Why is the noise gain of an op amp always constant at $$1+\frac{R_1}{R_2}$$ and independent of configuration, unlike signal gain?
Answer
There are two unwanted opamp properties - effective at the input nodes of the opamp: Noise and input offset voltage. When the opamp has negative resistive feedback, both voltages (noise, offset) are amplified with the factor (1+R2/R1).
This is obvious (typical non-inv. gain formula) if we allocate these voltages to the non-inverting terminal only (assuming no signal input voltage at this node). However, it is easy to show by calculation that the result is the same if such an unwanted noise or offset source is located directly at the inverting terminal (between the inv. input and the common node where the feedback resistors meet).
Calculation (edit): Assuming no signal input (non-inv. input node and R1 grouded) and a finite output voltage Vout, the corresponding voltage at the common point of R1 and R2 is Vo=Vout*R1/(R1+R2). Solving for Vout/Vo=1+R2/R1 gives the relation between Vout and a corresponding voltage Vo between the common point and the "virtual ground" potential at the inverting input node.
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