Question

Proportional band of an ON--OFF controller is

• A. zero
• B. 100%
• C. un defined
• D. Infinity

Hint:

Gain ($K_p$) and Proportional Band ($\text{PB}$) are inversely related.
Because an ON-OFF controller acts with infinite gain ($K_p = \infty$) to produce immediate full-scale output changes, its proportional band ($\text{PB}$) is exactly zero.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks for the proportional band value of a standard ON-OFF (two-position) controller.

Step 2: Key Formula or Approach:
The Proportional Band ($\text{PB}$) of a controller is inversely proportional to its proportional gain ($K_p$):
\[ \text{PB}\% = \frac{100}{K_p} \]

Step 3: Detailed Explanation:

• An ON-OFF controller is a two-position controller where the output switches abruptly between its maximum value (fully ON) and minimum value (fully OFF) based on a minute change in the error signal.

• This means that even an infinitesimally small error from the setpoint causes a full-scale swing in the controller output.

• This infinite sensitivity corresponds mathematically to an infinite proportional gain ($K_p \to \infty$).

• Substituting an infinite gain into the proportional band relationship:
\[ \text{PB} = \lim_{K_p \to \infty} \frac{100}{K_p} = 0 \]
• A proportional band of $0\%$ means that there is zero range of error over which the controller output varies proportionally; instead, it switches instantaneously.

Step 4: Final Answer:
The proportional band of an ON-OFF controller is zero.