Control

Second Order Filter Functions

The standard form of the transfer function for second order filters is
[pic]
Because the denominator is a second-degree polynomial, all such filters have two poles. Since the poles are the zeros of the denominator polynomial, we can find them by applying the quadratic formula. The location, and nature, of the poles depends only on the parameters [pic] and[pic]:
If [pic], then the two poles of [pic] are given by
[pic]
Note that when [pic] that the radicand is positive so that the radical is real. Also note that the radical is smaller than [pic] so that both poles lie on the negative real axis of the complex s-plane.
[pic]
If [pic], then the two poles of [pic] are coincident and are given by
[pic]
[pic]
If [pic], thenthe radicand above is negative so that the radical is pure imaginary. Thus the two poles of [pic] are given by
[pic]
or
[pic]
or
[pic]
For [pic], therefore, the poles form a complex conjugate pair and lie in the left half of the complex s-plane.
[pic]
Notice that as [pic], both poles approach the [pic] axis. (Poles on the [pic] axis correspond to constant amplitudeoscillations of the kind we will encounter when we consider oscillators.)
Although the denominator determines the poles of the filter transfer function, the type of filter is determined by the numerator polynomial. We consider several cases.

1. Low Pass Filter: n1 = n2 = 0

[pic]
If we let [pic] to obtain [pic] we see that for frequencies [pic] that
[pic]
For frequencies[pic],
[pic]
Thus, the transfer function with n1 = n2 = 0 is a low pass filter with a break frequency of [pic]. Because of the [pic] factor in the denominator, the gain in dB, [pic], falls off at an asymptotic rate of 40 dB/decade at low frequencies.
Any circuit that exhibits a transfer function of the form
[pic]
can act as a low pass filter. The equal component Sallen-Key circuit[pic]
has the transfer function
[pic]
Thus, this Sallen-Key circuit acts as a low pass filter with
[pic]
[pic]
[pic]
One of the advantages of an equal component Sallen-Key circuit for realizing a second order low pass filter is that [pic] and [pic] can be adjusted independently by adjusting [pic] and [pic], respectively. We’ll appreciate this advantage more,later.

2. High Pass Filter: n0 = n1 = 0

[pic]
If we let [pic] to obtain [pic] we see that for frequencies [pic] that
[pic], a constant
For frequencies, [pic]
[pic]
Thus, the transfer function with n0 = n1 = 0 is a high pass filter with a break frequency of [pic]. Because of the [pic] factor in the numerator, the gain in dB, [pic], increases at an asymptotic rate of 40dB/decade at frequencies below [pic].
Any circuit that exhibits a transfer function of the form
[pic]
can act as a high pass filter. The equal component Sallen-Key circuit
[pic]
has the transfer function
[pic]
Thus, this Sallen-Key circuit acts as a high pass filter with
[pic]
[pic]
[pic]
One of the advantages of an equal component Sallen-Key circuit forrealizing a second order high pass filter is that [pic] and [pic] can be adjusted independently by adjusting [pic] and [pic], respectively. We’ll appreciate this advantage more, later.

3. Band Pass Filter: n0 = n2 = 0

[pic]
It is straightforward to show that [pic] exhibits a maximum at [pic] and, for sufficiently large [pic], the passband of the filter is quite narrow. An equalcomponent Sallen-Key band pass circuit does not work well in practice. Later, we’ll consider a band pass filter circuit that contains more than one operational amplifier, a requirement necessary to achieve narrow (say a few per cent of the center frequency) passbands in practice.

4. Notch Filter: n1 = 0

[pic]
where [pic] is the notch frequency, a frequency corresponding to much reduced...