Dynamic analysis of dc machine

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Dynamic analysis of dc machine

  1. 1. PHYSICAL STRUCTURE 9/1/201311:48AM 1 PRB/SCE/Dept.ofEEE
  2. 2. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 2 PE 9211 Analysis of Electrical Machines
  3. 3. Dynamic Characteristics of Permanent Magnet DC Motor Modes of Dynamic operation 1. Starting from stall 2. Changes in load torque Condition: The machine supplied from a constant – voltage source 9/1/201311:48AM 3 PRB/SCE/Dept.ofEEE
  4. 4. Mathematical Model of a PMDC Motor: 9/1/201311:48AM 4 PRB/SCE/Dept.ofEEE
  5. 5. This motor consists of two first order differential equation and two algebraic equation Armature current equation, 9/1/201311:48AM 5 PRB/SCE/Dept.ofEEE
  6. 6. Speed equation, 9/1/201311:48AM 6 PRB/SCE/Dept.ofEEE
  7. 7. 9/1/201311:48AM 7 PRB/SCE/Dept.ofEEE
  8. 8. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 8 Simulink Model of PMDC Motor Motor Parameters
  9. 9. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 9 Solving armature current equation
  10. 10. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 10 Solving Speed equation
  11. 11. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 11
  12. 12. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 12 Dynamic performance during starting
  13. 13. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 13
  14. 14. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 14 Dynamic Characteristics of DC Shunt Motor
  15. 15. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 15
  16. 16. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 16 Simulink Model of DC Shunt Motor: Fig shows the Simulink model of DC Shunt Motor. It is constructed using subsystems for solving each differential equations (i.e.) armature current, field current and torque equation.
  17. 17. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 17
  18. 18. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 18
  19. 19. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 19
  20. 20. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 20
  21. 21. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 21 Time domain block diagrams and state equations Shunt connected dc machine W.K.T 𝒗 𝒂 = π’Š 𝒂 𝒓 𝒂 + 𝑳 𝑨𝑨 π’…π’Š 𝒂 𝒅𝒕 + 𝑳 𝑨𝑭 𝝎 𝒓 π’Š 𝒇 βˆ’ βˆ’ βˆ’βˆ’ 𝟏 𝒗 𝒇 = π’Š 𝒇 𝑹 𝒇 + 𝑳 𝑭𝑭 π’…π’Š 𝒇 𝒅𝒕 βˆ’ βˆ’ βˆ’βˆ’ 𝟐 𝑻 𝒆 = 𝑻 𝑳 + 𝑱 π’…πŽ 𝒓 𝒅𝒕 + 𝑩 π’Ž 𝝎 𝒓 βˆ’ βˆ’βˆ’ βˆ’ πŸ‘
  22. 22. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 22 Equations (1),(2) and (3) can be written in terms of its time constants 𝒗 𝒂 = 𝒓 𝒂 𝟏 + 𝑳 𝑨𝑨 𝒓 𝒂 𝒅 𝒅𝒕 π’Š 𝒂 + 𝑳 𝑨𝑭 𝝎 𝒓 π’Š 𝒇 𝒗 𝒂 = 𝒓 𝒂 𝟏 + 𝝉 𝒂 𝝆 π’Š 𝒂 + 𝑳 𝑨𝑭 𝝎 𝒓 π’Š π’‡βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ πŸ’ 𝑯𝒆𝒓𝒆, 𝝆 ⟢ 𝒅 𝒅𝒕 𝒗 𝒇 = 𝑹 𝒇 𝟏 + 𝑳 𝑭𝑭 𝑹 𝒇 𝝆 π’Š 𝒇 𝒗 𝒇 = 𝑹 𝒇 𝟏 + 𝝉 𝒇 𝝆 π’Š π’‡βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’(5) 𝑻 𝒆 βˆ’ 𝑻 𝑳 = ( 𝑩 π’Ž + 𝑱 𝝆) 𝝎 𝒓 βˆ’ βˆ’ βˆ’ βˆ’ πŸ”
  23. 23. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 23 𝝉 𝒂 ⟢ Armature time constant 𝝉 𝒇 ⟢ Field time constant π‘Ίπ’π’π’—π’Šπ’π’ˆ 𝒕𝒉𝒆 π’†π’’π’–π’‚π’•π’Šπ’π’π’” πŸ’ , πŸ“ 𝒂𝒏𝒅 πŸ” 𝒇𝒐𝒓 π’Š 𝒂 , π’Š 𝒇, 𝒂𝒏𝒅 𝝎 𝒓 π’šπ’Šπ’†π’π’…π’” π’Š 𝒂 = 𝟏 𝒓 𝒂 𝝉 𝒂 𝝆 + 𝟏 𝒗 𝒂 βˆ’ 𝑳 𝑨𝑭 𝝎 𝒓 π’Š 𝒇 βˆ’ βˆ’ βˆ’βˆ’ πŸ• π’Š 𝒇 = 𝟏 𝑹 𝒇 𝝉 𝒇 𝝆 + 𝟏 𝒗 𝒇 βˆ’ βˆ’βˆ’ βˆ’ πŸ– 𝝎 𝒓 = 𝟏 𝑱𝝆 + 𝑩 π’Ž 𝑻 𝒆 βˆ’ 𝑻 𝑳 βˆ’βˆ’ βˆ’ βˆ’ πŸ—
  24. 24. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 24 Time domain block diagram of a shunt connected dc machine
  25. 25. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 25 π‘Ίπ’π’π’—π’Šπ’π’ˆ 𝒕𝒉𝒆 π’†π’’π’–π’‚π’•π’Šπ’π’π’” 𝟏 , 𝟐 𝒂𝒏𝒅 πŸ‘ 𝒇𝒐𝒓 π’…π’Š 𝒂 𝒅𝒕 , π’…π’Š 𝒇 𝒅𝒕 𝒂𝒏𝒅 π’…πŽ 𝒓 𝒅𝒕 π’šπ’Šπ’†π’π’…π’” From (1) π’…π’Š 𝒂 𝒅𝒕 = βˆ’ 𝒓 𝒂 𝑳 𝑨𝑨 π’Š 𝒂 βˆ’ 𝑳 𝑨𝑭 𝑳 𝑨𝑨 π’Š 𝒇 𝝎 𝒓 + 𝟏 𝑳 𝑨𝑨 𝒗 𝒂— 𝟏𝟎 From (2) π’…π’Š 𝒇 𝒅𝒕 = βˆ’ 𝑹 𝒇 𝑳 𝑭𝑭 π’Š 𝒂 + 𝟏 𝑳 𝑭𝑭 𝒗 𝒇— 𝟏𝟏 From (3) π’…πŽ 𝒓 𝒅𝒕 = βˆ’ 𝑩 π’Ž 𝑱 𝝎 𝒓 + 𝑳 𝑨𝑭 𝑱 π’Š 𝒇 π’Š 𝒂 βˆ’ 𝟏 𝑱 𝑻 𝑳 βˆ’ βˆ’(𝟏𝟐) State equation of shunt dc machine
  26. 26. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 26 𝜌 π’Š 𝒇 π’Š 𝒂 πŽπ’“ = βˆ’π‘Ή 𝒇 𝑳 𝑭𝑭 𝟎 𝟎 𝟎 βˆ’π’“ 𝒂 𝑳 𝑨𝑨 𝟎 𝟎 𝟎 βˆ’π‘© π’Ž 𝑱 π’Š 𝒇 π’Š 𝒂 πŽπ’“ + 𝟎 βˆ’π‘³ 𝑨𝑭 πŽπ’“ 𝑳 𝑨𝑨 𝑳 𝑨𝑭 π’Š 𝒇 π’Š 𝒂 𝑱 + 𝟏 𝑳 𝑭𝑭 𝟎 𝟎 𝟎 𝟏 𝑳 𝑨𝑨 𝟎 𝟎 𝟎 βˆ’πŸ 𝑱 𝒗 𝒇 𝒗 𝒂 𝑻 𝑳 State equations in matrix form or vector matrix form Note: The second term on the right side contains the product of state variables causing the system to be nonlinear.
  27. 27. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 27 Permanent Magnet dc Machine 𝒗𝒇 π’Šπ’” π’†π’π’Šπ’Žπ’Šπ’π’‚π’•π’†π’… 𝑳𝑨𝑭 π’Šπ’‡ π’Šπ’” 𝒓𝒆𝒑𝒍𝒂𝒄𝒆𝒅 π’ƒπ’š π’Œπ’— π’Œπ’— π’Šπ’” π’…π’†π’•π’†π’“π’Žπ’Šπ’π’†π’… π’ƒπ’š Strength of the magnet Reluctance of the iron No. of turns in the armature winding
  28. 28. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 28 W.K.T Above eqns. (1) and (2) can be written in terms of its time constants
  29. 29. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 29Time domain block diagram of a permanent magnet DC machine
  30. 30. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 30 State Equation of a permanent magnet DC machine From (1) From (2)
  31. 31. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 31 The form in which the state equations are expressed in above eqn. is called the fundamental form. OR
  32. 32. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 32 Advantages to using the state space representation compared with other methods. 1.The ability to easily handle systems with multiple inputs and outputs; 2.The system model includes the internal state variables as well as the output variable; 3.The model directly provides a time-domain solution, the matrix/vector modeling is very efficient from a computational standpoint for computer implementation
  33. 33. 9/1/201311:48AMPRB/SCE/Dept.ofEEE 33
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