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I can specificy a minimum tPD, but without Fast Corner, the best and fastest case PVT is not considered.
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If you're doing Tpds and there's a lot of logic between pins, you might have multiple paths between those pins. This is an estimate on only one of those paths, most likely the longest, but if you have a shorter path, that's the one that should be used.
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"Minimum" in Quartus timing constraints and reporting does not mean fast PVT. Any number labeled as "minimum" applies to whatever PVT timing model is currently selected, whether a slow model or fast model.
For any single timing model (which means each model separately for a device family that has a fast model), what you are seeing reported for minimum tpd is the fastest path at that single model of potentially more than one path, not the fastest of min and max numbers for a single path. When there is more than one possible path between the specified endpoints as Rysc mentioned, the min and max tpd values will likely be different for any single timing model. If reported min and max are the same, that's usually because there is only one path between the specified endpoints.
This also applies in other places where Quartus timing reporting uses "minimum" and "maximum" for numbers shown for a single timing model, not just for tpd.
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Dividing by 4 from the fastest speed grade sounds indeed a lot. This would mean that if I'm using the slowest speed grade, I would need to consider the best case being something in the order of, possibly 10 times shorter than the worst case ???
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... it's certainly possible that you can get shipped something faster than your slowest speed grade, so that's going to cut into your margins quite a bit right there.
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For newer device families that have a fast timing model, Quartus uses the same fast model for all speed grades. (You can easily see this in TimeQuest, where the slow operating conditions have the speed grade in the name but the fast operating conditions do not.) This allows for the possibility of an actually fast device being marked with a slow speed grade. The device meets its specs, so this is OK, but it means that a slower speed grade has more potential min-to-max variation over process.
In case the same thing applies to the old FLEX families, I suggest that you base your fast-model estimate on slow-model reported timing of the fastest speed grade regardless of which speed grade you are actually using. Use your fast-speed-grade APEX results to estimate your FLEX fast-versus-slow-model derating factor, then run timing analysis on the fastest speed grade in FLEX to estimate your actual fast-PVT results.
Compile with the speed grade you are actually using selected in Quartus. You can run timing analysis on the fastest speed grade without recompiling. Unless for some reason this capability isn't supported for FLEX, you can change the speed grade and rerun the Classic Timing Analyzer in the GUI or run quartus_tan at the command line with the --speed argument.