When the system is first order:


Why is the bandwidth: $f_{-3db}=1/(2\pi \tau)$?

Where did the $2\pi$ come from? Why isn't it simply $1/\tau$? The unit of $2\pi$ is related to radian. But the unit of $f_{-3db}$ is $[1/second]$ so unit of $1/(2\pi\tau)$ does not seem to equate to Hz


1 Answer 1


Unit of frequency is not really $1/\mathrm{second}$.

$$ 1\mathrm{Hz} = 1 \frac{\mathrm{cycle}}{\mathrm{second}}=360\frac{\mathrm{degrees}}{\mathrm{second}}=2\pi\frac{\mathrm{radians}}{\mathrm{second}} $$

cycle, degrees, radians are kind of unitless since they all represent a fraction of a whole.

The unit of radians is kind of unitless since angle in radians is found by $\frac{\mathrm{arc length}}{\mathrm{radius}}$

Natural unit of frequency is radian/s. $$ \left|\frac{1}{1+s_{\text{(-3 dB)}}\tau}\right| = \frac{1}{\sqrt2}\\ \implies 1+\omega_{\text{(-3 dB)}}^2\tau^2=2 \qquad (\text{letting }s = j \omega)\\ \implies \omega_{\text{(-3 dB)}} = \frac{1}{\tau} $$ whre $\omega_{\text{(-3 dB)}}$ is the bandwidth frequency. Its unit is radians/second. To convert to hertz, $$ f_{\text{(-3 dB)}}=\frac{\omega_{\text{(-3 dB)}}}{2\pi} $$


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