Fun with filters!
Note: A basic knowledge of electrical engineering principles is helpful, but not vital, for understanding this node. Knowledge of the behaviour of capacitors and inductors is also desirable.
Band pass and band stop filters, although extremely similar in construction and use, perform very different functions. A band pass filter allows frequencies or ac voltages of a specific frequency (called the resonance frequency) to go through, and blocks out or at least severely attenuates any other frequency. Band stop filters are quite the opposite: they block specific frequencies from passing, and allow all others to go through. These things are used widely in various fields of engineering, from reducing noise to metal detectors.
Theory:
The relationship between voltage and current in a capacitor is given by:
dv Ic = capacitor current
Ic = C C = capacitance
dt v = rms voltage
And in an
inductor:
IL = inductor current
IL = 1/L∫vdt L = inductance
v = rms voltage
If the applied voltage is a perfect
sinusoid, i.e. v =
Vsin(ωt), then the respective currents will be:
Ic = ωCVcos(ωt) and I_{L} = 1/ωLVcos(ωt)
Thus, the
sine wave has produced a
cosine current. This is equivalent to a
phase shift of 90˚. For the capacitor this shift is +90˚, for the inductor this is 90˚. However,
fortuitously we can ignore the effects of phase shifting in these filters.
But where does that get us? I'm glad you asked,
Timmy. Now we have expressions for V and I, we can relate the two in accordance with
Ohm's Law, which produces:
V/I_{L} = ωL and V/Ic = 1/ωC
And since V/I is measured in ohms (Ω),
ωL and
1/ωC indicate the
impedance of an inductor or capacitor to the flow of a sinewave current which has frequency ω.
For a list of symbols used, please see the end of this node.
Basic circuit representations of filters:
^{Crappy ASCII art, go!}
KEY:
  
\   3
/ OR /\/\/\  OR  3 OR nnnn
   
Resistor Capacitor Inductor
Circuit 1: Band pass filter
__________ A
 /\/\/\+ /\
 /\ ____ 
 Input     
Oscillator  Vin  3  Vout
    3 
   _____ 
    
__________+ 
B
Circuit 2: Band stop filter

__________ ++ A
   + /\
 /\    
 Input   +nnnn+ \ 
Oscillator  Vin /  Vout
   \ 
   / 
    
__________+ 
B
Calculating the resonance frequency
Definition of resonance frequency: That frequency at which the capacitorinductor pair exhibits the highest impedance.
In the band pass filter, this is the frequency allowed to pass. In the band stop filter, this is the frequency blocked.
There are two ways to calculate the resonance frequency (f_{0}): mathematically with a formula, or through experimentation. Both are explained.. below.
Mathematically:
Both the inductance (L) and capacitance (C) need to be known. These can then be plugged into the following formula:
ω_{0} = √LC where ω_{0} = 2πf_{0}
Therefore
f_{0} = 1 / 2π√LC
Experimentation
Ah, the simple, oldfashioned method of plugging it all up and throwing the switch. The important readings to be taken are marked on the circuit diagrams as Vout
and Vin
.
Get readings for Vout
and Vin
for a range of different frequencies, say 50  500 Hz. For each reading, calculate Vout / Vin
. Now, plot these points on a graph, with Vout / Vin on the yaxis and log(frequency) on the xaxis. Something approaching the following should come out:
1 

0.9  x
 xx
0.8  x  x
 x  x
0.7  x  x
 x  x
0.6  x  x
Vo  x  x
 0.5  x  x
Vi  x  x
0.4  x  x
 
0.3  o  o
 o  o
0.2  o  o
 o  o
0.1  oo
 o
0 +X
10 100 f0 1000
Frequency (Hz) (log scale)
KEY:
x == band pass filter
o == band stop filter
The point where the peak and trough line up onto the xaxis is equal to
f_{0}. This line should be straight and vertical  if not, perhaps some experimental inaccuracy is to blame.
And that, as they say, is that. A quick note: For the band pass filter, a highvalue resistor (>1kΩ) is preferable, and for the band stop filter a lowvalue resistor (<1kΩ, ~100Ω) gives better results.
this node is almost over
List of variables, constants and other jobbies used
L = inductance, measured in Henrys.
C = capacitance, measured in Farads.
Ic = capacitor current, measured in amperes.
I_{L} = inductor current, measured in amperes.
v = r.m.s. voltage, measured in volts.
V = peak voltage, measured in volts.
Vout = output r.m.s. voltage, measured in volts.
Vin = input r.m.s. voltage, measured in volts.
ω = radian frequency, equal to 2πf.
f = frequency, measured in hertz.
f_{0} = resonance frequency, measured in hertz.
π = constant, approximately equal to 3.1415926535...
it's over