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1st Order Filter Stages

 

First order highpass

 

A first order highpass filter stage has one single pole on the negative real axis of the s-plane (p<0), as well as a single zero (z) at the origin:

 

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The 3dB point occurs at w = p. The following graphic is normalized to w = w/p.

 

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First order lowpass

A first order lowpass filter stage has one pole on the negative real axis of the s-plane (p<0), as well as an effective zero as s = jw approaches infinity:

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The 3dB point occurs at w = p. The following graphic is normalized to w = w/p.

 

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2nd Order Low Pass Filter Stage

A second order lowpass filter has a complex conjugate pole pair in the negative real region and two effective zeros as s = jw approaches infinity. The transfer function for this filter can be expressed as:

 

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or alternately in terms of Q and wp as:

 

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In the following pole-zero diagram, the pole pair is shown in two positions, both with the same wp value but with different Q values.

 

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For a given wp, as the pole pair approaches the imaginary axis Q increases and a more peaked gain effect results. The following graphic is normalized for w = w/wp. As Q approaches infinity, the frequency of the gain maximum approaches wp.

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2nd Order High Pass Filter Stage

A second order highpass filter has a complex conjugate pole pair in the negative real region and a zero pair at the origin. The transfer function for this filter can be expressed as:

 

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or alternately in terms of Q and wp as:

 

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In the following pole-zero diagram, the pole pair is shown in two positions, both with the same wp value but with different Q values.

 

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For a given wp, as the pole pair approaches the imaginary axis Q increases and a more peaked gain effect results. The following graphic is normalized for w = w/wp. As Q approaches infinity, the frequency of the gain maximum approaches wp.

 

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2nd Order Band Pass Filter Stage

 

A second order bandpass filter has a complex conjugate pole pair in the negative real region, a single zero at the origin, and an effective zero as s = jw approaches infinity. The transfer function for this filter can be expressed as:

 

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or alternately in terms of Q and wp as:

 

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In the following pole-zero diagram, the pole pair is shown in two positions, both with the same wp value but with different Q values.

 

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For a given wp, as the pole pair approaches the imaginary axis Q increases and a more peaked gain effect results. The following graphic is normalized for w = w/wp.

 

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2nd Order Low Pass Notch Filter Stage

 

A second order lowpass notch filter has a complex conjugate pole pair in the negative real region and a complex conjugate zero pair on the imaginary axis. The transfer function for this filter can be expressed as:

 

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or alternately in terms of Q and wp as:

 

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In the following pole-zero diagram, the pole pair is shown in two positions, both with the same wp value but with different Q values. Note that wz>wp (im[z]>im[p]).

 

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For a given wp, as the pole pair approaches the imaginary axis Q increases and a more peaked gain effect results. The following graphic is normalized for w = w/wp. The zero (z) forces the gain to zero at wz. As Q approaches infinity, the frequency of the gain maximum approaches wp.

 

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2nd Order High Pass Notch Filter Stage

 

A second order highpass notch filter has a complex conjugate pole pair in the negative real region and a complex conjugate zero pair on the imaginary axis. The transfer function for this filter can be expressed as:

 

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or alternately in terms of Q and wp as:

 

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In the following pole-zero diagram, the pole pair is shown in two positions, both with the same wp value but with different Q values. Note that wz<wp (im[z]<im[p]).

 

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For a given wp, as the pole pair approaches the imaginary axis Q increases and a more peaked gain effect results. The following graphic is normalized for w = w/wp. The zero (z) forces the gain to zero at wz. As Q approaches infinity, the frequency of the gain maximum approaches wp.

 

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2nd Order Notch (Band Stop) Filter Stage

 

A second order notch filter has a complex conjugate pole pair in the negative real region and a complex conjugate zero pair on the imaginary axis. The transfer function for this filter can be expressed as:

 

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or alternately in terms of Q and wp as:

 

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In the following pole-zero diagram, the pole pair is shown in two positions, both with the same wp value but with different Q values. Note that wz = wp (im[z] = im[p]).

 

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For a given wp, as the pole pair approaches the imaginary axis Q increases and a narrower gain notch results. The following graphic is normalized for w = w/wp.

 

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