Filters
Introduction
Low-Pass Filter

What Is A Filter and Why Would You Use One?

Electrical filters are like mechanical filters.  They are used to remove crud from signals.  A mechanical filter - like the fuel filter or oil filter in your car - might be built from a membrane that allows the fluid to flow through, but restricts the flow of contaminants that are particles.  Electrical filters can accomplish the same thing - removal of contaminating signals - but the physical actions are different.  In electrical filters, we take advantage of the filter's different responses at different frequencies, and the fact that many signals that are corrupted by noise have a signal and noise that have different frequency content.  For example, a low frequency signal corrupted by high frequency noise can often be "cleaned up" with a low-pass filter.

Filters come with different features.  For example, the low-pass filter mentioned above is an example of a filter that preferentially passes low frequency signals and does not pass higher frequency signals as well.  Other kinds of filters can include at least the following.

• High-pass filters - that preferentially pass high frequency signals.
• Band-pass filters - that preferentially pass signals with a strong frequency component within a band of frequencies.
• Band-reject filters - that preferentially reject signals in a certain frequency band.
In this lesson, we will examine some of these filters and the circuits that can be used to implement them.
Low-Pass Filters

A common type of filter is a low-pass filter.  Perhaps the simplest implementation of a low-pass filter is the simple RC filter shown below.

You should have encountered this filter in a previous lesson.  (Click here to go to that lesson.)  The primary results in that lesson give the ratio of the output amplitude and input amplitude and the phase shift between output and input.

• If the input, vin(t) is given by:
• vin(t) = A sin(wt)
• Then the output, vout(t) is given by:
• vout(t) = B sin(wt + f)
• Where
• A/B is given by:
• and, f = tan-1(wRC)
The principal conclusions gained from these expressions are:
• At low frequencies (much below w = 1/RC)
• The output amplitude is approximately equal to the input amplitude (B/A = 1).
• The output phase is approximately equal to the input phase (f = 0).
• At high frequencies (much above w = 1/RC)
• The output amplitude is attenuated so that B/A ~= 1/(wRC)
• The phase shift between input and output approaches -90o.

High Pass Filters

Filters come in many varieties.  Shown below is a high pass-filter.

This filter has the same components are the low-pass filter.  However, in this filter, the output is taken across the resistor instead of across the capacitor.  One simple explanation of how the circuit functions is as follows.

• In a low-pass filter, the output is taken across the capacitor, and the low frequency components appear across the capacitor.  That really means that low frequency components do not appear across the resistor.
• High frequency components do not appear across the capacitor.  However, since KVL must hold at any frequency, that means that the high frequency components appear across the resistor.
• So, high frequency components appear across the resistor, and low frequency components do not and the circuit must be a high-pass filter.
• If you re-do the analysis (as found in the first lesson on frequency response) you can determine that the ratio of output to input is:

What If?

When you use filters you can sometimes encounter loading problems.  For example, if you used a low-pass filter as a way of decreasing treble signals (because you really like to listen only to the bass!) then connecting a set of headphones directly to the filter would draw current from the filter (i.e. "load the filter") and the voltage would not be what you expected.  Here is the situation we are talking about.

Problem and Labs