Question

A temperature - measuring instrument is modeled as a first order system with a time constant of 5s. The sensor of the instrument is placed inside an oil bath whose tempreture has a sinisoidal variation with amlitude of 10${}^{o}C$ and a period of 20s around an average tempreture of 200 ${}^{o}C$. The sinusoidal component at the output of the instrument will have amplitude.

• A. 0${}^{o}C$
• B. 5.37 ${}^{o}C$
• C. 8.57 ${}^{o}C$
• D. 10${}^{o}C$

Answer 1

The correct option is B 5.37 ${}^{o}C$

The frequently response of a first order system when it is subjected to the sinusoidal input is given as :

$\frac{A_o}{A_i}=\frac{1}{\sqrt{(1+\omega \tau {)}^2}}$

Given, $\tau =$ 5 sec

$\omega =\frac{2\pi }{T}=\frac{2\pi }{20}=\frac{\pi }{10}$

$A_i={10}^{o}C$

$\therefore A_o=\frac{10}{\sqrt{1+{(\frac{\pi }{10}\times 5)}^2}}=\frac{10}{\sqrt{1+{\left(\frac{3.14}{2}\right)}^2}}$

$=\frac{10}{\sqrt{1+2.46}}=\frac{10}{1.86}={5.37}^o$