How to set Rin & Rfb to optimize input impedance / Johnson noise of an opamp circuit?

Question

This question follows great answers I got for my previous question about GBW. It teaches me I should use two stages in my microphone preamp project. This leads me to this question:

Considering the following sketch:

How do I choose the type (inverting or not) of opamp and resistor values to get 15dB & low noise without affecting input impedance of 2nd stage. Formulas (like Rfb/Rin) works with 43K/7K5 or 4K3/750R.

I understand that the lower the resistors, the lower the Johnson noise, but what is the ballpark? Does it affect impedance?

Thanks to you, knowledgeable people!

Please note that I don't care about phase revert since I can wire my input backwards.

Answer 1

If you use the first OpAmp as a Buffer (Av=+1) or Inverter(Av=-1), then the 2nd Stage also has to be a low-noise design. Also, adjusting the first gain-set pot will change the noise contribution of that stage and your total noise becomes unpredictable. Another approach is to design the first stage as a Low Noise Amplifier, with fixed gain so you know its output noise, and use the 2nd stage to vary the gain.

Here is a 2-stage fixed gain 20dB+20dB, to show the value of a final LowPassFilter to remove noise from the Rfeedback. The first OpAmp has thermal noise of 50_ohms (0.9 nanovolts/rtHz), one of the quietest OpAmps around. Note the use of 450 Ohms and 50 Ohms, to set the gain. Because of 10:1 gain in 1rst gain, the 2nd stage can be much more casual about KT noise. The final LPF wins us 9dB lower noise.

Here is a brief recipe for LNA. Read opamp datasheet and find the Noise Density. If 4nanoVolts/rtHz, we know the internal Rnoise is a 1Kohm resistor, located between Pin+ and Pin-. If 0.4nanoVolt/rtHz, Rnoise is 10 Ohms. If 40 nanoVolts / rtHz, Rnoise is 100,000 Ohms.

To find Rnoise of the two gain-set resistors (for stage #1), Rg & Rfb, just compute their parallel equivalent value.

Now simply add the OpAmp_Rnoise + gainset_Rnoise, and convert back to nanoVolts / rtHz. Or learn to use the Rnoise as your primary thinking method.

Suppose you have Rg=100 ohms, Rfb=900 ohms, with Gain=10. The gainset_Rnoise is just 100||900 or 90 ohms. Suppose the OpAmp has 4nanoV/rtHz, which is 1KOhm. We find the OpAmp is setting our Johnson Noise floor[our sum is 1090 ohms]. You can insert a discrete bipolar/NJFET preamp with low noise, or buy a Rnoise=60 ohms OpAmp, or just live with the noise.

What about that 2nd opamp, and its noise? We have much flexibility of choosing that opamp, because the 1rst gain stage has 10X gain. With 10X gain, the impact of any noise from stage 1 is reduced by (gain)^2. Thus OpAmp #2 can have 100,000 Ohms Rnoise before its random noise becomes as important as noise from stage #1.

Answer 2

I understand that the lower the resistors, the lower the Johnson noise, but what is the ballpark? Does it affect impedance?

If you look in any op-amp data sheet they will give you decent clues about how much "load" you can connect to the output. The section in the DS that talks about output voltage swing will usually specify load conditions and this might be 10 kohm, 1 kohm, 600 ohms for example.

My general rule of thumb is try and go for the typical figure (if several are specified) and don't load the output more heavily than this.

How do I choose the type (inverting or not) of opamp and resistor values to get 15dB & low noise without affecting input impedance of 2nd stage.

If you use an inverting stage the previous stage will be loaded by the input resistor. The 2nd stage would be operating in a virtual earth configuration and the output of the previous stage would see the input resistor as a load directly to the equivalent of ground. A non-inverting 2nd stage has hardly any loading effects on the first stage until you reach the MHz.