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:

`(Y(j\omega))/(X(j\omega))=(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 |(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*100)^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* 50* 0.0005)=-9^o`

At 100 Hz,

`\phi=tan^(-1)(-2\pi* 100 * 0.0005)=-17.4^o`