Page 299 - Analog and Digital Filter Design

296 Analog and Digital Filter Design

FU
Cutoff frequency Fc = -for Buttenvorth filters, where N is the filter order.
N

FU
Cutoff frequency Fc = for 0.5 dB Chebyshev filters.

Measurements on Filters

The frequency response of a filter is measured by applying a sine wave genera-
tor across the input terminals and an AC voltage-measuring device across the
output terminals. The signal generator should have the same impedance as the
filter under test; if the generator’s impedance is not the same as the filter’s
characteristic impedance the results will be wrong. Remember that the source
impedance is actually part of the filter design. The generator’s impedance
can be changed either by adding series resistance to increase it’s impedance or by
connecting a resistor across the signal generator’s output to reduce its impedance.

If the signal generator output is measured without any load, the voltage seen
is equal to the source EMF. If the source EMF is not constant with frequency,
or the generator’s internal impedance is not a constant resistance, the signal
generator output cannot be connected directly to the filter input. If it were, the
output response of the filter would be wrong. What should be done in this case
is the output of the generator should be monitored using an AC voltmeter and
kept to a constant voltage. A separate resistor of the required input impedance
should then be wired in series between the generator and the filter. By keeping
the output of the generator at a constant voltage it is in effect zero impedance
(since the load will not affect it). The source impedance will be equal to the series
resistance and the output response of the filter will now be correct.

The AC voltmeter across the filter’s output terminals must have a bandwidth
greater than the frequency range being measured. This may seem like an obvious
statement, but some meters have a bandwidth switch that is used to reduce the
noise. I, and many others, have been ”caught out” by forgetting to return this
switch to the wide bandwidth setting.

Another bandwidth problem is that of a spectrum analyzer; although a certain
resolution bandwidth may have been set, this is the 3 dB bandwidth and not the
noise bandwidth. That means that the signal-to-noise ratio appears worse than
it really is. Reputable spectrum analyzer suppliers provide information about
their equipment’s filter response.

The resistor used to terminate passive filters must have a value equal to the
filter’s characteristic impedance. Since an AC voltmeter has high impedance,
this means that a resistor must be physically placed across the filter’s output