A basic introduction to the theory, analysis, and design of bandstop 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.
Bandstop filters are a great tool in electronic design. Basically, they are the inverse of bandpass filters. Bandstop filters are variably known also as a bandreject or notch filter. Their usefulness comes from being able to filter out (attenuate) all frequencies in a given range or "band" or frequency, while passing all frequencies outside this range. Bandstop filters are also called Notch filters or wavetraps. A Notch filter has a slightly definition than a bandstop filter. Notch filters generally target a single frequency, such as 60Hz (the frequency of common walloutlet electricity). A bandstop filter generally rejects a range of frequencies. The two terms are often used interchangeably. Bandstop filters are used in telephone line noise reducers, DSL Internet technology, digital image processing, and many power amplification technologies. Their uses vary, but they pop up in a wide variety of different forms and complexities. Bandstop filters are also very common in electric guitar amplifiers. The "hum" that is produced in guitar amplifiers is 60Hz hum from the power outlet. A bandstop filter can significantly reduce this "hum," and can factor into the difference between a good amplifier, and very good amplifier. Bandstop filters are also very useful in communications electronics for eliminating harmonics of signals that are desired. Harmonics result from waves combining out of phase to produce other related frequencies. These related frequencies can cause interference or communications errors. Therefore, bandstop filters can be employed to eliminate residual harmonics.
If you read further, you'll see how bandstop filters help you reduce static on your radio, reduce interference in cell phones, and that they aren't as complicated as you might think. I guarantee you use them everyday and you never know it.
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 bandstop 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 topics.
Each type of filter has many particular applications, and multiple filters may be used to perform higherlevel filtering. Examples will be provided below.
Bandstop Filter Theory
Bandstop 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 bandstop filter rejects 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, bandstop 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.
Figure 1
Gain





1 ____________ ____________________
  
  
   
  
   
  
X_{1}ω_{o}X_{2}
Frequency
Figure 1 is a graph of an "ideal" bandstop 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 nonexistent (having no amplitude). This graph is somewhat normalized, where the gain is 1 for frequency outside the bandstop 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 bandstop filter will have an attenuated signal for frequencies within the frequency range "X_{1}", and "X_{2}." The frequency range is centered about the "center frequency", ω_{o}. Outside this range, the gain will be essentially unchanged, meaning the signal is passed. This relates the fundamental theory behind bandstop filtering. All the "high" and "low" frequencies, which are really determined by the application of the filter, will be passed. Any frequencies falling within the defined "band" of frequencies will be rejected.
To quickly summarize:
 At any frequency outside the defined "band" or range, bandstop filters have a theoretical gain of 1.
 At any frequency within the defined range, bandstop filters have a theoretical gain of 0.
 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.
Bandstop Filter Design and Functionality
The following digs a little deeper into real bandstop 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 bandstop filter is constructed from a capacitor, an inductor, and a resistor in series. The output is read across the capacitor and inductor series pair, and is referenced to ground. To make this a little clearer, I have included a simple circuit diagram:
Figure 2
Resistor
MWMWMWMWX (or Vo)
+  
 
 = Capacitor
 
Vs O 
 @
 @ Inductor
 @
  
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 resistor, but we are not altering the circuit. We are measuring the difference in signal between X and G. This difference will be the filtered signal (differing from the input signal in that it is not bandstop filtered).
This type of circuit is also commonly called a resonant circuit. 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:
Figure 3
Resistor
MWMWMWMWX (or Vo)
+  

 Capacitor
 
Vs O 
 
  Inductor
 
  
G
So in Figure 3, low frequencies see the capacitor and inductor as disconnected and pass freely to the output. Although the
inductor is very capable of allowing low frequencies to pass, the capacitor blocks them from passing to ground and this keeps them in the output.
The circuit is completely different at high frequency:
Figure 4
Resistor
MWMWMWMWX (or Vo)
+  
 
  Capacitor
 
Vs O 

 Inductor

  
G
Now, the capacitor appears as a short circuit to high frequencies, and the inductor acts as an open circuit. So the high frequencies will pass through the resistor, see and be available at the output. Because the inductor acts as an open circuit, signals are all present at the output (Vo).
Now you may be saying "If the inductor and capacitor retain all the frequencies, how do we filter 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 series combination of capacitor and inductor. 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. So by choosing a capacitor and inductor value that will make them both appear more like short circuits, we get Figure 5:
Figure 5
Resistor
MWMWMWMWX (or Vo)
+  
 
  Capacitor
 
Vs O 
 
  Inductor
 
  
G
So signals in the "middle" range see the
capacitor and
inductor as short circuits. These "middle"
frequency signals will pass directly across the
short circuit to
ground and be dispersed. This eliminates them from the
output signal that is passed through Vo to whatever is
hooked up to our filter. Figure 5 is how the circuit looks only to frequencies in the range to be
filtered out. Electronic signals always take
the path of least resistance. That is why a
short circuit takes all the "middle" signals away. These signals
take the easy way out and slip out of the circuit to ground.
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 in Figure 5, assuming we choose the right values for our circuit elements, we will see only the frequencies outside the defined range (X_{1} to X_{2}).
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) = j*(ωL  1/ωC) /( R + (j *(ωL  1/ωC)) = 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.
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 bandstop 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 1 (or (infinity)/(infinity) ). If we set ω = infinity, then the gain (H) becomes 1 (or (infinity)/(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, (ω_{o}) can also easily be determined:
ω_{o} = 1/SQRT(LC)
This is the same as the center frequency for bandpass filters.
SQRT is the
square root function.
When designing a bandstop 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 bandstop filter for any given application, you can determine where you want your cutoff frequencies, also called rolloff, and center frequency to be located.
Bandstop filters are commonly used, as I mentioned earlier, to eliminate particular frequencies of noise. If you've ever dealt with the phone company in preparing to install DSL service, they may have provided you with a filter for you household phone. This may be because your storebought phone can create noise on the phoneline that can interfere with the DSL Internet service. The filter they provide you generally helps eliminate interference and noise on the line that may degrade the DSL signal into your house.
Bandstop filters are used in many household appliances to reduce the 60Hz interference emanating from the walls of your house. If you have 60Hz electricity running through your whole house, you're bound to develop some serious interference in the 60Hz range. Therefore, bandstop filters at 60Hz have received a great deal of attention since the beginning of signal filtering.
Bandstop filters have become big news in the optical networking world as of late. There are often spurious frequencies of light that show up unwanted on the receiving end of fiberoptic cables. Bandstop filters provide a useful service in solidstate optoelectronic design.
There are many other circuits for bandstop filters. These can range from other simple examples, to very complicated filters used for wireless communication or highquality 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.