The fundamental problem is that a standardized method assumes a standard setup. It's a project that will require significant investment of time and effort, to say the least. Will never materialize, I'm afraid.
Continuity test is usually about a resistance measurement. It was suggested to use a generator. The generator generates a voltage pulse, not a resistance pulse. But it's a minor problem, it's easy to convert a voltage pulse to a resistance that the DMM can sense, with FET or something. Usually, input of such a device is quite delicate so a reasonable input protection circuit will be required (a bit more additional components).
In DMMs, the result of the continuity test is indicated by sound so the only non-intrusive way to convert it to a machine-readable form is to use a microphone. Therefore, a microphone amplifier will be required. I think it'll be better to stay away from any sort of digital audio (PC sound cards, etc.) because digital processing is associated with some latency and annoying calibration procedure must be followed to take the latency time into account. A simple analog amplifier seems a better solution (add a few more components).
Anyway the sound is generated by a beeper, internal location and characteristics of which is up to the DMM's manufacturer. That means that for each particular DMM, operator will have to identify optimal spatial location of the microphone relative to the DUT. The chances are in the optimal position, the microphone will be sensitive to a not related environmental sounds which can compromise the measurement accuracy and make the results debatable. That leads to the idea of a standardized fixture, e.g. in the form of wooden cabinet acoustically isolated from the environment. The microphone (or an array of them) are placed inside at a fixed position. The cabinet is of enough internal volume to accommodate a handheld or a bench-type DMM (if we would like to test the latter type as well).
The problem with a stationary DMMs is that many of them are fan-cooled. A high pass filter in the mic amp will be required to filter out a hum from the rotating impeller (even more additional components).
From the electrical perspective, there is nothing complex. A mic(s) followed by an amp followed by a rectifier followed by a comparator that produces a sharp edge at the moment of sound attack, so the time position can be easily measured. BTW the comparator output can be routed back to the input of the voltage-resistance converter device (perhaps through some delay circuit) to create a positive feedback oscillation that eliminates the need for external generator.
It can be more complex to define the test criteria. The DMMs can be very peculiar. The beep first starts for several hundreds of ms, then interrupts for tens of ms, and only then starts sounding continuously. No idea how typical it is but I've one evidence. Another DMMs, like my Fluke 189 generates not a tone, but a sound of quite weird cadence. I think it's a feature. In a really noisy environments, like a server room, that irregular cadence is easy to notice while a musically perfect tone would be a disadvantage. It will be nice if the test setup can differentiate between a well-tempered DMM and a poor one.
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