Chapter 6 Frequency Response & System Concepts

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1 hpte 6 Fequency esponse & ystem oncepts Jesung Jng stedy stte (fequency) esponse Phso nottion Filte

2 v v Foced esponse by inusoidl Excittion ( t) dv v v dv v cos t dt dt ince the focing fuction is sinusoid, the solution is ssumed we ( t) sin t cos t cos ( t φ ) ubstitute get v cos t inusoidl excittion ( t) into the diffeenti l eqution nd solving nd ( ) ( ) sin t ( ) ( ) NO cos t fo the to be of coefficien ts nd, n sinusoidlly excited line cicuit, ll bnch voltges nd cuents e sinusoids t the sme fequency s the excittion signl. he mplitudes of these voltges nd cuents e scled vesion of the excittion mplitude nd the voltge nd cuents my be shifted in phse with espect to the excittion signl. D the sme fom. Phso nottion to solve fo fequency, mplitude, nd phse with moe ese.

3 e ( cos sin ) : omplex phso nottion fo cos omplex lgeb f eithe sinusoidl voltge o cuent souce supplies line cicuit consisting of esistnce, inductnces, nd cpcitnces, ll bnch voltges nd cuents of those components will be sinusoids of the sme fequency. We need complex lgeb to pefom the ddition, subtction, division, nd multipliction of voltges nd cuents in excited cicuits. f n instntneous voltge is descibed by sinusoidl function of time such s v t cost ( t ) t { } {} v t ee e e e whee is mximum vlue, nd πf is n ngul fequency of voltges. fequency - domin (phso) fom ( t ) his simplifiction is ust mthemticl convenience. time - domin fom e e cos sin : Eule's dentity ( cos sin ) Phso concept hs no el physicl significnce. t is ust convenient mthemticl tool.

4 he combintion of el nd n imginy tem is clled complex numbe. omplex numbes cn be clculted s phsos. Gphiclly, the sum is the hypotenuse of the ight tingle fomed by the two phsos. e b b cos e Exmples > > > e( ) m( ) (in ctesin coodinte ) ( cos sin ) (in pol coodinte ) : mgnitude, tn ( ), b sin m( ),,, π omplex lgeb (cont.) π b : gument m b e b e ( ) m( ) : conugte of w ( ± w) ± w,( w) w, if w is not eo w

5 5, b e b e tesin fom is esie to use fo ddition nd subtction of complex numbes. b b b b b b b b b b b b b b b b b b b b b ± ± ± ± Pol fom is esie to use fo multipliction nd division of complex numbes. ± ± e e e e e e e e You hve to be ble to convet tesin fom to pol fom nd vice ves. omplex lgeb (cont.) Multipliction: multipliction of both mgnitudes & ddition of both guments Division: division of both mgnitudes & subtction of both guments

6 Hee, phso is defined s e. t hs the essentil infomtion: mplitude nd phse ngle of n c signl. he impednce is mesue of the opposition to the flow of cuent. ht is, it is complex esistnce. he element equtions tht define the voltge-cuent chcteistics of the pssive elements e (the Ohm s lw of sinusoidl cicuits) whee,, nd e phsos of voltge, cuent, nd impednce of n pssive element, espectively. / Y is clled dmittnce. he pssive elements hs impednces to sinusoids Fo esistnce, Fo inductnce, mpednce ( ) 9 X 9 whee X is the inductive ectnce. Fo cpcitnce, ` ( ) 9 X 9 whee X ( ) is the cpcitive he cpcitos nd inductos e fequency-dependent esistos. ectnce. 6

7 mpednce (cont.) he voltge coss esistnce is in phse with the cuent though it. he voltge coss n inductive ectnce leds the cuent though it by 9 o. (/) he voltge coss cpcitive ectnce lgs the cuent though it by 9 o. 7

8 mpednce (cont.) he totl impednces e clculted ust s esistnces e. n seies cicuit, the totl impednce is the sum of ll the impednces. n pllel cicuit, the ecipocl of the totl impednce is the ecipocl sum of individul impednces. lthough the impednce of single cicuit elements is eithe puely el (fo esistos) o puely imginy (fo inductos nd cpcitos), the genel definition of impednce should hve both el nd imginy pts since the pcticl cicuits consist of moe o less complex combintions of diffeent cicuit elements. 8

9 mpednce (cont.) - X Ω X Ω X 5 φ tn X 5 tn ( φ ) ( φ ) 6( φ ) ( φ ) X ( 9 ) 8( φ 9 ) Note: he vlue of the totl voltge () is less thn the sum of individul voltges ( 6 8), which is possible becuse these voltge vlues hve diffeent phse ngles. 9

10 tn 5 tn 5 φ φ φ φ φ φ X X

11 ummy he esistive (due to D) nd ective effects (due to ) must be combined by phsos (o complex numbes). Fo seies o pllel cicuits, ll the impednces e combined to find. he totl impednces e clculted ust s esistnces e. Fo line time-invint cicuit, the Netwok theoems (mesh nlysis, nodl nlysis, hevenin s theoem etc), K, o K tht wee used in solving D cicuits cn lso be used fo detemining sinusoidl () stedy-stte esponse using the phso nottions. Fo line time-vying cicuits o nonline cicuits, the stedy-stte esponse to sinusoidl input is usully not sinusoidl. icuit with (t) o (t) ime-invint cicuit: cicuit elements e time-invint (voltge o cuents of cicuit elements to inputs do not chnge with time) o independent souce.

12 Nodl nlysis with impednces K on the pinciple nodes nducto, cpcito, o esisto

13 Mesh nlysis with impednces Mesh cuent Mesh cuent K on the two meshes

14 Fequency esponse { } { } { } { } H whee Fequency esponse of cicuit povide mesue of how the cicuit esponds to sinusoidl inputs of bity fequency. Given the input signl mplitude, phse, nd fequency, knowledge of the fequency esponse of cicuit pemits the computtion of the output signl. o expess the fequency esponse of cicuit in tems of vition in output voltge s function of souce voltge, we use is phse-shifted nd mplitude-scled vesion of with the fequency unchnged.

Filtes Filttion system Filte: eliminte impuities fom dinking wte o outdoo i unglss: filte out eye-dmging ultviolet dition Electicl filte: ttenute (i.e. educe in mplitude) o eliminte signls of unwnted fequencies Filte types (in tems of mplitude esponse vs.

15 Filtes Filttion system Filte: eliminte impuities fom dinking wte o outdoo i unglss: filte out eye-dmging ultviolet dition Electicl filte: ttenute (i.e. educe in mplitude) o eliminte signls of unwnted fequencies Filte types (in tems of mplitude esponse vs. fequency) low-pss pss low fequencies nd ttenute high fequencies. high-pss pss high fequencies nd ttenute low fequencies. bnd-pss pss only specific bnd of fequencies fom its input to its output. bnd-stop block o seveely ttenute only specific bnd of fequencies. he filtes cn be thought of s fequency-dependent voltge divide, since the mount of output voltge is function of fequency. Pssive filtes: filtes mde up of cpcito, inducto, nd esistos, tht is, pssive components, which mens they do not genete voltges o cuents. oltge o cuent souce e not pssive but ctive. 5

16 ow-pss Filtes low-pss filte pss lowe fequencies (-> pss bnd) nd educe the mplitude of highe fequencies. he filte psses the udio signl (lowe fequency components, pssbnd) but ttenutes dio fequencies (highe fequency components, stopbnd). utoff fequency: the fequency t which the ttenution of filte educes the output mplitude to 7.7% of its vlue in the pss bnd. H Phse of ( ) ctn( ) o o i i ( ) o ctn( ) ctn( ) i o Mgnitude of i cutoff fequency of filte oltge divide 6

17 7 Exmple of ow-pss Filtes [ ] [ ] [ ] d/s 6 ctn ctn Phse of : :, : Mgnitude of ctn nd he mgnitude of fequency esponse of this filte is not equl to t the pssbnd.

18 High-Pss Filtes high-pss filte pss highe fequencies (-> pss bnd) nd educe the mplitude of lowe fequencies (-> stop bnd). H o i ( π ) ( ) ctn( ) o o i i ( ) o π ctn( ) i ( ) Mgnitude of Phse of 8

19 o o, : two fequencies :constnt Mgnitude of Phse of π nd-pss Filtes, & esonnce bnd-pss filte pss fequencies within cetin fequency nge (-> pss bnd) nd educe the mplitude of the othe fequencies (-> stop bnd). H o i o i ( )( ) ( ) ( ) o π ctn i ctn( ) ctn( ) tht detemine the pssbnd of o i ( ) ( ) ( ) the filte, 9

20 H nd-pss Filtes, & esonnce (cont.) he esponse of second-ode filtes cn be explined by ewiting the fequency esponse function in the following foms: o i ( ζ n ) ( ) ( ζ ) the shpness of Q n : dmping tio intesecting this hoiontl line. the powe t the output of the esonnt pek ( Qn ) ( ) ( Q ) n : ntul o esonnt fequency Q : Qulity fcto (Q fcto) ζ ζ n n n Q :(hlf - powe) bndwidth he fequency nge between (mgnitude) esponse points n Hlf powe stems fom the fct tht when the mplitude esponse is equl to.77, f the filte hs decesed by fcto of /. n n π

21 uning nd-pss filte: uning cicuits (-> choose desied fequency you wnt to he) employed in conventionl M dio by using vible cpcito. s illustted in the figue below, the vible cpcitnce cn be dusted to tune the seies cicuit to esonnce t ny one of five diffeent fequencies. -> he desied fequency component cn be collected by chnging the cpcitnce. f n ( ) π F π 6 ( 9 )( ) 5kH Fequencies with 5kH e visible nd the othes e eliminted o ttenuted t the output. mplitude fequency esponse function Five inputs

22 ode Plots ode plots: Fequency esponse plots of line systems displyed in the fom of logithmic plots. he mplitude tio is expessed in units of decibel (d), whee o o i ( ) ctn( ) ( ) cutoff, o hlf - powe, fequency of filte τ -d fequency ctn i ( ) d log o i,, o i o i d d log log d d

EECE 260 Electrical Circuits Prof. Mark Fowler

EECE 260 Electrical Circuits Prof. Mark Fowler EECE 60 Electicl Cicuits Pof. Mk Fowle Complex Numbe Review /6 Complex Numbes Complex numbes ise s oots of polynomils. Definition of imginy # nd some esulting popeties: ( ( )( ) )( ) Recll tht the solution