Not on the coldest day — then the unit runs almost continuously and barely cycles at all. Not on the mildest either, when it hardly runs. The worst cycling is at about 50% duty, which for a correctly sized system is ordinary spring and autumn weather. An oversized unit reaches that point on a milder day and spends more of the year there.
On-off control has no gain to tune. The only real parameter is the deadband — the gap between switch-on and switch-off — and everything else follows from the physics of the thing being heated.
Heating, the space rises towards ambient + authority; cooling, it falls towards
ambient. Both are first-order exponentials, so
ton = τ·ln((C − Tlo)/(C − Thi)) and
toff = τ·ln((Thi − Tamb)/(Tlo − Tamb)),
where C is the ceiling the heater could reach.
For a band small compared with the temperature differences, both logarithms are very nearly linear in the band, so the period is proportional to it. Halve the band and you halve the period: twice the cycles, half the temperature swing. That is the entire trade, and it is the only knob you have.
A house with a 4-hour time constant, 20 °C setpoint, 5 °C outside, a 1 K deadband and a boiler able to hold 25 K above ambient: 24 minutes on, 16 minutes off, about 1.5 cycles an hour at 60% duty. Double the boiler's authority to 50 K and the on-time collapses to 11 minutes while the off-time barely moves — over 2 cycles an hour, from buying a bigger boiler.
Start-up is where the wear is. A compressor draws several times its running current for the first seconds and runs without proper oil return until it is up to speed; a boiler condenses and purges on every start. Manufacturers commonly want a minimum run of about ten minutes and no more than roughly six starts an hour. Efficiency suffers too, because a system that never reaches steady state never reaches its rated efficiency.
Because you feel it. A 2 K band on a room thermostat is a noticeable swing, and comfort complaints follow. The band is a straight trade between temperature stability and equipment wear, and the right answer differs for a freezer (nobody cares about 2 K) and a living room (everybody does).
Because it crosses the band faster. Doubling the authority roughly halves the on-time while leaving the off-time alone — the space still cools at whatever rate the fabric allows. The period shortens, the cycle rate rises, and every start is another start. Oversizing is the most common cause of short-cycling and it is bought deliberately, usually for reassurance.
When the plant is slow relative to the switching, the actuator is genuinely two-state, and nobody needs the last fraction of a degree — which describes most heating, refrigeration and domestic hot water. It needs no tuning, cannot integrator-wind-up, and fails in an obvious way. Modulating control is better when you have a modulating actuator; fitting PID to a relay just gives you a slower relay.
A single well-mixed thermal mass with one time constant, a heater that is fully on or fully off with no modulation, instant switching, a symmetric deadband about the setpoint, and a constant ambient. Not modelled: thermal stratification and sensor placement, minimum on and off timers built into real controllers, anticipators, start-up transients and inrush, modulating or multi-stage plant, defrost cycles, solar gain and occupancy, or the difference between air temperature and what a person feels.
No changes to this tool’s own behaviour since the earliest archived release (v1.21). The full history for the site is in the changelog.