Introduction & Context

On-off (bang-bang) control is the simplest feedback strategy for maintaining a process variable near a desired set-point. A relay or contactor energises a heater (or chiller) until the measured variable reaches an upper trip point, then de-energises until it falls to a lower trip point. The resulting oscillation is characterised by a cycle time \( \tau \) and a differential band \( \Delta T \). Knowing \( \tau \) in advance allows the engineer to:

size switching elements for mechanical endurance,
predict thermal fatigue of heaters or compressors,
check that the frequency is low enough to avoid contactor chatter or compressor short-cycling.

Methodology & Formulas

Assume the process reacts with constant first-order rates while the actuator is either fully on or fully off.

Define the differential band \[ \Delta T = T_{\text{high}} - T_{\text{low}} \] which is symmetric about the set-point.
Heating leg: temperature rises from \( T_{\text{low}} \) to \( T_{\text{high}} \) at rate \( R_{\text{h}} \). \[ t_{\text{h}} = \frac{\Delta T}{R_{\text{h}}} \]
Cooling leg: temperature falls from \( T_{\text{high}} \) to \( T_{\text{low}} \) at rate \( R_{\text{c}} \). \[ t_{\text{c}} = \frac{\Delta T}{R_{\text{c}}} \]
Total cycle time \[ \tau = t_{\text{h}} + t_{\text{c}} = \Delta T \left( \frac{1}{R_{\text{h}}} + \frac{1}{R_{\text{c}}} \right) \]
Cycling frequency \[ f = \frac{1}{\tau} \]

Parameter	Symbol	Unit	Constraint
Differential band	\( \Delta T \)	°C	> 0
Heating rate	\( R_{\text{h}} \)	°C min⁻¹	> 0
Cooling rate	\( R_{\text{c}} \)	°C min⁻¹	> 0
Minimum cycle time	\( \tau_{\text{min}} \)	s	≥ 10 s (to avoid contactor chatter)

Start with the process time constant τ and the allowable temperature swing ΔT.

Measure the temperature rise rate R (°C/min) at full heater power.
Compute the heating interval t_on = ΔT / R.
Compute the natural cooling interval t_off using the lumped-capacitance model or plant data.
Total cycle time T = t_on + t_off.

For most air- or water-heated vessels, T ends up between 4–10 min; shorter cycles wear out relays, longer cycles let the temperature drift outside spec.

Industry rule-of-thumb is ≥ 10 s for electromechanical contactors; shorter pulses cause contact chatter and pitting.

Use solid-state relays (SSRs) or thyristors if you need sub-second modulation.
Verify the vendor’s cycle-life curve; 10 s pulses at rated load typically give >10⁶ operations.

Deadband ΔT_db widens the temperature span between switch-on and switch-off points.

Wider ΔT_db lengthens the cycle, reduces switch count, and extends relay life but increases overshoot.
Narrow ΔT_db shortens the cycle, improves control accuracy, but raises wear and may cause hunting.

Aim for ΔT_db ≈ 0.5–1 °C for sensitive processes, 2–3 °C for HVAC or bulk storage.

Yes, if you know the steady-state heat loss Q_loss and heater power P.

Duty ratio D = Q_loss / P.
Cycle time T is still set by thermal inertia; D only tells you the fraction of time the heater will be on.

Use this quick check during design to size heaters and estimate annual energy use.

Worked Example: On-Off Control Cycle for a Small Reactor Jacket

A process engineer is commissioning a 50-L batch reactor whose jacket temperature is regulated by an on-off controller. To avoid overshoot while maintaining throughput, the engineer needs to know how often the controller will switch the tempering water circulation between heating and cooling modes when the reactor is held at its nominal set-point.

Knowns

Set-point temperature: 85 °C
Differential (hysteresis) band: 1 °C
Heating rate (with steam valve open): 3 °C min⁻¹
Cooling rate (with chilled water valve open): 3 °C min⁻¹

Step-by-Step Calculation

Determine the temperature excursion \(\Delta T\) across the band. \[ \Delta T = 1\ ^\circ\text{C} \]
Compute the time required to traverse the band while heating. \[ t_{\text{heat}} = \frac{\Delta T}{\text{rate}_{\text{heat}}} = \frac{1\ ^\circ\text{C}}{3\ ^\circ\text{C min}^{-1}} = 0.333\ \text{min} \]
Compute the time required to traverse the band while cooling. \[ t_{\text{cool}} = \frac{\Delta T}{\text{rate}_{\text{cool}}} = \frac{1\ ^\circ\text{C}}{3\ ^\circ\text{C min}^{-1}} = 0.333\ \text{min} \]
Calculate the total cycle time (period) \(\tau\). \[ \tau = t_{\text{heat}} + t_{\text{cool}} = 0.333\ \text{min} + 0.333\ \text{min} = 0.667\ \text{min} \]
Compute the switching frequency \(f\). \[ f = \frac{1}{\tau} = \frac{1}{0.667\ \text{min}} = 1.5\ \text{min}^{-1} \]

Final Answer

The controller will complete one full on-off cycle every 0.667 minutes (≈40 s) and will switch approximately 1.5 times per minute.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle