Question

The characteristics of a thermometer measuring ambient temperature is $2\frac{d{T}_i}{dt}+T_i-T_a=0,$ where $T_{i}and{T}_a$ are the indicated and ambient temperatures, respectively, both in °C, and time is in seconds. The -3 dB cut-off frequency in the frequency response of the thermometer is

• A. $1/4\pi Hz$
• B. $1/2\pi Hz$
• C. 1 Hz
• D. $2\pi Hz$

Answer 1

The correct option is A $1/4\pi Hz$

Clearly it is a first order equation

$\frac{A_0}{A_1}=\frac{1}{\sqrt{1+(\omega \tau {)}^2}}$

Standard first order equation is
$\tau \frac{dy}{dt}+y=kx$

where,
$\tau =timeconstant$

k = sensitivity of the system

$\therefore \tau =2here$

$also,at3dB,\frac{A_0}{A_1}=\frac{1}{\sqrt{2}}$

$\therefore \frac{1}{\sqrt{2}}=\frac{1}{\sqrt{1+(2\omega {)}^2}}.\frac{1}{2}=\frac{1}{1+4{\omega }^2}$

$1+4{\omega }^2=2$

$\omega =\frac{1}{2}and2\pi f=\frac{1}{2}$

$\therefore f=\frac{1}{4\pi }$