2015 AULA 05 Suplementar

3. NORMAL AND INVERTED POLES AND ZEROS

How to choose the gain at any frequency as the Reference Gain

v.0.1 3/07 http://www.RDMiddlebrook.com 3. Norm & Inv Ps & Zs

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Thisformatiscommonlyconsideredtobe theanswer.

However,itismuchbettertoextracttheconstanttermfromboththenumeratoranddenominatorpolynomialsin :s


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This thepolynomials,andexposesazerofrequencygainandacornerfrequency.

normalizes

2 30 1 2 3

2 30 1 2 3

Thisisaspecialcaseofthegeneralresultasaratioofpolynomialsincomplexfrequencys:

b b s b s b s ...Aa a s a s a s ...

+ + + += + + + +

31 20 0 0

31 20 0 0

refbb b 2 3

b b bref aa a 2 3

a a a

ExtractionoftheconstanttermfromnumeratoranddenominatordefinesthezerofrequencyreferencegainA andnormalizesthepolynomials :

1 s s s ...A A

1 s s s ...

+ + + +=

+ + + +v.0.1 3/07 http://www.RDMiddlebrook.com

3. Norm & Inv Ps & Zs 5

( ) ( ) ( )z1 z2 z3p1 p2 p3

s s s

refs s s

Factorizationofthepolynomialsdefines thepolesandzeros,andhencethefinal(preferred)factoredpolezeroform:

1 1 1 ...A A

1 1 1 ...

+ + += + + +

Thereferencegainandthepolesandzerosshould,ofcourse,belowentropyexpressionsin termsofthecircuitelements.


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Returntotheexample:


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Exercise3.1Writefactoredpolezeroformsfromasymptotes


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Exercise3.1Solution


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Exercise3.1Solution


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Exercise3.1Solution


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Exercise3.1Solution


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Exercise3.2WritefactoredpolezeroformsfordifferentReferenceGains,

2 3 1.andwrite and intermsofA A A


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Exercise3.2Solution


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Exercise3.2Solution


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Exercise3.2Solution


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Exercise3.2Solution


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Ifyoudontuseinvertedpolesandzeros,youarestuckwiththezerofrequencygainasthereferencegain.

Theprincipalbenefitofusinginvertedpolesandzerosisthatyoucanchoosethegainat frequencyasthereferencegain.

any


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Writeinputandoutputimpedances and infactoredpolezeroforms.i oZ ZExercise3.3


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Exercise3.3Solution


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Exercise3.3Solution


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Exercise3.3Solution


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Exercise3.3Solution


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Exercise3.3Solution

2015 AULA 05 Suplementar

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