Open access peer-reviewed chapter

Analysis of Electrical Machines

Written By

Valéria Hrabovcová, Pavol Rafajdus and Pavol Makyš

Reviewed: 03 March 2020 Published: 20 May 2020

DOI: 10.5772/intechopen.91968

From the Monograph

Analysis of Electrical Machines

Authored by Valéria Hrabovcová, Pavol Rafajdus and Pavol Makyš

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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 Φμ creates linkage flux ψμ and corresponds to the magnetizing inductance Lμ. The leakage flux Φσ creates leakage linkage flux ψσ and 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.

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 αi is 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 δef includes 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 Lσ can 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 σsq is 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:

 υ 1 −17 19 −35 37 ksq 0.995 0.06 −0.05 −0.03 0.03

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/π is introduced. The rest of the equation represents air gap inductance (υ=1 is 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:

Λmag=μ0lh43b4.E18

The permeance factor λ is 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=λd is 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

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.

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

λz=k25δb15+4δb1,E34

where k2=134ε is 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.

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 lw and the product of lwλew can 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 λlew and λYew can 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=0.324 and 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=120 mm. 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 Δt is 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:

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 φξ and ψξ are 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 ω is 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>2 cm, 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).

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.

2.2.1.1 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 ψj is 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.

Transformation voltage created by the time variation of ψq is 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 ψd does 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.

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 ijdt and the result is:

ujijdt=Rjij2dt+ijdψj.E76

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

udiddt=Rid2dt+iddψdωψqiddt.E77

For the rotor winding in the q-axis, Eq. (74) multiplied by iqdt will 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 te is 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 JdΩdt with 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.

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.

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 id is 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=Ldfif shows that the linkage flux ψd in Eq. (94) is created by the mutual inductance Ldf between 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/p or n=60Ω/2π min−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 A Lq = 0.324 mH IfN = 6.4 A Rf = 13.2 Ω nN = 3200 min−1 Lf = 1.5246 H TN = 48 Nm Lqf = 0.0353 H PN = 16 kW J = 0.04 kg m2 p = 1 Te0 = 0.2 Nm

Table 1.

Nameplate and parameters of the simulated separately excited DC motor.

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.

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).

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:

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.

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.

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.

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.

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 V Rq = 1 Ω IqN = 5 A Lq = 0.005 mH PN = 925 W Rf = 1 Ω nN = 3000 min−1 Lf = 0.015 H MN = 3 Nm Lqf = 0.114 H IfN = 5 A J = 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.

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.

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)).

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.

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/kd and 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 ϑr angle, and the axis of the k-reference frame, which is shifted from the stator axis about an arbitrary ϑk angle. 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 ωk in the marked direction. The space vector of the stator voltage can be written as a sum of its real and imaginary components:

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 ejϑkϑr if 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 ϑkϑr instead 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:

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:

From Eq. (151), dψa/dt is selected:

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):

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 cosϑk yield:

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 ωk with 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 ϑk and from the rotor axis about the angle ϑkϑr. The angle of the k-system ϑk is 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,0 are employed. Initially the subscripts α,β,γ were introduced, but after some development the new system of the subscripts α,β,0 was 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=23 and 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=ϑr is chosen, the argument of the sinusoidal function ωstϑr is 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 ud in this system is in the form:

ud=23Umax32sinωstϑr=32Umaxsinωstϑr,E219

and the voltage uq after 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 ωk introduces ωs and changes the subscripts, the expression for ud is in the form:

ux=kdUmax32sinωsωstϑk0=kdUmax32sinϑk0=kdUmax32sinϑk0.E226

Similarly, for voltage uq at 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=23 were 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).

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).

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).

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).

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=0 and 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.

 PN = 1.1 kW Rs = 6.46 Ω UN = 230 V Ls = Lr = 0.5419 H IN = 2.4 A Rr = 5.8 Ω fN = 50 Hz Lμ = 0.5260 H nN = 2845 min−1 J = 0.04 kg m2 TN = 3.7 Nm p = 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).

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 kW Rs = 1.125 Ω UsN = 230 V , UrN = 64 V Ls = Lr = 0.1419 H IsN = 9.4 A, IrN = 47 A Rr = 1.884 Ω fN = 50 Hz Lμ = 0.131 H nN = 1370 min−1 J = 0.04 kg m2 TN = 30 Nm p = 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.

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.

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:

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:

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,LfD do 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.

2.16.2.1 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).

Then the mutual inductances can be expressed as follows:

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

similarly:

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 LaQ is 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.

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.

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

The magnitude of the self-inductance Laamax is 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 Lbc is maximal is shown in Figure 38a. Its waveform vs. rotor position is in Figure 38b.

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:

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,

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

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 Lfd and 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 ψd at the current iD (358) and ψD at the current id (355) are put into the equality, to get LDd=LdD. Then:

which results in kd=23.

In the q-axis it is done at the same approach: The expression at the current iq in the equation for ψQ (356) and the expression at the current iQ in 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
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
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:

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 Lff is 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) Lμd and 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.

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 ϑr is 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), difdt is expressed and introduced to Eq. (437), then after modifications it yields:

diddt=LffLdLffLdf2udRsid+ωrLqiqLdfLffuf+LdfLffRfif.E440

Similarly, if from (437) expression diddt is 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

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 kVA Rs = 0.0694 Ω UN = 231 V Lσs = 0.391 mH IN = 80.81 A Lμd = 10.269 mH fN = 50 Hz Lμq = 10.05 mH Uf = 171.71 V Rf’´= 0.061035 Ω If = 11.07 A Lσf = 0.773 mH nN = 1500 min−1 J = 0.475 kg m2 TN = 285 Nm p = 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.