Question

A thermometer with time constant $\tau ,$ initially at the ambient temperature, is used to measure the temperature of a liquid in a bath. The excess temperature of the thermometer and the liquid over the ambient are $\theta \left(t\right),{\theta }_i\left(t\right)$ respectively, where t denotes the time. If ${\theta }_i\left(t\right)=kt,$ where k is a constant, the steady state error, defined as $\underset{t\to \infty }{\lim }\left(\theta \left(t\right)-{\theta }_i\left(t\right)\right],$ is

• A. $\infty$
• B. 0
• C. -k
• D. $-k\tau$

Answer 1

The correct option is D $-k\tau$

Steady state error for ramp input of magnitude $k\tau$ and first order system.

$e_{\mathrm{ss}}=\underset{t\to \infty }{\lim }\left({\theta }_i\left(t\right)-\theta \left(t\right)\right]=\mathrm{k\tau }$

Here,

$\underset{t\to \infty }{\lim }\left(\theta \left(t\right)-{\theta }_i\left(t\right)\right]=-\mathrm{k\tau }$