On–off control is a special case of proportional control with extremely high gain. What is the approximate proportional band (band width) expressed as a percentage for ideal on–off control?

Introduction / Context:
On–off control is the simplest feedback strategy used in thermostats, level switches, and many binary actuators. It is often described as an extreme case of proportional control where the gain tends to infinity. Understanding its effective proportional band helps in comparing it with true proportional, PI, and PID controllers.

Given Data / Assumptions:

• Ideal on–off behavior with no intentional hysteresis or differential gap.
• Control output takes only two states: fully on or fully off.
• Proportional band is defined as the error range over which the output linearly changes from 0% to 100%.

Concept / Approach:
For proportional control, output varies linearly with error within the proportional band. In on–off control, there is no linear region: the output switches instantaneously between extremes at a threshold. Therefore, the effective proportional band is 0 percent. Practical systems may add hysteresis to avoid chattering, but the idealized theoretical band remains zero.

Step-by-Step Solution:
Define proportional band PB as the span of error yielding a continuous output change.For on–off, output is discrete; continuous span is nonexistent.Hence, PB ≈ 0% for the ideal case.

Verification / Alternative check:
Plotting controller output vs error reveals a step function for on–off, versus a sloped line for proportional control—confirming a zero-width band for on–off.

Why Other Options Are Wrong:
100%, 75%, 25%: These imply a nonzero linear region, which contradicts the binary nature of on–off control.

Common Pitfalls:

• Confusing practical hysteresis (added for stability) with proportional band.
• Assuming on–off is a low-gain proportional controller; it is effectively infinite gain at the switching threshold.
• Overlooking cycling and wear issues when no hysteresis is used.

Final Answer:
0