6.11s Notes for Lecture 3

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6.11s Notes for Lecture 3 PM Brushless DC
Machines Elements of Design June 14, 2006 J.L.
Kirtley Jr.
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Cross Section View Surface Magnet Machine Note
windings
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Alternate Surface Mount (Iron Free) Armature
Winding
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Magnets Inside the Rotor
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Machine Design for Very High (negative) Saliency
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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
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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) Rad
ial field But see the notes for this done
right
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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
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Current Capacity
This begs two questions How to establish
current density? How to establish slot fraction?
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Voltage Ratio
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Calculation of Inductance Start with a
Full-Pitch Coil Set
This current distribution makes the flux
distribution below
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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
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This is what we mean by short pitch see the
original drawing
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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
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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
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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
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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
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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
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The Process of design is a loop
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• 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

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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
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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)
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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
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Then per-unit torque is
Per-Unit Currents to achieve the maximum torque
per unit current are
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Note that per-unit flux achievable for a given
terminal voltage is
And this is related to current by
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Base speed
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Here is your basic three phase bridge
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Suppose we have this situation
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Here is one way of switching that circuit The
arrows designate when a switch is ON
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Here is what is on in State 0
Va V, Vb V, Vc 0 Vn 2V/3
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Here is what is on in State 1
Va 0, Vb V, Vc 0 Vn V/3
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Here is what is on in State 2
Va 0, Vb V, Vc V Vn 2V/3
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Here is what is on in State 3
Va 0, Vb 0, Vc V Vn V/3
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Here is what is on in State 4
Va V, Vb 0, Vc V Vn 2V/3
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Here is what is on in State 5
Va V, Vb 0, Vc 0 Vn V/3
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Voltages Line-Line Voltages are well defined
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• 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

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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)
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• 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

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Here is a comparison of switching strategies
180 degree six-pulse 120 degree six pulse Give
up a little timing between switch closings
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Switches Q_1 and Q_5 are on State0
Va V, Vb 0, Vc V/2
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Switches Q_1 and Q_6 are on State1
Va V, Vc 0, Vb V/2
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Switches Q_2 and Q_6 are on State2
Vb V, Vc 0, Va V/2
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Switches Q_2 and Q_4 are on State3
Va 0, Vb V, Vc V/2
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Switches Q_3 and Q_4 are on State4
Va 0, Vc V, Vb V/2
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Switches Q_3 and Q_5 are on State5
Vc V, Vb 0, Va V/2
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This switching pattern results in these voltages
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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
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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
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NAND (Not AND) Is the same as Negative Input OR
The 138 output is active low Matching
bubbles makes an OR function
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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
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Here is a good counter to use 74LS163 This is a
loadable counter dont 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
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And here are the counter states note how CL works
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We already detect state 5 with the 138
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The 138 is a simple selector use like this
And here are the pinouts of the 163 and 138
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Variable Voltage do the Pulse Width Modulation
thing
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Nomenclature Two more views of the machine
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This is a cut from the radial direction
(section BB)
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Here is a cut through the machine (section
AA) Winding goes around the core looking at 1
turn
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Voltage is induced by motion and magnetic
field Induction is Voltage induction rule Note
magnets must agree!
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Single Phase Equivalent Circuit of the PM
machine Ea is induced (speed)
voltage Inductance and resistance are as
expected This is just one phase of three
Voltage relates to flux
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PM Brushless DC Motor is a synchronous PM machine
with an inverter
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Induced voltages are
Assume we drive with balanced currents
Then converted power is
Torque must be
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Now look at it from the torque point of view
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Terminal Currents look like this
So torque is, in terms of DC side current
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Rectified back voltage is max of all six
line-line voltages
Va
Vc
Vb
ltEbgt
Vab
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Average Rectified Back Voltage is
Power is simply
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So from the DC terminals this thing looks like
the DC machine
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Magnets must match (north-north, south-south) for
the two rotor disks. Looking at them they should
look like this
End A
End B
Keyway
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Need to sense position Use a disk that looks
like this
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Position sensor looks at the disk 1white,
0black
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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)
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Control Logic Replace open loop with position
measurement
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Why do we need to PWM only the top switches? What
happens with you turn OFF switch Q1?