A basic introduction to the theory, analysis, and design of lowpass filters...
Basic Electric Filter Background
Electrical filters are an extremely integral part in the evolution of engineering, more specifically electrical engineering. Due to this importance, there has been an incredible amount of research and expansion on the design theory and construction of various types of filters.
Lowpass filters are used in a great variety of applications. They are integral in audio sampling and equalizing, they are used in data aquisition units, power lines are often lowpass filtered to eliminate noise, and they are extremely useful in digital image filtering. For these reasons, they are almost always included in introductory electrical engineering courses and texts. They are usually covered again in more depth during a circuit theory course, as well as in linear systems courses. Lowpass filters are needed when it is desireable to deal only with a low range of frequencies, and an incoming signal may have many frequencies or noise convolved together. They have the useful ability of stripping out all undesired frequency, leaving only the range that is useful.
As with any important aspect of technology, filters have been expanded from very simple, to extremely complex. The following information should be considered introductory with regards to theory and design of filter technology. An overview of the concepts, mathematics, and electrical principles of basic lowpass filtering will be covered. For more detailed information, consult a circuit theory or fundamental electrical engineering design textbook. There is much more to be learned on the subject, but it can quickly delve into more complicated math and electrical theory that would be extremely long and drawn out. There are very long textbooks written on the subject of electrical filtering. Needless to say, a more thorough treatment would be...well...boring for more casual readers. Obviously, I find this stuff extremely interesting.
Generally, there are four types of filters:
Beyond these filters, one can explore
digital filters,
electromechanical filters, and
microwave filters, to name a few, but these are generally much more
advanced topics.
Each type of filter has many particular applications, and multiple filters may be used to perform higher-level filtering. Examples will be provided below.
Lowpass Filter Theory
Lowpass filters, as well as the other three types of filters, can come in two generally varieties: passive and active. Passive filters consist of passive circuit elements: resistors, inductors, and capacitors (R, L, and C). These are more basic circuit elements. Active filters contain active components, such as transistors and operational amplifiers (op amps), in addition to passive elements (R, L, and C).
A lowpass filter passes low frequencies and stops high frequencies. Ideally, any frequency from DC (constant, no fluctuation) to a specific "cutoff frequency" is passed by the filter. The cutoff frequency is generally dependent on the filter design components and application.
Figure 1
|Gain|
|
|
|
|
|
1 |___________________
| |
| |
| |
| |
| |
| |
|-------------------X------------------
Frequency
Figure 1 is a graph of an "ideal" lowpass filter. For simplicity we will call the vertical axis "Gain." Think of this as the strength of the output signal. A high gain means you have a signal of significant amplitude. Conversely, a low gain means your signal is very weak, or non-existent (having no amplitude). This graph is somewhat normalized, where the gain is 1 for low frequency, meaning that the signal is passed. A higher gain, say 2, would mean the signal was amplified, where any gain lower than 1, means the signal is attenuated. For this example, the magnitude or absolute value of the gain, denoted by |Gain| is used. This notation is more complete as it takes into account the fact that an alternating signal of any given frequency may have positive and negative values with respect to a 'common' or "ground."
The horizontal axis is labeled "Frequency," and increases to the right. The lowest frequency will be constant, or DC, and the frequency increases theoretically to infinity along the horizontal axis.
Figure 1 shows that an ideal lowpass filter will have a strong signal for frequencies below the frequency labeled "X." Beyond that point, the gain will be essentially zero. This relates the fundamental theory behind lowpass filtering. All the "high" frequencies, which are really relative to the application of the filter, will be rejected. Any "low" frequencies will be passed.
To quickly summarize:
- At a frequency of 0 (DC), lowpass filters have a theoretical gain of 1.
- At a frequency of infinity, lowpass filters have a theoretical gain of 0.
- The "cutoff frequency," or the point at which the theoretical gain switches between 1 and 0, is determined by filter components and application.
Lowpass Filter Design and Functionality
The following digs a little deeper into real lowpass filter design and application. I will try to keep it as light on the math as possible. It's not easy to create more complicated graphs for the actual response of a lowpass circuit, so I will attempt to explain, rather than graph and calculate. Consider this "fat-free" lowpass filter design.
A typical, and simple, lowpass filter is formed when the output of an RC (resistor and capacitor) circuit is measured off the capacitor. To make this a little clearer, I have included a simple circuit diagram:
Figure 2
Resistor
______MWMWMWMW____
+ | |X (or Vo)
| |
| |
Vs O == Capacitor
| |
| |
- |__________________|G
In Figure 2, the label Vs on the left hand side is the "voltage source." This is essentially the signal you are filtering. The positive (+) side is the signal, and the ground (-) is the common ground of the filter and the signal. The common ground is just a reference for the circuit. This is similar to the third prong on appliance plugs. It grounds the system and allows signal levels between two different components (your wall socket and TV for example) to share voltage levels.
The output of the filter in Figure 2 is read across the points marked X and G. These are generally referred to as "terminals" and the measurement is taken with respect to ground (G). So the terminal marked X is the output of our filter (Vo means Output Voltage and is also common terminology). We are taking the "measurement" across the capacitor, but we are not altering the circuit. We are measuring the difference in signal between X and G.
The capacitor is the key to this circuit. As a circuit element, a capacitor behaves differently depending on frequency. To low frequencies, or DC (no frequency), a capacitor looks like an open circuit:
Figure 3
Resistor
______MWMWMWMW____
+ | |X (or Vo)
|
|
Vs O
|
|
- |__________________|G
So in Figure 3, low frequencies pass directly from the source to the output through the
resistor.
The circuit is completely different at high frequency:
Figure 4
Resistor
______MWMWMWMW____
+ | |X (or Vo)
| |
| |
Vs O |
| |
| |
- |__________________|G
Now, the capacitor appears as a short circuit to high frequencies. All high frequency signals will pass through the resistor to ground. This essentially eliminates them.
So a signal comes into the filter that is a composite of high and low frequencies. The low frequencies see the capacitor as an open circuit (Figure 3), and the high frequencies see the capacitor as a short circuit (Figure 4). Looking at the output at X, the low frequency signals will be present, and the high frequencies will not.
Theoretically, X would look similar to Figure 1 for different frequencies. We will soon see that this is not completely accurate, but is a good theoretical simplification.
Filters are compared and examined by means of a "transfer function". A transfer function is simply a ratio of the output voltage (resulting signal), to the input voltage (original signal). The transfer function, or ratio of output to input, for the circuit in Figure 2 is given by the following equation:
H(ω) = 1/(1 + jωRC) = Vo/Vs
The
derivation of this
formula is not difficult, but involves some basic circuit analysis. For information on how to derive this, check
voltage divider (replace R2 with
C =
jωC -- This is because capacitors are
complex circuit elements).
ω = 2 *
π *
f, where
f is the frequency and
π is just
Pi, or 3.14. This is more for
convenience than anything.
H is
convention for
transfer function, and is a
function of
ω.
H is
roughly equivalent to
gain, as mentioned in Figure 1. This basic transfer function describes the functionality of this lowpass circuit.
Plotting H versus ω will provide a
graph similar to Figure 1, but with much less of a hard
corner at the
cutoff frequency. The graph will take a much more gradual
slope at the cutoff frequency. Note, that in the transfer function equation, if we evaluate for
ω = 0 (frequency = 0), the gain (
H) is 1 (or 1/(1+0) ). If we set
ω =
infinity, then the gain (
H) becomes 0 (or 1/infinity ).
As was mentioned before, the actual graph of H will look more like a gradual slope, and less like Figure 1. Go ahead and graph it to see.
The cutoff frequency (ωc)can also easily be determined:
ωc = 1/(RC)
When designing a lowpass filter like this, you can choose your cutoff frequency by picking appropriate values for your resistor and capacitor. It's that easy. When designing a lowpass filter for any given application, you can determine where you want your cutoff, also called rolloff, frequency to be located.
An example of a lowpass filter in action can be found in your stereo equalizer. When you set the equalizer higher for the lower frequency sounds, you are essentially lowpass filtering the bass in your music, then amplifying the result, and outputting it again. This filtering allows you to adjust only the low frequency, or bass, by amplifying the output of your filter, which will only be the low frequencies. Similarly, you would use a highpass filter to do this to the higher frequency, or treble, in your music. The filters used in audio applications are much more complex than the example provided here.
There are many other circuits for lowpass filters. These can range from other simple examples, to very complicated filters used for power transmission or high-quality audio applications. If you are interested in these approaches and applications, I suggest you find a book on circuit theory and design, or more specifically on electric filter design.
I hope this provides a thorough and understandable overview. If you find that something has been omitted, or that something is unclear, please let me know and I will make an attempt to update or clarify. Look for additions here in the future when my ASCII graph skills improve.
Sources: My own brain. I have over six years of electrical engineering education under my belt (going for my masters currently). If you need some sources, I could name a few great textbooks for you to thumb through.