UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires heory of disribuions; someimes referred as hyperfuncion δ = if I mus saisfy he ideniy if = δ d = So, bulkiness of he uni impulse (hyper)funcion is equal o. The defining characerisic is () δ ( ) = f d f In differen areas, he uni impulse funcion has differen represenaions, bu in elecrical circui analysis is well suied following simple definiion: see figure. In pracice, i is impossible generae such impulse; usually i is enough, if for paricular circui is fulfilled he condiion (he shores ime consan in he circui) I has grea significance digial filers heory, where is represenaion is simple number - (uni impulse response of he FIR filer is equal o he se of he filer coefficiens). Symbol: Laplace ransform: L { w } = Pavel Máša, XE3EO2, lecure 9 page
Unie sep funcion (), < =, > Common used symbol is u (), bu in elecrical circui analysis i could be confused wih volage. Alhough commonly used mahemaical definiion is, < () =.5, = > we will simplify is descripion on ( ) =. We will draw i like: The simplified (from he poin of view of heory of disribuions) relaionships are d() δ () =, () δ( τ) d = Remember he laboraory measuremen las semeser measuremen of he volage across inducor and capacior and curren passing hese elemens; if he circui variable was derivaive of he recangular waveform, i exhibied Dirac-like pulse. Circui implemenaion connec he DC volage source wih magniude V. Laplace ransform: dτ Pavel Máša, XE3EO2, lecure 9 page 2
Uni impulse and uni sep response Consider linear circui, wih zero energeic iniial condiions. Using Laplace ransform, he relaionship beween inpu and oupu volage can be described by ransfer funcion P p U ( p) ( p) = 2. U Remember: we can define ransfer funcion using phasors (sinusoidal seady sae), or Fourier ransform, bu never in ime domain (u() u()). Suppose u () is DC volage;uu 2 () is, le say, single exponenial pulse; u2() 2 u () has differen value in each ime insan. Bu ransfer funcion is sill he same fracional raional funcion, independen on exciaion! In sinusoidal seady sae he ransfer funcion is jus frequency dependen complex number. Uni impulse response u ( ) = δ ( ), w = u ( ) Uni sep response u ( ) = ( ), a = u ( ) In Laplace domain he relaionship among ransfer funcion and ransforms of uni impulse response and uni sep response is: = = = U p W p P p P p 2 = P( p) W p 2 2 P p U2 ( p) = A( p) = P( p) = p p P p A( p) = P p = pa p p Pavel Máša, XE3EO2, lecure 9 page 3
The uni impulse / sep responses have informaion abou ime consans, ransfer funcion and frequency response as well as oher properies of he given circui. The Fourier ransform of uni impulse characerisic is analogously, Bu equivalen relaionship does no exis for uni sep response (Fourier ransform of he uni sep funcion does no exis). The relaion beween uni impulse and uni sep responses in ime domain we may find i from ransform of derivaive and inegral: d u = pu p u d L () ( ), u( ) = lim u( ) da P( p) = W( p) = p A( p) = L a d da w () = a d ( ) δ () ( ) () = ( ) a w τ dτ a Pavel Máša, XE3EO2, lecure 9 page 4
Convoluion How o qualify relaion beween volage u ( ) and u2 domain? jus in ime We know such relaion however jus for wo inpu waveforms Dirac dela funcion δ ( ) and uni sep funcion ( ). Oupu volage is uni impulse response or uni sep response. Any waveform may be fied by (infiniely many) Dirac dela funcions, or uni sep funcions, muliplied by value of a funcion a disinc ime insan. he sum of uni impulse (sep) responses. The disance beween pulses Impulse bulkiness x ( ) k Relaed oupu volage x ( ) w ( ) k k The waveform of oupu volage is hen he sum of responses on disinc Dirac dela pulses (uni impulse responses, weighed by value of funcion), n () = ( ) x x w 2 k = k When he sum urns ino inegral convoluion inegral k () = ( ) 2 x x τ w τ dτ Pavel Máša, XE3EO2, lecure 9 page 5
The symbol of convoluion is sar (*) and is valid: () = ()* () = () ( ) = ( ) () x x w x w τ dτ x τ w dτ 2 Geomerical meaning: u( τ ) u( τ) w(.75 τ), S = u 2(.75 ) w( τ ) u ( τ) w( τ), S = u 2( ) w ( τ) u ( τ) w( τ), S = u 2(.25 ).25 ( τ) w( τ), S = u 2(.25 ) u ( τ ) w ( τ ), S = u 2(.5 ) u.25.5 ( τ) w( τ), S = u 2 (.5) u ( τ) w( τ), S = u 2(.75 ) u.5.75 Pavel Máša, XE3EO2, lecure 9 page 6
Example: We have inegraing circui, excied by volage source u = Ue a Find he waveform of oupu volage u2 ( ). a) Laplace ransform ransfer funcion P( p) = p = p, U ( p) = U p a U U U2( p) = = p a p a p a p U a u2() = e e a b) Uni impulse response convoluion () = e, w u = Ue a ( τ aτ ) u () = u () * w () = Ue e dτ = 2 U = e d e e τ = a aτ U a c) Uni sep response Duhamel s inegral, see below. Pavel Máša, XE3EO2, lecure 9 page 7
Duhamel s inegral Insead of recangular pulses like in he case of convoluion we may approximae inpu variable by uni sep funcions: The disance beween jumps Pulse heigh Relaed oupu volage x ( ) a ( ) The waveform of oupu volage disinc weighed uni seps () ( ) () x = x a x a 2 k= n k k k k is hen he sum of responses on When, he sum urns ino inegral and we go Duhamel s inegral k 2 () = ( ) () ( ) x x a x τ a τ dτ Using Laplace ransform: = = X p X p P p X p p A p 2 dx ( ) L, pa( p) px p x d = da = L a d ( ) Pavel Máša, XE3EO2, lecure 9 page 8
dx X2( p) = L x( ) A( p) = d () da = L a X p d ( ) Using inverse ransform we ge anoher forms of Duhamel s inegral: Or () = ( ) () ()* () = ( ) () ( ) u x a x a x a x τ a τ dτ 2 () ( ) () ()* () ( ) () u = a x x a = a x x τ a τ dτ = 2 ( ) () = a x x τ a τ dτ (BIBO) sabiliy If a sysem is (BIBO) sable, hen he oupu will be bounded for every inpu o he sysem ha is bounded. Then valid 2 = lim w = u ( ) lim u u or, he ransien dies away. p Such circu is sable. Circuis are classified ino: sable limi of sabiliy unsable p Pavel Máša, XE3EO2, lecure 9 page 9
Sable circui (BIBO sabiliy) The limi of sabiliy (response o impulse) Unsable circui Pavel Máša, XE3EO2, lecure 9 page
BIBO Bounded-Inpu Bounded-Oupu If a sysem is BIBO sable, hen he oupu will be bounded for every inpu o he sysem ha is bounded. Passive circui is always sable, if i conains resisiviy greaer han zero, zero resisiviy limi of sabiliy. Acive circui (conaining some amplifier) need o have feedback. When we had been sudying ransiens we learned, he general soluion, describing he ransien iself doesn depend on he naure of exciing source The polynomial order in nominaor of he ransfer funcion has o be less han he polynomial order in denominaor: M p Q p W( p) = P( p) = = D N p N p Than - Q p w () = Dδ () L N p The form of inverse ransform of ransfer funcion is given by parial fracion decomposiion. I is affeced by polynomial N( p ); roos poles may be: p n real w = L Ke n repeaed ( n n2 ) complex conjugaed L sin( ω ψ ) L p n w = L K K L e L w K e α n = n n n In all cases he soluion is affeced by exponenially dumped funcion, so if he pole (is real par) is negaive, he circui is sable, when i is posiive he circui is unsable. p - plane L Sable circui Unsable circui Limi of sabiliy Pavel Máša, XE3EO2, lecure 9 page