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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)