Impulse Response Analysis of Second Order Systems

This detailed analysis explores the impulse responses of second order systems, focusing on underdamped and undamped scenarios. The visuals provide insights into the amplitude variations over time, showcasing different responses to varying input impulses.

• Second Order Systems
• Impulse Response Analysis
• Underdamped Response
• Undamped Response
• System Dynamics

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Uploaded on Apr 29, 2025 | 0 Views

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Presentation Transcript

1. Impulse Response analysis of second order systems

2. Second order impulse response Underdamped and Undamped Impulse response :

3. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 1 0.8 0.6 0.4 Amplitude 0.2 0 -0.2 -0.4 0 0.5 1 1.5 2 2.5 3 Time (sec)

4. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 1 0.8 0.6 0.4 Amplitude 0.2 0 -0.2 -0.4 0 0.5 1 1.5 2 2.5 3 Time (sec)

5. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 1 0.8 0.6 0.4 Amplitude 0.2 0 -0.2 -0.4 0 0.5 1 1.5 2 2.5 3 Time (sec)

6. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 1 0.8 0.6 0.4 Amplitude 0.2 0 -0.2 -0.4 0 0.5 1 1.5 2 2.5 3 Time (sec)

7. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 0.1 0.08 0.06 0.04 0.02 Amplitude 0 -0.02 -0.04 -0.06 -0.08 -0.1 0 0.5 1 1.5 2 2.5 3 Time (sec)

8. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 0.1 0.08 0.06 0.04 0.02 Amplitude 0 -0.02 -0.04 -0.06 -0.08 -0.1 0 0.5 1 1.5 2 2.5 3 Time (sec)

9. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 0.1 0.08 0.06 0.04 0.02 Amplitude 0 -0.02 -0.04 -0.06 -0.08 -0.1 0 0.5 1 1.5 2 2.5 3 Time (sec)

10. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 0.1 0.08 0.06 0.04 0.02 Amplitude 0 -0.02 -0.04 -0.06 -0.08 -0.1 0 0.5 1 1.5 2 2.5 3 Time (sec)

11. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 6 5 4 3 Amplitude 2 1 0 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec) 10 5 0 -5 -10 -6 -4 -2 0 2

12. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 6 5 4 3 Amplitude 2 1 0 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec) 10 5 0 -5 -10 -6 -4 -2 0 2

13. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 6 5 4 3 Amplitude 2 1 0 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec) 10 5 0 -5 -10 -6 -4 -2 0 2

14. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 6 5 4 3 Amplitude 2 1 0 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec) 10 5 0 -5 -10 -6 -4 -2 0 2

15. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 5 4 3 2 1 Amplitude 0 -1 -2 -3 -4 0 2 4 6 8 10 12 6 Time (sec) 4 2 0 -2 -4 -6 -5 0 5

16. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 5 4 3 2 1 Amplitude 0 -1 -2 -3 -4 0 2 4 6 8 10 12 6 Time (sec) 4 2 0 -2 -4 -6 -5 0 5

17. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 5 4 3 2 1 Amplitude 0 -1 -2 -3 -4 0 2 4 6 8 10 12 6 Time (sec) 4 2 0 -2 -4 -6 -5 0 5

18. Second order impulse response Underdamped and Undamped Increasing / Fixed Impulse Response 5 4 3 2 1 Amplitude 0 -1 -2 -3 -4 0 2 4 6 8 10 12 6 Time (sec) 4 2 0 -2 -4 -6 -5 0 5

19. Second order step response Underdamped and Undamped

20. Second order impulse response Underdamped and Undamped Higher frequency oscillations Faster response Slower response Unstable Lower frequency oscillations

21. Second order impulse response Underdamped and Undamped Less damping Unstable More damping

22. Second order step response Time specifications. 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3

23. Second order step response Time specifications. Steady state value. Time to reach first peak (undamped or underdamped only). % of in excess of . Time to reach and stay within 2% of . 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3

24. Second order step response Time specifications. Steady state value. More generally, if the numerator is not , but some :

25. Second order step response Time specifications. Peak time. Therefore, is the time of the occurrence of the first peak :

26. Second order step response Time specifications. Percent overshoot. Evaluating at , is defined as: Substituting our expressions for and :

27. Second order step response Time specifications. Settling time. Defining with , the previous expression for can be re-written as: As an approximation, we find the time it takes for the exponential envelope to reach 2% of . when

28. Typical specifications for second order systems. How many independent parameters can we specify? 3