1. 6.11s June 2006 L31 6.11s Notes for Lecture 3 PM ‘Brushless DC’ Machines: Elements of Design June 14, 2006 J.L. Kirtley Jr.

2. 6.11s June 2006 L32 Cross Section View: Surface Magnet Machine: Note windings

3. 6.11s June 2006 L33 Alternate: Surface Mount (‘Iron Free’) Armature Winding

4. 6.11s June 2006 L34 Magnets Inside the Rotor

5. 6.11s June 2006 L35 Machine Design for Very High (negative) Saliency

6. 6.11s June 2006 L36 Focus on Rating: Rating is number of phases times voltage times current Internal voltage is frequency times flux And flux is the integral of Flux density We will consider winding factor below

7. 6.11s June 2006 L37 Internal Voltage Construction: Here is flux Density from Magnets This is an approximation to the shape of the field in the air-gap (only an approximation) Radial field But see the notes for this done right

8. 6.11s June 2006 L38 Magnetic field can be found through a little field analysis The result below is good for magnets inside and p not equal to one. See the notes for other expressions Stator winding outside: Stator winding inside:

9. 6.11s June 2006 L39 Current Capacity This begs two questions: How to establish current density? How to establish slot fraction?

10. 6.11s June 2006 L310 Voltage Ratio

11. 6.11s June 2006 L311 Calculation of Inductance: Start with a Full-Pitch Coil Set This current distribution makes the flux distribution below

12. 6.11s June 2006 L312 Fundamental Flux Density Flux Linkage Idealized inductance of a full- pitch coil Taking into account phase-phase coupling (for 3 phase machine) and winding factor And for the PM machine the magneti is part of the magnetic gap

13. 6.11s June 2006 L313 This is what we mean by short pitch: see the original drawing

14. 6.11s June 2006 L314 Breadth Factor: Coils link flux slightly out of phase Here is a construction of the flux addition. It takes a bit of high-school like geometry to show that: The breadth factor is just the length of the addition of the vectors divided by the length of one times the number of vectors

15. 6.11s June 2006 L315 Slot Leakage: Suppose the slot were to look like this: It actually has two coils that have N c half turns each. Flux linked by one coil from one driven coil is: Use top of slot dimensions for tapered slots: very small error

16. 6.11s June 2006 L316 There are 2p(m-N sp ) slots with both coils in the same phase And 2p N sp slots with coils ineach of the different phases (in each phase) So slot leakage is

17. 6.11s June 2006 L317 Winding resistance is important So there are various ways of estimating winding length and area: Area is easier: Winding length must account for end turns and that is a geometric problem

18. 6.11s June 2006 L318 We have power conversion figure out Losses are: Armature conduction loss: I 2 R a Core Loss Friction, windage, etc To get core loss we use the model developed earlier, depending on the species of iron and fields calculated thus:

19. 6.11s June 2006 L319 The Process of design is a loop

20. 6.11s June 2006 L320 There are (at least) three types of performance specifications: Requirements are specifications that must be met a. Rotational Speed or frequency b. Rating Limits are specifications that must not be exceeded a. Tip Speed b. Maximum operating temperature Attributes are specifications that, all other things being equal, should be maximized or minimized So the design process consists of meeting the requirements, observing the limits and maximizing the attributes

21. 6.11s June 2006 L321 Multiple attributes make maximization iffy No simple way of telling if A is better than D (or C) But B is clearly superior to (dominates) E

22. 6.11s June 2006 L322 Novice Design Assistant: Is deliberately not an expert system Uses Monte Carlo to generate randomized designs Each variable in the design space is characterized by: Mean Value Standard Deviation Maximum value (limit) Minimum value (limit) Setup file (msetup.m) specifies Number of design variables For each: the above data Number of attributes to be returned function file called by nda.m: called attribut.m returns attributes and a go-no-go (limits not violated)

23. 6.11s June 2006 L323 Operation For the PM machine fluxes are given by simple expressions So torque is: Now normalize the machine in the following way: probably use field flux for normalization

24. 6.11s June 2006 L324 Then per-unit torque is: Per-Unit Currents to achieve the maximum torque per unit current are:

25. 6.11s June 2006 L325 Note that per-unit flux achievable for a given terminal voltage is: And this is related to current by:

26. 6.11s June 2006 L326

27. 6.11s June 2006 L327

28. 6.11s June 2006 L328 Base speed

29. 6.11s June 2006 L329 Here is your basic three phase bridge

30. 6.11s June 2006 L330 Suppose we have this situation:

31. 6.11s June 2006 L331 Here is one way of switching that circuit: The arrows designate when a switch is ON

32. 6.11s June 2006 L332 Here is what is on in State 0: V a = V, V b = V, V c = 0 V n = 2V/3

33. 6.11s June 2006 L333 V a = 0, V b = V, V c = 0 V n = V/3 Here is what is on in State 1:

34. 6.11s June 2006 L334 V a = 0, V b = V, V c = V V n = 2V/3 Here is what is on in State 2:

35. 6.11s June 2006 L335 V a = 0, V b = 0, V c = V V n = V/3 Here is what is on in State 3:

36. 6.11s June 2006 L336 V a = V, V b = 0, V c = V V n = 2V/3 Here is what is on in State 4:

37. 6.11s June 2006 L337 V a = V, V b = 0, V c = 0 V n = V/3 Here is what is on in State 5:

38. 6.11s June 2006 L338 Voltages: Line-Line Voltages are well defined

39. 6.11s June 2006 L339 To generate switching signals: Totem Pole A is High in states 0, 4 and 5 Totem Pole B is High in states 0, 1 and 2 Totem Pole C is High in states 2, 3 and 4 This allows us to use very simple logic: A = S0 + S4 + S5 B = S0 + S1 + S2 C = S2 + S3 + S4

40. 6.11s June 2006 L340 To generate switch signals Note that either top or bottom switch is on in each phase Generation of states: we will do this a bit later (see below)

41. 6.11s June 2006 L341 This ‘six pulse’ switching strategy: Makes good use of the switching devices Also requires ‘shoot-through’ delays Has very simple logic We propose an alternative switching strategy Makes minimally less effective use of switches Uses a little more logic But does not risk shoot through

42. 6.11s June 2006 L342 Here is a comparison of switching strategies 180 degree six- pulse 120 degree six pulse Give up a little timing between switch closings

43. 6.11s June 2006 L343 Switches Q_1 and Q_5 are on: State0 Va = V, Vb = 0, Vc = V/2

44. 6.11s June 2006 L344 Switches Q_1 and Q_6 are on: State1 Va = V, Vc = 0, Vb = V/2

45. 6.11s June 2006 L345 Switches Q_2 and Q_6 are on: State2 Vb = V, Vc = 0, Va = V/2

46. 6.11s June 2006 L346 Switches Q_2 and Q_4 are on: State3 Va = 0, Vb = V, Vc = V/2

47. 6.11s June 2006 L347 Switches Q_3 and Q_4 are on: State4 Va = 0, Vc = V, Vb = V/2

48. 6.11s June 2006 L348 Switches Q_3 and Q_5 are on: State5 Vc = V, Vb = 0, Va = V/2

49. 6.11s June 2006 L349 This switching pattern results in these voltages

50. 6.11s June 2006 L350 Switches turn on: Q1State_0 OR State_1 Q2State_2 OR State_3 Q3State_4 OR State_5 Q4State_3 OR State_4 Q5State_1 OR State_5 Q6State_1 OR State_2 Each switch is on for two states

51. 6.11s June 2006 L351 So here is how to do it 3 bit input to ‘138 selects one of 8 outputs Active low output! ‘138 has 3 enable inputs: two low, one high

52. 6.11s June 2006 L352 NAND (Not AND) Is the same as Negative Input OR The ‘138 output is ‘active low’: Matching bubbles makes an OR function

53. 6.11s June 2006 L353 Now we must generate six states in sequence If we have a ‘clock’ with rising edges at the right time interval we can use a very simple finite state machine This could be a counter, reset when it sees ‘5’

54. 6.11s June 2006 L354 Here is a good counter to use: 74LS163 This is a loadable counter: don’t need that feature Clear function is synchronous: so it clears only ON a clock edge Part is ‘edge triggered’: changes state on a positive clock edge P and T are enables: must pull them high

55. 6.11s June 2006 L355 And here are the counter states: note how CL works

56. 6.11s June 2006 L356 We already detect state 5 with the ‘138

57. 6.11s June 2006 L357 The ‘138 is a simple selector: use like this: And here are the pinouts of the ‘163 and ‘138

58. 6.11s June 2006 L358 Variable Voltage: do the Pulse Width Modulation thing

59. 6.11s June 2006 L359 Nomenclature: Two more views of the machine

60. 6.11s June 2006 L360 This is a ‘cut’ from the radial direction (section BB)

61. 6.11s June 2006 L361 Here is a cut through the machine (section AA) Winding goes around the core: looking at 1 turn

62. 6.11s June 2006 L362 Voltage is induced by motion and magnetic field Induction is: Voltage induction rule: Note magnets must agree!

63. 6.11s June 2006 L363 Single Phase Equivalent Circuit of the PM machine E a is induced (‘speed’) voltage Inductance and resistance are as expected This is just one phase of three Voltage relates to flux:

64. 6.11s June 2006 L364 PM Brushless DC Motor is a synchronous PM machine with an inverter:

65. 6.11s June 2006 L365 Induced voltages are: Assume we drive with balanced currents: Then converted power is: Torque must be:

66. 6.11s June 2006 L366 Now look at it from the torque point of view:

67. 6.11s June 2006 L367 Terminal Currents look like this: So torque is, in terms of DC side current:

68. 6.11s June 2006 L368 VaVa VbVb VcVc V ab Rectified back voltage is max of all six line-line voltages

69. 6.11s June 2006 L369 Average Rectified Back Voltage is: Power is simply:

70. 6.11s June 2006 L370 So from the DC terminals this thing looks like the DC machine:

71. 6.11s June 2006 L371 Magnets must match (north-north, south-south) for the two rotor disks. Looking at them they should look like this: End AEnd B Keyway

72. 6.11s June 2006 L372 Need to sense position: Use a disk that looks like this

73. 6.11s June 2006 L373 Position sensor looks at the disk: 1=‘white, 0=‘black’

74. 6.11s June 2006 L374 Some care is required in connecting to the position sensor Vcc GND Channel 1 Channel2 (you need to figure out which of these is ‘count’ and which is ‘zero’)

75. 6.11s June 2006 L375 Control Logic: Replace open loop with position measurement

76. 6.11s June 2006 L376 Why do we need to PWM only the top switches? What happens with you turn OFF switch Q1?