Lesson+10

# Lesson+10 - EE Lamentations Then there are days when I...

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EE Lamentations. Then there are days when I thank God I’m not one of those other engineers. 1 Lesson 10

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Laplace Transform Unit Step Function Impulse (Delta) Distribution Ch.12.1-4 Lesson 10 Lesson 10 2
Challenge 9 Lesson 10 Q. What differential equation does the circuit shown below satisfy, where s ( t ) is the input signal and y ( t ) is the output? s ( t ) y ( t ) a b IC 2 OP AMP circuit. 3 - IC 1 - -1

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Lesson 10 Assume the existence of d 2 y(t)/dt 2 -dy(t)/dt y(t) a(-1)(-dy(t)/dt) = a dy(t)/dt = b y(t) 4 s ( t ) y ( t ) a b IC 2 OP AMP circuit. - IC 1 - -1 y[0] = IC1,–dy(0)/dt = IC2
System Realization Experimental SPICE study (Homogeneous case s(t)=0) Frequency in r/s 0 =2.0 (b= 0 2 =4.0) a=0 ( undamped ) Frequency in Hz f 0 = 0 /2 = 0.32 Hz G=-4 y(t) -dy(t)/dt - ( )dt - ( )dt d 2 y(t)/dt 2 d 2 y(t)/dt 2 =4y(t) y(0)=1, dy(t)/dt| t=0 =0 s(t)=0 RC=1 5 Lesson 10 RC=1

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y(t)=cos( 0 t) d 2 y(t)/dt 2 -dy(t)/dt   0 =2; f 0 =0.32, T 0 =3.14 6 Lesson 10
Lesson 10 Continuous-time linear systems have been shown to be mathematically modeled by an ODE. In the real-world, how do engineers actually analyze such systems? Lesson 10 "Nature laughs at the difficulties of integration." Pierre-Simon de Laplace The slackers response: Laplace transform 7

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Lesson 10 8 What input x(t), |x(t)| 1, would maximize the integral’s value? d e t x t y M M s ) ( ) ( M d d e d e e t y e t x Let M M M M M M s s s 2 1 ) ( ) ( 0 Nevertheless, this seems to be a useful tool. What should we call it? Correlation Relevant question: Consider Why this is a relevant question will be addressed at a later time.
Laplace transform: Symbolic: ) ( ) ( s X t x L Lesson 10 dt e t x s X st Given for one instant an intelligence that could comprehend all the forces by which nature is animated and … sufficiently vast to submit these data to analysis – it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom. For it, nothing would be uncertain and the future as the past, would be present to its eyes. - Pierre Simon de Laplace. (X(s) is called the Laplace spectrum ) With this tool engineering became self-actualization ~ 1950 To be a transform, it must be invertible. 9

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There are 2 common forms for a Laplace transform: Lesson 10 st e t x s X 0 st e t x s X (bilateral or two-side form – non-causal signals) (unilateral or one-sided form – causal signals) Why did Laplace invent this strange formula and what’s with that e -st thing?
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