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Published byEugene Stokes Modified over 6 years ago

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Differential Equation Models Section 3.5

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Impulse Response of an LTI System

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H(s) H(s) is the the Laplace transform of h(t) With s=jω, H(jω) is the Fourier transform of h(t) Cover Laplace transform in chapter 7 and Fourier Transform in chapter 5. H(s) can also be understood using the differential equation approach.

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Complex Exponential

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RL Circuit Let y(t)=i(t) and x(t)=v(t) Differential Equation & ES 220

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n th order Differential Equation If you use more inductors/capacitors, you will get an nth order linear differential equation with constant coefficients

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Solution of Differential Equations Find the natural response Find the force Response – Coefficient Evaluation

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Determine the Natural Response 0, since we are looking for the natural response.

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Natural Response (Cont.) Assume y c (t)=Ce st

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Nth Order System Assume y c (t)=Ce st (no repeated roots) (characteristic equation)

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Stability ↔Root Locations (marginally stable) (unstable) Stable

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The Force Response Determine the form of force solution from x(t) Solve for the unknown coefficients P i by substituting y p (t) into

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Finding The Forced Solution

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Finding the General Solution (initial condition)

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Nth order LTI system If there are more inductors and capacitors in the circuit,

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Transfer Function (Transfer function)

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Summary (p. 125)

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Summary (p. 129)

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