The problem with these opamps is that they actually need a symmetrical supply voltage and all have a minimum supply voltage. The way it is used on The Flea, with a single supply voltage, is not common practise, but by carefully selecting the right parameters, it can be done. So the voltage cannot be too low, or you will run into stability problems. That*s because the inputs have to stay around 3V below the maximum supply voltage, this is called *Common Mode Input Range*. You can look for it in the datasheets.
Also, the output stage needs some room to operate. So if you need 5V output, you need a supply of maybe 7V minimum, depending on the opamp. Some of the boldface types can operate on such low voltages, you could try some of them and see what happens. Maybe if you use a 7805 instead of the 7812, then you*ll get around 7V on the opamp. Then you need 2V drop across the 7805 and 2V drop across the input gyrator. So that will be a total of 9V input voltage, which is half of what it is now
You have to reconfigure the voltage divider that sets the voltage, to get the right output voltage, and run some stability tests with an oscilloscope to see if it remains stable.
Some of the types below can run on lower supply voltages, I looked it up for you in the data, see below.
Type Min. supply voltage Input clearance to V+
AD797A +/-5V or 10V 3V
AD8597A +/-5V or 10V 3V
AD8675A +/-5V or 10V 1.5V
LME49710A
+/-2.5V or 5V 2V
NE5534A
+/-3V or 6V 3V
OPA211A
+/-2.25V or 4.5V 1.4V
OPA227P
+/-2.5V or 5V 2V
OPA1611A
+/-2.25V or 4.5V 2V
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