Question

A first-order low-pass instrument must measure hummingbird wing displacements (assume sinusoidal) with frequency content up to 100hz with an amplitude inaccuracy of less than 5%. What is the maximal allowable time constant for the instrument? What is the phase angle at 50 hz and at 100hz

Answer 1

Low pass sinusoidal transfer function is:

$\frac{Y(j\omega)}{X(j\omega)}=\frac{K}{j\omega \tau+1}$

Where, $\tau$ is time constant and $\omega=2\pi f=2\pi (100 Hz)$ where f is the frequency.

For 5% error, the magnitude must drop below 0.95 K

$\Rightarrow |\frac{K}{j\omega \tau}|=\frac{K}{sqrt{\omega^2\tau^2+1}}=0.95K$

$(\omega^2\tau^2+1)(0.95)^2=1$

$\Rightarrow \tau=[\frac{1-(0.95^2)} {(0.95)^2(2\pi\times100)^2}]^{1/2}=0.52 ms$

Thus, the maximal allowable time constant is 0.52 ms

Phase angle, $\phi=tan^{-1}(-\omega \tau)$

At 50 Hz,

$\phi=tan^{-1}(-2\pi\times 50\times 0.0005)=-9^o$

At 100 Hz,

$\phi=tan^{-1}(-2\pi\times 100 \times 0.0005)=-17.4^o$