Finding the Prototype Values

Calculations for the Prototype Filter

Calculating the prototypical filter requires knowing the order of the filter. It is possible to select an order based on past experience. There is a formula based on the corner frequency and the amount of attenuation desired at a selected frequency in the stop band.

order using log
where fc is the corner frequency, fs is the stop frequency, C and S are the attenuations (in dB) at the corner and stop frequencies respectively.

From time to time this version of the formula is given:

order using ln
where the variables have the same meaning. As it turns out, the ratio of the logs of any two numbers is constant as is shown by the following.
log ratios equal

Another variation has this denominator

omega and frequency
where the 2 * pi terms cancel and the denominator has the same ratio as before.

The formula for order determination is sometimes shown as

order with no denominator
with variables as described above. This formula can only be valid if the term 100.1*C - 1 equals 1.
finding 3.01 dB
or when the corner frequency is slightly different from the commonly accepted corner point of 3 dB.

The terms fc and fs are self explanatory in the case of low pass filters. Some allowances must be made for high pass filters so that fs is the larger quantity, so the variables are renamed.

In the case of band pass filters, fs is the stop bandwidth and fc is the pass bandwidth. This is done so that fs is the larger quantity. For band stop (and notch) filters, fs the stop bandwidth and fc is the the "attenuation bandwidth," called that for want of a better expression. See the information and illustrations contained in the Help - Data Entry section of the main menu.

Once the order, n, is known it is possible to generate a table of values si according to the following:

set of sine values
where n is the order and
set of i values

and finally the prototypes...

The prototype values for an nth order filter with equal input and output termination resistances is given by the following recursion formulae:

the recursion formula
 where Xi is the reactance value in either Henries or Farads, and where
set of i values
and n is the order of the filter.
Resistance Button LP Button HP Button BP Button BS Button


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