An introduction to the theory, analysis, and design of highpass 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.
Highpass filters are a very common construction, and they are extremely important in electronic design. They are found in myriad different applications. Some examples are TVs, digital image processors, AM/FM tuners, and many home appliances (to eliminate 60hz noise). A very common use in your home is to employ highpass filters as noise cancellation devices. The 60hz frequency of American wall outlets provides a large quantity of 60hz noise in many appliances. Highpass filters are a simple way to deal with this.
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 highpass 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.
Highpass Filter Theory
Highpass 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 highpass filter passes "high frequencies" and attenuates low frequencies. Ideally, any frequencies above a specified “cutoff frequency” are passed. The cutoff is determined by circuit components and can vary greatly. High and low frequency are completely relative, as the application can be within any frequency range. Therefore, when saying "high frequency" we generally mean in a relative sense.
Figure 1
Gain





1  ________________
 
 
 
 
 
 
X
Frequency
Figure 1 is a graph of an "ideal" highpass 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 high 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 highpass filter will have a strong signal for frequencies above the frequency labeled "X." Below that point, the gain will be essentially zero. This relates the fundamental theory behind highpass filtering. All the "low" frequencies, which are really determined by the application of the filter, will be rejected. Any "high" frequencies will be passed.
To quickly summarize:
 At a frequency of 0 (DC), highpass filters have a theoretical gain of 0.
 At a frequency of infinity, highpass filters have a theoretical gain of 1.
 The "cutoff frequency," or the point at which the theoretical gain switches between 1 and 0, is determined by filter components and application.
Highpass Filter Design and Functionality
The following digs a little deeper into real highpass 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 highpass circuit, so I will attempt to explain, rather than graph and calculate. Consider this "fatfree" highpass filter design.
A simple RC circuit highpass filter is constructed from a capacitor and resistor in series. The order of the capacitor and resistor is reversed from the lowpass filter design. 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:
Figure 2
Capacitor
X (or Vo)
+  
 
 W
 M
Vs O W Resistor
 M
 W
 
  
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 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 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
 X (or Vo)
+  
 
 W
 M
Vs O W Resistor
 M
 W
 
  
G
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.
The circuit is completely different at high frequency:
Figure 4
X (or Vo)
+  
 
 W
 M
Vs O W Resistor
 M
 W
 
  
G
Now, the capacitor appears as a short circuit to high frequencies. All high frequency signals will pass through the resistor to ground. We are reading the output across the resistor, and all we will see is the high frequencies. All the low frequency signals will be stopped before the capacitor.
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 high frequency signals will be present, and the low 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) = (jωRC)/(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 R1 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 highpass 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 0 (or 0/(1+0) ). If we set
ω =
infinity, then the gain (
H) becomes 1 (or infinity/(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 (ω_{c})can also easily be determined:
ω_{c} = 1/(RC)
When designing a highpass 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 highpass filter for any given application, you can determine where you want your cutoff, also called rolloff, frequency to be located.
An example of a highpass filter in action can be found in your stereo equalizer. When you set the equalizer higher for the higher frequency sounds, you are essentially highpass filtering the treble in your music, then amplifying the result, and outputting it again. This filtering allows you to adjust only the high frequency, or treble, by amplifying the output of your filter, which will only be the high frequencies. Similarly, you would use a lowpass filter to do this to the lower frequency, or bass, in your music. The filters used in audio applications are much more complex than the example provided here.
Another example of highpass filters can be found in digital signal processors (DSP). DSPs are found in everything from your cell phone to music sampling and mixing tools. DSPs need to take out all the low frequency noise in a signal, and will employ highpass filtering at the input to cut out any noise coming in along side the usable signal.
There are many other circuits for highpass filters. These can range from other simple examples, to very complicated filters used for power transmission 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.