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Hello Peter,
You can use the INA129 with a single 5V supply, but please understand that the linear operating region (Vcm vs. Vout) will be very limited due to the internal amplifiers' common-mode and output swing limitations. I recommend reading the articles in the following FAQ and downloading the Vcm vs. Vout calculator.
Vcm vs. Vout FAQ:
INA Common-mode Calculator download:
http://www.ti.com/tool/ina-cmv-calc
An alternate part that you may consider is the INA326. The INA326 has a unique topology that enables RRI/O operation.
I used the Vcm calculator and it looks like at +5Vin and gain of 1 V/V, Vout and Vin will only be linear from 1.87V max to 0.8V min with a common mode voltage of 2.66V. I'm a little confused about why the CM voltage bar only lets me go from 2.13V to 3.2V. Why can't I set the CM to 0V?
I think INA188 might be a better option. The datasheet specifically says it can operate with a single supply and goes rail to rail.
The INA common-mode calculator does not have INA326 part loaded. Is it because it's RR I/O and therefore doesn't have a plot?
I did a simulation in Tina with the INA326 in the circuit I plan to use and it behaves like I expected. But I also did the simulation using the INA333 instead, and it works the same. Vcm is 0V. Vdiff is less than 1V. V+ is +5V, V- is GND, and Gain = 1, Ref = 0.