect a short between the two bridge legs, as that causes the bridge to fail to the "wrong side" (i.e. indicate all is well, when the sensor says all is not well). One idea I had for this was to bias the in-amp that measures the bridge into its operating range from the input side, so that shorted inputs would cause the output common-mode voltage to shift dramatically, rendering the failure detectable by downstream circuitry.
The problem
So, I set to design a simple bias network to do this task:
simulate this circuit – Schematic created using CircuitLab
From this design, I went to solve the circuit for R by treating it as two voltage dividers with equations as follows:
$$V_1=5\frac{R+1M\Omega}{R+2.23M\Omega}$$ $$V_2=5\frac{1M\Omega}{R+2.23M\Omega}$$
However, when I plugged in a V1 of 2.2625V and a V2 of 2.2375V (2.25V +/- 12.5mV) and went to find my unknown R by treating the equations as a system, I was unable to come up with a solution, either by hand calculation or using Wolfram Alpha. Initially, this was because I had R1 set to 1MOhm, which was correct for an original center voltage of 2.5V, but wrong for the current center voltage of 2.25V. So, I solved a simple voltage divider for that center voltage and transposed the R1 I found (1.23MOhm, in E192) into my working circuit, but that didn't work either -- I'm getting far closer now, but it seems like I may have run out of resistor precision before I can get the small differential voltage drop I want?
