GATE 2023 BT Q.64: solution and answer

Official answer: 4Source: Official answer key, GATE 2023 BT (IIT Kanpur)

1. A proportional controller gives output $p=\bar{p}+K_c\epsilon$. Here $p$ is the controller output, $\bar{p}$ is the output when the error is zero (the bias), $K_c$ is the proportional gain and $\epsilon$ is the error. When the input and the output are both written as a percentage of their spans, $K_c$ has no units.

2. The proportional band (PB) is the percentage of the measured-variable span that drives the controller output through its full range (0 to 100%).

3. Span of the measured temperature: $130-60=70{}^{\circ }\text{C}$.

4. With PB = 25%, a temperature change of $0.25\times 70=17.5{}^{\circ }\text{C}$ drives the output through 100% of its range.

5. As a percentage of the input span, that change is $\frac{17.5}{70}\times 100\%=25\%$. So the gain is $$K_c=\frac{\Delta (\text{output, \%})}{\Delta (\text{input, \%})}=\frac{100\%}{25\%}=4$$

6. This is the standard relation $K_c=\frac{100}{\text{PB}(\%)}=\frac{100}{25}=4$. The $70{}^{\circ }\text{C}$ span cancels, so the temperature range does not affect the dimensionless gain. It only fixes how many degrees the 25% band covers ($17.5{}^{\circ }\text{C}$).