Open access peer-reviewed chapter

Analysis of Electrical Machines

By Valéria Hrabovcová, Pavol Rafajdus and Pavol Makyš

Reviewed: March 3rd 2020Published: May 20th 2020

DOI: 10.5772/intechopen.91968

Downloaded: 21

Abstract

The aim of this book is to present methods needed for the analysis of electrical machine performances in transients and steady-state conditions by means of simulations in MATLAB-Simulink and by means of FEM. The parameter determination based on the design procedure is presented, and some examples are given. The authors used outputs of their long-time experiences in research and scientists work as university professors and use their pedagogical skills to create a guide on how to investigate parameters of equivalent circuits and apply them for investigation of transients and steady-state conditions.

Keywords

  • general theory of rotating electrical machines
  • parameters of electrical machines and their calculation during their design
  • resistances
  • inductances
  • skin effect influence on the resistance and inductance
  • transients of electrical machines
  • parameters of equivalent circuits
  • FEM used to investigate parameters of electrical machines
  • induced voltage
  • electromagnetic torque
  • torque ripple
  • cogging torque

Introduction

In a coincidence with the title of this book, we can start with explaining what a term of analysis means: in general, it is a system of methods by means of which properties of investigated matters are gained. Here the properties of electrical machines are analyzed; therefore, it is welcomed to introduce the methods on how to do it.

To proceed in the investigation of transients and steady-state condition, it is necessary to know equivalent circuit parameters (resistances and inductances). The first method that is given below in Chapter 1 is the method of parameters calculated based on the design process in which geometrical dimensions and material properties must be known (see [1, 2, 3]). The other method on how to get the equivalent circuit parameters is to make measurements and testing, but this can be done only on the fabricated pieces. It is a very welcomed method on how to verify the calculated parameters gained during the design process. The method of measurement is not given here and can be found, e.g., in [4, 5, 6].

The other approach is if there is a real fabricated machine but without any documents and data. Then it is very useful to make the so-called inverse design calculation. It means to take all geometrical dimensions which can be seen on the real machine and get data from the real machine nameplate and catalogues, e.g., voltage, current, power, speed, pole numbers, slot shape, number of slots, number of conductors in the slot, etc. and to continue in the calculation of air gap magnetic flux density, etc. to the required parameters.

The general theory of electrical machines is presented in Chapter 2. It is possible to investigate transients and steady-state conditions of electrical machines by means of this theory, see [7, 8, 9]. The transients are solved on the basis of differential equations in which the parameters of equivalent circuits, i.e., resistances and inductances, are needed. The accuracy and reliability of the simulation results depend on the accuracy of the parameter values. Therefore, determination of the parameters must be done with the highest care.

Chapter 3 is formulated in this sense. This chapter is devoted to the modern computer method called finite element method (FEM), see also [10]. This method enables to investigate not only some parameters, mainly magnetizing inductances, but also the other properties such as losses, air gap developed torque, ripple torque, efficiency, etc., see [11, 12, 13].

It is important to add that verification of the calculated and simulated values and waveforms is made by measurement on a real machine if it exists. It is recommended to create a reliable simulation model. It means the gained simulated outputs verify on a real machine, and if the coincidence of measured and simulated values is satisfactory, such model can be employed to optimize geometrical dimensions or a concrete configuration, like slot shape, rotor barriers, etc., and the simulation outputs are considered as reliable. In such a way, it is possible in pre-manufacturing period to optimize the construction of the machine to the required properties, e.g., maximal torque, minimal ripple torque, maximum efficiency, and so on.

The authors would like to point out that all simulation models of electrical machines analyzed in this book are shown at the end as Appendixes A, B, C, and D, in the MATLAB-Simulink program. These models were employed in the appropriate chapters during the investigation of the simulation waveforms of all electrical machines.

Parameters of the Rotating Electrical Machines

1.1 Inductances

In the rotating electrical machines, the total magnetic flux can be divided into two components: main flux (air gap flux) and leakage flux. The main flux enables electromagnetic energy conversion, but a proportion of the total flux does not participate in energy conversion, and this part is called leakage flux. The main flux must cross the air gap of rotating machine and its function is electromagnetically connected to both stator and rotor windings. The leakage flux is linked only with this winding in which it was created.

The main magnetic flux in air gap Φμcreates linkage flux ψμand corresponds to the magnetizing inductance Lμ. The leakage flux Φσcreates leakage linkage flux ψσand corresponds to the leakage inductance Lσ. In the case of induction machines, stator inductance is the sum of magnetizing and stator leakage inductance: Ls=Lμ+Lσs. In the case of synchronous machines, this inductance is called synchronous inductance. In the next part, all components of the inductances are investigated.

1.1.1 Magnetizing inductance

Magnetizing inductance is the most important inductance of electrical machine and is defined by the winding, geometrical dimensions of the magnetic circuit, and the employed materials. It is derived from m-phase machine on the basis of magnetic flux density distribution on the surface of the rotor and its maximal value Bδmax(Figure 1), on the span of the pole pitch τp, and the reduced length of the machine l′ = lFe + 2δ.

Figure 1.

Distribution of the magnetic flux density fundamental harmonic in the air gap over a pole pitch and on the length of the lFe.

This relationship is valid for machines without ventilating ducts and takes into account the magnetic flux distribution on the end of the machines [1].

Maximal value of the air gap magnetic flux can be calculated as a surface integral of magnetic flux density B on one pole surface S:

Φmax=SBdS=αiτplBδmax,E1

where αiis a ratio of the arithmetic average value of the magnetic flux density Bav and maximal value of the magnetic flux density:

αi=BavBδmax.E2

The sinusoidal distributed waveform of the magnetic flux density is αi=2/π.

An expression “maximal value of the magnetic flux” means maximal flux which penetrates the surface created by the coil and therefore creates one phase maximal linkage flux of the winding on the stator (the subscript “ph” is used), with number of turns Ns and winding factor kws:

ψμph=NskwsΦμ=NskwsαiτplBδmax.E3

Magnitude of air gap magnetic flux density can be expressed on the basis of current linkage and the relationship between magnetic flux density and intensity of magnetic field. In the most simple case, B=μH,Hl=NI,H=NI/l, where the permeability of vacuum is taken and the length of the magnetic force line is only the length of air gap. The source of magnetomotive force (current linkage) is expressed for fundamental harmonic of rectangular waveform of the single-phase winding:

Umagmaxph=4πNskws2p2Is.E4

Then the magnitude of the magnetic flux density is:

Bδmax=μ0δefUmagmaxph=μ0δef4πNskws2p2Is,E5

and after the substitution to the expression of flux linkage, it yields:

ψμph=Nskwsαiτplμ0δef4πNskws2p2Is=αiτplμ0δef4πNskws22p2Is.E6

By dividing the result by the peak value of the current, which in this case is magnetizing current, we obtain the magnetizing inductance of a single-phase winding (the main inductance):

Lμph=αiμ0δef4π12pτplNskws2=αiμ0δef2πNskws2pτpl.E7

If the winding is multiphase, the magnetic flux is created by all the phases of the winding, with the corresponding instantaneous values of the currents. As it is known, the three-phase winding creates the magnitude of the air gap magnetic flux density equal to 1.5 multiple of that which is created by the single phase. The magnetizing inductance of an m-phase winding can be calculated by multiplying the main inductance by m/2:

Lμ=m2αiμ0δef2πNskws2pτpl=αimμ0δefτplπNskws2p,E8

and after the substitution of pole pitch, the expression is obtained, in which its dependence on the parameters and geometrical dimensions of the machine is seen:

Lμ=αimμ0δefDδπNskws2p2l.E9

Here it is seen that the magnetizing inductance depends on magnetic circuit saturation, i.e., αi, effective air gap δef, in which the Carter factor and saturation of magnetic circuit is included, on the length of the machine l′, phase number m, and quadrate of the effective number of the turns Nskws. In Eq. (9), it is seen that the magnetizing inductance is inversely proportional to p2, which means that in the case of asynchronous motors where it is important to get Lμ as high as possible, it is not suitable to employ multipole arrangement. In the case of the synchronous machines, the developed torque is inversely proportional to the synchronous inductance and also to the magnetizing inductance. Therefore, in the synchronous machines, there are multipole arrangements with lower inductance ordinary.

Effective air gap δefincludes Carter factor as well as the effect of the magnetic circuit saturation. This influence is in the interval of some to 10%. In such a case the magnetic circuit is already considerably saturated. In very precisely designed induction, motors can be the current linkage (magnetomotive force) needed for iron parts of magnetic circuit greater than for the air gap. On the other side in synchronous machines with permanent magnets, in which the equivalent air gap includes in d-axis also the length of permanent magnets, the value of current linkage needed for iron parts is very small.

Magnetizing inductance is not constant but depends on the voltage and the torque. The higher voltage activates the higher magnetic flux density; this activates higher saturation of magnetic circuit, and this requires higher magnetizing current.

1.1.2 Leakage inductance

Leakage inductances are described by the leakage magnetic fields, which are linked only with the turns of that winding by which they were created. It means they do not cross the air gap.

In greater detail, it can be said that leakage magnetic fluxes include the following:

  • All components of magnetic field that do not cross air gap

  • Those components of magnetic field that cross the air gap, but they do not take part in the electromechanical energy conversion

The leakage fluxes that do not cross the air gap can be divided into the next components:

  1. Slot leakage flux Lσd(slot leakage inductance)

  2. Tooth tip leakage flux Lσz(leakage tooth tip inductance)

  3. End winding leakage flux Lσew(end winding leakage inductance)

  4. Pole leakage flux Lσp(pole leakage inductance)

Leakage fluxes that do not cross the air gap are included into the air gap magnetic flux ΦδLδ(air gap leakage inductance). Air gap magnetic flux do not link completely windings of the machine because of short pitching, slot skewing, and spatial distribution of the winding, causing air gap harmonic components in the air gap, and do not contribute to the electromechanical energy conversion. The weaker linking between the stator and rotor windings caused by the short pitching and slot skewing is taken into account by means of the pitch winding factor kp and skewing factor ksq.

According to the electrical motor design tradition, leakage inductance Lσcan be divided into the following partial leakage inductance: skew leakage inductance Lσsq, air gap leakage inductance, slot leakage inductance, tooth tip leakage inductance, and end winding leakage inductance. The leakage inductance of the machine is the sum of these leakage inductances:

Lσ=Lσsq+Lσδ+Lσd+Lσz+Lσew.E10

1.1.2.1 Skew leakage inductances

Skewing slot factor defines skewing leakage inductance:

Lσsq=σsqLμ,E11

where the factor of the skewing leakage σsqis given by the skewing factor ksq:

σsq=1ksq2ksq2.

At the skewing by one slot, this factor is given by the expression:

ksq=sinπ21mqπ21mq,E12

where q is the number of slots per phase per pole, and it is possible to calculate for each υ—harmonic component:

ksqυ=sinυπ21mqυπ21mq.E13

Example 1. Calculate skewing factors for the fundamental stator slot harmonic component in four-pole rotor cage induction motor with 36 slots on the stator when the rotor slots are skewed by one stator slot.

Solution: As it is known, m-phase winding creates harmonic components of the order υ=1±2cm, where c = 0, 1, 2, 3, and so on.

The number of stator slots per phase per pole q = 36/(3·4) = 3. The first stator slot harmonics are 1±2mqc=1±2·3·3·c=17,19,35,37,if c = 1, 2. The skewing factor according to Eq. (13) for the fundamental and further harmonics is:

υ1−1719−3537
ksq0.9950.06−0.05−0.030.03

It can be seen that the lowest order stator harmonics (−17, 19, −35, 37) have very small skewing factors, and thereby their effects are eliminated to a great degree (3–6%). The fundamental harmonic is reduced only by 0.5%.

1.1.2.2 Air gap leakage inductance

Electromotive force (induced voltage) is given by the magnetizing inductance as a result of a propagating fundamental component of air gap flux density. Because of a spatial slotting and winding distribution, the permeance harmonics induce voltage of fundamental frequency of the winding. The air gap leakage inductance, i.e., the harmonic leakage inductance components, takes this into account. In integer slot machines per phase per pole q, the air gap leakage remains usually low. But in the case of fractional slot machines mainly in the machines with concentrating coils wound around the tooth, its influence can be even dominating. It relates to machines with permanent magnets. This case can be studied in greater details in [1].

The sum of all induced voltages from all harmonic components, fundamental included, gives the basis for the calculation of the total inductance, which is the sum of magnetizing and air gap leakage inductance:

Lμ+Lσδ=μ0πmδDlNp2υ=υ=+kυ2.E14

The expression for the fundamental harmonic, i.e., υ=1, represents fundamental component; it means magnetizing inductance of the machine Lμ, which is calculated on the basis of Eq. (9), if αi=2/πis introduced. The rest of the equation represents air gap inductance (υ=1is omitted):

Lσδ=μ0πmδDlNp2υ=υ1υ=+kυ2.E15

The air gap leakage inductance can be expressed also by means of the air gap leakage factor:

σδ=υ=υ1υ=+kυkw12.E16

Then the leakage inductance is given by:

Lσδ=σδLμ.E17

Of course, in Eq. (16), only the harmonics that are created by the given winding are used.

1.1.2.3 Slot leakage inductance

This inductance is created by a real leakage flux, which is closed through the space of the slot. Magnetic permeance of the magnetic circuit is taken as infinite; therefore the length of the magnetic circuit force line is taken only by the width of the slot (in the slot there is nonmagnetic material, i.e., there the vacuum permeability is used). For the rectangular slot, the magnetic permeance is derived in [1]. Gradually integrating the magnetic force lines and magnetic flux density along the slot height h in the area where the current flows (Figure 2), the magnetic permeance of the slot for the slot leakage is obtained:

Figure 2.

Geometrical dimensions of various slot shapes to define their permeance factors, the calculations of which are given in the text.

Λmag=μ0lh43b4.E18

The permeance factor λis defined, because in each slot the vacuum permeability and the length of the machine act:

λ=Λmagμ0l.E19

For the rectangular slot (Figure 2a) with the slot width b4 and the slot height h4, the permeance factor is:

λ4=h43b4.E20

In the slot area with the height h1, there is no current; therefore:

λ1=h1b1E21

The sum λ1+λ4=λdis the permeance factor of the whole slot, and the leakage inductance of the slot is:

Lσd=4mQμ0lN2λd=2N2pqμ0lλd,E22

where Q is the number of slots around the machine periphery, 2p is the number of poles, q is the number of slots per pole per phase, and N is the phase number of turns.

Equation (22) is derived from [1] and other references dealing with the machine design.

The expressions gained on the basis of magnetic permeance integration along the slot height of single-layer winding according to Figure 2 are given.

For the slot shapes b, c, d, e, and f, the permeance factor will be calculated as follows:

λd=h43b4+h3b4+h1b1+h2b4b1lnb4b1.E23

For the slot in Figure 2g, the next expression is known:

λd=h43b4+h1b1+0.685,E24

and for the round slot from Figure 2h, the next expression is valid:

λd=0.47+0.066b4b1+h1b1.E25

The slot leakage inductance of the double-layer winding, according to [1], on the basis of Figure 3, the appropriate expression is derived. It is necessary to consider that in some slots there are coil sides of different phases.

λd=k1h4h3b4+k2h1b4+h4b4,E26

Figure 3.

Geometry of the slot with double-layer winding.

where

k1=5+3g8,E27
k2=1+g2.E28

The factor g is linked with the fact that in double-layer winding with short pitching, the coil sides of upper and lower layers in some slots belong to different phases. If the phase shifting of the upper and lower layers is γ, the total current linkage must be multiplied by cosγ. Because the phase shifting can be different in each slot, the average value g on the 2q coil sides is:

g=12qn=12qcosγn,

but the factors k1 and k2 can be calculated also on the basis of short pitching (see below).

Similar to Eq. (26), also the equation for double-layer winding can be written for the slots in Figure 2 from (b) till (f):

λd=k1h4h3b4+k2h3b4+h1b1+h2b4b1lnb4b1+h4b4,E29

and for slots from (g) till (i):

λd=k1h4h3b4+k2h1b1+0.66+h4b4.E30

These expressions are valid also for the slots with single-layer winding, if h′= 0 and k1 = k2 = 1 are introduced.

If the winding is short pitching , factors k1 and k2 can be calculated by means of short pitching Y/Qp, where Y is a real pitch and Qp is a full pitch (the pole pitch is expressed by means of number of slots per pole), in this form:

For three-phase winding:

k1=1916ε,k2=134ε,E31

where

ε=1YQp.E32

For two-phase winding:

k1=134ε,k2=1ε.E33

Example 2. Calculate slot leakage inductance of a double-layer winding if 2p = 4, m = 3, Q = 24, Y/Qp = 5/6, and N = 40. The slot shape and dimensions are according to Figures 2c and 3 as follows:

b1 = 0.003 m, h1 = 0.002 m, h2 = 0.001 m, h3 = 0.001 m, h′ = 0.001 m, b4 = 0.008 m and h41 = h42 = 0.009 m, (h4 = 0.019 m), l′ = 0.25 m.

Compare the results with the slot leakage inductance of a corresponding double-layer full-pitch winding.

Solution: The short pitching is ε=1/6, and k1=191616= 0.906, k2=13416=0.875. The permeance factor is according to Eq. (29):

λd=k1h4h3b4+k2(h3b4+h1b1+h2b4b1lnb4b1)+h4b4=0.9060.0183·0.008+0.875(0.0010.008+0.0020.008+0.0010.0080.003ln0.0080.003)+0.0014·0.008=1.211

and slot leakage inductance is according to Eq. (22):

Ld=4mQμ0lN2λd=4·3244π·107·0.25·402·1.211=0.2513·103·1.211=0.304mH,

for a double-layer full-pitch winding k1 = k2= 1 and Eq. (29) yields:

λd=k1h4h3b4+k2h3b4+h1b1+h2b4b1lnb4b1+h4b4=0.0183·0.008+0.0010.008+0.0020.008+0.0010.0080.003ln0.0080.003+0.0014·0.008=1.352

The slot leakage inductance is now:

Lσd=4mQμ0lN2λd=0.2513·103·1.352=0.340mH

It is seen that the phase shift of the different phase coil sides in the double-layer winding causes a smaller slot leakage inductance for the short-pitched winding than the full-pitch winding. The slot leakage inductance in this case is about 10% smaller for the short-pitched winding.

1.1.2.4 Tooth tip leakage inductance

The tooth tip leakage inductance is determined by the magnitude of leakage flux flowing in the air gap outside the slot opening. This flux is illustrated in Figure 4. The current linkage in the slot causes a potential difference between the teeth on opposite sides of the slot opening, and as a result a part of the current linkage will be used to produce the leakage flux of the tooth tip.

Figure 4.

Tooth tip flux leakage around a slot opening, creating a tooth tip leakage inductance.

Tooth tip leakage inductance can be determined by applying a permeance factor:

λz=k25δb15+4δb1,E34

where k2=134εis given by Eq. (31). The tooth tip leakage inductance of the whole phase winding is given by applying Eq. (22):

Lσz=4mQμ0lN2λz.E35

In the machines with salient poles, the air gap is taken at the middle of the pole, where the air gap is smallest. If the air gap is selected to be infinite, a limit value of 1.25 is obtained, which is the highest value for permeance factor λz. If the air gap is small, as in the case of asynchronous machines, the influence of the tooth leakage inductance is insignificant. The above given equations are not valid for the main poles of DC machines. The calculation for synchronous machines with permanent magnets is in Example 3.

Example 3. Calculate the tooth tip leakage of the machine in Example 2. The machine is now equipped with rotor surface permanent magnets neodymium-iron-boron of 8 mm thickness. There is a 2 mm physical air gap. Compare the result with the value of inductance in Example 2.

Solution: As the permanent magnets represent, in practice, air with relative permeability μrPM = 1.05, we may assume that the total air gap in the calculation of the tooth tip leakage is:

δ=2+81.05=9.62mm.

The factor k2=13416=0.875.

Then the factor of tip tooth permeance is:

λz=k25δb15+4δb1=0.87550.009620.0035+40.009620.003=0.787,

and the tooth tip leakage inductance is:

Lσz=4mQμ0lN2λz=4·3244π·107·0.25·0.787·402=0.198mH.

In Example 2, the slot leakage inductance was 0.34 mH. As the air gap in a rotor surface magnet machine is long, the tooth tip leakage has a significant value, about 70% of the slot leakage inductance.

1.1.2.5 End winding leakage inductance

End winding leakage flux results from all the currents flowing in the end windings. The geometry of the end windings is usually difficult to analyze, and, further, all the phases of polyphase machines influence the occurrence of a leakage flux. Therefore, the accurate determination of an end winding leakage inductance would require three-dimensional numerical solution. On the other side, the end windings are relatively far from the iron parts; the end winding inductances are not very high. Therefore, it suffices to employ empirically determined permeance factor.

According to Figure 5, the end winding leakage flux is a result of influence of all coil turns belonging to the group coils q.

Figure 5.

Leakage flux and dimensions of the end winding.

If according to Eq. (22) this q-multiple of the slot conductors is put and instead of the length of the machine, the length of the end winding lew is introduced, the equation for the end winding inductance calculation is as follows:

Lσew=2pμ0N2lwλew.E36

The average length of the end winding lwand the product of lwλewcan be, according to Figure 5, written in the form:

lw=2lew+Yew,E37
lwλew=2lewλlew+YewλYew,E38

where lew is the axial length of the end winding, measured from the iron laminations, and Yew is a coil span according to Figure 5. Corresponding permeance factors λlewand λYewcan be changed according to the type of stator and rotor windings and are shown in, e.g., [1] in Table 4.1 and 4.2. At a concrete calculation of the real machines, it can be proclaimed that Eq. (36) gives the sum of the leakage inductance of the stator and leakage inductance of the rotor referred to the stator and that the essential part 60–80% belongs to the stator.

Example 4. The air gap diameter of the machine in Example 2 is 130 mm, and the total height of the slots is 22 mm. Calculate end winding leakage inductance for a three-phase surface-mounted permanent magnet synchronous machine with Q = 24, q = 2, N = 40, p = 2, lew = 0.24 m. The end windings are arranged in such a way that λlew=0.324and λYew=0.243.

Solution: Let us assume that the average diameter of the end winding is 130 + 22 = 152 mm. The perimeter of this diameter is about 480 mm. The pole pitch at this diameter is τp=480/4=120mm. From this it can be assumed that the width of the end winding is about the pole pitch subtracted by one slot pitch:

Yew=τpτd=0.120.4824=0.1m.

and the length of the end winding is:

lew=0.5lwYew=0.50.240.1=0.07m.

The product of the length and permeance factor is:

lwλew=2lewλlew+YewλYew=2·0.07·0.324+0.1·0.243=0.07m.

and end winding leakage inductance is:

Lσew=2pμ0N2lwλew=224π·107·402·0.07=0.1407mH.

The slot leakage inductance of the 5/6 short-pitched winding in Example 2 is 0.304 mH, so it is seen that the slot leakage inductance is much higher than end winding leakage inductance.

1.2 Resistances

Not only inductances but also resistances are very important parameters of electrical machines. In many cases winding losses are dominant components of the total loss in electrical machines. The conductors in electrical machines are surrounded by ferromagnetic material, which at saturation can encourage flux components to travel through the windings. This can cause large skin effect problems, if the windings are not correctly designed. Therefore, this phenomenon must be considered.

It is convention to define at first the DC resistance RDC, which depends on the conductivity of the conductor material σc, its total length lc , cross-section area of the conductor Sc, and the number of parallel paths a in the winding without a commutator, per phase, or 2a in windings with a commutator:

RDC=lcσcaSc.E39

Resistance is highly dependent on the running temperature of the machine; therefore a designer should be well aware of the warming-up characteristics of the machine before defining the resistances. Usually the resistances are determined at the design temperature or at the highest allowable temperature for the selected winding type.

Windings are usually made of copper. The specific conductivity of pure copper at room temperature, which is taken 20°C, is σCu= 58 × 106 S/m, and the conductivity of commercial copper wire is σCu= 57 × 106 S/m. The temperature coefficient of resistivity for copper is αCu= 3.81 × 10−3/K. Resistance at temperature t increased by Δtis Rt=R1+αCuΔt. The respective parameters for aluminum are σAl= 37 × 106 S/m, αAl= 3.7 × 10−3/K.

The accurate definition of the winding length in an electrical machine is a fairly difficult task. Salient-pole machines are a relatively simple case: the conductor length can be defined more easily when the shape of the pole body and the number of coil turns are known. Instead winding length of slot windings is difficult, especially if coils of different length are employed in the machine. Therefore, empirical expressions are used for the length calculation.

The average length of a coil turn of a slot winding lav in low-voltage machines with round enameled wires is given approximately as:

lav2l+2.4Yew+0.1m,E40

where l is the length of the stator stack and Yew is the average coil span (see Figure 5), both expressed in meters. For large machines with prefabricated windings, the following approximation is valid:

lav2l+2.8Yew+0.4m.E41

When the voltage is between 6 and 11 kV, the next can be used:

lav2l+2.9Yew+0.3m.E42

After the average length is determined, the DC resistance may be calculated according to Eq. (39), by taking all the turns and parallel paths into account.

1.3 Influence of skin effect on winding resistance and inductance

1.3.1 Influence of skin effect on winding resistance

The alternating current in a conductor and currents in the neighboring conductors create an alternating flux in the conductor material, which causes skin and proximity effects. In the case of parallel conductors, also circulating currents between them appear. The circulating currents between parallel conductors can be avoided by correct geometrical arrangement of the windings. In this chapter, the skin and proximity effects will be dealt together and called the skin effect.

Skin effect causes displacement of the current density to the surface of the conductor. If the conductor is alone in the vacuum, the current density is displaced in all directions equally to the conductor surface. But in the conductors embedded in the slots of electrical machines, the current density is displaced only in the direction to the air gap. In this manner, the active cross-section area of the conductors is reduced, increasing the resistance. This resistance increase is evaluated by means of resistance factor. It is the ratio of the alternating current resistance and direct current resistance:

kR=RACRDC.E43

The direct consequence of the resistance increase is loss increase; therefore, the resistance factor can be expressed also by the ratio of the losses at alternating current and direct current:

kR=ΔPACΔPDC.E44

Loss increase because of skin effect is the reason why it is necessary to deal with this phenomenon in the period of the machine design and parameter determination. Resistance and losses at alternating current can be calculated on the basis of Eq. (43) or (44), if the resistance factor is calculated by means of the equations given below.

In electrical machines the skin effect occurs mainly in the area of the slot but also in the area of end winding. The calculation in these two areas must be made separately, because the magnetic properties of the material in the slot and its vicinity and the end winding are totally different.

Analytical calculation of resistance factor which includes skin effect influence is given in many books dealing with this topic; therefore, here only expressions needed for resistance factor calculation are shown. An important role in the theory of skin effect is the so-called depth of penetration, meaning the depth to which electromagnetic wave penetrates into a material at a given frequency and material conductivity. The depth of penetration depends on the frequency of alternating current, specific electric conductivity of the conductor material σc, and vacuum magnetic permeability, because the conductor in the slot is a nonmagnetic material. For example, for cooper at 50 Hz, the depth of penetration yields approximately 1 cm. In Figure 6a, it is seen that bc is the conductor width in the slot and b is the total width of the slot. Then the depth of penetration is:

Figure 6.

Determination of reduced conductor height (a) if in the slot with width b there is only one conductor with the width bc and the height hc (b) if in the slot there are several conductors, za adjacent conductors and zt, conductors on top of each other. The width of one conductor is bc0, height hc0.

a=2ωμ0σcbbc.E45

The conductor height hc is obviously expressed in ratio to the depth of penetration. Then the conductor height is called the reduced conductor height ξ. It is a dimensionless number:

ξ=hca=hcωμ0σc2bcb.E46

Note that the product of specific electric conductivity and the ratio of conductor width to the slot width express the reduced electric conductivity of the slot area σcbcb, because not the whole slot width is filled with the conductor.

If in the slot there are rectangular conductors placed za adjacent and zt on top of each other, the reduced conductor height is calculated according to:

ξ=hc0a=hc0ωμ0σc2zabc0b.E47

The resistance factor of the kth layer is:

kRk=φξ+kk1ψξ,E48

where the functions φξand ψξare derived based on the loss investigation in the conductor placed in the slot of electrical machines and are given as follows [1]:

φξ=ξsinh2ξ+sin2ξcosh2ξcos2ξ,E49
ψξ=2ξsinhξsinξcoshξ+cosξ.E50

Equation (48) shows that the resistance factor is smallest on the bottom layer and largest on the top layer. This means that in the case of series-connected conductors, the bottommost conductors contribute less to the resistive losses than the topmost conductors. Therefore, it is necessary to calculate the average resistance factor over the slot:

kRd=φξ+zt213ψξ.E51

where zt is number of the conductors on top of each other.

If ξ is in the interval 0ξ1, Eq. (51) can be simplified:

kRd=1+zt20.29ξ4.E52

Equations above are valid for rectangular conductors. The eddy current losses (skin effect losses) of round wires are 0.59 times the losses of rectangular wire. If in the slot there are round conductors, resistance factor and also the eddy current losses are only 59% of that appeared in the rectangular conductors. Therefore, for the round conductors, Eq. (52) will have a form:

kRd=1+0.59zt20.29ξ4.E53

An effort of the designer is to reduce the resistance factor what would result in the reduction of the losses. Obviously, it is recommended to divide the height of the conductor: it means to make more layers zt. As shown in Eq. (52), the resistance factor is proportional to the square of the number of conductors on top of each other zt, which would look like the resistance factor would increase. But the reduced conductor height is smaller with the smaller conductor height, and according to Eqs. (52) and (53), the reduced conductor height ξ is with exponent 4. Therefore, the resistance factor finally will be lower.

If the conductors are divided into parallel subconductors, which are connected together only at the beginning and at the end of winding, they must be also transposed to achieve effect of the reduction of the resistance factor and of the eddy current (skin effect) losses. Transposition must be made in such a way that all conductors are linked with the equal leakage magnetic field. It means that the changing of the conductor positions must ensure that all conductors engage all possible positions in the slot regarding the leakage magnetic flux. Without transposition of the subconductors, a divided conductor is fairly useless. Further details are given in [1, 2, 3].

1.3.2 Influence of skin effect on the winding inductance

If the height of the conductor is large, or if the winding is created only by one bar, as it is in the case of the squirrel cage of asynchronous machines, in the conductors with alternating current, skin effect appears. The skin effect is active also at the medium frequencies and has considerable influence on the resistance (see Section 3.1) and on the winding inductance too. That part of winding, which is positioned on the bottom of the slot, is surrounded by the higher magnetic flux than that on the top of the slot. Therefore, the winding inductance on the top of the slot is lower than that on the bottom of the slot, and therefore the time-varying current is distributed in such a way that the current density on the top of the slot is higher than that on the bottom of the slot. The result is that skin effect increases conductor resistance and reduces slot leakage inductance. To express the decrease of the inductance, the so-called skin effect factor kL is introduced. This term must be supplemented to the equation for the magnetic permeance. Therefore, Eq. (20), which is valid for the slot on Figure 2a, must be corrected to the form:

λ4,L=kLh43b4.E54

To calculate the skin effect factor, we need to define the reduced conductor height:

ξ=h4ωμ0σbc2b4,E55

where bc is the conductor width in the slot, σ is the specific material conductivity of the conductor, and ωis the angular frequency of the investigated current. For example, in the rotor of induction machine, there is the angular frequency given by the slip and synchronous angular frequency. Then the skin effect factor is a function of this reduced conductor height and the number of conductor layers on top of each other zt:

kL=1zt2ϕξ+zt21zt2ψξ,E56

where

ϕξ=32ξsinh2ξsin2ξcosh2ξcos2ξ,E57
ψξ=1ξsinhξ+sinξcoshξ+cosξ.E58

In the cage armature, zt = 1; therefore, the skin effect factor is:

kL=ϕξ.E59

In the cage armature, it is usually h4>2cm, and for cooper bars, it is according to Eq. (55) ξ>2. Then sinh2ξsin2ξ,and cosh2ξcos2ξ, whereby sinh2ξcosh2ξ; consequently the kL is reduced to simple expression:

kL32ξ.E60

Example 5. Calculate the slot leakage inductance of aluminum squirrel cage bar zQ = 1 at cold start and 50 Hz supply. The slot shape is according to Figure 2a, b1 = 0.003 m, h1 = 0.002 m, b4 = 0.008 m, h4 = 0.02 m, l´= 0.25 m, and a slot at height h4 is fully filled with aluminum bar. The conductivity of aluminum at 20°C is 37 MS/m.

Solution: The permeance factor of that part of slot, which is filled by a conductor without skin effect, is:

λ4=h43b4=0.023·0.008=0.833.

The permeance factor of the slot opening is:

λ1=h1b1=0.0020.003=0.667.

The reduced height ξ of the conductor, which is a dimensionless number, is:

ξ=h4ωμ0σbc2b4=0.022π·50·4π·107·37·1060.0082·0.008=1.71.

Then the inductance skin effect factor is:

kL=1zt2ϕξ+zt21zt2ψξ=ϕξ+111ψξ=ϕξ=32ξsinh2ξsin2ξcosh2ξcos2ξ,
kL=32·1.71sinh3.42sin3.42cosh3.42cos3.42=0.838,

and permeance factor of the slot under the skin effect is:

λd=λ1+kLλ4=0.667+0.838·0.833=1.37.

The leakage inductance of a squirrel cage aluminum bar if skin effect is considered is:

Ld,bar=μ0lzQ2λd=4π·107·0.25·12·1.37=0.43·106H.

Principles of General Theory of Electrical Machines

2.1 Introduction to the general theory of electrical machines

The theory of individual types of electrical machines from the historical point of view was developed independently. Therefore, also terminology and signing of symbols and subscripts were determined independently. This theory was devoted to the investigation of steady-state conditions and quantities needed for design calculation of electrical machines.

However, the basic principles of electrical machines are based on common physical laws and principles, and therefore a general theory of electrical machines was searched. The first person, who dealt with this topic was Gabriel Kron, who asked the question: “Does a universal arrangement of electrical machine exist from which all known types of electrical machines could be derived by means of simple principles?” An answer to this question resulted in the fact that around the year 1935, G. Kron mathematically formulated general theory and defined universal electrical machine, which at various connections represented most of the known machines.

Kron’s theory employed tensor analysis and theory of multidimensional non-Euclidean spaces and therefore was misunderstood and non-employed by majority of the technical engineers. After the year 1950, the first works appeared, in which Kron’s theory was simplified and therefore better understood. But only after the personal computers (PC) were spread in a great measure and suitable software facilities were available, the general theory of electrical machines became an excellent working means for investigation of electrical machine properties. Nowadays it represents inevitable equipment of technically educated experts in electrical engineering.

A fundamental feature of the general theory of electrical machines is based on the fact that it generalizes principles and basic equations of all electrical machines on the common base, and in such a way it simplifies their explanation and study. Its big advantage is that it formulates equations of electrical machines in such a form that they are valid in transients as well as in steady-state conditions. In this theory the electrical machines are presented as a system of the stationary and moving mutual magnetically linked electrical circuits, which are defined by the basic parameters: self-winding and mutual winding inductances, winding resistances, and moment of inertia, see [7, 8, 9].

The general theory of electrical machines is general in such a sense that it is common for a majority of electrical machines and explains their basic properties and characteristics on the basis of common principles. Further it is applicable for various running conditions: steady state, transients, unsymmetrical, and if they are fed by frequency converter at a non-sinusoidal voltage waveform. On the other side, the electrical machines are idealized by simplifying assumptions.

These simplifying assumptions enable to simplify equations, mainly their solutions. Here are some of them:

  1. The saturation of the magnetic circuit is neglected. Then the relationship between currents and magnetic fluxes are linear. This assumption is needed to be able to use the principle of the magnetic fluxes’ superposition. On the other side, this assumption can have a considerable influence on the correctness of the results. In some cases, this assumption impedes the investigation of some problems, e.g., excitation of the shunt dynamo or running of asynchronous generator in island operation. Then the magnetizing characteristic must be taken into account.

  2. The influence of the temperature on the resistances is neglected. This assumption can be accepted only in the first approach. If accurate results are needed, which are compared with measurements, it is inevitable to take into account a dependence of the resistances on the temperature.

  3. The influence of the frequency on resistances and inductances is neglected. In fact, it means that the influence of skin effect and eddy currents is neglected. Again, it is valid that this fact is not involved in the equations of the general theory but at precise calculation this phenomenon is necessary to take into account. It is important mainly in the case of non-sinusoidal feeding from the frequency converters, when higher harmonic components with considerable magnitude appeared.

  4. It is supposed that windings are uniformly distributed around the machine periphery (except concentrating coils of field winding). In fact, the windings of the real machines are distributed and embedded in many slots, whereby slotting is neglected. In this way the real winding is replaced by current layer on the borderline between the air gap and this part of the machine where the winding is located, and calculation of magnetic fields, inductances, etc. is simplified.

    It should be noted that slotting is not ignored totally. In cooperation with the finite element method (FEM), it is possible to receive waveform of air gap magnetic flux density, where the influence of slotting is clearly seen. It is a distorted waveform for which harmonic analysis must be made and to determine components of the harmonic content. For each harmonic component, the induced voltage can be calculated, and the total induced voltage is given by the sum of all components. In this case the slotting influence is included in the value of the induced voltage.

  5. Winding for alternating currents is distributed sinusoidally. This assumption means that the real distributed winding with a constant number of the conductors in the slot, with finite number of slots around the machine periphery, is replaced by the winding with conductor (turns) density, varying around the periphery according to the sinusoidal function. This assumption can be used only for winding with many slots and can’t be used for concentrating coils of the field windings or for machines with permanent magnets. By this assumption sinusoidal space distribution of the current linkage around the periphery is received, with neglecting of the space harmonic components. In other words, non-sinusoidal waveform of the air gap magnetic flux density induces in such winding only the fundamental voltage component; it means the factor due to winding distribution for all harmonic components is zero.

Next, an arrangement of the universal machine, on the basis of which the general theory was derived, will be given.

2.2 Design arrangement and basic equations of the universal machine in the general theory

A two-pole commutator machine is taken. The theory spread to the multipole arrangement will be carried out if mechanical angles are converted to the electrical angles and mechanical angular speed to the electrical angular speed:

ω=,E61
ϑel=pϑmech.E62

The typical phenomenon of the universal machine is that its windings are located in two perpendicular axes to each other: The direct axis is marked “d” and quadrature axis marked q (see Figure 7).

Figure 7.

Design arrangement of the universal machine with marked windings.

Figure 8.

(a) Replacement of the rotor’s winding by solenoid, (b) creation of the quasi-stationary q-winding, and (c) creation of the quasi-stationary d-winding.

Stator has salient poles with one or more windings on the main poles in the d-axis and q-axis. In Figure 7, windings f and D are in the d-axis and windings g and Q in the q-axis. These windings can represent field winding (external, shunt, series, according to the connection to the armature), damping, commutating, compensating, and so on (see chapter about the DC machines) or, as we will see later, three-phase winding transformed into the two-axis system d, q.

The rotor’s winding with commutator expressed oneself as the winding of the axis that goes through the brushes. If the rotor rotates, conductors of the coils change their position with regard to the stator and brushes, but the currents in the conductors which are located in one pole pitch have always the same direction. In other words, there exists always such conductor which is in a specific position, and the current flows in the given direction.

Therefore, the rotor’s winding with commutator seems to be, from the point of view of magnetic effect, stationary; it means it is quasi-stationary. The magnetic flux created by this winding has always the same direction, given by the link of the given brushes. It is valid for the winding in d-axis and q-axis.

On the commutator, there are two sets of the brushes: one, on terminals of which is voltage uq, which is located on the q-axis, and the other, on terminals of which is voltage ud. This voltage is shifted compared with uq on 90° in the direction of the rotor rotating and is located in the direct d-axis.

2.2.1 Voltage equations in the system dq0

2.2.1.1 Voltage equations of the stator windings

The number of the equations is given by the number of windings. All windings are taken as consumer of the energy. Then the terminal voltage equals the sum of the voltage drops in the windings. The power in the windings is positive; therefore, the voltage and current have the same directions and are also positive. The voltage equations are written according to the 2nd Kirchhoff’s law and Faraday’s law for each winding. By these equations three basic variables of the given winding, voltage, current, and linkage magnetic flux, are linked. The voltage equation in general for j-th winding, where j = f, D, g, Q, is in the form:

uj=Rjij+dψjdt,E63

where uj is the terminal voltage, Rj is the resistance, ij is the current, and ψjis the linkage magnetic flux of the j-th winding. For example, for the f-th winding, the equation should be in the form:

uf=Rfif+dψfdt.E64

The linkage magnetic fluxes of the windings are given by the magnetic fluxes created by the currents of the respected winding and those windings which are magnetically linked with them. In general, any winding, including rotors, can be written as:

ψj=kψjk=kLjkikwherej,k=f,d,D,q,g,Q.E65

For example, for f-winding, the following is valid:

ψf=kψfk=kLfkik=Lffif+Lfdid+LfDiD+Lfqiq+Lfgig+LfQiQ.E66

Equations written in detail for all windings are as follows:

ψf=Lffif+Lfdid+LfDiD+Lfqiq+Lfgig+LfQiQ,
ψd=Ldfif+Lddid+LdDiD+Ldqiq+Ldgig+LdQiQ,
ψD=LDfif+LDdid+LDDiD+LDqiq+LDgig+LDQiQ,
ψq=Lqfif+Lqdid+LqDiD+Lqqiq+Lqgig+LqQiQ,
ψg=Lgfif+Lgdid+LgDiD+Lgqiq+Lggig+LgQiQ,
ψQ=LQfif+LQdid+LQDiD+LQqiq+LQgig+LQQiQ.E67

In these equations formally written in the order of the windings and their currents, it is shown also, which we already know, that their mutual inductances are zero, because their windings are perpendicular to each other, which results in zero mutual inductance.

2.2.1.2 Voltage equations of the rotor windings

Rotor winding is moving with an angular speed Ω; therefore, not only transformation voltage, which is created by the time varying of the magnetic flux, but also moving (rotating) voltage is induced in it. If the rotating-induced voltage is derived, sinusoidal waveform of the air gap magnetic flux density is assumed.

Rotor winding is composed of two parts, one is located in the d-axis, leading up to the terminals in the d-axis, and the second in the q-axis, leading up to the terminals in the q-axis. In Figures 9 and 10, it is shown that not both windings have both voltage components from both linkage magnetic fluxes.

Figure 9.

Illustration of the induced voltage in the q-axis.

Figure 10.

Illustration of the induced voltage in the d-axis.

Transformation voltage created by the time variation of ψqis induced in the winding in the q-axis, which is with it in magnetic linkage. This flux crosses the whole area of the q-winding turns:

utrq=dψqdt.E68

Linkage magnetic flux ψddoes not cross the area of any turns of the q-winding; therefore in q-winding there is no induced transformation voltage from ψd.

A movement of the q-winding in the marked direction (Figure 9) does not cause any rotating induced voltage from ψq, because the conductors of the q-winding do not cross magnetic force lines ψq; they only move over them.

Rotating voltage in q-winding is induced by crossing the magnetic force lines Φd:

urotq=CΦdΩ,E69

whereby there the known expression from the theory of electrical machines was used:

ui=urotq=CΦdΩ=paz2πΦdΩ.E70

This expression can be modified to the generally written mode; it means without regard which axis the winding is, in such a way that it will be written by means of electrical angular speed and linkage magnetic flux.

ui=za2πΦ=ωψ,E71

where half of the conductors z means number of the turns N. Then the linkage magnetic flux includes the next variables:

ψ=za2πΦ=NΦ=N2a2πΦ.E72

In this expression the effective number of turns of one parallel path N/2a is considered, on which the voltage is summed and reduced by the winding factor of the DC machines 2/π. Then the relationship for the induced voltage can be written in general form by means of electrical angular speed and linkage magnetic flux:

ui=za2πΦ=ωN2a2πΦ.E73

Now the relationship for the terminal voltage in q-axis can be written in the form:

uq=Riq+utrq+urotq=Riq+dψqdt+ωψd.E74

As shown in magnetic flux directions, movement and direction of currents in Figure 9 are in coincidence with the rule of the left hand, meaning for motor operation (consumer). Therefore, all signs in front of the voltages in Eq. (74) are positive.

In d-winding there is induced transformation voltage done by time varying of ψd, which is in magnetic link with it (this flux crosses the whole area of the d-winding turns). The rotating voltage in d-winding is induced only by crossing magnetic force lines of the ψq(Figure 10). Therefore, the equation for terminal voltage in the d-axis is next:

ud=Rid+utrdurotd=Rid+dψddtωψq.E75

In Figure 10, it is seen that current, magnetic flux, and moving directions are in coincidence with the right-hand rule, which is used for generator as a source of electrical energy.

Therefore, the sign in front of the rotating voltage is negative.

2.2.2 Power in the system dq0 and electromagnetic torque of the universal machine

Power and electromagnetic torque of the universal machine will be derived on the basis of energy equilibrium of all windings of the whole machine: We start with the voltage equations of stator and rotor windings, which are multiplied with the appropriate currents and time dt.

For the stator windings, Eq. (63) will be used for the terminal voltage of each winding. This equation will be multiplied by ijdtand the result is:

ujijdt=Rjij2dt+ijdψj.E76

For the rotor winding in the d-axis, Eq. (75) multiplied by iddtwill be used:

udiddt=Rid2dt+iddψdωψqiddt.E77

For the rotor winding in the q-axis, Eq. (74) multiplied by iqdtwill be used:

uqiqdt=Riq2dt+iqdψq+ωψdiqdt.E78

Now the left sides and right sides of these equations are summed, and the result is an equation in which energy components can be identified:

Σuidt=ΣRi2dt+Σidψ+ωψdiqψqiddt.E79

The expression on the left side presents a rise of the delivered energy during the time dt: Σuidt.

The first expression on the right side presents rise of the energy of the Joule’s losses in the windings: ΣRi2dt.

The second expression on the right side is an increase of the field energy: Σidψ.

The last expression means a rise of the energy conversion from electrical to mechanical form in the case of motor or from mechanical to electrical form in the case of generator: ωψdiqψqiddt.

The instantaneous value of the electromagnetic power of the converted energy can be gained if the expression for energy conversion will be divided by time dt:

pe=ωψdiqψqiddtdt=ωψdiqψqid=ψdiqψqid,E80

where p is the number of pole pairs. The subscript “e” is used to express “electromagnetic power pe,” i.e., air gap power, where also the development of instantaneous value of electromagnetic torque teis investigated:

pe=teΩ.E81

If left and right sides of Eqs. (80) and (81) are put equal, an expression for the instantaneous value of the electromagnetic torque in general theory of electrical machines yields:

te=pψdiqψqid.E82

If the motoring operation is analyzed, it can be seen that at known values of the terminal voltages (six equations) and known parameters of the windings, there are seven unknown variables, because except six currents in six windings there is also angular rotating speed, which is an unknown variable. Therefore, further equation must be added to the system. It is the equation for mechanical variables:

me=JdΩdt+tL,E83

in which it is expressed that developed electromagnetic torque given by Eq. (82) covers not only the energy of the rotating masses JdΩdtwith the moment of inertia J but also the load torque tL.

Therefore, from the last two equations, the time varying of the mechanical angular speed can be calculated:

dΩdt=1JtetL=1JpψdiqψqidtL.E84

For the time varying of the electrical angular speed, which is directly linked with the voltage equations, we get:

dωdt=pJpψdiqψqidtL.E85

These equations will be simulated if transients of electrical machines are investigated.

2.3 Application of the general theory onto DC machines

If the equivalent circuit of the universal machine and equivalent circuits of the DC machines are compared in great detail, it can be seen that the basic principle of the winding arrangement in two perpendicular axes is very well kept in DC machines. It is possible to find a coincidence between generally defined windings f, D, Q, g, d, and q and concrete windings of DC machines, e.g., in this way:

The winding “f” represents field winding of DC machine.

The winding “D” either can represent series field winding in the case of compound machines, whereby the winding “f” is its shunt field winding, or, if it is short circuited, can represent damping effects during transients of the massive iron material of the machines. However, it is true that to investigate the parameters of such winding is very difficult [1].

The windings “g” and “Q” can represent stator windings, which are connected in series with the armature winding, if they exist in the machine. They can be commutating pole winding and compensating winding.

Windings “d” and “q” are the winding of the armature, but in the case of the classical construction of DC machine, where there is only one pair of terminals, and eventually one pair of the brushes in a two-pole machine, only q-winding and terminals with terminal voltage uq will be taken into account. The winding in d-axis will be omitted, and by this way also terminals in d-axis, its voltage ud, and current id will be cancelled.

The modified equivalent circuit of the universal electrical machine applied to DC machine is in Figure 11.

Figure 11.

Modified equivalent circuit of the universal machine applied on DC machine.

2.3.1 Separately excited DC machine

The field winding of the separately excited DC machine is fed by external source of DC voltage and is not connected to the armature (see Figure 12). Let us shortly explain how the directions of voltages, currents, speed, and torques are drawn: The arrowhead of the induced voltage is moving to harmonize with the direction of the magnetic flux in the field circuit. The direction of this movement means the direction of rotation and of developed electromagnetic (internal) torque. The load torque and the loss torque (the torque covering losses) are in opposite directions. The source of voltage is on the terminals and current flows in the opposite direction. On the armature there are arrowheads of voltage and current in coincidence, because the armature is a consumer.

Figure 12.

Equivalent circuit of DC machine with separate excitation in motoring operation.

To solve transients’ phenomena, a system of the voltage equations of all windings is needed (for simplification g-winding is omitted):

uf=Rfif+dψfdt,E86
uD=RDiD+dψDdt,E87
uQ=RQiQ+dψQdt,E88
uq=Rqiq+dψqdt+ωψd,E89

where:

ψf=Lffif+LfDiD,E90
ψD=LDDiD+LDfif,E91
ψQ=LQQiQ+LQqiq,E92
ψq=Lqqiq+LqQiQ,E93
ψd=Lddid+Ldfif+LdDiD=Ldfif+LdDiD,E94

because the current in “d”-winding is zero, seeing that d-winding is omitted. In addition, the fact that mutual inductance of two perpendicular windings is zero was considered.

Then equation for electromagnetic torque is needed. This equation shows that electromagnetic torque in DC machine is developed in the form (again the member with the current idis cancelled):

te=pψdiqψqid=pψdiq,E95

and that it covers the energy of the rotating mass given by moment of inertia, time varying of the mechanical angular speed, and load torque:

te=JdΩdt+tL.E96

A checking of equation for electromagnetic torque of DC machines for steady-state conditions will be done if for ψd, Eq. (72) is applied to the d-axis:

ψd=N2a2πΦdE97

is introduced to Eq. (95) for the torque, whereby for the current the subscript “a” is employed and used for the armature winding and the number of the conductors z is taken as double number of the turns N:

Te=pψdiq=pN2a2πΦdIa=paz2πΦdIa=CΦdIa.E98

If the damping winding D, neither the windings in the quadrature axis Q, g, are not taken into account and respecting Eq. (90) for linkage magnetic flux, we get equations as they are presented below. The simplest system of the voltage equations is as follows:

uf=Rfif+Lfdifdt,E99
uq=Rqiq+Lqdiqdt+ωψd=Rqiq+Lqdiqdt+ωLdfif.E100

The expression ψd=Ldfifshows that the linkage flux ψdin Eq. (94) is created by the mutual inductance Ldfbetween the field winding and armature winding (by that winding which exists there and is brought to the terminals through the brushes in the q-axis).

We get from Eq. (96) the equation for calculation of the time varying of the mechanical angular speed:

dΩdt=1JpψdiqtL=1JpLdfifiqtL,E101

and the electrical angular speed ω, which appears in the voltage equations, is valid:

dωdt=pJpψdiqtL=pJpLdfifiqtL.E102

In this way, a system of three equations (Eqs. (99), (100), and (102)), describing the smallest number of windings (three), was created. The solution of these equations brings time waveforms of the unknown variables (iq = f(t), if = f(t), and ω= f(t)).

2.3.1.1 Separately excited DC motor

If a DC machine is in motoring operation, the known variables are terminal voltages, moment of inertia, load torque, and parameters of the motor, i.e., resistances and inductances of the windings.

Unknown variables are currents, electromagnetic torque, and angular speed. Therefore, Eqs. (99) and (100) must be adjusted for the calculation of the currents:

difdt=1LfufRfif,E103
diqdt=1LquqRqiqωψd.E104

The third equation is Eq. (102). It is necessary to solve these three equations, Eqs. (102)(104), to get time waveforms of the unknown field current, armature current, and electrical angular speed, which can be recalculated to the mechanical angular speed or revolutions per minute: Ω=ω/por n=60Ω/2πmin−1, at the known terminal voltages and parameters of the motor.

As it was seen, a very important part of the transients’ simulations is determination of the machine parameters, mainly resistances and inductances but also moment of inertia. The parameters can be calculated in the process of the design of electrical machine, as it was shown in Chapter 1. However, the parameters can be also measured if the machine is fabricated. A guide how to do it is given in [8]. The gained parameters are introduced in equations, and by means of simulation programs, the time waveforms are received. After the decay of the transients, the variables are stabilized; it means a steady-state condition occurs. The simulated waveforms during the transients can be verified by an oscilloscope and steady-state conditions also by classical measurements in steady state.

Designers in the process of the machine design can calculate parameters on the basis of geometrical dimensions, details of construction, and material properties. If they use the above derived equations, they can predestine the properties of the designed machine in transients and steady-state conditions. This is a very good method on how to optimize machine construction in a prefabricated period. When the machine is manufactured, it is possible to verify the parameters and properties by measurements and confirm them or to make some corrections.

2.3.1.2 Simulations of the concrete separately excited DC motor

The derived equations were applied to a concrete motor, the data of which are in Table 1. The fact that the motor must be fully excited before or simultaneously with applying the voltage to the armature must be taken into account. Demonstration of the simulation outputs is in Figure 13. In Figure 13a–d, time waveforms of the simulated variables if = f(t), iq = f(t), n = f(t), and te = f(t) are shown after the voltage is applied to the terminals of the field winding in the instant of t = 0.1 s. After the field current if is stabilized, at the instant t = 0.6 s, the voltage was applied to the armature terminals. After the starting up, the no-load condition happened, and the rated load was applied at the instant t = 1 s.

UqN = UfN = 84 V (in motoring)Rq = 0.033 Ω
IqN = 220 ALq = 0.324 mH
IfN = 6.4 ARf = 13.2 Ω
nN = 3200 min−1Lf = 1.5246 H
TN = 48 NmLqf = 0.0353 H
PN = 16 kWJ = 0.04 kg m2
p = 1Te0 = 0.2 Nm

Table 1.

Nameplate and parameters of the simulated separately excited DC motor.

Figure 13.

Simulation waveforms of the separately excited DC motor. Time waveforms of (a) field current, (b) armature current, (c) speed, and (d) developed electromagnetic torque. Dependence of the speed on the torque in steady state, for (e) various terminal voltages, (f) various resistances connected in series with armature, (g) various field currents, and (h) dependence of speed on the armature current for the various field currents.

In Figure 13e–g, basic characteristics of n = f(Te) are shown for the steady-state conditions. They illustrate methods on how the steady-state speed can be controlled: by controlling the armature terminal voltage Uq, by resistance in the armature circuit Rq, as well as by varying the field current if.

Figure 13h points to the fact that value of the armature current Iqk does not depend on the value of the field current if and also shows the typical feature of the motor with the rigid mechanical curve that the feeding armature current Iq is very high if motor is stationary; it means such motor has a high short circuit current. This is the reason why the speed control is suitable to check value of the feeding current, which can be ensured by the current control loop.

2.3.1.3 Separately excited generator

Equations for universal machine are derived in general; therefore, they can be used also for generating operation. If the prime mover is taken as a source of stiff speed, then the time changing of the speed can be neglected, i.e., /dt=0,and Ω= const is taken. In addition, the current in the armature will be reversed, because now the induced voltage in the armature is a source for the whole circuit (see Figure 14). According to Eq. (96), equilibrium occurs between the driving torque of prime mover Thn and electromagnetic torque Te, which act against each other, i.e., the prime mover is loaded by the developed electromagnetic torque. If the analysis is very detailed, it is possible to define dependence of the angular speed of the prime mover on the load by specific function according the mechanical characteristic and to introduce this function into the equation of the torque equilibrium.

Figure 14.

Equivalent circuit of the separately excited DC generator. The driving torque (Thn) in generating operation is delivered by prime mover.

The constant speed of the prime mover equations will be changed in comparison with motoring operation. The armature current is in the opposite direction, because now the induced voltage in the armature is a source and current flows from the source. An electrical load is connected to the terminals; therefore the voltage and current on the load are in the same directions. The induced voltage is divided between voltage drops on the resistances and inductances of the winding and on the terminal voltage. Terminal voltage is given by the resistance of the load and its current. Equations are created in this sense:

ui=ωψd=ωLdfif=u+Rqiq+Lqdiqdt,E105

and simultaneously the terminal voltage is given by equation of the load:

u=RLiq.E106

2.3.1.4 Simulations of a concrete separately excited DC generator

An electrical machine, the data of which are given in Table 1, is used also for simulation in generating operation. The dynamo is kept at constant speed and is fully excited before any loading occurs.

Simulation waveforms in Figure 15a–e show time dependence of the variables: if = f(t), ui = f(t) iq = f(t), uq=f(t), and te = f(t).

Figure 15.

Simulation of the separately excited dynamo: time waveforms of (a) field current, (b) induced voltage, (c) armature current, (d) developed electromagnetic torque, (e) terminal voltage, and (f) dependence of the terminal voltage on the load current in steady-state conditions (external characteristic).

Dynamo is rotating by the rated speed, and at the time t = 0.1 s is excited by the rated field voltage. After the field current is stabilized, at the instant t = 1 s, the dynamo is loaded by the rated current. In Figure 15f, there is a waveform of the armature voltage versus armature current uq = f(iq). It is the so-called stiff voltage characteristic, i.e., at the big change of the load, the voltage is almost constant. Its moderate fall is caused by voltage drops in the area of the rated load and by armature reaction.

2.3.2 Shunt wound DC machine

This machine is so called because the field circuit branch is in shunt, i.e., parallel, with that of the armature. Figure 16 shows equivalent circuits of the shunt machines, in motoring and generating operation. As it is seen, the shunt motor differs from the separately excited motor because the shunt motor has a common source of electrical energy for armature as well as for field winding. Therefore, the field winding is connected parallel to the armature, which results in the changing of equations. In Eqs. (99) and (100), the terminal voltages in both windings are identical:

Figure 16.

Equivalent circuits of the shunt DC machine in (a) motoring and (b) generating operation.

uf=Rfif+Lfdifdt=uq=u,E107
uq=Rqiq+Lqdiqdt+ωψdE108

The power input is given by the product of terminal voltage and the total current i, which is a sum of the currents in both circuits:

i=iq+if.E109

The power output is given by the load torque on the shaft and the angular speed. The developed electromagnetic torque is given by equation:

te=pψdiqψqid=pψdiq=pLdfifiq.E110

The time waveform of the electrical angular speed is given by Eq. (102).

2.3.2.1 Simulations of the concrete DC shunt motor

To get simulations of DC shunt motor transients, it is necessary to solve equations from Eq. (107) to Eq. (110) and Eq. (102). Terminal voltage and parameters are known; currents and speed time waveforms are unknown (see Figure 17). Here the investigated motor has the same data as they are in Table 1.

Figure 17.

Simulations of the DC shunt motor. Time waveforms of the (a) field current, (b) armature current, (c) speed, and (d) speed vs. torque for various terminal voltages and further in steady-state conditions waveforms of the (e) speed vs. torque for various rheostats connected in series with armature and (f) speed vs. torque for various field currents at constant terminal voltage.

Because this motor reaches its rated field current at the same field voltage uf as it is the armature voltage uq (at uq = 84 V, the field current waveforms reaches the value of if = 6.4 A), the simulated time waveforms do not differ from the waveforms of the separately excited DC motor (Figure 17a–c). At the other waveforms (Figures 17d–f), there are some differences, e.g., variation of the terminal voltage influences not only armature current but also the field current. This fact results in the almost constant speed if terminal voltage is changed. It is proven by the waveforms in Figure 17d, which show that in the region till 50 Nm, i.e., rated torque TN, there is no changing of the speed, even in no load condition. Therefore, this kind of speed control is not employed.

The control of the field current is carried out by variation of the field rheostat, which ensures decreasing of the field current if at the constant field and terminal voltage uq.

2.3.2.2 Shunt generator

A shut generator (dynamo) differs from the separately excited dynamo by an essential way because a source for the field current is its own armature, where a voltage must be at first induced. To ensure this, some conditions must be filled. They are as follows: (1) some residual magnetism must exist in the magnetic system of the stator, which enables building up of the remanent voltage, if dynamo rotates, (2) resistance in the field circuit must be smaller than a critical resistance, (3) speed must be higher than a critical speed, and (4) there must be correct direction of rotation and connection between polarity of the excitation and polarity of induced voltage in the armature.

Because the field current depends on the terminal voltage, and this terminal voltage on the induced voltage, which again depends on the field current, this mutual dependence must be taken into account in simulations by magnetizing curve of the investigated machine, i.e., induced voltage vs. field current U0 = Ui = f(If), which can be measured. The measurement of this curve can be made only with separate excitation. The speed of the prime mover is taken constant.

Equation (107) is valid, but Eq. (108) is changed, because the terminal voltage is smaller than induced voltage because of the voltage drops, or opposite, induced voltage covers terminal voltage as well as voltage drops:

ui=ωψd=ωLdfif=u+Rqiq+LqdiqdtE111

and armature current supplies field circuit as well as load circuit. Then the load current is:

i=iqif,E112

whereby the terminal voltage is given by the load current and load resistance:

u=uq=RLi.E113

2.3.2.3 Simulation of a shunt DC dynamo

A machine, in which its data are in Table 1, was used for simulations of transients and steady-state conditions. In addition it is necessary to measure magnetizing curve Ui = f(If), which is shown in Figure 18 for the investigated machine.

Figure 18.

Magnetizing curve Ui = f(If) for investigated machine.

During the simulation the machine is kept on the constant speed, and simulation starts with the connection of the armature to the field circuit. Because of remanent magnetic flux, in the armature there is induced small remanent voltage Uirem, which pushes through the armature circuit and field circuit small field current, by which the magnetic flux and induced voltage will be increased. This results in higher field current and higher induced voltage, which is gradually increased until it reaches the value of the induced voltage in no load condition uio. Simulation waveforms are in Figure 19a–e. They show the time waveforms of variables if = f(t), ui = f(t), iq = f(t), i = f(t), and uq = f(t) from the instant of connecting until the transients are in a steady-state condition in the time of t = 1.5 s.

Figure 19.

Simulations of the shunt dynamo: time waveforms of (a) field current, (b) induced voltage, (c) armature current, (d) load current, (e) terminal voltage, and (f) terminal voltage vs. load current in steady-state conditions at the constant resistance in the field circuit when the load resistance is changed.

A waveform of Uq = f(Iq) is shown in Figure 19f. It is terminal voltage Uq vs. load current Iq. As it was mentioned, it is a basic characteristic for all sources of electrical energy, and in the case of shunt dynamo, it is seen that there is also a stiff characteristic, similar to the case of the separately excited dynamo but only till the rated load. In addition, it is immune to the short circuit condition, because short circuit current can be smaller than its rated current IN. This performance is welcomed in the applications where this feature was required, e.g., in cars, welding set, etc.

2.3.3 DC series machine

A DC series machine has its field winding connected in series with its armature circuit, as it is seen in Figure 20 for motoring and generating operation. This connection essentially influences properties and shapes of characteristics of the series machine and also equations needed for investigations of its properties.

Figure 20.

Equivalent circuits of the series machine (a) in motoring and (b) in generating operation.

2.3.3.1 Series DC motor

For a series DC motor, it is typical that the terminal voltage is a sum of the voltages in the field circuit and in the armature circuit:

u=uq+uf=Rqiq+Lqdiqdt+ωLdfif+Rfif+Lfdifdt,E114

but because of only one current flowing in the whole series circuit, the next is valid:

i=iq=ifE115

and Eq. (114) is simplified:

u=uq+uf=Rq+Rfi+Lq+Lfdidt+ωLdfi.E116

Equation (110) for electromagnetic torque is also changed because of only one current:

te=pψdiqψqid=pψdi=pLdfi2E117

and angular speed is gained on the basis of the equation:

te=pLdfi2=JdΩdt+tL=Jpdωdt+tL.E118

2.3.3.2 Simulations of a DC series motor

The time waveforms of the current, developed electromagnetic torque and angular speed, which can be recalculated to the revolutions per minute, are based on Eqs. (116)(118). In Figure 20, there are simulated waveforms of the motor; the data of which are shown in Table 2.

UqN = 180 VRq = 1 Ω
IqN = 5 ALq = 0.005 mH
PN = 925 WRf = 1 Ω
nN = 3000 min−1Lf = 0.015 H
MN = 3 NmLqf = 0.114 H
IfN = 5 AJ = 0.003 kg m2
p = 1

Table 2.

Nameplate and parameters of simulated series motor.

Simulated waveforms in Figure 21a–f show time waveforms of the variables if = iq = f(t), te = f(t), and n = f(t) after the voltage is applied to its terminals. In Figure 21c, one of the basic properties of a series motor is seen, which is that in no load condition (here its load is only torque of its mechanical losses, which is about 10% of the rated torque), the field current is strongly suppressed, which results in enormous increasing of the speed.

Figure 21.

Simulations of series motor. Time waveforms of (a) armature current and field current, (b) developed electromagnetic torque, and (c) speed and then speed vs. torque in the steady-state conditions for (d) various terminal voltages; (e) various resistances in series with armature circuit, at UqN; and (f) various field currents.

For this reason, this motor in praxis cannot be in no-load condition and is not recommended to carry out its connection to the load by means of chain, or band, because in the case of a fault, it could be destroyed. In simulation the motor is after the steady condition at the instant t1 = 7 s loaded by its rated torque. In Figure 21d–f, mechanical characteristics n = f(Te) for steady-state conditions are shown, if speed control is carried out by terminal voltage Uq, resistance in the armature circuit Rq (in this case there is also resistance of field circuit), as well as field current if (there is a resistance parallelly connected to the field winding).

2.3.3.3 Series dynamo

The approach to the simulations is the same as in previous chapters concerning the generating operations: the constant driving speed is supposed, induced voltage is a source for the whole circuit, and this voltage covers not only the voltage drops in the field and armature windings but also the terminal voltage. The current is only one i = if = iq, and the terminal voltage is given also by the load resistance:

u=RLi=ωLdfiRq+RfiLq+Lfdidt.E119

The magnetizing characteristic, i.e., no load curve Ui = f(If), must be measured by a separate excitation.

2.3.3.4 Simulations of a DC series dynamo

Data and parameters of a machine which was simulated in generating operations are in Table 2. Dynamo is kept at constant speed; at first in the no load condition, it means terminals are opened, and no current flows in its circuit. A small voltage is possible to measure at its terminals at this condition. This voltage is induced by means of remanent magnetic flux (Figure 23b, ui = f(t)). For this purpose, it is necessary to measure magnetizing curve at separate excitation Ui = f(If). For the investigated machine, this curve is shown in Figure 22.

Figure 22.

Measured magnetizing curve for the investigated series machine Ui = f(If).

After the load is applied to the terminals at the instant t1 = 0.2 s, the current starts to flow in the circuit, because of the induced voltage (Figure 23a), if = iq = f(t), which flows also through the field winding and causes higher excitation of the machine, which results in higher induced voltage. Then the current is increased, which results again in the increasing of the induced voltage, etc. The transients are stabilized after the magnetic circuit is saturated. In this condition the voltage is increased with the increasing of the current, very slowly (Figure 22, Ui = f(If)). Similarly, as induced voltage, also the terminal voltage is increased with the increasing of the current but only till the saturation of the magnetic circuit. Then the terminal voltage can even sink, because the voltage drops on the armature and field resistances can increase quicker than induced voltage. In this simulated case, this did not appear, and the terminal voltage was increased with the increased current (see Figure 23d and the curve Uq = f(Iq)).

Figure 23.

Simulations of the series dynamo and time waveforms of (a) armature current, (b) induced voltage, (c) terminal voltage, and (d) terminal voltage vs. load current in steady-state conditions at the changing of the load resistance.

2.3.4 Compound machines

As it is known, compound machines are fitted with both series and shunt field windings. Therefore, also simulations of transients and steady-state conditions are made on the basis of combinations of appropriate equations discussed in the previous chapters.

2.3.5 Single-phase commutator series motors

These motors, known as universal motor, can work on DC as well as AC network. Their connection is identical with series DC motors, even though there are some differences in their design. At the simulations, it is necessary to take into account that there are alternating variables of voltage and current; it means that winding’s parameters act as impedances, not only resistances.

2.4 Transformation of the three-phase system abc to the system dq0

2.4.1 Introduction

Up to now we have dealt with DC machines, the windings of which are arranged in two perpendicular axes to each other. However, alternating rotating machines obviously have three-phase distributed windings on the stator, which must be transformed into two perpendicular axes, to be able to employ equations derived in the previous chapters.

In history, it can be found that principles of the variable projections into two perpendicular axes were developed for synchronous machine with salient poles.

A different air gap in the axis that acts as field winding and magnetic flux is created and, in the axis perpendicular to that magnetic flux, was linked with a different magnetic permeance of the circuit, which resulted in different reactances of armature reaction and therefore different synchronous reactances. It was shown that this projection into two perpendicular axes and variables can be employed much wider and can be applied for investigation of transients on the basis of the general theory of electrical machines.

On the other side, it is necessary to realize that phase values transformed into dq0 system have gotten into a fictitious system with fictitious parameters, where investigation is easier, but the solution does not show real values. Therefore, an inverse transformation into the abc system must be done to gain real values of voltages, currents, torques, powers, speed, etc. This principle is not unknown in the other investigation of electrical machines. For example, the rotor variables referred to the stator in the case of asynchronous machines mean investigation in a fictitious system, where 29 the calculation and analysis is more simple, but to get real values in the rotor winding a reverse transformation must be done.

Therefore, we will deal with a transformation of the phase variables abc into the fictitious reference k-system dq0 with two perpendicular axes which rotate by angular speed ωk with regard to the stator system. The axis “0” is perpendicular to the plain given by two axes d, q. As it will be shown, the investigation of the machine properties in this system is simpler because the number of equations is reduced, which is a big advantage. However, to get values of the real variables, it will be necessary to make an inverse transformation, as it will be shown gradually in the next chapters.

A graphical interpretation of the transformation abc into the system dq0 is shown in Figure 24. This arrangement is formed according to the original letters given by the papers of R.H. Park and his co-authors (around 1928 and later), e.g., [14], although nowadays it is possible to find various other figures, corresponding to the different position of the axes d, q, and corresponding equations.

Figure 24.

Graphical interpretation of the three-phase variable transformation abc into the reference k-system dq0, rotating by the speed ωk.

According to the original approach, if the three-phase system is symmetrical, the d-axis is shifted from the axis of the a-phase about the angle ϑk, and the q-axis is ahead of the d-axis by about 90°; then the components in the d-axis and q-axis are the projections of the phase variables of voltage, linkage magnetic flux, or currents, generally marked as x-variable, into those axes. In the given papers, there are derived equations of the abc into dq0 transformation as well as the equations of the inverse transformation dq0 into abc, because of the investigation of the synchronous reactances of the synchronous machine with salient poles. Also constants of proportions are given. Today this transformation is called “Park’s transformation” (see equations given below), even though this name is not given in the original papers. Next equations will be derived, and the constants of proportions kd, kq, and k0 will be employed. Later these constants will be selected according to how the reference system will be positioned, to apply the most profitable solutions. Employment of Park’s transformation equations is today very widespread, and they are used for all kinds of electrical machines, frequency convertors, and other three-phase circuits.

2.4.2 Equations of Park’s transformations abc into dq0 system

According to Figure 24, the d-component of the x-variable is a sum of a-, b-, and c-phase projections:

xd=xda+xdb+xdc,E120

where

xda=xacosϑk,E121
xdb=xbcosϑk120°,E122
xdc=xccosϑk+120°.E123

Also, projections into the q-axis are made in a similar way. It is seen that the projections to the q-axis are expressed by sinusoidal function of the phase variable with a negative sign, at the given +q-axis (see Eq. (125)).

The zero component is a sum of the instantaneous values of the phase variables. If the three-phase system is symmetrical, the sum of the instantaneous values is zero; therefore also the zero component is zero (see Eq. (126)). The zero component can be visualized in such a way that the three-phase variable projection is made in the 0-axis perpendicular to the plain created by the d-axis and q-axis, whereby the 0-axis is conducted through the point 0.

Then the equation system for the Park transformation from the abc to the dq0 system is created by Eqs. (124)(126). To generalize the expressions, proportional constants kd, kq, and k0 are employed:

xd=kdxacosϑk+xbcosϑk2π3+xccosϑk+2π3,E124
xq=kqxasinϑk+xbsinϑk2π3+xcsinϑk+2π3,E125
x0=k0xa+xb+xc.E126

It is true that R.H. Park does not mention such constants in the original paper,because he solved synchronous machine, which will be explained later (Sections 8, 10, and 16). For the purposes of this textbook, it is suitable to start as general as possible and gradually adapt the equations to the individual kinds of electrical machines to get a solution as advantageous as possible. Therefore, the constants can be whichever except zero, though of such, that the equation determinant is not zero (see Eq. (127)). Then the inverse transformation will be possible to do and to find the real phase variables.

The determinant of the system is as follows:

kdcosϑkkdcosϑk2π3kdcosϑk+2π3kqsinϑkkqsinϑk2π3kqsinϑk+2π3k0k0k0=kdkqk0332cosϑk2π3.E127

2.4.3 Equations for the m-phase system transformation

Equations for the three-phase system transformation can be spread to the m-phase system. Now the phases will be marked by 1, 2, 3, etc., to be able to express the mth phase and to see how the argument of the functions is created:

xd=kdx1cosϑk+x2cosϑk2πm+x3cosϑk4πm++xmcosϑk2m1πm.E128

Similarly, equations for the q- and 0-components are written. If a proportional constant 2/3 will be used for the three-phase system, then the corresponding constant for the m-phase system is 2/m [2].

xd=2mx1cosϑk+x2cosϑk2πm+x3cosϑk4πm++xmcosϑk2m1πm.E129

2.5 Inverse transformation from dq0 to the abc system

Equations for the inverse transformation are derived from the previous equations. Equation (124) is multiplied by expression cosϑk/kdand added to Eq. (125), which was multiplied by the expression sinϑk/kq. After the modification it is:

xdcosϑkkdxqsinϑkkq=xa12xb12xc=xa12xb+xc,E130

and from the third Eq. (126), the following is derived:

x0k0=xa+xb+xcx0k0xa=xb+xc,E131

which is necessary to introduce to Eq. (130):

xa12x0k0xa=32xa12x0k0=xdcosϑkkdxqsinϑkkq.E132

In this way, the equation for the inverse transformation of the a-phase variable is gained:

xa=231kdxdcosϑk231kqxqsinϑk+131k0x0.E133

In a similar way, equations for the inverse transformation and also for b-phase and c-phase are derived:

xb=231kdxdcosϑk2π3231kqxqsinϑk2π3+131k0x0,E134
xc=231kdxdcosϑk+2π3231kqxqsinϑk+2π3+131k0x0.E135

Equations (133)(135) create a system for the inverse transformation from dq0 to the abc system. These equations will be employed, e.g., for calculation of the real currents in the phase windings, if the currents in the dq0 system are known.

2.6 Equations of the linear transformation made by means of the space vectors of the voltage and currents

A space vector is a formally introduced symbol, which is illustrated in a complex plain in such a way that its position determines space position of the positive maximum of the total magnetic flux or magnetic flux density.

This definition is very important because as we know from the theory of electromagnetic field, neither current nor voltage is the vector. After the definition of the space vectors, it is possible to work with the currents and voltages, linked by Ohm’s law through impedance, but to image that it is a vector of the air gap magnetic flux density, which is by these currents and voltages created, which is very profitable. Therefore to distinguish a term “vector” as a variable which has a value and a direction, here the term “space vector” is used. The whole name “space vector” should be expressed and should not be shortened to “vector” because it can cause a misunderstanding, mainly between the people who do not work with investigation of transients.

To express that all three phases to which terminal voltages ua, ub, and uc are applied and contribute to the creation of the air gap magnetic field and magnetic flux density, it is possible to use the equation of the voltage space vector. In the complex plain, it will represent the value and position of air gap magnetic flux density magnitude:

u¯s=ksua+a¯ub+a¯2uc,E136

where unit phasors a¯mean a shift of the voltage phasor about 120° (note: phasor shows time shifting of variables):

a¯=ej2π3=cos2π3+jsin2π3,E137
a¯2=ej4π3=cos4π3+jsin4π3=ej2π3=cos2π3jsin2π3.E138

The subscript “s” means that it is a stator variable. Also, a proportional constant is marked with this subscript. In Figure 25, a complex plain with the stator axis is graphically illustrated, which is now identical with the axis of the a-phase winding. Then there is a rotor axis, which is shifted from the stator axis about the ϑrangle, and the axis of the k-reference frame, which is shifted from the stator axis about an arbitrary ϑkangle. Between the rotor axis and axis of the k-reference frame, there is an angle ϑkϑr. The axis of the k-reference frame is identical with its real component in the d-axis, and this system rotates by the angular speed ωkin the marked direction. The space vector of the stator voltage can be written as a sum of its real and imaginary components:

Figure 25.

Graphical illustration of the complex plain with the stator axis, rotor axis, and axis of the k-reference frame.

u¯s=ud±juq.E139

2.6.1 Stator variable transformation

The transformation of the stator variables into the k-reference frame (k, +jk) means to multiply stator variables by the expression ejϑk; it means the k-axis must be shifted back about the angle ϑk, to identify it with the stator axis:

usk¯=ksua+a¯ub+a¯2ucejϑk=ksuaejϑk+ubejϑk2π3+ucejϑk+2π3.E140

If this equation is split by means of the goniometrical functions into the real components, i.e., with cos-members, and into the imaginary components with sin-members,

ejϑk=cosϑkjsinϑk,E141

then in a coincidence with Eqs. (139) and (141), the next two equations are gained:

ud=ksuacosϑk+ubcosϑk2π3+uccosϑk+2π3,E142
uq=ksuasinϑk+ubsinϑk2π3+ucsinϑk+2π3.E143

As it is seen, these equations are identical with Eqs. (124) and (125), which were derived in general for three-phase circuits. In those equations separately marked constants in each axis were introduced, but here it is justified that it is enough to employ only one constant for both axes:

kd=kq=ks.E144

The third equation for the zero component, which is needed for investigation of the asymmetrical systems, can be added:

u0=k0ua+ub+uc.E145

2.6.2 Rotor variable transformation

The same approach is used for the rotor variables with the subscript “r.” To distinguish them from the stator variables, the subscripts DQ0 will be used for the k-axis and ABC for the phase variables. The rotor variables must be multiplied by the expression ejϑkϑrif rotor variables are transformed into the k-axis. It means the k-axis must be shifted back about the angle ϑkϑr, to identify it with the rotor axis. In general, space vector of the rotor voltage transformed into k-system can be written as follows:

urk¯=kruA+a¯uB+a¯2uCejϑkϑr=kruAejϑkϑr+uBejϑkϑr2π3+uCejϑkϑr+2π3.E146

The variables of the k-system expressed in two perpendicular d-axis and q-axis are:

urk¯=uD±juQ;E147

then Eq. (146) can be itemized into two equations:

uD=kruAcosϑkϑr+uBcosϑkϑr2π3+uCcosϑkϑr+2π3,E148
uQ=kruAsinϑkϑr+uBsinϑkϑr2π3+uCsinϑkϑr+2π3E149

and for the zero component:

uO=kOuA+uB+uC.E150

Here is a system of equations for rotor variables of the three-phase system ABC transformed into the k-system DQ0. They differ from the stator variables by the angle ϑkϑrinstead of ϑk.

2.7 Voltage equations of three-phase machines and their windings

The same equations as for the terminal voltage of the universal machine (63) can be written also for the terminal voltage of the three-phase machines. For example, the stator windings, where the phases are marked with a, b, and c equations of the terminal voltage, are in the next form:

ua=Raia+dψadt,E151
ub=Rbib+dψbdt,E152
uc=Rcic+dψcdt.E153

In the next only a-phase will be investigated. In Eq. (151), the variables ua, ia, and ψa will be introduced; after they are adapted according to Eq. (133), it means on the basis of inverse transformation (Section 5). For example, for the ψa it is expressed in the form:

ψa=231kdψdcosϑk231kqψqsinϑk+131k0ψ0.E154

Now a derivation by time dψa/dt is made:

dψadt=231kddψddtcosϑk231kdωkψdsinϑk231kqdψqdtsinϑk231kqωkψqcosϑk+131k0dψ0dtE155

From Eq. (151), dψa/dt is selected:

dψadt=uaRsia.E156

It was supposed that all three phases are identical and their resistances are equal:

Ra=Rb=Rc=Rs.E157

The expressions from the inverse transformation are introduced also for ua and ia, in Eq. (156):

dψadt=231kdudcosϑk231kquqsinϑk+131k0u0Rs231kdidcosϑk231kqiqsinϑk+131k0i0E158

The left sides of Eqs. (155) and (158) are equal; therefore, also right sides will be equal. Now the members with the same goniometrical functions and members without goniometrical functions will be selected and put equal, e.g., members at cosϑkyield:

231kddψddt231kqωkψq=231kdudRs231kdid.E159

The equation for ud is gained after modification, and in a very similar way, also the two other equations are obtained:

ud=Rsid+dψddtωkψq,E160
uq=Rsiq+dψqdt+ωkψd,E161
u0=Rsi0+dψ0dt.E162

Equations (160)(162) are the voltage equations for the stator windings of the three-phase machines, such as asynchronous motors in k-reference frame, rotating by the angular speed ωkwith the dq0-axis. As it can be seen, they are the same equations as the voltage equations in Section 2.1, which were derived for universal arrangement of the electrical machine. Here general validity of the equations is seen: if windings of any machine are arranged or are transformed to the arrangement with two perpendicular axes to each other, the same equations are valid. Of course, the parameters, mainly inductances, of the machine are different, and it is necessary to know how to get them.

2.8 Three-phase power and torque in the system dq0

2.8.1 Three-phase power in the system dq0

The instantaneous value of the input power in a three-phase system is a sum of instantaneous values of power in each phase (see also Section 2.2):

pin=uaia+ubib+ucic.E163

Instantaneous values of ua and ia will be introduced into this equation. These were derived in the inverse transformation chapter. They are Eqs. (133)(135). It means at first u is introduced for x, and it must be multiplied by expression, where i was introduced, and then the further phases in the same way are adapted. At the end all expressions are summed:

pin=[231kdudcosϑk231kquqsinϑk+131k0u0][231kdidcosϑk231kqiqsinϑk+131k0i0]+[231kdudcos(ϑk2π3)231kquqsin(ϑk2π3)+131k0u0].[231kdidcos(ϑk2π3)231kqiqsin(ϑk2π3)+131k0i0]+[231kdudcos(ϑk+2π3)231kquqsin(ϑk+2π3)+131k0u0].[231kdidcos(ϑk+2π3)231kqiqsin(ϑk+2π3)+131k0i0]E164

Now it is necessary to multiply all members with each other, including the goniometrical functions, and after a modification the result is:

pin=231kd2udid+231kq2uqiq+131k02u0i0.E165

Variables ud, uq, and u0 are given by Eqs. (160) to (162), which were introduced above, and after a modification, the result is:

pin=231kd2Rsid2+iddψddtωkψqid+231kq2Rsiq2+iqdψqdt+ωkψdiq+131k02Rsi02+i0dψ0dtE166

If an analysis in greater details is made, it is seen that an input power on the left side must be in equilibrium with the right side. It is supposed to be motoring operation. Therefore the input power applied to the terminals of the three-phase motor is distributed between the Joule’s resistance loss ΣRi2, time varying of the field energy stored in the investigated circuit Σidψdt, and the rest of the members’ mean conversion of electrical to mechanical energy and eventually to mechanical output power. If the resistance loss and power of magnetic field are subtracted from the input power on the terminals, the result is an air gap electromagnetic power, which is given by the difference of two rotating voltages in both axes:

pe=231kq2ωkψdiq1kq2ωkψqid.E167

Here it is seen that it is advantageous to choose the same proportional constants: kd=kq, to be able to set out it in front of the brackets, together with the angular speed:

pe=231kq2ωkψdiqψqidE168

and, eventually,

pe=231kdkqωkψdiqψqid.E169

This is the base expression for the power, which is converted from an electrical to a mechanical form in the motor or from a mechanical to an electrical form in the case of the generator. Next an expression for the electromagnetic torque is derived.

2.8.2 Electromagnetic torque of the three-phase machines in the dq0 system

As it is known, an air gap power can be expressed by the product of the developed electromagnetic torque and a mechanical angular speed, now in the k-system:

pe=teΩkE170

or by means of electrical angular speed:

pe=teωkpE171

where p is the number of pole pairs. An instantaneous value of the developed electromagnetic torque is valid:

te=pωkpe=pωk231kdkqωkψdiqψqidE172

and after a reduction the torque is:

te=p231kdkqψdiqψqid.E173

This is the base expression for an instantaneous value of developed electromagnetic torque of a three-phase machine. It is seen that its concrete form will be modified according to the chosen proportional constants. The most advantageous choice seems to be the next two possibilities:

  1. kd=kq=23,k0=13.

    Then:

    te=p2312233ψdiqψqid=p32ψdiqψqidE174

  2. kd=kq=23,k0=13.

    Then:

    te=p2312323ψdiqψqid=pψdiqψqid.E175

It will be shown later that the first choice is more advantageous for asynchronous machines and the second one for synchronous machine.

A developed electromagnetic torque in the rotating electrical machines directly relates with equilibrium of the torques acting on the shaft. During the transients in motoring operation, i.e., when the speed is changing, developed electromagnetic torque te covers not only load torque tL, including loss torque, but also load created by the moment of inertia of rotating mass JdΩdt. Therefore, it is possible to write:

te=JdΩdt+tL.E176

Unknown variables in motoring operation are obviously currents and speed, which can be eliminated from Eqs. (176) and (175). The mechanical angular speed is valid:

dΩdt=1JtetLE177

and electrical angular speed is:

dωdt=pJtetL.E178

The final expression for the time changing of the speed will be gotten, if for te Eqs. (174) and (175) according to the choice of the constants kd and kq are introduced:

  1. kd=kq=23,k0=13

    Then

    te=p32ψdiqψqidE179
    dωdt=pJp32ψdiqψqidtLE180

  2. kd=kq=23,k0=13.

    Then

    te=pψdiqψqidE181
    dωdt=pJpψdiqψqidtL.E182

If there is a steady-state condition, dωdt=0, and electromagnetic and load torque are in balance:

te=tL.E183

2.8.3 Power invariance principle

The expression for the three-phase power in dq0 system is:

pin=231kd2udid+231kq2uqiq+131k02u0i0E184

which was derived from the original expression for the three-phase power in abc system:

pin=uaia+ubib+ucic.E185

The expression can be modified by means of the constants kd and kq:

  1. If kd=kq=23,k0=13,

    then

    pin=32udid+32uqiq+3u0i0,E186

    in which the principle of power invariance is not fulfilled, because the members in dq0 axes are figures, although it was derived from Eq. (163), where no figures were employed.

  2. If kd=kq=23,k0=13,

    then

    pin=udid+uqiq+u0i0,E187

in which the principle of power invariance is fulfilled.

2.9 Properties of the transformed sinusoidal variables

In Section 7, the three-phase system abc into the dq0 system was transformed, and expressions for ud, uq, and u0 variables were derived. Now it is necessary to know what must be introduced for ud, uq, and u0, if variables ua, ub, and uc are sinusoidal variables (or also cosinusoidal variables can be taken). It means sinusoidal variables will be transformed from abc to dq0 followed by the rules given in Section 4.2.

Consider the voltage symmetrical three-phase system:

ua=Umaxsinωst,E188
ub=Umaxsinωst2π3,E189
uc=Umaxsinωst+2π3,E190

where ωs is the angular frequency of the stator voltages (and currents). In Figure 25, the relationship between the stator, rotor, and k-system is seen. As it was proclaimed, the stator axis is identified with the axis of the stator winding of phase a, the rotor axis is identified with the axis of the rotor winding of phase A, and this axis is shifted from the stator axis about angle ϑr. The axis of the reference k-system, to which the stator variables, now voltages, will be transformed, is shifted from the stator axis about the angle ϑkand from the rotor axis about the angle ϑkϑr. The angle of the k-system ϑkis during the transients expressed as integral of its angular speed with the initial position ϑk0:

ϑk=0tωkdt+ϑk0.E191

Equations for transformation (124) till (126), derived in Section 4 for the variable x, now are applied for the voltage:

ud=kduacosϑk+ubcosϑk2π3+uccosϑk+2π3,E192
uq=kquasinϑk+ubsinϑk2π3+ucsinϑk+2π3,E193
u0=k0ua+ub+uc,E194

For the phase voltages, expressions from Eqs. (188)(190) are introduced. At first, adjust expression for the voltage in the d-axis is as follows:

ud=kdUmaxsinωstcosϑk+sinωst2π3cosϑk2π3+sinωst+2π3cosϑk+2π3.E195

After the modification of the goniometrical functions and summarization of the appropriate members, in the final phase, it can be adjusted as follows:

ud=kdUmax(sinωstcosϑk+12sinωstcosϑk32cosωstsinϑk)=kdUmax(32sinωstcosϑk32cosωstsinϑk)=kdUmax32(sinωstcosϑkcosωstsinϑk)=kdUmax32sin(ωstϑk).E196

In the transients, if the speed is changing, the angle ϑk is given by Eq. (191). In the steady-state condition, when the speed is constant, ωk = const., the equation for the voltage is as follows:

ud=kdUmax32sinωstωktϑk0=kdUmax32sinωsωktϑk0.E197

Here it is seen that the voltage in d-axis is alternating sinusoidal variable with the frequency which is the difference of the both systems: original three-phase abc system with the angular frequency ωs and k-system, rotating with the speed ωk.

Now the same approach will be used for the q-axis:

uq=kqUmaxsinωstsinϑk+sinωst2π3sinϑk2π3+sinωst+2π3sinϑk+2π3.E198

The adjusting will result in equation:

uq=kqUmax32sinωstsinϑk+cosωstcosϑk,

which is finally accommodated to the form:

uq=kqUmax32cosωstϑk.E199

The voltage in the q-axis is shifted from the voltage in the d-axis about 90°, which is in coincidence with the definition of the d-axis and q-axis positions, which are perpendicular to each other. In transients when the speed is quickly changing, the angle ϑk is given by Eq. (191). In the steady-state condition, when the speed is constant, ωk = const., the equation for the voltage is in the form:

uq=kqUmax32cosωstωktϑk0=kqUmax32cosωsωktϑk0.E200

Finally, the equation for the zero component is adjusted as follows:

u0=k0Umaxsinωst+sinωst2π3+sinωst+2π3.E201

It is the sum of the voltage instantaneous values of the symmetrical three-phase system, which is, as it is known immediately, zero, or it is necessary to multiply all expressions for goniometrical functions, and after summarization of the appropriate members, the result is zero:

u0=k0Umaxsinωst+sinωst2π3+sinωst+2π3=0,E202

which is in coincidence with a note that the sum of the instantaneous values of variables, therefore also voltages, of the symmetrical three-phase system, is zero.

If the investigated three-phase system is not symmetrical, the zero component would have no zero value and would be necessary to add the equation for zero component to the dq0 system of equations. After the solution of dq0 variables, it would be necessary to make an inverse transformation on the basis of Eqs. (133)(135), where component x0 would appear.

Here the universality of the method of transformation is seen, because it is possible to investigate also unsymmetrical three-phase systems.

At the end of this chapter, the properties of the transformed sinusoidal variables are summarized, as shown in the above equations:

  1. Variables d and q are alternating variables with a frequency which is given by the difference of the frequency of both systems: original three-phase system abc with the angular frequency ωs and k-system rotating by angular speed ωk.

  2. Transformed variables d and q are shifted about 90°, unlike the three-phase system, in which the axes are shifted about 120°.

  3. Variables of the zero component, i.e., with the subscript 0, are in the case of the symmetrical system, zero. If the three-phase system is not symmetrical, it is necessary to take the zero component into account, to find its value and to employ it in the inverse transformation into the system abc.

  4. Magnitudes of variables dq0 depend on the choice of the constant of the proportionality.

The voltages in d-axis and q-axis are adjusted to the form:

ud=kdUmax32sinωstϑk=Udmaxsinωstϑk,E203

whereby

Udmax=kdUmax32,E204
uq=kqUmax32cosωstϑk=Uqmaxcosωstϑk,E205

whereby

Uqmax=kqUmax32.E206

Here it is seen that if:

  1. kd=kq=23, then Udmax=Uqmax=Umax, but the principle of the power invariance is not valid.

  2. kd=kq=23, then Udmax=Uqmax=32Umax, but the principle of the power invariance is valid (see Section 8 and Eqs. (186) and (187)).

Equations for the voltages ud and uq are adjusted to the final form not only on the basis of the constants of proportionality but also on the basis of the k-system position, i.e., how the angle ϑk is chosen (see Section 10).

Note that if there are supposed cosinusoidal functions of the three-phase system, i.e.,

ua=Umaxcosωst,E207
ub=Umaxcosωst2π3,E208
uc=Umaxcosωst+2π3,E209

after the same approach at derivation as for sinusoidal functions, equations for the variables in d-axis and q-axis are gotten:

ud=kdUmax32cosωstϑk,E210
uq=kqUmax32sinωstϑk.E211

As it will be shown in Section 19, this version of the voltage origin of the three-phase system definition is more suitable for a synchronous machine because of the investigation of the load angle.

2.10 Choice of the angle ϑk and of the reference k-system position

The final form of the voltage equations in the system dq0 does not depend only on the choice of the proportionality constants but also on the position of the reference k-system and the angle ϑk and the angular speed ωk.

The k-system can be positioned totally arbitrary, but some of the choices bring some simplicity in the investigation, which can be employed with benefit. Here are some of the most used possibilities, which are marked with special subscripts.

1.ϑk=0,ωk=0, subscripts α,β,0.

This choice means that the k-system is identified with the axis of the stator a-phase winding, i.e., the k-system is static and does not rotate, much like stator a-phase winding.

This choice is distinguished from all others by subscripts. Instead of the subscripts d, q, 0, the subscripts α,β,0are employed. Initially the subscripts α,β,γwere introduced, but after some development the new system of the subscripts α,β,0was introduced because it was more logical. The zero component is the same as in the system dq0.

Equations for the voltages and currents are adjusted to be able to see how this choice brings benefits. From the equation in the previous chapter:

ud=kdUmax32sinωstϑk=Udmaxsinωstϑk.E212

It is seen that if, simultaneously with the choice ϑk = 0, we take the proportional constants kd=kq=23and change the subscripts, then for the original voltage, ud is gotten:

uα=23Umax32sinωst0=Umaxsinωst=ua,E213

which are very important findings, in that an instantaneous value of the voltage (and current) in the transformed system is identical with the instantaneous value of the voltage (or current) in phase a. This brings very simple situation, because it is not needed to make any inverse transformation.

Have a look at the voltage in the β-axis. According to the equation from the previous chapter for the uq, and some accommodations, it results in the form:

uβ=23Umax32cosωst0=Umaxcosωst,E214

which means that this voltage is fictitious and such voltage does not exist in the real abc system and is shifted about 90° from the voltage uα.

It is the most important thing that in the same way the currents are transformed. It means that in the motoring operation, where the currents, together with the speed, are unknown, iα = ia is gotten, which means that the transformed system solution brings directly the current in a-phase and no inverse transformation is needed. The currents in the rest of two phases b and c are shifted about 120°, if there is a symmetrical system. In such system, it is valid that the zero component is zero. If there is an unsymmetrical system, where zero component is not zero, all variables must be investigated in great details and to find the real values in the abc system by inverse transformation.

Additionally here are equations for an electromagnetic torque and time varying of the angular speed. On the basis of Eqs. (179) and (180), derived for the kd=kq=23, after the changing of the subscripts, the following is gained:

te=p32ψαiβψβiα,E215
dωdt=pJp32ψαiβψβiαtL.E216

At the end it is necessary to say that this choice is not profitable only for the squirrel cage asynchronous motors (see Section 11) but also for asynchronous motors with wound rotor and for asynchronous generators.

2.ϑk=ϑr,ωk=ωr, subscripts d, q, 0

This choice means that the k-system is identified with the rotor axis and the speed of the k-system with the rotor speed.

This transformation is employed with benefit for synchronous machines, because in equations for the voltage, there is a so-called load angle (see Eq. (218)), which is a very important variable in the operation of the synchronous machines. On the rotor of the synchronous machine, there is a concentrating field winding fed by DC current, which creates DC magnetic flux. Here the d-axis is positioned. Therefore the rotor system is not necessary to transform because the field winding is positioned directly in d-axis, and if the rotor has damping windings, they are decomposed into two axes, d-axis and q-axis, perpendicular to each other. Finally, as it was mentioned before, this transformation was developed for synchronous machine with salient poles; therefore, the subscripts d, q are left in the form, in which they were used during the whole derivation.

As in previous case, equations for the voltages and currents are again adjusted to be able to see advantage of this choice.

ud=kdUmax32sinωstϑk=Udmaxsinωstϑk,E217

is seen that if ϑk=ϑris chosen, the argument of the sinusoidal function ωstϑris in fact the difference between the axis of rotating magnetic field and rotor position. This value is in the theory of synchronous machines defined as the load angle ϑL:

ωstϑr=ϑL.E218

In Section 16 and 18 there will be derived, why in the case of synchronous machines the proportionality constants are chosen in this form:

kd=kq=23, k0=13.

Then the original voltage udin this system is in the form:

ud=23Umax32sinωstϑr=32Umaxsinωstϑr,E219

and the voltage uqafter some accommodation is:

uq=23Umax32cosωstϑr=32Umaxcosωstϑr.E220

Equations for the electromagnetic torque and time varying of the speed (181) and (182), derived for kd=kq=23, are directly those equations, which are valid here, because the subscripts are not changed:

te=pψdiqψqid,E221
dωdt=pJpψdiqψqidtL.E222

Section 16 and others will deal with the synchronous machines in the general theory of electrical machines. These equations will be applied at the investigation of the properties of the synchronous machines in concrete examples, and also synchronous machines with permanent magnets will be investigated.

3.ϑk=0tωkdt+ϑk0E223

ωk=ωs, subscripts x,y,0

This choice means that the speed of k-system rotation ωk is identified with the synchronous speed of rotating magnetic field ωs, i.e., transformation axes rotate with the same speed as the space vector of the stator voltages.

Adjust equations for the voltages, in which the advantage of this choice will be visible. From the equations in the previous chapter:

ud=kdUmax32sinωstϑk=Udmaxsinωstϑk,E224
ud=kdUmax32sinωstωktϑk0=kdUmax32sinωsωktϑk0,E225

result that if ωkintroduces ωsand changes the subscripts, the expression for udis in the form:

ux=kdUmax32sinωsωstϑk0=kdUmax32sinϑk0=kdUmax32sinϑk0.E226

Similarly, for voltage uqat the changed subscripts, the following is gained:

uy=kqUmax32cosωsωstϑk0=kqUmax32cosϑk0=kqUmax32cosϑk0.E227

It is seen that both voltages in this system are constant DC variables, and it depends on the choice of the constants and initial value ϑk0 which value they will have. At the suitable initial position of the transformation axes, one of them can be zero.

If, for example, the initial position of the k-system is chosen to be zero, ϑk0 = 0, and constants of proportionality kd=kq=23; then equations are very simplified and are as follows:

ux=kdUmax32sinϑk0=23Umax32sin0=0,E228
uy=kqUmax32cosϑk0=23Umax32cos0=Umax.E229

If it looks uncomfortable that both voltages are negative values, it is enough, if derivation of transformation equations from abc to dq0 start with an assumption that:

ua=Umaxsinωst,E230
ub=Umaxsinωst2π3,E231
uc=Umaxsinωst+2π3.E232

In the steady-state condition, all variables on the stator and rotor are illustrated as DC variables. Therefore, the solution is very easy, but it is true that it is necessary to make an inverse transformation into the real abc system. This transformation system is very suitable for asynchronous motors.

The equation for torque is also very simplified, because the x-component of the current is also zero (ix = 0). Then together with the change of the subscripts, Eq. (174) for torque, where the constants of the proportionality kd=kq=23were used, is as follows:

te=p32ψxiyψyix=p32ψxiy,E233

and equation for time varying of the speed is:

dωdt=pJp32ψxiytL.E234

4.ϑk,ωk, are chosen totally generally, the position of the k-system is chosen totally generally, subscripts u, v, 0.

Although the whole derivation of transformed variables was made for the dq0 axis, because it was historically developed in such a way, and then the new subscripts were introduced by means of the special choice of the reference k-system position, it is seen that the subscripts dq0 are kept only for the synchronous machine, for which this transformation was developed. If it should be started now, perhaps two perpendicular axes to each other would be marked as u, v, 0. Nevertheless the original configuration of universal machine had windings in the axes d, q, and it is kept also for the future. However here introduced marking was not accepted by all experts dealing with this topic, and some authors used the system x, y, 0 instead of α, β, 0.

2.11 Asynchronous machine and its inductances

It is supposed that a reader is familiar with the basic design of asynchronous machine and its theory and properties. Now we will analyze the three-phase symmetrical system on the stator, marked abc and on the rotor, marked ABC, i.e., six windings together (Figure 26).

Figure 26.

Illustration figure of an asynchronous machine with three windings abc on the stator and three phase ABC on the rotor. They are shifted from each other about the angle ϑr.

Basic voltage equations for the terminal voltage can be written for each winding or by one equation, at which the subscripts will be gradually changed:

uj=Rjij+dψjdt,E235

where j = abc, ABC.

If the system is symmetrical, then it is possible to suppose that:

Ra=Rb=Rc=Rs,E236
RA=RB=RC=Rr.E237

Linkage magnetic flux can be also expressed by one equation as a sum of all winding contributions:

ψj=kψj,k=kLj,kikE238

where j, k = abc, ABC, but because of transparency here is the whole sum of the members in details:

ψa=Laaia+Labib+Lacic+LaAiA+LaBiB+LaCiC,
ψb=Lbaia+Lbbib+Lbcic+LbAiA+LbBiB+LbCiC,
ψc=Lcaia+Lcbib+Lccic+LcAiA+LcBiB+LcCiC,
ψA=LAaia+LAbib+LAcic+LAAiA+LAaBiB+LACiC,
ψB=LBaia+LBbib+LBcic+LBAiA+LBBiB+LBCiC,
ψC=LCaia+LCbib+LCcic+LCAiA+LCBiB+LCCiC,E239

where:

Laa=Lbb=Lcc=Lsareselfinductances of the stator windings.E240
LAA=LBB=LCC=Lrareselfinductances of the rotor windings.E241
Lab=Lac=Lba=Lbc=Lca=Lcb=Msaremutual inductances of the stator windings.E242
LAB=LAC=LBA=LBC=LCA=LCB=Mraremutual inductances of the rotor windings.E243

The others are mutual inductances of stator and rotor windings. It is necessary to investigate if they depend on the rotor position or not.

2.11.1 Inductances that do not depend on the rotor position

1. Self-inductances of the stator windings Ls

Self-inductance of stator single phase Ls without influence of the other stator phases and without influence of the rotor windings corresponds to the whole magnetic flux Φs, which is created by the single stator phase.

This flux is divided into two parts: leakage magnetic flux Φσs, which is linked only with the winding by which it was created and thus embraces only this phase, and magnetizing magnetic flux Φμ, which crosses air gap and enters the rotor and eventually is closed around the other stator or rotor windings. Inductances correspond with these fluxes according the permeance of the magnetic path and winding positions. Therefore it can be written:

Φs=Φσs+ΦμE244
Ls=Lσs+ME245

where M is the mutual inductance of single stator phase and single rotor phase if their axes are identical (see Figure 27a).

Figure 27.

(a) Illustration of mutual inductance M of the stator and rotor phase if their axes are identical, (b) justification of the mutual inductance value of the stator windings shifted about 120°, (c) magnetic flux directions if the phases are fed independently by positive currents, (d) sum of the magnetic fluxes if the windings are fed by three phases at the instant when a-phase has a positive magnitude and the b- and c-phases have half of negative magnitude.

2. Self-inductances of the rotor windings Lr.

These inductances are expressed similarly as the stator ones:

Lr=Lσr+M.E246

Have a note that in the whole general theory of electrical machines, rotor variables are referred to the stator side.

3. Mutual inductances of the stator windings—Ms

Take an image that two stator windings have an identical axis, e.g., b-phase is identified with a-phase. Then their mutual inductance is M. Now the b-phase is moved to its original position, i.e., about 120°. According to Figure 27b, the value of the mutual inductance in this position is:

cos120°=1/2M=MsE247

and this value is constant; it means it is always negative because the position of b-phase on the stator with regard to a-phase is stable.

4. Mutual inductances of the rotor windings—Mr.

The same analysis as in item 3 results in the finding that the mutual inductance of the rotor windings without the influence of the stator windings is always negative and equals (Figure 28).

Figure 28.

Equivalent circuit of the asynchronous machine (a) in d-axis and (b) in q-axis. All rotor variables are referred to the stator.

Mr=½ME248

2.11.2 Inductances depending on the rotor position

All mutual inductances of the stator and rotor windings are expressed as follows:

LaA=LAa=LbB=LBb=LcC=LCc=Mcosϑr,
LaB=LBa=LbC=LCb=LcA=LAc=Mcosϑr+2π3,
LaC=LCa=LbA=LAb=LcB=LBc=Mcosϑr2π3,E249

where M is the mutual inductance of the stator and rotor phase if their axes are identical.

All expressions are introduced into (239); therefore, the inductances, linkage fluxes, and currents can be written in the matrix form:

L=LsMsMsMcosϑrMcosϑr+23πMcosϑr23πMsLsMsMcosϑr23πMcosϑrMcosϑr+23πMsMsLsMcosϑr+23πMcosϑr23πMcosϑrMcosϑrMcosϑr23πMcosϑr+23πLrMrMrMcosϑr+23πMcosϑrMcosϑr23πMrLrMrMcosϑr23πMcosϑr+23πMcosϑrMrMrLr
ψaψbψcψAψBψC=LiaibiciAiBiCE250

After these expressions are introduced into (235), six terminal voltage equations are obtained, with nonlinear, periodically repeated coefficients Mcosϑr, Mcosϑr2π3, and Mcosϑr+2π3, where M is the mutual inductance of the stator and rotor phase if their axes are identical and ϑr is an angle between the axis of the same stator and rotor phase (Figure 29).

Figure 29.

Equivalent circuit of asynchronous machine for reference k-system: (a) in d-axis and (b) in q-axis for a chosen reference frame. All rotor variables are referred to the stator side to distinguish this case of transformation from the others, the axes are marked αβ0 (originally it was marked as αβγ), and also all subscripts of the currents, voltages, and linkage magnetic fluxes are with these subscripts. Then on the basis of the subscripts, it is possible to know what kind of transformation was used.

To solve such equations is very complicated; therefore, it is necessary to eliminate the periodically repeated coefficients. This is possible to do by various real or complex linear transformations. The most employed is Park linear transformation, mentioned in Section 4. In the next it is applied for this case.

2.12 Linkage magnetic flux equations of the asynchronous machine in the general theory of electrical machines

On the basis of the equations for transformation into dq0 system, the equation for ψd is written:

ψd=kdψacosϑk+ψbcosϑk2π3+ψccosϑk+2π3.E251

The expressions for ψa, ψb a ψc from (250) are introduced into it:

ψd=kd((LsiaMsibMsic)cosϑk+(M(cosϑr)iA+Mcos(ϑr+2π3)iB+Mcos(ϑr2π3)iC)(cosϑk))+kd((Msia+LsibMsic)cos(ϑk2π3)+(Mcos(ϑr2π3)iA+Mcos(ϑr)iB+Mcos(ϑr+2π3)iC)cos(ϑk2π3))+kd((MsiaMsib+Lsic)cos(ϑk+2π3)+(Mcos(ϑr+2π3)iA+Mcos(ϑr2π3)iB+Mcos(ϑr)iC)cos(ϑk+2π3))E252

If we consider that:

i0=k0ia+ib+ic,
i0k0=ia+ib+ic,
ib+ic=i0k0ia,
Msib+ic=Msi0k0ia,
Msia+ic=Msi0k0ib,
Msia+ib=Msi0k0ic,

and these expressions are introduced into the equation above for all three phases, after modifications, some expressions that are zero are found, e.g.:

Msi0k0cosϑk+cosϑk2π3+cosϑk+2π3=0,E253

and others in which transformed variables are seen, e.g.:

Mskdiacosϑk+ibcosϑk2π3+iccosϑk+2π3=Msid,E254

or:

Lskdiacosϑk+ibcosϑk2π3+iccosϑk+2π3=Lsid.E255

If these two expressions are summed, it results in:

Ls+Msid=Ldid,E256

where Ld is introduced as the sum of the self Ls and mutual inductance Ms of the stator windings in the d-axis. Then it is seen that Ld is a total inductance of the stator windings in the d-axis:

Ld=Ls+Ms=Lσs+M+M2=Lσs+32M=Lσs+Lμd=Lσs+Lμ.E257

This is evident also in Figure 27a, if a constant air gap of asynchronous machines is taken into account. Therefore, inductances in d-axis and q-axis are equal, and there is no need to mark separately magnetizing inductance in d-axis and q-axis. The members of Eq. (252), in which act rotor currents iA, iB, iC, can be also accommodated in a similar way as the stator currents, which results in the following:

32MkdiAcosϑkϑr+iBcosϑkϑr2π3+iCcosϑkϑr+2π3=LdDiD.E258

In this expression there is used a knowledge, that: (1) mutual inductance of the stator and rotor winding with contribution of all three stator phases is 3/2 M what is marked LdD, but it is known that in the equivalent circuit is marked as Lμ and (2) the angle between the axis of the rotor phase and the axis of the reference k-system is ϑkϑr. Therefore, the rotor variables are transformed into k-system by means of this angle.

Then it is possible to write that the transformed current of the rotor system is iD:

kdiAcosϑkϑr+iBcosϑkϑr2π3+iCcosϑkϑr+2π3=iD,E259

and the whole Eq. (252) can be written much more briefly:

ψd=Ldid+LdDiD,E260

where Ld is given by the (257) and

LdD=32M=Lμ.E261

The equation for ψq is obtained in a similar way and after accommodations is written in the form:

ψq=Lqiq32kqMiAsinϑkϑr+iBsinϑkϑr2π3+iCsinϑkϑr+2π3,E262

or briefly:

ψq=Lqiq+LqQiQ,E263

where:

LqQ=32M=Lμ,E264

and

kqiAsinϑkϑr+iBsinϑkϑr2π3+iCsinϑkϑr+2π3=iQ.E265

Considering that in the asynchronous machine the air gap is constant around the whole periphery of the stator boring, there is no difference in the inductances in d-axis and q-axis; therefore, the following can be written:

ψq=Ldiq+LdDiQ,E266

and also

LdD=LqQ=32M=Lμ.E267

Zero component is as follow:

ψ0=L0i0,E268

where:

L0=Ls2Ms=Lσs+M2M2=Lσs.E269

The fact that the zero-component inductance L0 is equal to the stator leakage inductance Lσs can be used with a benefit if Lσs should be measured. All three phases of the stator windings are connected together in a series, or parallelly, and fed by a single-phase voltage. In this way a pulse, non-rotating, magnetic flux is created. Thus a zero, non-rotating, component of the voltage, current, and impedance is measured.

Linear transformation is employed also at rotor linkage magnetic flux derivations in the system DQ0:

ψD=LDiD+LDdid,E270
ψQ=LQiQ+LQqiq,E271

eventually considering that the air gap is constant and the parameters in the d-axis and q-axis are equal:

ψQ=LDiQ+LDdiq.E272

The meaning of the rotor parameters is as follows:

LD=LQ=Lr+Mr=Lσr+M+M2=Lσr+32M=Lσr+Lμ.E273

Similarly, for the zero rotor component can be written as:

ψO=LOiO,E274

where:

LO=Lr2Mr=Lσr+M2M2=Lσr.E275

Take into account that all rotor variables are referred to the stator side; eventually they are measured from the stator side.

2.13 Voltage equations of the asynchronous machine after transformation into k-system with d-axis and q-axis

Voltage equations of the asynchronous machines in the dq0 system are obtained by a procedure described in Section 7. There are equations for the stator terminal voltage in the form:

ud=Rsid+dψddtωkψq,E276
uq=Rsiq+dψqdt+ωkψd,E277
u0=Rsi0+dψ0dt.E278

The rotor voltage equations are derived in a similar way as the stator ones but with a note that the rotor axis is shifted from the k-system axis about the angle ϑkϑr; thus in the equations there are members with the angular speed ωkωr:

uD=RriD+dψDdtωkωrψQ,E279
uQ=RriQ+dψQdt+ωkωrψD,E280
uO=RriO+dψOdt.E281

These six equations create a full system of the asynchronous machine voltage equations. Rotor variables are referred to the stator side; expressions for the linkage magnetic flux are shown in Section 12.

2.14 Asynchronous motor and its equations in the system αβ0

According to Sections 7 and 10, the reference k-system can be positioned arbitrarily, but some specific positions can simplify solutions; therefore, they are used with a benefit. One of such cases happens if the d-axis of the k-system is identified with the axis of the stator a-phase; it means ϑk=0,ωk=0. This system is in this book marked as αβ0 system.

This system is obtained by phase variable projection into stationary reference system, linked firmly with a-phase. It is a two-axis system, and zero components are identical with the non-rotating components known from the theory of symmetrical components.

The original voltage equations of asynchronous machine derived in Sections 7 and 13 are as follows:

ud=Rsid+dψddtωkψq,E282
uq=Rsiq+dψqdt+ωkψd,E283
u0=Rsi0+dψ0dt,E284
uD=RriD+dψDdtωkωrψQ,E285
uQ=RriQ+dψQdt+ωkωrψD,E286
uO=RriO+dψOdt,E287

where:

ψd=Ldid+LdDiD,E288
ψq=Lqiq+LqQiQ,E289
ψ0=L0i0,E290
Ld=Lq=Lσs+Lμ,E291
LdD=LDd=32M=Lμ,E292
LqQ=LQq=LdD=32M=Lμ,E293
ψD=LDiD+LDdid,E294
ψQ=LQiQ+LQqiq,E295
ψO=LOiO,E296
LD=LQ=Lr+Mr=Lσr+M+M2=Lσr+32M=Lσr+Lμ,E297
LO=Lr2Mr=Lσr+M2M2=Lσr.E298

Now new subscripts the following are introduced:

For currents and voltages:

d=αs,q=βs,D=αr,Q=βr,E299

For inductances:

Ld=Lq=Lσs+Lμ=LS,E300
LD=LQ=Lσr+Lμ=LR,E301
LdD=LqQ=32M=Lμ.E302

The original equations, rewritten with the new subscripts, with the fact that ϑk=0andωk=0and with an assumption that the three-phase system is symmetrical, meaning the zero components are zero, are as follows:

uαs=Rsiαs+LSdiαsdt+LμdiαrdtE303
uβs=Rsiβs+LSdiβsdt+LμdiβrdtE304
uαr=Rriαr+ωrLRiβr+ωrLμiβs+LRdiαrdt+LμdiαsdtE305
uβr=RriβrωrLRiαrωrLμiαs+LRdiβrdt+LμdiβsdtE306

If transients are solved for motoring operation, then stator terminal voltages on the left side of the equations are known variables and are necessary to introduce derived expressions for sinusoidal variables transformed into dq0 system, now α, β-axes ((196) for ud and (199) for uq). Rotor voltages are zero, if there is squirrel cage rotor. If there is wound rotor, here is a possibility to introduce a voltage applied to the rotor terminals, as in the case of asynchronous generator for wind power stations, where the armature winding is connected to the frequency converter. If the rotor winding is short circuited, then the rotor voltages are also zero.

In the motoring operation, the terminal voltages are known variables, and unknown variables are currents and speed. Therefore it is suitable to accommodate the previous equations in the form where the unknown variables are solved. From Eq. (303), the following is obtained:

LSdiαsdt=uαsRsiαsLμdiαrdt,E307

and from Eq. (305):

diαrdt=1LRuαrRriαrωrLRiβrωrLμiβsLμdiαsdt.E308

This equation is introduced into Eq. (307). Then it is possible to eliminate a time variation of the stator current in the α-axis:

diαsdt=LRLSLRLμ2uαsRsiαs+LμLRRriαr+ωrLμ2LRiβs+ωrLμiβrLμLRuαr.E309

The same way is applied for the other current components:

diαrdt=LSLSLRLμ2uαrRriαr+LμLSRsiαsωrLμiβsωrLRiβrLμLSuαs,E310
diβsdt=LRLSLRLμ2uβsRsiβs+LμLRRriβrωrLμ2LRiαsωrLμiαrLμLRuβr,E311
diβrdt=LSLSLRLμ2uβrRriβr+LμLSRsiβs+ωrLμiαs+ωrLRiαrLμLSuβs.E312

The last equation is for time variation of the speed. On the basis of Section 8, if in the equation for the electromagnetic torque the constants kd = kq = 2/3 are introduced and after changing the subscripts, the torque is in the form:

te=p231kdkqψdiqψqid=p32ψαsiβsψβsiαs=p32Lμiαriβsiβriαs,
te=p32Lμiαriβsiβriαs.E313

After considering Eq. (176), the electrical angular speed is obtained in the form:

dωrdt=pJp32LμiαriβsiβriαstL.E314

Mechanical angular speed is linked through the number of the pole pairs Ωr=ωrp, which directly corresponds to the revolutions per minute.

For Eqs. (309)(312), the next expressions are introduced for the voltages (see Sections 9 and 10):

uαs=Umsinωst=ua,E315
uβs=Umcosωst,E316

which is displaced about 90° with regard to the uαs. Rotor voltages in the most simple case for the squirrel cage rotor are zero:

uαr=uβr=0.E317

In the next chapter, solving of the transients in a concrete asynchronous motor with squirrel cage rotor and wound rotor is shown.

2.15 Simulation of the transients in asynchronous motors

2.15.1 Asynchronous motor with squirrel cage rotor

Equations derived in the previous chapter are applied on a concrete asynchronous motor with squirrel cage rotor. The rotor bars are short circuited by end rings; thus the rotor voltages uαr and uβr in Eqs. (309)(312) are zero.

In Figure 30, simulation waveforms of the starting up of an asynchronous motor when it is switched directly across the line are shown. Parameters of the investigated motor are in Table 3.

Figure 30.

Simulation waveforms of the asynchronous motor at its switching directly across the line, time waveforms of the (a) speed, (b) a-phase current Ia, (c) developed electromagnetic torque, and (d) torque vs. speed if the motor is fed by rated voltage.

PN = 1.1 kWRs = 6.46 Ω
UN = 230 VLs = Lr = 0.5419 H
IN = 2.4 ARr = 5.8 Ω
fN = 50 HzLμ = 0.5260 H
nN = 2845 min−1J = 0.04 kg m2
TN = 3.7 Nmp = 1
Tloss = 0.1 Nm

Table 3.

Nameplate and parameters of the investigated asynchronous motor.

Simulation waveforms in Figure 30a–c show time variations of the variables n = f(t), ia = f(t), and te = f(t) after switching the motor directly across the line. At the instant t = 0.5 s, the motor is loaded by the rated torque TN = 3.7 Nm. In Figure 30d, torque vs. speed curve Te = f(n) is shown. As it is seen from the waveforms, during the starting up, the motor develops very high starting torque, which could be dangerous for mechanical load of some parts of the drive system, and there are very high starting currents, which could be dangerous for the motor because of its heating and for the feeding part of the drive.

Relatively large starting current can cause an appreciable drop in motor terminal voltage, which reduces the starting current but also the starting torque. If the supply voltage drop would be excessive, some kind of across-the-line starter that reduces the terminal voltage and hence the starting current is required. For this purpose, a three-phase step-down autotransformer may be employed. The autotransformer is switched out of the circuit as the motor approaches full speed. The other method of starting is by a star-delta switch or by inserting resistances into the stator winding circuit. In the industry, a special apparatus is used, the so-called softstarter, which enables the starting of the defined requirement. Softstarter contains solid-state elements (thyristors), which enable to vary the terminal voltage of the motor. The start up is carried out by limitation of the maximal value of current, which will not be gotten over during the starting. This control is ensured by the possibility to change the terminal voltage of the motor. The more sophisticated way is a frequency starting during which not only voltage but also frequency is gradually increased, whereby the ratio U/f is kept constant. During start up, also maximum of the speed acceleration is defined.

Simulations of softstarter and frequency converter applications are shown in Figure 31. In both cases not only value of the starting torque is reduced, which is undesirable, but also the value of the starting current. The current does not cross the rated value and in this simulated case neither no-load current Ia0. It is seen in comparison waveforms in Figure 30b with waveforms in Figure 31(c) and (d).

Figure 31.

Simulations of the asynchronous motor starting up by means of softstarter, time waveforms of (a) speed n, (c) phase current Ia, and (e) developed electromagnetic torque, and by means of a frequency converter, again in the same order: time waveforms of the (b) speed n, (d) phase current Ia, and (f) developed electromagnetic torque.

2.15.2 Asynchronous motor with wound rotor

Equations in Section 14 are the basis for the simulations. In this case, it is possible to feed the terminals of the wound armature on the rotor. This possibility is employed in applications with asynchronous generators, where feeding to the rotor serves as stabilization of the output frequency of the generator. Previously, the rotor terminals of the asynchronous motor were used for variation of the rotor circuit resistance by external rheostats. Such starting up is shown in this part. The nameplate and parameters of the investigated motor are in Table 4.

PN = 4.4 kWRs = 1.125 Ω
UsN = 230 V , UrN = 64 VLs = Lr = 0.1419 H
IsN = 9.4 A, IrN = 47 ARr = 1.884 Ω
fN = 50 HzLμ = 0.131 H
nN = 1370 min−1J = 0.04 kg m2
TN = 30 Nmp = 2
Tloss = 0.1 Nm

Table 4.

Nameplate and parameters of the investigated wound rotor asynchronous motor.

Simulations are shown in Figure 32a–c. There are time waveforms of the variables n = f(t), ia = f(t), and te = f(t) after the switching directly across the line.

Figure 32.

Simulations of the rotor wound asynchronous motor during the switching directly across the line. Time waveforms of the (a) speed, (b) phase current Ia, (c) developed electromagnetic torque, and (d) torque vs. speed at rated voltage.

Figure 33.

Time waveforms of the simulations during the starting up of the wound rotor asynchronous motor by means of rheostats added to the rotor circuits: (a) speed n, (c) phase current Ia, (e) developed electromagnetic torque, and time waveforms during the starting up by means of frequency converter, again in the same order: (b) speed n, (d) phase current Ia, and (f) developed electromagnetic torque.

At the instant t = 0.5 s, the motor is loaded by the rated torque TN = 30 Nm. In Figure 32d there is a curve Te = f(n).

Simulation waveforms are very similar with those of the squirrel cage rotor (high starting current and torques). But in the case of wound rotor, there is a possibility to add external resistors and to control the current and the torque (Figure 33).

2.16 Synchronous machine and its inductances

It is supposed that a reader is familiar with the basic knowledge of a synchronous machine theory, properties, and design configuration. The synchronous machine with salient poles on the rotor; symmetrical three-phase system a, b, c on the stator; field winding f in the d-axis on the rotor; and damping winding, split into two parts perpendicular to each other (D and Q on the rotor), positioned in the d-axis and q-axis, as it is seen in Figure 34, is analyzed. The d-axis on the rotor is shifted about the angle ϑr from the axis of the a-phase on the stator.

Figure 34.

Synchronous machine with salient poles on the rotor and three-phase winding a, b, c on the stator, field winding f, and damping winding split into two parts (D and Q) perpendicular to each other, positioned in the d-axis and q-axis on the rotor. The d-axis on the rotor is shifted about the angle ϑr from the axis of a-phase on the stator.

Basic equations for terminal voltage can be written for each of the winding separately, or briefly by one equation, in which the subscripts are gradually changed for each winding:

uj=Rjij+dψjdtE318

where j = a, b, c, f, D, Q.

If symmetrical three-phase winding on the stator is supposed, then it can be supposed that their resistances are identical and can be marked by the subscript “s”:

Ra=Rb=Rc=RsE319

Linkage magnetic flux can be also expressed briefly by the sum of all winding contributions:

ψj=kψj,k=kLj,kikE320

where j, k = a, b, c, f, D, Q. For a better review, here are all the equations with the sum of all members:

ψa=Laaia+Labib+Lacic+Lafif+LaDiD+LaQiQ,ψb=Lbaia+Lbbib+Lbcic+Lbfif+LbDiD+LbQiQ,ψc=Lcaia+Lcbib+Lccic+Lcfif+LcDiD+LcQiQ,ψf=Lfaia+Lfbib+Lfcic+Lffif+LfDiD+LfQiQ,ψD=LDaia+LDbib+LDcic+LDfif+LDDiD+LDQiQ,ψQ=LQaia+LQbib+LQcic+LQfif+LQDiD+LQQiQ.E321

Although it is known that mutual inductances of the windings that are perpendicular to each other are zero:

LfQ=LQf=LDQ=LQD=0,E322

for computer manipulation is more suitable if the original structure is kept and all inductances appear during the analysis:

Laa,Lbb,Lccareselfinductances of the stator windings.E323
Lff,LDD,LQQareselfinductances of the rotor windings.E324
Lab,Lac,Lba,Lbc,Lca,Lcbaremutual inductances of the stator windings.E325
LfD,LfQ,LDf,LDQ,LQf,LQDaremutual inductances of the rotor windings.E326

The rest of the inductances are mutual inductances of the stator and rotor windings:

Laf,Lbf,Lcf,LaD,etc.

It is important to investigate if inductances depend on the rotor position or not.

2.16.1 Inductances that do not depend on the rotor position

Self- and mutual inductances of the rotor windings Lff,LQQ,LDD,LfDdo not depend on the rotor position because the stator is cylindrical, and if the stator slotting is neglected, then the air gap is for each winding constant. Thus, the magnetic permeance of the path of magnetic flux created by these windings does not change if the rotor rotates.

2.16.2 Inductances depending on the rotor position

2.16.2.1 Mutual inductances of the rotor and stator windings

Investigate, for example, a-phase winding on the stator and field winding f on the rotor, as it is shown in Figure 34.

When sinusoidally distributed windings are assumed, i.e., coefficients of higher harmonic components are zero, then the waveform of mutual inductance is cosinusoidal, if for the origin of the system such rotor position is chosen in which the a-phase axis and the axis of the field winding are identical (see Figure 35).

Figure 35.

(a) Illustration to express mutual inductance of the a-phase on the stator and field winding f on the rotor, (b) waveform of the mutual inductance Laf versus rotor position ϑr.

Then the mutual inductances can be expressed as follows:

Laf=Lfa=LafmaxcosϑrE327
Lbf=Lfb=Lafmaxcosϑr2π3E328
Lcf=Lfc=Lafmaxcosϑr+2π3E329

similarly:

LaD=LDa=LaDmaxcosϑrE330
LbD=LDb=LaDmaxcosϑr2π3E331
LcD=LDc=LaDmaxcosϑr+2π3E332

Expressions for Q-winding positioned in the q-axis are written according to Figure 36a, where it is seen that the positive q-axis is ahead about 90° of the d-axis. Hence if the d-axis is identified with the axis of the a-phase, the q-axis is perpendicular to it, and mutual inductance LaQis zero. To obtain a position in which LaQ is maximal, it is necessary to go back about 90°, to identify q-axis with the a-phase axis. There the LaQ receives its magnitude. The magnitudes of the mutual inductances between Q-winding and b- and c-phases are shifted about 120°, as it is seen in Figure 36b.

Figure 36.

(a) Illustration to express mutual inductance of the Q-winding on the rotor and a-phase on the stator and (b) waveform of the mutual inductances LaQ, LbQ, LcQ, versus rotor position ϑr.

LaQ=LQa=LaQmaxcosϑr+π2=LaQmaxsinϑr,E333
LbQ=LQb=bLaQmaxsinϑr2π3,E334
LcQ=LQc=cLaQmaxsinϑr+2π3E335

2.16.2.2 Self-inductances of the stator

Self-inductances of the stator depend on the rotor position if there are salient poles. Self-inductance of the a-phase is maximal (Laamax), if its axis is identical with the axis of the pole. In this position the magnetic permeance is maximal. The minimal self-inductance of the a-phase (Laamin) occurs if the axis of the a-phase and axis of the pole are shifted about π/2. Because the magnetic permeance is periodically changed for each pole, it means north and south, the cycle of the self inductance is π, as it is seen in Figure 37.

Figure 37.

(a) Illustration to express self-inductance Laa of the a-phase on the stator and (b) waveform of the self-inductance Laa versus rotor position ϑr.

Laa=La0+L2cos2ϑr,E336
Lbb=La0+L2cos2ϑr2π3,E337
Lcc=La0+L2cos2ϑr+2π3.E338

The magnitude of the self-inductance Laamaxis obtained if the axis of the salient pole is identical with the axis of the stator a-phase; it means ϑr=0. Then:

Laamax=La0+L2.E339

The minimal value of the self-inductance is obtained if the axis of the salient pole is perpendicular to the axis of the stator a-phase, i.e., ϑr=π/2. Then:

Laamin=La0L2.E340

If the rotor rotates about ϑr=π, the self-inductance obtains again its maximal value, etc.; accordingly self-inductance does not obtain negative values, as it is seen in Figure 37b.

2.16.2.3 Mutual inductance of the stator windings

Mutual inductances of the stator windings depend on the rotor position only in the case of the salient poles on the rotor. These inductances are negative because they are shifted about 120° (see explanation in Figure 27b). The rotor is in a position where mutual inductance Lbcis maximal is shown in Figure 38a. Its waveform vs. rotor position is in Figure 38b.

Figure 38.

(a) Illustration to express mutual inductance of the stator windings a, b, c and (b) waveform of the mutual inductance Lbc vs. rotor position ϑr.

It is possible to assume that for the sinusoidally distributed windings, the magnitudes of harmonic waveform L2 are the same as in the case of the self-inductance of the stator windings. In the windings embedded in the slots, with a final number of the slots around the rotor periphery and the same number of the conductors in the slots, this assumption is not fulfilled; thus magnitudes of self and mutual waveforms can be different. Here a source of mistakes can be found and eventually discrepancies between the calculated and measured values. The waveforms in Figure 38b can be written as follows:

Lbc=Lab0L2cos2ϑrE341
Lca=Lab0L2cos2ϑr2π3=Lab0L2cos2ϑr+2π3E342
Lab=Lab0L2cos2ϑr+2π3=Lab0L2cos2ϑr2π3E343

or

Lab=Lab0+L2cos2ϑr+2π3=Lab0+L2cos2ϑr2π3E344

which better corresponds to the waveform in Figure 38.

Now all the expressions of these inductances are introduced into Eq. (65) and Eq. (318). They are equations with nonlinear periodically changed coefficients. To eliminate these coefficients, it is necessary to transform the currents, voltages, and linkage magnetic fluxes. The most suitable is Park linear transformation, which was explained in Section 4 and is applied again in the next chapter.

2.17 Terminal voltage equations of the synchronous machine after a transformation into k-system with the axes d, q, 0

Terminal voltage equations of the synchronous machine stator windings in a system d, q, 0 are obtained by means of the procedure described in Section 7. The next equations were derived:

ud=Rsid+dψddtωkψq,E345
uq=Rsiq+dψqdt+ωkψd,E346
u0=Rsi0+dψ0dt.E347

Equations (160)(162) are voltage equations of the three-phase stator windings, in this case synchronous machine but also asynchronous machine, as it was mentioned in Section 13. They are equations transformed into reference k-system rotating by angular speed ωk, with the axes d, q, 0. As it is seen, they are the same equations as in Section 2.1, which were derived for universal configuration of an electrical machine.

Terminal voltage equations of the synchronous machine rotor windings are not needed to transform in the d-axis and q-axis, because the rotor windings are embedded in these axes, as it is seen in Figure 34, and are written directly in the two-axis system d, q, 0:

uf=Rfif+dψfdt,E348
uD=RDiD+dψDdt,E349
uQ=RQiQ+dψQdt.E350

The next the expressions for linkage magnetic flux are investigated.

2.18 Linkage magnetic flux equations of the synchronous machine in the general theory of electrical machines

In Eq. (65) of linkage magnetic fluxes, expressions for inductances as they were derived in Section 16 are introduced. For example, for field winding with a subscript “f,” the equation for linkage magnetic flux is written as follows:

ψf=Lfaia+Lfbib+Lfcic+Lffif+LfDiD+LfQiQ,E351
ψf=Lafmaxiacosϑr+ibcosϑr2π3+iccosϑr+2π3+Lffif+LfDiD+LfQiQ.E352

If this equation is compared with Eq. (124), written for a general variable x, it is seen that the expression in the square bracket is equal to id/kd,, if ϑk=ϑr:

iacosϑr+ibcosϑr2π3+iccosϑr+2π3=1kdid.E353

After this modification in Eq. (352), the following is obtained:

ψf=1kdLafmaxid+Lffif+LfDiD+LfQiQ=Lfdid+Lffif+LfDiD,E354

where it was taken into account that mutual inductance of two perpendicular windings f and Q is zero.

On the same basis, the linkage magnetic flux for damping rotor windings D and Q is received:

ψD=1kdLaDmaxid+LDfif+LDDiD+LDQiQ=LDdid+LDfif+LDDiD,E355
ψQ=LaQmax[iasinϑr+ibsin(ϑr2π3)+icsin(ϑr+2π3)]+LQfif+LQDiD+LQQiQ=1kqLaQmaxiq+LQfif+LQDiD+LQQiQ=LQqiq+LQQiQE356

A derivation for the stator windings is made in the like manner. It is started with a formal transformation equation from system a, b, c into the d-axis and then into the q-axis. The equation in the d-axis is as follows:

ψd=kdψacosϑr+ψbcosϑr23π+ψccosϑr+23π.E357

If into this equation expressions from Eq. (321), for linkage magnetic fluxes of a, b, c phases, are introduced, and for inductances appropriate expressions from Section 16 are introduced, then after widespread modifications of the goniometrical functions and for a rotor position in d-axis, i.e., if

ϑk=ϑr=0,

the following is received:

ψd=Ldid+32kdLafmaxif+32kdLaDmaxiD=Ldid+Ldfif+LdDiD.E358

Here:

Ld=La0+Lab0+32L2E359

is a direct synchronous inductance. The other symbols are for mutual inductances between the stator windings transformed into the d-axis and rotor windings, which are also in the d-axis:

Ldf=32kdLafmax,E360
LdD=32kdLaDmax.E361

The linkage magnetic flux in the q-axis is derived in a similar way, which results in:

ψq=Lqiq+32kqLaQmaxiQ=Lqiq+LqQiQ,E362

where:

Lq=La0+Lab032L2E363

is a quadrature synchronous inductance and

LqQ=32kqLaQmax,E364

is the mutual inductance between the stator winding transformed into the q-axis and rotor winding which is in the q-axis . From Eqs. (359) and (363), it is seen that if there is a cylindrical rotor, then L2 = 0, and inductances in both axes are equal:

Ld=Lq,E365

which is a known fact.

Finally, a linkage magnetic flux for zero axis is derived in a similar way:

ψ0=L0i0,E366

where:

L0=La02Lab0E367

is called zero inductance. It is seen that this linkage magnetic flux and inductance are linked only with variables with the subscript 0 and do not have any relation to the variables in the other axes. Additionally, also here a knowledge from the theory of the asynchronous machine can be applied that zero impedance is equal to the leakage stator inductance that can be used during the measurement of the leakage stator inductance. Here can be reminded equation Section 12:

L0=Ls2Ms=Lσs+M2M2=Lσs.E368

Namely, if all three phases of the stator windings (in series or parallel connection) are fed by a single-phase voltage, it results in the pulse, non-rotating magnetic flux (see [8]).

If there is a request to make equations more simple, then it is necessary to ask for equality of mutual inductances of two windings, for example, inductance Lfd for the current id should be equal to the inductance Ldf for the current if:

Lfd=Ldf.E369

Therefore from Eq. (354) for ψf,take the expression at the current id, which was marked as Lfdand put into the equality with the expression at the current if in Eq. (358) for ψd, which was marked as Ldf:

1kdLafmax=32kdLafmax.E370

Then:

kd2=23,E371

and

kd=23.E372

The same value is obtained if expressions for ψdat the current iD(358) and ψDat the current id(355) are put into the equality, to get LDd=LdD. Then:

1kdLaDmax=32kdLaDmax,E373

which results in kd=23.

In the q-axis it is done at the same approach: The expression at the current iqin the equation for ψQ(356) and the expression at the current iQin equation for ψq(362), put into equality to get LQq=LqQ, are as follows:

1kqLaQmax=32kqLaQmax.E374

It results in the value:

kq=23.E375

Hence a choice for the coefficients suitable for synchronous machines flows:

kd=kq=±23,

but it is better to use the positive expression:

kd=kq=23.E376

If the next expressions are introduced:

Ldf=3223Lafmax=32Lafmax=Lfd,E377
LdD=3223LaDmax=32LaDmax=LDd,E378
LqQ=3223LaQmax=32LaQmax=LQq,E379

then equations for linkage magnetic fluxes of the synchronous machines in the d, q, 0 system have the form as follows:

ψd=Ldid+Ldfif+LdDiD,see358
ψq=Lqiq+LqQiQ,see362
ψ0=L0i0,see366E380
ψf=Lfdid+Lffif+LfDiD,see354
ψD=LDdid+LDfif+LDDiD,see355
ψQ=LQqiq+LQQiQ,see356

where:

Ld=La0+Lab0+32L2,see359
Lq=La0+Lab032L2,see363
L0=La02Lab0,see367
Ldf=Lfd=32Lafmax,see377E381
LdD=LDd=32LaDmax,see378
LqQ=LQq=32LaQmax,see379.

By this record it was proven that not only terminal voltage equations of the stator and rotor windings but also equations of the linkage magnetic fluxes are identical with the equations of the universal electrical machine. Of course, expressions for inductances and a mode of their measurements are changed according the concrete electrical machine.

To complete a system of equations, it is necessary to add the equation for angular speed and to derive expression for the electromagnetic torque.

2.19 Power and electromagnetic torque of the synchronous machine

The instantaneous value of electrical input power in the a, b, c system can be written in the same form as it was derived in Section 8:

pin=uaia+ubib+ucic

and also in the d, q, 0 system:

pin=231kd2udid+231kq2uqiq+131k02u0i0.

If the constants recommended in the previous chapter are employed,

kd=kq=23k0=13

The power of synchronous machine is obtained in the form:

pin=udid+uqiq+u0i0,

for which the principle of the power invariancy is valid.

An expression for the instantaneous value of electromagnetic torque of synchronous machine is derived from the energy balance of the machine. If the stator variables are transformed and the rotor variables remained in their real form, on the basis of Eq. (184), the following can be written:

pin=231kd2udid+231kq2uqiq+131k02u0i0+ufif+uDiD+uQiQ.E382

This electrical input power for motor operation equals the Joule resistance losses, time varying of magnetic field energy, and internal converted power from the electrical to the mechanical form. If this internal converted power is given only by rotating members of the voltages, which are seen in the equations for ud and uq, then after introduction and modification, the same equation as in (173) is received:

te=p231kdkqψdiqψqid.

If the constants as they were derived above are introduced, the next expression for the electromagnetic torque is obtained:

te=pψdiqψqid.

This expression is identical with that derived for universal electrical machine in Section 2, Eq. (82).

In the next chapters, here derived equations will be applied on a concrete synchronous machine, and transients will be investigated. Again, this reminds that all rotor variables are referred to the stator.

2.20 Synchronous machine in the dq0 system

In Section 17 there are derived terminal voltage equations of the synchronous machine stator windings, transformed into k-reference frame, rotating by angular speed ωk, with the d, q, 0 axis. It was not needed to transform the rotor voltage equations, because the rotor windings are really embedded in the d-axis and q-axis, as it is seen in Figure 34 and is written directly in the d, q, 0 system.

Then it looks suitable to investigate transients of synchronous machine in the system d, q, 0. It means that it is necessary to identify the reference k-system with the rotor windings in such a way that the d-axis is identical with field winding axis and q-axis, which is perpendicular to it, as it is known from the arrangement of the classical synchronous machine with salient poles.

This reminds that an analysis of the synchronous machine armature reaction, with a splitting into two perpendicular d-axis and q-axis, is the basis for the general theory of the electrical machine. Therefore, it is justified to mark the subscripts d, q, according the axes. Then the k-system rotates by the rotor speed, and the angle between the positive real axis of the k-system and stator a-phase axis is identical with the angle of the rotor d-axis:

ωk=ωr
ϑk=ϑr.E383

Respecting these facts the stator voltage equations from Section 17 are in the form:

ud=Rsid+dψddtωrψq,E384
uq=Rsiq+dψqdt+ωrψd,E385
u0=Rsi0+dψ0dt.E386

The rotor equations are without any change:

uf=Rfif+dψfdt,E387
uD=RDiD+dψDdt,E388
uQ=RQiQ+dψQdt.E389

Linkage magnetic fluxes and inductances were given in Equations (380) and (381).

2.20.1 Relation to the parameters of the classical equivalent circuit

Now is the time to explain a relation between the terminology and inductance marking of the classical theory of synchronous machine and its equivalent circuit and general theory of electrical machines. As it is known, a term “reactance of the armature reaction,” and also inductance of the armature reaction, corresponds to the magnetizing reactance (inductance). The sum of the magnetizing reactance Xμd(inductance Lμd) and leakage reactance Xσs(inductance Lσs) creates synchronous reactance (inductance) in the relevant axis; thus for the synchronous reactance (inductance) in the d-axis, the following can be written:

Xd=Xad+Xσs=Xμd+Xσs,
Ld=Lad+Lσs=Lμd+Lσs,E390

and for the synchronous reactance (inductance) in the q-axis, the following can be written:

Xq=Xaq+Xσs=Xμq+Xσs,
Lq=Laq+Lσs=Lμq+Lσs.E391

For the rotor windings, it is valid that the self-inductance of the field winding Lffis a sum of a mutual inductance of the windings on stator side in the d-axis (because the field winding is also in d-axis) Lμdand leakage inductance of the field winding Lσf. Both windings must be with the same number of turns; in other words both windings must be on the same side of the machine. It is suitable to refer the rotor windings to the stator side and mark them with the “′(prime),” as it is known from the theory of transformer, where secondary variables are referred to the primary side, or rotor variables of the asynchronous machine referred to the stator side. The mutual (magnetizing) inductance in the d-axis is defined and measured from the stator side; therefore, it is not necessary to refer it:

Lff=Lσf+Lμd.E392

The same principle is applied also for the damping windings in both axes. For damping winding in the d-axis, the following is valid:

LDD=LσD+Lμd,E393

and for damping winding in q-axis:

LQQ=LσQ+Lμq.E394

A factor by which the rotor variables are referred to the stator is 32g2, where g is a factor needed to refer variables between the stator and rotor side, known from the classical theory of the synchronous machine. Its second power can be justified by the theory of the impedance referring or inductances in transformer or asynchronous machine theory. The constant 3/2 is there because of the referring between three-phase and two-axis systems. Then the leakage inductance of the field winding referred to the stator side is:

Lσf=32g2Lσf.E395

For a more detailed explanation, it is useful to mention that the factor g is defined for referring the stator current to the rotor current, e.g., the stator armature current Ia referred to the rotor side is Ia′ = gIa, which is needed for phasor diagrams.

As it is known from the theory of transformers and asynchronous machines, the current ratio is the inverse value of the voltage ratio and eventually of the ratio of the number of turns. If the impedance of rotor is referred to the stator, it is made by the second power of the voltage ratio (or number of turns ratio) of the stator and the rotor side. Because in the synchronous machine, a current coefficient (subscript I) from the stator to the rotor (subscript sr) is obviously applied and marked with g, it will be shown that the voltage ratio (subscript U) is its inverse value:

gIsr=IfoIk0=1gUsr=11If0/Ik0=1Ik0If0
gUsr=Ik0If0.E396

Shortly to explain, the If0 is the field current at which the rated voltage is induced in the stator winding at no load condition, and Ik0 is the stator armature current which flows in the stator winding, if the terminals are applied to the rated voltage at zero excitation (zero field current), or in other words, it is the current which flows in the stator winding at short circuit test, if the field current is If0.

If rotor variables should be referred to the stator (subscript rs; in Eqs. (392)(395) these variables are marked with ′(prime)), it will be once more inverse value of the voltage ratio, and hence it is again current ratio g, which is possible to be written shortly as follows:

gUrs=If0Ik0=gIsr=g.E397

Similar to Eq. (395), also the other variables of the rotor windings would be referred to the stator:

LσD=32g2LσD,E398
LσQ=32g2LσQ,E399
Rf=32g2Rf.E400

The mutual inductance of the field winding and windings in the d-axis is in the classical theory of the synchronous machine called magnetizing inductance in the d-axis; also the mutual inductance of the damping winding in the d-axis is the same magnetizing inductance; therefore, it can be expressed (see also Section 12, where the expressions for asynchronous machine are derived):

Ldf=Lfd=LdD=LDd=Lμd.E401

Likewise it is valid in the q-axis:

LqQ=LQq=Lμq.E402

Zero inductance was derived in Section 12 for asynchronous machine:

L0=Ls2Ms=Lσs+M2M2=Lσs.E403

Then it is clear that the zero inductance for synchronous machine is equal to its leakage inductance of the stator winding and is measured in the same manner:

L0=Lσs.E404

Equations for the linkage magnetic fluxes can be rewritten in the form, where the parameters of the synchronous machine known from its classical theory are respected, with a note that all rotor variables are referred to the stator side:

ψd=Ldid+Lμdif+LμdiD=Lσs+Lμdid+Lμdif+iD,E405
ψq=Lqiq+LμqiQ=Lσs+Lμqiq+LμqiQ,E406
ψ0=L0i0=Lσsi0,E407
ψf=Lμdid+Lσf+Lμdif+LμdiD=Lffif+Lμdid+iD,E408
ψD=Lμdid+Lμdif+LσD+LμdiD=Lμdid+if+LDDiDE409
ψQ=Lμqiq+LσQ+LμqiQ=Lμqiq+LQQiQ.E410

The rotor terminal voltage equations written for the rotor variables (resistances, linkage magnetic fluxes, terminal voltages) referred to the stator side are as follows:

uf=Rfif+dψfdt,E411
uD=RDiD+dψDdt,E412
uQ=RQiQ+dψQdt.E413

In this way, a system of six differential equations is obtained, namely, three for stator windings (384)(386) and three for rotor windings, the parameters of which are referred to the stator (411)(413). The linkage magnetic fluxes are written in Eqs. (405)(410).

2.20.2 Equations for terminal voltages of the stator windings

If the motoring operation is investigated, then it is necessary to derive what the terminal voltage on the left side of Eqs. (384) and (385) means. The terminal voltages are known variables in the motoring operation, but they are sinusoidal variables of the three-phase system, which must be transformed into the system dq0. Therefore it is necessary to go back to Section 9, where the expressions for the transformed sinusoidal variables were derived. If the voltages of the three-phase system were assumed as sinusoidal functions, the voltage in the d-axis was derived in the form:

ud=kdUmax32sinωstϑkE414

and in the q-axis in the form:

uq=kqUmax32cosωstϑk.E415

The voltage u0 for the symmetrical three-phase system is zero.

It is seen that concrete expressions for these voltages depend on the kd, kq choice and a choice of the reference system position. In Section 18 there were derived expressions, in which the coefficients suitable for synchronous machines were defined as follows:

kd=kq=23.E416

These coefficients are introduced to the equations for voltages in the d and q axes, and simultaneously it is taken into account that the reference frame dq0 is identified with the rotor, i.e., ϑk=ϑr:

ud=23Umax32sinωstϑr=32Umaxsinωstϑr,E417

and for the q-axis voltage:

uq=23Umax32cosωstϑr=32Umaxcosωstϑr.E418

These expressions can be introduced to the left side of Eqs. (384) and (385).

As it was mentioned in Section 9, for the synchronous machine, it is more suitable to employ voltages of the three-phase system as the cosinusoidal time functions (see (207)(209)) and for the d-axis and q-axis voltages to use the derived expressions (210) and (211). Together with the mentioned coefficients and after the reference frame is positioned to the rotor, the next expressions could be employed:

ud=23Umax32cosωstϑr=32Umaxcosωstϑr,E419
uq=23Umax32sinωstϑr=32Umaxsinωstϑr.E420

In Figure 39a, there are sinusoidal waveforms of i, B, H, ϕ, and in Figure 39b, there is a phase distribution in the slots of the cylindrical stator. The phasor of the resultant magnetic field at the instant t0, when the a-phase current is zero, is seen in Figure 39c, and at the instant t1, when a-phase current is maximal, it is seen in Figure 39d.

Figure 39.

(a) Sinusoidal waveforms of the variables i, B, H, ϕ of the symmetrical three-phase system, (b) phase distribution in the slots of the cylindrical stator, (c) phasor of the resultant magnetic field at the instant t0, and (d) phasor of the resultant magnetic field at the instant t1, when the a-phase current is maximal.

In Figure 39d, the position and direction of the resultant magnetic field phasor mean magnitude of the rotating magnetic field, identical with a contribution of the a-phase. The load angle, as an angle between the magnitudes of the rotating magnetic field, in this case identical with the a-phase axis and instantaneous rotor position, is measured from this point.

If stator voltage (or current) is investigated from an instant where the current in a-phase is maximal, then the current is described by cosinusoidal function. The other two phases are also cosinusoidal functions, shifted by about 120°. Transformation into the d, q-system results in Eqs. (419) and (420). The load angle is given by the calculation based on Eq. (436).

However, if the origin of the a-phase current waveform is put into zero, it means at instant t0, the waveform is described by the sinusoidal function, and the phasor of the resultant magnetic field is at the instant t0, which is the magnitude of the rotating magnetic field, shifted about 90° from the a-phase axis, as it is seen in Figure 39c. Since the load angle is investigated from the a-phase axis, it is necessary to subtract that 90° from the calculated value. Then the result will be identical with that gained in the previous case.

2.20.3 Equation for the mechanical variables

Equations for the developed electromagnetic torque and rotating speed are given in the previous chapter:

te=pψdiqψqid.E421

This electromagnetic torque covers two components of the torque:

te=JdΩrdt+tL,E422

where tL is load torque on the shaft of the machine, including torque of the mechanical losses, J is moment of inertia of the rotating mass, and dΩ/dt is time varying of the mechanical angular speed.

In the motoring operation, the rotating speed is an unknown variable which is calculated from the previous equation:

dΩrdt=1JtetL=1JpψdiqψqidtL.E423

In equations of the voltage and current, there is an electrical angular speed. Therefore it is necessary to make a recalculation from the mechanical to the electrical angular speed:

dωrdt=pJtetL=pJpψdiqψqidtL.E424

This is the further equation which is solved during the investigation of the synchronous machine transients.

2.21 Transients of the synchronous machine in the dq0 system

At first the transients of the synchronous motor without damping windings are investigated, i.e., machine is without D, Q windings and on the rotor; there is only field winding f. On the stator, the three-phase winding fed by the symmetrical voltage system is distributed. It means that the zero voltage component is zero: u0=0. Then equations for stator windings are in the form (the variables with ′ (prime) sign mean rotor variables or parameters referred to the stator):

ud=Rsid+dψddtωrψq,E425
uq=Rsiq+dψqdt+ωrψd,E426
uf=Rfif+dψfdt,E427

where:

ψd=Ldid+Lμdif=Lσs+Lμdid+Lμdif,E428
ψq=Lqiq=Lσs+Lμqiq,E429
ψ0=L0i0=Lσsi0,E430
ψf=Lμdid+Lσf+Lμdif=Lffif+Lμdid.E431

In motoring operation, the terminal voltages are known variables, and unknown parameters are currents and angular speed. To investigate the currents, it is necessary to derive expressions from Eqs. (425) and (426), and Eq. (423) is valid for speed. On the left side of Eqs (425) and (426), there are voltages transformed into the d-axis and q-axis, which were derived in the previous chapter:

ud=32Umaxcosωstϑr,E432
uq=32Umaxsinωstϑr,E433

if the cosinusoidal functions of the phase voltages a, b, c were accepted. The rotor position expressed as the angle ϑris linked with the electrical angular speed by equation:

ϑr=ωrdt+ϑr0;E434

eventually:

dϑrdt=ωr.E435

In Eqs. (432) and (433), it is seen that load angle ϑL, which is defined as a difference between the position of the rotating magnetic field magnitude, represented by the expression ωst, and position of the rotor axis, represented by the angle ϑr, is present directly at the expression for the voltages:

ωstϑr=ϑL,eventuallydϑLdt=ωsωr.E436

Because the load angle is a very important variable of the synchronous machine, its direct formulation in the voltage expressions should be employed in simulation with benefit.

For motor operation the currents, as unknown variables, are derived from the voltage equations after the linkage magnetic fluxes are introduced. In the next text, it is taken as a matter of course that rotor variables are referred to the stator and this fact is not specially marked.

ud=Rsid+Lddiddt+LdfdifdtωrLqiq,E437
uq=Rsiq+Lqdiqdt+ωrLdid+ωrLdfifE438
uf=Rfif+Lfddiddt+Lffdifdt.E439

If from Eq. (439), difdtis expressed and introduced to Eq. (437), then after modifications it yields:

diddt=LffLdLffLdf2udRsid+ωrLqiqLdfLffuf+LdfLffRfif.E440

Similarly, if from (437) expression diddtis introduced to the (439), then after some modifications it yields:

difdt=LdLdLffLdf2ufRfifLfdLdud+LfdLdRsidωrLfdLdLqiq.E441

From Eq. (438), an expression for current in the q-axis is received:

diqdt=1LquqRsiqωrLdidωrLdfif.E442

Consequently at transient investigation, a system of Eqs. (432), (433), (435), (437), (440), (441), (442), and (424) must be solved. The outputs are the currents of the stator windings in the form of id, iq, which are fictitious currents. The real stator phase currents must be calculated by inverse transformation:

ia=is=23123idcosϑr23123iqsinϑr=23idcosϑr23iqsinϑr.E443

This is current in the a-phase, and currents in the other phases are shifted about 120°. In Section 23 there is an example of a concrete synchronous motor with its nameplate and parameters, equations, and time waveforms of the investigated variables at transients and steady-state conditions.

Following the same equations, it is possible to investigate also generating operation with that difference that mechanical power is delivered, which requires negative load torque in the equations and to change sign at the current iq. Then the electromagnetic torque is negative.

2.22 Transients of synchronous machine with permanent magnets

If synchronous machine is excited by permanent magnets (PM), this fact must be introduced into equations, which are solved during transients. See [15, 16, 17].

At first it is necessary to determine linkage magnetic flux of permanent magnets ψPM, by which an electrical voltage is induced in the stator winding. Obviously, it is measured on a real machine at no load condition in generating operation. From Eq. (73) it can be derived that in a general form, the PM linkage magnetic flux is given by equation:

ψPM=Uiω.

In Eq. (428), there is instead the expression with a field current directly ψPM. As it is supposed that this magnetic flux is constant, its derivation is zero, and Eq. (437) is in the form:

ud=Rsid+LddiddtωrLqiq,E444
uq=Rsiq+Lqdiqdt+ωrLdid+ωrψPMdq.E445

In that equation the PM linkage magnetic flux is transformed into the dq0 system, because also the other variables are in this system. To distinguish it from the measured value, here a subscript “dq” is added. It can be determined as follows: In no load condition at the rated frequency, the currents id, iq are zero; therefore also the voltage ud is zero according to Eq. (444), and the voltage in q-axis, according to Eq. (445), is:

uq=ωrNψPMdqE446

and at the same time according to Eq. (418), in absolute value, is:

uq=32Umax,E447

because also load angle is, in the no-load condition, zero. Then:

ψPMdq=32UmaxωrN.E448

This value is introduced into Eq. (445), to calculate the currents id, iq. The real currents in the phase windings are obtained by an inverse transformation according to Eq. (443). Examples of these motor simulations are in Section 23.2.

2.23 Transients of a concrete synchronous motor

2.23.1 Synchronous motor with field winding

Equations from Section 21 are used for transient simulations of a concrete synchronous motor with field winding. The nameplate and parameters of this motor are shown in Table 5.

SN = 56 kVARs = 0.0694 Ω
UN = 231 VLσs = 0.391 mH
IN = 80.81 ALμd = 10.269 mH
fN = 50 HzLμq = 10.05 mH
Uf = 171.71 VRf´= 0.061035 Ω
If = 11.07 ALσf = 0.773 mH
nN = 1500 min−1J = 0.475 kg m2
TN = 285 Nmp = 2
Tloss = 1 Nm

Table 5.

Nameplate and parameters of the investigated synchronous motor with field winding.

Seeing that the simulation model is created in the d, q-system, linked with the rotor position, also the terminal voltages must be given in this system. It is made by Eqs. (419) and (420). The field winding voltage is a DC value and is constant during the whole simulation.

As it is known from the theory of synchronous motor, the starting up of synchronous motor is usually not possible by directly switching it across the line. If the synchronous motor has a damping winding, which is originally dedicated for damping of the oscillating process during motor operation, this winding can act as a squirrel cage and develop an asynchronous starting torque sufficient to get started. After the motor achieves the speed close to synchronous speed, it falls spontaneously into synchronism.

The damping winding is no longer active in torque development. The investigated motor has no damping winding; therefore a frequency starting up is carried out, which means continuously increasing voltage magnitude and frequency in such a way that their ratio is constant. In Figure 40, simulation waveforms of frequency starting up of this motor are shown, which means time dependence of variables n = f(t), ia = f(t), te = f(t), and ϑL = f(t). Motor is at the instant t = 3 s loaded by rated torque TN = 285 Nm.

Figure 40.

Simulation time waveforms of the synchronous motor with field winding starting up: (a) speed n, (b) current of the a-phase ia = is, (c) developed electromagnetic torque te, and (d) load angle ϑL.

The mechanical part of the model is linked with Eq. (423); however, it had to be spread by the damping coefficient of the mechanical movement of the rotor. This coefficient enabled implementation of the real damping, which resulted in a more stable operation of the rotor.

2.23.2 Synchronous motor with PM

Equations derived in Section 22 are employed in transient simulation of the synchronous motor with PM. The equation for the developed electromagnetic torque (221) is modified to the form:

te=pLdid+ψPMiqLqiqid.E449

Also this simulation model is created in the system d, q linked with the rotor position; therefore, it is necessary to adjust the terminal voltage to this system. The nameplate and parameters of the investigated motor are in Table 6.

PN = 2 kWRs = 3.826 Ω
UphN = 230 V, star connection YLd = 0.07902 H
IN = 10 ALq = 0.16315 H
fN = 36 HzψPM = 0.8363 Wb
nN = 360 min−1J = 0.02 kg m2
TN = 54 Nmp = 6
Tloss = 0.5 Nm

Table 6.

Nameplate and parameters of the investigated synchronous motor with PM (SMPM).

In Figure 41, there are simulation waveforms of the simulated variables during the frequency starting up of the synchronous motor with PM, if stator voltage and its frequency are continuously increasing in such a way that their ratio is constant. It is true that this way of the starting up is not typical for this kind of the motors. Such motors are usually controlled by field-oriented control (FOC), but its explaining and application exceed the scope of this textbook.

Figure 41.

Simulated time waveforms of the synchronous motor with PM: (a) speed n, (b) current of the a-phase ia, (c) electromagnetic developed torque te, and (d) load angle ϑL.

Simulation waveforms show time dependence of the n = f(t), ia = f(t), te = f(t), and ϑL = f(t). The motor is at the instant t = 3 s loaded by rated torque TN = 54 Nm. The mechanical part is widened by damping of the mechanical movement of the rotor.

Analysis of Electrical Machines Using Finite Element Method

3.1 Introduction

Design of electrical machines is a very important task from several points of their qualitative parameters. The design methods were and still are developed to obtain best results in electrical and magnetic optimal utilization of machine circuits. One of the main tasks of electrical machine design is to maximize output power and efficiency and to decrease its volume and losses. For this design, analytical and empirical terms, verified by years of experience in designing the electrical machines, are used. The best-known designs are listed in [1, 2, 3].

With the development of computer technology, however, the old-new method for analyzing the distribution of electromagnetic fields in electrical machines, for determining the parameters of the machine equivalent circuit and for calculating other properties, is also possible. It is the finite element method (FEM), which was previously derived as a mathematical method, but it has not been established until recently due to its complexity in the calculations. Its use enabled the development of computer technology. It is currently used to solve problems not only in most industries but, for example, also in medicine.

In electrical machines, FEM is used in conjunction with the numerical solution of Maxwell’s equations in the analyzed electrical machine. The solution can be carried out in plane (2D) or in space (3D). In cylindrical electrical machines where the diameter of the machine is negligible in relation to the active length of the machine, it is sufficient to solve the problem in 2D, i.e., in the cross section of the machine. In some electrical machines, the deviation of the calculation is of the order of a percentage compared to measurements (reluctance machines), in some up to 10–20%, e.g., in asynchronous machines. Various types of commercial programs that use FEM can be used for the calculation, e.g., Ansys, Ansoft, Quickfield, Opera, Maxwell, etc. This chapter uses a program that is freely distributed on the web and is called the finite element method magnetostatics (FEMM), see [10].

The advantage of FEM analysis of an electrical machine is that the FEM can also solve an electrical machine that does not yet exist or is only under the design process. Therefore, it is profitable to use FEM in the design of electrical machines. First, the electrical machine is designed by years of verified design calculations and equations of the electrical machine design, if known, and then its dimensions and other parameters are fine-tuned using FEM.

The calculation of the electrical machine parameters from its dimensions and the properties of the materials used (using analytical methods and equations) take into account the distribution of the magnetic field in the machine but in a simplified way. In opposite, the FEM program uses this magnetic field distribution as the basis for its calculations, including the saturation of magnetic circuits and the magnetic flux density distribution in the air gap, taking into account the stator and rotor slots. Therefore, the parameters of the electrical machine equivalent circuit, which are determined by FEM, should be more accurate than from the classical design calculation.

3.2 Physical basis of FEMM calculations

This section describes the physical basis of FEMM calculations [10]. For low-frequency tasks, including electrical machines that are solved in FEMM, only some Maxwell equations are sufficient. Low-frequency tasks are those tasks in which shifting currents can be neglected. The shifting currents are important at radio frequencies only. Therefore, magnetostatic analysis can be used. The solution will be based on four Maxwell equations. In some publications, Maxwell equations have various rankings, because the authors use their own labeling, e.g., [1]. Therefore, they are ranked on the basis of physical laws and their discoverers. More detailed information on the different shapes of Maxwell’s equations can be found, e.g., in [12]. It can be started with the Ampère’s circuital law, which is in integral form as follows:

lHdl=I+dψdtE450

where H is vector of magnetic field strength, l is the mean length of the field line, I is total current, and ψ is magnetic flux linkage.

Another law is the magnetic flux density law, known as Faraday’s law of induction, which is defined in integral form:

lEdl=NdϕdtE451

where E is electric field strength, N in number of turns, and ϕ is magnetic flux.

The next equation corresponds to magnetic flux linkage law, which is defined as:

SBdS=0E452

where B is vector of magnetic flux density of magnetic field and S is cross section, through magnetic flux passes.

The last law that belongs to Maxwell’s equations is Gauss’s law of electrostatics, which can be applied using finite element method only to solve electrostatic field. Its integral form is:

SDdS=QE453

where D is electric flux density and Q is amount of electric charge.

To analyze properties of electrical machines using FEM, first magnetostatic analysis is applied, in which magnetic vector potential A has an important role. The definition of magnetic vector potential value is analogous to the definition of electric field potential. The essential difference is that magnetic potential is vector, while electric potential is scalar. Enclosed conductor of length l with current I creates magnetic vector potential, which is defined as:

A=μ04πlIdlrE454

where r is the value of r-vector, of which end point remains in place where the potential is being evaluated, while being integrated. Magnetic flux density vector B can be derived from magnetic vector potential, using derivations. We will show the derivations using the defined vector product of nabla operator and magnetic vector potential A, according to [10]. Note that in Eq. (455), r is the nominator and r the denominator:

×A=μ04πlIr×dl=μ04πlIrr3×dl=μ04πlIdl×rr3=B.E455

This equation between magnetic vector potential A and magnetic flux density B often utilizes FEM to evaluate magnetostatic field. It will be shown in further chapters where various types of electrical machines are analyzed.

If the magnetostatic problem is solved, magnetic fields are constant in time. In this case, equations for magnetic field strength H and magnetic flux density B are defined by the following equations:

×H=JE456

where J is the current density.

The relationship between B and H for each material is given by the equation:

B=μH.E457

If the material is nonlinear (e.g., saturated iron or Alnico magnets), permeability μ is the function of B:

μ=BHB,μ=fB.E458

The FEMM program analyzes the electromagnetic field; thus, it solves equations from (455) to (458) utilizing magnetic vector potential approach. Then it can be written as:

×1μB×A=J.E459

For linear isotropic material and supposed validity of Coulomb’s criterion ∇A = 0, Eq. (459) is reduced to:

1μ2A=J.E460

FEMM retains Eq. (459), allowing to solve also magnetostatic tasks with nonlinear B-H equation. These equations are used, when the no load condition of electrical machine is solved.

In overall three-dimensional case, vector A consisted of three components. Before we will show how to calculate forces, torques, inductances, etc., utilizing FEMM, we can say that some tasks can be solved based on the time variable field. Such approach is called harmonic task. It can be used, e.g., to evaluate asynchronous motor in no load condition. Harmonic task evaluation will be shown in Section 5.

Harmonic task can be considered, when field is time variable (e.g., harmonically changing current). In materials with non-zero permeability, eddy currents can be induced. Basic equation for electric field strength E and current density J is defined as:

J=σEE461

where σ is specific conductivity.

Integral form of electromagnetic flux density law (induction law) can be rewritten as follows:

×E=Bt.E462

where B is substituted with vector potential expression:

×E=×A.E463

In the case of solving 2D problem, (463) can be rewritten as:

E=AVE464

and by substituting to Eq. (461), it is:

J=σAσV.E465

By substituting to Eq. (459), differential expression is obtained:

×1μB×A=σAσV+JsrcE466

where Jsrc represents applied current sources. Component ∇V is additional voltage gradient, which is constant in 2D problem. FEMM applies this voltage gradient in some harmonic tasks to constrain current in conductive areas.

Equation (466) is considered in case, when field oscillates at one constant frequency. In such case, a steady-state equation is obtained, which is solved for amplitude and phase of vector potential A. This transformation is:

A=Reacosωt+jsinωt=ReaejωtE467

in which a is a complex number. Substituting to Eq. (466) and its separation, we can get an equation, which can be used by FEMM when solving harmonic tasks:

×1μB×a=jωσaσV+JsrcE468

in which Jsrc represents transformed phasor of applied current sources.

For harmonic analysis, permeability μ should be constant. However, FEM takes into account nonlinear equation also in harmonic formulation, which allows solving nonlinear tasks similar to magnetostatic analysis. FEMM also includes complex and frequency-dependent permeability in time-harmonic tasks. These properties enable the program to model materials with thin sheets and also to approximately model hysteresis in ferromagnetic materials.

3.2.1 Calculation of forces and torques using FEM

Calculation of forces and torques of electrical machines is one of the most important properties of FEMM, which can be used toward this purpose. We can use different means, but most known are these three methods of force and torque calculation:

  • Maxwell stress tensor

  • Coenergy method

  • Lorentz force equation

3.2.1.1 Maxwell stress tensor

Using of this tensor is very simple from force and torque calculation point of view, because it requires only local distribution of magnetic field density along straight line or curve, which can be selected in the analyzed problem. In linear motion systems (e.g., linear motors), force calculation is applied. In rotating electrical machine, we can define circle in the middle of air gap, on which the electromagnetic torque of machine can be calculated. This is written in more details in Sections 5, 6 and 7. Total force can be calculated as:

F=1μ0BB·n12μ0B2ndSE469

where normal component of the force is:

Fn=lFe2μ0Bn2Bt2dlE470

and tangential component of the force is:

Ft=lFeμ0BnBtdlE471

where n is normal vector, S is the area, lFe is active length of the iron in rotating electrical machines or z-axis, Bn is normal component of magnetic flux density, and Bt is tangential component of the magnetic flux density.

The torque can be calculated as follows:

T=r×FE472

Then

T=lFeμ0lrBnBtdlE473

where r is radius of the circle, where the torque is calculated, and l is its length. The accuracy of this method to calculate the force or torque depends on finite elements number in the air gap of the machine, where the force or torque is calculated. The greater the number of elements, the more accurate the calculation is.

3.2.1.2 Coenergy method for force and torque calculation

Coenergy method of force or torque calculation is based on electrical to mechanical energy conversion principle in systems with variable air gap and ferromagnetic core. Among such systems are, e.g., electromagnet of contactor and electrical device based on variable reluctance principle (reluctance synchronous machine, switched reluctance machine, see Section 7 and [18, 19, 20]). Force calculation is defined by small difference (derivation) of coenergy W´ with respect to small difference (derivation) of position (deflection) x in linear motion or ϑ in calculation of torque. Thus for force, it is defined as:

F=dWdxΔWΔxE474

and the torque is defined as:

T=dWdϑΔWΔϑ.E475

As it was mentioned above, for calculation of one force or torque value, two FEM calculations are needed, because difference between two coenergies is required. Setup of this position or deflection step is important. When step is too big, calculated value of force or torque can be inaccurate. Setting of step must be adjusted predictably according to the problem, which is being solved. In linear motion, it can be, e.g., Δx = 1 mm, in rotating machines single mechanical degree (Δϑ = 1°). Instantaneous value of torque can be calculated as:

T=Wiϑdϑi=const=Wiϑdϑψ=const.E476

3.2.1.3 Lorentz force equation for force and torque calculation

By using Lorentz’s law to calculate force or torque, instantaneous value of torque can be obtained as function of phase induced voltage values and phase current values for three-phase system as sum of products of instantaneous values in all three phases:

T=1ΩuiAtiAt+uiBtiBt+uiCtiCtE477

where ui(t) are instantaneous induced voltages of three phases, i(t) are instantaneous values of the current, and Ω is mechanical angular speed.

3.2.2 Calculation of inductance by means of FEM

The method of steady-state inductance calculation is shown in this chapter. In electrical machines, it can be calculation of the self-inductance of phase, leakage inductance, armature reaction inductance, or magnetizing inductance. We can use two ways to calculate these:

  • Inductance calculation for linear systems from magnetic field energy

  • Inductance calculation for nonlinear systems from flux linkage

In linear systems, inductance calculation can be carried out by electromagnetic field energy evaluation, or by coenergy, because these are equal to the linear cases. It is assumed that electrical machine represents linear system, when operating in linear area of B-H characteristic. This means that ferromagnetic circuit is not saturated. Calculation is done by equation:

W=W=12LI2L=2WI2E478

where electromagnetic field energy is defined as:

W=V0BH·dBdVE479

and V is volume, where the electromagnetic field energy is stored.

System is considered as nonlinear, if the machine operates in nonlinear area of B-H characteristic. In nonlinear systems, inductance calculation can be carried out based on flux linkage evaluation, using Stokes theorem and magnetic vector potential:

L=ψI=S×A·dSI=A·dlI=A·JdVI2E480

Utilization of these equations and calculation of all parameters are shown in the next chapters.

3.2.3 Procedure of FEM utilization

In general, most software for FEM analysis consists of three main parts:

  • Preprocessor: preparation and evaluation of the analyzed model

  • Solver (processor): assemblage of differential equation system and its solving.

  • Postprocessor: analysis of results and calculation of further required parameters

All parts here are described and applied to the FEMM program.

3.2.3.1 Preprocessor

It is a mode or part of the program, where the user creates a model with finite elements. This module contains more sub-modules or other parts. At first, geometrical dimensions of model (mm or English units) are defined. It is chosen between 2D and 3D modeling. In this chapter, FEMM program is able to do only 2D analysis. This is satisfactory in most cases of electrical machines analysis, and calculation time of evaluation is significantly shorter than in 3D. If the user disposes software with 3D, it is required to regard on the manual provided by the software.

Options can be selected between planar x, y coordinates (2D) and axisymmetric system, where it can work with polar coordinates. This system is suitable to solve, e.g., cylindrical coils. In most of the electrical machines, the planar system is used, and also the z-coordinate is selected.

  • Drawing mode: This mode is used to draw models, which will be solved. It uses specific points given by their x, y coordinates. These points are connected with lines or curves, aiming to obtain required geometrical shape. In this part of preprocessor, the model can be imported also from CAD programs in dxf format.

  • Materials definition: In this mode, used materials of drawn models are defined by specific quantities. Each part of the model must be outlined; thus it is clearly specified which space responds to the given material. In electrical machines materials as: air, insulation, ferromagnetic materials for magnetic circuits of cores, stators and rotors (mostly defined by B-H- characteristic), electric current conductors are mostly made of copper or aluminum materials, ferromagnetic materials for shafts and permanent magnets, are employed. The definition of these materials is shown in each chapter, related to particular examples. FEMM program offers a library, which contains various materials, and it is possible to define these individually, according to user’s requirements and needs.

  • Excitation quantities definition: In this mode, excitation quantities are defined. These are current density, currents, or voltages, which respond to the respective parts of model (slot, winding, coil). These values must be calculated directly or obtained for a particular state of the electrical machine (e.g., no load condition).

  • Boundaries definition: Analysis and solving of models in FEMM use two important boundary conditions: Dirichlet boundary conditions, which are defined as the constant value of magnetic vector potential A = const. This is used mostly on the surface of electrical machines, where A = 0 is supposed (Figure 42a). Neumann boundary conditions are used for solving symmetrical machines, where we can analyze one quarter only or smaller part which replicates in the model. It is suspected for this condition that normal derivation of magnetic vector potential is zero: ∂A/∂n = 0 (Figure 42b).

  • Finite element mesh creation: This comprises the FEMM method principle. As titled, the created model is split into too many parts by finite elements. This finite number of elements covers the whole model. The more complicated the shape of the model is, the higher number of elements is required. These elements are mostly triangles, yet some software can use other shapes. By first mesh generating, the program creates a certain number of elements of certain size. This number and size can be set up by the user. It is important that dependence between number and size of elements, calculation speed, and accuracy is not linear. It is not obvious to get more accurate solution by high number of small-size elements. This can be compared to the magnetization curve of ferromagnetic material, where the number of elements is on x-axis and accuracy of solution is on y-axis. This setting is to be done by the user according to the experiences and requirements of a particular model. In each of finite elements, magnetic vector potential A is calculated.

Figure 42.

Boundary conditions: (a) homogenous Dirichlet condition and (b) Neumann condition.

These modes are defined as preprocessor, and we can now continue to the solver part.

3.2.3.2 Solver (processor)

In this part of the program, a system of partial differential equations is created in each node of triangle, i.e. finite element, where vector magnetic potential values are being solved. Based on the numerical solution of equations utilizing defined iteration methods, the program solves this system of partial differential equations. Result can converge (favorable case), where suspected solution can be obtained, or diverge, where program cannot calculate successful solution. In such case, it is necessary to revise the model and make corrections. As stated above, this part of program will take time according to the selected number and size of finite elements.

3.2.3.3 Postprocessor

In this last part, results of FEMM can be analyzed. The first result is the distribution of magnetic field in the model by means of magnetic field flux line mapping (equipotential lines). This mode allows to calculate other quantities on a particular point, defined line, or curve (by means of space integral calculation). Thus, the following can be obtained: magnetic flux density (normal and tangential component), magnetic field strength, potential, torques, energies, inductances, losses, etc.

According to this description, FEMM program will be used to analyze and calculate parameters in individual electrical devices and electrical machines. We will start with Section 3 in this chapter, where we calculate the force and inductance of a contactor electromagnet. In Section 4, we will calculate equivalent circuit parameters of single-phase transformer (mostly inductances). In Section 5, we will show calculation of parameters and torques of asynchronous machines. In Section 6, synchronous machines are analyzed, and finally in Section 7, switched reluctance machines are analyzed.

3.3 Analysis of electromagnet parameters

It is shown here how the parameters of electromagnet, or more precisely “of ferromagnetic circuit with variable air gap, fed by DC voltage,” can be investigated by means of the FEMM. The purpose of this is to inform the reader how practical calculations and analyses by means of FEMM are carried out.

Such circuits are employed in electrical engineering praxis, as it is electromagnet of the relay, or rotating electrical machine with variable reluctance (Section 7). Calculation of the phase inductance and developed electrodynamic force for one general condition (moving armature is in certain position) is shown. There are two important positions of the electromagnet: switched-on position with the minimal air gap and switched-out position with maximal air gap. Examples of such circuits are in Figure 43b,c.

Figure 43.

Ferromagnetic circuits with an air gap: (a) constant air gap and (b, c) variable air gap.

Expressions from Sections 2.1 and 2.2 are employed at the inductance and force calculations. Section 2.3 is the base for calculation procedure. At the first step (preprocessor), the base settings are: Planar Problem Type and Length Units (mm). Frequency is zero, because a magnetostatic problem is investigated, and there is given DC voltage. Set z-coordinate, i.e., active length of iron, marked in the program as Depth (Figure 44). After these settings, the model is drawn in such a way that the points are put in the node mode and by means of tabulator the coordinates x, y of the points are put in. After the points are drawn, the program can be changed to the segment mode and join the points to get the required shape of the ferromagnetic circuit. Into the center of the air gap, an auxiliary line is drawn, on which a value of the developed force is investigated (see Figure 44).

Figure 44.

Drawn mode with demonstration of the basic settings.

After the drawing mode is ready, the materials are entered. Here three materials are present: air, copper (conductors of the coil), and ferromagnetic material with the known B-H curve. These materials can be entered by the Materials Library, from which the basic material constants can be copied by the employed materials here (Figure 45).

Figure 45.

Parameter setting of the materials.

Materials as air or conductor material (copper, aluminum) have relative permeability in x- and y-axis equal 1, μx = 1, μy = 1, and then set up electrical conductivity of a conductor (e.g., the electrical conductivity for copper is 58 MS/m). If the winding is excited by a field current, then set up the excited variables, e.g., current density of the winding. Choose, e.g., the field current I = 2.25 A. The number of coil turns is N = 1000. Cross-section area of the coil is (50 × 9) mm. The setting up is made as follows: According to the current value I and cross-section area of the conductor, calculate the current density:

J=NIS=1000·2.250.05·0.009=5MA/m2.E481

In the FEMM program, the current density is given in MA/m2. There is also the other way where only the number of turns and current value is entered. These settings can be made by Properties/Circuits. In the case of ferromagnetic materials, it is possible to take from the library a material present there with its B-H curve (Figure 46). In the case here, a B-H curve from the library with indication US Steel Type 2-S 0.018 inch is employed.

Figure 46.

The B-H curve of a ferromagnetic material taken from the Materials Library: US Steel Type 2-S 0.018 inch.

If all materials are defined, allocate them to drawn objects in such a way that the program is switched over to the mode Block, and mark gradually the objects. In this case, these are the next parts: air, right part of the coil, left side of the coil, fixed part of the ferromagnetic circuit, and moving part of the ferromagnetic circuit (armature). Materials are allocated to these parts by means of the key space and from the open window take a correspondent material (see Figure 45).

The last but one step in preprocessor is the definition of the boundary conditions. In this task, the homogenous Dirichlet condition can be chosen, if the investigated electromagnet is limited by the boundaries, where zero magnetic vector potential can be assumed. The setting up is made in Properties/Boundary/Prescribed A, whereby all values are zero (Figure 47).

Figure 47.

Definition of the boundary condition.

The last step in preprocessor is generation of the finite element mesh. By means of the command Mesh/Create Mesh, the program generates a random mesh with a relatively low number of the finite elements, which can have an important influence on the accuracy of investigation. This mesh can be refined if the correspondent material is marked and set up Mesh size (Figure 48). If, e.g., Figure 1 is chosen, it means that the size of a finite element triangle is 1 mm, which is less than that created by an automatic program.

Figure 48.

Definition of the finite element size in materials.

This way, how to set up a size of the finite elements in all materials, eventually all regions of the analyzed task, can be employed.

In this way, the preprocessor was finished, and a solver, eventually processor for construction and solving a system of partial differential equations, can be started. This can take certain time depending on the number of finite elements and complexity of the investigated circuit.

After the solving is finished, the calculated values can be analyzed. The program shows distribution of the force lines and offer three ways how to present the outputs of the calculation:

  1. In the exactly defined point (Figure 49) (e.g., in the point with coordinates x, y, black point). In this case a magnetic vector potential A; magnetic flux density (absolute value |B|, the values in direction of the x-axis, Bx, and y-axis, By); magnetic field intensity (absolute value |H|, values in x-axis, Hx, and y-axis, Hy); relative permeability in x-axis, μx, and in y-axis, μy; circuit energy E; and current density J are obtained. Because the point in which the calculation was made is in ferromagnetic material, the current density is zero: J = 0.

  2. By means of integration along the defined line or curve. In this case, it is force calculation in the air gap.

  3. By means of the marked area integration. In this case, it is inductance calculation of the magnetic circuit.

Figure 49.

Presentation of the results in the defined point x, y.

3.3.1 Electromagnet force calculation

In this case the calculation is made by means of the Maxwell stress tensor based on the equations in Section 2.1. An auxiliary line is marked, which is in the middle of the air gap between the fixed part of the electromagnet and its armature. The force is calculated by means of integration as follows: Integrate/Line Integrals/Force form Stress Tensor. The outputs are presented in both directions x and y (Figure 50). As it is seen in this figure, the force acts mainly in the y-axis direction, while in the x-axis direction, it is nearly neglected.

Figure 50.

Output presentation: Values of the force obtained by integration in the x-direction and y-direction on the auxiliary line in the middle of the air gap.

By this way, the force value can be investigated for various size of the air gaps and various currents which correspond to various feeding voltages.

On the defined auxiliary line, it is possible to obtain also other variables, which can help during the performance analysis, e.g., a normal component of magnetic flux density, eventually magnetic flux density in y-axis (Figure 51), whereby the value 0 corresponds to the length axis to the origin of the auxiliary line in the air gap and the value 70 corresponds to the end of the auxiliary line. This possibility is used in the next chapters. In the figure, there is this component that is always marked on the right side in the form B.n (marking), Tesla (here are units—see Figure 51), because in this form the figure is generated from the program.

Figure 51.

Illustration of the normal component of the magnetic flux density on the defined auxiliary line.

3.3.2 Electromagnet inductance calculation

The calculation is made by two ways, as it was described in Section 2.2: by means of coenergy (linear case is supposed) and by means of linkage magnetic flux.

  1. Calculation by means of coenergy

    In the program, all areas of the materials are colored by green (Figure 52). By means of Integrate/Block Integrals, calculate the magnetic field coenergy. Here the other calculation is also made, to be able to compare the inductances obtained by both ways. In Figure 52 it is seen that the calculated value of coenergy is W = 0.416682 J.

    This value is introduced into Eq. (478) and it is received:

    W=W=12LI2L=2WI2=2·0.4166822.252=0.164H

As it was written before, the electromagnet coil was fed by the current I = 2.25 A.

  • For the calculation by means of the linkage magnetic flux, Eq. (480) is used. Here only the blocks that correspond to the coil (Figure 53) are highlighted, and by means of the surface integral, the value of the A.J is calculated: It is 0.8334454 HA2 (Henry Amperes2). Then the inductance can be calculated as:

    L=ψI=S×A·dSI=A·dlI=A·JdVI2=0.8344542.252=0.164H

  • Figure 52.

    Inductance calculation by means of coenergy.

    Figure 53.

    Inductance calculation by means of linkage magnetic flux.

    It is seen that the results obtained by both ways of calculation are identical. If the ferromagnetic circuit was saturated, the results would be different.

    Other calculations can be made for any armature position and any current. It depends on the reader needs.

    3.4 Analysis of the single-phase transformer parameters

    The FEM is used for the analysis of the single-phase transformer parameters. As it is known, the single-phase transformer can be designed into two main configurations: core-type construction (Figure 54a) and shell-type construction (Figure 54b). In the transformers of small powers and low voltage, the primary winding can be wound on the core and on it the secondary winding, as it is seen in Figure 54. Transformers for higher voltage have usually the secondary winding wound closer to the core and on it the primary winding.

    Figure 54.

    Winding and core arrangements of the single-phase transformers: (a) core-type construction and (b) shell-type construction.

    The no load test can be simulated by means of FEM. The parameters of the square branch of the equivalent circuit, mainly magnetizing inductance, can be calculated. Second, also the short circuit test can be simulated by FEM. The parameters of the direct axis of the equivalent circuit, mainly leakage inductances, can be calculated. The usual equivalent circuit of the single-phase transformer is in Figure 55.

    Figure 55.

    Single-phase transformer equivalent circuit.

    The calculation of the equivalent circuit parameters is made for a real transformer, nameplate and rated values of which are in Table 7. An illustration figure of ¼ of transformer cross-section area is in Figure 56.

    Figure 56.

    An illustration figure of ¼ of transformer cross-section area.

    3.4.1 Simulation of the single-phase transformer no load condition

    The purpose of the simulation in no load condition is the calculation of the magnetizing inductance Lμ.

    The procedure is the same as in the case of electromagnet analysis. The calculation is started with a preprocessor, where the magnetostatic analysis and the planar type of the problem is set up. Frequency is zero, because only one time instant is solved. The z-coordinate (depth) is an active thickness of ferromagnetic core. The total thickness of the ferromagnetic core is obtained from the measurement lFetotal = 49.6 mm. This must be reduced by the value of sheet insulation thickness and air layers between them. This reduction is respected by correction factors kFe = lFe/lFetotal = 0.866 - > lFe = 43 mm, and this value was used in the calculation.

    Based on the transformer dimensions, a cross-section area is drawn and the materials are allocated to the blocks as follows: air around the coils; ferromagnetic material of the core; and primary and secondary windings. These are divided into two parts, right and left, and their indication is as follows: primary winding right side (pwr), primary winding left side (pwl), secondary winding right side (swr), and secondary winding left side (swl). The geometry of the cross-section area with the marked parts and setting up of the parameters are in Figure 59. Magnetic energy Wm is absorbed mainly into ferromagnetic core; therefore it is suitable to increase density of the mesh nodes in the core. In the other parts, the magnetic energy is neglected.

    In the no load condition, the secondary winding is open-circuited, and rated voltage at rated frequency is applied to the primary terminals. Under this condition, the primary current, the so-called no load current I0 (Figure 55), flows in the primary winding. It is not necessary to draw the individual turns but the whole coil side is replaced by one block in the FEMM program. It is seen in Figure 59 that the block (pwl) corresponds to all conductors of the left side of the primary winding. A calculation based on the magnetostatic analysis is very popular, and many authors recommend this kind of simulation [11].

    A constant value of the no load current, which is the magnitude of the sinusoidal waveform, is entered in this analysis. The secondary winding is opened, in which no current flows ; therefore, a zero value of the current density is allocated to the blocks corresponding to the secondary winding. The value of the current density in the primary winding can be obtained from the analytical calculation during the transformer design or by measurements on a real transformer, which is this case.

    From the no load test, the no load current at rated voltage UN is I0 = 0.274 A, which is an effective (rms) value. Its magnitude at the sinusoidal waveform is I0max=2I0=0.387A, but it is true that the waveform of the no load current at rated voltage is not sinusoidal, which is caused by the iron saturation [1, 13]. For illustration in Figure 57, there is no load current waveform of the analyzed transformer.

    Figure 57.

    No load current waveform of the analyzed transformer.

    The current density Jp of the magnitude of the no load current in the primary winding is calculated based on the primary winding number of turns:

    Jp=N1I0maxSp=354·0.3870.00065=0.210766MAm2

    where Sp is a surface of one part of primary winding, right or left side (pwr or pwl), and can be calculated by means of the geometrical dimensions (Figure 59).

    Then the current density is introduced to the block which corresponds to the right side of primary winding (pwr) Jpwr = +0.210766 MA/m2 and to the block corresponding to the left side of the primary winding (pwl) Jpwl = −0.210766 MA/m2. In the section where materials are defined, B-H curve is introduced, which was obtained by a producer, by the values of magnetic flux density, and by the magnetic field intensity of the employed sheets. The B-H curve is shown in Figure 58.

    Figure 58.

    B-H curve of the sheets employed in the investigated transformer, given by its producer.

    After a definition of all geometrical dimensions, materials, and current densities, it is necessary to define boundary conditions (Figure 59). Here Dirichlet boundary conditions can be applied, where constant value of the magnetic vector potential A = 0 is defined. It is defined on the circumference of the transformer, and the dialog window of the FEMM program is given as Boundary Property, as seen in Figure 60.

    Figure 59.

    Geometry of the investigated transformer with marked blocks and settings of the parameters.

    Figure 60.

    Definition of the transformer boundary conditions.

    In the last step before starting the calculation, it is necessary to create a mesh of the finite elements. As was mentioned before, in the ferromagnetic core, it is necessary to create a denser mesh, because there is concentrated dominant part of electromagnetic energy. The mesh is created by means of the command Mesh – Create mesh. After a modification, there were 22,158 finite elements created (see Figure 61).

    Figure 61.

    Created mesh of the finite elements.

    Now the Processor can be started, and calculation of the transformer in the no load condition is launched. After the calculation, a distribution of magnetic flux lines in the cross-section area of transformer is seen in postprocessor. Also distribution of magnetic flux density by means of command View density plot can be displayed (see Figure 62).

    Figure 62.

    Magnetic flux lines and magnetic flux density distribution in the cross-section area of the transformer.

    A magnetizing inductance calculation is done by means of linkage magnetic flux, according to Eq. (480). It must be done this way, because in the no load condition at rated voltage, the B-H curve is in nonlinear region, which means that energy and coenergy is not the same [4], because usually coenergy is higher, e.g., in this case the electromagnetic energy is 0.163 J and coenergy 0.263 J. It corresponds also to the non-sinusoidal waveform of the no load current, as it is seen in Figure 57.

    Therefore, it is better to employ Eq. (480), in which linkage magnetic flux appears. In the program FEMM value A.J, is obtained in such a way that only the blocks corresponding to the coils of primary winding are marked (Figure 63) and by means of surface integral calculate the A·J = 0.425975 H/A2:

    Figure 63.

    Distribution of the magnetic flux lines and calculation of the surface integral A.J (integral result) of the investigated transformer.

    LμFEMM=ψI0max=S×A·dSI0max=A·dlI0max=A·JdVI0max2=0.4259750.3872=2.84H.

    If this value is compared with the value obtained by no load measurement (Lμ = 2.82 H, see Table 7), it is seen that the difference is less than 1%.

    As it is known, during the measurements, it is possible to make measurement also at other voltages, not only rated, which results in various no load currents vs. voltage and also magnetizing inductances vs. no load currents. Such characteristics (Lμ vs. I0) can be employed in transient investigation during the no load transformer switch onto the grid. For comparison in Figure 64, there are waveforms of such characteristics obtained by measurements and simulations. In the region of the rated voltage and corresponding no load currents I0N, the coincidence is almost perfect, but at the lower currents, the discrepancy is higher because of lower saturation and perhaps lower precision of the B-H curve.

    Figure 64.

    Comparison of the magnetizing inductances obtained by measurement and FEMM simulation.

    3.4.2 Simulation of single-phase transformer short circuit condition

    According to the theory of electrical machines, the short circuit test is made at transformer, if the secondary terminals are short circuited and a fraction of the rated voltage sufficient to produce rated currents, at rated frequency is applied to the primary terminals. At the simulation of this condition, both windings are fed by their rated currents; it means that magnetomotive forces are equal and current density corresponds to the values of currents and the next equation, at which the magnetizing current is neglected Iμ ≈ 0 A, is valid, N1I1 = − N2I2, which results in the fact that no magnetizing flux is created in the core. Then only leakage flux occurs, which is closed through the leakage paths, which means by air, insulation, and nonmagnetic materials [1]. The 2D analysis is done in a different way in comparison with no load condition, when the whole magnetic flux was in the ferromagnetic core.

    A procedure described in [11] is applied in the analysis of the short circuit condition. The value of the leakage inductance is calculated from the magnetic field energy, because in short circuit condition no saturation of ferromagnetic circuit occurs. The value of the energy is calculated as follows:

    W=lavS12μH2dS

    where S is a surface of the whole transformer cross-section area and lav is an average length of the conductor or a half of the average length of the turn. It can be calculated as an average length between both windings of the primary and secondary coils (Figure 65). Nevertheless, based on experience, this calculated value should be increased about 5 till 10%, because during the manufacturing, the coils are not wound exactly and this dimension is very important for leakage inductance calculation. An increase of about 7.5% is used here. Based on Figures 56 and 65, this value can be calculated as lav = 1.075·(50 + 2·2 + 2·10 + 2·0.25 + 49.6 + 2·2 + 2·10 + 2·0.25) = 159.7 mm, and the measured value is lav = 161 mm, which is employed during further calculation. The calculation of the leakage inductance can be made by two ways:

    1. The first one is setting of the z-coordinate in Problem (Depth) on the value lav. The calculation is then made as follows:

      Lσ1+Lσ2=Lσ=2WI1N2

    or similarly as in the no load condition:

    Lσ1+Lσ2=Lσ=ψI1N2=A·JdVI1N2

  • The second approach is such, that z-coordinate is set in Problem (Depth) on the value 1 mm and the values obtained in the postprocessor must be multiplied by the average length of the conductor or by one half of the average length of the coil turn lav. Then the calculation is as follows:

    Lσ1+Lσ2=Lσ=lav2WI1N2

    or

    Lσ1+Lσ2=Lσ=lavψI1N2=lavA·JdVI1N2.

  • Figure 65.

    Illustration figure of the transformer cross-section area to define the average length of the turn.

    The purpose of this simulation is to calculate the total value of the leakage inductance Lσ1+Lσ2=Lσ. In transformers that have only small number of the turns on the secondary side, e.g., there is only one layer, it is needed to draw individual turns and, in each turn, to define current or current density. If that few turns are replaced by only one turn to simplify it, a significant error could appear, because of leakage flux is flowing around the individual turns. In investigated transformer here, the secondary side is created by some layers; therefore the solution is made by one block of the turns.

    A procedure of the calculation is as follows: The current densities J, corresponding to the windings, are calculated. It must be valid N1I1 = − N2I2. Then the current density is equal in both windings, but with opposite signs. The rated current in the primary winding is I1N = 2.75 A, and current density for the windings is as follows:

    Primary winding, left side:

    Jpwl=N1I1NSp=354·2.750.00065=1.497693MA/m2

    Primary winding, right side:

    Jpwr=N1I1NSp=354·2.750.00065=1.497693MA/m2

    In coincidence with the equation N1I1 = −N2I2, the secondary winding, left side, is:

    Jswl=N1I1NSp=354·2.750.00065=1.49769MA/m2

    and secondary winding, right side, is:

    Jswr=N1I1NSp=354·2.750.00065=1.49769MA/m2

    These values are introduced to the preprocessor in the corresponding coils. It is recommended to refine the mesh and to increase the number of finite elements around the windings and in the air because there is a main part of the leakage flux. Then calculation and analysis of the results can be made. In Figure 66, the distribution of the flux lines, which correspond to the short circuit condition, is seen. As it was supposed, the flux lines are in the surrounding of the coils because the main flux is neglected. In this condition, no saturation occurs; therefore the calculation of the leakage inductance can be made by means of the energy of electromagnetic field and for comparison also from the linkage magnetic flux. It is recommended to do it by both ways.

    Figure 66.

    Magnetic flux distribution in transformer under the short circuit condition.

    The first way starts at setting of the z-coordinate on the value lav = 161 mm. Then the calculation is launched to calculate the total leakage inductance. At the calculation based on the magnetic field energy, the whole cross-section area of transformer is marked in postprocessor. Then it is valid that:

    Lσ1+Lσ2=Lσ=2WI1N2=2·0.01931072.752=5.1mH

    or based on the linkage magnetic flux, but in postprocessor only the areas corresponding primary and secondary windings are marked and calculate A.J. Then it is valid that:

    Lσ1+Lσ2=Lσ=ψI1N2=A·JdVI1N2=0.03862152.752=5.1mH

    The second way is based on the fact that z-coordinate is set on 1 mm. Therefore, the value of energy must be multiplied by the average length of the conductor lav = 161 mm. For the calculation from the magnetic field energy, it is valid that:

    Lσ1+Lσ2=Lσ=lav2WI1N2=1612·0.0001199422.752=5.1mH

    or from the linkage magnetic flux by means of integral A.J:

    Lσ1+Lσ2=Lσ=lavψI1N2=lavA·JdVI1N2=1610.0002398852.752=5.1mH

    The measured value of total leakage inductance is 5.2 mH (see Table 7), which means a very good coincidence of the results.

    In this FEMM program, it is possible to calculate approximately the resistance of primary and secondary winding. For more precise calculation, the 3D program would be needed.

    The resistance depends on the electrical conductivity of copper from which the windings are made. In simulating the value of copper, specific electrical conductivity σ = 58 MS/m is used or can be set based on the material library in the FEMM program. The calculation starts from the short circuit simulation, whereby z-coordinate is set on 1 mm. Now the average length of the turn of primary lavp and secondary lavs winding must be calculated. The calculation is made based on the geometrical dimensions in Figures 56 and 65. Then lavp = 255.2 mm and lavs = 339.2 mm. After the calculation, in postprocessor, the blocks must be marked, which correspond to the primary winding, and by means of the command Resistive losses, the Joule losses in the primary winding ΔPj are calculated. In the same way, the losses in the secondary winding are calculated. Nevertheless, it is the same value because the cross-section area of the winding and current density is the same. The resistance is then calculated at 20°C for both windings as follows:

    Rp=lavpΔPjI1N2=255.20.0502762.752=1.71Ω
    Rs=lavsΔPjI2N2=339.20.05027626.32=24.6

    For comparison the measured values are Rp = 1.91 Ω and Rs = 20 mΩ.

    In the end the simulated and measured values of equivalent circuit parameters are summarized in Table 8. It can be proclaimed that the values obtained by simulation and measurement are in good coincidence.

    Rated voltage of the primary side U1N230 V
    Rated voltage of the secondary side U2N24 V
    Rated power SN630 VA
    Rated frequency f50 Hz
    Rated current of the primary side I1N2.75 A
    Rated current of the secondary side I2N26.3 A
    Number of turns of the primary side N1354
    Number of turns of the secondary side N239
    Shell-type construction
    Total thickness of the core lFetotal49.6 mm
    Transformer sheets EI 150 N
    No load current I0 obtained from no load measurement0.274 A
    Magnetizing inductance Lμ obtained from no load measurement2.82 H
    Total leakage inductance Lσ obtained from short circuit measurement5.2 mH

    Table 7.

    Nameplate and rated values of the investigated transformer.

    MeasurementFEMMDeviation
    Magnetizing inductance Lμ [H]2.822.851.05%
    Total leakage inductance Lσ [mH]5.25.11.9%
    Primary winding resistance Rp [Ω]1.911.7110.4%
    Secondary winding resistance Rs [Ω]0.020.024618.6%

    Table 8.

    Comparison of the equivalent circuit parameters.

    3.5 Analysis of asynchronous machine parameters

    Asynchronous motor parameters are simulated based on the no load test and locked rotor test in accordance with the equivalent circuit parameters [21]. Also calculation of the air gap electromagnetic torque and its ripple is made. Analysis is made for a real three-phase squirrel-cage asynchronous motor (its type symbol is 4AP90L); the nameplate and rated values are in Table 9. A picture of its geometrical parts and their dimensions are in Figures 67 and 68.

    Rated stator voltage U1N400 V
    Stator winding connectionY
    Rated power PN1500 W
    Rated frequency f50 Hz
    Rated speed n1410 min−1
    Phase number m3
    Rated slip sN6%
    Number of pole pairs p2
    Rated torque TN10.15 Nm
    Number of one-phase turns Ns282
    Number of slots per phase per pole q3
    Winding factor kw0.959
    Active length of the rotor lFe98 mm
    Number of conductors in the slot zQ47
    Magnetizing current I0 obtained from no load measurement2.3 A
    Lμ magnetizing inductance obtained from no load measurement0.32 H
    Rated stator current IsN3.4 A

    Table 9.

    Nameplate and parameters of the investigated three-phase asynchronous motor.

    Figure 67.

    Cross-section area of the stator sheet and detail of the stator slot with its geometrical dimensions.

    Figure 68.

    Cross-section area of the rotor sheet and detail of the rotor slot with its geometrical dimensions.

    3.5.1 Simulation of the no load condition

    An ideal no load condition is defined at synchronous speed of the rotor, when rotor frequency is zero. In fact, at real no load condition, the rotor rotates at speed lower than synchronous speed, but the difference is not significant. Therefore, an ideal no load condition can be investigated without a big error. A magnetizing inductance of the equivalent circuit and also fundamental harmonic of the magnetic flux density in the air gap can be calculated by means of FEM. Here is a procedure how to do it:

    • Draw a model of the investigated motor in a cross-section area in a program of FEMM (Figure 69).

    • Enter the three-phase currents to the stator windings, materials, boundary conditions, and a mesh density. (The rotor currents are in the ideal no load condition zero).

    • After the calculation, analyze in postprocessor distribution of the air gap magnetic flux density

    • To make a Fourier series of the air gap magnetic flux density, calculate its fundamental harmonic, electromotive force (induced voltage), and from it the magnetizing inductance.

    Figure 69.

    The 1/4 of the cross-section area of the squirrel-cage asynchronous motor together with basic settings.

    3.5.1.1 Drawing of the asynchronous motor model

    From Figure 69 it is seen that the asynchronous motor cross-section area is much more complicated than the transformer. There are more possibilities how to draw this model, either directly in FEMM or in other graphical program (e.g., AUTOCAD, CAD, with the suffix *.dxf), and then to import it into FEMM. Here a drawing of the editor of the FEMM program is explained.

    The first step is the setting of the task type which is investigated. In the beginning, when a new problem is investigated, the program calls up the user to define the type of the problem. For the no load condition, the Magnetostatic Problem is set. In the block problem, the units of the geometrical dimensions are set, usually in mm. Stator and rotor frequency is zero, because only one instant is investigated. In the block Depth, the active length of iron lFe is set. Problem Type is planar (see Figure 69).

    The drawing starts with changing over the drawing editor to the point mode, and by means of tabulator, the points based on the x- and y-coordinates are set. It is recommended to draw the model in such a way that the center of the machine has coordinates 0,0. After the points are drawn, change the program into the line mode or arc mode, and join the points by straight lines or curves. If there is curve mode, the user is asked, which angle should have the curve, e.g., for semicircle it is 180o, and what the accuracy should be. According to the accuracy of the calculation, it is recommended to enter the number 1, higher number means lower accuracy.

    If the same objects are drawn several times, e.g., stator or rotor slots, it is possible to apply copying, which is in the block Edit and Copy.

    The next step is the setting of the materials and currents into the appropriate blocks. All bordered areas created during the drawing present the blocks into which it is necessary to input the materials. On the toolbar, it is necessary to change over to block label, group mode, to mark all blocks, and to define them. But before that, the materials must be designated and defined.

    Materials are defined in Properties – Add properties, where the constants for materials are entered (Figure 70). In asynchronous machine, there is air in the air gap and ventilating channels; ferromagnetic circuit, which is defined by a nonlinear magnetizing B-H curve (in this case the employed sheet is Ei70, the thickness of the sheet is 0.5 mm, block name in the program is core); and material, from which the stator and rotor conductors are produced. It is usually copper or aluminum. These materials can be copied from the program library. For stator slots, the current density, corresponding to no load condition, must be entered. A calculation of the current density for stator slots is as follows:

    Figure 70.

    Material designation and definition.

    Think one instant of the three-phase no load current for all three phases. For example, if phase A crosses to zero, then phases B and C have the values equaled to sin60o from the magnitude. If the next phase sequence is assumed around the stator circumference: +A, -C, +B, -A, +C, -B, etc., then A = 0, −A = 0, -C = -Jmax sin 60o, C = +Jmax sin 60o, B = -Jmax sin 60o, and −B = +Jmax sin 60o, where Jmax is the magnitude of the current density. It can be calculated as follows:

    Jmax=zQImaxSd=47·2·2.349.6=3.082MA/m2E482

    where Imax is the magnitude of the no load current, which flows along the conductors, zQ is a number of the conductors in the slot, and Sd is a cross-section area of the stator slot (Figure 67). In calculation there is a no load current I0 = 2.3 A used (see Table 9).

    The calculated current densities are entered to the appropriate slots. In this case the number of slots per phase per pole is q = 3.

    If all materials are defined, then it is necessary to allocate them to the appropriate blocks. The demanded block is designated by the right mouse button, and by pushing the space key, it is possible to allocate the material to the block. In this way, all blocks are defined. If it happens that a block is forgotten, it is not possible to make calculation until the block is not designated.

    3.5.1.2 Setting of the boundary conditions

    The last task before the calculation is launched, which is definition of the boundary conditions. Because the whole cross-section area of the asynchronous motor is analyzed, the Dirichlet boundary condition on the outer circumference, which is zero magnetic vector A = 0, can be applied (Figure 71).

    Figure 71.

    Definition of the boundary conditions.

    In the block Properties, click on the Boundary and define the boundary condition according to Figure 71. Then change over to the curve mode, choose the stator external circle by right mouse button, and by means of space key the appropriate boundary condition is allocated to the circle.

    Before starting the calculation, click on the icon Mesh, and create the demanded mesh of the finite elements. If it looks in some parts to be widely spaced, it can be densified in the next way: It is necessary to change over to the block mode, by means of the right mouse button, choose the demanded block, mark Let triangle choose Mesh size, and set the required value of the finite elements. Then the very calculation can be started by means of the icon solve.

    3.5.1.3 Calculation of the air gap magnetic flux density

    After the calculation is finished, a distribution of the magnetic flux lines in the cross-section area of the investigated motor appears on the screen (see Figure 72). For illustration there is the whole cross-section area of the four-pole asynchronous motor. If a distribution of the magnetic flux density in the whole cross-section area is needed, it is possible to see its value in each point of the cross section and accord these values to optimize construction of the motor.

    Figure 72.

    Distribution of the magnetic flux lines in four-pole asynchronous motor in no load condition.

    Most important is to know the shape of the waveform of the air gap magnetic flux density. Therefore, after the calculation in postprocessor, it is necessary to mark a circle in the middle of air gap (see Figure 73a) and on this circle in dialog window, to mark calculation of the normal component of the magnetic flux density. These values can be saved in data file and their elaboration in the other program continued. In such a way based on the Fourier series, the fundamental harmonic component can be calculated. In Figure 73b, a waveform of the air gap magnetic flux density is shown. After the Fourier series is made, it is possible to calculate electromotive force (induced voltage) and from this value the magnetizing inductance.

    Figure 73.

    (a) Marking of the circle in the middle of the air gap to calculate a normal component of the magnetic flux density and (b) distribution of the normal component of the air gap magnetic flux density in the investigated asynchronous motor.

    Based on the geometrical dimensions, a calculation of the induced voltage can be made. The induced voltage in the three-phase alternating rotating machines is given by an expression as follows:

    Ui=2πfΦavNskw=2πf2πBπD2plFeNskwE483

    where D = 84 mm, lFe = 98 mm, Ns = 282 turns, and kw = 0.959. From the waveform of the air gap magnetic flux density in Figure 73b, the magnitudes of the harmonic components Bδν can be calculated by means of the Fourier transformation. In Figure 74, the harmonic analysis (Fourier series) of the air gap magnetic flux density for 60 harmonic components is shown. To make a Fourier transformation, the program MATLAB can be employed.

    Figure 74.

    Magnitude spectrum of the air gap magnetic flux density harmonic components in no load condition.

    The magnitude of the air gap magnetic flux density of the fundamental harmonics is Bδ1max = 0.94 T. After this value is introduced to Eq. (483), the rms value of phase induced voltage fundamental harmonic is obtained:

    Ui1=2πf2πBπD2plFeNskw=2π2π0.94π·0.08440.098·282·0.959=232.4V

    The total voltage could be calculated by the sum of all harmonic components. The terminal voltage given in the nameplate is 3 x 400 V at star connection. Then the phase value is 230 V, which is in very good coincidence with the value obtained by FEMM. For illustration in Figure 75, there is a distribution of the magnetic flux density in the cross-section area of the investigated motor at no load condition.

    3.5.1.4 Calculation of the magnetizing inductance at no load condition

    Figure 75.

    Distribution of the magnetic flux density in no load condition.

    At no load condition, also magnetizing inductance can be calculated. If the induced voltage, frequency, and no load current (from the no load measurement) are known, the magnetizing inductance is as follows:

    Lμ=UiωI0=232.42·π·50·2.3=0.321H

    It was supposed that at no load condition, the magnetizing current is almost identical with no load current. The measured value of the magnetizing inductance is Lμ = 0.32 H (see Table 9), which is in very good coincidence of the results obtained from FEMM and measurement.

    A calculation of magnetizing inductance can be done also based on the linkage magnetic flux and integral A.J, because in this no load condition, the ferromagnetic circuit is in the nonlinear area. A procedure can be described as follows: After the calculation in postprocessor, all slots corresponding with the stator winding are marked, and the integral A.J is calculated (Figure 76). It must be realized that the flux is created by all three phases; therefore the value of no load current is multiplied by 3, or linkage magnetic flux is divided by 3. Then the magnetizing inductance is:

    Figure 76.

    Marking of stator slots for magnetizing inductance calculation.

    Lμ=ψ3I0=A·JdV3I02=5.183453·2.32=0.326H

    It is seen that a coincidence with the values obtained by other methods is very good.

    3.5.2 Simulation of locked rotor condition

    During the no load simulation by means of the FEMM, a magnetostatic field was employed. It is possible to do it also during the simulation of the locked rotor condition, but it is necessary to enter to the rotor bars actual rotor currents, induced there from the stator currents. Such setting of the rotor currents is time-consuming, mainly if there is bigger number of the rotor slots. For completeness this way is described in Section 5.7, where the rated condition and rated torque are calculated.

    Simulation of the locked rotor condition means to solve a harmonic task. It means to enter the frequency corresponding with the actual rotor frequency at locked rotor condition. Because in FEMM program there is no possibility to enter more various frequencies in one solution, it is necessary to enter to the base settings (Figure 69) the slip frequency. At the locked rotor condition, the slip is equal to 1, and corresponding frequency is frequency of the stator current f = 50 Hz. During the analysis, the same cross-section area of the asynchronous motor as in the no load condition is employed. In stator slots, the material of the stator slots (Cu, Al) and its conductivity and permeability, which is close to the vacuum, is entered. Usually the stator conductors in asynchronous motors are copper (copper electrical conductivity is 58 MS/m) and rotor cage is aluminum, the electrical conductivity of which is 24.59 MS/m. It is also the case of motor simulated here.

    To define the currents, a block Circuits is employed. It is in the block below Properties, under Property definition and Circuit Property (see Figure 77).

    Figure 77.

    Currents definition in locked rotor condition.

    In Figure 78, there are phasors of stator currents Is for corresponding phases, eventually circuits. The circuits can be set on the Parallel or Series (Figure 77). Here the circuit Series is employed, and then the currents definition is in coincidence with Figure 78. It is a product of the complex figure which corresponds to the current phasor position, number of the conductors in the slot, and magnitude of the current. In this case a current is set on its rated value IN = 3.4 A, and then its magnitude is INmax = 4.8 A. The phases are excited as follows:

    1. Phase A+: 1+j0zQINmax= 1+j0472·3.4=226+j0

    2. Phase A−: 1+j0zQINmax=1+j0472·3.4=226+j0

    3. Phase B+: 12j3247·2·3.4=113j195.7

    4. Phase B−: 12+j3247·2·3.4=113+j195.7

    5. Phase C+: 12+j3247·2·3.4=113+j195.7

    6. Phase C−: 12j3247·2·3.4=113j195.7

    Figure 78.

    The current phasors in complex plain for the locked rotor condition.

    From the winding theory, it is known:

    q=QS2pm=362·2·3=3E484

    where q is number of slots per phase per pole and p is number of pole pairs. It means that the three adjacent stator slots are defined by the same phase current, e.g., A+. These triplets of the slots are changed in the order, A+, A+, A+, C-, C-, C-, B+, B+, B+, A-, A-, A-, C+, C+, C+, B-, B-, B-, which is seen also in the Figure 78, where this changing of the phases creates clockwise rotating magnetic field.

    As the number of the pole pairs p = 2, this order is repeated two times around the stator inner circumference. If there is double-layer winding, it must be taken into account. In here investigated motor, there is a single-layer winding.

    In rotor conductors, the aluminum with its electrical conductivity is set, because at the harmonic task the currents are induced in the rotor conductors.

    3.5.2.1 Calculation of the equivalent circuit parameters from the locked rotor simulation

    The equivalent circuit parameters needed for the calculation of its parameters in locked rotor condition are in Figure 79. The resistance and leakage inductance depend also on the frequency. In locked rotor condition, the rotor frequency is identical with the stator frequency; therefore in starting settings of the FEMM program, (problem) f = 50 Hz is entered.

    Figure 79.

    Modified equivalent circuit of the asynchronous motor.

    In the 2D program, it is not possible to calculate resistance and leakage inductance of the end connectors and rotor end rings. For a correct calculation, these parameters must be calculated in another way or employ a 3D program. In Figure 79 there is an adapted equivalent circuit, where the dashed line shows which parameters can be calculated by 2D program.

    Parameters out of dashed line are caused by 3D effect. They are the following: Rs, stator winding resistance; Lσs,3D, stator leakage inductance of end windings; Lσ,2D, stator and rotor leakage inductance without end windings and rotor rings; Rr,2Ds, rotor resistance referred to the stator without rotor rings; Rr,3Ds, resistance of rotor rings referred to the stator; and Lσr,3D, leakage inductance of the rotor rings. Supplied current in the simulation is rated current which corresponds also to locked rotor measurement. Magnetic flux lines distribution in locked rotor state is shown in Figure 80. The following calculations are carried out in accordance with [11].

    Figure 80.

    Distribution of the magnetic flux lines of the asynchronous motor in the locked rotor condition.

    During locked rotor simulation, the following parameters can be obtained: Lσ,2D and Rr,2D. Usually, locked rotor test is done with low supply voltage, so no saturation effect is present. Then, leakage inductance Lσ,2D can be calculated from stored energy:

    Lσ,2D=2W3IsN2=2·0.3673·3.42=21.1mHE485

    From (485), it can be seen that the current is multiplied by 3. It is caused by three-phase supplying and all three phases create energy of magnetic field. The value of energy W can be obtained from whole cross-section area of the motor in postprocessor. To obtain total leakage inductance of the stator and rotor, analytical calculation of stator end winding leakage inductance and rotor ring leakage inductance must be taken into account, e.g., from [20], which is 26.8 mH. The value obtained from the locked rotor measurement is 29.5 mH, which is appropriate coincidence of the results.

    The value of the rotor resistance is calculated from the losses in the rotor bars. In postprocessor, all blocks belonging to the rotor bars are marked (Figure 81) and by means of the integral, the ΔPjr are calculated. Then the resistance of the rotor bars without the end rings is calculated as follows:

    Figure 81.

    Calculation of the losses in rotor bars.

    Rr,2D=ΔPjr3IsN2=128.7553·3.42=3.71ΩE486

    The resistance of the rotor end rings is calculated from the expressions known from the design of electrical machines according to [1]. The total rotor resistance, which includes a bar and corresponding part of the end rings, referred to the stator side is 3.812 Ω. The measured value from the locked rotor test is 3.75 Ω, which is an appropriate coincidence of the parameters.

    3.5.3 Calculation of the rated torque

    The rated condition can be analyzed into two ways:

    1. Magnetostatic task, at which instantaneous values of current densities are entered to the stator and rotor slots, which correspond to the same time instant at investigated load, without a frequency. Then an electromagnetic torque is calculated around the circle in the middle of the air gap, according to Eq. (473).

    2. Harmonic task, at which the rated currents are entered only to the stator slots. The currents in the rotor are calculated. There are two possibilities how to do it: (a) In the settings of FEMM program problem, the frequency is set, corresponding to the rotor frequency at the investigated load. For example, at the rated condition with the slip sN = 6%, then the frequency f = 3 Hz is set. (b) A conductivity of the rotor bar proportional to the slip of investigated load is set, to simulate a changing of the current following the load. For example, if the rotor bars are aluminum, the electrical conductivity is σAl = 24.59 MS/m at the slip 1. At the rated load (sN = 0.06), the conductivity is Al = 0.06·24.59 = 1.475 MS/m, which corresponds to the lower current than for the slip 1.

    3.5.3.1 Magnetostatic task

    The procedure is similar as for the no load condition. It means to the stator slots the current densities corresponding to the phases and to the instantaneous values of the rated current are entered. An instant at which phase A crosses zero is taken into account. At this instant, phases B and C have an equal value but with opposite polarity. Then the current density in a slot belonging to the particular phase is as follows:

    1. Phase A+: JA=INzQ2Sd, seeing that the current in phase A crosses zero, then JA = 0.

    2. Phase A−: JA=INzQ2Sd, seeing that the current in phase A crosses zero, then - JA = 0.

    3. Phase B+: JB=INzQ2Sdsin60o, because the phases are shifted about 120o, instantaneous value of the current density in phase B+ is negative.

    4. Phase B−: JB=INzQ2Sdsin60o, because the phases are shifted about 120o, instantaneous value of the current density in phase B- is positive.

    5. Phase C+: JC=INzQ2Sdsin60o, because the phases are shifted about 120o, instantaneous value of the current density in phase C+ is positive.

    6. Phase C−: JC=INzQ2Sdsin60o, because the phases are shifted about 120o, instantaneous value of the current density in phase C- is negative.

    The J is current density, IN is rated current, zQ is number of conductors in the slot, and Sd is slot cross-section area.

    In the magnetostatic task, the currents must be entered also to the rotor, to be able to calculate the torque in the air gap. The current is referred from the stator to the rotor side and to the corresponding current density at particular time instant. The expression is known from the electrical machine design theory [1]:

    Jr=IN2mNskwQrSdsinαE487

    where m is stator phase number, Ns is number of turns of the stator phase, kw is its winding factor, and Qr is number of rotor bars. The number of the rotor turns is ½; Sd is cross-section area of the rotor slot. An angle α represents angular rotation of the rotor bar currents, and sin(α) corresponds to the instantaneous value of the current density in each rotor slot. The angle α is calculated as follows:

    α=2p180QrnE488

    where n is a number of the rotor bar. Numbering of n is started from zero in that rotor slot, where current density starts from zero. After all current densities are entered, the calculation is launched. After the calculation is carried out, and the magnetic flux lines are depicted, the value of electromagnetic torque in the air gap can be calculated as follows:

    1. In postprocessor a circle in the middle of the air gap is marked by a red line.

    2. The value of the torque is calculated based on the Maxwell stress tensor (Section 3.2, according to Figure 82).

    The obtained value of the electromagnetic torque at the rated current 3.4 A is 9.89 Nm (Figure 82a). For comparison, the rated torque on the shaft is TN = 10.15 Nm and the value of the loss torque, obtained from the no load test, is Tloss = 0.7 Nm. Then the value of developed electromagnetic torque is:

    Figure 82.

    Illustration figure for electromagnetic torque calculation based on the (a) magnetostatic task and (b) harmonic task.

    Te=TN+Tloss=10.15+0.7=10.85Nm,

    which is an appropriate coincidence of the results. The calculation can be done also based on the harmonic task.

    3.5.3.2 Harmonic task

    The other way how to analyze the rated condition of the asynchronous motor is calculation by means of harmonic task. The entered frequency is slip frequency, e.g., if the frequency of the stator current is 50 Hz and the rated slip is 6%, then the slip frequency is 50·0.06 = 3 Hz, which is employed for setting. In this case the calculated electromagnetic torque in the air gap is 10.69 Nm, which is better coincidence in comparison with calculation of magnetostatic task.

    The developed electromagnetic torque can be calculated in such a way, that electrical conductivity proportional to the slip is entered to the rotor slots. The feeding frequency in the stator is 50 Hz. It is supposed that the electrical aluminum conductivity is σAl = 24.59 MS/m. It means at locked rotor condition, at slip equal 1, the conductivity is Al = 1 24.59 = 24.59 MS/m and for the rated condition is sNσAl = 0.06·24.59 = 1.47 MS/m. In this case the calculated electromagnetic torque in the air gap is 10.71 Nm (Figure 82b), which is very close to the measured value. For the calculation of the electromagnetic torque, it is recommended to employ the harmonic task, from the point of view of accuracy and time demanding.

    By means of the harmonic task, it is possible to calculate also other conditions of the asynchronous motor along the mechanical characteristic. It is possible also to parameterize the investigated problem and to program by means of LUA script in the FEMM program [10]. It means that the whole model is assigned in parameters mode and then it is possible to change its geometrical dimensions and optimize its properties.

    3.5.4 Calculation of the torque ripple

    The torque ripple is caused by the stator and rotor slotting and mainly by the stator and rotor slot openings. The calculation of the torque ripple is made in the air gap. A procedure is similar like in the case of the harmonic task during the calculation of the torque in rated condition. The stator slots are fed by rated current in coincidence with Figure 78 (Section 5.5), and electrical conductivity of the rotor slots is proportional to the rated slip sNσAl = 0.06·24.59 = 1.47 MS/m. The rotor position is changed gradually by 1° mechanical. The calculation was carried out for positions from 0o till 45o mechanical, which means one half of the pole pitch τp/2. The calculated torque values vs. rotor position ϑ are shown in Figure 83. The average value of all calculated values is:

    Figure 83.

    Electromagnetic torque of the asynchronous motor in the air gap vs. rotor position, calculated based on the harmonic task.

    Teav=ϑTedϑ=10.975Nm.E489

    The torque ripple in the air gap presented in percentage is:

    Tripp=TemaxTeminTeav100[%]E490

    From Figure 83 it is seen that the ripple torque is moving in the interval from 1.45% till 4.88%.

    3.6 Analysis of the synchronous machine parameters

    From the point of view of operation principles and construction, the synchronous machines can be divided into three basic groups:

    1. Synchronous machine with wound field coils creating electromagnetic excitation, which can be designed with salient poles or with cylindrical rotor (Section 6.1).

    2. Synchronous reluctance machine with salient poles on the rotor without any excitation (Section 6.2).

    3. Synchronous machine with permanent magnets on the rotor, creating magnetic flux excitation. The permanent magnets can be embedded in the surface of the rotor, or on the surface of the rotor (Section 6.3).

    Here, parameter investigation by means of FEM of all three types of the synchronous machines is presented in the chapters as it is written above.

    3.6.1 Synchronous machine with wound field coils on the rotor salient poles

    In synchronous machine with wound rotor, an analysis of the no load condition, short circuit condition, synchronous reactances in d- and q-axis, and air gap electromagnetic torque is carried out by means of the FEM. The nameplate and parameters of the investigated generator are shown in Table 10. Its basic geometrical dimensions are in Figure 84.

    Rated voltage U1N400 V
    Stator winding connectionY
    Rated power SN7500 VA
    Rated frequency f50 Hz
    Rated speed n1500 min−1
    Rated power factor cosφ1
    Rated stator current IN10.8 A
    Rated field voltage Uf32 V
    Rated field current If7.4 A
    Number of pole pairs p2
    Number of field coil turns Nf265
    Number of turns of single stator phase Ns174
    Number of slots per pole per phase q3
    Number of stator slots Qs36
    Number of conductors in the slot zQ29
    Cross-section area of the rotor winding Sf0.00081 m2
    Winding factor kw0.9597
    Active length of the rotor lFe80.2 mm
    Diameter of the boring D225 mm
    Air gap δ2 mm
    Magnetizing current measured from the rotor If04.6 A
    Magnetizing current measured from the stator side Iμ12 A
    Factor g = If0/Iμ0.383
    Leakage stator reactance (measured) Xσs1.4 Ω
    Magnetizing reactance in d-axis (measured) Xμd21.6 Ω
    Magnetizing reactance in q-axis (measured) Xμq6.27 Ω

    Table 10.

    Nameplate and parameters of the analyzed synchronous generator.

    Figure 84.

    Cross-section area of the investigated four-pole synchronous machine with salient poles and its geometrical dimensions.

    3.6.1.1 Simulation of the no load condition of synchronous machine

    The goal of the simulation in no load condition is to calculate induced voltage in the no load condition if the field current needed for this voltage is known. This current can be calculated during the design procedure or by a measurement on the real machine in generating operation. Here the measured value If0 = 4.6 A, at the induced rated voltage Ui = UphN = 230 V, is used.

    A procedure is the same as in the case of asynchronous machine. It is started with preprocessor, where it sets magnetostatic analysis, planar problem, and zero frequency, because only field winding is fed by DC. Z-coordinate depth represents active length of the rotor, lFe = 80.2 mm, obtained by measurement.

    Then based on the geometrical dimensions, a cross-section area is drawn as it is in Figures 84 and 85. The names of materials are allocated to the blocks. In this condition, when the stator currents are zero (no load condition), the rotor is in random position. However, it is better to set it to the d-axis, which is used also at the investigation of the short circuit condition. This rotor setting is made in such a way that the rotor pole axis is in coincidence with those slots in the stator, in which it would be zero current during the short circuit test. In this case, they are the slots corresponding with phase A (Figure 86).

    Figure 85.

    Stator slot with its geometrical dimensions.

    Figure 86.

    Setting of materials and blocks of analyzed machine.

    Stator and rotor sheets are marked as ferromagnetic material. The stator winding is in single layer, and then in the stator slots, there are gradually phases A+, C-, B+, A-, C+, B-, which are now zero currents. The slots are defined only by copper permeability and conductivity. In the middle of the air gap is drawn an auxiliary circle on which distribution of magnetic flux density and electromagnetic torque is calculated. The field coils are marked as F1 and F2 (Figure 86), in which current density is set, corresponding with the current If0. The current density is calculated as follows:

    Jf=NfIf0Sf=265·4.60.00081=1.504938MA/m2,E491

    where Sf is cross-section area of the field coil on the single rotor pole, e.g., F1 Figure 86. With regard to the fact that the field current is DC, the current density in the block F1 is positive and in the block F2 negative. All defined materials and blocks are shown in Figure 86. Magnetizing characteristic is taken from the program library, because the real characteristic of the investigated machine is not known. Therefore, calculation for other field currents, as it is obvious during the measurement, is not made. If in other cases B-H characteristic is known, there is possibility to make calculations in its whole scope.

    The boundary conditions are set in a similar way like in the case of asynchronous machine in the previous chapter. It is supposed that at zero magnetic vector potential, it means set A = 0. Then the mesh can be created and calculation launched. In the surroundings of the air gap, the mesh can be refined, to get more accurate results. After the calculation is finished, a distribution of the magnetic flux lines is described (Figure 87a). In postprocessor an auxiliary circle in the middle of the air gap is marked and asked to show a waveform of the magnetic flux density around the whole circumstance (Figure 87b). Then harmonic components are calculated by means of Fourier series (Figure 88). The rms value of the induced voltage is calculated based on the magnitude of the fundamental harmonic component in the air gap Bδ1max.

    Figure 87.

    Outputs obtained from postprocessor, (a) distribution of the magnetic flux lines in the no load condition and (b) magnetic flux density waveform along the whole air gap (by means of rotor field current If0).

    Figure 88.

    Fourier series of air gap magnetic flux density in no load condition.

    Now it is possible to calculate a phase value of the fundamental harmonic of the induced voltage:

    Ui1=2πf2πBδ1maxπD2plFeNskw=2π502π0.69π0.22540.08·174·0.959=230.78VE492

    where Bδ1max = 0.69 T (Figure 88). The calculated value of the induced voltage is in very good coincidence with the measured value, which is 230 V.

    3.6.1.2 Induced voltage calculation by means of the stator current

    If the field current is zero If = 0 A and the synchronous machine is applied to the grid with rated voltage UN, then a magnetizing current Iμ flows in the stator winding to induce rated voltage equal to the terminal voltage. The same current flows in the stator winding if the terminals are short circuited and the machine is excited by the field current If0, in generating operation. The investigated synchronous machine was measured in short circuit condition in the generating mode, and the measured value is Iμ = 12 A.

    To get the induced voltage on the terminals, the stator slots are fed by the current <