Determining DC gain of single supply op-amp (no AC component)

Question

I'm building a homebrew PIC programmer, and want to support devices with varying Vpp voltages, so I'm using the DAC output of a PIC16F1825. But the PIC is supplied with a +5V Vdd, so that's the maximum output of the DAC. I can use an opamp to increase that voltage with an available +12V rail using the following circuit (DAC buffer opamp stage not shown):

^{simulate this circuit – Schematic created using CircuitLab}

What has me mystified is how to determine the voltage gain in this circuit apart from trial and error with R1 and R2 values. Ideally, in this case, I'd like a gain of about 2.5 with signals referenced to ground (the real ground, not a virtual one as is typical in single supply AC amplifiers).

For normal dual-supply AC amps, I have a formula of 1+(R2/R1), but that doesn't seem to be valid here.

So my question is: is there a formula to determine the gain produced with a given R1 and R2 or, conversely, to determine R1 and R2 to produce a given gain?

(I could probably accomplish this with a simple transistor, but since I need the DAC buffer anyway, I have a spare opamp available)

Answer 1

I have a formula of 1+(R2/R1), but that doesn't seem to be valid here.

It's perfectly valid; \$\dfrac{V_O}{V_I} = 1+\dfrac{R_2}{R_1}\$

Think about how an op-amp works when negative feedback is applied; the output, via R2 and R1 sets the inverting input to equal the voltage on the non-inverting input. In doing so the output must be higher because of the potential dividing effect of R1 and R2.

The op-amp is like a servo system. You set a target voltage on Vin and the output finds a level to make -Vin almost exactly match +Vin. Ignoring DC offset errors (a mV or so) the closeness that the output pushes the two inputs together is determined by open-loop gain which, for an op-amp is generally about 1,000,000 (this can be more or it can be less depending on the op-amp chosen).

If R2 is 5 times bigger than R1 then Vout is 6 times bigger than -Vin. If +Vin is 1 volt then the output is 6 volts in order to maintain the two inputs at the same voltage (with an error because of small dc offsets and the op-amp gain not being infinite).