Second order low pass filter

First order filters have a roll-off of 20 decibel per decade.
If it is desired to have a steeper roll-off in order to attenuate a certain frequency range more effective, a second order filter should be used.
A second order low pass filter is, for example, obtained by combining a capacitor and an inductor, as shown in the figure below:
If the second order low pass filter is analysed below in a more mathematical way, one obtains the following transfer function:

Such a circuit does not contain a resistor anymore. As both the inductor and the capacitor are non-dissipating components, the system has no ability to lose energy. Although this is an idealised situation as in reality systems are never lossless, because of the resistance of the cables, for example, those losses are negligible and so the assumption is valid. However, power is continuously fed into the system, which implies the system has a resonance. The presence of resonance at a certain frequency corresponds to what we know about second order systems.

As the denominator of the transfer function has both a fixed positive part and a variable negative part, for some frequency w it will be zero. Such behaviour implies a resonance, as that is where the total transfer function will have an infinite magnitude. The frequency at which the denominator equals zero depends on the properties of the capacitor and the inductor:

The bode plot of such a transfer function type is shown in the figure below. For this figure we used L = 40 mH and C = 50 µF. The resonance is indicated by the peak.

The next figure presents a second order low pass filter, in which a resistor is placed parallel to the capacitor. This creates a damping term, which can be used to attenuate the resonance.

In the last figure on this page the frequency response of both a first order low pass filter and a damped second order low pass filter is plotted. For both filters the properties of the components have been chosen such that a 440 Hz corner frequency was obtained. This figure clearly shows difference between a first and second order filter in the attenuation of higher frequencies.