bodeplot - Introduction to Bode Plot 2 plots both have...

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Introduction to Bode Plot 2 plots – both have logarithm of frequency on x-axis o y-axis magnitude of transfer function, H(s), in dB o y-axis phase angle The plot can be used to interpret how the input affects the output in both magnitude and phase over frequency. Where do the Bode diagram lines comes from? 1) Determine the Transfer Function of the system: ) ( ) ( ) ( 1 1 p s s z s K s H + + = 2) Rewrite it by factoring both the numerator and denominator into the standard form ) 1 ( ) 1 ( ) ( 1 1 1 1 + + = p s sp z s Kz s H where the z s are called zeros and the p s are called poles. 3) Replace s with j? . Then find the Magnitude of the Transfer Function. ) 1 ( ) 1 ( ) ( 1 1 1 1 + + = p jw jwp z H If we take the log 10 of this magnitude and multiply it by 20 it takes on the form of 20 log ( H(jw) ) = + + ) 1 ( ) 1 ( log 20 1 1 1 1 p jwp z = ) 1 ( 20 20 20 ) 1 ( 20 20 20 1 10 10 1 10 1 10 1 10 10 + - - - + + + z p z z K Each of these individual terms is very easy to show on a logarithmic plot. The entire Bode log magnitude plo the result of the superposition of all the straight line terms. This means with a little practice, we can quickly the effect of each term and quickly find the overall effect. To do this we have to understand the effect of the different types of terms. These include: 1) Constant terms K 2) Poles and Zeros at the origin | j? | 3) Poles and Zeros not at the origin 1 1 p j w + or 1 1 z j w + 4) Complex Poles and Zeros (addressed later)
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Effect of Constant Terms: Constant terms such as K contribute a straight horizontal line of magnitude 20 log 10 ( K ) H = K Effect of Individual Zeros and Poles at the origin: A zero at the origin occurs when there is an s or j? multiplying the numerator. Each occurrence of this causes a positively sloped line passing through ? = 1 with a rise of 20 db over a decade. H = | w j | A pole at the origin occurs when there are s or multiplying the denominator. Each occurrence of this causes a negatively sloped line passing through ? = 1 with a drop of 20 db over a decade. H = w j 1 Effect of Individual Zeros and Poles Not at the Origin Zeros and Poles not at the origin are indicated by the (1+ j? /z i ) and (1+ j? /p i ) . The values z i and p i in each of these expression is called a critical frequency (or break frequency). Below their critical frequency these terms do not contribute to the log magnitude of the overall plot. Above the critical frequency, they represent a ramp function of 20 db per decade. Zeros give a positive slope. Poles produce a negative slope. H = i i p j z j w w + + 1 1 20 log ( K ? 0.1 1 10 100 (log scale) 20 log (H) 0.1 1 10 100 (log scale) 20 log( H -20 db dec. dec. +20 db z i p i -20 db 0.1 1 10 100 (log scale) 20 log( H dec 0.1 1 10 100 (log scale) 20 log(H) 20 db
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To complete the log magnitude vs. frequency plot of a Bode diagram, we superposition all the lines of the different terms on the same plot.
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bodeplot - Introduction to Bode Plot 2 plots both have...

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