Sounds like you are losing current flow through the machine. I would check my workclamp cable at both ends. Make sure you clamp to the work and not the table. With Fine Cut you are working at much lower currents and gap volts.Randomly the arc will bounce up form the steel and not cut this may happen for an inch or two or it may five or six and then continue to cut fine for a while and then do the same thing not cutting through.
In one circumstance we found that the workclamp connection inside the plasma was loose and had built up corrosion (insulator) so current flow was unpredictable.
Since it happens with no THC and with the same cut file (?) and is not a total arc failure then the only thing that is missing is sufficient current flow in the cut to maintain penetration. That can come from an excessive cut gap but I assume the THC300 is tracking when it's turned on.
Make sure your THC300 has the "Fine Cut" switch because the base target volts won't go below about 85 or 90 volts (true tip volts). You meter being out of cal makes it difficult to see what the real voltage is. All the cal does is set the meter scaling (not the THC internal voltage).
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