Differential Equation Models Section 3.5. Impulse Response of an LTI System.
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Transcript of Differential Equation Models Section 3.5. Impulse Response of an LTI System.
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)