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## Differential Equation Models

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**Differential Equation Models**Section 3.5**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.**RL Circuit**Let y(t)=i(t) and x(t)=v(t) Differential Equation & ES 220**nth order Differential Equation**• If you use more inductors/capacitors, you will get an nth order linear differential equation with constant coefficients**Solution of Differential Equations**• Find the natural response • Find the force Response • Coefficient Evaluation**Determine the Natural Response**• Let L=1H, R=2Ω & =2 • (0≤t) • Condition: y(t=0)=4 • Assume yc(t)=Cest • Substitute yc(t) into • What do you get? 0, since we are looking for the natural response.**Natural Response (Cont.)**• Substitute yc(t) into Assume yc(t)=Cest**Nth Order System**Assume yc(t)=Cest (characteristic equation) (no repeated roots)**Stability ↔Root Locations**(unstable) Stable (marginally stable)**The Force Response**• Determine the form of force solution from x(t) Solve for the unknown coefficients Pi by substituting yp(t) into**Finding the General Solution**(initial condition)**Nth order LTI system**• If there are more inductors and capacitors in the circuit,**Transfer Function**(Transfer function)