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Poles and zeros

 

The frequency response of any filter can be expressed as a a ratio of poynomials in s , a complex number :

 

                                               _bm96                _bm97

 

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:

 

                                               _bm98                                 _bm99

 

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.

 

                               _bm100

 

The transfer function is:

                                                   _bm101

and  looks as follows in 3 dimensions:

 

                                _bm102

 

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:

 

                               _bm103

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.

 

 

               _bm89

 

All of the second order stages have a pole pair, resulting in the general transfer function:

 

                               _bm90        _bm91

 

This can be expanded as:

 

                                               _bm92

 

The denominator polynomial can also be expressed as:

                               

                                               _bm93

 

where Q, the ‘Quality Factor’, is:   

                                                                _bm94

 

and wp, the pole frequency, is:                

                                                               _bm95

 

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):

 

                               _bm174

 

All of these bandstop filters have a pole pair and a zero pair, resulting in the general transfer function:

 

                               _bm175        _bm176

 

As with the pole pair, the numerator can be expressed as:

 

                               _bm177_bm178

 

Since the zeros lie on the imaginary axis, Q is infinite and the numerator reduces to:

 

                               _bm179

 

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.