1. Differential Equation Models Section 3.5

2. Impulse Response of an LTI System

3. 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.

4. Complex Exponential

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

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

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

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

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

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

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

12. 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

13. Finding The Forced Solution

14. Finding the General Solution (initial condition)

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

16. Transfer Function (Transfer function)

17. Summary (p. 125)

18. Summary (p. 129)