Differential Equation Models

Section 3.5

Impulse Response of an LTI System

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 FourierTransform in chapter 5.

H(s) can also be understood using the differential equation approach.

Complex Exponential

RL Circuit

Let y(t)=i(t) and x(t)=v(t)

𝐿𝑑𝑦 (𝑡)𝑑𝑡

+𝑅𝑦 (𝑡 )=𝑥 (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

(no repeated roots)

(characteristicequation)

Stability ↔Root Locations

(marginally stable)

(unstable)

Stable

The Force Response

• Determine the form of force solution from x(t)

𝐿𝑑𝑦 (𝑡)𝑑𝑡

+𝑅𝑦 (𝑡 )=𝑥 (𝑡 )

Solve for the unknown coefficients Pi by substituting yp(t) into

𝐿𝑑𝑦 (𝑡)𝑑𝑡

+𝑅𝑦 (𝑡 )=𝑥 (𝑡 )

Finding The Forced Solution

Finding the General Solution

(initial condition)

Nth order LTI system

• If there are more inductors and capacitors in the circuit,

Transfer Function

(Transfer function)

Summary (p. 125)

Summary (p. 129)