Mathematical simulation, determination and analysis of the magnetic field occurred by the armature

reaction of the synchronous machine

Aleksandrs Mesņajevs Riga Technical University

RTU Riga, Latvia

kbl@inbox.lv

Andrejs Zviedris Riga Technical University

RTU Riga, Latvia

aaazzz@eef.rtu.lv

Abstract— To solve partial derivative equations for vector magnetic potential by method of mathematical simulating finite element method is used. Designed, theoretically based, and described method which allows from stationary magnetic field simulation results to achieve vector potential as the spatial coordinate’s functions so obtaining the flux leakage with the armature winding as a function of time. Calculations are made for a particular synchronous machine and from simulation results the synchronous inductive reactance values are obtained.

Keywords- Mathematical simulation, synchronous machine, magnetic field.

I. INTRODUCTION Notion “Parameters of synchronous machines” generally

means inductive reactance of these machines (or in some cases of inductance) in different operating conditions.

In this work the basic statements and the methodological aspects connected with inductive resistance of synchronous motor as input and use of this physical quantity are considered. Different approaches and methods of definition of these resistances are described and analyzed.

II. INDUVTIVE REACTANCES OF SYNCHRONOUS MACHINES Armature reaction’s direct-axis reactance (Xad) and

quadrature-axis reactance (Xaq), and direct-axis field’s synchronous reactance (Xd) and quadrature-axis field’s synchronous reactance (Xq) are widely applied to the description of symmetric static mode of the synchronous motor. Active resistance of motors with power 5... 10 kW can be ignored, as it is relatively small and practically does not influence on energy transformation process.

Electric machines parameters and processes are depending on character of an electromagnetic field, i.e. spatial and time distribution of a field.

In the theory of alternating current electric machines winding’s EMF E is induced by alternating current’s I

magnetic field is determined by inductive reactance of this winding [2]

IEX = . (1)

The biggest part of magnetic field’s lines creates main magnetic flux which is, while crossing an air gap, completing through the stator’s and rotor’s magnetic cores and, simultaneously, is linked to stator’s and rotor’s windings.

Work of an alternating current machines is determined by a fundamental harmonic’s magnetic induction B1 and induced by this harmonic EMF E1, which (see (7)) is possible to describe by inductive reactance

,1

IE

X m = (2)

which is called basic inductive reactance.

Leakage fields are possible to describe by inductive leakage reactance σX , that considers slots, frontal connection and differential leakage fluxes.

The stator winding’s main inductive reactance can by calculate by equation [2]:

.4 21

210

pkw

kklmfX w

m δτμ

π δμ

δ= (3)

Inductive reactance Xσ is practically constant, because flux leakage is completing through the air gap.

Accordingly, main factors which define synchronous machines inductive reactance are elements of magnetic system, the geometrical sizes and a magnetic circuit’s saturation.

III. DEFINITION OF SYNCHRONOUS MACHINES PARAMETERS Taking into account magnetic asymmetry of salient-pole

synchronous machines, the main inductive reactance according to the two-reaction theory is a variable quantity

978-1-4244-8782-0/11/$26.00 ©2011 IEEE

which depends on position of a rotor concerning an axis of a stator’s magnetic field.

where kμd and kμq - in order, quadrature-axis and direct-axis armature reaction factors of magnetic flux’s saturation; kad and kaq - quadrature-axis and direct-axis armature reaction factors of the field’s form.

Figure 1. Voltage phasor diagram of saturated synchrnous motor.

Figure 2. Salient pole synchronous generator’s EMF determination (with saturation taken into account).

Precisely to take into account saturation’s influence on parameters Xad and Xaq of the salient-pole synchronous machine impossible, because of the quadrature-axis armature reaction of magnetic flux Φq influences a direct-axis armature reaction Φd. On this reactance Xad depends not only on Φd, but also on Φq. The same is distributed on Xaq. Saturation factors kad and kaq in expressions (4) - (5) are different, as well as variables and mutual dependent. Taking into account that there are no comprehensively checked up recommendations for the account of these factors in calculations using the simplified and approximate methods.

For a basis for this method application it is possible to use the vector diagrams shown on Fig. 1 and Fig. 2 [2].

The methods which are described, as example, [5], without paying attention to not quite correct theory, in many cases gives satisfactory results.

This method is based on two assumptions: 1) the main harmonic of a direct-axis field practically depends only on a resultant MF of armature Fd; 2) the main harmonic of a quadrature-axis field is possible to considering separately, but it depends on both Faq and Fad (the similar approach is shown in [6]).

In this case a field of direct-axis reaction induced EMF Eaq in the saturated regime is possible for calculating as well as non-saturated regime, i.e. as aqqaq XIE = , only to find Xaq it is necessary to take into account, that the size of this reactance depends on the main harmonic of a quadrature-axis field.

Hence, constructing of vector diagrams salient pole synchronous machines will not change, if reactance

σXXX aqq +=∞∞

replace with σXXX aqq += . Researches [5] show, that influence of saturation on reactance X is defined with a point on the characteristic of non-load which the resultant MF is equal aqa FFF += . Only the same point and consequently the reactance Xaq is possible to determine only when Fd and Faq are known, which are possible to find only by means of the vector diagram. While carrying out practical calculations, are necessary to build these diagrams using iterative procedures. As show such calculations, for synchronous machines with the non-load characteristic nearest to the normal non-load characteristic, two iterations are sufficient, besides in the first iteration reactance

∞aqX can be accepted. In the second iteration value Xaq is determined as

∞= aqaq XkX μ where kμ is found, using a point of the non-

load characteristic )(FfE = which corresponds to the certain value of a voltage, for example to the rated voltage NUE = .

It is necessary to note, that the mentioned methods of taking into account the saturation are not precise and to use them proving theoretical researches are necessary for which are based on numerical methods of mathematical modeling.

As is known, the operating mode of the synchronous machine can be described in four parameters: an armature

adw

dad k

pkw

kklmfX

21

2104

δτμ

π δμ

δ= , (4)

aqw

qaq k

pkw

kklmfX

21

2104

δτμ

π δμ

δ= , (5)

voltage U, an armature current I, a phase angle between these values ϕ and an excitation current If. Analyzing and studying characteristics of synchronous machines it is useful to use these four values, further on not connecting them with two-reaction parameters Xad and Xaq. The mentioned values are easily measured and experimentally checked. Taking into account these reasons it is possible to use the vector diagram shown on Fig. 3. (see [2]) on which simultaneously with time-

dependent sizes ϕδ ,,,•••EIU , excitation windings, an

armature winding and the resultant MF of the main harmonic

space vectors fa FF••

, and δ•F are shown.

This vector diagram corresponds to the synchronous machine’s main equations, which flows from the eclectic and circuit’s theory, when in these equations use "integrated" quantities (voltage, current, MF, magnetic flux, EMF, ect.).

Let's examine more detailed the main statement which allow finding above mentioned synchronous machine’s characteristics, using results of magnetic field’s mathematical modeling, by means of numerical methods. These methods are realized by means of accessible, effective, modern computer programs which, besides, using additional programs-postprocessors allow to adapt the received results to familiar methods of synchronous machine characteristic’s definition.

From numerical methods of magnetic fields calculation the most effective and widespread is a finite elements method (FEM). According to this method the main set quantity [8] is the machine’s geometrical sizes and its separate elements, excitation winding’s and all armature slot’s current density that unequivocally defines an angle ψ between quadrature-axis q and the anchor’s MF main harmonic’s maximum. It is possible to set medium characteristics, a material’s magnetization diagrams of a )(HfB = kind.

For the mathematical simulation of the magnetic field and obtaining results the complex multi-functional program QuickField [3] is used.

Figure 3. : Phasor diagram ofsynchronous motor (engine mode).

IV. DETERMINATION OF THE SYNCHRONOUS REACTANCE USING THE SYNCHRONOUS MACHINE’S MATHEMATICAL

SIMULATION RESULTS As known [2], the magnetic asymmetry of the salient pole

synchronous machine’s rotor ensures reluctances, which are bigger in quadrature axis (q direction) than the reactance in the direct axis (d direction). Therefore, for the synchronous machines with rotor’s magnetic asymmetry, it is appropriate to use a two-reaction method based on the superposition principle. According to this principle, the direct axis ( dΦ flux) and quadrature axis ( qΦ flux), where the magnetic flux is operating, are mutually independent. It should be noted, that this assumption is correct only for machines with unsaturated magnetic system. However, making additional adjustments based on the magnetic field mathematical simulation results; two-reaction method can also be used for machines with a saturated magnetic system.

According to the two-reaction method theory, the synchronous inductive reactances dX and qX , can be determined similar to (1).

In these equations, EMF dE and qE are the EMFs induced by the fundamental harmonic armature winding of the direct field and the quadrature field; dI and qI - armature current’s direct and quadrature components.

The effective value of the EMFs dE and qE can be determined using the following formulas

1144,4 wdd kfE Ψ= , 1144,4 wqq kfE Ψ= (6) where 1dΨ and 1qΨ - fundamental harmonic ( 1=ν ) amplitude values of flux linkages generated by the direct field and the quadrature field , 1wk - the fundamental harmonics winding factor.

The necessary sequence of operations is the following: first, determine 1dΨ and 1qΨ (see (6)), which, in turn, are required for reactance dX and qX determination.

To determine the flux linkage )( id tωΨ with sufficient accuracy, it is preferably that the time step is chosen small enough. As experience of numerical experiments shows, the rotor rotation angle step is appropriate to be chosen so, that

2zt=Δα , where zt - the armature tooth pitch.

The magnetic flux of one pole can be obtained with different armature current instantaneous values according to (7) with the rotor position angle iα . After a series of magnetic field calculations for time moments it

lAAt iiidid )()()( 21 −=Φ=Φ αω (7) and flux linkage with armature winding phase as a function of time

wlAt iid )()( 1=Ψ ω (8)

where iA1 and iA2 - the vector potential values on the surface of the armature in 180 degree opposite points, w - number of turns per phase, l - the machine length in axial direction.

Numerical harmonic analysis of the function )( id tωΨ allows to obtain the fundamental harmonic amplitude value

1dΨ of the flux linkage’s, which is required according to formula (6) for determination of EMF dE .

Similarly to the expression (6), the EMF qE can be

obtained if α is replaced with 90−α while simulating quadrature axis magnetic field.

V. RESULTS The simulation of the synchronous machine magnetic field

is made for the experimental three-phase machine with 22 =p , in which a single layer full step ( τ=y ) winding is

located in armature 36 slots. Calculations were made for five armature current values NII )2.1;0.1;8.0;6.0;4.0(= , where the effects of saturation on the magnetic field’s character, on parameters dX and qX values are evaluated.

Because of the periodicity and symmetry of the function )( ti ω , it is sufficient if the calculations are made in one

quarter of the period of current changes, that is 90== tωα (electric degree).

Figure 4 shows the direct axis and quadrature axis magnetic field’s picture, acquired for machine, if the armature current components dI and qI in relative units 1== qd II , i.e. equal to the rated current.

Flux linkage curves )( id tωΨ and )( iq tωΨ are obtained with calculations carried out for different rotor rotation angles, observed with the armature phase currents angle for moments

it ( ii tωα = ).Various armature currents *dI and *

qI values for one half of the period are shown on Figure 5.

The flux linkage fundamental harmonics 1dΨ and 1qΨ are determined during numerical harmonic analysis of )( id tωΨ and )( iq tωΨ . The EMFs dE and qE values for various armature currents are calculated and presented in table 1.

Direct-axis reaction a) Quadrature-axis reaction b)

Figure 4. Armature direct-axis reaction (a) and quadrature-axis reaction (b) magnetic field pictures for angle in moment tω , when the phase A current is

at maximum value.

TABLE I. EMF Ed AND Eq FUNDAMENTAL HARMONICS DEPENDENCE ON THE ARMATURE CURRENTS.

*I )(VEd )(VEq

0.4 44.44 17.45 0.6 65.09 26.16 0.8 80.94 34.58 1.0 93.29 42.60 1.2 103.49 49.90

Reactances dX and qX calculated from formula (2) are

given in table 2. Table 2 also indicates the saturation coefficients dkμ and qkμ values corresponding to the rated

current 1* =I .

TABLE II. REACTANCES DEPENDENCE FROM SATURATION LEVEL INDUCED BY ARMATURE CURRENTS

*I )(ΩdX )(ΩqX qd XX dkμ qkμ

0.4 16.93 6.65 2.55 0.6 16.53 6.64 2.49 0.8 15.42 6.59 2.34 1.0 14.22 6.49 2.19 1.22 1.02 1.2 13.14 6.34 2.07

VI. CONCLUSIONS The classical synchronous machines theory is based on a

superposition principle and in which it is entered direct-axis and quadrature-axis reaction’s inductive reactances adX and

aqX is not correct for the saturated machines and does not give those advantages, because of what it was developed.

Saturation effect on parameters dX and qX can be

estimated by the saturation factors dkμ and qkμ . Saturation factors values are depending on magnetic system geometric dimensions and on level of saturation ( 5,1...2,1=dk μ , 2,1...0,1=qk μ ). Estimation results are also confirmed by the experimental results of examined machine.

REFERENCES

[1] Zviedris . Electromagnetic calculations in electrical machines. Riga, RPI, 1973.

[2] Voldek A. Electrical machines. Leningrad, Energija, 1978.

[3] QuickField. Finite element analysis system. Version 5.7. User’s guide. Svendborg. Tera Analysis Ltd, 2009.

[4] Bianchi N. Electrical machines analysis using finite elements. Boca Raton, CRC Press, 2005.

[5] Vazhnov Electrical machines. Leningrad, Energija, 1969.

[6] Ivanov-Smolenskij . Electrical machines. Moscow, Energija, 1980.

[7] A. Blondel. Complements a la theorie de alternateurs a deu reactions, RGE, 1922.