Are these low frequency FFT components due to insufficient op-amp slew rates?

Question

In my circuit I have three op-amps in (direct) series connection:

1. AD8422 acting as an input buffer (a large input impedance is needed) with a fixed gain (slew rate: 0.8 V/us)
2. LTC6910 being used as a PGA (slew rate: 16 V/us)
3. AD8476 acting as ADC driver for single-ended to differential conversion (slew rate: 10 V/us)

All op-amps have a +5 V to -5 V dual supply. I directly inject the following (single-ended) signal into the first OP amp:

It is a 50 Hz sine wave with a peak of 1 V (2 Vpp) and in addition to that steep pulses (representing corona discharge) at both peaks of the half waves (approx. 1 Vp), leading to a total of approx. 4 V peak-to-peak. According to an oscilloscope reading, the rise time of each corona pulse is approx. 50 V/us (1V/20 ns) and the width is approximately 1 us each.

The ADC has a very steep analog and digital filter at around 9 kHz: What I do not understand is that with this input at the ADC (yellow) my DSP behind the ADC tells me that my signal peaks are +- 4 V (8 Vpp) -> hence the Corona signal is not filtered out! Why is this?

On the ADC signal I perform an FFT. When I add these corona pulses, I get a significant contribution to 150 Hz amplitude in the FFT output. When I have corona pulses in only one half wave, I get 100 Hz contribution.

At first I thought the real-life corona pulses may overdrive my front end concerning signal magnitudes (railing of input op-amp,) but from my test described above, I know that 4 Vpp is well within the output range of all op-amps (and input range of my ADC: 8 Vpp.)

Question:

Can it be that a too slow slew rate of one or multiple OP amps in my analog front end causes the modulation of a low frequency (150 Hz) signal (some kind of OP amp dynamic saturation due to the pulses?) If this is the case, what would help?

Answer 1

This would be expected. Your Corona discharges are not random, they occur near the positive and negative peaks of the sine, and they always add to the amplitude. So, the Corona pulses have a 150 Hz component to them.

Answer 2

Assuming each pulse is fast enough to look like a Dirac pulse, its spectrum is a flat constant across all frequencies. Realistically, it won't go to "infinity Hertz" but you can expect a wideband spectrum.

Now imagine a train of pulses happening at random times. The spectrum of that is the sum of spectra of the individual pulses, taking into account the time when each pulse occurs.

In the frequency domain, a time delay is equivalent to multiplying the whole spectrum by \$ e^{j\omega t} \$.

So the spectrum of the above signal is the sum of \$ e^{j\omega t_n} \$ with \$ t_n \$ being the arrival time of each pulse, which is random.

On average, this spectrum also has constant power across frequency. In the frequency domain, the sequence of random pulses above is indistinguishable from white noise. Signal on top, spectrum on the bottom:

Now let's multiply this pulse train with the gating signal below:

This gating signal is 50Hz periodic, so it has harmonics of the 50Hz fundamental. If the two pulses are symmetric, it'll only be odd harmonics, but if they are not you'll get a mix of even and odd. This is what you're seeing here:

When I add these corona pulses, I get a significant contribution to 150 Hz amplitude in the FFT output. When I have corona pulses in only one half wave, I get 100 Hz contribution.

Now let's multiply the pulse train and the gating signal.

Time domain multiplication is frequency domain convolution, so the spectrum is the convolution of the pulse signal spectrum (white noise) and the gating signal.

And the result is... white noise.

The question you're not asking is: why aren't you getting energy at all the other frequencies, say between 100Hz and 150Hz?

If you're only doing a 1-period FFT then these frequencies are not in the result so it's expected, but if you're using more than 1 period you should be getting lots of noise in all the bins. It's absolutely normal, that's what your signal actually is.

Can it be that a too slow slew rate

No, but if you want to measure the energy of these pulses, then the opamp must not slew limit. If it does, then it will miss a portion of the pulse:

Black is the input pulse. Blue is the output of the opamp, with severe slew limiting. In this case it is not possible to know what the amplitude of the pulse was, because the opamp never got there.

Low pass filtering averages the signal, it is linear, so the information you want to measure is preserved within the filter bandwidth. Slew rate limiting ignores the signal so whatever happens during slewing is simply lost information.

So you should definitely do enough filtering before your opamps to make sure they don't slew limit.

But that won't remove the 100Hz and 150Hz components, because these are actually your signal.

Answer 3

If your corona pulses were bipolar, that is added signals above and below the 50 Hz waveform, then you would not expect to see 150 Hz signals.

However, your corona pulses are only positive during the positive half cycle, and only negative during the negative half cycle. This only adds to the amplitude, and so looks like odd harmonic distortion of the fundamental 50 Hz waveform. It's the opposite of what happens to a mains waveform that gets clipped in voltage by all of the rectifier loads on it drawing current, which adds odd harmonics, principally the 3rd. As the additional amplitude is symmetrical, there are no even multiples of 50 Hz.

A similar thing occurs when you have corona pulses only on one half of the waveform. You now have an asymmetric pulsey waveform at a 50 Hz fundamental, which has all harmonics, even and odd.

It would be quite easy to see that the corona pulses have this energy at harmonics of 50 Hz by zeroing the amplitude of the 50 Hz sinewave, while leaving the pulses present. FFT, and you'll see the harmonic energy.

Answer 4

In addition to existing answers (bobflux's in particular), there is another effect that may be worth considering.

Many op-amps, and other semiconductor circuits, are susceptible to RF rectification effects.

About Bandwidth

It's not perfectly clear just how wideband your signals are -- for example, if you merely inspected them at a slow sweep rate (on the oscilloscope), you might get only one sample during a given peak, and considerable aliasing (most of the peaks missing, some runted, few resolved to full amplitude).

It's reasonable to suspect this, because gas discharge phenomena in general can be extremely fast. For example, ESD has a risetime under 1ns. It's quite possible the individual corona spikes are similarly fast (and far taller), and you're already seeing either a smoothed out, or aliased, mush of what's really there.

It's also possible you've ruled this out, having inspected the signal at high sample rate (with adequate scope bandwidth, and controlled-impedance probing technique, etc.); but given the data presented here, the above remains a possibility.

(On that note, the edit including the oscilloscope waveform was made while composing this answer. Notice the far lower density of peaks compared to the original screenshot. This seems to further confirm my above concern.)

Rectification

As for this, the general effect is, a semiconductor junction gets forward-biased (or avalanche too) during the peaks, causing signal clipping, but also an unexpected current flow, which is modulated by the sum of intended (here, the sine wave) and noise (the corona interference*) components. The current pushes against the baseline value of the signal, shifting it.

Thus, a mixing effect occurs, and harmonics and intermodulation products are generated. Which includes signal-frequency and DC components. Which can even be exaggerated, by knock-on effects like charge storage and reverse recovery.

*Well, maybe it's intended too, but point being it's a very random signal, and constitutes significant EMI and RFI if present in a real-world environment.

Besides direct mixing, nonlinear current flows can result, for example diode junctions charging the capacitance of internal gain nodes during the peaks, which are restored to normal levels only gradually by internal bias currents. An op-amp's input transistors (or elsewhere internally) can act like capacitor-input rectifiers, causing the internal nodes to follow the peak voltage of the signal. Thus, an incident RF envelope can manifest as an input bias voltage.

Op-amp slew rate can also be asymmetrical, so that when repetitive limiting occurs, the output (or internal gain node(s)) become biased off-center from the real mean voltage. Thus apparently reading an input offset voltage, when everything is actually working perfectly normally (i.e. even in the absence of rectification effects per se).

Mitigation

It can pay to apply even quite modest (e.g. single-pole) filtering, ahead of an analog front-end (AFE) circuit where these effects can occur.

The most susceptible op-amps tend to be bipolar types, where the input base-emitter junctions themselves act to rectify RF. Lower transconductance types, JFET, and especially MOSFET types, mitigate the effect at the direct input by biasing the junction to a higher voltage, or eliminating the junction entirely.

Many precision amps these days also boast onboard RFI filtering. More or less, an RC filter is integrated on-chip, effective perhaps from 100s of MHz, to well into the GHz where effective on-board filtering would even be a challenge.

Fast enough amps, of course, can also handle high slew rates without limiting or rectification.

The most general (and perhaps important) take-away, is this:

Use only what bandwidth you require.

Limit bandwidth early, and as often as necessary to maintain immunity, noise level, etc. (Mostly, internal bandwidth can be left un-limited, as there are fewer extenuating effects; but there may be special cases where you need additional immunity to external fields (onboard filtering cheaper than shields?), or to prevent device noise from mixing (and accumulating) with the signal.) If you need to resolve the spikes, you must have at least enough bandwidth to resolve the pulse width, or depending on exactly how much detail you want to resolve them to, the risetime or further details as well. How this bandwidth is arranged, doesn't matter -- you might well for example split it into a low-pass path for the mains-frequency stuff, and a band-pass path for the spikes.