EECE 301 Signals & Systems Prof. Mark FowlerSystem ModelingLinear Constant-Coefficient Differential Equations1/15EECE 301 Signals & SystemsProf. Mark FowlerNote Set #6• System Modeling and C-T System Models• Reading Assignment: Sections 2.4 & 2.5 of Kamen and Heck2/15Ch. 1 IntroC-T Signal ModelFunctions on Real LineD-T Signal ModelFunctions on IntegersSystem PropertiesLTICausalEtcCh. 2 Diff EqsC-T System ModelDifferential EquationsD-T Signal ModelDifference EquationsZero-State ResponseZero-Input ResponseCharacteristic Eq.Ch. 2 ConvolutionC-T System ModelConvolution IntegralD-T System ModelConvolution SumCh. 3: CT Fourier SignalModelsFourier SeriesPeriodic SignalsFourier Transform (CTFT)Non-Periodic SignalsNew System ModelNew SignalModelsCh. 5: CT Fourier SystemModelsFrequency ResponseBased on Fourier TransformNew System ModelCh. 4: DT Fourier SignalModelsDTFT(for “Hand” Analysis)DFT & FFT(for Computer Analysis)New SignalModelPowerful Analysis ToolCh. 6 & 8: Laplace Models for CTSignals & SystemsTransfer FunctionNew System ModelCh. 7: Z Trans.Models for DTSignals & SystemsTransfer FunctionNew SystemModelCh. 5: DT Fourier System ModelsFreq. Response for DTBased on DTFTNew System ModelCourse Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).3/15System ModelingTo do engineering design, we must be able to accurately predict the quantitative behavior of a circuit or other system. CircuitsDevice RulesR: v(t)=Ri(t)L: v(t)=L(di(t)/di)C: dv(t)/dt=1/Ci(t)Circuit Rules-KVL -KCL-Voltage Divider-etc.Differential EquationMechanicalDevice RulesMass: M(d2p(t)/dt2)Spring: kxp(t)Damping: kd(dp(t)/dt)System Rules-Sum of forces-etc.Differential EquationSimilar ideas hold for hydraulic, chemical, etc. systems…“differential equations rule the world”This requires math models:4/15Simple Circuit Example:Sending info over a wire cable between two computersComputer#2Computer#1Two conductors separated by an insulator⇒ capacitance“Twisted Pair” of Insulated WiresTypical values: 100 Ω/km50 nF/kmcoaxial cableconductors separated by insulatorRecall: resistance increases with wire lengthTwo practical examples of the cable5/15Simple Model:Cable Model0 0 1 1 0 1 0 1 …Effective Operation:x(t)t5vx(t)y(t)Receiver’s TheveninEquivalent Circuit (Computer #2)Infinite Input Resistance (Ideal)Driver’s TheveninEquivalent Circuit (Computer #1)Zero Output Resistance (Ideal)6/150 0 1 1 0 1 0 1 …x(t)t5vx(t)Use Loop Equation & Device Rules:This is the Differential Equation to be “Solved”:Given: Input x(t) Find: Solution y(t)Recall: A “Solution” of the D.E. means…The function that when put into the left side causes it to reduce to the right sideDifferential Equation & System… the solution is the outputdttdyCtitRitvtytvtxRR)()()()()()()(==+=)(1)(1)(txRCtyRCdttdy=+y(t)7/15Now… because this is a linear system (it only has R, L, C components!) we can analyze it by superposition.0 0 1 1 0 1 0 1 …t5vx(t)t5vt-5vt5vt-5v+++Decompose the input…8/15t5vt-5v+t5v+t-5v+Input Components Output Components (Blue)Standard Exponential Response Learned in “Circuits”:t5v+t-5v+t5v+t-5v9/15t5vt-5vt5vt-5v+++Output Componentst5vOutput0 0 1 1 0 1 0 1 …t5vx(t)InputOutput is a “smoothed” version of the input… it is harder to distinguish “ones” and “zeros”…it will be even harder if there is noise added onto the signal!10/15Computer#2Computer#1x(t)y(t)Physical System:Schematic System:)(1)(1)(txRCtyRCdttdy=+Mathematical System:t5vOutputMathematical Solution:Progression of Ideas an Engineer Might Use for this Problem11/15Automobile Suspension System ExampleM2Auto FrameM1Roadx(t) = Input: Tire’s Positiony(t) = Output: Frame’s PositionwheelSuspension springShock absorberkskdktTire’s spring effectResults in 4thorder differential equation:)]([)()()()()(0122233344txFtyadttdyadttydadttydadttyd=++++Some functionof Input x(t)The aiare functions of system’s physical parameters: M1, M2, ks, kd, kt12/15Again… to find the output for a given input requires solving the differential equationEngineers could use this differential equation model to theoretically explore: 1. How the car will respond to some typical theoretical test inputs when different possible values of system physical parameters are used2. Determine what the best set of system physical parameters are for a desired response3. Then… maybe build a prototype and use it to fine tune the real-world effects that are not captured by this differential equation model13/15So… What we are seeing is that for an engineer to analyze or design a circuit (or a general physical system) there is almost always an underlying Differential Equation whose solution for a given input tells how the system output behavesSo… engineers need both a qualitative and quantitative understanding of Differential Equations.The major goal of this course is to provide tools that help gainthat qualitative and quantitative understanding!!!14/15Linear Constant-Coefficient Differential EquationsGeneral Form: (Nth-order)Input:x(t)Output:y(t)Solution of the Differential Equation∑∑=−==+MiiiNiiiNtxbtyaty0)(10)()()()()(Indicatesithorder derivativetNttZINeCeCeCtyλλλ+++=⇒ 2121)(tttNeeeλλλ,,,21…N “modes”: Assuming distinct roots…Then: y(t) = yZI(t) + yZS(t) (yZS(t) is our focus, so we will often say ICs = 0)Recall:Two parts to the solution(i) one part due to ICs with zero-input (“zero-input response”)(ii) one part due to input with zero ICs (“zero-state response”)“Homogeneous Solution”See Video ReviewCharacteristic Polynomial: λN+ an-1 λN-1+ … + a1 λ + a0N roots: λ1 , λ2 , λ3 , … , λN15/15So how do we find yZS(t)?If you examine the zero-state part for all the example solutions of differential equations we have seen you’ll see that they all look like this:So we need to find out:1. Given a differential equation, what is h(t-λ)See Ch. 3, 5, 6, 8See Ch. 2See Ch. 3, 5, 6, 8Really just need to know h(t)… it is called the system’s “Impulse Response”λλλdxthtyttZS)()()(0∫−=This is called “Convolution”(We’ll study it in Ch. 2)InputOutput when “in zero state”2. How do we compute & understand the convolution integral3. Are there other (easier? more insightful?) methods to find