# Converting time constant to bandwidth of 1st order system

When the system is first order:

$$1/(1+s\tau)$$

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

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, $$f_{\text{(-3 dB)}}=\frac{\omega_{\text{(-3 dB)}}}{2\pi}$$