1. Permanent magnet synchronous machines as “brushless
DC drives”
1.1.5 Equivalent circuit of PM synchronous machine
Source: Siemens AG, Germany
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/1
Institut für Elektrische
Energiewandlung • FB 18
Equivalent circuit of PM synchronous machines
dis (t )
u s (t )  Rs  is (t )  Ld
 u p (t )
dt
Voltage equation per phase:
- back EMF up(t)
Ld  Lh  L
- self-induced voltage ~ dis/dt
Synchronous inductance:
- resistive voltage drop
main and leakage inductance:
- voltage from feeding inverter: us(t)
Leakage:
Q: slot; b: overhang, o: harmonic
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/2
Institut für Elektrische
Energiewandlung • FB 18
Air gap flux density excited by the
distributed stator winding
Resulting air-gap:
 res    hB  hM
mechanic air gap
bandage thickness
  "1 / 2"
of phase V
magnet height
  "1"
Fundamental of this flux density induces back into stator winding, thus linking phases U,
V, W, generating a self-inductance (main inductance) Lh, which is equal for all three
phases (here is shown flux excited by phase V, linked with coil of phase U)
2ms  p l Fe
Lh 
 0  ( N s  k ws )  2 
s  I s
  p  res
U s ,s
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/3
Institut für Elektrische
Energiewandlung • FB 18
Fundamental air gap flux linkage between the three
phases – Consideration for phase U
U  UU UV UW  LUphiU  M UV iV  M UW iW
 self  L phi ~   mutual  M  i ~  / 2
  "1"
M   L ph / 2
  "1 / 2"
iU  iV  iW  0  iU  iV  iW
U  LUphiU  L phiV / 2  L phiW / 2  (3 / 2)  L phiU  LhiU
Lh  (3 / 2)  L ph
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/4
Institut für Elektrische
Energiewandlung • FB 18
Alternative derivation: Self inductance of the stator
fundamental air gap field wave
Self induced voltage:
u s , s , 1 (t )   s  N s  k w1 
2

  p l Fe B , 1  sin( s t )  2  U s , s  sin( s t )
Fundamental stator field amplitude, excited by the stator sine current Is:
0
2 ms
B , 1 


 N s  k w1  I s
 res  p
Fundamental stator field winding factor: k w1
 k ws
Self inductance Lh of the fundamental stator field:
 p l Fe
Lh 
  0  ( N s  k ws )  2

s  I s
  p  res
U s,s
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2
2 ms
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/5
Institut für Elektrische
Energiewandlung • FB 18
Schematic drawing of stray flux lines
Slot leakage flux
Winding overhang leakage flux
Leakage flux density of winding
overhangs
Slot leakage flux, rising
linear from bottom to top of
slot according to Ampere´s
law
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/6
Institut für Elektrische
Energiewandlung • FB 18
Equivalent circuit per phase of
synchronous PM machine
dis (t )
u s (t )  Rs  is (t )  Ld
 u p (t )
dt
• Considering only time fundamentals = use complex phasors Us , Up and Is !
• Field oriented operation = current in phase with back EMF: q-axis current Iq.
• Surface magnets: inductivity for d- and q-axis identical (Ld = Lq = Ls)
U s  Rs I s  jLd I s  U p
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/7
Institut für Elektrische
Energiewandlung • FB 18
Phasor diagram per phase of synchronous PM machine
at operation with sinusoidal voltage and current
field-oriented control = current in phase
with back EMF Up (”brushless DC
drive”)
arbitrary current phase shift
between back EMF Up and
phase current
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/8
Institut für Elektrische
Energiewandlung • FB 18
1. Permanent magnet synchronous machines as “brushless
DC drives”
1.1.6 Stator current generation
Source: Siemens AG, Germany
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/9
Institut für Elektrische
Energiewandlung • FB 18
Stator current generation
DC link voltage source inverter with switching
transistors and free-wheeling diodes
Rs neglected:
U d  U p , LL  Ls  dis / dt
a) Equivalent switching
scheme of DC link voltage
source inverter, connected to
the two phases with
switching transistor and freewheeling diode,
b) Current ripple and
chopped inverter voltage
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/10
Institut für Elektrische
Energiewandlung • FB 18
Hysteresis band current control
block commutation
sine wave commutation
Shaping of current
with hysteresis
band
Current commutation from phase U to V etc.:
Determination of current phase shift (= firing
angle) by encoder to get rotor position. Current
shall be in phase with back EMF !
Block commutation: Six step encoder:
A rotor disc and three stator-fixed sensors U, V, W, spaced by
120°/p (p: number of pole pairs), are sufficient for rotor
position sensing for block commutation (here: 2p = 4)
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/11
Institut für Elektrische
Energiewandlung • FB 18
Measuring rotor position for sine wave
commutated synchronous PM machine
Rotor position must be known at every moment,
as frequency might change at every moment,
hence changing sine wave shape !
Position measurement:
a) Resolver: Continuous measurement of
position (analogue electromagnetic device)
Source: Heidenhain, Traunreut,Germany
Optical incremental encoder,
b) Incremental encoder: High resolution
necessary (e.g. 1024 x 4 counts per revolution),
hence optical sensors !
to be mounted on motor nondrive shaft end
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/12
Institut für Elektrische
Energiewandlung • FB 18
Rotor position encoder at NDE motor side
Stator
Encoder
NDE
bearing
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PM rotor
Source: Engel
Elektroantriebe GmbH,
Germany
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/13
Institut für Elektrische
Energiewandlung • FB 18
Inverter operation with q-current
At a given current we get the maximum possible torque at
we do not have a reluctance torque component!
Me 

ms
 U p  I sq  ( X d  X q )  I sd  I sq
 syn
I s  I sq , I sd  0
, if

No reluctance difference between dand q-axis: Xd = Xq:
M e  ms U p  I sq /  syn
 = -
The torque is at Is = Isq at maximum,
because due to Ld = Lq only the qcurrent Isq produces a torque.
Back EMF of PM machine: U p  js p / 2
Me  p  ms p  Isq / 2
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/14
Institut für Elektrische
Energiewandlung • FB 18
Air gap power at q-current operation
Up
Is
Us
P  ms  U p  I sq  M e   syn
 At Isq-current operation current Is and back EMF Up are in phase.
 The air gap power is therefore at maximum.
 Hence also the torque is for a given current at maximum: Maximum Torque per Ampere MTPA
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/15
Institut für Elektrische
Energiewandlung • FB 18
Inverter-operated PM synchronous machine
Ld  Lq :
U s  ( s Lq I sq ) 2  ( Rs I sq   s p / 2 ) 2
For not too small stator angular frequency s we get:
2
U s   s L2q I sq
 ( p / 2 ) 2
U s ~ s
 Control law for the inverter
(similar to induction machine operation):
U s ~ s
 Influence of stator resistance Rs at low
stator frequency:
U s  (s Ld I sq )2  ( Rs I sq  s p / 2 )2
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/16
Institut für Elektrische
Energiewandlung • FB 18
1. Permanent magnet synchronous machines as “brushless
DC drives”
1.1.7 Operating limits of brushless DC drives
Source: Siemens AG, Germany
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/17
Institut für Elektrische
Energiewandlung • FB 18
Operating limits of brushless DC drive
Phasor diagram per phase of synchronous PM machine at
high speed with neglected stator resistance; field-oriented
control with current in phase with back EMF, no saliency
assumed Ld = Lq
Speed-torque curve limit for
synchronous PM machine with
field-oriented control (current in
phase with back EMF)
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/18
Institut für Elektrische
Energiewandlung • FB 18
Operation limits for brushless DC drive
- Steady state torque: temperature limit of insulation material of stator winding.
E.g.: Temperature limit (IEC 34-1): 105 K for insulation class F (ambient
temperature 40°C).
Stand-still torque: M0 at n = 0: only resistive losses
Rated torque an rates speed: MN at nN: Resistive losses, friction and iron losses,
additional losses.
For constant temperature and self-cooled machine: Total losses must be constant
 Resistive losses must decrease at nN, hence current decreases: MN < M0.
- Dynamic torque (Overload up to about 4M0 ): Accelerating and braking: short time
operation (several seconds). Temperature rise according to thermal time constant
Tth of the motor winding stays below temperature limit. So dynamic torque overload
up to about 4M0 is only possible for.
- Maximum torque: inverter current limit.
- Demagnetization limit: Inverter current limit must be below the critical motor current
which would cause irreversible demagnetization of the hot rotor magnets (at 150°C).
- Mechanical maximum speed limit nmax > rated speed nN !
- Inverter voltage limit: Internal motor voltage reaches inverter voltage limit, hence
current input and torque decreases
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/19
Institut für Elektrische
Energiewandlung • FB 18
No-load and load radial air gap field component
- Stator slot opening influence INCLUDED, constant air gap
Critical magnet
edge
Source: 2D Finite element program FEMAG,
ETH Zurich
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/20
Institut für Elektrische
Energiewandlung • FB 18
Surface mounted PM synchronous machines:
Load dependent saturation
1,0
0,8
Leerlauf
Nennpunkt
0,6
0,4
0,2
B
0,0
T
-0,2
-0,4
-0,6
-0,8
-1,0
0,00
0,02
0,04
0,06
0,08
0,10
Umfangskoordinate x / m
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/21
Institut für Elektrische
Energiewandlung • FB 18
Example: Load dependent saturation = Loss in torque
Source: TU Darmstadt, Germany
IN = 94.6 A
MIN = 477.4 Nm
M-I Kennlinie
M-I-characteristic
2IN = 189.2 A
M2IN = 695.1 Nm
Calculation FEMAG
800
700
M2IN / MIN = 1.46
M [Nm]
600
Measurement
500
400
300
Me  m  p 
200
100
2p = 16
q=½
2
 Iq
0
0
25
50
75
100
125
150
175
200
I [A]
m=3
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 p (Is )
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/22
Institut für Elektrische
Energiewandlung • FB 18
225
Torque formula from air gap power at q-current operation
m  U p I sq
P
P


Me 
s / p
2n s / p
Me 
m  (s p ( I s ) / 2 )  I sq
s / p
Me  m  p 
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 p (Is )
2
 m p
 p (Is )
 Iq
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/23
2
 Iq
Institut für Elektrische
Energiewandlung • FB 18
Voltage limit for brushless DC drive
 Inverter-output voltage Umax defines maximum
possible motor speed nmax.
At high speed neglect Rs << Xq:
I s ,lim
1

 U s2,max  ( s p / 2 ) 2
 s Ld
M lim  m  p 
p
2
I lim (n)
 Maximum possible motor speed = No-load speed at voltage limit:
U s ,max   s ,max  p / 2
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nmax
U max
 nN
U pN
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/24
Institut für Elektrische
Energiewandlung • FB 18
Typical M(n) characteristic of a PM servo drive
Different number of turns per
phase:
variants AC7, AF7, AH7
Admissible torque:
- S1 steady state operation,
- S3 short term operation
Continuous duty torque must
stay below S1-line
For acceleration maximum
possible torque is used, but only
Source: Groß/Hamann/Wiegärtner: Elektrische
for short time !
Vorschubantriebe in der Automatisierungstechnik,
Publicis Verlag, Munich, Germany
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/25
Institut für Elektrische
Energiewandlung • FB 18
Different „Number of turns“ for one motor
Motor A
- Identical are:
Copper mass
Iron mass
Motor size
Torque
Current density
I2R-losses
Magnet flux
Motor B
Slot cross
section
100 A
50 A
Conductor
cross
section
50 A
Number of turns per phase:
N
2N
Conductor cross section:
qCu
qCu/2
Current:
Is
Is/2
Back EMF at speed n:
U i ~ N . n. 
Maximum speed:
nmax ~ Umax/(N. )
Inverter power at nmax:
S = 3UmaxIs
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/26
2Ui ~ 2N.n.
nmax/2 ~ Umax/(2N. )
S/2 = 3UmaxIs/2
Institut für Elektrische
Energiewandlung • FB 18
Example: M(n) characteristic of a PM servo drive
Source: Siemens motor catalogue SIMODRIVE,
Erlangen, Germany
The same closed motor, but
a) External fan for forced cooling
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b) Self-cooling = no fan
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/27
Institut für Elektrische
Energiewandlung • FB 18
1. Permanent magnet synchronous machines as “brushless
DC drives”
1.1.8 Use of reluctance torque to increase the torque
Source: Siemens AG, Germany
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/28
Institut für Elektrische
Energiewandlung • FB 18
No-load rotor air gap field with parallel PM magnetization No reluctance variation
- Surface mounted magnets, stator slot opening influence neglected
Pole coverage ratio e = 1
Rotor air gap field amplitudes:
 = 1, 3, 5, 7, …

1
3
5
7
9
B / T
0.8829 = 100%
0.1992 = 23%
0.0969 = 11%
0.0564 = 6%
0.0362 = 4%
x
- Parallel magnetization yields at low pole
count considerable difference to radial
magnetization
- A better approximation of sinusoidal field is
achieved, higher harmonics B are reduced
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Source:
2D Finite element
program FEMAG,
ETH Zurich
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/29
Institut für Elektrische
Energiewandlung • FB 18
No-load rotor air gap field with buried magnets –
Rotor reluctance variation occurs
- Stator slot opening influence neglected, constant air gap
Pole coverage ratio e < 1
Rotor air gap field amplitudes:
 = 1, 3, 5, 7, …

B / T
1
3
5
7
9
11
0.7690 = 100%
0.1287 = 17%
0.0351 = 5%
0.0889 = 12%
0.0817 = 11%
0.0439 = 6%
x
- Depending
a) on the shape of air gap
b) on the saturation of iron bridge
the air gap flux density may vary.
Source:
2D Finite element
program FEMAG,
ETH Zurich
- Numerical calculation is recommended.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/30
Institut für Elektrische
Energiewandlung • FB 18
Torque from power balance at Ld  Lq
Stator flux excited in
direction of the d-axis
Stator flux excited in
direction of the q-axis
 q (t )  Lq iq (t )
 d (t )  p  Ld id (t )
Electromagnetic torque at Ld  Lq: m = 3 phases
M e (t )  p  m  ( d (t )iq (t )   q (t )id (t )) / 2
M e (t )  p  m  ( p iq  ( Lq  Ld )id iq ) / 2
PM torque
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reluctance torque
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/31
Institut für Elektrische
Energiewandlung • FB 18
Example: Rotor geometries A, B, C for a stator with distributed
integer slot winding
Rotor A
Rotor B
Rotor C
Motor A:
Surface magnet motor A (three phases m = 3, four poles 2p = 4, 15000/min, over-speed:
18000/min)
M e  p  m  ( p I q  ( Lq  Ld ) I d I q ) / 2
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/32
Source:
2D Finite element
program FEMAG,
ETH Zurich
Institut für Elektrische
Energiewandlung • FB 18
Numerically calculated torque for motors A, B, C
Motor
Average torque Me
(Nm)
PM torque
(Nm) / (percent %)
Reluctance torque
Me - Mp
(Nm) / (percent %)
A
7.95
7.95 / 100%
0 / 0%
B
7.88
6.14/ 78%
1.74 / 22%
C
7.76
5.42 / 70%
2.34 / 30%
Comparison of
- the average total torque,
- the PM torque and
- the reluctance torque of motors A, B, C for the same stator geometry, winding,
sinusoidal current 44.5 A rms per phase, calculated with FEMAG.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/33
Institut für Elektrische
Energiewandlung • FB 18
Reduction of PM material due to reluctance torque
Motor
Voltage ph.
Us / V rms
Angle

Inductances
Ld / Lq (H)
App. Power S
(kVA / %)
Less magnet
volume
A
95.17
0
127.5 / 136.0
12.7
-
B
99.7
-26°
198.0 / 575.0
13.31 / +5.0%
-34%
C
99.92
-30°
200.0 / 650.0
13.39/+5.5%
-53.1%
- Comparison of inverter and motor data of motors A, B, C for the same stator
geometry, winding
- Calculated with FEMAG.
- Motors B, C have for the same torque 7.5 Nm less winding losses and
temperature rise than motor A.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/34
Institut für Elektrische
Energiewandlung • FB 18
1. Permanent magnet synchronous machines as “brushless
DC drives”
1.2 Brushless DC drive systems
Source:
Lehmann R.: Technik bürstenloser Servoantriebe, Elektrotechnik 21: 96-101, 1989
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/35
Institut für Elektrische
Energiewandlung • FB 18
Block current commutated
drive system
- Inner loop current control as in
DC machines (“brushless DC
drive”)
- Outer loop speed control
- Master position control
- PWM inverter with block phase
currents
- Simple rotor position
measurement (H : Hall sensors),
additional speed sensor (T: Tacho)
and high resolution position sensor
(E: encoder) for position control
Source:
Lehmann R.: Technik bürstenloser Servoantriebe, Elektrotechnik 21: 96-101, 1989
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/36
Institut für Elektrische
Energiewandlung • FB 18
Sine wave
commutated drive
system
- Inner loop space
vector current control in
rotor reference frame
(d-q-system)
Source:
Lehmann R.: Technik bürstenloser Servoantriebe, Elektrotechnik 21: 96-101, 1989
- Outer loop speed
control
- PWM inverter with
sinusoidal phase
currents
- Resolver ( R ) or
encoder for high
resolution rotor position
measurement
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/37
Institut für Elektrische
Energiewandlung • FB 18
Application of PM servo drive in milling machine
Milling
working piece
movable table
screw ball spindle
Transmission wheels and belt n2 < n1
PM Servo motor
Source: Groß/Hamann/Wiegärtner: Elektrische Vorschubantriebe in
der Automatisierungstechnik, Publicis Verlag, Munich, Germany
Transfer of rotating high speed motor to linear low
speed movement via belts and screw spindle
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/38
Institut für Elektrische
Energiewandlung • FB 18
1. Permanent magnet synchronous machines as “brushless
DC drives”
1.2.1 Comparison of block and sine wave commutated
motors
Source:
Lehmann R.: Technik bürstenloser Servoantriebe, Elektrotechnik 21: 96-101, 1989
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/39
Institut für Elektrische
Energiewandlung • FB 18
Steady state torque of block and sine commutated motors
• For same thermal limit, only copper losses, same stator geometry, identical
winding, identical magnet material and magnet height:
• Sine wave and block commutated motors give for the same copper losses the
same output power.
PM machine
Block commutation (B)
Air gap flux density
amplitude
B  B p
Back EMF
Uˆ pB  2 N s  2 f p  B p  l Fe
2
Uˆ pS  2f  N s  k w   p l Fe B ,1
Stator copper losses
2
PCu  2  Rs  IˆsB
2
PCu  (3 / 2)  Rs  IˆsS
Air gap power
PB  2  Uˆ pB  IˆsB
PS  (3 / 2)  Uˆ pS  IˆsS
Equal copper losses:
IˆsS / IˆsB  2 / 3
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Sine wave commutation (S)
B ,1 
  
 sin  e 

 2 
4B p

PS (3 / 2)  Uˆ pS  IˆsS
3 2
  e 

 k w  sin 


ˆ
ˆ

PB
2  U pB  I sB
 2 
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/40
Institut für Elektrische
Energiewandlung • FB 18
Comparison of block and sine wave commutated motors
Example A: Operation at same copper losses :
Winding factor kw = 0.933, pole coverage ratio e = 0.85:
PS
32
 0.85  

 0.933  sin 

PB
 2

  1.00035

Example B:
Operation at inverter current limit Is,max : Block or sine wave commutated motor
delivers the higher short term torque ?
M e, S
M e, B
PS (3 / 2)  Uˆ pS  Iˆs ,max
32
1
  e  3



 k w  sin 


ˆ
ˆ
PB

2  U pB  I s ,max
 2  2 1.15
The block commutated motor is able to deliver at the SAME current
amplitude 15% higher maximum torque, whereas for the same copper losses
the thermal steady state torque for block and sine wave commutated PM
machine is equal.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/41
Institut für Elektrische
Energiewandlung • FB 18
1. Permanent magnet synchronous machines as “brushless
DC drives”
1.2.2 Torque ripple of brushless DC motors
Source:
Lehmann R.: Technik bürstenloser Servoantriebe, Elektrotechnik 21: 96-101, 1989
TECHNISCHE
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/42
Institut für Elektrische
Energiewandlung • FB 18
Torque ripple of brushless DC motors
•
Electromagnetic air gap torque me(t): May exhibit a considerable ripple,
which might be also visible at the shaft as a torque ripple of the shaft torque
ms(t)
•
Cogging torque: No-load torque ripple due to rotor magnets and stator slot
openings. The stator current is zero.
•
Pulsating torque at ideal sine wave current: Torque variation at load due to
interaction between stator and rotor field. Step-like stator m.m.f. distribution due
to distributed stator winding may be regarded as FOURIER sum of space
harmonics (ordinal number ), causing pulsating torque components with the
rotor magnet field space harmonics.
•
Pulsating torque due to current ripple: Inverter switching causes current
ripple = current time harmonics. Each current harmonic causes a stator
fundamental field, which interacts with rotor PM fundamental field  = 1.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/43
Institut für Elektrische
Energiewandlung • FB 18
Cogging torque Mcog and pulsating load torque
Cogging torque frequency:
f Q  n  Qs
Cogging effect at no-load (is = 0):
Unaligned position: rotor tangential
magnetic pull Ft on stator tooth sides
generates torque,
Aligned position: sum Ft = 0, no torque
Typical good values:
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wˆ M 0 ~ 0.5%...1%
Pulsating load torque:
Quantification of torque ripple from
measured torque time function,
e.g. measured with strain gauge
torque-meter:
wˆ M 
Mˆ cog
M av
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/44
( M max  M min ) / 2

( M max  M min ) / 2
Institut für Elektrische
Energiewandlung • FB 18
Load torque of fundamental fields, calculated via internal
power
• Internal power varies with time, leading to torque and speed variation
• Speed variation much smaller than torque variation due to rotor inertia,
hence we assume CONSTANT speed
• Internal power gives electromagnetic torque:


me (t )  u p ,U (t )  iU (t )  u p ,V (t )  iV (t )  u p ,W (t )  iW (t ) /( 2n)
•
Ideal sine wave current feeding: NO rotor field space harmonics =
Back EMF is ideally sinusoidal; NO inverter current ripple:
p (t )  Uˆ p cos(t )  Iˆ cos(t )  Uˆ p cos(t  2 / 3)  Iˆ cos(t  2 / 3)  Uˆ p cos(t  4 / 3)  Iˆ cos(t  4 / 3)
p (t ) 
p (t )  m
Uˆ p Iˆ 
8
4
 Uˆ p Iˆ 

 cos(2t )  1 
 cos(2t  )  1 
 cos(2t  )  1
3
3
2 
2 


No load torque ripple occurs
(3 / 2)  Uˆ p  Iˆ
 const. due to ideally sinusoidal back
 const. M e 
2   n
EMF & current time function !
Uˆ p Iˆ
2
Uˆ p Iˆ
2
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/45
Institut für Elektrische
Energiewandlung • FB 18
Load torque of fundamental fields, calculated via air gap fields
Air gap torque:
M e (t )  z  Fc (t )  d si / 2 
z
 ic ( x j , t )  Br ( x j , t )  l  d si / 2
j 1
Air gap torque via current loading:
M e (t ) ~ l 
Determination of stator current loading from
stator air gap field:
2 p p
 As ( xs , t )  Br ( xs , t )  dxs
0
Bs ( x s , t )  0 H s ( x s , t )  0Vs ( x s , t ) /   0   As ( x s , t )  dx s / 
As ( xs , t )  ( /  0 )  dBs ( xs , t ) / dxs
Air gap torque of stator and rotor fundamental field wave  = 1 and  = 1 for Is = Iq:
2 p p
M e (t ) ~
 As1 cos( xs /  p  t )  Br1 cos( xs / p  t )  dxs
0
Air gap torque is constant: M e (t ) ~ As1Br1  p p  const.  M e
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/46
Institut für Elektrische
Energiewandlung • FB 18
Torque ripple due to space
harmonics (1)
 Current supply is purely sinusoidal
 Field wave changes shape six times per
period T = 1/fs
 Torque pulsates with 6fs
Example:
4-pole motor, q =3,
semi-closed slots,
surface magnets
180°
Source:
2D Finite element
program FEMAG,
ETH Zurich
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/47
Institut für Elektrische
Energiewandlung • FB 18
Torque ripple due to space harmonics (2)
Air gap torque of stator and rotor 5-th field wave  = -5 and  = 5 for Is = Iq:
2 p p
me,5 (t ) ~
 As5 cos(5 xs / p  t )  Br 5 cos(5xs / p  5t )  dxs
2 p p
0
me,5 (t ) ~ ( As 5 Br 5 / 2) 
 cos(10 xs / p  4t )  cos(6t ) dxs
0
me,5 (t ) ~ As 5 Br 5  p p  cos(6t )
Air gap torque of stator and rotor 7-th field wave  = 7 and  = 7 for Is = Iq:
2 p p
 As7 cos(7 xs / p  t )  Br 7 cos(7 xs /  p  7t )  dxs
2 p
0
me,7 (t ) ~ ( As 7 Br 7 / 2)   cos(14 xs /  p  8t )  cos(6t ) dxs
me,7 (t ) ~
p
0
me,7 (t ) ~ As 7 Br 7  p p  cos(6t )
Air gap torque of 5th and 7th space harmonics add to a pulsating torque with 6fs.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/48
Institut für Elektrische
Energiewandlung • FB 18
Alternative torque ripple calculation via air gap power
Rotor 5-th field wave  = 5 induces stator back EMF with 5fs: u p ,5 (t )  Uˆ p 5  cos(5t )
2  ˆ 
2  ˆ
4  ˆ 
4 


p 5 (t )  Uˆ p 5 cos(5t )  Iˆ cos(t )  Uˆ p 5 cos 5t  5 
  I cos t 

  I cos t 
  U p 5 cos 5t  5 
3
3
3
3








4  ˆ 
2  ˆ
2  ˆ 
4 


p 5 (t )  Uˆ p 5 cos(5t )  Iˆ cos(t )  Uˆ p 5 cos 5t 
I
t
U
t
I
t









cos
cos
5
cos







p5
3 
3 
3 
3 




Uˆ p 5  Iˆ 
2 
4 


 cos(6t )  cos( 4t )  cos(6t )  cos 4t 
p 5 (t ) 
  cos(6t )  cos 4t 

2
3
3





p 5 (t ) 
3Uˆ p 5  Iˆ
2
 cos(6t )  me5 (t )  2n
Rotor 7-th field wave  = 7 induces stator back EMF with 7fs: u p ,7 (t )  Uˆ p 7  cos(7t )
4 
2  ˆ 
2  ˆ
4  ˆ 


p 7 (t )  Uˆ p 7 cos(7t )  Iˆ cos(t )  Uˆ p 7 cos 7t  7 
  U p 5 cos 7t  7 
  I cos t 

  I cos t 
3 
3 
3 
3 




Uˆ p 7  Iˆ 

4 
2 


 cos(8t )  cos(6t )  cos 8t 
p 7 (t ) 
  cos(6t )  cos 8t 
  cos(6t )
3 
3 
2




p 7 (t ) 
3Uˆ p 7  Iˆ
2
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 cos(6t )  me 7 (t )  2n
Air gap torque of 5th and 7th space harmonics
add to a pulsating torque with 6fs.
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/49
Institut für Elektrische
Energiewandlung • FB 18
Load torque ripple in block commutated
brushless DC machines
Generation of load
torque ripple due to
block current
commutation with
finite current rise time
tr (corresponding
angle r)
Typical block
commutation torque
ripple values:
wˆ M ~ 3%...5%
Facit:
The generated load torque ripple is with six times fundamental frequency.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/50
Institut für Elektrische
Energiewandlung • FB 18
Two typical reasons for load torque ripple with
block commutated brushless DC motors
a) deviation of block current from
ideal rectangular shape (finite rise
time tr),
b) deviation of trapezoidal back
EMF from ideal shape (slope
increased by td)
a)
Facit:
The sine wave commutated motor
has a lower load dependent torque
ripple (~ 1%) than the block
commutated brushless DC drive
(ca. 4 ... 5%).
b)
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/51
Institut für Elektrische
Energiewandlung • FB 18
Pulsating torque due to current ripple
• Inverter switching causes current ripple = current time harmonics.
Each current harmonic causes a stator fundamental field, which
interacts with rotor fundamental PM field  = 1.
Influence of inverter switching and current
controller time constant
Source:
Henneberger G, Schleuter W (1989) Elektrotechn. Z. 110: 274-279
Measured block wave commutated stator phase current at low, medium and
rated speed.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/52
Institut für Elektrische
Energiewandlung • FB 18
Block commutated PM servo: Torque ripple at voltage limit
- At very high speed and frequency fs, when voltage limit is reached, only sixstep operation is possible, if inverter switching frequency is too low with respect
to maximum fs.
- Phase current a) loses block shape, and torque b) shows MAXIMUM ripple
with 6fs !
(34  25) / 2
ˆ
wM 
 15.3%
(34  25) / 2
Source:
Huth G, etz-Archiv 11:
401-408, 1989
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/53
2p = 6
Institut für Elektrische
Energiewandlung • FB 18
Block commutated PM servo: Torque ripple at very low speed
- At very low speed and frequency fs, due to fast switching with respect to fs,
phase current looks perfectly block-like.
- Torque ripple is MINIMUM, being determined by (1) cogging with slot
frequency, (2) current commutation with 6fs !
wˆ M  3.6%
Source: Siemens AG, Germany
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/54
Institut für Elektrische
Energiewandlung • FB 18
Comparison of torque ripple of block and sine wave
commutated PM motors at low speed
- Measured FFT-analysis of torque signal me(t) at 20/min
- Block commutated motor has higher torque ripple due to 6fs component
Block commutation
Slot harmonic cogging
Sine wave commutation
Single layer fractional slot winding
instead of more expensive two-layer
winding q = 3/2
q=2
wˆ M  1.0%
wˆ M  3.6%
Source: Siemens AG, Germany
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/55
Institut für Elektrische
Energiewandlung • FB 18
1. Permanent magnet synchronous machines as “brushless
DC drives”
1.2.3 Significance of torque ripple for motor and drive
performance
Source:
Lehmann R.: Technik bürstenloser Servoantriebe, Elektrotechnik 21: 96-101, 1989
TECHNISCHE
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/56
Institut für Elektrische
Energiewandlung • FB 18
Torsion resonance

- Rotor of motor coupled to
rotating load via an elastic
coupling
  M   L in mech. degrees!
- Coupling stiffness c
- Inertia of motor and load JM, JL
- Exciting k-th harmonic air gap
torque me,k
J LL  ms  0
L  ms / J L
,
J M M  ms  me,k
, M  (me,k  ms ) / J M
Differential equation:
ms  c   M   L 
me,k
 1
1 
  ms 
 M  L  

JM
 JM JL 
Homogeneous solution leads to
me,k
c  (J M  J L )
 
 
JM  JL
JM
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,
torsion resonance frequency:
1
JM  JL
 c
f 0  0 /( 2 ) 
2
JM  JL
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/57
Institut für Elektrische
Energiewandlung • FB 18
Excitation of torsion vibrations
Pulsating torque excites
torsion vibrations:
me,k (t )  Mˆ  sin(t )
  k  (2  n  p)
Solution of differential equation
with exciting torque ripple yields
vibration angle  and oscillating
shaft torque ms :
Mˆ
c
ms (t )  c   (t ) 
 2
 sin(t )
2
J M o  
Mˆ
1
 (t ) 
 2
 sin(t )
2
J M o  
1. It must be avoided that the dominant pulsating torque frequency excites the torsion
resonance of the drive system. This can be achieved by designing the drive with a stiff coupling
(c: high value) to stay with the pulsating torque frequency below the resonance.
2. The high frequency pulsating air gap torque ( > 0) is filtered and not visible in the shaft
torque ripple.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/58
Institut für Elektrische
Energiewandlung • FB 18
Speed ripple due to torque pulsation
Speed ripple definition:
n(t )  n  n(t )
From solution of torsion oscillation we know oscillation angle:
ˆ 

Angular acceleration is: M (t )  me (t )  ms (t )  M  1  c / J M   sin(t )
JM
J M  o2   2 
Speed ripple:

c / J M 
Mˆ

n(t )  
 cos(t )
 1 2
n(t )  M (t ) /( 2 )
2


2    J M  o   
Staying below the resonance   0 , we observe that especially at low speed the
speed ripple amplitude, expressed as percentage of actual speed, increases with
DECREASING speed:
n
Mˆ

n
(2 ) 2  k  p  n 2  J M

1
c / JM 
Mˆ

 1  2

~
2 
2
2
2









(
2
)
k
p
n
(
J
J
)
n
o
L
M


At low speed (servo operation !) the speed ripple is big and of importance!
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/59
Institut für Elektrische
Energiewandlung • FB 18
Block commutated brushless DC drive systems
- permanent magnet motor with three phase, single layer winding,
skewed slots and 100% pole coverage ratio for the rotor magnets
- simple rotor position sensor with rotor disc according to pole number,
- brushless DC tachometer for speed measurement,
- encoder with high resolution for precise positioning,
- voltage source inverter system (power transistors), speed and current
control, implemented in a micro-computer system; motor current
measurement devices such as shunts or DC transformers with Hallsensors,
- motion control system, implemented in a second microcomputer. It
allows position control, but also different motion control such as special
velocity profiles according to the needs of the driven load.
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Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/60
Institut für Elektrische
Energiewandlung • FB 18
Sine commutated brushless DC drive systems
- permanent magnet motor with three phase, double layer winding
(or special single layer winding with non-integer q), skewed slots
and 80%...85% pole coverage ratio for the rotor magnets,
- high resolution rotor position sensor (optical encoder or magnetic
resolver), which acts also as speed sensor and sometimes as
position sensor for precise positioning,
- voltage source inverter system (power transistors), speed and
current control, implemented in a micro-computer system and
motor current measurement devices such as shunts or DC
transformers with Hall-sensors,
- motion control system, which is implemented in a second
microcomputer and allows position control, but also different motion
control such as special velocity profiles according to the needs of
the driven load.
TECHNISCHE
UNIVERSITÄT
DARMSTADT
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/61
Institut für Elektrische
Energiewandlung • FB 18
Block and sine commutated brushless DC drives
Block commutation
Sine commutation
Advantages
- cheap rotor position and speed sensor,
- cheap motor winding
- 15% higher overload at inverter current limit
- 10% ... 15% reduced amount of magnets,
- very low torque ripple below 1%,
- reduced additional losses at high speed,
- reduced torque ripple sensitivity due to
misalignment of rotor position sensor.
Disadvantages
- extra encoder for accurate drive positioning,
- higher minimum torque ripple (2% ... 4% at
low speed),
- increased additional losses in the motor due to
extra eddy currents especially in the rotor due
to the rapid chance of stator flux at commutation
- Strongly increased load torque ripple at misaligned rotor position sensor.
TECHNISCHE
UNIVERSITÄT
DARMSTADT
- expensive encoder for current commutation and speed measurement,
- expensive stator winding, if two layer
winding is used,
- 15% lower overload capability at
inverter current limit,
- more complex mathematical model
for motor control.
Prof. A. Binder : Motor Development for Electrical Drive Systems
1.2/62
Institut für Elektrische
Energiewandlung • FB 18