Even on the same scale, there are several answers to this simple question. The length of your cars, the space between parallel tracks, the use of easements, and your personal style as you lay down the tracks all factor into this answer.
Length of Your Train
The length of your equipment, even more than the scale of the model, is the most important factor in setting a minimum radius. The list below includes conventionally accepted minimums for each of the major modeling scales. Note that there does not appear to be much difference between them. Large-scale models have compromises built-in to allow operation on some very tight curves. The trade-off is that they do not always look very good. Special note about O scale: O scale trains come in two forms, there are traditional three-rail toy trains that often feature major compromises in body length, articulation, and couplers that allow them to make incredibly sharp turns. More to-scale models better replicate the prototype, including a more demanding turn radius. And, there are small “scale” prototypes and large “toy” models meaning there is still no definitive minimum radius.
Parallel Tracks
If you have parallel tracks in a curve, the spacing between tracks is important. The sharper the radii, the greater the distance between rails will need to be. Depending on the model, trains may overhang the track on both the inside and outside of a curve. Space tracks too close and they could sideswipe.
Easements
Easements are a gradual transition in radius. Used at the entrance or exit of curves, an easement can make even a tight minimum radius operate more effectively. Easements can be easily added using either flexible or sectional track.
“S” Curves
“S” or reverse curves look amazing when done correctly. When the radii are tight, this can pose problems. Adding easements and a short section of tangent (straight) track between the curves can reduce operating problems.
Personal Preference
Just because a model train can negotiate a minimum radius does not mean it should. Most layout curves are already tighter than the prototype could manage. Larger curves prevent unrealistic overhang, allow close-coupling multiple unit locomotives and passenger cars, and are usually less prone to derailments. In addition to easements, super-elevation, or banked curves also help make a curve look more realistic. There is nothing like watching a long model train lean into a broad sweeping super-elevated curve. Ultimately, the minimum radius for your railroad is up to you and your personal aesthetic preference.
