RLC low pass filter type 1
The magnitude of the voltage transfer function is derived from the potential divider equation: -
\(\dfrac{V_{OUT}}{V_{IN}} = \dfrac{\frac{1}{j\omega C}}{R + j\omega L +\frac{1}{j\omega C}} =\dfrac{1}{j\omega RC + j^2\omega^2 LC + 1}=\dfrac{\frac{1}{LC}}{\frac{1}{LC}-\omega^2+j\omega\frac{RC}{LC}}\)
For a 2nd order filter \(\frac{1}{LC} = \omega_n^2\) and \(RC\omega_n = \frac{1}{Q} = 2\zeta\) we get: -
\(\dfrac{V_{OUT}}{V_{IN}} =\dfrac{1}{1-\dfrac{\omega^2}{\omega_n^2}+j2\zeta\dfrac{\omega}{\omega_n}}\) this yields the TF's phase angle: \(-\arctan\left[\dfrac{2\zeta\dfrac{\omega}{\omega_n}}{1-\dfrac{\omega^2}{\omega_n^2}}\right]\)
Multiplying numerator and denominator by the conjugate of the denominator we get: -
\(\dfrac{V_{OUT}}{V_{IN}} =\dfrac{1-\dfrac{\omega^2}{\omega_n^2}-j2\zeta\dfrac{\omega}{\omega_n}}{(1-\dfrac{\omega^2}{\omega_n^2})^2+4\zeta^2\dfrac{\omega^2}{\omega_n^2}}\)
The magnitude is found by taking the square root of the squared numerator terms: -
\(|H(j\omega)| = \dfrac{\sqrt{(1-\dfrac{\omega^2}{\omega_n^2})^2+4\zeta^2\dfrac{\omega^2}{\omega_n^2}}}{(1-\dfrac{\omega^2}{\omega_n^2})^2+4\zeta^2\dfrac{\omega^2}{\omega_n^2}} = \dfrac{1}{\sqrt{(1-\dfrac{\omega^2}{\omega_n^2})^2+4\zeta^2\dfrac{\omega^2}{\omega_n^2}}} \)
\(|H(j\omega)| = \dfrac{1}{\sqrt{1+\dfrac{\omega^2}{\omega_n^2}\cdot(4\zeta^2-2)+\dfrac{\omega^4}{\omega_n^4}}}\)
As a sanity check, if \(\omega\) is set to the natural frequency we get: -
\(|H(j\omega)| = \dfrac{1}{\sqrt{1+(4\zeta^2-2) +1}} = \dfrac{1}{2\zeta} = Q\) as expected.
