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TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/1
Energy Converters –
Computer-Aided Design (CAD)
and
System Dynamics
A. Binder
Institut für Elektrische Energiewandlung
Technische Universität Darmstadt
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/2
Lecturer
M. Sc. Sascha NeusüsInstitut für Elektrische Energiewandlung
TU Darmstadt 64283, Landgraf-Georg-Strasse 4, Darmstadt
tel.: +49-6151-16-24192fax.:+49-6151-16-24183
e-mail: sneusues@ew.tu-darmstadt.de
Tutorial
Prof. Dr.-Ing. habil. Dr. h.c. Andreas BinderInstitut für Elektrische Energiewandlung
TU Darmstadt 64283, Landgraf-Georg-Strasse 4, Darmstadt
tel.: +49-6151-16-24181 o. 24182fax.:+49-6151-16-24183
e-mail: abinder@ew.tu-darmstadt.de
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/3
Learning outcomes
Understanding of design rules for electrical machines
- scaling laws (typical for power engineering)
i.e. AC and DC motors and generators
Knowledge of design of AC machinery, here: cage induction machine
- Magnetic circuit and winding topology
- Calculation of resistances and inductances
- Losses and efficiency
Knowledge of basics in cooling systems and temperature calculation
- applications in electrical motors and generators
Understanding of dynamics of DC and AC machinery
- system dynamics of variable speed drives & space vector theory
- transients in generator systems & power stability
Calculation examples for self-training
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/4
Contents
1. Basic design rules for electrical machines
2. Design of Induction Machines
3. Heat transfer and cooling of electrical machines
4. Dynamics of electrical machines
5. Dynamics of DC machines
6. Space vector theory
7. Dynamics of induction machines
8. Dynamics of synchronous machines
Source:
SPEED program
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/5
Organization
- Moodle: Down-load of slides, text book & collected exercises
- CityCopies, Holzstrasse 5: Paper copies of slides, text book & collected exercises
- Demo videos:Via Moodle platform link
- Introduction of two computer programs:
- SPEED program: Induction machine design
- Matlab/SIMULINK for dynamic calculations
- Excursion offered
Source:
SPEED program
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/6
Examination
- Examination: In written form
Details see - Moodle or
- “Blue-board” (Institute)
a) Three calculation examples
b) Theoretical questions
- Homework:Dynamics examples – add-on points for examination
Source:
SPEED program
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/7
Type of examination
Written examination
Winter term: 2 partial exams
Summer term: 1 exam
Calculation examples & List of theory questions: see “Collection of Exercises”
Examination
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/8
Energy Converters – CAD and System Dynamics
1. Basic design rules for electrical machines
2. Design of Induction Machines
3. Heat transfer and cooling of electrical machines
4. Dynamics of electrical machines
5. Dynamics of DC machines
6. Space vector theory
7. Dynamics of induction machines
8. Dynamics of synchronous machines
Source:
SPEED program
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/9
1. Basic design rules for electrical machines
Source:
Winergy, Germany
Energy Converters - CAD and System Dynamics
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/10
Energy Converters – CAD and System Dynamics
1. Basic design rules for rotating machines
1.1 Torque generation and internal power
1.2 Electromagnetic utilization
1.3 Thermal utilization
1.4 Overload capability of AC machines
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/11
Air-gap between
stator and rotor
Stator
tooth
Rotor
tooth
1. Basic design rules for electrical machines
Cross section of a cage induction machine
Stator
yoke
Rotor
yoke
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/12
1. Basic design rules for electrical machines
Air gap field, excited by slot conductors
sQsQ
ΘQ
A = ΘQ/ sQ A = 0
Air gap δ
Assumption: µFe → ∞ : HFe = 0
Ampere´s law: ∫∫∫∫ ⋅=⋅+⋅=⋅=δδ
δδΘCCC
Fe
C
Q sdHsdHsdHsdH
Fe
rrrrrrrr
Case 1: Tangential air gap field along path 1:
Case 2: Radial air gap field along path 2:
QQQ
C
Q sHsHsdH /11
1
ΘΘ δδδ
δ
=⇒⋅=⋅= ∫rr
)2/(2 22
2
δΘδΘ δδδ
δ
Q
C
Q HHsdH =⇒⋅=⋅= ∫rr
Current loading A introduced to replace
the slot geometry
The same air gap field is obtained at µFe → ∞a) for the real slot geometry and
b) for the current loading model.
Stator
RotorSource:
H. Kleinrath, Studientext 1975,
Wiesbaden
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/13
1. Basic design rules for electrical machines
Resulting air gap field at the slot
ΘQ ΘQ=0Stator
Rotor
- Only stator self field
of slot conductor
- No external rotor field
δ
Θµδ
2
0 QsB =
0=rBδ
δsBδ− sBδ+
- No stator self field
of slot conductor
- Only external rotor field
0=sBδ
0>rBδ
rBδ+ rBδ+
Source:
H. Kleinrath, Studientext 1975,
Wiesbaden
Superposition of:- Stator self field
of slot conductor
and external rotor field
δ
Θµδ
2
0 QsB =
0>rBδ
sr BB δδ +sr BB δδ −
Note: For simple superposition we have to
assume µFe(x,y) = const.
x
y
ΘQ
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/14
1. Basic design rules for electrical machines
Magnetic force Fe on a slot conductor
ΘQ=0
δ
External rotor field is at the slot
conductor location very small:
rrc BB δ<<,
rBδ+ rBδ+
Source:
H. Kleinrath, Studientext 1975,
Wiesbaden
sr BB δδ +sr BB δδ −
LORENTZ-force Fc on the
slot conductor is very small:
rcB ,
rcQc BlF ,⋅⋅= Θ
Magnetic pull force FM due
to MAXWELL stress fn on
magnetized iron dominates!
ΘQ
)( yBleftx
y
)( yBright
)2/( 02 µnnFe
A
nM BfdAfF
Fe
≈⋅= ∫ leftMrightMM FFF ,, −=
∫ ⋅−⋅=y
leftrightM dyBBl
F )(2
22
0µ
Fc
FMResulting magnetic force Fe:
Mce FFF +=
ca. 10 % ca. 90 %
rQe BlF ,δΘ ⋅⋅=
Mathematical proof:
Williams-Mamak: IEE Trans. C, 1961,
Monograph no. 456U
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/15
1. Basic design rules for electrical machines
Simplified model for magnetic force Fe on a conductor
δ
Source:
H. Kleinrath, Studientext 1975,
Wiesbaden
sr BB δδ +sr BB δδ −
LORENTZ-force Fe on a
SURFACE conductor yields
the same force as the exact
model of a slot conductor:
ΘQ
)( yBleft )( yBright
Fc
FM
Mce FFF += rQe BlF ,δΘ ⋅⋅=
rBδ
ΘQ
rQe BlF ,δΘ ⋅⋅=
The same force is obtained
a) for the real slot geometry and
b) for the surface conductor model, leading again to a current loading model A(x).
Slotting replaced by “discrete” current loading
e.g.: m = 3, q = 2, W/τp = 1
Slotting
Current loading
sQ
sQ → 0
⊗Fe
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/16
1. Basic design rules for electrical machines
m.m.f. V(x) and current loading A(x) in AC machines
)/sin(ˆp1 txV ωτπ −⋅
Distributed three-phase winding excites a distributed stator field:
δγµγδ /)()( 0 VB ⋅=
Example:
q = 2, m = 3,
full-pitched coils at time t = 0
Fundamental of m.m.f. distribution V(x) is a
sine wave:
Current loading is derivative of m.m.f.:
))/sin(ˆ(),( p11 txVdx
dtxA ωτπ −⋅=
)/cos(ˆ
),( p
p
11 tx
VtxA ωτπ
τπ
−⋅=
pw1p11 /2
/ˆˆ τππ
τπ ⋅
⋅⋅⋅⋅=⋅= IkN
p
mVA
Fundamental current loading A1(x) is a continuous function,
not any longer “discrete” like single conductors!
V1(x)
0 πp/τπγ x=
δδ ⋅= )()( xHxV
∫ ⋅= dxxAxV )()(
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/17
1. Basic design rules for electrical machines
Fundamental m.m.f. V1(x), air-gap field Bδ1(x) & current loading A1(x)
)/sin(ˆ),( p11 txVtxV ωτπ −⋅=
)/sin(ˆ/),(),( 10 txBtxVtxB p ωτπδµ δδ −=⋅=
Stator winding:
Fundamental m.m.f. distribution V(x) as sine wave:
Current loading is derivative of m.m.f.:
)/cos(ˆ/),(),( p111 txAdttxdVtxA ωτπ −⋅==
V1(x)
-τp 0 τp x
Bδ1(x)
-τp 0 τp x
t = 0:
Stator fundamental air-gap field distribution Bδ(x) as sine
wave at µFe →∞:
A1(x)
-τp -τp/2 0 τp/2 τp xFundamental current loading A1(x) is a continuous function,
not any longer “discrete” like single conductors!
1V̂
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/18
1. Basic design rules for electrical machines
• LORENTZ force: Stator field and rotor current
OR
rotor field and stator current: e.g.:
Number of conductors per element dx:
With current loading As:
ers ltxBtxidztxdF ⋅⋅⋅= ),(),(),( ,δ
pp
dxzdz
τ2⋅=
z: total number of conductors
Total tangential force:(acting on stator)
p
sss
p
txiztxA
τ2
),(),(
⋅=
dxtxBtxAltF
pp
rse ⋅⋅⋅= ∫τ
δ
2
0
, ),(),()(
dγ ~ dx
r
s: stator
r: rotor
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/19
1. Basic design rules for electrical machines
Self-field does not produce resulting tangential force
0),(),()(
2
0
, =⋅⋅⋅= ∫ dxtxBtxAltF
pp
sse
τ
δ
( ) dxtxBtxBtxAldxtxBtxAltF
pp p
rsse
p
rse ⋅+⋅⋅=⋅⋅⋅= ∫∫τ
δδ
τ
δ
2
0
,,
2
0
, ),(),(),(),(),()(
dxtxBtxAltF
pp
se ⋅⋅⋅= ∫τ
δ
2
0
),(),()(Tangential force on stator:
Tangential force on rotor: “Actio est reactio” (Newton´s 3rd law):
dxtxBtxAltF
pp
se ⋅⋅⋅−= ∫τ
δ
2
0
),(),()(
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/20
1. Basic design rules for electrical machines
Fundamental sine wave magnetic air gap travelling field in AC machines
1ˆ2δτ
πΦ Bleph ⋅⋅=
Magnetic flux per pole:
1
Resulting air gap field
fundamental:
)/cos(ˆ),( 1, txBtxB p ωτπδδ −⋅=
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/21
1. Basic design rules for electrical machines
Torque generation in AC machines by fundamental fields
dxtxBtxAltF
pp
se ⋅⋅⋅= ∫τ
δ
2
0
),(),()( 2/)()( sie dtFtM ⋅=
dxtxBtxAlF
pp
se ⋅⋅⋅= ∫τ
δ
2
0
1,1,1 ),(),( 2/1, siACe dFM ⋅=
Generally the tangential force depends on time due to slotting and phase
bands +U, -W, +V, -U, +W, -V. Therefore we have a torque ripple ∆Me(t):
Me(t) = Me,av + ∆Me(t)
Considering only fundamental fields we get a CONSTANT torque:
Me(t) = Me,av ∆Me(t) = 0
2
sisi
dr =
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/22
1. Basic design rules for electrical machines
Torque generation in AC machines by fundamental fields
πϕτ δδ /cosˆˆ)( 1,1
2
, ⋅⋅⋅⋅= BAplM speACe
γπ
τωγϕωγ
π
δ dpltBtAM
p
s ⋅
⋅⋅⋅−⋅−−= ∫
2
0
2
p
eδ,11ACe, )cos(ˆ)cos(ˆ
px τπγ /⋅=
dxtxBtxAldM
pp
sesiACe ⋅⋅⋅⋅= ∫τ
δ
2
0
1,1,, ),(),()2/(Torque on stator:
psi pd τπ 2=
απ
ταϕα
πω
ωδ dplBAM
pt
t
s ⋅
⋅⋅⋅⋅−= ∫
+−
−
2 2
p
eδ,11ACe, )cos(ˆ)cos(ˆ
αωγ =− t
[ ] ααϕαϕαπ
τ πω
ωδδ dplBAM
pt
t
s ⋅+⋅
⋅⋅⋅= ∫
+−
−
22
p
eδ,11ACe, cossinsincoscosˆˆ2/))2cos(1(cos2 αα +=
2/)2sin(sincos ααα =
Torque on rotor: “Actio est reactio” (Newton´s 3rd law):
πϕτ δδ /cosˆˆ)( 1,1
2
, ⋅⋅⋅⋅−= BAplM speACe
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/23
1. Basic design rules for electrical machines
Torque generation in AC machines by fundamental fields
πτ δ /315cosˆˆ)( 1,1
2
, °⋅⋅⋅⋅= BAplM speACe
Example:
Phase shift between stator current loading and air-gap field: ϕδ = 315°
Bδ
ϕδπ
τ δ
1
2
1ˆˆ)( 1,1
2
, ⋅⋅⋅⋅⋅= BAplM speACe
Positive torque on the stator = in counter-clockwise direction
)cos(ˆ))2(cos(ˆ)cos(ˆ111 δδδ ϕωγπϕωγϕωγ −−=−−−=−− tAtAtA sss
πϕϕ δδ 2−=
Note: Due to
we also get: °−= 45δϕ πτ δ /)45cos(ˆˆ)( 1,1
2
, °−⋅⋅⋅⋅= BAplM speACe
πτ δ
1
2
1ˆˆ)( 1,1
2
, ⋅⋅⋅⋅⋅= BAplM speACe
δϕ
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/24
Internal phase angle between internal
voltage and stator current:
1. Basic design rules for electrical machines
Example: Torque generation in an induction machine
Bδ,1
As(γ)
ϕδ
Main flux Φh ~ Bδ,1 ~ Im and internal voltage Uh ~ j.Φh:
Bδ,1 Φh Im
Uh
Bδ,1,s ~ Is
Is
As
ϕδ
ϕi
πϕϕ δ −=i
Torque on rotor: πϕτπϕτ δδδ /cosˆˆ)(/cosˆˆ)( 1,1
2
1,1
2
, ispespeACe BAplBAplM ⋅⋅⋅⋅=⋅⋅⋅⋅−=
)coscos( iϕϕδ =−
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/25
1. Basic design rules for electrical machines
πϕτππ δδ /cosˆˆ)(22 11
2
, ispes
ACesyn BAplp
fMnP ⋅⋅⋅⋅⋅==
Phase shift ϕi between Is and Uh!
Internal phase shift ϕi in an induction machine
Internal power:
2/2 1 hswssh NkfU Φπ ⋅=1
ˆ2δτ
πΦ Bleph ⋅⋅=
ishs IUmP ϕδ cos⋅=
Example: 0 < ϕi < π/2: Me,AC > 0 on rotor = motor operation
Phase shift ϕs between Is and Us!
Uh
sss
s IkNp
mA ⋅⋅⋅⋅= ws1
p1
2ˆτ
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/26
As(γ)ϕδ
1. Basic design rules for electrical machines
Internal phase shift ϕi of a synchronous machineBδ1
Main flux Φh ~ Bδ1
Bδ1 ~ Im , Uh ~ j.Φh
Bδ1 ~ Im
⊗Bδ,1,s ~ Is
As(γ)ϕδ
πϕϕ δ −=i
Bδ1 ~ Im
⊗
Is
•
Uh
Example: Cylindrical rotor, generator operation, over-excited
•
Source:
Siemens AG
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/27
1. Basic design rules for electrical machines
Phasor diagram of a synchronous machine
Rotor pole axis
ϕi
Field axis
Phase shift ϕi between Is and Uh
Phase shift ϕs between Is and Us
Example: -π/2 > ϕi > -π: Me,AC < 0 on rotor = generator operation
πϕτππ δδ /cosˆˆ)(22 11
2
, ispes
ACesyn BAplp
fMnP ⋅⋅⋅⋅⋅==
Internal power:
2/2 1 hswssh NkfU Φπ ⋅=1
ˆ2δτ
πΦ Bleph ⋅⋅=
ishs IUmP ϕδ cos⋅=
Uh
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/28
1. Basic design rules for electrical machines
Torque generation in DC machines
dxtxBtxAlF
pp
sre ⋅⋅⋅= ∫τ
δ
2
0
, ),(),( 2/sie dFM ⋅=
πατ δ /ˆ)(2 ,2
, serpeDCe BAplM ⋅⋅⋅=Φτ
⋅⋅⋅=⋅⋅⋅⋅⋅= ∫ rr AdpdxxBAld
pM si
0
sδ,esi
e
p
)(2
2
.)(si
constd
IzAxA c
rsr =⋅
⋅==
πsδ,pee
0
sδ,eˆ)(:
p
BldxxBlxx s ⋅⋅⋅=⋅== ∫ ταΦτ
Bδ,s
s
∧
r
s s
s
Tangential force on rotor:
s
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/29
1. Basic design rules for electrical machines
Internal power in DC machines
sδ,pee B̂l ⋅⋅⋅= ταΦ
πατππ δδ /ˆ)(222 ,2
, serpeDCe BAplnMnP ⋅⋅⋅⋅=⋅=
π⋅⋅
=sid
IzA c
rΦ⋅⋅⋅
= na
pzUi ca IaI ⋅= 2
ai IUP ⋅=δ
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/30
1. Basic design rules for electrical machines
• Torque generation
dxtxBtxAlF
pp
⋅⋅⋅= ∫τ
δ
2
0
),(),(
πατ δ /ˆ)(2 ,2
, serpeDCe BAplM ⋅⋅⋅=
DC machines:
Current load and air gap flux density constant along αe~0.7 of pole pitch:
2/sie dFM ⋅=
AC machines: Current loading As and air gap flux density Bδ are sinusoidal distributed, phase shift ϕi
between Is and Uh:
πϕτ δ /cosˆˆ)( 112
, ispeACe BAplM ⋅⋅⋅⋅=Rotor torque:
Rotor torque:
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/31
1. Basic design rules for electrical machines
)/( ldF sie ⋅⋅= πτ
Specific air gap thrust
Specific air gap thrust: Force per surface:
2/cosˆˆ11 isAC BA ϕτ δ ⋅⋅=)2/(cosˆˆ
)( 2211
2
lpBApl
pisp
AC τπϕπ
ττ δ ⋅⋅⋅⋅
⋅=
Surface: lpldArea psi τπ 2=⋅⋅=
)/()2//( pesiee pMdMF τπ== )2/( 22 lpM pe τπτ =
AC machines:
DC machines:
)2/(ˆ)(2 22
,
2
lpBApl
pserp
DC τπαπ
ττ δ ⋅⋅⋅
⋅= serDC BA ,
ˆδατ ⋅=
ell =
TECHNISCHE UNIVERSITÄT
DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG
Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics
1/32
1. Basic design rules for electrical machines
Effective current loading
)/cos(ˆ
),( pp
1s1 πϕωτπ
τπ
−−−⋅= is tx
VtxA
swsss
ss AkIkNp
mVA ⋅⋅=⋅⋅⋅⋅⋅=⋅= 1pws1p11 2/
2/ˆˆ τπ
πτπ
p2
2
τ⋅⋅⋅⋅
=p
INmA sss
s
AC machines:
AC: „Fictive“ effective current loading is constant along the air gap circumference!
DC machines:
.2
)2/()(
si
constp
aIz
d
IzAxA
p
acrsr =
⋅⋅
=⋅
⋅==
τπ
DC: Current loading is constant along the air gap circumference!
Ia: Armature current
Ic = Ia/(2a): Coil current
2/cosˆˆ11 isAC BA ϕτ δ ⋅⋅=
serDC BA ,ˆδατ ⋅=
AsAr
2/cosˆ11 iwsAC BkA ϕτ δ ⋅⋅⋅=
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Electromagnetic torque Me is determined by air gap flux density Bδ and
current loading A ( = ampere-turns per unit length) and corresponds with
internal power Pδ (air gap power).
- Air gap flux density peak value
- Typical maximum current loading for air cooling with open
ventilation:
A = 700 A/cm (DC-machines), corresponding amplitude for
AC-machines
a) DC-machines: αe = 0.7:
N/m2 ≅ 0.5 bar
b) AC-machines: kws1 ≈ 0.95, = 940 A/cm, maximum thrust at :
N/m2 ≅ 0.5 bar
In reality:
cosϕi ~ 0.9, so thrust for AC lower than for DC machines.
sws AkA ⋅⋅= 11 2ˆ
490000.17.070000ˆ, =⋅⋅=⋅= serDC BA δατ
470002/11940002/cosˆˆ1,1 =⋅⋅=⋅⋅= isAC BA ϕτ δ
1. Basic design rules for electrical machines
1ˆ
sA
Specific air gap thrust
T,0.1ˆ:DCT,0.1ˆ:AC ,1, == sBB δδ
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Summary: Torque generation and internal power
- Radial and tangential magnetic forces on magnetized iron and on conductors
- Tangential forces lead to torque
- Equivalent current loading represents slot conductor arrangement
- Fundamental waves for torque in AC machines
- Internal phase angle between resulting field wave and current loading
- DC machines have bigger torque at same peak current loading and field
- Specific air gap thrust as force per area in the range of 0.5 N 1 bar
Energy Converters – CAD and System Dynamics
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Energy Converters – CAD and System Dynamics
1. Basic design rules for rotating machines
1.1 Torque generation and internal power
1.2 Electromagnetic utilization
1.3 Thermal utilization
1.4 Overload capability of AC machines
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1. Basic design rules for electrical machines
Air gap torque Me, air gap power Pδ, internal apparent power Sδ
Air gap torque: Me
Air gap power (internal power):
AC machines:
DC machines:
)/(22 pfMnMP sesyne ⋅⋅=⋅= ππδ
nMP e ⋅⋅= πδ 2n: Rotor speed !
Internal apparent power:
AC machines:
DC machines:
DC:
si IUS ⋅= 3δ
ai IUS ⋅=δ Ui, Ia: Induced armature voltage
& armature current
Ui (or Uh), Is: Internal stator phase voltage
& stator phase current (r.m.s. values!)
δδ PS =
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DC machines:
Induced armature voltage:
With and current loading
we get:
Internal apparent power Sδ : Induced voltage x current:
AC machines: Induced phase voltage r.m.s:
With , r.m.s. current loading and
we get:p
ssss
p
INmA
τ2
2 ⋅⋅⋅=
1. Basic design rules for electrical machines
hwsssi kNfU Φπ ⋅⋅⋅⋅= 12
sis IUmS ⋅⋅=δ
1ˆ2δτ
πΦ Blph ⋅⋅=
sepeh Bl ,ˆδταΦ ⋅⋅=
hi na
pzU Φ⋅⋅
⋅=
p
ar
p
aIzA
τ2
)2/(⋅=
sreesi BAnldPS ,22 ˆ
δδδ απ ⋅⋅⋅⋅⋅⋅==
ai IUPS ⋅== δδ
Internal apparent power Sδ
swss
esishwsssssis ABkp
fldIkNfmIUmS 11
22
1ˆ
22 δδ
πΦπ ⋅⋅=⋅⋅⋅⋅⋅==
πτ /2 psi pd =
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Electromagnetic utilization (Esson´s number) C:
AC machines:
11
2
2ˆ
2δ
δ πBAk
nld
SC sws
synesi
AC ⋅⋅⋅=⋅⋅
=
1. Basic design rules for electrical machines
sre
esi
DC BAnld
PC ,
2
2ˆδ
δ απ ⋅⋅⋅=⋅⋅
=
Electromagnetic utilization C = Sδ per volume & speed
ACACC τπ ⋅= 2
DC machines:
at cosϕi = 1
DCDCC τπ ⋅= 2
Rotor volume: esiFesir ldldV ⋅≈⋅⋅≈ 22 4/π
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The internal apparent power Sδ per stator bore volume (usually
neglecting π/4) and per speed nsyn is called electromagnetic utilization or Esson´s number C. It increases with current loading A and air gap flux density Bδ.
ldsi ⋅)4/(2 π
11
2
2ˆ
2δ
δ πBAk
nld
SC sws
synesi
AC ⋅⋅⋅=⋅⋅
= sre
esi
DC BAnld
PC ,
2
2ˆδ
δ απ ⋅⋅⋅=⋅⋅
=
p
ssss
p
INmA
τ2
2 ⋅⋅⋅=
p
ar
p
aIzA
τ2
)2/(⋅=
32 ~/~ LldCnSM siN ⋅⋅=δ
Torque determines size of machine, NOT power !
r.m.s. current loading
1. Basic design rules for electrical machines
Electromagnetic utilization (Esson´s number) C
Machine volume (roughly): 3LV ≈
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Mounting of air-air heat exchanger on slip ring
induction wind generatorSource:
Winergy
Germany
Doubly-fed induction wind generator
1500 kW at 1800/min
Air inlet fan
Air-air heat
exchanger
Generator terminal
box
1. Basic design rules for electrical machines
- Internal “open”
ventilation
- Closed air-circuit to
reduce air acoustic noise L
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Example:
AC induction machines:
four poles, winding temperature rise 80 K,
air-cooled, open ventilated, for larger rated power
rated apparent power SN kVA 100 1000 10000
current loading As A/cm 300 550 1000
air gap flux density Bδ,1 T 1.0 1.05 1.1
Esson´s number C kVA.min/m3 3.3 6.4 12.2
Facit:
At given rotational speed: Bigger power = bigger torque = bigger machine size
= bigger rotor diameter = higher rotor surface speed = better air cooling!
With a better cooling higher current loadings are possible,
so electromagnetic utilization increases with rated power.
1. Basic design rules for electrical machines
∧
11
2ˆ
2δ
πBAkC swsAC ⋅⋅⋅=
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Example: AC induction machine:four poles, winding temperature rise 80 K, air-cooled, open ventilated
rated apparent power SN 10 000 kVA
current loading As 1000 A/cm
air gap flux density Bδ,1 1.1 T
Esson´s number C 12.2 kVA.min/m3
1. Basic design rules for electrical machines
23 N/m732000VAs/m732000 ==C
ACC τπ 22N/m732000 ==
bar67.09.074.0:9.0cosAt
1cosatbar74.0N/m74176/ 22
=⋅==
====
ACi
iAC C
τϕ
ϕπτ
∧
3352
11
2
min/mkVA2.12VAs/m7320001.11095.02
ˆ2
⋅=≈⋅⋅⋅≅⋅⋅⋅=ππ
δBAkC sws
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Summary: Electromagnetic utilization
- Internal apparent power = apparent air gap power Sδ
- Apparent air gap power per rotor volume and speed = electromagnetic utilization
- ESSON´s utilization C = „figure of merit“
- Utilization C ≈ 10 x Specific air gap thrust τ- Utilization C increases with current loading A and air-gap flux density Bδ
- Better cooling = higher current loading = higher utilization
Energy Converters – CAD and System Dynamics
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Energy Converters – CAD and System Dynamics
1. Basic design rules for rotating machines
1.1 Torque generation and internal power
1.2 Electromagnetic utilization
1.3 Thermal utilization
1.4 Overload capability of AC machines
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AC machines: ms phase winding: 22 )(2s
ca
bFesssssCu I
Aa
llNmIRmP ⋅
⋅⋅+⋅
⋅=⋅=κ
1. Basic design rules for electrical machines
Armature copper losses
sssibFe
si
sss
c
assibFeCu AJ
dll
d
INm
A
aIdllP ⋅⋅
⋅+=
⋅⋅⋅⋅
⋅+=
κπ
πκπ )(2/)(
Current density per conductor: cc AIJ /= )/( asc aII =
DC machines: z armature conductors:
222
2
⋅=⋅==a
IRzIRzIRP a
cccaaCu
AJdll
d
Iz
A
IdllI
A
llzP sibFe
si
c
c
csibFec
c
bFeCu ⋅⋅
⋅+=
⋅⋅⋅
⋅+=
⋅+
⋅=κ
ππκ
πκ
)()(2
AJdll
P sibFeCu ⋅⋅
⋅+=
κπ)(
(c: “conductor”)
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Heat transfer coefficient αk, W/(m2K): skkCu AP ϑ∆α ⋅⋅=
sskbFesik
Cus AJ
lld
P⋅⋅
⋅=
+⋅⋅=
καπαϑ∆
1
)(
Facit:
Temperature rise in armature winding for given
machine torque is determined by product of current
density J and current loading A.
It may be reduced by superior cooling (increased αk)
or decreased losses (increased κ).
sss AJ ⋅~ϑ∆
1. Basic design rules for electrical machines
Thermal utilization
Stator resistive losses
Cooling surface Ak
dsi
Steady state temperature rise:
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1. Basic design rules for electrical machines
Thermal scaling effectIncreasing motor size: Increasing surface of conductors
Example:
Surface cc ld ⋅π for round conductor wire (wire diameter dc & length lc),
Heat transfer coefficient at conductor surface: αc
Losses per conductor:
222, J
lAI
A
lIRP
c
ccc
cc
ccccCu ⋅=⋅==
κκ ⇒ 23
2
~ JLPJ
lA
P
V
PCu
ccc
Cu
c
Cu ⋅⇒=⋅
=κ
Temperature rise in conductor:
222
,
4
)4/(J
dJ
ld
ld
ld
P
cc
c
cccc
cc
ccc
cCu ⋅⋅⋅
=⋅⋅⋅⋅
⋅=
⋅⋅=
καπακπ
παϑ∆ ⇒ 2~ JL ⋅ϑ∆
Facit:
Admissible current density for SAME temperature rise ∆ϑ and cooling lower for bigger machines:
LJ /1~
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Source:
Winergy
Germany
Totally enclosed doubly-fed induction wind generator
Surface air-cooled with iron-cast cooling fin housing
600 kW at 1155/min
Fan hood Cooling fins Feet Power terminal box Slip ring
Shaft mounted fan inside terminal box
1. Basic design rules for electrical machines
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Lower admissible winding temperature rise means lower thermal utilization !
a) winding temperature rise 105 K:
rated power PN kW 5 650
current loading As A/cm 280 430
current density Js A/mm2 7.6 5.0
thermal utilization As.Js A/cm.A/mm2 2100 2150
b) winding temperature rise 80 K:
rated power PN kW 4 570
current loading As A/cm 240 380
current density Js A/mm2 6.6 4.4
thermal utilization As.Js A/cm.A/mm2 1580 1670
1. Basic design rules for electrical machines
Example: AC machines thermal utilization
totally enclosed, surface cooled, for smaller rated power
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Lower admissible winding temperature rise means lower utilization !
1. Basic design rules for electrical machines
Example:
Thermal utilization at Class F (105K) and Class B (80K)
totally enclosed, surface cooled
76.0105
80==
∆∆
F
B
ϑϑ
76.02100
1580
)(
)(==
⋅⋅
F
B
JA
JA
8.05
4
,
, ==FN
BN
P
P86.0
280
240
)(
)(
,
, ==≈=F
B
F
B
FN
BN
A
A
AB
AB
P
P
δ
δ
87.076.0~ 2 ==⇒=⋅F
B
A
AAAJA
Rough estimate of power scaling with lower admissible winding temperature:
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Air-cooled, four-pole induction machines, internal air flow
temperature rise ∆ϑ = 80 K (atϑamb = 40 °C), rated speed of about 1450/min.
Rated power / kW 100 1000 10000
Bδ,1 / T 1.0 1.0 1.0
As / A/cm 300 550 1000
C / kVA.min/m3 3.3 6.1 11.1
machine volume ~ L3 / p.u. 1.0 5.4 29.7
machine size L / p.u. 1.0 1.75 3.1
1.0 0.75 0.55
/A/mm2 6.8 5.1 3.7
/ (A/cm).(A/mm2) 2040 2850 3700
L/1LJ s /1~
ss JA ⋅
1. Basic design rules for electrical machines
Example: AC machines thermal utilizationopen ventilated
Facit:
Thermal utilization A.J rises with increased power.
Increased current density J for big machines needs improved cooling.
∧
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Summary: Thermal utilization
- I2.R losses per cooling surface ~ thermal utilization A.J
- Reduced current density J at bigger machine size L for same cooling system
- Thermal scaling laws
- Increased thermal utilization A.J leads to increased ESSON´s number C ~ A.B
- Thermal Class (B, F, H, N) of insulation system determines thermal utilization
Energy Converters – CAD and System Dynamics
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Energy Converters – CAD and System Dynamics
1. Basic design rules for rotating machines
1.1 Torque generation and internal power
1.2 Electromagnetic utilization
1.3 Thermal utilization
1.4 Overload capability of AC machines
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Rated stator impedance x of AC machines
pshd A
B
xx τδδ ⋅≈ 1
ˆ~
11
1. Basic design rules for electrical machines
N
dd
sN
sNN
N
ss
Z
Xx
I
UZ
Z
Xx === or
hshs xxxx ≈+= σ
δ
τ
πµπ
l
p
mkNfX
pswssh 2
210
2)(2 ⋅=
11
10
ˆ~
ˆ
2
2
2
δδ
δ
τδ
τ
τ
πµ
B
A
B
p
INm
kx
ps
p
p
sNss
wsh
⋅⋅≈
1
10
11
2
210
ˆ
2
2
2
ˆ22
2)(2
δδ
δ
τ
τ
πµ
τπ
π
δ
τ
πµπ
B
p
INm
k
BlkNf
Il
p
mkNf
U
IX
U
IXx
p
p
sNss
ws
pwss
sNps
wss
h
sNh
sN
sNhh ⋅=
⋅
⋅
=≈=
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Per-unit pull-out power
1. Basic design rules for electrical machines
Synchronous machines:
d
p
d
sNsN
sN
p
sNsN
dsNp
N
pqds
x
u
X
IU
U
U
IU
XUU
S
PXXR =⋅=
°===
/
3
/)90sin(3:,0
0
Induction machines:
sssN
sN
sNsN
ssN
sN
sN
sNsN
bsN
N
b
N
bs
xXI
U
IU
XU
p
p
IU
Mp
S
P
S
PR
⋅−
=−
⋅=
−⋅
=⋅
=≈=σ
σσ
σσσ
ωωω
δ
2
1
2
1
3
2
13
3:0
2
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Overload capability of AC machines:
psdd
p
N
p
A
B
xx
u
S
P
τδδ ⋅≈= 10
ˆ~
1
psssN
b
A
B
xxS
P
τδ
σσσ δ ⋅⋅⋅
−≈ 1
ˆ~
11~
2
1
1. Basic design rules for electrical machines
Synchronous machines: pull out torque Induction machines: Breakdown torque
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Summary: Overload capability of AC machines
- Overload capability = Maximum vs. rated torque Mmax/MN
- DC machine :
Maximum torque simply limited by maximum armature current Ia,max
- AC machines:
Maximum torque given by breakdown torque Mb or pull-out torque Mp0
- Small stray reactance Xσ and small synchronous reactance Xd
lead to increased maximum torque
Energy Converters – CAD and System Dynamics
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