University of California at Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences

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1 A Uiversity of Califoria at Berkeley College of Egieerig Departmet of Electrical Egieerig ad Computer Scieces U N I V E R S T H E I T Y O F LE T TH E R E B E LI G H T C A L I F O R N I A EECS : Sigals ad Systems Fall Semester Examples of Aalysis of Cotiuous-Time LTI Systems Usig Laplace Trasform Example : First-Order Highpass Filter Differetial Equatio dy - dx + y() t -, >. dt dt (If the system is implemeted usig a RC circuit, the RC ). Trasfer Fuctio Takig the bilateral LT of the differetial equatio: sy( s) + Y( s) sx( s) Ys ( ) s - Xs ( ) s + Frequecy Respose Sice the system is causal, the impulse respose is right-sided, ad ROC H { s Re( s) > }. Sice the ROC icludes the axis, the system is stable ad the frequecy respose exists. First Order Highpass Filter, s H ( ) - + log 3 Note that for this system, i, the degree of the umerator equals the degree of the deomiator, i.e., M N. Before ivertig, we should rewrite it i the form: M N f l s l + H ( s), l

2 Examples of Aalysis of Cotiuous-Time LTI Systems Usig Laplace Trasform of 5 where H ( s) is a ratioal fuctio i which the degree of the umerator is less tha the degree of the deomiator. s s + s + s + I this case, we see that M N, f ad H ( s) ( ) ( s + ). Takig the iverse bilateral LT of : δt () e t ut (). We have chose the right-sided iverse of H ( s) sice the system is kow to be causal. Step Respose We kow that the step respose st () satisfies Takig the bilateral LT: st () ut ()*h() t Ss ( ) -. s + I Ss ( ), the degree of the umerator is less tha the degree of the deomiator, i.e., M adn,so we ca immediately ivert Ss ( ) to obtai: We have chose the right-sided iverse of s - s s s + st () e t ut (). Ss ( ) sice the system is kow to be causal. Example : Secod-Order Lowpass Filter Differetial Equatio d y dy - + dt ζ - + dt yt () xt (), >, ζ <. (If the system is implemeted usig a RLC circuit, the LC ad ζ ( R ) C L). Trasfer Fuctio Takig the bilateral LT of the differetial equatio: s Ys ( ) + ζ sy( s) + Ys ( ) Xs ( ) Ys ( ) -, Xs ( ) s - d + ζ s + ( s d )( s d ) ζ + ζ Case (a): Overdamped, ζ >, two distict, real poles Sice ζ >, d ad d are distict ad real. As ζ, d. As ζ, d ad d.

3 Examples of Aalysis of Cotiuous-Time LTI Systems Usig Laplace Trasform 3 of 5 - ( s d )( s d ) A A s d s d A ( s d ) -, A s d ζ ( s d ) s d - ζ Takig the iverse bilateral LT of : ζ s d s d We have chose the right-sided iverse of - e ( ζ ζ )t e ( ζ + ζ )t ut (). ζ sice the system is kow to be causal. Case (b): Uderdamped, < ζ <, two distict, complex-cojugate poles Sice ζ <, let ζ j ζ. The d ζ + j ζ d ad d are distict ad complex cojugates of each other. As ζ, d ±. As ζ, d. We use exactly the same mathematics as for the overdamped case, but after we obtai substitutio ζ j ζ., we make the e t. ζ j e ζ t e ζ t u() t e t si ζ t ut () ζ Case (c): Critically Damped, ζ There is a sigle real pole of multiplicity two., oe real pole of multiplicity two,. s + s + ( s + ) d There is o eed to do partial fractio expasio. We ca fid the iverse bilateral LT of We choose a right-sided iverse, sice we kow this is a causal system. t te ut () i the table.

4 Examples of Aalysis of Cotiuous-Time LTI Systems Usig Laplace Trasform of 5 Frequecy Respose (For Cases (a), (b) ad (c)).77.5 ζ (poles).77.5 Sice the system is causal, is right-sided, ad ROC H { s Re( s) > Re( rightmost pole of )}. For < ζ <, both poles are i the left half-plae, the ROC icludes the axis, the system is stable, ad the frequecy respose exists. H ( ) s - ( ) + ζ ( ) + Secod Order,, ζ.5 Secod Order,, ζ.77 log log 3 3 Secod Order,, ζ Secod Order,, ζ log log 3 3

5 Examples of Aalysis of Cotiuous-Time LTI Systems Usig Laplace Trasform 5 of 5 Example 3: Nth-Order Butterworth Lowpass Filters See HV Sectio 8.5 for a detailed discussio. The Nth-order Butterworth lowpass filter is geerated by N poles i the left half-plae, placed o a semicircle of radius,where is the cutoff frequecy. The poles are separated by agles of N. The pole(s) closest to the axis are separated by agle(s) N from the axis. The Nth-order Butterworth lowpass filter is the uique Nth-order system havig maximally flat magitude respose: d k H ( ) d k, k,..., N Note that log H( ) db, log H ( c ) 3dB, ad that at large, the rolloff rate is N db/decade. Also, ote that the secod-order Butterworth respose is equivalet to a secod-order system with ad ζ. These filters ca be realized by cascadig first- ad secod-order lowpass filters (the former are ecessary oly for odd N). N N N 3 N First Order Butterworth, Secod Order Butterworth, log 3 log 3 Third Order Butterworth, Fourth Order Butterworth, log 3 log 3