The Engineering Behind the Terms

The pulling force a locomotive can put to the railhead before its drivers slip.

A driven wheel can only push on the rail as hard as friction lets it before it spins, so the ceiling is tractive effort = μ × weight on the drivers. μ is the wheel–rail adhesion coefficient (clean, dry nickel–silver model track sits around 0.2–0.3). Nothing about the motor raises this ceiling — only driver weight or a grippier interface does. That is why a heavier model loco out-pulls a lighter one with the same mechanism, and why traction tires exist.

Check a pull on a grade in RailGrade →

The fraction of a wheel’s downward load that friction can turn into pulling force.

Friction between two surfaces is F = μ × N, where N is the normal (downward) load. On rails, N is the weight carried by the driven wheels, so μ sets how much of that weight becomes usable pull. Rearranging tractive effort shows μ = TE / weight-on-drivers — measure the slip point and the weight and you have measured μ for your track. Oil, dust, and oxide drop it; a touch of sand or a clean, slightly textured railhead raises it.

Put adhesion to work in RailGrade →

The force needed to keep a free-rolling car moving on level, straight track.

Even with no grade, a car resists motion through bearing drag, wheel–rail deformation, and axle friction. Model it as a fraction of the car’s weight: resistance ≈ C_rr × weight. The train total is the sum over every car, which is why pulling capacity is really “tractive effort minus the summed rolling resistance minus the grade force.” Lower-friction wheelsets (metal, free-rolling trucks) shrink C_rr and let one loco pull more cars.

See it inside a pull check →

How steeply track rises, as rise per unit of run, expressed as a percent.

A grade is just slope: grade% = (rise / run) × 100. A 2% grade climbs 2 units for every 100 travelled. The force it adds to the load is weight × sin(θ), and for the shallow grades a layout can tolerate, sin(θ) ≈ grade/100 — so a 2% grade effectively asks the loco to carry about 2% of the train’s weight as extra pull, on top of rolling resistance. Curves on a grade add resistance too (see compensated grade).

Size a grade in RailGrade →

A gradual transition that smooths the hard corner where one grade meets another.

At a grade break the track direction changes by an angle Δθ = |atan(g_in) − atan(g_out)|. Force that change into a sharp corner and a rigid car bridges it on its two truck points — couplers pull apart on a crest or jam on a sag. Spread Δθ over a vertical curve of length L and each car only sees Δθ × (truck-centers / L), so a longer easement means a gentler per-coupler angle. That ratio is exactly what sets the minimum easement length.

Size an easement →

The tightest curve a given car or loco can negotiate without binding or uncoupling.

On a curve of radius R, a coupled car’s coupler face swings off the tangent line by asin(p / R), where p is how far the coupler reaches past the truck. Set that swing to the most the coupler can take and solve: R_min = p / sin(swing). Longer cars (bigger p, wider truck centers) demand broader curves — the geometry, not the motor, decides what survives the loop.

Check min radius & turnouts →

How a car body swings off the track centerline on a curve — out at the ends, in at the middle.

A rigid car rides on the straight chord between its trucks, but the curve keeps bending away from that chord. The body corners therefore poke outside the railhead (end outswing) while the body middle cuts inside toward curve center by R − √(R² − (truck-centers/2)²) (mid inswing). Add both to a neighbor’s swing and you get the track-center spacing two trains need to pass without touching.

Visualize overhang & clearance →

The slight taper of a wheel tread that lets a wheelset steer itself through curves.

Tread is coned, so when a wheelset shifts sideways by y the two wheels roll on different diameters. The rolling-radius difference is about 2 × conicity × y, so the outer wheel travels farther per turn and the axle naturally steers back toward center — self-centering with no moving parts. The same effect lets a solid axle round a curve, the outer wheel rolling on a larger radius than the inner.

Explore it in the wheel-rail tool →

The point where a wheel flange rides up the rail and derails instead of staying down.

At the flange contact the rail pushes with a lateral force L and a vertical force V. The flange stays seated only while the L/V ratio is below what the flange-face angle can hold: the Nadal criterion, L/V ≤ tan(flange-face angle) (friction-adjusted). A steeper flange face tolerates more lateral force before climbing, which is why fine-scale and pizza-cutter profiles behave so differently on the same curve.

See the L/V limit →

The rail spacing and the spacing between wheel backs that together set tracking play.

Track gauge is the rail-to-rail distance (HO = 16.5 mm); back-to-back is the gap between the inner faces of the two wheels. The free lateral play — how far a wheelset can shift before a flange touches rail — is gauge − back-to-back − flange thickness. Too little play binds in curves; too much lets the set hunt and pick points. Standards (NMRA RP-25) exist to keep that difference in a sane band.

Measure play in the wheel-rail tool →

The fixed proportion between a model dimension and its real-world prototype.

HO is 1:87.1 — every model length is the prototype length divided by 87.1 (and vice-versa, multiply by 87.1). A 50-foot prototype boxcar becomes 50 ft × 12 / 87.1 ≈ 6.9 in. The ratio is linear, so it applies to any length directly; areas scale by the square of it and volumes (hence weight of a solid part) by the cube — which is why scaled-down mass never matches a scaled-down size.

Convert at any scale →

How fast a model’s motion represents in real-world prototype terms.

Speed scales with length, so scale speed = model speed × ratio. Time a car crossing a measured distance d in t seconds: its model speed is d/t, and the prototype equivalent is d/t × 87.1 for HO. Convert the units and a few inches a second becomes a believable scale mph — the basis for both realistic running and speed-matching two locos.

Time a pass for scale speed →

A recommended rolling-stock weight that keeps cars tracking reliably.

Too-light cars stringline and derail; the NMRA RP-20.1 recommendation for HO is 1 oz + 0.5 oz per inch of car body length. It scales weight to length because a longer car has more leverage to be shoved around in a curve, so it needs proportionally more mass low in the frame to stay planted. It is a tracking guideline, not a prototype-mass figure (which the cube law would make absurd).

Compute RP-20.1 weight →

A numbered memory slot in a decoder that stores one setting as a 0–255 value.

A DCC decoder is a tiny computer; each CV is one byte of its settings. CV1 holds the short address, CV2/5/6 shape the speed curve, CV3/4 set momentum. Some CVs are bit fields — CV29 packs direction, speed-step mode, and long-vs-short address into individual bits of one number, so you compose it by adding the bit values rather than picking from a list. Decode it by reading the number back into its bits.

Build or decode CVs →

Tuning two locos so they run at the same scale speed for smooth consisting.

Two mechanisms rarely run alike at the same throttle, so a consist fights itself. Measure each loco’s scale speed at low/mid/high, then scale the slave’s governing CV (CV2 Vstart, CV6 Vmid, CV5 Vhigh) by target ÷ measured × its current value, clamped 0–255 and kept monotonic. It is a strong first pass — motors are not perfectly linear with voltage, so you re-measure and nudge.

Match two locos →

The track voltage lost over the length of the power bus feeding a district.

Wire has resistance, and current flows out and back, so the loss is drop = 2 × length × (ohms/ft) × current. Thinner wire (higher gauge number) and longer runs both raise it; past a few percent of the ~14.5 V DCC bus, decoders brown out and shorts stop tripping reliably. The fix is heavier bus wire or shorter feeders — sizing it is just that equation against a 3% budget.

Size a bus run →

A section of track fed by its own booster or breaker so faults stay contained.

One booster supplies a fixed current; demand is locos × per-loco draw plus accessory load. Divide the layout so no district’s demand exceeds its booster (with ~20% headroom), and a short in one district trips only that section — the rest keeps running. The count falls straight out of total demand ÷ usable amps per booster, rounded up.

Plan districts & boosters →