Poles and zeros
The frequency response of any filter can be expressed as a a ratio of poynomials in s , a complex number :
Any value of s for which T(s) = 0 is termed a zero, while any value of s for which T(s) is infinite is termed a pole. For filters designed by this software, poles are confined to the negative real region (-Re[s]) and zeros are confined to the imaginary axis. A transfer function T(s) can be expressed as the product of linear terms:
Two poles that have the same real value but opposite imaginary values are termed a conjugate pole pair. Similarly two zeros that have opposite sign but equal values are termed a zero pair.
This software assembles filters by cascading first and second order filter stages. A lowpass filter of order 7 will therefore be formed by one first order stage in series with three second order stages.
There are two possible first order stages, lowpass and highpass. There are four second order stage types, lowpass, highpass, bandpass and bandstop. Bandstop (notch) stages are further classified as lowpass notch, highpass notch, or notch.
As an example, the first order highpass stage has one pole and one zero.
The transfer function is:
and looks as follows in 3 dimensions:
Notice that the tranfer function response in the region of the pole has been truncated, as the function is infinite at the pole. Also the zero forces the function to zero at the origin, resulting in a first order highpass response along the s = +jw axis (+Im[s]). A two dimensional representation of the highpass response follows. Note that the frequency axis here is logarithmic:
Pole pairs and Qp,wp
A pole pair consists of two poles (p, p*), each of the same real and imaginary magnitudes, but of opposite imaginary sign.
All of the second order stages have a pole pair, resulting in the general transfer function:
This can be expanded as:
The denominator polynomial can also be expressed as:
where Q, the ‘Quality Factor’, is:
and wp, the pole frequency, is:
Clearly wp is the magnitude of the radius of the arc that passes through the two poles, and as the second order filter stages reveal, Q is a measure of the ‘peakedness’ of the filter reponse. As the poles move closer to the imaginary axis, the Q value increases, as does the peaking of the frequency response.
Zero pairs and Qz,wz
Azero pair consists of two zeros of equal magnitude and opposite sign, and are found in the second order bandstop filter stages (lowpass notchn highpass notch and notch):
All of these bandstop filters have a pole pair and a zero pair, resulting in the general transfer function:
As with the pole pair, the numerator can be expressed as:
Since the zeros lie on the imaginary axis, Q is infinite and the numerator reduces to:
For any circuit topology that implements a bandstop filter, component values are automatically calculated to make Q infinite. Nevertheless, component spread will alter this value. If the circuit is to be tuned, the gain notch at frequency wz is adjusted for null gain. The program indicates the component to adjust for null gain for both the lowpass and highpass notch filters. In the case of the notch filter type (wp = wz) tuning information is not provided as these circuits do not readily lend themselves to adjustment.