What is the gain equation for an op amp amplifying only white noise?

I have a white noise source (zener diode in full avalanche mode) and I've been amplifying it with an op amp. The op amp is a common device (TL081) with resistors set for a gain of 4. So there's an equation for the gain of -Rfeedback / Rin. Unremarkable, and this works for signals like a fixed frequency sine wave. I've tested this on a breadboard with a 1 kHz sine wave and the formula holds as expected.

But.

The -Rfeedback / Rin equation does not work for amplifying pure white noise. If you amplify a white noise signal, the gain is always a lot less than the formula predicts. In my case a 275mVp-p noise source is only amplified to 455mVp-p using the exact set up as for the sine wave. That's only a gain of 1.7. The standard gain equation fails badly. And the relationship get's worse for higher gains, so for a gain of 20, you might only raise the peak to peak level fourfold.

I encountered this a while ago. Then I theorised that it was due to the Arduino's ADC input impedance being incorrectly specified. I asked Is the ADC input impedance actually 3750 Ohms? and promptly was shown to be a fool :slightly_frowning_face:

I have a better theory, but it's complicated. Please indulge me. White noise has a flat spectrum. So the signal spectrum covers the whole bandwidth of the op amp, 3 MHz in my case. The amplitude of white noise is proportional to the square root of the measurement bandwidth. More bandwidth means more amplitude. That's standard stuff. As you amplify white noise, you hit the gain bandwidth product of the op amp which attenuates the higher frequencies. So the signal amplitude drops. Yet you're still amplifying an ever more attenuated signal. Hence the traditional gain equation doesn't work as it makes no account of the op amps' bandwidth.

Does this make sense? I think I might be right this time, but I'm looking for educated opinions and perhaps a gain equation for noise only amplification...

cossoft:
So the signal spectrum covers the whole bandwidth of the op amp, 3 MHz in my case.

Slight correction to make here. 3 MHz is the unity gain bandwidth. It's also called the gain-bandwidth product (product as in multiplication). Increasing the amplifier's closed-loop gain decreases the bandwidth proportionally, so your actual bandwidth here is 750 kHz. For a gain of 20, it's 150 kHz.

How are you measuring the noise magnitude? What is the bandwidth of your source and measuring device? If your source and measurement bandwidth are too high, it's very likely you're running afoul of the op amp's natural low pass filter.

Every real (practical) circuit having an input and an output is a filter ..... of some sort.

Real filters have ranges of frequency for which the filter has particular defined behaviours... like bandpass, bandstop, highpass, lowpass.

White noise....as you correctly say.... has a noise power spectral density value that is constant (versus frequency).

A practical op-amp amplifier's bandwidth is limited. So it's not going to amplify the amplitude (across an limited frequency range) by the same amount.

Checkout gain bandwidth product details.

click here....

Hi,

I don't believe OpAmps lie, or exaggerate their own wonderfulness. I find electronics a welcome relief from Politics.

So, (Theoretically Perfect) Opamps amplify according to the values in their feedback circuit.

BUT Reality enters the room.

If the Opamp you are using has a bandwidth less than your measuring device(s) it will appear to have less "gain" for (True, actual) White Noise. If the "real" feedback circuit of your opamp has capacitances or inductances, that will cause some frequency vs gain dependence, so again your measurement equipment may see a difference.

So you may have been attacked by CNN (Capacitance Not Negligible) or other 3-letter annoyances.

Southpark

Exactly, so is there a gain equation? I'm not the first person in the three Universes to have done this. All white noise sleep machines and random number generators must have this issue. But my most important question is whether this is a plausible explanation?

Jiggy-Ninja

I don't know what the diode's signal bandwidth is - I don't have a signal analyser. It must be lot's more than 10 MHz. It shows up on my oscilloscope down to 500 ns/div. The measuring device is a Rigol @ 100 MHz but that doesn't really seem to matter. I usually run it limited to 20 MHz and this makes no difference to breaking the traditional gain equation.

cossoft:

Southpark

Exactly, so is there a gain equation? I'm not the first person in the three Universes to have done this. All white noise sleep machines and random number generators must have this issue. But my most important question is whether this is a plausible explanation?

Audible noise is quite a bit lower in frequency than what you're dealing with here.

Jiggy-Ninja

I don't know what the diode's signal bandwidth is - I don't have a signal analyser. It must be lot's more than 10 MHz. It shows up on my oscilloscope down to 500 ns/div. The measuring device is a Rigol @ 100 MHz but that doesn't really seem to matter. I usually run it limited to 20 MHz and this makes no difference to breaking the traditional gain equation.

So the problem isn't with the gain equation, it's that you are using the op amp far past its frequency limitations. The 20 MHz bandwidth of the scope front end is still a lot more than the 750 kHz bandwidth your op amp circuit has. All op amps have frequency limitations. You need to keep your input signal within that limit if you want the simple gain equation to stay valid. Otherwise you have to consider it a low pass filter.

Try putting a low pass filter on your noise source to limit it to something like 100 kHz, and you'll probably get more sensible results comparing the input to the output. If you really want to amplify stuff up in the tens of MHz range, you need to find an op amp with high enough GBWP, probably 100 MHz or more.

Let me rephrase then...

I have a white noise source which measured at 20MHz bandwidth is 275 mVp-p.
I want 1 Vp-p at 20KHz so that I can hear it.
What gain should I set on my TL081 op amp..?
3.6 is not the answer.

Put a 20 kHz low pass filter on your noise source, and measure the magnitude of the output of that. It will likely be significantly smaller than 275 mV. Use that measurement to calculate the gain you need.

4.7 k ohm resistor and 1.8 nF capacitor should work by my calculations.

cossoft:
Let me rephrase then...

I have a white noise source which measured at 20MHz bandwidth is 275 mVp-p.
I want 1 Vp-p at 20KHz so that I can hear it.
What gain should I set on my TL081 op amp..?
3.6 is not the answer.

You can set the gain to whatever you want, as long as the region of frequency that you're working in is within your closed loop bandwidth.

To measure the closed loop bandwidth ..... set whatever gain you want at low frequency.... eg...gain of 5, or 10, or whatever. Then set a constant amplitude sinusoid for the input at low frequency. Then increase the frequency....and measure the closed loop gain as a function of frequency. This gain 'map' versus frequency will tell you something about the frequency range for which your 'constant' gain will stay 'relatively' constant.....provided that you don't work above a certain frequency.

Your gain map (closed loop gain versus frequency) will have a -3 dB point.... where the gain (relative to your low frequency gain) will be 0.7071, or -3 dB. Record this frequency....and call it f_3dB.

The equation you're talking about (or after) will look like......

Closed loop Gain of your op amp circuit at frequency f in Hertz = MAGNITUDE OF { closed_loop_DC_gain * (2pif_3dB) / (j2pif + 2pi*f_3dB) }

Closed loop DC gain is the closed loop gain at relatively low frequency. There is the 'j' or imaginary operator hanging in the equation.

The important thing is..... if the low frequency gain you started with is significantly reduced..... and if you do the same bandwidth test....... then the you will get higher bandwidth when smaller gains are used.

White noise is measured as nV/sqrt(Hz), which is just another way of saying the power per unit
spectrum is constant.

Noise doesn't have a true peak-to-peak value (the longer the timescale you look at the noise signal the larger the
peak-to-peak range becomes - if you notice every peak that is)

Use rms noise measurements if you want accuracy, that is well defined.

If you have x volts at 20MHz, you'll see about x/32 volts at 20kHz,
so you need a gain of 115 for that 275mV source as its actually only producing 8.7mV in the audio range.

MarkT:
...rms noise measurements...

Isn't that just Vp-p /6.6? Therefore entirely proportional the the peak value?

I'd very much love to know how to measure it as V/sqrt(Hz) but I can't figure out how with only a budget digital oscilloscope. I don't think that it can be done without other specialised kit like perhaps a spectrum analyser. The really annoying thing is that I can physically see it with my own myopic eyes. Why can't I measure something that I can actually see? >:(

cossoft:
Isn't that just Vp-p /6.6? Therefore entirely proportional the the peak value?

I'd very much love to know how to measure it as V/sqrt(Hz) but I can't figure out how with only a budget digital oscilloscope. I don't think that it can be done without other specialised kit like perhaps a spectrum analyser. The really annoying thing is that I can physically see it with my own myopic eyes. Why can't I measure something that I can actually see? >:(

Bandpass filter and an RMS measuring AC Voltmeter.

If your digital scope can export waveforms, you might be able to get a value with analysis software. How accurate do you need to be?

cossoft:
Isn't that just Vp-p /6.6? Therefore entirely proportional the the peak value?

No, thats a fairly arbitrary way to define pk-to-pk in terms of rms - pk-to-pk just isn't
well defined for a noise signal, that same way the maximum wave height on a beach isn't
well defined - different days the maximum will vary, despite the same random distribution of
waves.

The rms value is the most well defined as it samples the entire waveform, not just some
extreme values, so more rapidly converges to a stable value when measuring noise.

Your scope will have a FFT facility - what does that show?

Allan.

Allan, to tell you the raw, naked and embarrassing truth, I don’t know. It’s this type of display:-

but I’m not sure what it’s saying. This one is copied of the internet. Mine’s the same scope and useless display, with a peak value of 600mV on the left @ the 0 Hz point, which then rolls off to 0V on the right @ 3MHzish. And to complicate things, the x scale and bandwidth of the FFT changes in proportion to the horizontal time base.

Clearly V/root(Hz) is a singular quantity, not a curve, so…?

But this distraction doesn’t help with amplifying it. I’d have thought that any op amp gain equation must have to take into account the bandwidth product of the particular device. Perhaps I’m misunderstanding badly. I’m starting to suspect it’s too complicated and will have to judge it experimentally… :frowning:

What you have there is a odd-function square wave. And the fast fourier transform provides frequency info. The lowest frequency spike is the square wave's frequency. Then there are the spikes at frequencies that are ODD integer multiples of the lowest non-zero frequency.

Ah :slight_smile: That's a grab from the internet to illustrate the FFT function on a basic Rigol. I did that as I can't post my own screen grab due to the weirdness with images on this forum. I tried to textually describe my FFT display, and clearly did a poor job. I have a peak value of 600mV on the left @ the 0 Hz point, which then rolls off to 0V on the right @ 3MHzish. This question has always been about white noise (or specifically avalanche noise), and it's unpredictable amplification.

Can’t you just take a picture of the screen of your scope with a camera and post the photo?

I don’t think experimenting with the FFT function is a distraction as you said before. Using the FFT allows you to determine whether or not your input signal really is AWGN, the input signal bandwidth, etc. If you connect the scope to the output, it will show you the bandwidth of the output signal (not to mention the gain of the signal at different frequencies compared to the FFT of the input).

I think in order to measure the gain of an AWGN signal, you need a front end filter (as already mentioned by another forum member) that feeds to the amplifier circuit and then graph both the output of the front end filter AND output of the amplifier traces in the time domain with your scope. You can do some quick calculations from there to determine the gain.

cossoft:
I’d have thought that any op amp gain equation must have to take into account the bandwidth product of the particular device. Perhaps I’m misunderstanding badly. I’m starting to suspect it’s too complicated and will have to judge it experimentally… :frowning:

Quite the opposite. The formulas are made based on calculating the behavior of an ideal op amp with the following assumptions:

  • VIN+ = VIN-
  • IIN+ = IIN- = 0
  • VOUT = anything

Real op amps have an input offset voltages and input bias currents. They have bandwidth and slew rate limitations that slow down how fast the output can change. The equations will be accurate enough if you use the op amp properly, well within its pass-band, and with component values where the small input imperfections aren’t a big deal.

There are ways to analyze op amp behavior outside the passband, but it’s been a while and I didn’t bring my network analysis textbook to my parent’s house this week. You don’t need it anyway.

It’s not too complicated. I and other posters have been telling you from the beginning what you’ve been doing wrong. If you’re just concerned with audible frequencies, filter out all the high frequency crap so that it stops confusing you. You don’t need all the signal up to 3 MHz, and it’s outside what the op amp can handle. GET RID OF IT. That’s what filters are for.

Okay, Okay. You can stop shouting. :slight_smile: If you don't know the equation then it's fine to say so, or even to not say anything - I won't mind. Filtering down a signal prior to amplifying it so that I can shoehorn it into an inappropriate equation isn't what I was looking for. I'm clearly not explaining myself adequately. Perhaps I'll try Stack Exchange...