B - ECSE 210: Electric Circuits 2 ECSE 210: Electric...

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Unformatted text preview: ECSE 210: Electric Circuits 2 ECSE 210: Electric Circuits 2 hapter hapter 3 3 Chapter Chapter 13 13 requency Response requency Response Frequency Response Frequency Response Complex Poles and Zeros In general, network functions can be expressed as the ratio of two polynomials in s : H ( s ) N ( s ) D ( s ) b m s m b m 1 s m 1 L b 1 s b o a s n a s n 1 L a s a n n 1 1 o K (s - z 1 ) (s - z 2 )..... (s - z m ) The roots of the numerator and denominator can be (s - p 1 ) (s - p 2 )..... (s - p n ) complex, resulting in complex poles and zeros . Complex Poles and Zeros 1 ) ( s H 2 2 2 o o s s The roots of the denominator are: 1 2 2 , 1 o o p 1 Two real poles wo real and equal poles 1 Two real and equal poles (a pole of multiplicity 2) wo complex poles 1 Two complex poles Complex Poles and Zeros Example 10 o H ( s ) 1 00 5 . s 2 10 s 100 25 . 1 10 5 1 2 2 , 1 j p o o 1 p Im H 1 6 . 8 5 2 , 1 j p Re ( s ) ( s 5 j 8.6)( s 5 j 8.6) Treat the two complex conjugate poles 2 p simultaneously as a pair. Complex Poles and Zeros Low frequency asymptote: o 2 1 log 20 ) ( o dB j H dB 40 dB 20 dB dB 20 dB 40 1 01 . 0.1 1 1 1 10 1 100 Complex Poles and Zeros High frequency asymptote: o ) log( 40 1 log 20 ) ( 2 dB j H dB 20 dB 40 dB-40dB/dec dB 20 B 1 01 . 1 1 . 1 1 10 1 100 dB 40 Complex Poles and Zeros Note that the two asymptotes intersect at = o dB 20 dB dB 20 B -40 db/decade 01 . 1 . 10 100 dB 40 dB 60 1 o o o o Example H ( s ) 1 s 2 2 o s o 2 o 10 rad ./sec.-40-30-20 =0.05 =0.2 =0.5 - 0-60-50 =0.707 -90-80 70-1 1 2 3-120-110-100 10 10 10 10 10 (rd/s) Complex Poles Maximum Maximum value of | H ( j )| H ( s ) 1 s 2 2 o s o 2 1 2 2 , 1 o o p 1 2 2 , 1 1 o o j p 2 2 1 1 1 ) ( j j j j j H o o o o Complex Poles Maximum value of | H ( j )| H ( s ) 1 s 2 2 o s o 2 o o j j 2 1 ) ( 2 2 H 1 ) ( j H 2 2 2 2 2 2 4 o o ake derivative and set it to zero to find t maximum value Take derivative and set it to zero to find at maximum value....
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B - ECSE 210: Electric Circuits 2 ECSE 210: Electric...

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