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INA126: INA126P or other instrumentation amplifier for single supply amplifier of a signal between 0V and 10mV

Part Number: INA126
Other Parts Discussed in Thread: INA122, , INA326


I need to amplify a pyrheliometer signal ranging from 0 to 10mv, in the range of an STM32F429ZI ADC that is 0 to 3.3V .

We already have produced some PCB where we have a single 5V supply so INA126P is supplied between 0V and 5V.

Using some resistor I've added 2.5 volt to my signal on Vin and 1.25V to Vref according to following schematics:

It is almost OK, maybe it is still possible to add 4 flying resistors on the PCB, but I've the following questions:

1) why close to zero is not linear

2) how I can determine the transfer function in this case

Best solution for me could be to use an instrumentation amplifier with same pin but that is single supply,

maybe like INA122, but simulating a similar circuit the result is not good and I can't understand why

so, my question is where is my error in this circuit with INA122

and also if you know any other TI instrumentation amplifier that could be suitable for this application.