Asymptote. 2 Problems 2 Methods
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1 Asymptote Problems Methods
2 Problems Assume we have the ollowing transer unction which has a zero at =, a pole at = and a pole at =. We are going to look at two problems: problem is where >> and problem is where >>. We will look at two methods to solve these two problems: one where the coeicient o the zero is / and the other where it is /. The goal is to determine the best way to calculate the Bode magnitude plot o the transer unction using asymptotes. A = j ( + j )( + j ) The process we will use is irst to determine the asymptotes associated with this transer unction. Then divide the requency spectrum into segments where these asymptotes deined. Then determine the composite asymptote by adding the asymptotes deined within each segment.
3 Problem : >> Method : Coeicient o the Zero = / A db = = ( ) + ( ) log[ ] log( ) log ( ) log ( ) = + + log( ) log( ( ) ) log( ( ) ) where >> 3 Asymptotes can be determined rom the logarithmic orm Asymptote=log( ) Zero at = Asymptote= log( ) Pole at = Asymptote3= log( ) Pole at =
4 Problem : >> Method : Coeicient o the Zero = / To determine the composite, let's look are three regions (see the ollowing drawing): For > Composite = Asymptote = log( ) Zero slope=+db/decade For > > Composite = Asymptote + Asymptote = log( ) log( ) Zero + Pole at slope= For > Composite = Asymptote + Asymptote + Asymptote3 = log( ) log( ) log( )Zero + Pole at + Pole at slope=-db/decade
5 Problem : >> Method : Coeicient o the Zero = / 6 4 Zero Pole - Pole Zero - Pole -4 Pole -6 Composite Magnitude Composite
6 Problem : >> Method : Coeicient o the Zero = / To make the coeicient o the zero = /, we multiply the transer unction by / and we get. j j A ( + j )( + j ) ( + j )( + j ) + ( ) + ( ) = π = = tan ( ) tan ( ) Adb = log[ ] = log( ) + log( ) log + ( ) log + ( ) + ( ) + ( ) = log( ) + log( ) log( + ( ) ) log( + ( ) ) where >>
7 Problem : >> Method : Coeicient o the Zero = / A db = log( ) log( ) log( ( ) ) log( ( ) ) We now ind there are 4 Asymptotes Asymptote= log( ) Constant Asymptote=log( ) Zero at = Asymptote3= log( ) Pole at = Asymptote4= log( ) Pole at =
8 Problem : >> Method : Coeicient o the Zero = / For > Composite = Asymptote+ Asymptote = log( ) + log( ) Constant + Zero slope=-db/decade For > > Composite = Asymptote + Asymptote + Asymptote3 = log( ) + log( ) log( ) For > Constant + Zero + Pole at slope= Composite = Asymptote + Asymptote + Asymptote3 + Aysmptote4 = log( ) log( ) log( ) log( ) Constant + Zero + Pole at + Pole at slope=-db/decade +
9 Problem : >> Method : Coeicient o the Zero = / Constant Zero Pole Pole Constant 5 Zero Pole Pole -5 Composite Magnitude Composite
10 Problem : >> Method : Coeicient o the Zero = / A db = = ( ) + ( ) log[ ] log( ) log ( ) log ( ) = + + log( ) log( ( ) ) log( ( ) ) where >> 3 Asymptotes Asymptote=log( ) Zero at = Asymptote= log( ) Pole at = Asymptote3= log( ) Pole at =
11 Problem : >> Method : Coeicient o the Zero = / For > Composite = Asymptote = log( ) Zero slope=+db/decade For > > Composite = Asymptote + Asymptote = log( ) log( ) Zero + Pole at slope= For > Composite = Asymptote + Asymptote + Asymptote3 = log( ) log( ) log( )Zero + Pole at + Pole at slope=-db/decade
12 5 Problem : >> Method : Coeicient o the Zero = / Zero Pole Pole 5 Zero Pole Pole Composite Magnitude Composite
13 Problem : >> Method : Coeicient o the Zero = / j j A ( + j )( + j ) ( + j )( + j ) + ( ) + ( ) Adb = = ( ) + ( ) = + ) log( + ( ) ) log( + ( ) ) = π = = tan ( ) tan ( ) log[ ] log( ) log( ) log ( ) log ( ) log( ) log( where >> 4 Asymptotes Asymptote=log( ) constant Asymptote=log( ) Zero at = Asymptote3= log( ) Pole at = Asymptote4= log( ) Pole at =
14 Problem : >> Method : Coeicient o the Zero = / For > Composite = Asymptote+ Asymptote = log( ) log( ) Constant + Zero slope=-db/decade For > > Composite = Asymptote + Asymptote + Asymptote3 = log( ) log( ) log( ) For > Constant + Zero + Pole at slope= + + Composite = Asymptote + Asymptote + Asymptote3 + Aysmptote4 = log( ) log( ) log( ) log( ) Constant + Zero + Pole at + Pole at slope=-db/decade +
15 Problem : >> Method : Coeicient o the Zero = / 5-5 Constant Zero Pole Pole 5 Constant Zero Pole Pole -5 Composite Magnitude - Composite
16 Which one is easier? Whichever approach you chose you must be consistent and calculate the composite using all o the asymptotes in the region where they exist.
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