Low Pass Filters

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Low-Pass Filters

A Low-Pass Filter is a circuit that only allows low-frequency signals to pass, and attenuates or reduces signals whose frequencies exceed its cut-off frequency. It is also referred to as a 'high-cut filter' or, when used in audio applications, as a 'treble-cut filter'. One common application of low-pass filters is for driving subwoofers (speakers designed for bass sounds) and other loudspeakers that don't efficiently broadcast sounds of high pitches. The low-pass filter is the opposite of the high-pass filter.

An ideal low-pass filter is one that completely blocks all frequencies above a given frequency, while allowing all those with lower frequencies to pass unchanged. Of course, an ideal low-pass filter doesn't exist in the real world, so ways to quantify or describe the effectiveness and efficiency of a low-pass filter have been devised.

Figure 1. Low-Pass Filters

The quality of a low-pass filter may be expressed in terms of its n-order. An n-order filter reduces the signal strength by 6n dB for every octave increase in frequency, i.e., every time the frequency doubles.

Thus, a first-order low-pass filter (n=1) will reduce the signal strength by 6 dB every time the frequency doubles. Mathematically, -6dB = 20 log P2/P1, which yields P2/P1 = 0.501. This means that a first-order filter reduces the strength of the signal by about 50% every time the frequency doubles.

As further illustration, a second-order low-pass filter (n=2) will reduce the signal strength by 12 dB every time the frequency of the signal doubles (-12 dB per octave). Thus, a second-order low-pass filter will reduce a signal to just 1/4 its original level every time its frequency increases by an octave.

Note that in each of the low-pass filters shown above, the inductors are in series with the input while the capacitors are in shunt with the input. This is because the reactance XL of an inductor increases with the signal frequency, i.e., XL = 2πfL, while the reactance XC of a capacitor decreases with the signal frequency, i.e., XC = 1 / 2πfC. Thus in these low-pass filters, the inductors resist the passing of an ac signal as the frequency increases, while the capacitors shunt them towards the ground as the frequency increases. Either way, the effect is to attenuate the signal as frequency increases.

The following equations apply to the low-pass filters in Figure 1 above:

1) L = Zo / πf

2) C = 1 / (πf Zo)

3) Zo = sqrt(L/C)

4) f = 1 / (π sqrt(LC))

where Zo is the line impedance and f is the cut-off frequency of the filter.