I've been designing them, teaching about them and fixing them for about 20 years. Does that count?
In all that time though I have never worked directly with a spectrometer specifically designed for Carbon ratio measurement. However the method by which any isotope is measured is completely the same as for any other isotope. The only difference is in the ion source.
The most simplified version of a ratio measurement mass spectrometer would be something like this
The accelerated ions are passed through a magnetic field where the ions of different mass (momentum mV technically but mass is near enough) are separated and then each is sent into a different detector where the number of ions arriving over a period of time is counted. This kind of spectrometer is known as a magnetic sector instrument. It will have other electrical filters which also focus the beam by kinetic energy (1/2 mV squared). An ion beam typically contains a spread of kinetic energy that follows a Gaussian bell curve so these are used to reject ions with energy outside of an arbitrarily set acceptable range.
The AMS works a little differently in that it uses acceleration energy in the region of MV rather than the usual 6 to 10 KV. In an AMS the ions start out as KV level negatively charged ions which can often include charged molecules. They are passed through a sector magnet to separate the ions out by mass before combining the beams with a MV particle beam created in a nuclear particle accelerator. This effectively smashes all the molecular species into atoms and also reverses their charge to positive. The ions are now traveling extremely fast. A series of carefully tuned 'filters', either magnetic, electrostatic or both, are used to remove unwanted species leaving only the isotopes that we are interested in. The isotopes will then be measured in one of several different ways. They may be slowed and captured in Faraday cups. They may be fired into a TOF (Time Of Flight) lens or their kinetic energy can be directly measured to determine the mass of each ion. This wiki article describes AMS pretty well. https://en.wikipedia.org/wiki/Accelerator_mass_spectrometry
As for the issue of background measurements and empty targets.... Background noise has always been the analytical chemist's worst nightmare. Carbon is everywhere. Little tiny bits of it will work their way into every place they possibly can so we always have to clean, clean some more and then clean again a few more times just for luck before we analyze anything. Atmospheric C02 will stick to samples just because it can. Sodium is even worse but that's kind of beside the point in this topic.
So we always have to measure the instrumental backgrounds before, during and after every analysis of any kind in order to determine the BEC (Background Equivalent Concentration) of whatever isotopes we are measuring.
Does that help at all? So my image didn't show up. Not too sure why