EECE 301 Signals & Systems Prof. Mark Fowler

Transcription

1 EECE 30 Sigal & Sytem Prof. Mark Fowler Note Set #8 C-T Sytem: Laplace Traform Solvig Differetial Equatio Readig Aigmet: Sectio 6.4 of Kame ad Heck /

2 Coure Flow Diagram The arrow here how coceptual flow betwee idea. Note the parallel tructure betwee the pik block C-T Freq. Aalyi ad the blue block D-T Freq. Aalyi. New Sigal Model Ch. Itro C-T Sigal Model Fuctio o Real Lie Sytem Propertie LTI Caual Etc Ch. 3: CT Fourier Sigal Model Fourier Serie Periodic Sigal Fourier Traform CTFT No-Periodic Sigal Ch. Diff Eq C-T Sytem Model Differetial Equatio D-T Sigal Model Differece Equatio Zero-State Repoe Ch. 5: CT Fourier Sytem Model Frequecy Repoe Baed o Fourier Traform New Sytem Model Ch. Covolutio C-T Sytem Model Covolutio Itegral Ch. 6 & 8: Laplace Model for CT Sigal & Sytem Trafer Fuctio New Sytem Model New Sytem Model D-T Sigal Model Fuctio o Iteger New Sigal Model Powerful Aalyi Tool Zero-Iput Repoe Characteritic Eq. Ch. 4: DT Fourier Sigal Model DTFT for Had Aalyi DFT & FFT for Computer Aalyi D-T Sytem Model Covolutio Sum Ch. 5: DT Fourier Sytem Model Freq. Repoe for DT Baed o DTFT New Sytem Model Ch. 7: Z Tra. Model for DT Sigal & Sytem Trafer Fuctio New Sytem Model /

3 6.4 Uig LT to olve Differetial Equatio I Ch. we aw that the olutio to a liear differetial equatio ha two part: y y y total z zi Ch. Ch. Ch. 5 We ve ee how to fid thi uig: covolutio w/ impule repoe or uig multiplicatio w/ frequecy repoe We ve ee how to fid thi uig the characteritic equatio, it root, ad the ocalled characteritic mode Here we ll ee how to get y total uig LT get both part with oe tool!!! 3/

4 Firt-order cae: Let ee thi for a t -order Diff. Eq. with a caual iput ad a o-zero iitial coditio jut before the caual iput i applied. The t -order Diff. Eq. decribe: a imple RC or RL circuit. The caual iput mea: we witch o ome iput at time t 0. The iitial coditio mea: jut before we witch o the iput the capacitor ha a pecified voltage o it i.e., it hold ome charge. Iput: Time-Varyig Voltage e.g., guitar, microphoe, etc. x t 0 V IC R Output: Time-Varyig Voltage C y Aume that for t<0 thi ha bee witched o for a log time Thu the cap i fully charged to V IC volt 4/

5 Thi circuit i the decribed by thi Diff. Eq.: Cap voltage jut before x tur o dy dt With IC y0 - V IC RC y x RC x 0, t < 0 For thi ex. we ll olve the geeral t -order Diff. Eq.: dy dt ay bx Now the key tep i uig the LT are: take the LT of both ide of the Differetial Equatio ue the LT propertie where appropriate olve the reultig Algebraic Equatio for Y fid the ivere LT of the reultig Y Laplace Traform: Differetial Equatio tur ito a Algebraic Equatio Hard to olve Eay to olve 5/

6 We ow apply thee tep to the t -order Diff. Eq.: dy L ay L dt { bx } Apply LT to both ide dy L al dt { y } bl{ x } [ Y y0 ] ay b y0 b Y a a Part of ol drive by IC Zero-Iput Sol Part of ol drive by iput Zero-State Sol Note that /a play a role i both part Hey! a i the Characteritic Polyomial!! Ue Liearity of LT Ue Property for LT of Derivative accoutig for the IC Solve algebraic equatio for Y Now the hard part i to fid the ivere LT of Y 6/

7 Example: RC Circuit Now we apply thee geeral idea to olvig for the output of the previou RC circuit with a uit tep iput. x u dy dt RC y RC x y0 / RC Y / RC / RC Thi trafer the iput to the output Y We ll ee thi later a The Trafer Fuctio Now we eed the LT of the iput From the LT table we have: x u y0 / RC Y / RC / RC Now we have jut a fuctio of to which we apply the ILT 7/

8 So ow applyig the ILT we have: L - { Y } L - y0 / RC / RC / RC Apply LT to both ide y L - y0 / RC L - / RC / RC Liearity of LT Thi part zero-iput ol i eay Jut look it up o the LT Table!! Thi part zero-tate ol i harder It i NOT o the LT Table!! L y 0 / RC - t / RC y 0 e t / RC u y0 e u t So the part of the ol due to the IC zeroiput ol decay dow from the IC voltage 8/

9 Now let fid the other part of the olutio the zero-tate ol the part that i drive by the iput: y L - y0 / RC L - / RC / RC We ca factor thi fuctio of a follow: L - / RC - L / RC / RC Ca do thi with Partial Fractio Expaio, which i jut a fool-proof way to factor L - - L Now each of thee term i o the LT table: u / RC t / RC e u Liearity of LT [ t / RC e ] u 9/

10 So the zero-tate repoe of thi ytem i: [ t / RC e ] u t / RC [ e ] u t Now puttig thi zero-tate repoe together with the zero-iput repoe we foud give: y y0 e t / RC u [ t / RC e ] u IC Part Iput Part Notice that: The IC Part Decay Away but The Iput Part Perit 0/

11 Here i a example for RC 0.5 ec ad the iitial V IC 5 volt: Zero-Iput Repoe Zero-State Repoe Total Repoe /

12 Secod-order cae Circuit with two eergy-torig device C & L, or C or L are decribed by a ecod-order Differetial Equatio d y dt a dy dt a w/ IC 0 & y0 0 y b dx dt b 0 x Aume Caual Iput y x 0 t < 0 x 0 0 We olve the d -order cae uig the ame tep: Take LT of Diff. Equatio: [ Y y0 y 0 ] a [ Y y0 ] a Y b b 0 0 From d derivative property, accoutig for IC From t derivative property, accoutig for IC From t derivative property, caual igal /

13 Solve for Y: y0 y 0 a y0 b b 0 Y a a 0 a a 0 Part of ol drive by IC Zero-Iput Sol Note thi how up i both place it i the Characteritic Equatio Part of ol drive by iput Zero-State Sol Note: The role the Characteritic Equatio play here! It jut pop up i the LT method! The ame happeed for a t -order Diff. Eq ad it happe for all order Like before to get the olutio i the time domai fid the Ivere LT of Y 3/

14 To get a feel for thi let look at the zero-iput olutio for a d-order ytem: [ y 0 a y0 ] y0 y0 a y0 y0 Y zi a a0 a a0 which ha either a t -order or 0 th -order polyomial i the umerator ad a d -order polyomial i the deomiator For uch ceario there are Two LT Pair that are Helpful: Ae Ae ζω t ζω t i i [ ω ζ t] where : A [ ω ζ t φ] where : A β φ ta ω α ζω ω ζ α u ζ u ω ζ α ζω β ω ζω α α ζω ω For 0< ζ < Thee are ot i your book table but they are o the table o my webite! Otherwie Factor ito two term 4/

15 Note the effect of the IC: [ y 0 a y0 ] y0 y0 a y0 y0 Y zi a a0 a a0 Ae ζω t i [ ] ω ζ t u ω ζω α If y0-0 Thi form give y zi 0 0 a et by the IC Ae ζω t i [ ] ω ζ t φ u α ζω ω Otherwie 5/

16 Example of uig thi type of LT pair: Let y 0 y 0 4 The Y zi a 4 a a a 0 a a Now aume that for our ytem we have: a 0 00 & a 4 0 Pulled a out from each term i Num. to get form jut like i LT Pair. The Y zi Compare to LT: β α ζω ω Ad idetify: α 6 β ω 00 ω 0 ζω 4 ζ 4 / ω 4 / /

17 So ow we ue thee parameter i the time-domai ide of the LT pair: Ae ζω t i where : α ζ 6 ω 0 0. β [ ω ζ t φ] A β α ζω ω u ζ A β φ ta α ζω ω ζ ω ζ α ζω ta Aumig output i a voltage! 0..6 volt rad φ ta ω ζ α ζω t yzi.6e i [ 9.80t.8] u Notice that the zero-iput olutio for thi d -order ytem ocillate t -order ytem ca t ocillate d - ad higher-order ytem ca ocillate but might ot!! 7/

18 Here i what thi zero-iput olutio look like: t yzi.6e i [ 9.80t.8] u Zoom I Slope of 4 Notice that it atifie the IC!! y 0 y 0 4 8/

19 N th -Order Cae Diff. eq of the ytem N d y N dt N M d y dy dx dx an... a a0 y b b b0 x N M M dt dt dt dt i d x For M N ad 0 i 0,,,..., M i dt where t 0 Takig LT ad re-arragig give: IC B Y A A LT of the olutio i.e. the LT of the ytem outpu N N A an... a a0 output-ide polyomial M B bm... b b0 iput-ide polyomial IC polyomial i that deped o the IC Recall: For d order cae: IC y0 [ y 0 a y0 ] 9/

20 0/ Coider the cae where the LT of x i ratioal: D N D N A B A IC A B A IC Y The Thi ca be expaded like thi: D F A E A IC Y for ome reultig polyomial E ad F D F A E A IC Y So for a ytem with A B H D N ad iput with ad iitial coditio you get: Zero-Iput Repoe Zero-State Repoe Traiet Repoe Steady-State Repoe Decay i time domai if root of ytem char. poly. A have egative real part Decay i time domai if root of ytem char. poly. A have egative real part

21 If all IC are zero zero tate C 0 The: Coectio To ect. 6.5 B Y A H Called Trafer Fuctio of the ytem ee Sect. 6.5 Y E A Zero-State Repoe F D Traiet Repoe Steady-State Repoe /

22 Summary Commet:. From the differetial equatio oe ca eaily write the H by ipectio!. The deomiator of H i the characteritic equatio of the differetial equatio. 3.The root of the deomiator of H determie the form of the olutio recall partial fractio expaio BIG PICTURE: The root of the characteritic equatio drive the ature of the ytem repoe we ca ow ee that via the LT. zero-iput rep. zero-tate rep. We ow ee that there are three cotributio to a ytem repoe:. The part drive by the IC a. Thi will decay away if the Ch. Eq. root have egative real part. A part drive by the iput that will decay away if the Ch. Eq. root have egative real part Traiet Repoe 3. A part drive by the iput that will perit while the iput perit Steady State Repoe /