Linear System Analysis I
Linear System Analysis I ECE 311
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This 4 page Study Guide was uploaded by Sarina Wintheiser on Tuesday September 22, 2015. The Study Guide belongs to ECE 311 at Colorado State University taught by William Eads in Fall. Since its upload, it has received 102 views. For similar materials see /class/210285/ece-311-colorado-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Colorado State University.
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Date Created: 09/22/15
EE 311 Linear Systems Midterm Review October 2008 Test will be Open Books Open Notes Calculator no computer Allowed 1 hour 15 minutes about 3 questions REVIEW Complex Numbers 39e s XJy reJ r X2 y212 6 tan391yX eJ e cos6 jsinG ODE Solution quot dquot z m dkxt Zak dyk 217k k k0 t k 0 dt yt yea ypt Total Complementary Particular Response Function Integral Characteristic Equation ansnan1sn391a1sao 0 anS39SlS39SZS39Sn Roots of equation s1s2sIl y0 Cleslt Cze 52 Cnesnt System modes Particular Integral Solution total Solution Initial Conditions solve for coef cients Dynamic Sytem Models ElectricalMechanical Components Mass Inductor Spring Capacitor Damper Resistor Interconnect Laws Newton Kirchoff MassSpringDamper Example Md2ytdt2 ft Kyt detdt Md2ytdt2 detdt Kyt ft ODE Model Op Amp Circuits idealizing assumptions input voltage input current Nonlinear Differential Equations Linearize e g sinx x x33z x55z z x for small x Signals and Systems Time transformations reversal scaling shifting Amplitute transformations Even and Odd signals X0 Xet KOO Xet 12Xt Xt KOO WM 7 Xt average of Xt average of Xet evenodd properties Periodic signals Xt Xt T Vt Period T Fundamental period To f0 lTo 030 21th Eat030 Common periodic signals sin0t cos0t ej Complex exponential signals Xt CeSt C Ael l s 60 jOJo Xt Ae soejwot 1 SPlane Chart 3901 A E a Larger Icol faster growthdecay where so Res Larger Imol faster oscillation rate where no lms 50 lt 0 decaying so gt 0 growing so 0 oscillatingconstant Stable Signals Step function ut 60 lt 0 ut 0 for tlt0 1 for tgt1 Impulse function 5t Ift5t todt on Properties of stepsimpulses Building Signals from impulses steps ramps Systems Linear X1t gt y1t X2t gt y2t 3 a1X1t a2X2t gt a1y1t a2y2t Time Invariant Xt gt yt 3 X0 t0 gt W t0 LTI Systems ODE dytdt 2yt Xt Maybe non zero initial conditions I G implicit time domain SOLV N Transfer Function Hs 1s2 Zero initial conditions amp forced only explicit frequency domain MULTIPLICATION Impulse Response ht e 21 ut Zero initial conditions amp forced only explicit time domain CONVOLUTION Convolution yt ht Xt IxTht TdT xt ht j hma TdT ODETransfer FunctionImpulse Response quot dk t m dkxt Zak yk 217k k k0 dt k0 dt Xe gt Hsest 71 bus bm4s b1s 120 H Rational Polynomial form ans aHs als a0 Hs Djihte dt x0 aye x0 aye x0 ya Xejmt gt Hjco Xejmt application of sinusoid to system Hslsjo xt System Properties Memoryless static Causal Invertible Stable frequency response Hjo ZHGLO ZXkew gt MS ZXkHGka at Sk ijk H k1 ODE Transfer Function and Convolution H LTI System ya KXO Hs K ht 2 K50 ht0fortlt0 Hs PROPER ie ODE has n 2 In No information lostiformal method later His 1Hs Characteristic equation of ODE has all roots strictly in LHP Resi lt 0 for all roots Transfer function has all poles strictly in LHP Resi lt 0 for all roots Oil10M lt co
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