Question

If Proportional band(PB) increases then, $K_p$

• A. increases
• B. decreases
• C. constant
• D. PB and $K_p$ are not related

Hint:

Proportional gain ($K_p$) and Proportional Band ($\text{PB}$) are always inversely proportional.
$\text{PB} = 100/K_p$.
An increase in one always leads to a decrease in the other.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks about the mathematical relationship between the Proportional Band ($\text{PB}$) and the Proportional Gain ($K_p$) of a proportional controller.

Step 2: Key Formula or Approach:
The proportional band ($\text{PB}$) is defined as the percentage of the error range that causes the controller output to change over its full scale ($0-100\%$).
The Proportional Gain ($K_p$) and Proportional Band ($\text{PB}$) are inversely related by the following standard equation:
\[ K_p = \frac{100}{\text{PB}\%} \quad \text{or} \quad \text{PB}\% = \frac{100}{K_p} \]

Step 3: Detailed Explanation:

• According to the inverse relation:
\[ K_p \propto \frac{1}{\text{PB}} \]
• If the Proportional Band ($\text{PB}$) is increased, the controller becomes less sensitive to error inputs.

• Mathematically, as the value in the denominator ($\text{PB}$) increases, the resulting value of the Proportional Gain ($K_p$) must decrease.

• For example, a narrow proportional band (low $\text{PB}$) corresponds to high controller gain ($K_p$), making the controller highly sensitive and responsive.

• A wide proportional band (high $\text{PB}$) corresponds to low controller gain ($K_p$), making the controller less aggressive.

Step 4: Final Answer:
If the Proportional Band ($\text{PB}$) increases, the proportional gain $K_p$ decreases.