Question

In a piezo-electric transducer a flat frequency response within 5% is required. Find the value of minimum frequency in terms of time constant for which it can be used. If the time constant of the transducer is 1.5 ms, find the phase shift at minimum frequency.

• A. $2027rad/sec,{18.2}^o$
• B. $2.027rad/sec,{12.8}^o$
• C. $2027rad/sec,{1.82}^o$
• D. $2.027rad/sec,{1.28}^o$

Answer 1

The correct option is A $2027rad/sec,{18.2}^o$

In order to obtain a flat response within 5%, the amplitude ratio should be:

$M=1-0.05=0.95$

$Now,M=\frac{1}{\sqrt{1+(1/\omega \tau {)}^2}}=0.95$

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

$\Rightarrow \omega \tau =3.04$

$\therefore$ Minimum value of frequency

$\omega =\frac{3.04}{\tau }=\frac{3.04}{1.5\times {10}^{-3}}$

$=2.0267\times {10}^{3}rad/s\approx 2027rad/sec$

$\text{Phase shift}\left(\varphi \right)=(\pi /2-ta{n}^{-1}\omega \tau )rad.$

$={90}^o-ta{n}^{-1}3.04={90}^o-{71.8}^o={18.2}^o$