How does R4 prevent the output voltage drifting towards one of the supply rails to the op amp? I understand that R4 has to have a high resistance but I do not know why?
Here is the schematic which is puzzling me:
The link to the CircuitLab schematic:
https://www.circuitlab.com/circuit/x66cq6/basic-frequency-control-circuit/
Without R4, you do not have any DC feedback because the only negative feedback path is through a capacitor. Without DC feedback, you don't have a stable DC operating point.
If R4 has a low resistance (let's say, for the sake of argument, zero). In this case, all the negative feedback goes through R4 (path of least resistance) and the R2/C2 branch of the circuit does nothing.
Intuitively, though, the 4.7 Megohm choice for this resistor seems quite high. For this to work well, you need an op-amp with a very high input impedance (e.g. JFET input).
The idea in this circuit is that R4 conveys the DC output voltage from the output of the op-amp to the non-inverting input, and the impedance at that input is so high that it draws nearly zero current, allowing such a high valued resistor to be used.
Even without doing any calculations, I think a much lower resistance for R4 would still work there.
With R4 removed, there is no DC feedback path from the op-amp output to the input. So if the b node drifts away from ground, there's nothing the op-amp can do to drive it back towards ground (which it will try to do to keep the two inputs equal). If b drifts high, the output will tend to rail negative, and if b drifts low the output will tend to rail positive, according to the open-loop gain of the op-amp.
With R4 in place, you have a DC feedback path. If b drifts high, the op-amp output can go a little bit low and pull it back to ground. If b drifts low, the op-amp output can go a little bit high and pull it back to ground.
Put in more jargonistic terms, with R4 removed the DC circuit is an open-loop amplifier. With R4 in place, the DC circuit is a voltage follower.
I agree with what the others have said about what R4 does in the circuit. However, I would put it accross C2 only, not where it is now. That way the high pass filter rolloff frequency it causes won't change with other parameters in the system. For example, if you wanted the high pass rolloff to be 20 Hz, then a resistor just accross C2 would be 8.0 kΩ, so 10-15 kΩ would be good if this is audio.
Having the high pass filter rolloff too low causes long startup transients and glitches. You want to block frequencies below what you care about, but you also don't want the time constant to get to steady state operation to be too long. 1 µF and 4.7 MΩ is a time constant of 4.7 seconds, and it could take multiple time constants before the system settles to steady state operation. That's definitely unacceptable if this is a ordinary audio device.
Your Op Amp has > 100dB of DC gain and your design will integrate to the supply rails every time, because you have almost designed an "Integrator"
added An interesting idea has come out of your oversight. If capacitors C1 & C2 come from the same batch and are matched, they may have the same leakage and thus your DC output gain will be unity but with a really large RC time constant depending on the leakage values. If one wanted to perfect this oblique application, consider Panasonic Polyurethane plastic caps. Now any change in leakage of C2 will result in a gain change, likely in orders of magnitude, so it has in fact become an extremely sensitive uA meter or leakage meter, which has all sorts of applications such as smoke detectors. Using C1 itself as an instrument probe, your tester with,C2 as your matched benchmark for leakage, you now have a very sensitive Leakage controlled DC gain for measuring changes in leakage. You would have to periodically dump the charge on C2 to null the integrator. You might consider a leakage detector useful for measuring soil moisture or fluid contamination measurements. Of course there other ways to instrument this, but this method wil match values on C1,C2 has an interesting unity gain condition for DC. an Offset adjustment pot would be useful across the +/- inputs to Vdd. YOu could consider this as an alternative for those expensive medical devices that measures the leakage in plasma using an OmegaMeter or leakage in distilled water from contamination, or looking at the leakage of fresh solvents after the last rinse to verify contamination free of aerospace exposed material in a clean room. Either way, you have nullified the huge DC gain now by matching component values with extreme low loss Polyurethane caps or better yet Teflon Caps but made it sensitive to leakage changes if you expose one part C1 to the test condition. Using it as a GO/NoGo tester one can use a small DC voltage, say 1.5V to force a leakage no go test relative to a fixed resistance across C2 such as 100MΩ or 1000MΩ. If the output saturates in the opposite direction ( biased by external battery source ) One can say the new leakage across C2 probe meets or fails the defined reference Rf test criteria. a.k.a Wheatstone bridge with gain of 10 billion or more comparator to reference resitor Rf. You would also want to null common mode noise so the Op Amp's output could be used as a shield or twisted pair.
Of course the prefered design with bypass feedback resistor could also work across C2, it just becomes a HPF pole that is affected also by gain control Pot, not recommended.
