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This lecture notes provide an overview of the design elements of brushless DC machines and focus on performance optimization. It covers topics such as winding design, current capacity, slot leakage, losses, and the design process.
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6.11s Notes for Lecture 3 PM ‘Brushless DC’ Machines: Elements of Design June 14, 2006 J.L. Kirtley Jr.
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
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
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:
Current Capacity This begs two questions: How to establish current density? How to establish slot fraction?
Calculation of Inductance: Start with a Full-Pitch Coil Set This current distribution makes the flux distribution below
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
This is what we mean by short pitch: see the original drawing
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
Slot Leakage: Suppose the slot were to look like this: It actually has two coils that have Nc half turns each. Flux linked by one coil from one driven coil is: Use top of slot dimensions for tapered slots: very small error
There are 2p(m-Nsp) slots with both coils in the same phase And 2p Nsp slots with coils ineach of the different phases (in each phase) So slot leakage is
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
We have power conversion figure out Losses are: Armature conduction loss: I2 Ra 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:
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
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
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)
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
Then per-unit torque is: Per-Unit Currents to achieve the maximum torque per unit current are:
Note that per-unit flux achievable for a given terminal voltage is: And this is related to current by:
Here is one way of switching that circuit: The arrows designate when a switch is ON
Here is what is on in State 0: Va = V, Vb = V, Vc = 0 Vn = 2V/3
Here is what is on in State 1: Va = 0, Vb = V, Vc = 0 Vn = V/3
Here is what is on in State 2: Va = 0, Vb = V, Vc = V Vn = 2V/3
Here is what is on in State 3: Va = 0, Vb = 0, Vc = V Vn = V/3
Here is what is on in State 4: Va = V, Vb = 0, Vc = V Vn = 2V/3
Here is what is on in State 5: Va = V, Vb = 0, Vc = 0 Vn = V/3
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
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)
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
Here is a comparison of switching strategies 180 degree six-pulse 120 degree six pulse Give up a little timing between switch closings
Switches Q_1 and Q_5 are on: State0 Va = V, Vb = 0, Vc = V/2
Switches Q_1 and Q_6 are on: State1 Va = V, Vc = 0, Vb = V/2
Switches Q_2 and Q_6 are on: State2 Vb = V, Vc = 0, Va = V/2
Switches Q_2 and Q_4 are on: State3 Va = 0, Vb = V, Vc = V/2
Switches Q_3 and Q_4 are on: State4 Va = 0, Vc = V, Vb = V/2
Switches Q_3 and Q_5 are on: State5 Vc = V, Vb = 0, Va = V/2
Switches turn on: Q1 State_0 OR State_1 Q2 State_2 OR State_3 Q3 State_4 OR State_5 Q4 State_3 OR State_4 Q5 State_1 OR State_5 Q6 State_1 OR State_2 Each switch is on for two states
