A basic introduction to the theory, analysis, and design of bandpass 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.
Bandpass filters are extremely useful tools in electronic design. A bandpass filter is designed to pass all frequencies within a band of frequencies, such that there is a high and low cutoff. For example, a bandpass filter could be designed specifically to pass only those frequencies found in the human voice range. What may surprise you is that you use bandpass filters every day. Most telephone systems, including cellular and landline phones, use bandpass filtering. If you try and play sounds above the 3000Hz (3kHz) range, you will find that the person on the other end will not hear it. Now try playing some really low tones, below 300Hz, and you won't hear them on the other end either. This is designed to make telephone conversation more intelligible at human voice frequencies. People have said that the human voice (male and female) carries the majority of power and emotion at frequencies below this level, but that is an argument for another node...
If you read further, you'll see how bandpass filters help you listen to the radio (ever wonder how you can just magically pick up a new FM station on your dial?) and that they aren't as complicated as you might think. I guarantee you use them everyday and you never know it.
As you can see, bandpass filters are extremely useful. They are also found in DSL and cable Internet technologies, fiber optic communication, types of digital image processing, noise cancellation, and myriad other examples. Understanding them is important for both engineers, and hobbyists.
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 bandpass 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. It's obvious that 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
Each type of filter has many particular applications, and multiple filters may be used to perform higher-level filtering. Examples will be provided below.
Bandpass Filter Theory
Bandpass filters, as well as the other three types of filters, can come in two general 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 bandpass filter passes all frequencies within a given band of frequencies. The frequency band, or range, is determined by the design of the circuit, which dictates the given cutoff frequencies. Unlike highpass and lowpass filtering, bandpass filters have two distinct cutoff frequencies that provide the high and low ends of the range. Any frequency falling outside this range will be attenuated. Bandpass filters can be formed by cascading a highpass and lowpass filter together to cutoff both the low and the high frequencies surrounding the desired range. Typically, this results in a circuit less suitable for specific design tweaking, so designs typically are made using a single circuit instead of two cascaded designs.
1 | _______________
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Figure 1 is a graph of an "ideal" bandpass 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 frequency inside the bandpass range, 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 bandpass filter will have a strong signal for frequencies within the frequency range "X1", and "X2." The frequency range is centered about the "center frequency", ωo. Outside this range, the gain will be essentially zero, meaning the signal is attenuated. This relates the fundamental theory behind bandpass filtering. All the "high" and "low" frequencies, which are really determined by the application of the filter, will be rejected. Any frequencies falling within the desired "band" of frequencies will be passed.
To quickly summarize:
- At any frequency outside the desired "band" or range, bandpass filters have a theoretical gain of 0.
- At any frequency within the desired range, bandpass filters have a theoretical gain of 1.
- The "cutoff frequencies," or the points at which the theoretical gain switches between 1 and 0 or 0 and 1, is determined by filter components and application.
Bandpass Filter Design and Functionality
The following digs a little deeper into real bandpass 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 bandpass circuit, so I will attempt to explain, rather than graph and calculate.
A simple RLC circuit bandpass filter is constructed from a capacitor, an inductor, and resistor in series. The output is read across the resistor, and is referenced to ground. To make this a little clearer, I have included a simple circuit diagram:
|---@@@@@----------||----|X (or Vo)
+ | |
Vs O W Resistor
- | |
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 common terminology). We are taking the "measurement" across the resistor, but we are not altering the circuit. We are measuring the difference in signal between X and G.
The capacitor and inductor are the keys to this circuit. As circuit elements, capacitors and inductors behave differently depending on frequency. To low frequencies, or DC (no frequency), a capacitor looks like an open circuit and an inductor acts as a short circuit:
|--------------- ----|X (or Vo)
+ | |
Vs O W Resistor
- | |
So in Figure 3, low frequencies see the circuit disconnected. They never pass to the output, and we will never see them on the other side of the capacitor
. Although the inductor
is very capable of allowing low frequencies to pass, the capacitor blocks them.
The circuit is completely different at high frequency:
|--- ----------------|X (or Vo)
+ | |
Vs O W Resistor
- | |
Now, the capacitor appears as a short circuit to high frequencies, and the inductor acts as an open circuit. No high frequency signals will pass through inductor, and therefore our output will be zero. So we've established that at "high" and "low" frequencies, we will see no output at Vo.
Now you may be saying "If the inductor and capacitor block all the frequencies, how do we pass anything through to the output?" Well, by choosing the values of our circuit elements (resistors, capacitors, and inductors), we can specify a band of frequencies between the "low" and "high" range that will pass through the circuit. Remember that when we say low and high, we mean relative to the application of the circuit. The application dictates the frequency range, and in turn, the resistor, capacitor, and inductor values required. As an example, for POTS telephone systems (land-line phones) high frequency is anything over 3kHz. For cable Internet service, high frequency can be in the MHz range (varies by provider and channel allocation).
So a signal comes into the filter that is a composite of many frequencies. The low frequencies see the capacitor as an open circuit and the inductor as a short circuit (Figure 3), and the high frequencies see the capacitor as a short circuit, and the inductor as an open circuit (Figure 4). Looking at the output at X, assuming we choose the right values for our circuit elements, we will see only the frequencies within our desired range (X1 to X2).
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 determination of the values for our circuit elements comes from the transfer function, so it is important to determine this for any circuit you design. The transfer function, or ratio of output to input, for the circuit in Figure 2 is given by the following equation:
H(ω) = R /( R + (j *(ωL - 1/ωC)) = Vo / Vs
of this formula
is not difficult, but involves some basic circuit analysis. For information on how to derive this, check voltage divider
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 bandpass circuit. Plotting H versus ω will provide a graph similar to Figure 1, but with much less of a hard corner at the cutoff frequencies. The graph will take a much more gradual slope at the cutoff frequencies. Note, that in the transfer function equation, if we evaluate for ω = 0 (frequency = 0), the gain (H) is 0 (or R/(R+infinity) ). If we set ω = infinity, then the gain (H) becomes 0 (or R/(R+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 center frequency, as mentioned earlier in reference to Figure 1, (ωo) can also easily be determined:
ωo = 1/SQRT(LC)
SQRT is the square root
When designing a bandpass filter like this, you can choose your center frequency by picking appropriate values for your resistor, capacitor, and inductor. It's that easy. When designing a bandpass filter for any given application, you can determine where you want your cutoff frequencies, also called rolloff, and center frequency to be located.
An example of a bandpass filter in action can be found in your stereo equalizer. When you set the equalizer higher for the middle frequency sounds, you are essentially bandpass filtering the mid-range in your music, then amplifying the result, and outputting it again. This filtering allows you to adjust only the middle frequency, between treble and bass, by amplifying the output of your filter, which will only be the middle frequencies in the audible human frequency range. Similarly, you would use a lowpass filter to do this to the lower frequency, or bass, in your music, and a highpass filter to do this to the treble. The filters used in audio applications are much more complex than the example provided here.
Another example of bandpass filters can be found in any AM/FM radio. When you change the station on your AM or FM dial, you are essentially dynamically changing the values of the circuit elements in your bandpass filter to tune in a different frequency of radio signal. So the radio transmitter at station WXYZ sends out a signal at 100.1 MHz FM (frequency modulated). You change your car stereo receiver to 100.1 MHz and the internal filter selects only the 100.1MHz frequency from your antenna. Pretty neat huh? Analog TV tuning (rabbit ears) uses a very similar method. We have so many different frequencies running through the atmosphere that picking up just the frequency you want isn't easy without a trusty bandpass filter.
There are many other circuits for bandpass filters. These can range from other simple examples, to very complicated filters used for wireless communication 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.