HANDOUT E.17 - EXAMPLES ON BODE PLOTS OF FIRST AND SECOND ORDER SYSTEMS
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1 Lecture 7,8 Augut 8, 00 HANDOUT E7 - EXAMPLES ON BODE PLOTS OF FIRST AND SECOND ORDER SYSTEMS Example Obtai the Bode plot of the ytem give by the trafer fuctio ( We covert the trafer fuctio i the followig format by ubtitutig ( ( We call <<, the break poit So for, ie, for mall value of Therefore takig the log magitude of the trafer fuctio for very mall value of, we get 0 log 0 log( 0 Hece we ee that below the break poit the magitude curve i approximately a cotat For, >>, ie, for very large value of Similarly takig the log magitude of the trafer fuctio for very large value of, we have 0log 0log 0log 0 log( 0 log( 0 log( So we ee that, above the break poit the magitude curve i liear i ature with a lope of 0 db per decade The two aymptote meet at the break poit The aymptotic bode magitude plot i how below
2 Lecture 7,8 Augut 8, 00 0log 05 0 Slope of 0 db per decade The phae of the trafer fuctio give by equatio ( i give by φ 0 ta ( ta ( So for mall value of, ie, 0, we get φ 0 For very large value of, ie,, the phae ted to 90 degree To obtai the actual curve, the magitude i calculated at the break poit ad oiig them with a mooth curve The Bode plot of the above trafer fuctio i obtaied uig MATLAB by followig the equece of commad give um ; de [ ]; y tf(um,de; grid; bode(y The plot give below how the actual curve
3 Lecture 7,8 Augut 8, 00 3 Example Obtai the bode plot of the ytem give by the trafer fuctio ( Subtitutig i the above trafer fuctio, we get ( ( From the above trafer fuctio, it ca be cocluded that, o therefore reducig the above trafer fuctio by dividig both the umerator ad deomiator by, we get 05 ( ζ I thi cae the break poit i Therefore for
4 Lecture 7,8 Augut 8, 00 <<, ie, for mall value of, Takig the log magitude, we get 0 log 0 log( 0 Therefore the magitude i approximately a cotat below the break poit For larger value of ie, for >>, we get Takig the log magitude, we get 0log 0log 0 log( 0log 0log 0log From the above relatio, it ca be cocluded that the magitude plot i liear i ature with a lope of 0 db per decade The aymptotic plot i a how below 0log 0 Slope of 0 db per decade The trafer fuctio ca be rewritte a, ( a( b where a ad b are the root of the deomiator
5 Lecture 7,8 Augut 8, 00 Subtitutig ( a(, we get b The phae of the above trafer fuctio i give a φ 0 ta ( ta ( a b So therefore for 0, we get φ 0 For very large value of, ie,, the phae ted to 80 degree The actual bode plot i obtaied by followig the give MATLAB equece um ; de [ ]; y tf(um,de; grid bode(y The plot i attached below 5
6 Lecture 7,8 Augut 8, 00 Example 3 Plot the Bode magitude ad phae for the ytem with trafer fuctio 000( 05 ( 0( 50 Step : We covert the fuctio to the form give below by ubtitutig 000( ( ( 0( Step : We ote that the term i i firt order ad i the deomiator, o - Therefore, the low frequecy aymptote i defied by the firt term: ( Thi aymptote i valid for < 0 becaue the lowet break poit i at 05 The magitude plot of thi term ha a lope of or 0 db per decade We locate the magitude by paig through the value at eve though the compoite curve will ot go through thi poit becaue of the break poit at 05 Step 3: We obtai the remaider of aymptote a how i the figure Firt we draw a lie with 0 lope that iterect the origial lope at 05 The we draw a lope lie that iterect the previou oe at 0 Fially, we draw a lope lie that iterect the previou lope at 50 Step : We the ketch the actual curve by calculatig the value of the magitude at the break poit ad oiig thoe poit by a mooth curve We ee that the actual curve i approximately tagetial to the aymptote whe far away from the break poit ad are a factor of ( 3 db above the aymptote at 05 break poit ad a factor of 07 (-3 db below the aymptote at 0 ad 50 break poit Step 5: Sice the phae of i frequecie -90, the phae curve tart at -90 at the lowet Step 6: The idividual phae curve are how i the form of dahed lie Note that the compoite curve approache each idividual term 6
7 Lecture 7,8 Augut 8, 00 The followig plot depict the bode magitude plot of the idividual term i the trafer fuctio 0log Slope of 0 db per decade 0 05 Bode plot of trafer fuctio 05 0log 0 0 Slope of 0 db per decade 0log Bode plot of trafer fuctio Bode plot of trafer fuctio Slope of 0 db per decade 50 7
8 Lecture 7,8 Augut 8, 00 0log 0log( 0 0 Slope of 0 db per decade Bode plot of trafer fuctio Combiig the above bode diagram, the compoite aymptotic curve i a how below 0log Slope of 0 db per decade Slope of 0 db per decade Slope of 0 db per decade The actual bode magitude curve i obtaied by evaluatig the actual magitude at the break poit ad oiig thee poit with a mooth curve The actual bode plot i how below Similar procedure i adopted to plot the phae curve 8
9 Lecture 7,8 Augut 8, 00 9 Example Draw the frequecy repoe of the ytem give by the trafer fuctio 0 ( 0 ( Rewrite the above trafer fuctio a 0 0 ( Subtitutig i the above trafer fuctio, we get 0 0 (
10 Lecture 7,8 Augut 8, 00 The breakpoit for the above trafer fuctio i at Followig the ame procedure a i example 3, the compoite aymptotic bode magitude curve i a how below 0log Slope of 0 db per decade 0 Slope of 0 db per decade The actual bode magitude plot i obtaied by evaluatig the magitude of the trafer fuctio at the break poit ad oiig it with a mooth curve The imilar procedure i adopted to obtai the phae curve for the ytem The compoite actual curve i a how below 0
11 Lecture 7,8 Augut 8, 00 Aigmet Draw the bode plot for each of the followig ytem Compare your ketche with the plot obtaied uig the bode commad i MATLAB a b c 000, ( 0, ( ( 0( ( ( 5 5 Recommeded Readig Feedback Cotrol of Dyamic Sytem th Editio, by ee F Frakli etal pp Recommeded Aigmet Feedback Cotrol of Dyamic Sytem th Editio, by ee F Frakli etal problem 63
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