IMPULSE RESPONSE OF SECOND ORDER SYSTEM
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The document is a case study on the impulse response of second order systems in electrical engineering, focusing on characteristics like damping ratio and different categories of damping. It discusses the mathematical representation of second order systems, their impulse responses, and resulting behaviors under varying conditions. Additionally, it references MATLAB coding to analyze damping, along with various academic sources for further reading.
IMPULSE RESPONSE OF SECOND ORDER SYSTEM
- 1. Department of Electrical & Electronics Engineering Raghu Engineering College (Autonomous) CASE STUDY ON IMPULSE RESPONSE OF SECOND ORDER SYSTEM BACHELOR OF TECHNOLOGY IN ELECTRICAL AND ELECTRONICS ENGINEERING Under the Supervision of Mr.P.SIVA RAMA RAO Assistant Professor Mr P.Eswar sai 18981A0237 Mr P.Mohan 18981A0238 Ms P.Supriya 18981A0239 Mr.P.Nitish 18981A0240 Mr.P.Sanjay Kumar 18981A0241 By Head of the department Dr.P.SASI KIRAN Professor
- 2. ANALYSIS OF SECOND ORDER SYSTEM Damping Ratio : The damping is measured by a factor called as damping ratio. It is denoted by 𝜀 . So when zeta is maximum ; it produces maximum opposition to the oscillatory behaviour of system.
- 3. BLOCK DIAGRAM OF SECOND ORDER SYSTEM FOR UNIT STEP UNIT
- 19. A general second-order system is characterized by the following transfer function: We can re-write the above transfer function in the following form (closed loop transfer function):
- 20. According the value of ζ, a second-order system can be set into one of the four categories: 1. OVER DAMPED : when the system has two real distinct poles (𝜖 >1). 2. UNDER DAMPED : when the system has two complex conjugate poles (0 <𝜖 <1) 3. UN-DAMPED : when the system has two imaginary poles (𝜖 = 0). 4. CRITICALLY DAMPED : when the system has two real but equal poles (𝜖 = 1).
- 21. IMPULSE RESPONSE OF THE SECOND ORDER SYSTEM: „Laplace transform of the unit impulse is R(s)=1 „Impulse response :- „Transient response for the impulse function, which is simply is the derivative of the response to the unit step: derivative of the response to the unit step:
- 22. „Time Responses The characteristic equation has two roots: Where 𝑠1 and 𝑠2 are the poles of the second order system.
- 23. 𝑐 𝑠 = 1 𝑠 − 𝑠 + 𝜔 𝑛 𝑠 + 𝜔 𝑛 − 𝜔 𝑛 𝑠 + 𝜔 𝑛 2 When 𝜀 = 1 C(s) = 1 𝑠 - 1 𝑠+𝜔 𝑛 − 𝜔 𝑛 𝑠+𝜔 𝑛 2 By taking the inverse Laplace transform C(t) = 1 - 𝑒−𝜔𝑛𝑡 − 𝜔 𝑛 𝑒−𝜔𝑛𝑡 ∗ 𝑡 At t =0 𝑐 𝑡 = 1 − 𝑒0 1 + 0 = 1 − 1 = 0 At t = ∞ 𝑐 𝑡 = 1 − 𝑒∞ ( 1 +∞ ) = 1 – 0 = 1
- 24. When 𝜀 = UNDERDAMPED :-
- 25. When 𝜀 > 1, OVERDAMPING :- 0,1)( 1 ))(( )( 1 )( ))(( 2 )( 21 11 21 2 21 2 21 2 22 2 tekek pp ty spsps sY s sX psps ss sC tptpn n n nn n
- 26. • The transfer function can be rewritten as: 22 2 222222 2 22 2 2 2 )( dn n nnnn n nn n s ss ss sC
- 27. • The step response, after some simplification, can be written as: • Hence, the response of this system eventually settles to a steady- state value of 1. However, the response can overshoot the steady-state value and will oscillate around it, eventually settling in to its final value. n d d t n n where ttety n 1 tan 0,sin1)(
- 28. IMPULSE RESPONSE: The first pulse has a width T and height 1/T, area of the pulse will be 1. If we halve the duration and double the amplitude we get second pulse. The area under the second pulse is also unity The pulse for which the duration tends to zero and amplitude tends to infinity is called impulse. Impulse function also known as delta function.
- 29. MATLAB CODING :-
- 30. RESULTS FOR DAMPING UNDER DAMPING AND OVER DAMPED
- 31. REFERENCES :- 1. AUTOMATIC CONTROL SYSTEM KUO & GOLNARAGHI 2. 2. CONTROL SYSTEM ANAND KUMAR 3. 3. AUTOMATIC CONTROL SYSTEM 4. CLASS NOTES