How to get the crossover frequency from a Laplace oquasion
Cross-over frequency, in the context of control / signal-systems, is defined as the frequency where the magnitude of transfer function crosses the 0 dB axis (amplification = 1). For the first-order system placed in standard form as below
$$H(s) = \frac{1}{\frac{s}{a}+1}$$
the cross-over frequency is zero (see the figure). In other hand, when a gain \$K\$ is present, that value can be determined as:
$$H(s) = \frac{K}{\frac{s}{a}+1}$$
For the sinusoidal steady-state response:
$$H(j\omega)=\frac{K}{j\frac{\omega}{a}+1}$$
With magnitude:
$$|H(j\omega)|=\frac{K}{\sqrt{(\frac{\omega}{a})^2+1}}$$
Making the above expression equal to 1 and \$\omega=\omega_c\$, where \$\omega_c\$ is the cross-over frequency:
$$\omega_c=a\sqrt{K^2-1}$$
The unit is radians/second. In Hertz, divide by \$2\pi\$.
This expression is valid for \$K&gt;1\$. But if \$K\$ is very large, then the result can be approximated by
$$\omega_c\approx aK$$
in this case coinciding with the high frequency asymptote on the Bode diagram.