SLOA049D July 2000 – February 2023

4 Math Review

A second-order polynomial using the variable $s$ can be given in two equivalent forms.

The coefficient form:

Equation 2.

s^{2} + a_{1} s + a_{0}

Or the factored form:

Equation 3.

(s - z_{1}) (s - z_{2})

In summary,

Equation 4.

P (s) = s^{2} + a_{1} s + a_{0} = (s - z_{1}) (s - z_{2})

where $z_{1}$ and $z_{2}$ are the locations in the s-plane where the polynomial is zero.

The three filters being discussed here are all-pole filters, meaning that their transfer functions contain all poles and no zeros. The polynomial, which characterizes the response of the filter, is used as the denominator of the transfer function of the filter. The zeros of the polynomial are thus the poles of the filter.

All even-order Butterworth, Bessel, or Chebyshev polynomials contain complex-conjugate zero pairs. This means that Equation 20 and Equation 6 are true, where $Re$ is the real part and $Im$ is the imaginary part.

Equation 20.

z_{1} = Re + Im

Equation 6.

z_{2} = Re - Im

In typical mathematical notation, $z_{1}$ indicates the conjugate zero with the positive imaginary part and $z_{1}^{*}$ indicates the conjugate zero with the negative imaginary part. Odd-order filters have a real pole in addition to the complex-conjugate pairs.

Some filter books provide tables of the zeros of the polynomial which describes the filter, others provide the coefficients, and some provide both. Since the zeros of the polynomial are the poles of the filter, some books use the term poles. Zeros and poles are used with the factored form of the polynomial, and coefficients are used with the coefficient form. No matter how the information is given, conversion between the two forms is routine.

Expressing the transfer function of a filter in factored form makes it easy to quickly see the location of the poles. Conversely, a second-order polynomial in coefficient form makes it easier to correlate the transfer function with circuit components. This is seen later when examining the filter-circuit topologies. Therefore, an engineer typically wants to use the factored form, but needs to scale and normalize the polynomial first.

The coefficient form of the second-order equation shows that when $s ≪ a_{0}$ , the equation is dominated by $a_{0}$ ; when $s ≫ a_{0}$ , s dominates. $a_{0}$ is the break point where the equation transitions between dominant terms. To normalize and scale to other values, divide each term by $a_{0}$ and divide the $s$ terms by $ω c$ . The result is shown in Equation 7:

Equation 7.

P (s) = {(\frac{s}{ω c \sqrt{a_{0}}})}^{2} + \frac{a_{1} s}{a_{0} ω c} + 1

This scales and normalizes the polynomial so that the break point is at $s = ω c \sqrt{a_{0}}$ .

By making the substitutions $s = j 2 π f$ , $ω c = 2 π f c$ , $a_{1} = \frac{1}{Q}$ , and $\sqrt{a_{0}} = F S F$ , the equation becomes:

Equation 8.

P (f) = - {(\frac{f}{F S F \times f_{c}})}^{2} + (\frac{1}{Q} \times \frac{j f}{F S F \times f_{c}}) + 1

This is the denominator of Equation 17—standard form for low-pass filters. Throughout the rest of this article, the substitution $s = j 2 π f$ is used.