There is a thread about a formula for research costs in Civ 3 Vanilla, that can be found here:
https://forums.civfanatics.com/threads/research-cost-formula-v1-29f.29485/
There are many factors influencing the settings for research and a modder has to find the best balance of these settings for the aims that mod or scenario should achieve, among them besides minimum and maximum research per example there are the Tech Rate, the Movement Rate and the production costs for units for the used map sizes.
In my eyes the most important functions of the minimum research time are to help that the units in the mod/scenario have "enough room for breathing" in the game and that the time scale of the game can come closer to the historical scale. The worst settings are units that are obsoleted when being produced or when moved out of the producing cities very shortly after being produced.