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)