Laplace Transforms in Linear Circuits, Exams of Introduction to Sociology

Download Laplace Transforms in Linear Circuits and more Exams Introduction to Sociology in PDF only on Docsity! MAE140 Linear Circuits 107 Laplace Transforms – recap for ccts What’s the big idea? 1. Look at initial condition responses of ccts due to capacitor voltages and inductor currents at time t=0 Mesh or nodal analysis with s-domain impedances (resistances) or admittances (conductances) Solution of ODEs driven by their initial conditions Done in the s-domain using Laplace Transforms 2. Look at forced response of ccts due to input ICSs and IVSs as functions of time Input and output signals IO(s)=Y(s)VS(s) or VO(s)=Z(s)IS(s) The cct is a system which converts input signal to output signal 3. Linearity says we add up parts 1 and 2 The same as with ODEs MAE140 Linear Circuits 108 Laplace transforms The diagram commutes Same answer whichever way you go Linear cct Differential equation Classical techniques Response signal Laplace transform L Inverse Laplace transform L-1 Algebraic equation Algebraic techniques Response transform Ti m e do m ai n (t do m ai n) Complex frequency domain(s domain) MAE140 Linear Circuits 111 Laplace Transform Pair Tables damped cosine damped sine cosine sine damped ramp exponential ramp step impulse TransformWaveformSignal )(t! 22)( !" " ++ + s s 22)( !" ! ++s 22 ! ! +s 22 !+s s 1 s 1 2 1 s !+s 1 2)( 1 !+s )(tu )(ttu )(tue t!" )(tu t te !" ( ) )(sin tut! ( ) )(cos tut! ( ) )(sin tutte !"# ( ) )(cos tutte !"# MAE140 Linear Circuits 112 Laplace Transform Properties Linearity – absolutely critical property Follows from the integral definition Example { } { } { } )()()()()()( 2121 sBFsAFtBtAtBftAf +=+=+ 21 fLfLL [ ] ( ) ( ) 22 1 2 1 2 222 ))cos(( ! !! ! !!!! + = + + " = +=# $ % & ' ( += "" s As js A js A e A e A ee A tA tjtjtjtj LLLL MAE140 Linear Circuits 113 Laplace Transform Properties Integration property Proof Denote so Integrate by parts ( ) s sF df t = ! " # $ % & ' 0 )( ((L dtste t df t df !" # "=" $ % & ' ( ) * + , - . / 0 0 )( 0 )( 0000L )(and, 0 )(and, tf dt dy e dt dx t dfy s ste x st == != "" = " ## !!! " # " # + $ $ % & ' ' ( )# = $ $ % & ' ' ( ) 0000 )( 1 )()( dtetf s df s e df st tstt ****L MAE140 Linear Circuits 116 Laplace Transform Properties Initial Value Property Final Value Property Caveats: Laplace transform pairs do not always handle discontinuities properly Often get the average value Initial value property no good with impulses Final value property no good with cos, sin etc )(lim)(lim 0 ssFtf st !"+" = )(lim)(lim 0 ssFtf st !"! = MAE140 Linear Circuits 117 Rational Functions We shall mostly be dealing with LTs which are rational functions – ratios of polynomials in s pi are the poles and zi are the zeros of the function K is the scale factor or (sometimes) gain A proper rational function has n≥m A strictly proper rational function has n>m An improper rational function has n<m )())(( )())(( )( 21 21 01 1 1 01 1 1 n m n n n n m m m m pspsps zszszs K asasasa bsbsbsb sF !!! !!! = ++++ ++++ = ! ! ! ! L L L L MAE140 Linear Circuits 118 A Little Complex Analysis We are dealing with linear ccts Our Laplace Transforms will consist of rational function (ratios of polynomials in s) and exponentials like e-sτ These arise from • discrete component relations of capacitors and inductors • the kinds of input signals we apply – Steps, impulses, exponentials, sinusoids, delayed versions of functions Rational functions have a finite set of discrete poles e-sτ is an entire function and has no poles anywhere To understand linear cct responses you need to look at the poles – they determine the exponential modes in the response circuit variables. Two sources of poles: the cct – seen in the response to Ics the input signal LT poles – seen in the forced response MAE140 Linear Circuits 121 Inverse Laplace Transforms – the Bromwich Integral This is a contour integral in the complex s-plane α is chosen so that all singularities of F(s) are to the left of Re(s)=α It yields f(t) for t≥0 The inverse Laplace transform is always a causal function For t<0 f(t)=0 Remember Cauchy’s Integral Formula Counterclockwise contour integral = 2πj×(sum of residues inside contour) ( )[ ] ∫ ∞+ ∞− − == j j stdsesF j tfsF α απ )( 2 1)(1L MAE140 Linear Circuits 122 Inverse Laplace Transform Examples Bromwich integral of On curve C1 For given θ there is r→∞ such that Integral disappears on C1 for positive t x R→∞ s-plane pole a t≥0 t<0 !" ! # $ < % = + = & '+ '& ( 0for0 0for 1 )( t te dse as tf at j j st ) ) as sF + = 1)( C1 C2 !"<<+= r j res , 2 3 2 , # $ #$ % 0foras0 0cos)Re( )Im()Re( >!""= <+= treee rs tsjtsst #$ MAE140 Linear Circuits 123 Inverting Laplace Transforms Compute residues at the poles Example )()(lim sFas as ! " !" # $% & ' ' ' (' )()( 1 1 lim )!1( 1 sF m as m ds m d asm ( ) ( ) ( ) ( )31 3 21 1 1 2 31 3)1(2)1(2 31 522 + ! + + + = + !+++ = + + ssss ss s ss 3 )1( )52()1(lim 3 23 1 −= + ++ −→ s sss s 1 )1( )52()1(lim 3 23 1 =         + ++ −→ s sss ds d s 2 )1( )52()1(lim !2 1 3 23 2 2 1 =         + ++ −→ s sss ds d s ( ) )(32 )1( 52 2 3 2 1 tutte s ss t −+=         + + −− L MAE140 Linear Circuits 126 Example 9-12 Find the inverse LT of )52)(1( )3(20)( 2 +++ + = sss ssF 21211 )( * 221 js k js k s k sF ++ + !+ + + = ! 4 5 2555 21)21)(1( )3(20 )()21( 21 lim2 10 152 2 )3(20 )()1( 1 lim1 j ej jsjss s sFjs js k sss s sFs s k =""= +"=+++ + ="+ +"# = = "=++ + =+ "# = )() 4 5 2cos(21010 )(252510)( 4 5 )21( 4 5 )21( tutee tueeetf tt jtjjtj t !" # $% & ++= ! ! " # $ $ % & ++= '' '''++' ' ( (( MAE140 Linear Circuits 127 Not Strictly Proper Laplace Transforms Find the inverse LT of Convert to polynomial plus strictly proper rational function Use polynomial division Invert as normal 34 8126)( 2 23 ++ +++ = ss ssssF 3 5.0 1 5.0 2 34 2 2)( 2 + + + ++= ++ + ++= ss s ss s ssF )(5.05.0)(2 )( )( 3 tueet dt td tf tt !" # $% & +++= ''( ( MAE140 Linear Circuits 128 Multiple Poles Look for partial fraction decomposition Equate like powers of s to find coefficients Solve )())(()( )())(( )( )( 1222121 2 211 2 2 22 2 21 1 1 2 21 1 pskpspskpskKzKs ps k ps k ps k psps zsK sF !+!!+!=! ! + ! + ! = !! ! = 11222112 2 1 22212121 211 2 )(22 0 Kzpkppkpk Kkppkpk kk =!+ =++!! =+