ECSE- Lecue. Paial facio expasio (m<) ypes of poles Simple Real poles Real Equal poles Complex cojugae poles sawyes@pi.edu www.pi.edu/~sawyes Mehod fo fidig f() fom F(s) wihou akig he ivese laplaceasfom (wihou iegaio) Cocep: Expad F(s) io a sum of simple ems whose ivese laplace asfoms we kow Ca he use ables Use lieaiy popey Expad F(s) = F (s) + F (s) + F (s) +... Fom Lieaiy Popey: => = = + + + f() L {F(s)} f () f () f ()... Whee L {F (s)} = f () L {F (s)} = f () ec. Fid f (), f (), ec. } Fom Tables N(s) b s + b s +... + b s + b F(s) = = D(s) a s a s... a s a m m m m + + + + (s z )(s z )...(s z ) m F(s) = K (s p )(s p )...(s p ) Facos of N(s) = zi => Zeos of F(s) Facos of D(s) = p j => Poles of F(s) Thee ae oly Types of Poles: Simple, Real Poles : (s 4), => p = 4 Real, E es + => = = qual Pol : (s ), p p Complex Cojugae Poles: => p,p = 4 ± (s 8s 5) + +
Thee is a diffee way of doig Paial Facio Expasio fo Each Type of Pole Le's Fis Look a Simple Real Poles The Complex Cojugae Poles Fially, Real, Equal Poles (Muliple Poles) Will Fis Look a Cicuis whee m < : N(s) s Example: F(s) = = D(s) Simple Real Poles a p =, p = 4 To Fid L {F(s)} fo Simple, Real Poles: s + s + 4 s s + s + 4 α Kow ha L [ ] = e s + α => = + 4 f() e e ; Need Oly o Fid ad I Geeal: = (s p )F(s) s = p "Cove-Up Rule" s + s + 4 s F(s) = To Fid Coefficies,...; Use "Cove-Up Rule": )F(s)] s = p )F(s)] s = p p = p = 4 s s + s + 4 )F(s)] s = p s => = [(s + ) ] s ( ) => = [ ] = = (s + 4) + 4
s s + s + 4 )F(s)] s = p s => = [(s + 4) ] 4 s ( 4) => = [ ] = = + 8 (s + ) 4 + 4 s s + s + 4 => =, = + 8 f() fo Simple Poles = e + e ; p p f() e => = + 8e 4 ; I Geeal: s p s p s p + +... p )F(s)] ; Cove-Up Rule => f() = ) e + e + e +...) p p p Complex Cojugae Poles s s Example: F(s) = = s + 8s + 5 s + α s + ω α = 4, ω = 5, β = 5 = ; Complex Cojugae Poles a p, p = 4± Complex Cojugae Poles F(s) = s s + 8s + 5 Complex Cojugae Poles a p, p = 4± To fid {F(s)} fo Complex Cojugae Poles: L Expad F(s) = + s ( 4+ ) s ( 4 ) = + s + 4 s + 4+ Expad F(s) = + s + 4 s + 4+ = Complex Coefficie = + j = Complex Cojugae of = ji Le's Look a a Picue i = /φ = / φ
Complex Space Imagiay / φ = a a agle of φ Expad F(s) = + s + 4 s + 4+ = + i φ = a i j i φ = + ji = /φ Real = Complex Coefficie = + j = /φ = Complex Cojugae of = ji = / φ => = + ( 4+ ) ( 4 ) f() e e => = + 4 f() e cos( φ) i Expad F(s) = + s + 4 s + 4+ To Fid Coefficies, Use "Cove-Up Rule": )F(s)] s = p )F(s)] s = p Good News: Need o Fid Oly Bad News: Mus use Complex lgeba s F(s) = = + s + 8s + 5 (s + 4 ) (s + 4+ ) s = [(s + 4 ) ] (s + 4 )(s + 4 + ) 4+ ( 4 + ) 8 + j j j8 = = = ( 4 + + 4 + ) j j + j8 j8 = = + j8 = = + j + 4 = = => = + α f() e cos( β φ) α = 4; β = i 8 φ = a = 5. 4 => f() = e cos( + 5. ) I Geeal: Expad F(s)... = + + + s p s + α jβ s + α + jβ p α f() e... e co Fid ad = / φ fom Cove-Up Rule = > = + + s ( β + φ) Simple Poles Complex Poles 4
Real, Equal Poles: p, p = α s Example: F(s) = (s + ) Real, Equal Poles a p, p = To Fid L {F(s)} fo Real, Equal Poles: s + (s + ) Real, Equal Poles: s + (s + ) => f() = e + e s F(s) = (s + ) Need Oly o Fid ad s Real, Equal Poles: F(s) = (s + ) To Fid, Use Cove-Up Rule: => = + [(s ) F(s)] s = = s = Cao Use Cove-Up Rule fo s Real, Equal Poles: F(s) = (s + ) s + (s + ) () F() = = = + => = ( + ) + ( + ) f() e = e Real, Equal Poles Double Pole: Expad F(s) = +.. + [ + ] s p s p (s p ) = (s p ) F(s) ; Cove-Up Rule p Usually Fid fom evaluaig F() o F() => f() = ( e +... + e + e ) p p p Simple Poles Repeaed Poles Real, Equal Poles Double Pole: Ca lso Use Diffeeiaio: = (s p ) F(s) ; Cove-Up Rule p d = (s p ) F(s) ; Diffeeiaio ds p => f() = ( e +... + e + e ) p p p Simple Poles Repeaed Poles 5
Real, Equal Poles Tiple Pole: Expad F(s) = +.. + [ + + ] s p s p (s p ) (s p ) = (s p ) F(s) ; Cove-Up Rule p Usually Fid fom F() o F() Usually Fid fom lim sf( s) s p p p p => f() = (e +... + e + e + e )! Real, Equal Poles Tiple Pole: Usig Diffeeiaio = (s p ) F(s) ; Cove-Up Rule p d = [(s p ) F(s)] ; Diffeeiaio ds p d = [(s p ) F(s)] ; Diffeeiaio! ds p p p p p => f() = (e +... + e + e + e )! Wha happes whe m =? (s z )(s z )...(s z ) m F(s) = K (s p )(s p )...(s p ) Use Log Divisio F(s) = K + Remaide Remaide will have m < m = f() = L {K} + L {Remaide} Use Paial Facio Expasio o Fid L {K} = K δ () δ f() = K () + L {Remaide} Mos Cicuis will have m < L {Remaide} Will Do i Seps: Mehod Fis Fid Diffeeial Equaio Tasfom o a lgebaic Equaio Take Ivese Laplace o Fid y() Mehod Defie s-domai Cicuis No Moe Diffeeial Equaios! Same Resul as Solvig Diffeeial Equaio: No Clea ha his is Easie fo s Ode Cicuis wih Swiched DC Ipus Sill have o fid Diffeeial Equaio dvaage?: Ca ow Solve Cicuis of NY Ode Ca ow Solve Cicuis wih NY Ipu IF we ca fid he Diffeeial Equaio
Le s Pacice wih civiy -: This is a d Ode Cicui: Need d Ode Diffeeial Equaio Igoe Poblem Saeme abou sdomai diagam ad iiial codiio souces Igoe Seies-Paallel Impedace Reducio 7