L6 – Energy Conversion
Industrial Electrical Engineering and Automation
Previous lecture
L6: Energy conversion
a potential for causing a change
Industrial Electrical Engineering and Automation
Lund University, Sweden
RΩ1
Rδ
Rμ
Lσ1
w
Lσ2
RΩ2
Ex(y), i
y
d
x
N2i 2
N1i1
u1
e1
e2
i1
z
u2

Ex(y), i
Bz
d 2  dBz 
pec t  


12    dt 
i2
2
• electromagnetism – electro- and magnetomotive force
• Induction action – transformed or motional voltage
• Electromagnetic energy conversion
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Design of Electrical Machines
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Today’s goal
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Industrial Electrical Engineering and Automation
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-0.5
• a potential for causing a change
• Formulation of energy conversion and field coupling
• Meaning of shear stress in design of electrical
machines
• Electromechanical and electromagnetic energy
conversion
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inductance
energy
torque
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20
40
60
80
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L6 – Energy Conversion
Mathematic formulation
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Energy conversion
• Energy is the capacity of a system to do work
• Energy cannot be created or destroyed, but only
converted from one form into another
• Coupling between the different fields obeys to the
principle of energy conversion
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 E  
D  
Gauss’s Law,
Electricity
B  0
Gauss’s Law,
Electricity
F   tm  ds
Magnetic stress
per unit of area
F    Wm
Change of system
energy
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Ampere’s circuital
law
• Electromagnetism by
Maxwell’s equations
• Electro-mechanism by
electromagnetic stress
tensor or virtual work
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Energy conversion principle
• Electromechanic energy converter via intermediate
magnetic field
• Equivalent circuit representation of the different fields
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Faraday’s law
 H  J
S
Electromechanical energy converter
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B
t
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• The energy conservation principle says that the sum of
electrical and mechanical energy input to a device at
each time instant has to be equal to the sum of
accumulated electromagnetic and mechanical energy
and losses.
N ph
Fdx   ik  uk dt  dWmag  dWmech  dWloss
k 1
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Design of Electrical Machines
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2
L6 – Energy Conversion
Electrical energy
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Mechanical energy
• In a freely and linearly moving part with a mechanic
support we are considering only kinetic energy and
neglecting potential energy
d 2x
dWmech  M  2 dx
dt
• In case the moving body is connected to the elastic
spring the stiffness of the spring has to be taken into
consideration
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• Considering Kirchhoff’s voltage equations the
differential of the electric energy consists of resistive
voltage term and electromotive voltage term
ik uk dt  ik ik Rk dt  e dt  dWconductor  ik d k
• According to Faradays law the opposing
electromotive voltage can be either a transformed
voltage or/and a motional voltage
e
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• Losses in an electromechanical energy converter can
be separated according the loss origin: electrical,
magnetic and mechanical
m
dWloss   i 2j R j dt  dWcore  dW friction  dWconductor  dWcore  dW friction
j 1
• Electromagnetic losses shear often the same loss
mechanism.
• Common for the losses is a phenomenon of friction
that opposes (current) flow, magnetisation, motion,
etc and causes (irreversible) heat energy loss.
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Energy balance
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Losses
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d
d
di
dx dL
  L i  L i
dt
dt
dt dx
dt
• By separating resistive voltage term and induced
voltage term, and dividing loss term between the
electric, magnetic and mechanic origin the energy
balance can be rewritten
m
F  Floss dx   ik d k  M  d
k 1
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2
x
dx  dWmag  dWcore
dt 2
Design of Electrical Machines
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3
L6 – Energy Conversion
Magnetic energy and losses
• The magnet energy is distributed in all passive elements
• The differential of flux is expressed through the
permeance and the mmf drop in the magnetic element
• The permeances are: linear, parametric nonlinear and
inherently nonlinear ones
• These permeances independent and dependent of
motion and displacement x
Ne
Ne


dWmag   Fk dk   Fk dGk  Fk Gk dFk  dWstat 
k 1
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k 1
2
F
2
param
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dG
dx
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Magnetic energy
13
• The differential of magnetic energy in non-parametric
elements are expressed in terms of currents and flux
linkages
k 1
Wcore  Wcore x  
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• Only the parametric nonlinear permeances gives the
force and the force does not depend in the state of
core or leakage paths, but on conditions in the air-gap
only.
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dG
d 2x
 F2
dx
dt 2
Design of Electrical Machines
k 1
dWcore
0
dx
Design of Electrical Machines
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Force components
• Electromechanical energy conversion is possible if
the motion is present dx≠0
• The applied force is loaded with the reaction force of
the system.
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N ph
• Core losses are dissipated in inherently nonlinear
permeances only, hence they are independent on
displacement
Force equation for translation
F  Floss  m 
Ne
dWstat   Fk dk   ik d k
• Interaction between ferrous material and magnetic
fields (permanent magnet(s) or/and coil(s)) resulting in
variation of the magnetic energy due to a position
dependent reluctance
– Reluctance force/torque
– Detent or cogging force/torque
• Interaction between electromagnet(s) or/and
permanent magnet(s) cause magnetic force/torque
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L6 – Energy Conversion
Electromechanical energy conversion
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Electromagnetic vs Reluctance
M
M
F
M
M
F
F
M F
M
F
M
M
• The basic goal in the analysis of every electromechanical energy converter is to compute the amount
of energy transformed from one form to another.
• The common methods to calculate force (or torque)
– Lorentz force
– Virtual work
– Maxwell’s stress tensor
M
M
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F
• The energy conversion between the electric and
mechanic energy takes place in presence of the
magnetic field
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Virtual work
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Lorentz force
• Force that is applied
to a current
conducting wire in a
magnetic field
F
i
B
l
F  qE  v  B    J  BdV   JBdV  
iB
Adl Bli
A
0
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• The force computation,
according to the virtual
work method, considers
the rate of the change of
the total co-energy
against the virtual
displacement.
F
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WC  x 
x
d to
relate distance
a
e
r
a
Work ental force
m
e
r
inc Design of Electrical Machines
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5
L6 – Energy Conversion
Force calculation I
Force calculation II
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0.02
0.03
0.04
0.2
0
0.01
0.02
x [m]
x 10
0.6
1.2
2
200
0.6
250
0.6
0.4
150
150
200
current
250
0.2
0
0.01
0.02
0.03
change of energy
1
1.2
100
100
0.02
x, [m]
0.8
Ec, [J]
0.
06
6
flux
4
0.01
0.03
200
150
100
0.04
x, [m]
-30
-2
5
0
-2
6
220.4
66
00.2.0
1.4
250
202.
6
02.8
1
10
1.4
0.4
6
0.04
energy
-3
12
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0.03
x, [m]
-15
0.01
-10
0
100
-5
100
Ec /x, [J/m]
100
150
-5
0.01
Depth along z-axis: 0.02 m
0.02
0.03
Industrial Electrical Engineering and Automation
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-10 -15
-5
Ec /x, [J/m]
150
5
Ec, [J]
-10
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d/dx coenergy
200
-15
-5
-15
200
-30
250
0.8
5
-3
150
change of coenergy
1
-25
-20
-15 10
200
Fx [N]
coenergy
1.6 1.4 .2
1
250
-10
force
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0
5
-3 0
-3
5
-2
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-15
-10
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Shear stress
250
d1
G1
21
ro2
s
t
p
2d1/π
G1 G2 G3
d1
g
G6
G7
2d2/π
d2
ro2
x
W
1
 1 G 
   2  2

x
2
 G x 
1
W
 1 G 
T
   2  2

2

 G  
F
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Maxwell’s Stress Tensor
B
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Force integration lines
forces for 1 m long mover stator
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d2
x
Force calculation III
Flux density, B
2d2/π
G9
x, [m]
Design of Electrical Machines
g
G3
G6
d/dx energy
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G2
• Magnetic force on a
surface according to
magnetic pressure

1
tn 
 Bn2  Bt2
2  0
B B
tt  n t

Bn
tn
0
t
α
α
Bt t t
mover
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L6 – Energy Conversion
B
• The value is generally
<70000 N/m2 for normally
cooled machines
I
h
torque of RF machine, T
emRF
emAF
20 0
30 0
40 0
60
50 0 0
10
0
5
0
5
10
15
20
25
30
length, l [cm]
35
40
emAF
[kW]
1
7
6
5
4
3
2
4
40
30
3
5
4
7
3
4
6
5
7
2
2
25
6
1
1
radius, r [cm]
35
20
2
15
3
4
3
2
1
1
10
5
0
0
5
10
15
20
25
30
length, l [cm]
weight of RF machine, M
35
emRF
40
45
50 0
5
10
[kg]
20
25
30
length, l [cm]
weight of AF machine, M
35
emAF
40
45
50
[kg]
50 0
400
100
200
50 0
400
100
200
30 0
50
45
15
40
30
20 0
10
0
30 0
0
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5
10
15
20
25
30
length, l [cm]
35
40
45
50 0
5
10
15
20
25
30
length, l [cm]
50
0
20
0
10
0
10 0
5
0
40
10 0
15
0
30
50
0
0
20
0
40
20
0
30
100
25
0
20
radius, r [cm]
35
35
40
30
0
45
50 0
5
10
15
20
25
30
length, l [cm]
20
0
– σ=5000 N/m2
70
50 60 0 0
4
0
30 0 0
0
35





40
45
– Coreless machines
50
Design of Electrical Machines
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Design Target
• Based on the
previous torque
capability map the
power capability at
50 rpm is expressed
• By estimating the
thickness for the
machine h=10 cm
and average density
of 7kg/dm3 the
weight is expressed
z
w
B
K
• Energy conversion
Pel  
F
T
T
η - efficiency
1
u t i t dt  T  Pmech

T 0
• Torque per rotor volume
2F
T
Fr


 2
VRT r 2l Agap
• Air-gap shear stress

r
l
F
BIz BKwz


 BK
Agap Agap
Agap
• Product of magnetic and
electric loadings
σ=kBK
50
Design of Electrical Machines
45
Avo R
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power of AF machine, P
7
6
5
3
2
1
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[kW]
1
80 90 0 0 00
0
0
10 0
10
60 70 0
0
00
15
10 0
20
9 10
70 800 0 0 00
0
20
25
60
40 500 0
0
5
0
40
0
30
emRF
00
10 0
90 0
80
30
3
• Selected value of
magnetic shear stress
35
10 0
power of RF machine, P
45
[Nm]
40
Low speed & high power
50

torque of AF machine, T
300
200
100
45
[Nm]
60 00
50
400
50
0
20
25
o
 r 
2
Tem AF  2  r 2 dr  ro3 1   i 
  ro 
3
ri

2
3
3
  r  0.5l   r  0.5l 
3
r
l
0
30
Design of Electrical Machines
Tem EF  2r 2 l
r
r
0
Avo R
• Radial flux machine vs
Axial flux machine
l
00
10 0
90 0
80 0
70
• Gives the approximate
size of the machine
σshear
h
radius, r [cm]
• σs=F/A, the average thrust
F per unit area A of gap
surface
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Torque capability
50 0
400
300
200
100
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Magnetic shear stress
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L6 – Energy Conversion
Lorenz force on permanent magnets
• Specific force:
fs=BK<600kN/m2
• Magnetic induction: B<1.8T
• Sheet current density:
K<350kA/m
• DC machine:
fs<20-40kN/m2
• Synchronous machine:
fs<30-300kN/m2
• Induction machine:
fs<20-40kN/m2
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Design of Electrical Machines
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Lorentz force
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• Specific force:
fs=BHChm/τp<100kN/m2
• Magnetic induction: B<1.2T
• Coercitive force:
HC<800kA/m
• Pole length to magnet
thickness: τp/ hm=10
• DC machine:
fs<20-40kN/m2
• PMSM, BLDC:
fs<30-80kN/m2
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Design of Electrical Machines
Summary
• Specific force: fs=1/2i2dL/dx≈
B2/μ0δ/ τp <150kN/m2
• Magnetic induction: B<1.5T
• Sheet current density:
K<0.1kA/m
• Pole length to gap length:
τp/ δ =10
• SRM, stepping machines:
fs<30kN/m2
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Reluctance force
30
• Maxwell stress.. estimate B
• Interaction
– Estimation of linking flux 
– Determination of current I
– Deriving torque from I…
• Attraction
– Estimation of gap
permeances G
– Estimation of linking flux 
– Deriving torque from energy
Avo R
• Energy conversion,
coupled circuits, work
and force: electromagnet
example
• Force computation,
Maxwell shear stress,
types of machines
• Learning through
simulations – the second
home assignment
Design of Electrical Machines
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