energy converters – cad and system dynamics · prof. dr. a. binder, energy converters - cad and...

TECHNISCHE UNIVERSITÄT

DARMSTADTINSTITUT FÜRELEKTRISCHE ENERGIEWANDLUNG

Prof. Dr. A. Binder, Energy Converters - CAD and System Dynamics

8/1

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




8/2


Energy Converters - CAD and System Dynamics

Source:

H. Kleinrath, Springer-Verlag




8/3



8.1 Basics of steady state and significance of dynamic performance of

synchronous machines

8.2 Transient Flux Linkages of Synchronous Machines

8.3 Set of dynamic equations for synchronous machines

8.4 Park transformation

8.5 Equivalent circuits for magnetic coupling in synchronous machines

8.6 Transient performance of synchronous machines at constant speed operation

8.7 Time constants of electrically excited synchronous machines with damper cage

8.8 Sudden short circuit of electrically excited synchronous machine with damper

cage

8.9 Sudden short circuit torque and measurement of transient machine parameters

8.10 Transient stability of electrically excited synchronous machines




8/4


Electrically excited synchronous machine

a) Three phase stator winding, rotor field and damper winding

b) DC excitation via slip rings from a DC voltage (e.g. controlled rectifier)

a) b)

Source:

H. Kleinrath, Studientext




8/5


Permanent magnet (PM) excited synchronous machinea) Cylindrical rotor, surface mounted rotor rare earth high energy magnets

b) Application of inverter fed PM machine as adjustable speed drive for driving robots

a) b)

6 pole rotor, 4 magnets per pole

Source:

a) Siemens AG, b) Kuka, Germany




8/6


a) Cylindrical rotor synchronous machine with constant air gap

b) Salient pole rotor synchronous machine with non-constant air gap

a) b)

Air gap of synchronous machines

Rotor oriented d- and q-axis

reference frame




8/7


Cylindrical rotor Salient pole rotor

Source:

Siemens AG

Source:

Andritz Hydro, Austria




8/8


Steady state Me(ϑ) characteristic

- neglected stator losses (Rs = 0) - constant r.m.s. phase value Us

- constant frequency fs - constant rotor flux

Cylindrical rotor synchronous machine Salient pole synchronous machine




8/9


Transient operation of synchronous machine

Large transient disturbance: e. g. generator operation at the grid:

- sudden short circuits,

- voltage dips or oscillations due to switching,

- load steps on turbine or grid side etc.

2,,

2

1

2α

ππω ϑ −

⋅==

J

cpf

mdmd

Small transient disturbances lead to natural oscillations of load angle and rotor speed

ϑϑd

dMc e=Stiffness” of ”torsion spring”:

Damper cage:Damping of natural

oscillations by induced

damper bar currentsd,m




8/10

Summary: Basics of steady state and significance of dynamic performance of synchronous machines

- Repetition of bachelor course contents

- Salient versus cylindrical pole rotors

- Synchronous versus reluctance torque

- Electrically versus permanent magnet excited rotors

- Pull-out torque and stability limit

- Natural oscillations of rotors damped by damper cage





8/11












cage






8/12

Rotor air gap field of cylindrical rotor synchronous machine

• Rotor field winding is excited by the DC field current If

• Rotor m.m.f. Vf(xr) and rotor air gap field distribution Bp(xr) have steps due to rotor slots

• FOURIER series gives the fundamental rotor field (µ = 1):

ffdfpf

f Ikkp

NV ⋅⋅⋅== )(

2ˆ:1 ,,πµ

δµ f

p

VB

ˆˆ

0= fcrf NqpN ⋅⋅= 2

2

3)3/sin(

2sin, ==

⋅= ππ

τ p

fp

Wk

))6/(sin(

)6/sin(,

rr

fdqq

kππ

= dfpfwf kkk =

,

,

Example: qr = 2


Facit:

The fundamental rotor air-gap field is sinusoidal distributed.

It may described by a space vector, fixed in the rotor reference frame.




8/13

Cylindrical rotor synchronous excitation-stator transfer ratio


f

If

f Iü

I1

=′

fwsss

fwff

f Ikp

NmIk

p

NV ′⋅⋅⋅=⋅⋅⋅=

ππ22ˆ

wff

wsssIf

kN

kNmü

2/=

• Fundamental of rotor m.m.f. Vf(xr):

üIf: field current transfer ratio

Ufs

Uff

sIf ü

mü

m

mü ⋅=⋅=

1

fUff UüU ⋅=′

üUf: field voltage transfer ratio

wff

wssUf

kN

kNü

2/=




8/14

Cylindrical rotor synchronous machine: Rotor magnetic field

Magnetic rotor field, no-load (Is = 0, If > 0):

- Field winding excited by If

- Stator winding without current (no-load)


f

If

f Iü

I1

=′

fwsss

fwff

f Ikp

NmIk

p

NV ′⋅⋅⋅=⋅⋅⋅=

ππ22ˆ

wff

wsssIf

kN

kNmü

2/=

• Fundamental of rotor m.m.f. Vf(xr):

• Expressed with stator winding data (ms = 3):

N

fphNfws

sf

I

IVIk

p

NV

ˆ

2ˆ

2

332ˆ,1

′⋅⋅⋅=′⋅⋅⋅=

π

)(ˆ2

3)( ,1 tiVtV fphNf

′⋅⋅=

• Rotor m.m.f. space vector:

wsNs

phN kIp

NV ⋅⋅⋅=

π4ˆ

2ˆ

,1

N

ff

I

Ii

ˆ

2 ′⋅=′

Source: E. Fuchs, IEEE-PAS

Example:

2p = 2, qs = 6, qr = 6

d-axis(Re-axis)

q-axis

Vf(t)




8/15


)()(ˆ

)()()(ˆ

2

3)(

1,1 titi

V

tVtvtiVtV ff

N

fffphNf

′=′==′⋅⋅=

Cylindrical rotor synchronous machine: Stator and rotor m.m.f. and current space vectors

[ ])()()(ˆ)(2

,1 tiatiatiVtV WVUphNs ⋅+⋅+⋅=Stator space vectors:

phNN VV ,1,1ˆ

2

3ˆ ⋅=

[ ])()()(3

2

ˆ

)()(

2

1

tiatiatiV

tVtv WVU

N

ss ⋅+⋅+⋅==

[ ])()()(3

2)(

2tIatIatItI WVUs ⋅+⋅+⋅= [ ] )()()()(

3

2

ˆ

)()(

2tvtiatiati

I

tIti sWVU

N

ss =⋅+⋅+⋅==

Rotor space vectors: in direction of d-axis! (= Re-axis)

)(ˆ

)(2)( tv

I

tIti f

N

ff =

′⋅=′




8/16

Magnetizing current space vector

Source:

E. Fuchs, IEEE-PAS


Magnetic field at load (Is > 0, If > 0):

Field winding excited by If ,

Stator winding excited by Is

Example:

2p = 2,

qs = 5, qr = 9rotor pole axis = d-axis

field axis

In per-unit:

)()()( τττ mfs iii =′+

Magnetizing current

space vector im

)()( ττψ mhhix ⋅=

Main stator flux linkage space

vector:

q-axis




8/17


Stator flux linkage equation in rotor reference frame(d- and q-axis) without damper cage

Stator flux linkage:

Stator winding inductance:

(“synchronous inductance”)dshs xxxx =+= σ

We assume constant iron saturation, so xh is constant.

)()()()()( τττψττψ σσ mhsshsssixixix ⋅+⋅=+⋅=

fhshsfshsssixixxiixix ′+⋅+=′+⋅+⋅= )()( σσψ

.etc)()()()( )()(τττψτψ rssrss

ii ==

fhqsdsqdsixijxixj ′+⋅+⋅=+= ψψψ

fhdsd ixix ′+⋅=ψ qsq ix ⋅=ψ

qds ijii ⋅+=

( ),...)()(

γψψ j

ssrse−⋅=

In rotor reference frame:




8/18


Space vector diagram in rotor reference frame (d- and q-axis)

q

d

iq

id

is

ψh

ψdh

ψqh

i´f

0

)´( fdhdhiix +⋅=ψ

qhqhix ⋅=ψ

fhdsd ixix ′+⋅=ψ

qsq ix ⋅=ψ

qhdhhj ψψψ ⋅+=

d-axis = Re-axisq-axis = Im-axis

)( fdhdh iix ′+⋅=ψ

qhqh ix ⋅=ψ




8/19


Dynamic equations of cylindrical rotor synchronous machine without damper cage

)()()()()()(

)()()(

)()()()(

)()()(

)()(

)()()(

)()(

ττψττψτττω

τ

ττψ

τττψ

τψτωττψ

ττ

τψτωττψ

ττ

σ

σ

sqddqm

J

qshq

fhdshd

dmq

qsq

qmd

dsd

miid

d

ixx

ixixx

d

diru

d

diru

−⋅−⋅=⋅

⋅+=

′⋅+⋅+=

⋅++⋅=

⋅−+⋅=

Stator zero sequence voltage system:

Separate zero-sequence flux linkage equation (not discussed here)τψd

diru ssss

000 +⋅=

rotor reference frame = d- and q-axis qds ijii ⋅+= qds ujuu ⋅+=

qdsj ψψψ ⋅+=

)(),(),(),(),( τωτψτψττ mqdqd ii

Unknowns:

Given:

)(),(),(),( ττττ sfqd miuu ′




8/20

Rotor field of salient pole synchronous machine

• Bell shaped rotor air gap field curve Bδ(xr): A constant m.m.f. Vf = IfNf,Pol excites

with a variable air gap δ(xr) a bell shaped field curve. Fundamental of this “bell-shape” (µ = 1):

)/cos(ˆ)(:1)(

)( 1r1r

0r prpf

xBxBx

VxB τπµ

δµ δδ ⋅==→=


• FOURIER-fundamental wave: Amplitude proportional to If

• Field factor β (numerically determined):

00rmax )0(δ

µδδfV

xBB ===

1ˆ

pB

1:usually/ˆmax1 <= ββδBBp

Polff NpN ,2 ⋅=

rx0δ

1ˆ

pB

fundamental field wavemaxδB

p

rr τ

πγ

x=

)( rxBδ




8/21

- Fundamental stator m.m.f. leads to non-sinus Bd(x), Bq(x) !

- Stator field at cylindrical rotor synchronous machine (air gap δ0):

- Amplitudes of FOURIER-fundamental waves of Bd(x), Bq(x):

Saliency: Stator air gap field larger in d- than in q-axis

Air gap is LARGER in neutral zone (inter-pole gap of q-axis) than in pole axis (d-axis). Hence for equal m.m.f. Vs (sinus fundamental ν = 1) the corresponding air gap field is SMALLERin q-axis than in d-axis and NOT SINUSOIDAL


Stator field in d-position

Stator field in q-position

swsss IkNp

V ⋅⋅⋅=32ˆ

π

sqsd BBBB ˆˆ,ˆˆ11 <<<




8/22

Saliency: Different main inductances in d- and q-axis

195.0ˆ/ˆ1 <≈= sdd BBc

hds

hd

d

dhdh Lc

I

c

IL ⋅=

⋅==

2/2/ ΨΨ

16.0...5.0ˆ/ˆ1 <≈= sqq BBc


- Stator main inductance of cylindrical rotor synchronous machine (air gap δ0):

s

s

s

h

C

epwsshFe

I

B

Ik

l

pkNL

ˆ~

2/6)(:at

02

20

Ψδ

τ

πµµ =⋅⋅⋅=∞→

Stator field in d-position:Is = Id

Stator field in q-position:Is = Iq

- Saliency: d-axis: hqs

hq

q

qhqh Lc

I

c

IL ⋅=

⋅==

2/2/ ΨΨq-axis:




8/23

Stator current space vector:

decomposed:

Rotor current space vector:

(in d-axis!)

Magnetizing current space vector:

In d-axis:

In q-axis:

Flux linkage space vector:

Saliency: Stator current and flux linkage space vectors

sqqhqh IL 2⋅=Ψ

qds III +=

)(2 sdfdhdh IIL +′⋅⋅=Ψ


fI ′

dfmd III +′=

qmq II =

dhΨ hΨ

qhΨ

dI

qI

fI

(Re-axis)

qhdhh ΨΨΨ +=




8/24


Saliency: Stator flux linkage equations in d- and q-axis

qqqqhsq

fdhddfdhddhsd

ixixx

ixixixixx

=⋅+=

′+⋅=′+⋅+=

)(

)(

σ

σ

ψ

ψStator flux linkage in per-unit:

Stator inductances:

Synchronous inductance of d-axis xd and of q-axis xq:

qdsqhqsdhd xxxxxxxx >+=+= σσ

We assume constant iron saturation, so xdh, xqh are constant.

By calculating the magnetizing current in d- and q-component,

the resulting variable iron saturation due to

is usually not correctly considered!mqmdm III +=




8/25


Set of dynamic equations of salient rotor synchronous machine without damper cage

)()()()()()(

)()()(

)()()()(

)()()(

)()(

)()()(

)()(


τ

ττψ

τττψ

τψτωττψ

ττ

τψτωττψ

ττ

σ

σ

sqddqm

J

qsqhq

fdhdsdhd

dmq

qsq

qmd

dsd

miid

d

ixx

ixixx

d

diru

d

diru

−⋅−⋅=⋅

⋅+=

′⋅+⋅+=

⋅++⋅=

⋅−+⋅=


Separate zero-sequence flux linkage equation (not discussed here)τψd

diru ssss

000 +⋅=

rotor reference frame = d- and q-axis qds ijii ⋅+= qds ujuu ⋅+=

qdsj ψψψ ⋅+=

)(),(),(),(),( τωτψτψττ mqdqd ii

Unknowns:

Given:

)(),(),(),( ττττ sfqd miuu ′




8/26

Summary: Flux linkage in cylindrical and salient rotor synchronous machines

- Repetition of bachelor course contents

- Also with space vectors:

One magnetizing stator inductance in cylindrical rotor

machines

- In salient pole rotors:

d-axis magnetizing stator inductance bigger than in q-axis

- So far: Influence of damper cage neglected





8/27


Fluxes in salient pole synchronous machine with damper cage

a) Stator magnetic field component, excited in parallel with rotor d-axisb) Stator magnetic field component, excited in parallel with rotor q-axis

a) d-axis flux linkage b) q-axis flux linkage

Stray fluxes:

Φsσ⇔ xsσ

Φfσ⇔ xfσ

ΦDσ⇔ xDσ

ΦQσ⇔ xQσ

Main fluxes:

Φdh ⇔ xdh

Φqh ⇔ xqh




8/28

Dynamic inductances and reactances

h Magnetic equivalent circuit of synchronous machine in dynamic state of operation:

– “Transformer”: Magnetizing inductance: d-axis Ldh , q-axis Lqh

– Secondary leakage inductance:

Excitation winding: Φfσ, Xfσ = ωsLfσ,

– Damper cage:

– d-axis ΦDσ: XDσ = ωsLDσ,

– q-axis ΦQσ : XQσ = ωsLQσ

h Damper bars are short circuited by end rings!

h Exciting DC voltage source of excitation winding has small internal resistance,

which may be regarded for induced transient voltages and currents as short circuit.

h So secondary windings are short circuited!





8/29

q-axis transformer equivalent circuit of synchronous machine

2-Winding-transformer:

Stator winding with damper cage = like induction machine

Magnetic linkage in q-axis of synchronous machine:


Simplification: Bundled flux lines:

Main flux is identical for stator, field and damper winding

Voltage and current transfer ratios for field and damper winding

different!QsQqqq IMIL +=Ψ

QQqQsQ ILIM +=ΨIQUQ üü ≠

)/( IQQsQIQqqq üIMüIL ⋅+=Ψ

)/( IQQQIQUQqQsUQQUQ üILüüIMüü ⋅+=Ψ

Qqhqqhsq ILILL ′⋅+⋅+= )( σΨ

QqhQqqhQ ILLIL ′⋅+′+=′ )( σΨ

)( Qqqhqsq IILIL ′+⋅+= σΨ

)( QqqhQQQ IILIL ′+⋅+′′=′ σΨ

QIQUQQ RüüR =′

We skip the notation ´ in the following for transferred values!

)( Qqqhqsq IILIL +⋅+= σΨ

)( QqqhQQQ IILIL +⋅+= σΨ

1)2/1( ⋅== wss

wrr

wssUQ

kN

kN

kNü

rUQ

r

sUQIQ

Qü

m

müü

3⋅=⋅=

σσ QIQUQQ LüüL =′

QhQQ LLL += σ




8/30

d-axis transformer equivalent circuit of synchronous machine

3-Winding-transformer:Electrically excited rotor with damper cage

Magnetic linkage in d-axis of synchronous machine:


Simplification: Bundled flux lines:

Main flux is identical for stator, field and damper winding

Voltage and current transfer ratios for field and damper winding

different:

We skip the notation ´ in the following for transferred values!

DsDfsfddd IMIMIL ++=ΨDfDffdfsf IMILIM ++=Ψ

DDfDfdDsD ILIMIM ++=Ψ

IDUDIfUf üüüü ≠≠≠

)/()/( IDDsDIDIffsfIfddd üIMüüIMüIL ⋅+⋅+=Ψ)/()/( IDDfDIDUfIfffIfUfdfsUffUf üIMüüüILüüIMüü ⋅+⋅+=Ψ

)/()/( IDDDIDUDIffDfIfUDdDsUDDUD üILüüüIMüüIMüü ⋅+⋅+=Ψ

Ddhfdhddhdd ILILILL ′+′+⋅+= )( σΨ

Ddhfdhfddhf ILILLIL ′+′⋅+′+=′ )( σΨ

DdhDfdhddhD ILLILIL ′⋅+′+′+=′ )( σΨ

)( Dfddhdsd IIILIL ++⋅+= σΨ)( Dfddhfff IIILIL ++⋅+= σΨ

)( DfddhDDD IIILIL ++⋅+= σΨ

fIfUff RüüR =′ DIDUDD RüüR =′σσ fIfUff LüüL =′ σσ DIDUDD LüüL =′

fhff LLL += σ

DhDD LLL += σ




8/31


Example: Rotor DC field winding: Voltage step

If only transient change of current for very short time t << T after disturbance is of

interest, only the resulting inductance for dynamic condition must be known!

( )TteR

U

R

Uti /1 1)( −−⋅+=

∆ tL

U

R

Utiti

Tt

⋅≈−=<<

∆∆ 1)()(

Understanding transients




8/32

Magnetic coupling in d-axis

No damper cage:

transient reactance X´d

No damper cage, no field winding:

synchronous d-reactance Xd

Damper cage and field winding:

subtransient d-reactance X”d


Id

If ID




8/33

No damper cage, no field winding:

synchronous q-reactance Xq

Damper cage:

subtransient q-reactance X”q

Magnetic coupling in q-axis


Iq

IQ




8/34

)(σσσσ

σσσωω

DfDdhfdh

Dfdhssdsd

LLLLLL

LLLLLX

+++⋅=′′=′′

)(σ

σσωω

Qqh

Qqhssqsq

LL

LLLLX

++⋅=′′=′′

15.01.01.01.09.01.09.0

1.01.09.01.0/ =

⋅+⋅+⋅⋅⋅

+=′′=′′ dNd xZX

h „Subtransient reactance“ of d-axis (secondary resistances neglected):

• "Subtransient reactance of q-axis" :

Example: Salient poles: Xd/ZN = xd = 1 p.u., Xq/ZN = xq = 0.6 p.u.,

Xsσ = Xfσ = XDσ = XQσ = 0.1.ZN:

Subtransient inductances and reactances


18.01.05.0

1.05.01.0/ =

+⋅

+=′′=′′ qNq xZX

symmetrynt subtransie:

18.015.0

:but,6.01

qd

qd

qd

xx

xx

xx

′′≈′′

=′′<=′′

=>=




8/35

h During subtransient state of synchronous machine not the synchronous reactances

Xd, Xq are active,

but the much smaller subtransient reactances:

h Note: Even if Xd > Xq like in salient pole machines,

the subtransient reactances are nearly the SAME:

h Winding resistances Rs, Rf, RD and RQ cause a decay of dynamic currents, which flow

due to induced transient voltages, in the stator and rotor windings

h In damper cage the dynamic current in damper bars decays much faster

than in field winding

h Typical time constant of decay:

in damper winding 20 ... 50 ms,

in excitation winding 0.5 s ... 2 s

Subtransient performance of synchronous machine

qd XX ′′′′ ,


qdqd XXXX ′′≈′′′′<′′ but,




8/36

Example: Calculation of rs, rf, rD (1)Two-pole turbine generator (= cylindrical rotor) with complete damper cage,

Copper wedges of rotor field winding slots are damper bars

m9.2,mm920Hz,50A,6880,YkV5.10,MVA125 ,NNNN ====== rFera ldfIUS

Ω88.0/ NphNphN == IUZ

Stator winding resistance per phase:11/222,2,1,3,11,66,mΩ56.1Cs,75 =⋅⋅=⇒=======° sscsssscsss aNqpNpaNmqQR

9166.0,9553.00.9595,,33/27/ ==== wsdspsp kkkW τp.u.00236.0/,mΩ078.256.133.1 NACs,DCs,ACACs, ===⋅=⋅= ZRrRkR s

Field winding resistance:

144/2,1,9,1,8,32,mΩ2.94Cf,75 =⋅⋅=⇒======° ffcrfffcfrf aNqpNaNmqQR

8276.0,9556.0/2,3 === wfdfpf kkk

01518.0,1795.02

,0846.01448276.0

119166.0UfIfUfIfUf =⋅=

⋅⋅==

⋅⋅

== üüm

müü

Nk

Nkü

f

s

fwf

sws

p.u.00162.0/,mΩ43.1 NfffUfIff =′==⋅⋅=′ ZRrRüüR





8/37

Example: Calculation of rs, rf, rD (2)

µΩ87)10440405.002.0/(1.3)/(

m1.31.029.22

mm,5.402/)4932(mm,20

6Cbar,75

,

DD

=⋅⋅⋅==

=⋅+=⋅+=

=+==

° CuDDbar

rFebar

bhlR

lll

bh

κ

∆

hD

bD

Nfc = 7

wDwrDrDrr kkNNmmQ ======= 1,2/1,48

µΩ15.6104.1035.01.0

0301.0

2/mm,2.60/mm,35

6steelstainlessendcap

endcapCendcap,75

endcapendcap

=⋅⋅⋅

=⋅⋅

=

≈===

° κ∆

∆∆

τ∆πτ

hl

bR

bQdh QrrraQr

µΩ718))48/(sin(2

15.6

))/(sin(2 22

Cendcap,75*Cendcap,75 =

⋅=

⋅= °

°ππ

∆∆

rQp

RR

µΩ80571887*Cendcap,75Cbar,75 =+=+= °° RRRD ∆

414.25,26.1,165.205.01

119166.0UDIDUDIDUD =⋅=⋅==

⋅⋅

== üüm

müü

Nk

Nkü

D

s

DwD

sws

p.u.020736.0/,mΩ247.18 NDDDUDIDD =′==⋅⋅=′ ZRrRüüR

Damper winding resistance:Ideal symmetrical damper cage = no difference between d- and q-axis considered





8/38

Example: Calculation of rs, rf, rD (3)

Per-unit stator / field / damper winding resistance, with respect to stator-side:

02074.0,00162.0,00237.0 Dfs === rrr

Stator / field / damper winding resistance in physical units:

µΩ805,mΩ2.94,mΩ078.2 Dfs === RRR

Facit:

• In physical units the field winding has the biggest ohmic value

• The damper bar has the smallest ohmic value, although the retaining stainless steel cap

(as “end-ring”) increases the resistance strongly.

• In p.u. the damper resistance is the biggest value, the field winding resistance is the smallest.

• As the p.u. inductance values xs, xf, xD (with respect to the stator side) are similar, the time constant for the separated windings

is for the damper the shortest ⇒ The dynamic damper currents decay fastest!

Dsf /// rxrxrx DDssff =>=>= τττ





8/39

„Transient“ state of synchronous machineh Intermediate state in d-axis: Dynamic currents in damper cage have already decayed,

whereas in field winding still a big dynamic field current is flowing.

h Equivalent reactance for this “transient”: "transient reactance” of d-axis Xd´! Is only slightly bigger than subtransient reactance !

h After decay of all dynamic currents stator reactance becomes again Xd resp. Xq

)(σ

σσωω

fdh

fdhssdsd

LL

LLLLX

++⋅=′=′

Example: xdh = 0.9, xsσ = xfσ = 0.1: 19.0/ =′=′ dNd xZX


19.01.09.0

1.09.01.0 =

+⋅

+=′dx

19.015.0 =′<=′′ dd xx




8/40


Subtransient and transient state

• Time constant of damper winding for decay of transient damper current is much shorter

(factor 10) than time constant of field winding for decay of transient field current

• So we can distinguish “transient” and “sub-transient” state

Subtransient time section 0 .... ≈ 0.5 s

Dynamic current flow in stator, damper and field winding

Transient time section 0.5 s .... ≈ 2 s

Dynamic current flow in stator and field winding

Steady state (synchronous) > 2 ... 3 s

Steady state current flow in stator and field winding




8/41


Steady state Transient Sub-transient

Direct axis Synchronous inductance Ld , xd

Transient inductance

dd xL ′′ ,

Subtransient inductance

dd xL ′′′′ ,

Quadrature axis Synchronous inductance Lq , xq

Synchronous inductance Lq , xq

Subtransient inductance

qq xL ′′′′ ,

d-axis q-axis

x´q = xq

Summary 1: Stator inductance per phase for steady state and dynamics




8/42


Subtransient inductance of d-axis: (physical units: L, per unit value: x)

σσσσ

σσσ

σσσσ

σσσ

DfDdhfdh

Dfdhsd

DfDdhfdh

Dfdhsd

xxxxxx

xxxxx

LLLLLL

LLLLL

+++=′′

+++=′′

Subtransient inductance of q-axis:

σ

σσ

σ

σσ

Qqh

Qqhsq

Qqh

Qqhsq

xx

xxxx

LL

LLLL

++=′′

++=′′

Transient inductance of d-axis:

σ

σσ

σ

σσ

fdh

fdhsd

fdh

fdhsd

xx

xxxx

LL

LLLL

++=′

++=′

Transient inductance of q-axis:

qqqq xxLL =′=′

Summary 2: Stator inductance per phase for steady state and dynamics




8/43

synchronous Xd transient X´d sub-transient X´´d

d-axis: Field lines for dynamic and steady state

Sinusoidal distributed (= fundamental) stator m.m.f. Vs(1)(x)

- synchronous: no reaction of rotor windings

- transient: reaction of rotor field winding

- sub-transient: reaction of rotor field and damper winding

Source:

K. Bonfert, Springer-Verlag

DC excitation

If not shown





8/44

- synchronous: no reaction of rotor windings- NO transient state, because no linkage with rotor field winding- sub-transient: reaction of damper winding

q-axis: Field lines for dynamic and steady state

synchronous Xq NO transient X´q = Xq sub-transient X´´q

DC excitation

If not shown

Source:

K. Bonfert, Springer-Verlag





8/45

d-axis

q-axis

synchronous transient sub-transient

Field lines for dynamic and steady state

Sinusoidal distributed (= fundamental) stator m.m.f. Vs(1)(x)

- synchronous:

No reaction of rotor

windings

- transient:

Reaction of

rotor field winding

- sub-transient:

Reaction of rotor field

and damper winding

Xd , Xq X´d , Xq X´´d , X´´q

DC excitation

If not shown

Source: K. Bonfert, Springer-Verlag





8/46

Synchronous reactance of direct axis

dNd

Ndd

LX

ZXx

ω== /

0.8 ... 1.2: Salient pole synchronous machines

with high pole count 1.2 ... 2.5: 2- and 4-pole cylindrical rotor

synchronous machines with high utilization

synchronous reactance of quadrature axis

qNq

Nqq

LX

ZXx

ω=

= /

dx⋅)6.0...5.0( : Salient pole synchronous machines

with high pole count

dx⋅)9.0...8.0( : 2- and 4-pole cylindrical rotor


transient reactance of direct axis

dNd

Ndd

LX

ZXx

′=′

′=′

ω/

0.2-0.25 ... 0.35-0.4

transient reactance of quadrature axis

Nq

qq

ZX

xx

/=

==′

Identical with synchronous reactance

subtransient reactance of direct axis

dNd

Ndd

LX

ZXx

′′=′′

′′=′′

ω/

0.1-0.12 ... 0.2-0.3

subtransient reactance of quadrature axis

qNq

Nqq

LX

ZXx

′′=′′

′′=′′

ω

/

usually dq xx ′′>′′ , as field winding is missing in q-axis:

0.1 ... 0.3





8/47

Summary (1): Dynamic Flux Linkages of Synchronous Machines

- d- and q-axes separation of flux linkages in synchronous machines

- Electrically excited synchronous machines:

- Three- winding transformer in d-axis

- Two-winding transformer in q-axis

- Sudden change in flux linkage leads to transient rotor currents

in damper and field winding

- Subtransient d- and q-axis inductance (in p.u.: „reactance“)

- Transient d-axis inductance, when transient rotor current only in field winding





8/48

Summary (2): Dynamic inductances in synchronous machines

- Steady-state and dynamic inductances are mostly given in p.u.

- p.u.-values as inductances or reactances identical

- „p.u.-reactances“ is the standardized wording

- Subtransient reactances smaller than transient reactance

- Transient reactance smaller than steady-state reactance

- Sub-transient q-reactance slightly larger than subtransient d-reactance

- In subtransient reactances machines are nearly symmetrical,

although at steady-state (esp. salient poles!) not!





8/49











cage







8/50


Equations are formulated in rotor reference frame:

Real axis = d-axis, Imaginary axis = q-axis

d-axis

q-axis

sjj

sj

ssj

ssrs ueeueueuu msm =⋅⋅=⋅=⋅= ⋅−⋅⋅−⋅− τωτωτωτγ)(

)()()(

In synchronous steady state condition (ωs = ωm) stator voltage / current space vectors at

sinus stator three-phase voltage feeding DO NOT MOVE in rotor reference frame:

Space vectors in rotor reference frame




8/51

τ

ψ

ψωτ

ψ

d

dir

jd

diru

rrrrr

rsmrs

rssrs

)()(

)(

)()()(

0′

+′⋅′=

⋅⋅++⋅=- Subscript (r) denotes:

rotor reference frame

- We skip subscript (r) for simplicity

Decomposition of stator space vectors in d- and q-components:

qdsqdsqds jijiiujuu ψψψ ⋅+=⋅+=⋅+= ,

Transformation ratio for voltage and current:

QDrQDr jijii ψψψ ⋅+=′⋅+=′

2/1,

)2/1(

swsUQUD

r

sswsIQID

Nküü

Q

mNküü

⋅==

⋅⋅⋅

==

Ifff üii /=′ fUff ü ψψ ⋅=′ fIfUff rüür ⋅⋅=′

Voltage equations of stator and damper winding in rotor reference frame

- Stator :

- Damper :

Field winding: Different transformation ratio: WE SKIP THE ´ FURTHER

Damper current space vector:





8/52

Dynamic voltage equations for synchronous machines

τ

ψτ

ψτψ

ψωτ

ψ

ψωτψ

d

diru

d

dir

d

dir

d

diru

d

diru

ffff

QQQ

DDD

dmq

qsq

qmd

dsd

+⋅=

+⋅=

+⋅=

⋅++⋅=

⋅−+⋅=

0

0

d, q: Stator voltage equations

D, Q: Damper voltage equations

f : Field voltage equation

Equations are valid in rotor reference frame !





8/53

Flux linkage equations in d- and q-axis

ffDdhddhf

QQqqhQ

fdhDDddhD

Qqhqqq

fdhDdhddd

ixixix

ixix

ixixix

ixix

ixixix

++=

+=

++=

+=

++=

ψ

ψ

ψ

ψ

ψStator flux linkage

Damper flux linkage

Field winding flux linkage

Stator inductance

σ

σσ

σσ

fdhf

QqhQDdhD

sqhqsdhd

xxx

xxxxxx

xxxxxx

+=

+=+=

+=+=

Damper inductance

Field inductance

We assume constant iron saturation, so xdh, xqh are constant





8/54

Electromagnetic torque

qddqsse iiim ψψψ ⋅−⋅=⋅= *Im

Complete set of dynamic equations in rotor reference frame for

a) electrically excited synchronous machines with damper cage (and rotor saliency):11 equations, 11 unknowns

b) permanent magnet synchronous machines: NO damper cage, NO field winding:5 equations, 5 unknowns

mfQDqdfQDqd iiiii ωψψψψψ ,,,,,,,,,,

mqdqd ii ωψψ ,,,,

sfqd muuu ,,,

spqd muu ,Perm.Flux,, f ψψ =

Stator current and flux linkage space vector

components in d-q-reference frame

)()()()()()(


τ sqddqm

J miid

d−⋅−⋅=⋅ Mechanical equation





8/55

Electrically excited synchronous machines- with damper cage and rotor saliency

)()()()()()(

)()()()()(

)()()()()()()()()(

)()()()()()()()()(

)()()(

)()(0

)()(0

)()()(

)()()()()(

)()(


τ

ττττψ

τττψττττψ

τττψττττψττψ

ττ

ττψ

τττψ

τ

τψτωττψ

τττψτωττψττ

σ

σσ

σσ

sqddqm

J

ffdhDdhddhf

QQqhqqhQfdhDDdhddhD

QqhqsqhqfdhDdhdsdhd

ffff

QQQ

DDD

dmq

qsqqmd

dsd

miid

d

ixxixix

ixxixixixxix

ixixxixixixx

d

diru

d

dir

d

dir

d

diru

d

diru

−⋅−⋅=⋅

⋅+++=

⋅++=+⋅++=

⋅+⋅+=++⋅+=

+⋅=

+⋅=+⋅=

⋅++⋅=⋅−+⋅=





8/56

Permanent magnet synchronous machinesno damper / field winding

)()()()()()(

)()()(

)()()(

)()()(

)()(

)()()(

)()(

ττψττψτττωτ

ττψ

ψττψ

τψτωττψ

ττ

τψτωττψ

ττ

σ

σ

sqddqm

J

qsqhq

pdsdhd

dmq

qsq

qmd

dsd

miid

d

ixx

ixx

d

diru

d

diru

−⋅−⋅=⋅

⋅+=

+⋅+=

⋅++⋅=

⋅−+⋅=


Separate zero-sequence flux linkage equations (not discussed here)τψd

diru ssss

000 +⋅=





8/57

Steady state operation of synchronous machine

fffQD

fff

QQ

DD

dmqsq

qmdsd

ruiii

iru

ir

ir

iru

iru

/,0

0

0

===

⋅=

⋅=

⋅=

⋅+⋅=

⋅−⋅=

ψω

ψωqqqfdhddd ixixix =+= ψψ

The steady state back emf up is directed in q-axis:

fdhmp ixu ⋅=ω

fdhmddmqsqqqmdsd ixixiruixiru ⋅+⋅+⋅=⋅−⋅= ωωω

- In rotor reference frame: steady state: d./dτ = 0

- DC values = d- and q-phasor amplitudes of d-q-phasor diagram





8/58

fdhmddmqsqqqmdsd ixixiruixiru ⋅+⋅+⋅=⋅−⋅= ωωω

Example:

Under-excited motor operation

Conventional phasor diagram

Steady state DC values diagram

Space vector diagram for steady state:

corresponds

directly to

conventional

phasor diagram

IDENTICAL





8/59

Do you remember ? Motor operation – under-excited

Xd ≠ XqXd = Xq

Cylindrical rotor

Salientrotor





8/60

Summary: Set of dynamic equations for synchronous machines

- Formulation in rotor reference frame

- Due to damper and field winding:

11 dynamic equations for d- and q-axis

- PM machines:

Only 5 dynamic equations, if rotor eddy currents are neglected

- Steady state solutions are in rotor reference frame DC values

- Steady state DC values correspond

with

Re- and Im-part of steady state AC complex phasor results

(see: Bachelor´s course)





8/61












cage






8/62

Transformation from stator (s) three-phase U, V, W-system

into rotor (r) two-axis d-q-system : e.g. stator voltage space vector us:

( ) γ

γ

⋅−

⋅−

⋅⋅+⋅+⋅=

=⋅==⋅+

jWVU

jssrsqd

euauau

euuuju

2

)()(

3

2

( ) 003

1uuuuu WVUs =++⋅=

⋅

−⋅−−⋅−⋅−

−⋅−⋅⋅

=

W

V

U

q

d

u

u

u

u

u

u

3

1

3

1

3

1

)3

4sin(

3

2)

3

2sin(

3

2sin

3

2

)3

4cos(

3

2)

3

2cos(

3

2cos

3

2

0

πγ

πγγ

πγ

πγγ

Park transformation (1)


3/2π⋅= jea




8/63

Park transformation (2)

⋅=

⋅

−⋅−−⋅−⋅−

−⋅−⋅⋅

=

W

V

U

W

V

U

q

d

u

u

u

T

u

u

u

u

u

u

)(

3

1

3

1

3

1

)3

4sin(

3

2)

3

2sin(

3

2sin

3

2

)3

4cos(

3

2)

3

2cos(

3

2cos

3

2

0

πγπγγ

πγπγγ

⋅=

⋅

−−−

−−−

−

=

−

0

1

0

)(

1)3

4sin()

3

4cos(

1)3

2sin()

3

2cos(

1sincos

u

u

u

T

u

u

u

u

u

u

q

d

q

d

W

V

U

πγπγ

πγπγ

γγ





8/64

Summary: Park transformation

- Alternative formulation to KOVAC´s complex space vectors

- Matrix transformation from U,V,W to d,q,0-system

- Historically older, since ca. 1930

- CLARKE´s transformation:

From U,V,W to stator α,β,0-system

- PARK´s transformation:

From U,V,W to rotor d,q,0-system





8/65












cage






8/66

q-axis:

2 windings transformer

Magnetic coupling in d- and q-axis

d-axis:

3 windings transformer

Stator leakage inductance xdσ ≠ xqσ due to saliency slightly different

Here: Simplification: xdσ = xqσ = xsσ





8/67

ffdhDdhddhf

fdhDDdhddhD

fdhDdhdsdhd

ixxixix

ixixxix

ixixixx

⋅+++=

+⋅++=

++⋅+=

)(

)(

)(

σ

σ

σ

ψ

ψ

ψd-axis: 3 windings transformer

q-axis: 2 windings transformer

Equivalent circuits for magnetic coupling in d- and q-axis in per-unit

xsσ =

xsσ =


QQqhqqhQ

Qqhqsqhq

ixxix

ixixx

⋅++⋅=

⋅+⋅+=

)(

)(

σ

σ

ψ

ψ




8/68

Blondel stray coefficients 0 ≤ σ ≤ 1

Df

dhfD

fd

dhdf

Dd

dhdD

xx

x

xx

x

xx

x 222

1,1,1 −=−=−= σσσ

Qq

qhqQ

xx

x2

1−=σ

d-axis:

q-axis:

Describes degree of magnetic coupling between two windings.

σqQ

σdf

σdD

σfD


QQqqhQ

Qqhqqq

ixix

ixix

⋅+⋅=

⋅+⋅=

ψ

ψ




8/69

σ

σσ

σ

σ

fdh

fdhs

f

dh

f

fdhs

f

dhd

fd

dhddfdd

xx

xxx

x

x

x

xxx

x

xx

xx

xxxx

++=

=−+=−=

=

−⋅=⋅=′

22

2

1

(xdσ = xsσ)

(xqσ= xsσ)

σ

σσ

σ

σ

Qqh

Qqhs

Q

qh

Q

Qqhs

Q

qhq

Qq

qhqqQqq

xx

xxx

x

x

x

xxx

x

xx

xx

xxxx

++=

=−+=−=

=

−⋅=⋅=′′

22

2

1

Expression of dynamic inductances with BLONDEL coefficients





8/70

Electrically excited salient pole synchronous machine with damper cage: 1.0,1.0,2.0,15.0,6.0,2.1 ====== σσσσ QDfsqhdh xxxxxx

32.02.02.1

2.02.115.0

24.075.031.024.01.06.0

1.06.015.0

21.01.02.01.02.12.02.1

1.02.02.115.0

=+⋅

+=+

+=′

=⋅===+⋅

+=+

+=′′

=⋅+⋅+⋅

⋅⋅+=

+++=′′

σ

σσ

σ

σσ

σσσσ

σσσ

σ

fdh

fdhsd

qqQQqh

Qqhsq

DfDdhfdh

Dfdhsd

xx

xxxx

xxx

xxxx

xxxxxx

xxxxx

32.035.1238.0 =⋅=⋅=′ ddfd xx σ

35.115.02.1

75.015.06.0

=+=+=

=+=+==′

σ

σ

sdhd

sqhqq

xxx

xxxx

Example: Stator inductance per phase for steady state and dynamics

- Calculation via BLONDEL coefficients

=

⋅−=−= 31.0

7.075.0

6.011

22

Qq

qhqQ

xx

xσ

=

⋅−=−= 238.0

4.135.1

2.111

22

fd

dhdf

xx

xσ





8/71

Summary: Equivalent circuits for magnetic coupling in synchronous machines

- Three-winding transformer in d-axis yields three BLONDEL stray coefficients

- Two-winding transformer in q-axis yields one BLONDEL stray coefficient

- Subtransient q-axis and transient d-axis reactance:

May be calculated with BLONDEL coefficients

- BLONDEL coefficients are extensively used to write formulas shorter

- Here:

Only ONE main flux in d-axis, but in reality:

Main fluxes between d and f resp. d and D different (see: 8.9.3)





8/72












cage






8/73

- Electrical time constants of stator and rotor windings are much shorter than

mechanical time constant

- Operation at constant speed during electrical transient is assumed.

Examples: - sudden short circuits,

- electric load steps with only change of reactive power,

- switching during inverter operation

– 10 LINEAR DYNAMIC EQUATIONS: Laplace transformation may be used !

00000 ,,,, QDfqd ψψψψψ

Dynamic performance at constant speed operation

- Only electrical equations are needed, as n = const.

Initial conditions for dψ/dτ for flux linkages:

const.=mω





8/74

dmqqsqq

qmddsdd

siru

siru

ψωψψ

ψωψψ((((

((((

⋅+⋅+⋅=+

⋅−⋅+⋅=+

0

0

fffff

QQQQ

DDDD

siru

sir

sir

ψψ

ψψ

ψψ

(((

((

((

⋅+⋅=+

⋅+⋅=

⋅+⋅=

0

0

0

Linear dynamic equations in Laplace domain, n = const.

(1) Voltage equations: Unknowns:

⋅

=

⋅

=

Q

q

Qqh

qhq

Q

q

f

D

d

fdhdh

dhDdh

dhdhd

f

D

d

i

i

xx

xx

i

i

i

xxx

xxx

xxx

(

(

(

(

(

(

(

(

(

(

ψψ

ψψψ

(2) Flux linkage equations:

⋅

=

⋅

=

0

0

0

0

0

0

0

0

0

0

Q

q

Qqh

qhq

Q

q

f

D

d

fdhdh

dhDdh

dhdhd

f

D

d

i

i

xx

xx

i

i

i

xxx

xxx

xxx

ψψ

ψψψ

(3) Initial conditions:

fQDqdfQDqd iiiii ψψψψψ ,,,,,,,,,





8/75

Eliminating rotor quantities: Flux linkages & currents:

10 – 6 = 4 remaining stator equations!

fQDfQD iii ,,,,, ψψψ

s

isx

s

iisx

s

s

isx

s

i

r

usx

s

iisx

s

QQ

qqq

qq

DD

f

f

ff

ddd

dd

000

0000

)()(

)()()(

⋅−

−⋅=−

⋅−

−⋅+

−⋅=−

((

(((

ψψ

ψψ

⋅

=

⋅

=

Q

q

Qqh

qhq

Q

q

f

D

d

fdhdh

dhDdh

dhdhd

f

D

d

i

i

xx

xx

i

i

i

xxx

xxx

xxx

(

(

(

(

(

(

(

(

(

(

ψψ

ψψψ

fffff

QQQQ

DDDD

siru

sir

sir

ψψ

ψψ

ψψ

(((

((

((

⋅+⋅=+

⋅+⋅=

⋅+⋅=

0

0

0

“Reactance operators”





8/76

8.6.1 “Reactance operators”

( )( ) 1

)(2

2

++⋅+⋅⋅⋅

+⋅+⋅⋅⋅+⋅⋅⋅′′⋅=

DfDffD

dDdDfdfdDffDdd

ss

xxsxssx

ττττσ

τστσττσ

( ) dh

DfDffD

f

f

D xss

r

xs

sx ⋅++⋅+⋅⋅⋅

+⋅

=1

1

)(2 ττττσ

σ

( ) dh

DfDffD

D

D

f xss

r

xs

sx ⋅++⋅+⋅⋅⋅

+⋅=

1

1

)(2 ττττσ

σ

Q

QqQqq

s

s

xsx

τ

τσ1

1

)(

+

⋅+

⋅′′=

qh

Q

QQ x

s

sx ⋅+

=

τ

τ1

1

)(

Important: xd(s), xq(s)

Open-circuit time constants of damper and field winding:

f

ff

Q

QQ

D

DD

r

x

r

x

r

x=== τττ ,,





8/77

“Reactance operators”: )(),(),(),(),( sxsxsxsxsx fQDqd

−⋅=−

−⋅=−

s

iisx

s

s

iisx

s

qqq

qq

ddd

dd

00

00

)(

)(

((

((

ψψ

ψψ

Initial condition: Synchronous operation:

s

i

rs

u

r

u f

f

f

f

f 00 =⋅

=(

0,0 00 == QD ii

( )( ) 1

)(2

2

++⋅+⋅⋅⋅

+⋅+⋅⋅⋅+⋅⋅⋅′′⋅=

DfDffD

dDdDfdfdDffDdd

ss

xxsxssx

ττττσ

τστσττσ

Q

QqQqq

s

s

xsx

τ

τσ1

1

)(

+

⋅+

⋅′′=

At transient operation:

- Most important are “the reactance operators” xd(s), xq(s)!

- The others xf(s), xD(s), xQ(s) are only

needed, if iD0 ≠ 0, iQ0 ≠ 0, if0 ≠ uf0/rf

00 =−

s

i

r

u f

f

f(





8/78

Determination of “reactance operator” xq(s) (1)

QQQQ sir ψψ ((⋅+⋅=0

( )QQ

qqhQQQQqqhQQQ

xsr

ixsiixixsir

⋅+

⋅⋅−=⇒+⋅+⋅=

((((( 0

0

ψψ

QQ

Qqhq

QQ

qhq

QQ

qqhQqhqqQqhqqq

xsr

xi

xsr

xsx

xsr

ixsxixixix

⋅+

⋅+⋅

⋅+

⋅−=

⋅+

⋅⋅−⋅+=+= 0

20 ψψ

ψ(

(((((

QQ

QqQq

QQ

QqqQQq

QQ

qhQQqq

xsr

xxsrx

xsr

xxsrx

xsr

xsxsrxsx

⋅+

′′⋅+=

⋅+

⋅⋅+=

⋅+

⋅−⋅+=

σ2)()(

Voltage equation: Flux linkage equation:

QQqqhQ ixix((( +=ψ

Calculation of damper current:

Calculation of q-flux linkage:

Determination of reactance operator xq(s): xq(s)





8/79

By using we get with the abbreviation :

QQ

QqQqq

xsr

xxsrxsx

⋅+

′′⋅+=)(

q

Q

Qq

Q

Q

QqQ

Q

QQ

QqQqQ

q

q x

s

s

x

x

rs

x

rs

xsr

xxsrx

sx ′′⋅+

+

=′′⋅+

+

=⋅+

′′⋅+′′

=

τ

τσσ σ

1

1

)(

Determination of “reactance operator” xq(s) (2)

Reactance operator in a more convenient form:

s

isx

s

iisx

s

QQ

qqq

qq

000)()( ⋅−

−⋅=−

(( ψψ

qh

Q

QQ x

s

sx ⋅+

=

τ

τ1

1

)(000 QQqqhQ ixix +=ψ

)( QqQQ

QqQQ

r

xτσστ σ ⋅=⋅=





8/80

“Reactance operator of q-axis damper winding” xQ(s)


000 QQqqhQ ixix +=ψ 000 Qqhqqq ixix +=ψ

s

isx

sxsr

x

s

iisx

s

qq

q

QQ

Qqhqqq

qq

00000)()( ⋅+−

⋅+

⋅+

−⋅=−

ψψψψ

((

s

i

xsr

xsxsrx

xsr

xsr

s

ixix

xsr

ixixx q

QQ

qhQQq

QQ

QQQqhqq

QQ

QQqqhqh 02

0000 )()(⋅

⋅+

⋅−⋅+⋅+

⋅+

⋅+⋅

+−

⋅+

+⋅

QQ

qhQQqq

xsr

xsxsrxsx

⋅+

⋅−⋅+=

2)()(

QQ

qqhqqQqqQ

QqhQqqQQqhQqqQ

QQqhqqh

xsr

ixixxs

ixrixxixx

s

ixr

s

ixrixxix

⋅+

+++−−−−+ 02

00

0000

002

)(:00

sxs

i

xsr

xr

s

i

Q

Q

QQ

qhQQ⋅−=

⋅+

⋅⋅− qh

Q

Qqh

Q

Q

Q

Q

QQ

qhQQ x

s

x

sx

r

x

r

xsr

xrsx ⋅

+=⋅

+=

⋅+=

τ

τ1

1

)(




8/81

Reactance operators at short and long time scale

qqqddd isxisx(((( ⋅=⋅= )()( ψψ

Reactance operators are flux transfer functions !

0,0,0,0 0000 ==== qqdd ii ψψWe assume

Example: Current steps

sisxssiiL ddd /)()(/)( ⋅== ψ(

ixisxss dds

ds

d ⋅′′=⋅=⋅=∞→∞→

)(lim)(lim)0( ψψ (

For τ→ ∞ : ixisxss dds

ds

d ⋅=⋅=⋅=∞+→+→

)(lim)(lim)(00

ψψ (

q-axis:

For τ = 0:

id(τ), iq(τ)

id = i

τ

iq = i

ixisxss qqs

qs

q ⋅′′=⋅=⋅=∞→∞→

)(lim)(lim)0( ψψ (

For τ→ ∞ : ixisxss qqs

qs

q ⋅=⋅=⋅=∞+→+→

)(lim)(lim)(00

ψψ (


sisxssiiL qqq /)()(/)( ⋅== ψ(

d-axis:

For τ = 0:




8/82

8.6.2 Electrical rotor time constants for flux change in d- and q-axis

qqq

ddd

isx

isx((

((

⋅=

⋅=

)(

)(

ψ

ψ

- Usually voltages are given, defining the flux linkages.

- The currents have to be calculated !

)()(

)1

()1

(

)(21 dd

dddd

ss

ss

xsxααττ

+⋅+′′

+⋅′

+⋅′′=qqq

ddd

isx

isx((

((

=

=

)(/

)(/

ψ

ψ

ddd

d

DffDd

d

DffD

DdDfdfssRoots

x

x

x

xss

ττττσττσ

τστσ′′

−=′

−=⇒=′′

⋅+′′

⋅+

⋅+1

,1

:01

212

22112 ,:0

1dd

DffDDffD

DfssRootsss αα

ττσττσ

ττ−=−=⇒=+

+⋅+

Numerator

Denominator

Numerator:

Denominator:





8/83

Electrical rotor time constants for the d-axis

)1

()1

(

)()(1

)(

1 21

dd

dd

dd ss

ss

xsx

ττ

αα

′′+⋅

′+

+⋅+⋅′′

=

d

dd

d

dddd s

s

xxs

s

xxxsx

ττ ′′+

⋅

′

−′′

+

′+

⋅

−

′+=

1

11

1

111

)(

1

dd

d s

sC

s

sBA

sx

ττ ′′+

⋅+

′+

⋅+=

11)(

1

Determine A, B, C by comparison of numerators !

Result:

)()(

)1

()1

(

)(21 dd

dddd

ss

ss

xsxααττ

+⋅+′′

+⋅′

+⋅′′=

′′+⋅

′−+

′−′′

⋅′′=′

d

d

ddd

dddd

x

xxx

11

11

21 ταα

ττwhere:





8/84

q

qqqq

d

dd

d

dddd

s

s

xxxsx

s

s

xxs

s

xxxsx

τ

ττ

′′+

⋅

−′′

+=

′′+

⋅

′

−′′

+

′+

⋅

−

′+=

1

111

)(

1

1

11

1

111

)(

1

“Reactance operators” are flux transfer functions:

a) transient time constant dτ ′ in d-axis,

b) subtransient time constants qd ττ ′′′′ , in d- and q-axis.

Inverse “reactance operators” and electrical rotor time constants

- The stator current in d- and q-axis is changing with changing stator flux linkage with a

subtransient (very short) and a transient (short) time constant, caused by rotor winding

transients !

a) Open-circuit transient time constant fdd τατ ==′ 10 /1 in d-axis,

b) Open-circuit subtransient time constants Qqdd ττατ =′′=′′ 020 ,/1 in d- and q-axis.

- The stator flux linkage in d- and q-axis is changing with changing stator current with a

short and a long “open-circuit time constant, caused by these rotor winding transients !





8/85

q

q

q

q

q

Q

q

q

q

Q

Q

qq s

s

xs

s

xs

xs

s

xs

s

xsx

τττ

τ

ττ

τ

σ ′′+

⋅+

′′+

⋅−

′′+

⋅′′

+

′′+

⋅′′

=+

+

⋅′′

=1

1

1

1

1

1

1

1

1

1

1

1

)(

1

Calculation of inverse reactance operator 1/xq(s): Subtransient time constant of q-axis

q

qqqq s

s

xxxsx

τ ′′+

⋅

−′′

+=1

111

)(

1

qQ ττ σ ′′=

QqQ

q

QqQ

Q

qqQ

q

qqq s

s

xs

xs

s

xxsx

τστσ

τ

στ

1

1

1

1

1

1

11

)(

1

+⋅+

+⋅+

′′+

⋅

−′′

=

qx

1

0





8/86

Calculation of d-axis “short circuit time constants” τ´d, τ´´d

012 =

′′⋅+

′′⋅

+⋅+

d

d

DffDd

d

DffD

DdDfdf

x

x

x

xss

ττσττστστσ

)()(0 212 ssssqpss −⋅−==+⋅+ qppss −±−= 2

21 )2/(2/,

+⋅

⋅⋅⋅′′−⋅

⋅⋅⋅⋅′′

+⋅=′′′ 2)(

411

2

)(1,

1

DdDfdfd

DffDd

DffDd

DdDfdfd

dd x

x

x

x

τστσ

ττσττστστσ

ττm

−⋅=

′′′ 2

411

2

1,

1

p

qp

dd

mττ 21

1,

1ss

dd

−=′′

−=′ ττ





8/87

DfDd

dfd

fdf

fdfd

DfDd

DfDd

dfd

fdfd

DfDd

DfDd

dfd

DdDfdfd

DffDd

DffDd

DdDfdfd

dd

x

x

x

x

x

x

x

x

x

x

x

x

x

x

τσσ

τσ

τστσ

τσσ

τστσ

τσσ

τστσττσ

ττστστσ

ττ

⋅⋅′′⋅

≅⋅

=

=

⋅

⋅⋅′′−⋅

⋅⋅⋅′′⋅

≈

⋅

⋅⋅′′−⋅

⋅⋅⋅′′⋅

≈

≈

+⋅

⋅⋅⋅′′−⋅

⋅⋅⋅⋅′′+⋅

=′′′

,1

211

2

411

2

)(

411

2

)(1,

1

22

2

mm

m

Calculation of d-axis “short circuit time constants” for τD << τf

dfd

DfDdd

x

x

σ

τστ

⋅

⋅⋅′′≅′′b) Short circuit time constant of damper winding in d-axis =

= subtransient time constant of d-axis

(with for a << 1)2/11 aa −≅−a

fD ττ <<


fdfd τστ ⋅≅′a) Short circuit time constant of field winding =

= transient time constant of d-axis




8/88

d-axis: Sub-transient and transient time constants τ´d, τ´´d

dfd

DfDdd

x

x

σ

τστ

⋅

⋅⋅′′≅′′

fdfd τστ ⋅≅′

- Subtransient time constant of d-axis:

- Transient time constant of d-axis:

Example:

σfD = σdf = 0.1, xd = 1, x”d = 0.15, τf = 10τD

DD

dfd

DfDdd

x

xτ

τσ

τστ ⋅=

⋅⋅⋅

=⋅

⋅⋅′′=′′ 15.0

1.01

1.015.0

DDfdfd τττστ =⋅⋅=⋅=′ 101.0

If open-circuit time constant of the field winding τf is much bigger than of the

damper winding τf >>τD, then also transient time constant is much bigger than

subtransient time constant:

715.0/1/ ==′′′ dd ττ

Exact values:

Dd

Dd

ττττ⋅=′⋅=′′

94.0

16.0

dd ττ ′′>>′





8/89

q-axis: Sub-transient time constant τ´´q

Subtransient time constants are much shorter than the transient time constant !

Q

QqQQQqQq

r

x⋅==⋅=′′σ

ττστ σ- Subtransient time constant of q-axis

Example: σqQ = 0.1, τQ = τD DQqQq ττστ ⋅=⋅=′′ 1.0

Dd ττ ⋅=′′ 15.0Result: Dq ττ ⋅=′′ 1.0

7/11/15.0/ ==′′′ dd ττ 10/11/1.0/ ==′′′ dq ττ





8/90

d-axis open circuit damper time constant of damper much shorter than of field winding:

fD ττ <<

⋅−

⋅⋅≈

+

⋅⋅−

⋅⋅⋅

+=

−⋅=

f

DfD

DfDDf

DffD

DffD

Dfdd

p

qp

τ

τσ

τσττ

ττσ

ττσ

τταα

411

2

1

)(

411

2

411

2,

2221 mmm

With 2/11 aa −≅− for a << 1:

DfDff

DfD

DfDf

DfD

DfDd τσττ

τστστ

τστσ

α⋅

≅=

⋅−

⋅⋅≈

⋅−

⋅⋅≈

1,

1211

2

1411

2

12,1 mm .

a) Open circuit time constant of field (= field winding time constant) 10 /1 dfd αττ ==′

b) Open circuit time constant of damper in d-axis 20 /1 dDfDd ατστ =⋅=′′

Calculation of “open circuit” d-axis time constants for τD << τf

0)()(1

2122 =+⋅+=+⋅+=+

+⋅+ dd

DffDDffD

Dfssqpssss αα

ττσττσ

ττ

a





8/91

Example: Rotor time constants for d-axis

02.0,002.0,1.0,2.0,15.0,2.1 ====== DfDfsdh rrxxxx σσσ

3.11.02.1,4.12.02.1,35.115.02.1,21.0 =+==+==+==′′ Dfdd xxxx

179.0)3.135.1/(2.11,209.0)4.13.1/(2.11,238.0)4.135.1/(2.11 222 =⋅−==⋅−==⋅−= dDDfdf σσσ

Given:

6502.0/3.1,700002.0/4.1 ==== Df ττ

59.1365209.0,879.8238.035.1

65209.021.00 =⋅≅′′=

⋅⋅⋅

≅′′ dd ττ

12032.0/00619.0)65179.0700238.0(35.1

65700209.021.0411

65700209.0221.0

)65179.0700238.0(35.11,

12=

⋅+⋅⋅

⋅⋅⋅⋅−⋅

⋅⋅⋅⋅⋅+⋅⋅

=′′′

m

dd ττ

700,6.166700238.0 0 =≅′=⋅≅′ fdd τττA) For τf >>τD:

B) For arbitrary τf ,τD:

64.12,31.8,4.752,5.161 00 =′′=′′=′=′ dddd ττττ

07091.0/00133.0)65700(

65700209.0411

65700209.02

657001,

12

00

=

+

⋅⋅⋅−⋅

⋅⋅⋅+

=′′′

m

dd ττ

dτ ′0dτ ′

dτ ′′0dτ ′′

A) B)4.7525.161

64.1231.8

0.7006.166

59.1388.8


For τf >> τD: Simple formulas A) sufficient !




8/92

Example: Rotor time constants for q-axis


75.015.06.0,236.075.0314.0 =+==⋅=′′ qq xx

314.0)7.075.0/(6.01 2 =⋅−=qQσ

Given:

75.808.0/7.0 ==Qτ

For arbitrary τQ:

75.275.8314.0,75.80 =⋅=′′==′′ qQq τττ

6.0=qhx 1.0=σQx 08.0=Qr

75.200 , qd ττ ′′′′ 75.864.12

qd ττ ′′′′ , 31.8


In the q-axis the damper current iQ vanishes faster than the damper current iD in the d-axis !




8/93

Proof: Transient inductance of stator winding in d-axis may

be described by a simple equivalent circuit !dx′

Due to τf >> τD the transient inductance may be easily calculated from an equivalent circuit, where transient current in damper winding has already vanished

(xdσ = xsσ)

σ

σσ

σσ

fdh

fdhs

f

dh

f

fdhs

f

dhd

fd

dhddfdd

xx

xxx

x

x

x

xxx

x

xx

xx

xxxx

++=

=−+=−=

−⋅=⋅=′

222

1


σ

σσσ

τσ

τσσ

τστστ

τστσσ

ταα

ττ

fdh

fdhsdfd

DfD

DfDd

dfd

d

d

d

fdfDfDf

fdfDfDd

dfd

d

d

d

ddd

dddd

xx

xxxx

x

x

x

x

x

x

x

x

x

xxx

++=⋅=

=⋅⋅′′

⋅

⋅′′≈

′′+⋅

⋅−+

⋅−

⋅⋅′′

⋅

⋅′′≈

′′+⋅

′−+

′−′′

⋅′′=′1

1111

1

11

11

21

fD ττ <<fD ττ <<




8/94

Example: Transient reactance x´d


35.115.02.1,21.0 =+==′′ dd xx 179.0,209.0,238.0 === dDDfdf σσσ

Given:

3213.035.1238.0 =⋅≅′dx

3683.0

35.1

21.01

5.161

107091.0001329.0

5.161

1

31.8

1

21.0 =

+⋅−+

−⋅=′dx

A) For τf >>τD:

B) For arbitrary τf ,τD:

A) B)

dx′ 3213.0 3683.0


Due to τf >> τD the transient inductance may be calculatedfrom an equivalent circuit with sufficient accuracy!




8/95

8.6.3 Electric excitation: Dynamic equations for constant speed

Electrically excited synchronous machine

- Field winding

- Damper cage

- Saliencys

isx

s

iisx

s

s

isx

s

i

r

usx

s

iisx

s

siru

siru

QQ

qqq

qq

DD

f

f

ff

ddd

dd

dmqqsqq

qmddsdd

000

0000

0

0

)()(

)()()(

⋅−

−⋅=−

⋅−

−⋅+

−⋅=−

⋅+⋅+⋅=+

⋅−⋅+⋅=+

((

(((

((((

((((

ψψ

ψψ

ψωψψ

ψωψψ

000 ,, fQD iii,, 00 qd ψψInitial conditions:

,, 00 qd ii

qdqd ii ψψ ,,,4 unknowns:

fqd uuu ,,3 given values:

q

qqqq s

s

xxxsx

τ ′′+

⋅

−′′

+=1

111

)(

1

”Reactance operators” for

d

dd

d

dddd s

s

xxs

s

xxxsx

ττ ′′+

⋅

′

−′′

+

′+

⋅

−

′+=

1

11

1

111

)(

1

fffQD ruiii /,0 000 ===





8/96

d

dd

d

dddd s

s

xxs

s

xxxsx

ττ ′′+

⋅

′

−′′

+

′+

⋅

−

′+=

1

11

1

111

)(

1

5 reactance operators for electrically excited synchronous machines with damper cage

q

qqqq s

s

xxxsx

τ ′′+

⋅

−′′

+=1

111

)(

1Reactance operators:

( )( ) 1

)(2

2

++⋅+⋅⋅⋅

+⋅+⋅⋅⋅+⋅⋅⋅′′⋅=

DfDffD

dDdDfdfdDffDdd

ss

xxsxssx

ττττσ

τστσττσ

( ) dh

DfDffD

f

f

D xss

r

xs

sx ⋅++⋅+⋅⋅⋅

+⋅

=1

1

)(2 ττττσ

σ

( ) dh

DfDffD

D

D

f xss

r

xs

sx ⋅++⋅+⋅⋅⋅

+⋅=

1

1

)(2 ττττσ

σ

Q

QqQqq

s

s

xsx

τ

τσ1

1

)(

+

⋅+

⋅′′=

qh

Q

QQ x

s

sx ⋅+

=

τ

τ1

1

)(





8/97

8.6.4 Permanent magnets: Dynamic equations for constant speed

Permanent magnet excited synchronous machine

- No field winding

- Often no damper cage

- Often no saliency

- Permanent rotor flux linkage due to permanent magnets

“Reactance operators”:

(if no damper cage)

0000

0

0

qqqpddd

qqqdd

p

d

dmqqsqq

qmddsdd

ixix

ixixs

siru

siru

⋅=+⋅=

⋅=⋅=−

⋅+⋅+⋅=+

⋅−⋅+⋅=+

ψψψ

ψψ

ψ

ψωψψ

ψωψψ

((((

((((

((((

dddd xxxsx =′=′′=)(

qqq xxsx =′′=)(

dq xx =

00 fdhfp ix ⋅=⇔ψψ

00 , qd ii,, 00 qd ψψInitial conditions:

qdqd ii ψψ ,,,4 unknowns:pqd uu ψ,,3 given values:





8/98

Gearless permanent magnet wind generator Scanwind / Norway 3 MW, 17/min

Source:

Siemens AG

Germany

Wind rotor diameter 90 m

Three-blade rotor

Pitch control

Variable speed operation

10 ... 20/min

Gearless drive

IGBT inverter 690 V





8/99

“Multibrid” - permanent magnet wind generator – dual stage gear

Source:

Multibrid,

Germany





8/100

Rotor of first built Permanent Magnet Synchronous Motor for Submarine propulsion

Source:

Siemens AG, Germany

Skewed rotor magnets to

reduce cogging torque

Cooling fins to dissipate

rotor magnet eddy current

losses and to enhance

internal air flow

High pole count (32 poles)

for slow speed operation





8/101

Summary: Transient performance of synchronous machines at constant speed operation

- At constant speed and constant iron saturation:

Linear voltage and flux linkage equations

- Eliminating of rotor flux linkages lead to “reactance operators” as abbreviation

- Five reactance operators, but only two (stator d- and q-axis) mainly of interest

- „Reactance operators“ give the dynamic change of e.g. stator inductances from

the small subtransient to the big steady state values

- In PM machines:

Reactance operators are simply the steady-state inductances

(if eddy currents in the rotor magnets are negligible)

- In voltage-fed machines the inverse reactance operators are needed


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