# What is the Q factor of a Transfer Function?

#### Fredje

Joined Jun 1, 2022
15

#### Papabravo

Joined Feb 24, 2006
18,846
A 2nd order filter is described by a rational polynomial with a second order polynomial in the denominator. The set of polynomials that can be in a transfer function for a stable system is NOT the set of all 2nd order polynomials. They have some unique characteristics:
1. All the coefficients are real numbers
2. All the coefficients are positive
3. All the roots are either real AND identical or are complex conjugate pairs.
We can describe such polynomials by:
1. Writing the three explicit terms of the polynomial
2. By writing the values of the two roots.
3. By writing the polynomial in such a way that the properties of interest are manifestly obvious.
That 3rd alternative is what is happening when we say that:

$$as^2+bs+c\;=\;s^2+(\omega_0/Q)s+(\omega_0)^2$$

Then when you substitute jω for s, you get the expression that you posted. Q is just a letter that is used to represent the location of the pole with respect to the negative real axis, nothing more, and
$$\omega_0$$
is just the radius of the circle on which the poles are located in the left half-plane.

Last edited:
• Fredje