Where

(16)

STATE VARIABLE ACTIVE FILTER

State variable active filters are circuits that can simulta-

neously represent high pass, band pass, and low pass fil-

ters. The state variable active filter uses three separate

amplifiers to achieve this task. A typical state variable active

filter is shown in Figure 13. The first amplifier in the circuit is

connected as a gain stage. The second and third amplifiers

are connected as integrators, which means they behave as

low pass filters. The feedback path from the output of the

third amplifier to the first amplifier enables this low frequency

signal to be fed back with a finite and fairly low closed loop

gain. This is while the high frequency signal on the input is

still gained up by the open loop gain of the first amplifier. This

makes the first amplifier a high pass filter. The high pass

signal is then fed into a low pass filter. The outcome is a

band pass signal, meaning the second amplifier is a band

pass filter. This signal is then fed into the third amplifiers

input and so, the third amplifier behaves as a simple low

pass filter.

The transfer function of each filter needs to be calculated.

The derivations will be more trivial if each stage of the filter is

shown on its own.

The three components are:

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For A

(17)

This relationship depends on the output of all the filters. The

input-output relationship for A

(18)

And finally this relationship for A

(19)

Re-arranging these equations, one can find the relationship

between V

V

V

relationships are as follows:

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FIGURE 12. Band Pass Filter Transfer Function

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FIGURE 13. State Variable Active Filter

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