01 J Pyrhonen - Synch Machn

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Electrical Drives Juha Pyrhönen, LUT, Department of Electrical Engineering 7.1

7. SYNCHRONOUS MACHINE .................................................................................................... 1

7.1 Synchronous Machines Models ........................................................................................... 4

7.2 Equivalent Circuits and Machine Parameters of a Synchronous Machine ........................ 11

7.3 Measurement of Motor Parameters by an Electrical Drive (DTC Drive) .......................... 16

7.4 Finite Element Method (FEM) for Determining the Inductances of a Synchronous

Machine.......................................................................................................................................... 20

7.5 The Relationship between the Stator Excitation and the Rotor Excitation of aSynchronous Machine .................................................................................................................... 25

7.5.1 Non-Salient Pole Machine ......................................................................................... 25

7.5.2 Salient-Pole Machine ................................................................................................. 26

7.5.3 Referring Factor ......................................................................................................... 30

7.5.4 Referring (to the Stator) When Applying Space Vectors .......................................... 31

7.6 Vector Diagram of a Synchronous Machine ...................................................................... 32

7.7 Torque Production of a Synchronous Machine ................................................................. 37

7.8 Simulation of an Electrically Excited Salient-Pole Machine by Constant Parameters ...... 38

7.9 Current Equations of a Synchronous Machine .................................................................. 40

7.10 Simulation of a Synchronous Machine in a Discrete-Time System .................................. 41

7.11 Implementation of the Vector Control of a Synchronous Machine ................................... 42

7. SYNCHRONOUS MACHINE

The subject of rotating-field machines is approached by starting from synchronous machines, as

they represent the most versatile machine type and furthermore, they share basically all the charac-

teristics of asynchronous machines. For instance the equivalent circuits determined for synchronous

machines are only simplified when we change over to the analysis of asynchronous machines.

Nowadays, there are several types of synchronous machines on the market, and their role as drive

motors in industrial applications is constantly increasing. New applications are emerging both for

permanent magnet synchronous machines and synchronous reluctance (SR) machines.

A poly-phase synchronous machine is a rotating-field machine, in which the rotor, that is, the pole

wheel, rotates synchronously with the rotating magnetic field generated by the armature winding of

the machine when the machine is in a steady state. The stator of the synchronous machine is com-

posed of a laminated stack comprised of electric sheets and mounted to a steel frame. In the stator

stack, there are slots for the stator winding, which is usually a three-phase construction. The rotor of

the machine can be implemented in various ways: either as a cylindrical non-salient pole rotor, as a

salient pole rotor with separate magnetic poles on the rotor axis, as a reluctance rotor, or as a per-

manent magnet rotor with magnetic poles generated by permanent magnets. The rotors can be im-

plemented either as laminated constructions or as solid ones. In the non-salient pole machines and

salient pole machines, the field current (rotor excitation) is conducted to the rotor either throughslip rings or brushes, or there is a separate excitation generator mounted on the shaft of the machine

to produce the field current. The latter case represents the so-called brushless synchronous machine.

In the permanent magnet machine, excitation is based on permanent magnets, and therefore excita-

tion cannot be controlled similarly as in the two other cases. The final excitation state is determined

by the magnitude of the supply voltage; the flux of the machine can be influenced by armature reac-

tion, that is, by magnetizing the machine with stator current.

In the non-salient pole machine, there are slots in the rotor for the field winding, similarly as there

are slots in the stator for the stator winding. The field winding of the salient-pole machine is wound

around the iron core of the magnetic poles. This assembly is mounted on the rotor shaft. On the

outer surface of the magnetic poles, there may be a damper winding, constructed as a cage winding.

It comprises bars on the surface of the pole shoe, the bars being connected together at both ends by

short-circuit rings.




Damper windings are used in synchronous machines to improve the stability of he drive. Direct-on-

line machines in particular should be equipped with damping, however, damper windings are useful

also in controlled synchronous machine drives. In speed-controlled drives, it is possible to reach

stability also without damping. Due to the effect of the damper windings, the flux linkage of the

machine changes more slowly, whereas on the other hand, the stator current changes faster than in a

machine without damper windings. Since the stator current can be adjusted in a short time, also thetorque of the machine can be altered rapidly, and thus the dynamics of the machine is improved

with damping.

The structure and vector diagram of a salient pole machine is illustrated in Figure 7.1.

Figure 7.1 Salient pole synchronous machine and its vector

diagram. Stator voltage us, stator current is, field current iF,sum current isum, stator flux linkage s, air gap flux linkage

m. It is typical of a salient pole machine that the directions

of the vector sum of currents and the air gap flux linkage

deviate from each other in such a way that the flux turns

towards the d-axis.

s

m

Lsis

is

us

iF

s

isum

Synchronous machines constitute a significant family of electrical machines among AC machines.

Different machine types can be categorized for instance according to the schematic diagram illus-

trated in Figure 7.1.SYNCHRONOUS MACHINES

SEPARATELY

MAGNETIZED

SYNCHRONOUSMACHINES SM

SYNCHRONOUSRELUCTANCE

MACHINESSyRM

PERMANENT

MAGNET SYNCHR.

MACHINES PMSM

SyRM Ld/ Lq

> 10

SyRM, permanentmagnets

Ld/ Lq

> 10

PMSM,

withsaliency

PMSM,

non-salient

Lq/ Ld

> 1 Ld/ Lq = 1

PMSMembeddedmagnets

PMSM

pole shoes

Salient pole m.

Ld/ Lq

> 1

Non-salient pole m. Ld/ Lq = 1

Brushless:-ext. pole SM

(+PM-gen)

-axial

transformer

-radial

transformer

-rotating fieldmagnetizing

With brushes-rotating

magnetizingmachine-thyristor bridge

With brushes-rotatingmagnetizing

machine

-thyristor bridge

Brushless:

-ext. pole SM(+PM-gen)

-axialtransformer

-radial

transformer

-rotating field

magnetizing

Figure 7.1 Synchronous machine family

Slow drives are often salient-pole configurations, and thus the machine is magnetically asymmetric.A Finnish tradition has been to construct salient pole machines both for motor and generator use.

The cooling of a salient pole machine is easier to manage than the cooling of a non-salient pole ma-




chine, since there is plenty of room for the circulation of the cooling air. The implementation of a

salient pole construction is also technically reasonable. The salient-pole configuration leads to

magnetic asymmetry, and consequently, to considerable problems in modelling the machine. The

problems in modelling are one reason why for instance Siemens manufactures non-salient pole ma-

chines also for slow drives. Therefore, the purpose of use is no longer a relevant basis to categorize

the machines into salient pole and non-salient pole ones. However, high-speed synchronous ma-

chines are typically non-salient pole configurations, generally implemented with solid rotors. If such a rotor does not include a separate damper winding, problems arise in modelling again, when

we try to model the damping effects of the eddy currents in the solid rotor frame.

In steam power plants, the machine powers may reach up to 1500 MW. The rotation speed of such

machines is often 3000 or 1500 min-1, and the machines are typically non-salient pole constructions.

In that case, a solid rotor is employed, which, even as a long and slim construction, is able to with-

stand a high rotation speed without the critical speeds being exceeded. Excitation is often imple-

mented by a brushless solution. However, when aiming at good dynamics, constructions with

brushes are selected (e.g. the Loviisa nuclear power plant).

Hydropower machines are typically salient-pole constructions. The world’s largest hydropower sta-

tion is located in Itaipu, upon the Paraná River on the border between Brazil and Paraguay. The to-

tal capacity is 14000 MW divided between 20 generating units of 700,000 kW each. There are 715

MW Francis turbines in the power plant, the water intake of one single turbine being 700 m³/s. The

height of the dam reaches 196 m. The weighted efficiency of such a large turbine is 93.8 %. The

generator efficiency of the salient-pole machines is 98.6 %. The rotor diameter is 16 m and the ro-

tating mass is 2650 t. There are 66 poles in the 50 Hz type generators, and the rated output is 823.6

MVA (90.9 min-1); the power factor is 0.85. The corresponding technical data of 60 Hz type genera-

tors are: 78 poles, 737 MVA (92.3 min-1) 0.95. The output voltage of the machines is 18 kV.

The powers of diesel generators range from a few megawatts to a few dozen megawatts. The ma-chines manufactured by ABB are salient-pole machines with 4 ... 8 ... 12 poles (the maximum speed

of a large machine being for instance 750 min-1, 8 poles, 50 Hz).

Synchronous machines may be either direct-on-line machines or converter-fed ones. Slow-speed

machines are often salient-pole constructions; however, Siemens manufactures non-salient pole ma-

chines for instance for rolling mill drives. The applications vary from wood grinders to pumps and

blowers. High-speed machines are typically non-salient pole machines; for example natural gas is

pumped from Norway into the European markets by high-speed LCI-fed ca. 40 MW, 4000 min-1

non-salient pole synchronous machines. The largest power electronics-fed synchronous motor drive

is probably the 101 MW LCI drive for a wind tunnel installation of NASA, supplied by ABB.

Considering the motor characteristics, the damper winding is of central importance. Direct-on-line

operation is impossible without a damper winding: a torque disturbance would make an undamped

machine to oscillate like a spring in the grid. However, in many vector controlled converter drives a

damper winding is not required, since the state of the machine can be controlled adequately also

without damping. Nevertheless, the machine’s ability to respond to fast torque steps is improved by

a damper winding.

If the damper winding has to be dimensioned to have a very high resistivity due to the direct-on-line

starting, the machine’s properties in synchronous operation may be impaired correspondingly.

Therefore, a synchronous motor is often started up with an auxiliary drive.




As shown in Figure 7.1, there is a wide selection of synchronous machines; however, the same fun-

damental theory holds for them all, and therefore, only the most versatile example of synchronous

machines, that is, the salient pole machine, is addressed in this material. The equations presented

for salient pole synchronous machines hold for other machine types as well. When analyzing differ-

ent machine types, appropriate terms can be omitted from the equations to achieve the machine-

specific results. In the case of a permanent magnet synchronous machine, the field current of the

rotor can, if desired, be expressed as a virtual field current source.

7.1 Synchronous Machines Models

Nowadays, a two-axis model is employed for the synchronous machine; this model can be derived

by applying space vector theory. First, we start with the frames of reference required for the analy-

sis of the model, Figure 7.2. In the figure, the windings are illustrated as concentrated, in other

words, the real winding has been replaced by a bar-shaped equivalent winding depicted on the

magnetic axis of the real winding. The magnetic axes of the stator phase windings a, b, and c are

fixed to the respective phase windings. The stator reference frame is fixed in the direction of the

phase winding a and in the direction perpendicular to the phase winding. The axes of this two-phase

stator reference frame are denoted x and y. The rotor reference frame is fixed aligned with the

magnetic pole of the rotor, and in the direction perpendicular to the magnetic pole. The axes of this

two-phase stator reference frame are denoted d and q. The rotation angle between the rotor and sta-

tor reference frame is equal to the rotor electric angle r .

qd

x

y

a

c

b

r

Figure 7.2 Frames of reference related to a synchronous machine: a, b, and c indicate the directions of the magnetic

axes of the phase windings of a three-phase stator. The xy reference frame is a two-phase reference frame, the axes of

which are fixed in the direction of the stator phase winding a and perpendicular to it. The dq reference frame is a two-

phase reference frame fixed on the rotor, the axes being in the direction of the magnetic pole and perpendicular to it.

The angle between the xy and dq frames of reference is equal to the rotor position angle r .

The core idea of the current vector is that the winding current is determined as a vector parallel to

the magnetization axis of the winding. Figure 7.3 depicts the positive directions of the magnetiza-

tion axes and the current vectors of different windings. In a three-phase machine, there are angles of

120 electric degrees between the magnetic axes. The current vector is constructed by geometrically

summing up the phasors of different phases, as shown in the previous chapter. The selected positivecurrent and voltage directions are valid for motor operation.




a-phase

b-phase

c-phase

120°

120°

positive and negative direction

of the magnetic axis of the

winding

120°

isb

isa

usb

usausc

isc

Figure 7.3 The directions of the magnetic axes of different phase windings of an AC machine, when the windings are

illustrated as concentrated, as well as the directions of the current vectors of different phase windings determined on the

basis of these aforementioned directions. isa, isb, and isc are the stator currents and usa, usb, and usc are the stator voltages.

In a three-phase machine, there are angles of 120 electric degrees between the magnetic axes of different phases.

Now we are ready the introduce also the flux linkage reference frame fixed to the air gap flux link-

age vector; Figure 7.4. The axes of the reference frame are the flux linkage axis and the torque

axis T.

q

d

m

r

T

x

y

m

m

Figure 7.4 The dq reference frame fixed to the rotor, the T reference frame fixed to the air gap flux linkage, and the

xy reference frame fixed to the stator. The angle between the dq and T reference frames is the pole angle m of the air

gap flux linkage. The angle between the dq and ab reference frames is the rotor position angle r . The angle m between

the T and xy reference frames is the position angle of the air gap flux linkage vector in the xy reference frame.

In the case of a synchronous machine, it is extremely important to operate in a reference frame

fixed to the rotor. This is illustrated by Figure 7.5, which shows the behaviour of the measuredmagnetizing inductance depending on the rotor position. If we stick to a fixed stator reference

frame, the magnetizing inductance will vary, which in turn complicates the determination of the




equations considerably. The term d Lm/dt has to be taken into account in the voltage equation of the

stator.

Lm

Lmd

Lmq

rotor position with respect to stator

d q

Figure 7.5 The behaviour of the magnetizing inductance as a function of the rotor position. If the inductance is meas-

ured at the direct axis, a direct magnetizing inductance Lmd is obtained, and respectively at the quadrature axis, the

quadrature magnetizing inductance Lmq.

We avoid the modelling of the inductance variation as a function of rotor position when we change

over to the rotor reference frame. Equation (7.1) is represented in a stationary reference frame fixedto the stator. When computing the operation of the synchronous machine, it is advisable to select a

reference frame in which the equations can be expressed as simplified as possible. When transform-

ing a vector initially represented in the stator reference frame into the dq reference frame that ro-

tates fixed to the rotor, or into the T flux linkage reference frame fixed to the air gap flux linkage

vector, the vector has in both cases to be turned by the position angle of the rotating reference

frame. Figure 7.6 depicts the varying vector components when performing coordinate transforma-

tion. The reference frames employed here are the rectangular xy reference frame fixed to the stator,

shortly the stator reference frame, and the rectangular dq reference frame fixed to the rotor, that is,

the rotor reference frame.

q

d

x

y

r

isisy

isx

isd

isq

Figure 7.6 The components of the current vector in different reference frames. The xy reference frame is a stationary

reference frame fixed to the stator, i.e., the stator reference frame. The dq reference frame, i.e., the rotor reference

frame is fixed to the rotor; the d-axis of the reference frame is aligned with the magnetic pole of the rotor. is is the stator

current vector, isx and isy are its components in the stator reference frame, and isd and isq are the components in the rotor

reference frame. r is the rotor position angle, 1 are 2 the angles of the stator current vector in the stator and rotor

reference frames.

The voltage equation of the stator of the machine in its own frame of reference is recapitulated here




u is

s

s s

s s

sd

d R

t

. (7.1)

The stator current vector

is

s je 1 i , (7.2)

where i is the magnitude of the vector, that is, its length, and α1 is its angle in the stator reference

frame. Since the rotor of a synchronous machine is both magnetically and electrically asymmetric,

it is advisable to represent the equations in the rotor reference frame. First, the stator current is

transferred to the rotor reference frame

is

r j je e2 1 r

i i . (7.3)

Here 2 is the vector angle in the rotor reference frame and r is the rotor position angle in the stator

reference frame. The superscript r refers to the rotor reference frame.

i is

r

s

s je r , (7.4)

i is

s

s

r je r . (7.5)

To transform Eq. (7.1) into the rotor reference frame, we have thus to make the substitutions

i i u us

s

s

r j

s

s

s

r j= e , = er r and s

s

s

r je r . Consequently, we obtain

u is

r j

s s

r j s

r j

e ed( e

dr r

r

Rt

), (7.6)

r r r r jr

sr j

r

s jr

ss

jr

s ed

d je

d

dee

t t R iu . (7.7)

Finally, both sides are divided by the term e j r , which yields

u is

r

s s

r s

r

r

s

r d

d j

d

d R

t t

. (7.8)

Now we may conclude that the first derivative term is the voltage generated by the change in the

magnitude of the flux linkage, in other words, the induction voltage, and the latter is the rotatingvoltage caused by rotation.

The vector model according to Eq. (7.8) is a complex single-axis model, and therefore it is not ca-

pable of easily taking into account the magnetic asymmetry of the salient-pole machine. Therefore,

it is advisable to divide the quantities into two components on the magnetic axes of the machine;

this model is known as the two-axis model. The model is represented in the rotor reference frame,

since the inductance parameters of the flux linkage equations are there not dependent on the rotor

position angle. The same result is obtained also by investigating the structure of the machine. Fig-

ures 7.7−7.9 illustrate different cases, in which all the windings are concentrated. In other words, an

equivalent winding is depicted on the magnetic axis, the effect of which is the same as the effect of

the real winding, since the magnetic axes of the windings are now clearly detectable.




Initially, there is an ordinary three-phase winding in the stator, Figure 7.7. The direction of the

magnetic pole of the rotor, or the direct direction, is called the d-axis. The direction perpendicular

to the d-axis is known as the quadrature direction, or the q-axis. The field winding magnetizes the

magnetic circuit in the direction of the d-axis. The damper winding is illustrated by the two short-

circuited equivalent windings, one of which magnetizes the machine together with the field winding

in the d-direction, and the other in the q-direction. The equivalent damper windings are denoted D-

and Q-damper windings. The angle r between the d-axis of the rotor and the direction of the a- phase winding is the rotor angle with respect to the stator.

Figure 7.7 Representation of the synchronous ma-

chine, in which the three-phase stator winding is

illustrated by three concentrated phase windings. In

the rotor, there is a field winding and two equivalent

damper windings. isa, isb, and isc are the stator cur-

rents, and usa, usb, and usc are the stator voltages. iD and iQ are the damper winding currents. iF and uF are

the field current and voltage. r is the rotor position

angle. The abc axes and the xy reference frames are

fixed to the stator, and the dq reference frame is

fixed to the rotor.

iQ

iD r

qd

x

y

isb

a

c

b

usb

usa

usc

isa

isc

iF

uF

The rotating magnetic field generated by the three-phase stator winding can also be created by a

two-phase winding, in which the magnetic axes of the windings are perpendicular to each other, asshown in Figure 7.8. The windings of the three-phase winding and the coordinate axes in the direc-

tions of their magnetic axes are denoted a, b, and c. In the two-phase case, the symbols x and y are

employed.

Figure 7.8 The representation of the synchronous

machine, in which the three-phase winding of the

stator is replaced by a two-phase stationary winding.

In the rotor, there are a field winding and two

equivalent damper windings. isx are isy the stator cur-

rents and usx and usy are the stator voltages. iD and iQ

are the damper winding currents. iF and uF are the

field current and voltage. r is the rotor position an-

gle.

qd

x

y

usx

usy

isx

isy

uF

r iQ

iD

When the rotor rotates, the magnetic connection between the stator and rotor windings changes,

which means in practice that the inductance coefficients in the flux linkage equations depend on the




rotor angle r . To eliminate this dependence on the rotor angle, the two-phase winding fixed to the

stator is replaced by an fictitious winding rotating with the rotor, as shown in Figure 7.9. The d-

direction of this winding is congruent with the d-axis of the rotor, and the direction of the q-winding

is perpendicular to this direction, that is, aligned with the q-axis. There is another advantage with

the dq reference frame rotating along with the rotor: as the rotor rotates at the same speed with the

magnetic field in the steady state, the vectors remain stationary in the dq reference frame, whereas

in the xy reference frame, the vectors rotate at the synchronous speed.

iF

iQ

uF

iD

r

isd

usd

qd

x

y

usqisq

Figure 7.9 Representation of the synchronous machine, in which the three-phase winding has been replaced by a two-

phase rotating winding. In the rotor, there are a field winding and two equivalent damper windings. isd and isq are the

stator currents and usd and usq are the stator voltages. iD and iQ are the damper winding currents. iF and uF are the fieldcurrent and voltage. r is the rotor position angle.

The following symbols are employed for the resistances and inductances represented in the rotor

reference frame and referred to the stator:

- Ld direct synchronous inductance

- Lq quadrature synchronous inductance

- Lmd direct magnetizing inductance

- Lmq quadrature magnetizing inductance

- Ls stator leakage inductance

- LF total inductance of the field winding

- LF leakage inductance of the field winding

- LdF mutual inductance between the stator equivalent

winding on the d-axis and the field winding (in practice Lmd)

- LdD mutual inductance between the stator equivalent

winding on the d-axis and the direct equivalent damper winding

- LqQ mutual inductance between the stator equivalent

winding on the q-axis and the quadrature equivalent

damper winding (in practice Lmq)

- LD total inductance of the direct damper winding

- LD leakage inductance of the direct damper winding- LQ total inductance of the quadrature damper winding

- LQ leakage inductance of the quadrature damper winding




- Rs stator resistance

- RF resistance of the field winding

- RD resistance of the direct damper winding

- RQ resistance of the quadrature damper winding

Sometimes, the somewhat artificial Canay inductance is employed; this inductance may obtain even

negative values. It has thus to be interpreted as a corrective factor in the model, which can in manycases be omitted altogether.

- Lk mutual leakage inductance between the field winding and

the direct damper winding, i.e., the Canay inductance

Next, the same situation is approached by using the vector model. We obtained previously for the

voltage equation of the synchronous machine in the rotor reference frame

u is

r

s s

r s

r

r

s

r d

d

jd

d

R

t t

. (7.9)

The current, voltage, and flux linkage vectors are decomposed into their real and imaginary parts on

the d- and q-axes of the rotor reference frame. We obtain

us

r

d q s

r

d q s

r

d q j j j u u i i i; ; ; (7.10)

The equations for the real and imaginary parts of the voltage equation

u R i

t d s d

d

q

d

d

, (7.11)

u R it q s q

q

d

d

d . (7.12)

The voltage equations of the rotor circuits referred to the stator become

u R it F F F

Fd

d , (7.13)

0 R it D D

Dd

d, (7.14)

0 R it Q Q

Qdd

. (7.15)

The inductances of the synchronous machine model are determined in the rotor reference frame:

Ld = Lmd + Ls , (7.16)

Lq = Lmq + Ls , (7.17)

LF = Lmd + LF + Lk , (7.18)

LD = Lmd + Lk + LD , (7.19)

LQ = Lmq + LQ . (7.20)

In the literature, the following equations are given for stator flux linkages and other flux linkages;

the equations are written by using the inductances and currents referred to the stator.




d d d dF F dD D L i L i L i , (7.21)

q q q qQ Q L i L i , (7.22)

F dF d F F FD D L i L i L i , (7.23)

D dD d FD F D D L i L i L i , (7.24)

Q qQ q Q Q L i L i . (7.25)

In the traditional two-axis model, it is assumed that the stator circuit (i.e., the armature circuit), the

damper windings, and the field winding are magnetically interconnected only through the magnetiz-

ing inductances Lmd and Lmq. However, the measurements have shown that in the transients, the al-

ternating component of the field current may be triple to the calculated value. Therefore, the pa-

rameter Lk known as the Canay inductance is added to the model, since the traditional two-axis

model describes only the armature circuit correctly. The Canay inductance takes into account the

deviation of the magnetic connection of the damper winding and the field winding from the direct

magnetizing inductance. Thus we obtain for the different mutual inductances:

LdD = LdF = Lmd , (7.26)

LFD = Lmd + Lk , (7.27)

LqQ = Lmq . (7.28)

Based on the above assumptions, the equations of the flux linkages can be expressed in the rotor

reference frame in the following form:

The stator flux linkage:

d md d F D s d L i i i L i( ) , (7.29)

q mq q Q s q L i i L i( ) . (7.30)

The field winding flux linkage:

F md d F F md k D L i L i L L i( ) . (7.31)

Damper flux linkages:

D md d md k F D D L i L L i L i( ) . (7.32)

Q mq q Q Q L i L i . (7.33)

7.2 Equivalent Circuits and Machine Parameters of a Synchronous Machine

The equivalent circuits of Figures 7.10−7.11 can be represented for a synchronous machine in the

rotor reference frame, since there the inductance coefficients of the flux linkage equations no longer

depend on the rotor position, and thus, the coefficients are constants. The equivalent circuits are

given separately for the d- and q-directions, since the salient pole machine is magnetically asym-

metric. Although the non-salient pole machine is in principle magnetically symmetric, it also in-

volves asymmetry to such degree that it is advisable to employ the two-axis model. Furthermore,

the field winding is usually a single-phase construction, which also justifies the application of thetwo-axis model. Only a slip-ring asynchronous machine, which can also be used as a synchronous




machine by supplying direct current to the rotor, is in principle magnetically completely symmetric,

and thus does not necessarily require a two-axis model fixed to the rotor.

ud

id

Rs

q

d

Ls

Lmd

uF

iFiD

RD RF

id

+iD

+iF

LDLF

md

Lk

Figure 7.10 The equivalent circuit of the synchronous machine in the d-direction. id and ud are the direct components of

the stator current and voltage. q and q are the direct and quadrature components of the stator flux linkage. iD is the

current of the direct damper winding. iF is the field current. Rs is the stator resistance, RD is the resistance of the direct

damper winding and RF is the resistance of the field winding. Ls is the leakage inductance of the stator, Lmd is the direct

magnetizing inductance, Lk is the Canay inductance, LD is the leakage inductance of the direct damper winding and

LF is the leakage inductance of the field winding. uF is the voltage of the field winding.

uq

iqRs

d

q

Ls

Lmq

iQ

RQ

iq+iQ

LQ mq

Figure 7.11 The equivalent circuit of the synchronous machine in the q-direction. iq and uq are the quadrature compo-

nents of the stator current and voltage. q and q are the direct and quadrature components of the stator flux linkage. iQ

is the quadrature current of the damper winding. Rs is the stator resistance, RQ is the resistance of the quadrature damper

winding. Ls is the leakage inductance of the stator, Lmq is the quadrature magnetizing inductance, LQ is the leakage

inductance of the quadrature damper winding.

Usually the parameters of the synchronous machine are obtained from the manufacturer as well as

by measurements carried out by the user of the machine. The parameters given for the machine are

not very applicable to the construction of the equivalent circuits, since the parameters defined by

traditional methods represent different magnetic states of the machine. For instance, the direct syn-chronous inductance is determined in no-load operation; the transient and subtransient inductances,

on the other hand, are defined by short-circuit tests, whereas the quadrature synchronous inductance

and the quadrature transient inductances are determined by various methods in different loading

situations. As a result, the parameters determined for the machine are not simultaneously valid, but

they all represent different magnetic states of the machine. In their design software, the manufactur-

ers apply experimentally defined coefficients of their own. The frequency converter technology

provides some solutions for determining the machine parameters. The parameters can be updated

on-line, or they can be determined in the initial identification run of the drive.

Let us next consider the traditional parameters and time constants of synchronous machines as well

as their determination. The machine parameters of Figure 7.12 are based on the results of three-

phase short-circuit tests. It is usually assumed that the mutual inductances between the windings on

the direct axis of the machine are equal, and of the magnitude of the direct magnetizing inductance




LdF = LdD = LFD = Lmd. The figure clearly illustrates the parameters traditionally given by the manu-

facturer:

The direct synchronous inductance is the sum of the stator leakage inductance and the direct

magnetizing inductance

L L Ld md s (7.34)

Correspondingly, we may write for the quadrature synchronous inductance

L L Lq mq s (7.35)

The direct transient inductance is the sum of the stator leakage inductance and the parallel con-

nection of the direct magnetizing inductance and the field winding leakage inductance

L L L L

L Ld s

md F

md F

'

(7.36)

The direct subtransient inductance Ld'' is the sum of the stator leakage inductance and the parallel

connection of the direct magnetizing inductance, the damper winding leakage inductance and the

field winding leakage inductance

L L

L L L

L L

L L L

L L

d s

md

D F

D F

md

D F

D F

''

(7.37)

There is no field winding on the quadrature axis, and therefore the quadrature subtransient induc-

tance is there

Qmq

Qmq

s

''

q L L

L L L L (7.38)

In the figure, we can also find the equivalent circuits for the time constants d0

'' , d'' , d0

' , d' , q0

'' ,

and q'' . The time constants are known as

d0

'' direct subtransient time constant, the stator winding open

d'' direct subtransient time constant

d0

' direct transient time constant, the stator winding open

d' direct transient time constant

q0

'' quadrature subtransient time constant, the stator winding open

q'' quadrature subtransient time constant.

Now we can easily first determine the machine parameters in the commissioning of the machine.

The subtransient inductance of the quadrature axis is determined by the equivalent circuit

L L L

Lq

''

q

mq

Q

2

, (7.39)

where




L L LQ mq Q . (7.40)

The magnetizing inductance Lmq of the quadrature axis is notably larger than the leakage inductance

of the quadrature damper winding LQ. Without a serious error being made, we may state that LQ

Lmq. Thus, the subtransient inductance becomes

L L L

LL L L Lq q

mq

Q

s mq mq s

'' 2

. (7.41)

Ld

Ls

LmdLq Lmq

L' d

Ls

Lmd

LF

L'' d

Ls

Lmd

LF

LD L'' q

Ls

Lmq LQ

Ls

Lmd

LF RF

' d0

Ls

Lmd

LF

'' d0

RD

LD

Ls

Lmq '' q0

RQ

LQ

Ls

Lmd

LF

'' d RD

LD

L s

Lmq '' q RQ

LQ

Ls

Lmd

LF RF

' d

Ls

Figure 7.12 The traditional machine parameters of a synchronous machine and the respective equivalent circuits.




The obtained subtransient estimate can be used to estimate the stator leakage inductance

L k Ls ls q '' , (7.42)

where the coefficient k ls varies typically between 0.4−0.6. The exact measuring of the stator leak-

age inductance is carried out without the rotor, and therefore such a measurement is not possible at

the actual operating location of the electrical drive; therefore, an estimate has to suffice. The other

parameters required for the current model are obtained by employing the equivalent circuits of Fig.

7.12. Thus, we obtain

L k Ls ls q '' , (7.43)

L L Lmd d s ,

L L Lmq q s ,

L L L L L

L LF

d md s md

d d

'

' , (7.44)

L L

L LLQ

mq

q q

mq

2

'' ,

L L L L L L L

L L L L L L L L L LD

d md F s md F

s F s md md F d F d md

''

'' '' .

When all the parameters according to the two-axis model are known, the resistances RD and RQ of

the damper windings can be calculated

R L

L L

L L L

LD

D

md F

md F

d

d

d

''

''

' , (7.45)

R

L L L

L L

Q

Q

mq s

mq s

q

''

In order to compute the damper currents of the direct and quadrature axes, the following four terms

are required

k modD

D Sd1

, k L

L Lmod

md

md Dd2

, (7.46)

k mod

Q

Q Sq1

, k

L

L Lmod

mq

mq Qq2

.

The current model can be determined by employing Eqs. (7.44)−(7.46). The successful accom-

plishment of this task depends on the accuracy of the traditional motor parameters. Next, we discuss

briefly how these parameters can be found.

The laboratory measurement technology for synchronous machines is defined in the standards IEC

34-4 “Methods for determining synchronous machine quantities from tests” and IEEE Std 115-1983“Test procedures for synchronous machines”. The measurements comprise a DC resistance meas-

urement, a no-load test, a steady-state short-circuit measurement, a slip test, a short-circuit test of




the field winding, a sudden three-phase short-circuit measurement, and the measurement of V-

curves. Table 4.1 lists the technical measurement data of a test motor obtained by these measure-

ments at the Laboratory of Electrical Drives at LUT.

Table 4.1: The machine parameters of a test motor determined by standard measurement procedures. The test machine:

14.5 kVA, 400V/21A, 50Hz/1500 rpm.

Parameter value Notes

relative stator resistance r s 0.048

relative field winding resistance r F 0.0083* *in the stator voltage level

reduction factor k ri

4

relative direct synchronous inductance l d 1.19

relative quadrature synchronous inductance l q 0.56

relative direct transient inductance l d’ 0.33

relative direct subtransient inductance l d’’

0.105

relative quadrature subtransient inductance l q’’ -* * cannot be measured

time constant of the field winding do’ 0.236 s

transient time constant of the direct axis d’ 0.054 s

subtransient time constant of the direct axis d’’ 0.024 s

subtransient time constant of the quadrature axis q’’ -* * cannot be measured

The data from the supplier, probably based on a calculation program, are respectively:

Table 4.2 Motor parameters given by the supplier: 14.5 kVA, 400V/21A, 50Hz/1500 rpm.

Parameter Value Notes

relative stator resistance r s 0.048

relative field winding resistance r F 0.00793*

*in the stator voltage level

reduction factor k ri 4.63

relative direct synchronous inductance l d 1.196

relative quadrature synchronous inductance l q 0.475

relative direct transient inductance l d’

0.129

relative direct subtransient inductance l d’’ 0.09

relative quadrature subtransient inductance l q’’

0.109

time constant of the field winding do’ do

’ 0.284 s

transient time constant of the direct axis d’ 0.031 s

subtransient time constant of the direct axis d’’ 0.006 s

subtransient time constant of the quadrature axis q’’ 0.008 s

7.3 Measurement of Motor Parameters by an Electrical Drive (DTC Drive)

The DC resistance of the stator, the direct synchronous inductance at no load at different voltage

steps, and the subtransient inductance both in the direct and quadrature directions can be measured

by a frequency converter. A modern frequency converter has a good measuring and computing ca-

pacity. Therefore, various measurements can be carried out automatically in the commissioning of the machine.




If the drive can be loaded in the commissioning, and it can be run similarly as in the case of DTC,

i.e., by utilizing both the current model and the model based on the voltage integral in the torque

estimation, the parameters of the current model can be updated both on the direct and quadrature

axes. The torque obtained by the current model and the voltage model has to be equal, and there-

fore, the inductance parameters have to be selected such that this condition is met.

The measurement of the transient inductance is straightforward; in the test, the motor is supplied byshort voltage pulses from the frequency converter, Dd t . By switching on the voltage vectors of

different directions, it is possible to get quite a good picture of the transient inductance of the ma-

chine in different directions. When supplying the machine in the direction of the direct axis of the

machine, the direct-axis subtransient inductance can be calculated

Lu t

i i id

'' sd

sd

sd

sd

sd

sd

d

d

d

d

. (7.47)

Correspondingly, when supplying the machine with a voltage pulse in the direction of the quadra-

ture axis of the machine, d Qt we obtain the quadrature subtransient inductance

Lu t

i i iq

'' sq

sq

sq

sq

sq

sq

d

d

d

d

. (7.48)

In the laboratory, the machine can be measured also in the intermediate positions; Figure 7.13 illus-

trates the subtransient inductance of a synchronous machine as a function of the rotor angle

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.180.19

0.2

0 10 20 30 40 50 60 70 80 90

rotor angle/ o

l ''

direct position quadrature position

Figure 7.13 The measurement result of the relative subtransient inductance as a function of rotor angle. The angle is

given in electrical degrees.

Note that in this case, the subtransient inductance is larger in the quadrature direction than in the

direct direction. Evidently, the activity preventing the penetration of the flux of the damper windingto the rotor is not as efficient on the q-axis as it is on the d-axis. Therefore, a short voltage pulse




generates a larger flux linkage when the rotor is in the quadrature position than when it is in the di-

rect position.

The direct-axis synchronous inductance of the motor can also be measured when the motor is oper-

ating at no load. The measurement is based on the flux linkage, obtained by integrating from the

voltages and comparing to the measured currents. In the inductance measurement, the motor has to

run at a relatively high speed in order to be able to reduce the uncertainty of the measurement. Themeasurement cannot be carried out before all the transients have settled down, in order for the

damper winding currents to have been completely damped out. The measurement of the synchro-

nous inductance can be performed by varying the field current of the stator and rotor. The first

value for the synchronous inductance can be obtained by completely omitting the direct current of

the rotor, ss0s u Li s . The no-load current is0 is in that case chiefly inductive field current.

As the rotor current is increased, the stator current reduces being finally completely resistive. The

low current is required to overcome mechanical frictions.

The measurement of the quadrature-axis magnetizing inductance is not possible in no-load opera-

tion. Therefore, to be able to measure the quadrature axis, the machine has to be loaded. The mag-netizing inductance and the stator leakage inductance can be calculated by applying the information

on the stator flux linkage in the rotor reference frame.

Li L

i i i

Li L

i i

Li

L

i

i

i i

md

sd d s

d F D

mq

sq q s

q Q

s

sd

d

md ,

D

Q

F D

0

0

0 0.

(7.49)

Figure 7.14 illustrates the measured no-load saturation curves in the d-direction.

0

0.3

0.6

0.9

1.2

1.5

0 0.4 0.8 1.2 1.6 2

imd [pu]

[pu]

Lmd

Figure 7.14 The measured no-load saturation curves for the direct-axis magnetizing inductance and the stator leakage

inductance.

The inductance measurements of the loaded machine were carried out by setting the sum current of

the d-axis imd = id + if + iD at the desired value and by varying the torque, which in turn had an im-

pact on the sum field current of the q-axis imq = iq + iQ. The measurement was repeated at the flux

linkage range ref = 0.3−1.3 pu, the torque varying between t ref = 0−2.5 pu. At each measured point,the measuring result was computed in the control program of the inverter. As a result, we obtained




the direct L i t i t md md mqf , and quadrature magnetizing inductance L i t i t mq md mqf , as a

function of imd and imq. The stator leakage inductance was assumed constant. The measured induc-

tance surfaces are illustrated in Fig. 7.15. The figure shows that the current of the quadrature axis

has an effect on the magnetizing inductance of the quadrature axis and vice versa. The effect of the

so-called cross-saturation is thus clearly visible.

0.000.47

0.94 1.42 1.901.55

1.19

0.92

0.710.530.350.180.060.00

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

Lmd [pu]

imd[pu]

imq [pu]

a)

0.00 0.48 0.71 0.95 1.19 1.43

1.67 1.90

1.55

1.19

0.92

0.71

0.530.35

0.00

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

L mq [pu]

imd [pu]

imq [pu]

b)Figure 7.15 a) The magnetizing inductance surface of the d-axis and b) the magnetizing inductance surface of the q-

axis as a function of the sum field currents.

Figure 7.16 illustrates the same inductance surfaces as two-dimensional curves.

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

0.00 0.50 1.00 1.50 2.00

imq = 1.5 pu

imq = 0 pu Lmd[pu]

imd[pu]

0.00

0.20

0.40

0.60

0.80

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Lmq

[pu].

imq[pu]

imd = 2.0 pu

imd = 0.5 pu

Figure 7.16 a) The magnetizing inductance of the d-axis and b) the magnetizing inductance of the q-axis as a function

of the field currents.

Table 4.3 presents a comparison of the results obtained by different methods. The DTC measure-

ment data have been gathered at the nominal operation point. The transient values in particular are

notably different in different methods.

Table 4.3 Comparison of the different measurement methods. Machine: 14.5 kVA, 400V/21A, 50Hz/1500 rpm. *not

measuredx

no-load value,+

measuring point not determined.

Parameter Manufacturer Standard measure-

ment

DTC meas.

pu relative stator resistance r s 0.048 0.048 0.051

relative field winding resistance r F

0.00793 0.0083 -*

reduction factor k ri 4.63 4 3.96

relative direct synchronous inductance l d 1.196x

1.19x

1.066

relative quadrature synchronous inductance l q 0.475+ 0.56x 0.439




relative direct transient inductance l d’ 0.129 0.33 -*

relative direct subtransient inductance l d’’ 0.09 0.105 0.125

relative quadrature subtransient inductance l q’’ 0.109 -* 0.194

time constant of the field winding do’ do

’ 0.284 s 0.236 s -*

transient time constant of the direct axis d’ 0.031 s 0.054 s -*

subtransient time constant of the direct axis d’’ 0.006 s 0.024 s -*

subtransient time constant of the quadrature axis q’’ 0.008 s -* -*

7.4 Finite Element Method (FEM) for Determining the Inductances of a Synchronous Ma-

chine

By means of magnetic calculation, the above inductance surfaces can be determined already in the

machine design. Next, the results of FEM calculation are investigated. Magnetic calculation is a

useful tool in the determination of the saturation behaviour of the direct and quadrature magnetizing

inductance. The definition of the inductances at a certain loading point is also possible. However, at

the present, magnetic calculation is still a rather laborious method, and therefore not even the larg-est suppliers calculate all their machines by the FEM, but simpler methods are applied to the ma-

chine calculation.

Figure 7.17 illustrates the cross-section of a small test motor. In the machine, there is a three-phase

winding, the stator slot number Qs = 24, the number of slots per pole and per phase qs = 2, the

length of the stator stack l = 140 mm, the inner diameter of the stator D = 196 mm, and the number

of turns in the stator winding per slot N' s = 56. The field winding is a four-pole construction. The

number of turns per pole is N rp = 220. The direct-axis magnetizing inductance L i i imd Ff d q

0 is

calculated by using different values of the rotor current. In the calculation, also a completely quad-

rature magnetizing can easily be carried out, and thus it is possible to determine L i i imq qf F d

0 .

At the loading points, the values are obtained as qf dmqqf dmd ,, and ,, iii f Liii f L

d-axis

q-axis

+a

+b

+c

+a

-a

-a+a

+a

-a

-a

-b-b

-b-b

+b

+b

+b -c

-c

-c

-c

+c+c

+c

Figure 7.17. The cross-section of the analyzed synchronous machine and its windings.




In the FEM calculation, in the two-dimensional case in particular, the vector potential A is utilized;

in this case, it is obtained from the equations B = A and A = 0. Figure 7.18 illustrates the

machine running at no load and loaded at the nominal operating point.

d/x

q/y

+a

+a

-b

-c

-a

-aa

a

-b

-c

+b

+b

+c+c

+c+c

-b

-b

sif

+a

s

q/y

d/x

is

if

+a

-a

-a

-b

-b

-c

-c

+b

+b

+c+c-b

-b

+c

+c

m

Figure 7.18 The flux diagram of the machine at no load and at the nominal point. Calculation by MAGNET software

The finite element method yields for instance the air gap flux density, which is naturally distorted

due to the slot openings and the armature reaction. To calculate the magnetizing inductances, it is

necessary to perform the Fourier analysis for the curve, since the space vector theory is based on the

assumption of a sinusoidal curve form. Figure 7.19 depicts the air gap flux density distribution and

its fundamental harmonic at no load.

-1.20

-0.90

-0.60

-0.30

0.00

0.30

0.60

0.90

1.20

0 90 180

periphery angle

B [T]

d-axis q-axis d-axis q-axis

360270

Figure 7.19 Air gap flux density and its fundamental harmonic at nominal no-load operation, calculated by the FEM.

To calculate the inductances, B1 and the phase angle are searched from the result of the Fourier

analysis. The air gap flux is now obtained from the equation




'ˆ2'ddcosˆˆ

p1

'

0

2

p1m

p

p

L B L x x

B

L

, (7.50)

where the effective machine length is defined as L’= l +2 is the air gap in the middle of the pole

shoe, p D2

p is the pole pitch of the machine, D is the stator boring and p is the number of pole

pairs. The air gap flux linkage of the phase A is now

' mA p f12

1B L N , (7.51)

f1 is the winding factor of the fundamental harmonic and N is the number of turns of the stator

phase in series. The flux linkages of the phases B and C are calculated accordingly, and thus we can

determine the stator flux linkage vector of the machine

jmA

34π j

mC3π2 j

mBmAm ˆ3

2eet et t

. (7.52)

The flux linkage is divided into direct and quadrature components

md m

mq m

cos

sin(7.53)

finally, inductances are calculated

Li i i i

Li i

i

mdFEM

md

d F D D

mqFEM

mq

q QQ

0

0

(7.54)

Figure 7.20 illustrates the comparison of the measured and calculated results.




a)

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35

imd [A]

calculated

measured

Lmd [mH]

b)

20

22

24

26

28

30

32

34

36

38

40

0 5 10 15 20 25 30 35

Lmq [mH]

imq [A]

Figure 7.20 a) The measured and calculated direct-axis inductance, when there is magnetization only on the direct axis.

b) Calculated quadrature inductance.

From our point of view, the most interesting issue are the results of the FEM analysis for a machine

running under load. Figure 7.21a depicts the behaviour of the direct-axis inductance at the nominal

stator flux linkage at different loads, and the item b illustrates the behaviour of the quadrature in-

ductance. Again, the results deviate somewhat from the measured ones.




MeasuredFEM

20

22

24

26

28

30

32

34

36

38

40

0 5 10 15 20 25 30 35

Lmd [mH]

imq [A]

imd = 19 A

a)

Measured

FEM

imd =19 A

0 5 10 15 20 25 30 35

Lmq [mH]

imq [A]

25

20

15

10

5

0

b)

Figure 7.21 a) The measured and calculated behaviour of the direct-axis inductance and b) the behaviour of the quadra-

ture-axis inductance when the torque varies between 0−250 %.

The above presentation shows clearly how difficult it is to obtain exact results from the machine.

However, there is a good congruence between the results measured by the DTC inverter and the re-

sults calculated by the FEM method; we may thus conclude that a modern frequency converter is

capable of performing the measurements required for the machine.




7.5 The Relationship between the Stator Excitation and the Rotor Excitation of a Syn-

chronous Machine

7.5.1 Non-Salient Pole Machine

Let us first consider a non-salient pole synchronous machine. The poly-phase winding, depending

on the number of turns N 1 of the phase, the winding factor of the fundamental harmonic, the

number of pole pairs p, and the phase number m, produces a peak value of the total magnetomotive

force ( s is the amplitude of the fundamental)

s11

sˆ

2π

4ˆ2 i p

N m

. (7.55)

For a three-phase machine, the above is rewritten as

ss1sss1s

ss1s

sπ

262

2π

43ˆ

2π

43ˆ2 I N I

p

N i

p

N

. (7.56)

In a synchronous machine, there is a single-phase rotor, into which direct current is supplied. The

peak value of the total mmf is now

rfDCr 1r r π

4ˆ2 I N . (7.57)

If we want to express the stator current with the direct current of the rotor, the magnetomotive

forces are written to be equal

rfDCs1s

r 1r

s1s

rfDCr 1r s

23

2

π

4

26

π I

N

N

N

I N I

. (7.58)

The relationship between the fundamental of the stator current and the magnetizing DC current is

thus

s1s

r 1r rirfDC

s32

N N k

I I

. (7.59)

Here we must note that the magnetomotive forces given by the equations, if written as equal, do not

really give an equal flux density in the air gap, since the leakage fluxes of the windings are not

equal, and thus inequal parts of the magnetomotive forces of the stator and rotor are exerted to the

air gap.

The same result is obtained by assuming a situation illustrated in Figure 7.22, where one stator

phase is currentless. Thus, at a sinusoidal supply, in the other two phases there flows a current

22

3s I i . (7.60)




The sum current of the instantaneous currents of the phases that corresponds to the common magne-

tomotive force may thus be written as

i I I yht s

o

s 23

22 30

3

22cos . (7.61)

Figure 7.22 Currents of a symmetric three-phase system,

when one phase current is zero.

The mmf of a "single-phase" stator at that moment is thus

24 3

22 s s 1s s

I N , (7.62)

and the mmf of the rotor is the same as previously, and thus the relationship of the currents is real-

ized as previously.

7.5.2 Salient-Pole Machine

Next, the three different air gaps of a salient-pole internal-pole machine are investigated. The air

gaps seen by the rotor pole excitation are usually shaped by the pole shoes so that as sinusoidal flux

density distribution as possible is obtained by the rotor magnetization on the direct axis.

The pole field has to be shaped so that the length of the density line of the field graph is inversely

proportional to the cosine of the electrical angle, when the frame of reference is fixed in the middle

of the pole shoe. The pole shoes shaped this way generate a sinusoidally distributed magnetic flux

density in the air gap. The stator winding is constructed so that also its magnetomotive force is si-

nusoidally distributed when proceeding along the stator surface. Also this mmf distributed in the air

gap creates a flux in the air gap. Since the air gap is shaped so that the flux generated by the poleexcitation is sinusoidal, it is clear that the flux generated by the stator is not sinusoidal. The three-

phase winding of the stator creates a magnetomotive force of its own, and correspondingly, an own

flux density component to the air gap. In a normally running machine, the flux density in the air gap

is composed of the flux components generated by all the windings of the machine. Next we investi-

gate how the stator magnetizes a shaped air gap.

Assume that the part ' d of the amplitude of the mmf of the stator is exerted to the centre line d of

the pole. Along the pole pitch, the magnetic voltage varies thus as follows

cosˆ'

d

'

d . (7.63)

The permeance d of the passage from this point to the rotor surface is




'0

00

cos

2

d

Δ

dd

p

DL

n

A . (7.64)

The magnetic density at the point is thus

2'

d'

0

0d cos

d

d

A B . (7.65)

The proportion of the flux density created by the mmf of the stator current to the air gap is thus pro-

portional to the square of the cosine. However, this density function is often replaced by its funda-

mental harmonic, that is, by a cosine function with an equal flux. The condition for keeping the

magnitude of the flux equal is

π/2

π/2

π/2

π/2

d2'

d'0

0 dcosˆdcosˆ

B . (7.68)

The amplitude of the cosine function is thus

'

d'

d

0'

d

0

0d

ˆ'4

ˆ

B . (7.69)

In the latter form of Eq. (4.59), the air gap d' is an fictitious air gap, which the magnetomotive

force of the stator meets. Its theoretical value is

' ' /d 4 0 . (7.70)

Figure 7.23a illustrates this situation. Equation (7.70) describes the air gap experienced by the stator

in the case of a salient-pole machine, when the air gap is shaped to produce a sinusoidal distribution

by the rotor magnetizing. In reality, the distance at the pole edge from the stator to the rotor cannot

obtain an infinite value, and therefore the theoretical value of Eq. (7.70) is not realized exactly as

such. The most exact value for this air gap can be determined by the FEM method, however, it is

possible to reach quite accurate results also by manual calculation.




p

Bd1

Bd

s

p

s

Bq1

Bq

Figure 7.23 a). The cos2-shaped density wave Bd created to an air gap shaped by the pole shoe; the density wave is gen-erated by a cosinusoidal mmf s1 occurring at the direct axis of the stator; the figure illustrates also the corresponding

equivalent cosinusoidal fundamental harmonicd1

ˆ B . b) The cosinusoidal mmf distribution s1 of the stator occurring at

the quadrature axis generates the curve Bq. The peak value of the corresponding equivalent density curve isq

ˆ B .

Figure 7.23b illustrates the definition of the quadrature air gap. The axis of the magnetic voltage of

the stator is imagined at the quadrature axis of the machine. The flux density curve on the quadra-

ture axis is sketched and the flux q is calculated. The density amplitude corresponding this flux is

written as

q

q

0q

q 'ˆ'

ˆ

DL p B . (7.71)

'q is the fictitious quadrature air gap. All the magnetomotive forces acting upon the air gap are set

equal: ' ' F d q , in which case the equivalent air gaps behave like the inverses of the density

amplitudes

qd0

qd'

1:

'

1:

'

1ˆ:ˆ:ˆ

B B B . (7.72)

The direct and quadrature equivalent air gap is calculated from this proportion. Note that the direct

and quadrature magnetizing inductances of the stator are inversely proportional to the above air

gaps. As a result, the quadrature synchronous inductance of the traditional salient-pole machine is

usually notably lower than the direct synchronous inductance.

The mmf of the rotor creates thus a sinusoidal flux density distribution on the stator surface, the

peak value of which converges with the smallest air gap 0. The slotting of the stator reduces the air

gap by an amount of the Carter’s coefficient.

0 0

'

k C . (7.73)




Observed from the stator, the direct air gap appears as regards to the fundamental harmonic theo-

retically according to Eq. (7.70). The length of the air gap has thus become 4/-fold.

Since the field winding is located only on the direct axis, the current ratio of the stator and rotor ex-

citation is required in this direction only. The current ratio is determined by defining the fundamen-

tal harmonics of the flux densities caused by the stator and rotor currents to be equal. Here it is as-

sumed that there are no leakage fluxes and the reluctance of iron is zero, and thus the entire mmf isexerted to the air gaps. The amplitude of the mmf of the stator is now written as

s 1s s s3

2 p

N I

. (7.74)

The fictitious flux density can be calculated accordingly

Bp

N I N I

p1s

1s s s1s s s

32

4

3 2

40

00

0

. (7.75)

The mmf of the rotor is of the shape of a square wave, and its height is

rFDCr

r 2

1 I

p

N . (7.76)

Since the air gap is shaped to produce a sinusoidal density distribution, the peak value of the ficti-

tious flux density becomes now

0

0

rFDCr

1r

2

1

ˆ

I

p

N

B . (7.77)

By comparing Eqs. (7.75) and (7.77) we obtain the current ratio

s1s

r

rFDC

sri

2

23 N

N

I

I k

. (7.78)

This is equal to the result we obtained previously. The ratio is thus the same both for a salient-pole

and non-salient pole machine, assuming that the pole shoe of the rotor of the salient-pole machine is

shaped to produce a sinusoidal distribution. If this is not the case, the current ratio has to be recon-

sidered, and we have to employ the partly empirical form factors given in literature. Equation (7.78)

is not quite exact either, since it is based on the assumption that the air gap met by the stator mmf in

the quadrature direction is infinite; the error is yet not significant, since the direct mmf is small

when approaching the quadrature axis and eventually reduces to zero.

If it is possible to run the permanent short-circuit test by the machine, the current ratio can easily be

determined with the short-circuit test. As we know, in the short circuit, the mmf of the stator currentshould cancel the mmf of the rotor current, and thus the short-circuit test directly produces the cur-




rent ratio. A certain magnetizing direct current of the rotor has a corresponding short-circuit current

of the steady state of the stator.

7.5.3 Referring Factor

When referring the resistances from the single-phase rotor winding to the three-phase stator wind-

ing or vice versa, the single-phase rotor DC can be considered to form a current vector of an ficti-tious three-phase rotor. The rotor is imagined to be replaced by a non-salient pole sheet rotor with a

three-phase winding. The three-phase current flowing in the winding generates an equal flux den-

sity in the air gap as the rotating pole wheel magnetized by direct current.

According to the energy principle, an equal power loss has to be generated both in the real rotor and

in the space vector equivalent circuit. When operating with three-phase quantities, the rotor power

is determined as

P R I r r3v r3v 3 2 . (7.79)

This power has to be equal to the power calculated with real rotor quantities

P I RF FDC FDC 2 . (7.80)

We obtain thus

R I

I Rr3v

FDC

r3v

FDC2

23. (7.81)

When we substitute the effective value I r3v of the rotor phase current by the respective stator current I s, we obtain

R I

I R

k Rr3v

FDC

s

FDC

ri

FDC

' 2

2 23

1

3. (7.82)

The same result is obtained also by the following procedure: we replace the single-phase rotor

winding by a three-phase winding with an equal number of turns. The magnetomotive forces of the

single-phase and the fictitious three-phase rotor are written equal per pole pair, and thus we obtain

r3v FDC

rp

r3v

rp

FDC

3

2

4

2 2

4

2

1 1

N

I

N

I . (7.83)

N rp is the number of turns in the rotor pole pair. This yields for the fictitious three-phase current of

the rotor

I I r3v FDC2

3. (7.84)

According to the energy principle, there has to occur an equal power loss both in the real rotor and

in the fictitious three-phase circuit. When operating with three-phase quantities, the rotor power is

determined as

P R I R I R I r r3v r3v r3v FDC r3v FDC

3 3

2

3

2

32

22 . (7.85)




This power has to be equal to the power calculated with the real rotor quantities, and thus we obtain

the resistance of the fictitious three-phase rotor

R Rr3v FDC3

2. (7.86)

Since the number of phases is equal both in the fictitious rotor and in the stator, the resistance is

now referred to the stator by employing the current ratio determined by the turns and winding fac-

tors

R RN

N R k

r3v FDC

1s s

1r r

FDC ri

'

3

2

1

3

2

2

. (7.87)

7.5.4 Referring (to the Stator) When Applying Space Vectors

When operating with the fictitious three-phase quantities in the rotor, we obtained for the effective

value of the three-phase current of the rotor

I I r3v FDC2

3. (7.88)

When operating with sinusoidal quantities, the peak value of this current is now

i I r3v FDC2

3. (7.89)

When we apply the above to construct the space vector of the rotor current, we obtain the result

shown in Figure 7.24 for the length of the rotor current vector, for instance in a situation where the

current of the phase A is at its positive maximum, and the currents of the phases b and c are a half

of their negative maximum values.

a

a2

1

3 I FDC

1

3 I FDC

2

3I FDC

I I r FDC2

3

Figure 7.24 The formation of the space vector of the current of an fictitious three-phase rotor, when the current of the

phase A is at its positive maximum.

Now, the magnitude of the space vector is thus (2/3) I FDC. By writing the magnetomotive forces of

the space vectors of the stator and rotor equal, we obtain




2

3I N N FDC rp r s1 sp i

' . (7.90)

This yields the ratio of the magnitudes of the current vectors

ir

FDC

rp

sp

riav

'

I

N

N k

2

3 1 . (7.91)

The result deviates from the DC reduction by an amount of 2 . In this system, the phase resistance

of the rotor is R Rr3v FDC3

2. When we now calculate the powers by applying space vectors and the

effective value of the direct current and when we write them equal, we obtain

P

N

N I R I R

k I R I R

3

2

3

2

2

3

3

2

2

2

2 2 2

ui

* '

'

rp

1s spFDC r FDC rfDC

riav FDC r FDC FDC

. (7.92)

The above yields for the reduction when operating with space vectors

Rk

Rr

riav

FDC

' 2

3

12 . (7.93)

7.6 Vector Diagram of a Synchronous Machine

Often when analysing the operation of a machine, a vector diagram is employed; the vector diagram

is nearly identical with the effective value phasor diagram, however, unlike the phasor diagram, it is

rotating. Figure 7.25 illustrates the vector diagram of a synchronous machine in the rotor oriented

dq reference frame. The machine operates as a motor and rotates counter-clockwise.




j LmqiQ

q

d

is m

iD+ jiQ

m

j Lmqiq

j LmdiD

j Lmdid

j LmdiF = emF

s

sLs

is

Lmdid + j Lmqiq

LmdiF

iF

LmdiD

LmqiQ

j s = es

j m = em

Rsis

us

j Lsis

-es

Figure 7.25 The vector diagram of a synchronous machine in a general case in the rotor reference frame. is the elec-

tric angular speed. s is the pole angle of the stator flux linkage, m is the pole angle of the air gap flux linkage, is the

angle between the stator flux linkage and the stator current, and is the angle between the stator current and stator

voltage. is is the stator current vector and us is the stator voltage vector. s is the vector of the stator flux linkage and m

is the vector of air gap flux linkage. es is the back electromagnetic force induced by the stator flux linkage. The figure

also illustrates a small damper winding current iD + j iQ. The entire vector diagram rotates counter-clockwise at an elec-

tric angular speed .

The flux linkage s of the machine is produced as an effect of the stator voltage vector u's from

which the resistive voltage loss is eliminated, and the flux linkage is thus 90 behind the vector. The

perpendicularity is realized in a steady state; however, in a dynamic state, in which there occur for

instance damper winding currents, the condition of perpendicularity is not met. The air gap flux

linkage m differs from the stator flux linkage by the amount of the leakage flux linkage s = Lsis.

In the machine, there occurs an fictitious flux linkage LmdiF created by the field winding; this flux

linkage is modified towards the air gap flux linkage by the armature reactions in the direct ( Lmdid),

and quadrature directions ( Lmqiq). Each flux linkage – also the leakage flux linkage – causes an in-

duction of its own. Considering a motor control, the inductions es = j s and em = j m are of

central importance. The induction es induced by the stator flux linkage is the induced voltage

which, added to the supply voltage, determines the current of the machine i u es s s s / R . The

induction created by the field winding j LmdiF is fictitious, since a flux linkage of such a high

value as represented by LmdiF cannot usually be generated to a machine due to the saturation of iron.

The inductions caused by the armature and damping reactions j Lmdid and j Lmqiq as well as

j LmdiD and j LmqiQ modify this fictitious flux linkage into the real air gap flux linkage. The




induction caused by the air gap flux linkage becomes observable if the stator of the machine sud-

denly becomes currentless. Particularly in the field weakening, it is possible that the magnitude of

the air gap flux linkage is notably larger than the magnitude of the stator flux linkage. If the control

of such a machine is terminated for some reason, the situation becomes extremely dangerous, since

the terminal voltage of the machine attempts to rise excessively – first to a value corresponding to

the air gap flux linkage, and finally to a maximum value allowed by the iron of the machine, this

value corresponding to the field current. Figure 7.26 recapitulates the vector diagram of the syn-chronous machine without damper currents.

q

d

ism

m

j Lmqiq

j Lmdid

j Lmd

iF

= emF

s

s

Lsis

Lmqiq

LmdiFiF

j s = es

j m = em

Rsisus

j Lsis

-es = u' s

Lmdid

id

iq

Figure 7.26 vector diagram of the synchronous machine when the damper currents have zero values.

For practical control purposes, the vector diagram can be simplified by omitting several inductions,

Figure 7.27




q

d

is m m s

s

Lsis

iF

Rsis

us u' s

Figure 7.27 Simplified vector diagram of a synchronous machine. The machine runs at the power factor cos = 1.

Figure 7.28 depicts the currents of a synchronous machine in different reference frames, and also

shows that the current of the field winding iF takes part in the torque production. In the air gap flux–

torque reference frame T, the field current has namely a component iFT clearly decelerating the

machine.

x

y

r

imm

r

d

Tq

f

iFiFTis

isT

iF

isd

isq

m

m

isy

isx

is

Figure 7.28 Stator current in different reference frames. The reference frames are the stator reference frame xy, the ro-

tor reference frame dq, and the air gap flux–torque reference frame T.

When applying the two-axis model, the different coordinate transformation equations have to pre-

sented in a component form. These component forms are next given by using stator current as the

quantity to be investigated; however, the equations hold also for stator voltages and stator flux link-

ages. When transferring from three-phase quantities to two-phase ones, the following equations are

applied to:

i i i isx sa sb sc

2

3

1

2, (7.94)

i i isy sb sc 1

3, (7.95)

We may write for the zero current




i i i is sa sb sc0

1

3 . (7.96)

The transformation formulas into opposite direction are

i i isa sx s0 , (7.97)

i i i isb sx sy s0 1

2

3

2, (7.98)

i i i isc sx sx s0 1

2

3

2. (7.99)

When changing over from the stator reference frame to the rotor reference frame, also the angle be-

tween the reference frame, that is, the rotor position angle r has to be known, as stated previously.

Since the zero component is not included in the current vector, it has to be taken individually into

account.

i i id sx r sy r cos sin , (7.100)

i i iq sx r sy r sin cos , (7.101)

i is0 s0 . (7.102)

The equations into opposite direction are written as

i i isx d r q r cos sin , (7.103)

i i isy d r q r sin cos , (7.104)

i is0 s0 . (7.105)

When transferring directly from the phase quantities to the rotor reference frame, the following

equations are applied to:

i i i id sa r sb r sc r

2

3cos cos cos

(7.106)

i i i iq sa r sb r sc r

2

3sin sin sin

(7.107)

i i i is sa sb sc0

1

3 (7.108)

When transferring to the opposite direction, the equations are

i i i isa d r q r s0 cos sin + (7.109)

i i i isb d r q r s0

cos sin

(7.110)

i i i isc d r q r s0

cos sin

4(7.111)




7.7 Torque Production of a Synchronous Machine

The synchronous machine operates according to the cross-field principle. The air gap flux linkage

in particular is significant for torque production; previously, we have seen that the current compo-

nent perpendicular to the air gap flux linkage produces torque. As shown in the previous chapter,

the equation for the electric torque T e can be given in various forms; let us briefly recapitulate these

different expressions of torque again. The general equation for the torque of any rotating-field ma-chine can be given as a vector representation

kiiT

sin2

3

2

3sssse p p , (7.112)

where

- p number of pole pairs

- s stator flux linkage vector

- is stator current vector

- angle between vectors- k unit vector aligned with the shaft of the machine

When we analyze the vector diagrams of a synchronous machine, we may state that there is no con-

tradiction between the above representation and the role of the aforementioned air gap flux linkage.

The torque equation can also be expressed in the scalar form

sin2

3sme i pT . (7.113)

According to these equations, the magnitude of torque depends on the magnitudes of the vectorsand the angle between them. The sign of the torque expressed in scalar form depends on the angle

between the vectors. Thus, in the case of synchronous machines, changing over from motor opera-

tion to generator operation or vice versa is accomplished by only changing the sign of the angle be-

tween the vectors.

When Eq. (7.112) is represented by its components in the rotor reference frame, it can be written in

the scalar form:

dqqde2

3ii pT . (7.114)

Further, when we substitute the components of the stator flux linkage to the equation by applying

the inductances and currents, we obtain

dQmqqDFmdqdqde2

3ii Liii Lii L L pT . (7.115)

Finally, we decompose the middle term and rewrite the equation in the form

dQmqqDmdqFmdqdqde 2

3ii Lii Lii Lii L L pT (7.116)

The equation shows that the torque comprises the following parts:




the reluctance torque ( Ld Lq)idiq resulting from the magnetic asymmetry; occurs only in salient-

pole machines

a component related to the field current and the quadrature stator current; this is the actual torque

of the machine

two terms describing the torque components of the damper winding currents; these components

occur only in transients

7.8 Simulation of an Electrically Excited Salient-Pole Machine by Constant Parameters

In the context of a synchronous machine, per unit values are often employed. Using per unit values

brings certain benefits; the relative scale of a parameter can be seen directly from the per unit value.

For instance, if the relative synchronous inductance is l d = 2, the value is high. Correspondingly, if

l d = 0.5, the value is low. Hence, it is possible to compare machines, the rated values of which devi-

ate from each other. The role of per unit values diminishes when we change over to the application

of floating-point processors. The fixed-point signal processors require proportioning of all calcula-

tions in the range of 0 ... 1. In that case, the application of per unit values alone does not suffice, butthe figures have to be scaled considerably lower to ensure that for instance the result of a sum or a

product will never exceed one.

Per unit values are obtained by dividing each quantity by its basic value. The basic values are usu-

ally determined as follows:

the peak value of the rated stator phase current in

the peak value of the rated stator phase voltage un

the rated flux linkage, which also corresponds to the rated angular speed /n n n u

the rated impedance Z u in n n

/

The time in which the angle of one radian in electric degrees is travelled at the rated frequency

t n nrad 1 / . The relative time is thus measured as an angle nt

the apparent power corresponding to the rated current and voltage S i un n n3

2

the apparent torque corresponding to the rated power and rated frequency T i un

n

n n3

2

The rotor quantities are referred to the stator voltage level before the per unit values are con-

structed. For example, the stator voltage equation is expressed in the rotor reference frame as

u is

r

s s

r s

r

s

r d

d j R

t

, (7.117)

the above is divided by the basic value of the voltage and it is grouped appropriately, and thus we

obtain

u is

r

n

s n

n

s s

r

n

s

r

n

n

n n

n s

r

n

d

d

j

u

R i

u

R

i t u u

. (7.118)

The equation clearly shows the per unit values




uu

s,pu

r s

r

n

u

, (7.119)

ii

s,pu

r s

r

n

i

, (7.120)

r

R i

us

s n

n

, (7.121)

s,pu

r n s

r

n

u, (7.122)

pu

n n

pu n

f n (7.123)

nt . (7.124)

The voltage equation given in per unit values becomes thus

u is,pu

r

s s, pu

r s,pur

pu s,pu

r d

d j r

. (7.125)

The form is equal with the original equation, however, the relative time may cause some problems.

If normal time is used in the per unit value equations, the equation will be rewritten as

u is,pu

r

s s, pu

r

n

s,pu

r

pu s,pu

r d

d j r

t

1

. (7.126)

The per unit values of inductances are equal to the per unit values of reactances. We obtain for in-stance

l L

L

L

u

i

i

uX xm

m

b

m

n

n n

n

n

m m

. (7.127)

The stator flux linkage equation

s s s m r L Li i (7.128)

is divided by the basic value of the flux linkage

b

n

n

u

, (7.129)

which yields

n s

n

n n s

n

s

n

n n m

n

r

n

u

i L

u i

i L

u i

i i. (7.130)

Thus, the flux linkage equation given in per unit values becomes




s,pu s s,pu m r,pu l l i i . (7.131)

The motion equation of an electrical machine is written as

3

2

2

2 pJ

p t T s s m

d

d i

. (7.132)

When the motion equation is divided by the basic value of torque, we obtain

3

23

2

3

2

3

2

2

2 p

u i

p

J

p t

u i

p

T

u i

p

s s

n n

n

n n

n

m

n n

n

d

d

i

/

/

/

. (7.133)

This yields the per unit value equation

s,pu s,pu m,pu

d

d i T T J

2

2

. (7.134)

In the above equation, we have a mechanical time constant

T p

J

u i J

n

n

n n

22

3 . (7.135)

The mechanical time constant is the ratio of the kinetic energy of a rotor rotating at synchronous

speed to the apparent power of the machine.

7.9 Current Equations of a Synchronous Machine

The relationship between the flux linkages and currents of a synchronous machine has to be inves-

tigated in the rotor reference frame in order to avoid the division of the direct and quadrature quan-

tities into their components.

The dependence between the flux linkages and currents is determined by an inductance matrix; weomit the pu (per unit value) notations.

L i , (7.136)

or

F

Q

D

q

d

FFDmd

Qmq

FDDmd

mqsq

mdmdsd

F

Q

D

sq

sd

00

000

00

000

00

i

i

i

i

i

l l l

l l

l l l

l l

l l l

, (7.137)




where

l l l

l l l

l l l l

l l l

l l l l

l l l

sd md s

sq mq s

D md D k

Q mq Q

F md F k

FD md k

(7.138)

Now the currents can be determined from the flux linkages by applying the inverse inductance ma-

trix K L 1 .

i

i

ii

i

k k k

k k

k k k k k

k k k

d

q

D

Q

F

sd dD dF

sq qQ

dD D FD

qQ Q

dF FD F

sd

sq

D

Q

F

0 0

0 0 0

0 00 0 0

0 0

, (7.139)

where

k l l l

k l

k l l l

k l

k l l l

sd

F D FD

sq

Q

D

F sd md

Q

sq

F

sd D md

2

1 2

2

1 2

2

1 , , , ,

k l l l

k l l l

k l

k l l l

dD FD F

md

dF FD D

md

qQ

mq

FD

md FD sd

1 1 2

2

1

, , , (7.140)

the denominator components 1 and 2 are written as

1

2 2 2 2

2

2

2

l l l l l l l l l l l

l l l

md FD md D md F sd D F FD sd

Q sq mq

(7.141)

7.10 Simulation of a Synchronous Machine in a Discrete-Time System

With the mathematical representation of a synchronous machine, it is possible to compile a simula-

tion model, the inputs of which are the stator voltage us,pu , the excitation voltage uF,pu , and the me-

chanical angular speed n (for the integration of the rotor angle) of the synchronous machine; theoutputs are the stator current is,pu , the field current iF,pu, the currents of the damper windings iD,pu

and iQ,pu, and finally, the electric torque te of the machine. The simulation presumes naturally a

knowledge of the inductances and resistances of the synchronous machine.




The simulation progresses according to the flowchart below:

0 Appropriate initial values are initialized for the flux linkage and current parameters,

and the inductance matrix is calculated

1 The derivatives of the stator flux linkage s,pu (k) are calculated in the stator refer-

ence frame in the x and y directions by applying the stator voltage us,pu (k)

s pusx, pusx,n

pusx,1-k k

d

k dr iu

t

s pusy, pusy,n

pusy,1-k k

d

k dr iu

t

(7.142)

2 The derivatives of the flux linkages of the damper winding and the field winding

are calculated in the rotor reference frame

F puF, puF,n puF,

1-k k k

d

dr iu

t

D puD,n puD,

1-k k d

dr i

t

Q puQ,n

puQ,1-k

d

k dr i

t

(7.143)

3 New values are integrated for the flux linkages by applying the obtained derivatives

T t

T t

k d

d1-k k

k d

d1-k k

sysysy

sxsxsx

T t

T t

T t

k d

d1-k k

k d

d1-k k

k d

d1-k k

FFF

QQQ

DDD

(7.144)

4 New values are determined for the currents by applying the inverse inductance ma-

trix k k Ki (7.145)

5 The electric torque is calculated for the machine

k k k k k re

s

im

s

im

s

re

se iit (7.146)

6 The cycle is repeated from item 1

7.11 Implementation of the Vector Control of a Synchronous Machine

Figure 7.29 illustrates the basic signal processing diagram of a synchronous machine. The control-lers in the diagram are usually PI controllers. First, the currents and the rotor position of the motor

are measured; next, a two-phase rotor-oriented representation is adopted. Then, the current compo-




nents producing flux linkage and torque are calculated by applying the motor model. The speed

controller produces a torque reference, which in turn yields the respective current reference. The

voltage limiter is an essential element when considering the field-weakening control. The elimina-

tion of cross-effects takes place in a circuit designed for the purpose. The angular speeds in the fig-

ure are electric angular speeds.

U DC

U DC

pulse encoder or absolute encoder

i1

i2

i3

iF

r

ix

ix

ix

iyiy

iy

t et eim

im

im

iyref t e ref

im ref ix ref

us0

ûx

ûyref

uxref

uyref

uxref

uy ref

Excitationcontrol

uFref

PWMmodulator-

S A

S B

S C

S1,S2,S3,

32

ux

uy

ix

iy

iF

r

motor model

curr.contr.

current

contr.

torque

contr.speed

contr.

flux

link.

contr

volt.

limiter

removal of.

cross eff.

r

r,ref

r

r

r

r

e j r

ref

Figure 7.29 The current vector control for a synchronous machine in a flux oriented system.

First, we investigate separately the determination of the reference values for the field current com-

ponents and for the current components producing electric torque. These current references are cal-

culated in the flux linkage–torque reference frame. The axes of this reference frame are known as

the flux linkage and torque axes; these axes were introduced in Figure 7.4. Thus, the current refer-

ences are from now on designated as the flux linkage-aligned current reference and the torque axis-

aligned current reference.

The torque reference is obtained from the output of the speed controller; the input of the controller

comprises the difference of the speed reference and the actual value. The speed controller is a PI

controller. The speed reference is obtained form the system controlling the process. Since the elec-tric torque is proportional to the magnitudes of the stator current and the stator flux linkage, the cur-

rent reference aligned with the torque axis is obtained when the torque reference is divided by the

flux linkage reference.

The flux linkage axis-oriented current reference is obtained from the output of the flux linkage con-

troller. The flux linkage controller is also a PI controller. The input consists of the difference of the

field current reference and the actual value. The flux linkage reference depends on the actual value

of the rotation speed. The flux linkage is kept at the rated value when operating below the rated

speed. At speeds above the rated speed, the flux linkage is reduced inversely proportional to the

speed; the latter speed range is known as the field weakening range.




When the frequency rises above the rated frequency of the machine, the terminal voltage of the ma-

chine is limited to the rated value by utilizing the field weakening range. Let us first consider the

equation for the rotating voltage in the stator reference frame:

es s j . (7.147)

It suffices to investigate only the magnitudes:

es s . (7.148)

For a synchronous machine, the following is known about the mechanical angular speed:

2n n f

p, , (7.149)

mechanical angular speed

n rotation speed [1/s]

f frequency

p number of pole pairs in the machine.

Bearing in mind that the magnitude of the stator voltage |us| differs from the magnitude of the rotat-

ing voltage |es| only for the amount of the resistive voltage losses, we may write after the substitu-

tions:

us s 2 p

f . (7.150)

If we consider the case from the point of view of the flux linkage, and take only the proportional

relationships into account, we obtain

s| ~us

f . (7.151)

Here we see that the flux linkage remains constant, if the ratio of the terminal voltage and the fre-

quency is kept constant. We can also see that the voltage remains constant, if the flux linkage is re-

duced inversely proportional to the frequency. Figure 7.30 illustrates the stator flux linkage and the

voltage as a function of frequency. If the flux linkage of the machine were not allowed to diminish,

it would be necessary to increase the stator voltage proportional to the frequency also in future (the

dashed extension of the upward line in Figure 7.30).




s

f, , f N , ,

field weakening rangeconstant flux range

us

us s

sus

0 1 2 3

1

0

Figure 7.30 The constant flux linkage range and the field weakening range of a synchronous machine. On the horizon-tal axis, there are the frequency f and the electric angular frequency or the mechanical angular frequency . f N, N,

and N are their rated values. On the vertical axis, there are the stator voltage us and the stator flux linkage s. U N and

N are their rated values. In the constant flux range, the voltage and frequency are allowed to increase in equal propor-

tion, and thus the magnitude of the stator flux linkage is kept constant. Above the nominal operating point, the flux

linkage is reduced inversely proportional to the speed and frequency, and thus the voltage can be kept constant.

However, the stator voltage cannot be increased as described above, since the voltage rectified from

the line voltage cannot be raised above this limit in a reasonable way. On the other hand, the deter-

mination of the field weakening point is an essential part of electrical machine design, since it is

usually advisable to dimension the motor in such a way that at high speeds, the field weakening is

applied. Otherwise, the motor should be dimensioned unfavourably large to reach the desired torque

at low rotation speeds. Selecting a correct field weakening point is essential in dimensioning the

machine to suit the purposes.

The actual values of the stator and air gap flux linkage are determined by applying the flux linkage

equations of the machine; therefore, measurement data of the phase currents, the field current, the

rotor position angle and the calculated damper winding currents are required. The phase currents

are transformed into currents in the rotor reference frame; for the transformation, information on the

rotor position angle is required.

sd md s d md d D F s d L i L i i i L i ( ) , (7.152)

sq mq s q mq q Q s q L i L i i L i ( ) . (7.153)

The obtained direct-axis and quadrature axis components are employed in calculating the magni-

tude of the flux linkage and the pole angle.

s sd

2

sq

2 , (7.154)

m md

2

mq

2 , (7.155)

s

sq

sd arctan , (7.156)




m

mq

md

arctan (7.157)

This calculation method for the flux linkages is commonly known as the current model. Since it is

impossible to measure the damper winding currents, they have to be eliminated, and appropriate

time constant representations are employed instead. Let us consider this elimination task in the rotor

reference frame; first, the direct and quadrature flux linkage components are written as

md md md md F d D i L L i i i , (7.158)

mq mq mq mq q Q i L L i i , (7.159)

and the voltage equation of the direct-axis damper winding of the rotor

u R it

R it

i LD D D

D

D D md D D

d

d

d

d 0 . (7.160)

Next, the time constants describing the rotor damping are adopted

rDmD D

D

L L

R

, (7.161)

rDD

D

L

R. (7.162)

The equation for the flux linkage is substituted to the voltage equation

0 R it

L i i i i LD D md F d D D D

d

d , (7.163)

the expression is developed somewhat further

0

i

t

L

Ri i i i

L

RD

md

D

F d D D

D

D

d

d

, (7.164)

0

i

t

L

Ri i i

L L

RD

md

D

F d D

D md

D

d

d

. (7.165)

Now we obtain

1

d

d

d

d

D md

DD

md

DF dt

L L

Ri

t

L

Ri i

, (7.166)

1

rD D rD F d rD F d

d

d

d

d

d

dt i

t i i

t i i , (7.167)

it

i i it

i iD rD F d D rD F d

d

d

d

d . (7.168)

Both sides of the equation are multiplied by the magnetizing inductance, and the equation for the air gap flux linkage is added to both sides




rD md F d D md F d D md F d rD md F d

d

d

d

dt L i i i L i i i L i i

t L i i . (7.169)

Now, the flux linkage equation can be rewritten as

rD md md md F d rD md F d

d

d

d

dt L i i t L i i , (7.169)

the air gap flux linkage can be solved from the above equation

md md F d

rD

rD

d

dd

d

L i i t

t

1

1

. (7.170)

Correspondingly, it is possible to determine a representation for the quadrature-axis air gap flux

linkage components that is dependent on the quadrature-axis time constants of the damper winding

mq mq q

rQ

rQ

d

dd

d

L i t

t

1

1

. (7.171)

If the information on the damper winding currents is required, these currents can be solved by di-

viding the flux linkage terms by the respective magnetizing inductances.

i i i L

i i t

t

D F d

md

mdF d

rD

rD

d

dd

d

1

1

, (7.172)

i i i t

t

D F d

rD

rD

d

dd

d

1

1

1

, (7.173)

and

i i L

i t

t

Q q

mq

mq

q

rQ

rQ

d

dd

d

1

1

. (7.174)

i i t

t

Q q

rQ

rQ

d

dd

d

1

1

1

. (7.175)

The time constants for the leakage components are so short when compared with the damping time

constants in the denominators that these components can be neglected in certain cases. The damper

winding currents are in then calculated of the other currents of the dq plane in the current model,

since the damper winding currents cannot be measured. The equations are transformed to the

Laplace domain and the effect of Canay inductance is added




i K i K i sD td d tf F

rD

1

11

, (7.176)

i K i sQ tq q

rQ

1

11

. (7.177)

The block diagrams corresponding to the equations are presented in Figure 7.31.

i

i i

i

i

K

F

d

q Q

D

tq

K tF

K td

Figure 7.31 Determining the damper winding currents from other currents in the dq plane. iF is the field current, id and

iq are the stator current components in the rotor reference frame, and iD and iQ are the calculated damper winding cur-rents. The time constants of the filter blocks are rD and rQ. K tF, K td, and K tq are the constant factors of the figure.

The parameters required in determining the damper winding currents, i.e., the coefficients and time

constants of the filters, are obtained from

K L

L L Ltd

md

md k D

, (7.178)

K L L

L L LtF

md k

md k D

, (7.179)

K L

L Ltq

mq

mq Q

, (7.180)

rDmd k D

D

L L L

R

, (7.181)

rQmq Q

Q

L L

R

. (7.182)

In the following example, a stator flux linkage oriented control is investigated. The current refer-

ences that are aligned with the flux linkage axis and the perpendicular effective power axis as well

as the sine and cosine of the pole angle of the stator flux linkage are obtained by the method pre-sented in Fig. 7.32.




abc

3

2

dq

i

Calcul.

of

stator

flux link.

with

curr.

model

sin s

| |F

id

nref

sref

Flux controller

Function block for generating

flux linkage reference

cos s

nact

T ref iT ref

i ref

sact

iq

isa

isb

isc

r

Figure 7.32 Calculation of the current references. iTref is the current reference aligned with the effective power axis and

iref is the current reference aligned with the stator flux linkage axis. s is the pole angle of the stator flux linkage, | s| is

the magnitude and | sref | is the reference value. nref is the speed reference and n is the actual value. iF is the actual value

of the field current, isa, isb, and isc are the actual values of the stator phase currents. The block 32 performs the trans-

formation from the phase currents into the currents in the rotor reference frame. id and iq are the components of the ac-

tual value of the stator current in the rotor reference frame. r is the rotor position angle.

In addition to the current model, the stator flux linkage can be determined also by applying the volt-age model. In the voltage model, the flux linkage is integrated from the supply voltage. The model

is not applicable to zero speed or low-speed operation, when the voltage is too low. Furthermore,

also the rotor angle has to be known in order to be able to transfer the flux linkage to the rotor ref-

erence frame, where its magnitude and pole angle would be known.

The next phase in the vector control is the construction of the current references in the rotor refer-

ence frame. Now, the pole angle of the stator flux linkage is required, since we change over from

the flux linkage reference frame to the rotor reference frame. The required equations are given be-

low.

Tref sref sdohje sincos iii (7.183)

Tref sref sqohje cossin iii (7.184)

The above information is expressed in the block diagram of Figure 7.33.




sin s

cos s

iT ref

i ref

iqT re f

idT re f

id ref

iq re f

idref

iqref

Figure 7.33 Calculation of the current references of the rotor reference frame from the current references of the stator

flux linkage reference frame. iTref is the quadrature-axis current reference and iref is the current reference aligned with

the stator flux linkage axis. s is the pole angle of the stator flux linkage, idyref is the reference of the quadrature-axis

current and idref is the reference of the direct-axis current that magnetizes the machine in the direction of the stator flux

linkage. idref is the direct-axis current reference and iqref is the quadrature-axis current reference.

The next phase is the current control, Figure 7.34. An integral control is performed for the currents

of the rotor reference frame. Next, the feedforward from the rotating voltages is added to outputs of

these current signals. Since the rotating voltages of the machine are quite close to the terminal volt-

ages, it is easier to adjust the output of the controller to the correct range, and the model has to cor-

rect only the errors caused by the non-ideality of the model and the non-linearity of the cyclocon-

verter bridges. Then, the obtained reference values are transformed into phase values. Since the in-

tegration is carried out in the rotor reference frame, the signals to be integrated are DC signals and

we avoid for instance phase errors.

Before the proportional control (P-control), the dq current references are transformed into phasecurrent references, since the P-control is performed for phase currents unlike the I-control. Because

the P-control is performed for phase quantities, it is easier to prevent the occurrence of the zero

component. The zero component is not detectable in the currents of the rotor reference frame, as it

was shown previously. The P-controllers yield the other components for the reference values. These

components are summed up for different phases, and thus, we obtain the reference values, which

are by nature chiefly voltage references for different phases. The references are modified further to

produce the ignition angle reference for the thyristor igniters.




i

dq

2

3abc

dref

iqref

id

iqd

q

isa, ref

isa

isb, ref

isc, ref

isc

isb

u sa, ref

u sb, ref

u sc, ref

sa

r

r

sb

sc3abc

dq

2

Figure 7.34 The current control of the stator. Part I in the dq plane, the feedforwards from the rotating voltages and the

P part for the phase quantities. idref is the current reference aligned with the d-axis and iqref is the current reference

aligned with the q-axis. r is the rotor angle, q is the quadrature component of the stator flux linkage. - q is the di-

rect-axis rotating voltage. d is the quadrature-axis rotating voltage. is the electric angular speed. idref and iqref are

the current references in the rotor reference frame and isa,ref , isb,ref and isc,ref are the references of the stator phase currents.

id, iq, isa, isb, and isc are the respective actual values of the currents. 2/3 blocks perform the transformation from the rotor

reference frame to the stator phase quantities. usa,ref , usb,ref and usc,ref are the voltage references for different phases and

sa, sb, and sc are the control angles for the bridges of different phases.

The field current is an essential quantity when considering the operation of the machine. In the tran-

sient states, it has to compensate the negative effect of the direct-axis stator current and also to

maintain the desired power factor of the stator. If the target is to keep the power factor of the stator

one, the field current reference can be calculated as follows:

md

dref dssref Fref

cos

L

i Li

, (7.185)

where

- iFref is the reference value of the field current- | sref | is the reference value of the stator flux linkage

- s is the pole angle of the stator flux linkage

- Ld is the direct-axis synchronous inductance

- Lmd is the direct-axis magnetizing inductance

Also a PI controller is applied in the control of the field current, and the controller yields chiefly the

voltage reference, from which the control angle reference is modified for the excitation bridge as

shown in Figure 7.35. The control of the field current has to be tuned correctly to avoid a control

that would oscillate in the transients. The problem with the reference value calculated above is that

the magnetizing inductance does not remain constant, and therefore it should be estimated beforethe calculation process.




i

i

Calc. of the

reference for

field

current

uF

dref

sref

Fref F

s

Field current controller

iFref

Figure 7.35 The field current controller. | sref | is the reference value of the magnitude of the stator flux linkage, idref is

the reference value of the current id, s is the load angle and iF is the field current. uFref is the voltage reference and F is

the control angle for the excitation bridge.

In the above case, the vector control comprises only the current model and the respective coordinate

transformations of the currents, the flux controller, and the generation of the direct and quadraturecurrent reference. The rest belongs to the ordinary control system with the controllers. This example

of a vector control of a synchronous machine is based on a concrete cycloconverter drive manufac-

tured by ABB. The control has thus been implemented as stator flux linkage oriented. We may now

ask whether it is necessary to implement the vector control always as flux linkage oriented, since, as

we know, the leakage flux linkage of the stator flux linkage does not produce torque. However, in

the voltage source drives – also the cycloconverter is of this type – it is necessary to ensure that the

voltage is sufficient, and therefore a control based on the stator flux linkage is a justified choice.

Figure 7.32 depicts the current reference iTref , although we may well claim that this current alone

does not produce the torque, since it is not perpendicular to the air gap flux linkage. A control based

on the air gap flux linkage would best correspond to the control of a fully compensated DC ma-

chine; however, the control is based on the stator flux linkage due to its voltage dependence. Figure

7.36 illustrates the difference between the controls based on the stator flux linkage and on the air

gap flux linkage.

q

d

ism

m

s

s Lsis

iF

us

cos = 1

q

d

ism

m

s

s Lsis

iF

us

cos < 1

id id

Figure 7.36 Controls based on the stator flux linkage and on the air gap flux linkage.

The figure shows that when operating at a rated voltage and keeping the power factor and the mag-

nitude of the stator current at the value one, a certain torque is obtained (e.g. rated torque). When

the stator current is kept perpendicular to the air gap flux linkage, and the voltage determines the

length of the stator flux linkage, the same current produces less power than in the previous case,

since the stator flux linkage has now decreased. The power has now decreased in accordance with

cos = 0.85. The same applies to the torque. Although the magnitude of the current remains thesame, the air gap flux linkage is now 15 % shorter, although the torque has decreased by 85 % of

the value of the left-hand illustration. Note also the respective reduction in the field current. Thus




we may state that when operating at a certain voltage, it is advisable to select a control based on the

stator flux linkage, and to apply the power factor cos = 1.

Figure 7.37 illustrates the schematic control diagram of the cycloconverter based on the stator flux

linkage.

abc

3

2dq

i

Calc. of stator flux

linkagewith

currentmodel

| |

F

i sa

isb

isc

i q

i d

s

sin

cos s

s

n

nref T ref

sref

iTref

iref

r

Flux controller

Speed PI controller

Functtion bloc for flux link. ref.

i dTref

i qTref

idref

iqref

idq

2

3

abc

dref

iqref i

d

iq d

q

isa, ref

isaisb, ref

isc, ref

isc

isb

usa, ref

usb, ref

usc, ref

sa

r

r

sb

sc

3

abc

dq

2

i

i

K

Q

D

tq

K tF

K td

i

i

Calc. of theref. for field current

u

F

dref

sref

Foref F

s

Field current controller iFref

s

usa

sb

sc

u

u

Current controllers

Figure 7.37 A schematic diagram of a stator flux linkage based cycloconverter drive.

A cycloconverter drive can also operate as a generator, and therefore we introduce the vector dia-

gram of a synchronous machine also in generator operation. The representation follows the same

logic as in the case of motor operation.

q

d

i = 0 m, s, iF

us

q

d m

iF

us

cos = 1

id

01 J Pyrhonen - Synch Machn

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