June 9, 2005 6.11s 2005 1

Massachusetts Institute of Technology

Department of Electrical Engineering and Computer Science

6.11s Design of Motors, Generators and Drive Systems

June 9, 2005

J.L. Kirtley Jr.

©2005, J.L. Kirtley Jr.

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Coils are wound on a toroidal core ( a magnetic circuit)

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Axial View: Magnets are interacting with stator current

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Magnetic materials exhibit hysteresis

For core materials you want a narrow curve

But for PM materials you want a wide curve

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Hard permanent magnet materials have B-H curves like this

Remanent Flux density can be as high as 1.4 T

Incremental permeability is like free space

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€

By = Br + μ0Hm = μ0Hg

€

gHg + hmHm = 0

Magnet Characteristic

Geometry

Combination

€

By = Brhmg+ hm

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This graphic shows that calculation

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s = −μ0hmg

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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!

€

E '= E + v × B

€

V = ulB =ωRlB =Cω

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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

€

Ea =ωλ 0Voltage relates to flux:

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PM Brushless DC Motor is a synchronous PM machine with an inverter:

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Ea = E cos ωt +θ( )

Eb = E cos ωt +θ −2π

3

⎛

⎝ ⎜

⎞

⎠ ⎟

Ec = E cos ωt +θ +2π

3

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Ia = I1 cosωt

Ib = I1 cos ωt −2π

3

⎛

⎝ ⎜

⎞

⎠ ⎟

Ic = I1 cos ωt +2π

3

⎛

⎝ ⎜

⎞

⎠ ⎟

€

P =1

2EI1 cosθ =

1

2λ 0ωI1 cosθ

Induced voltages are:

Assume we drive with balanced currents:

Then converted power is:

Torque must be:

€

T =p

ωP =

p

2λ 0I1 cosθ

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Now look at it from the torque point of view:

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T = IlBR =CI

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I1 =4

πsin

120o

2I0 =

4

π

3

2I0

Terminal Currents look like this:

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T =3

2pλ 0I1 = p

3 3

πλ 0I0

So torque is, in terms of DC side current:

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Va VbVc

Vab

0 12

Rectified back voltage is max of all six line-line voltages

<Eb>

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< Eb >=3

π3

−π

6

π

6

∫ ωλ 0 cosωtdωt

=3 3

πωλ 0

Average Rectified Back Voltage is:

Power is simply:

€

Pem =3 3

πωλ 0I0 =KI0

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So from the DC terminals this thing looks like the DC machine:

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T =KI

Ea =Kω

K = p3 3

πλ 0

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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: 1=‘white, 0=‘black’

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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