Two things reject common-mode interference and only one of them is on a datasheet. The electrodes and the amplifier's input impedance form a divider, and any mismatch between the two electrodes converts part of the interference into a differential signal the amplifier cannot tell from your biosignal.
| Term | Residual at the input | Equivalent rejection | Share |
|---|---|---|---|
| Amplifier rejection | 15.8 uV | 110 dB | 39% |
| Electrode imbalance | 25.0 uV | 106 dB | 61% |
| Total at the input | 40.8 uV | 102 dB effective | 100% |
Rejection from the electrode divider alone
What the electrode path gives you before the amplifier contributes anything, by impedance mismatch and input impedance. Compare a row against the CMRR on your amplifier's datasheet: wherever this figure is lower, the electrodes are the limit and a better amplifier buys nothing.
| Mismatch | 100 Mohm input | 500 Mohm input | 1 Gohm input | 10 Gohm input |
|---|---|---|---|---|
| 0.5 kohm | 106 dB | 120 dB | 126 dB | 146 dB |
| 1 kohm | 100 dB | 114 dB | 120 dB | 140 dB |
| 2 kohm | 94 dB | 108 dB | 114 dB | 134 dB |
| 5 kohm | 86 dB | 100 dB | 106 dB | 126 dB |
| 10 kohm | 80 dB | 94 dB | 100 dB | 120 dB |
| 20 kohm | 74 dB | 88 dB | 94 dB | 114 dB |
| 50 kohm | 66 dB | 80 dB | 86 dB | 106 dB |
Red is below 90 dB, amber below 110, which is roughly where a good instrumentation amplifier sits. A cell in red means the amplifier specification is irrelevant: the electrodes have already decided the outcome.
A worked example
A wearable EEG front end specified with a 110 dB instrumentation amplifier and a 1 gigohm input impedance, recording 50 microvolt signals in a room with about 5 volts of mains coupling on the body. The bench results are clean and the part selection is not in question.
The amplifier contributes 15.8 microvolts of residual. A 5 kilohm electrode mismatch, which is unremarkable for dry contact partway through a session, contributes 25 microvolts on its own. The two together give an effective CMRR of 102 dB, so eight decibels have gone before anyone wrote firmware, and the electrode path is already the larger term.
The number that matters clinically is the last one: 41 microvolts of interference against a 50 microvolt signal is 1.8 dB of signal to interference. The physiology is not swamped, but it is not comfortably above the noise either, and the mismatch is the term that grows as the session runs.
Which makes the design decision a materials and contact one rather than a semiconductor one. Improving the amplifier to 120 dB moves the total from 41 to 30 microvolts. Halving the mismatch moves it to 28. Doing both is not twice as good as either, because the two terms add and the smaller one stops mattering.
The arithmetic, so you can check it
The amplifier contributes Vcm / 10^(CMRR / 20). The electrode path contributes Vcm x mismatch / input impedance, because the divider formed by the two electrodes against the input impedance is unbalanced by exactly that ratio. The two residuals add at the input, and the effective rejection is 20 x log10(Vcm / total residual).
This is the resistive case at a single frequency, which is the right first calculation and not the whole story. Real electrodes are complex impedances that vary with frequency, so the mismatch at 50 hertz is not the mismatch at 500, and cable capacitance adds a path this does not model. Treat the result as the floor on your interference rather than the total, and measure the real thing on the real electrodes before you commit the bill of materials.
The physical reasoning is in common-mode rejection explained, and the constraint a driven reference introduces in the driven reference electrode. If you are adding one, check its margin with the return-path calculator before the board exists.
Questions
Because the electrodes and the amplifier's input impedance form a divider, and any mismatch between the two electrodes converts part of the common-mode interference into a differential signal before the amplifier sees it. That path has its own rejection figure, set by the ratio of input impedance to mismatch, and the two combine. A very good amplifier behind a poor electrode balance behaves like a much worse amplifier.
It depends on your input impedance, which is why the ratio matters rather than the absolute figure. A 5 kilohm mismatch against a 1 gigohm input gives about 106 dB of rejection from that path alone, which is already comparable to a good instrumentation amplifier. Against a 100 megohm input the same mismatch gives 86 dB, and the electrodes are then unambiguously the limit.
During, and at the end. Electrode impedance mismatch grows through a session as gel dries and contact degrades, so the figure at setup is the best case and the one at the end of the longest expected recording is the one to design against. A front end that meets its specification only in the first ten minutes will pass a bench test and fail in clinic.
It reduces the common-mode voltage on the body, which lowers the interferer term rather than improving the rejection ratio, and that helps both paths at once. It also introduces its own constraint: the current it drives has to return through the electrode, and the amplifier doing the driving has to be able to sink it without railing. That is a separate calculation and it is worth doing before the board exists.
Only partly, and it costs you signal. A notch at the mains frequency removes physiology sitting at that frequency along with the interference, and the harmonics remain. Filtering is a reasonable last layer over a front end with adequate rejection; it is not a substitute for one, because it cannot recover a signal that has already been swamped at the input.