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IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 3, MARCH 2017 8101011

Time- and Spatial-Harmonic Content inSynchronous Electrical Machines

Bert Hannon, Peter Sergeant, and Luc Dupré

Electrical Energy Laboratory, Department of Electrical Engineering Students Association,Ghent University, 9000 Ghent, Belgium

The use of power electronics has led to a growing importance of higher time-harmonic content in electrical machines. To gaininsight in phenomena related to these higher harmonics, such as noise and losses, a good understanding of the magnetic field’sharmonic content is mandatory. Moreover, the development of fast and accurate, harmonic-based, analytical models requires aqualitative knowledge of the machine’s time- and spatial-harmonic content. Although the harmonic content of electric machines isan extensively studied topic, previous publications tended to focus on one type of synchronous machines and often did not considerhigher time-harmonic orders. This paper complements the existing theory by providing a more general approach, thereby coveringmachines and operating points that were not covered until now. It considers both three-phase and multi-phase machines with anodd number of phases. The winding distribution can either have an integer or a fractional number of slots per pole and per phase,and higher time-harmonic content is regarded as well. Note that the saturation is neglected. Despite its general validity, this papersucceeds at providing one simple equation to determine the machine’s time-and spatial-harmonic content. Moreover, this paper alsoextensively discusses the physical causes of the harmonic content. The combination of this general validity, the simple result, andthe insight in the physics makes that this paper is a strong tool to both study harmonic-related phenomena in electric machinesand to develop harmonic-based analytical models.

Index Terms— Analytical models, electric machines, harmonic analysis.

I. INTRODUCTION

IN MODERN industry, the effect of harmonic content inelectrical drives is increasingly important. Indeed, electri-

cal motors are often controlled using pulsewidth modulatedvoltage-source inverters, which result in higher time-harmonicorders in the current signal. The rising importance of theseharmonics has led to a growing interest for harmonic-relatedphenomena in electrical machines, such as noise [1], [2] andlosses [3], [4]. To better understand these phenomena, it ismandatory to have a good understanding of which time- andspatial-harmonic combinations exist in the machine and whattheir physical cause is.

On the other hand, a variety of studies and applicationsrequire fast and accurate simulation of the magnetic fieldin an electrical machine. In that light, a lot of research onFourier-based (FB) models has been done lately [5]–[16].Such models analytically compute the machine’s magneticfield by expressing it as a summation of harmonic components.The computational time of these models can be significantlyreduced if the present time- and spatial-harmonic content isknown in advance.

A. Literature

As follows from the above, a preliminary knowledge ofwhich time- and spatial-harmonic combinations exist in anelectric machine’s magnetic field is of great interest. Moreover,it is particularly useful to have physical insight in the origin of

Manuscript received September 18, 2016; revised November 7, 2016and December 4, 2016; accepted December 5, 2016. Date of publicationDecember 7, 2016; date of current version February 15, 2017. Correspondingauthor: B. Hannon (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2016.2637316

the machine’s harmonic content. This is of course not new, andharmonic analysis has been one the most important tools tostudy electrical machines since long before the rise of powerelectronics and FB modeling. It is, therefore, not surprisingthat there are a lot of publications on the topic. However, thegreat majority of these publications uses harmonic analysis ofthe magnetic field as a postprocessing tool, i.e., the machine’smagnetic field is first calculated and then decomposed inits harmonic content to better understand the results of thecalculation [17], [18]. Evidently, these articles give only littleinsight in the harmonic content of electric machines in general,let alone the physical cause for those harmonics.

Publications that do present a broader discussion on theharmonic content of electric machines can mainly be classifiedaccording to the type of machines that are studied and towhether, or not, they consider higher time-harmonic orders.In the following, some of the most interesting publicationsare used to sketch the evolution of the literature on harmoniccontent. Evidently, these publications are only a small portionof the large amount of literature on the topic.

The most basic publications, such as basictextbooks [19], [20], only consider the spatial-harmoniccontent of three-phase machines with an integer q , q beingthe number of slots per pole and per phase. Usually, thesepublications study the magnetomotive force (MMF) to statethat the spatial-harmonic orders (k) have to satisfy

k = p(6c + 1) (1)

with p the number of pole pairs and c an integer.However, as early as the 1950’s, several authors published

on spatial-harmonic content of machines with an arbitrarynumber of phases and higher time-harmonic orders. BothKron [21] and White and Woodson [22] considered such

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8101011 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 3, MARCH 2017

multi-phase machines, and Kron [21] also considered highertime-harmonic orders. However, neither Kron [21] nor Whiteand Woodson [22] presented clear equations to determinewhich harmonic orders are present in the electric machine.Such equations can be found in some textbooks on electricmachines, such as Pyrhönen’s book on the design of electricmachines [23] and Boldea’s work on induction machines [24].None of the latter publications regarded the harmonic contentin machines with a fractional q .

In 1983, Klingshirn [25] presented a relation between theMMF’s spatial-harmonic orders and what he calls orders ofsequence sets. The latter are similar to time-harmonic orders.Although Klingshirn [25] does account for machines withmore than three phases, his work is restricted to diametri-cally wound machines. Latter, Toliyat et al. [26] presented apaper that does consider machines with concentrated windings.However, his study is restricted to three specific cases andhe does not give a preliminary discussion of the presentharmonic orders. Neither Klingshirn [25] nor Toliyat et al. [26]discussed the harmonic orders other than those originatingfrom the machine’s stator-current density. Moreover, none ofboth papers extensively discusses the physical cause of theharmonic orders.

In 2000, Atallah et al. [27] described the current densityon the stator using a Fourier series in his paper on rotorlosses. Although he only considers the fundamental time-harmonic order, he presents a clear equation for the spatial-harmonic orders. The equation is valid for machines withinteger-slot windings and for machines with fractional-slotwindings (FSWs). However, it will be shown in Section IVthat Atallah’s findings can be made more strict for machineswith an even number of slots per period, i.e., for machineswith Ns/τ even. Atallah et al. [27] do not discuss the physicalcauses for the present harmonic orders, neither does he discussthe harmonic content under no-load conditions.

Similar to Atallah et al. [27], Zhu et al. [28] described thestator’s current density in 2004. He does account for highertime-harmonic content, but he only considers machines withthree phases. Moreover, his formula is not generally valid formachines with FSWs.

In 2006, Bianchi et al. [29] and Bianchi and Pré [30]discussed machines with FSWs. He is the first to differentiatebetween machines with Ns/τ even and Ns/τ odd, and healso regards fractional-slot single-layer windings. However, theequations he proposes for spatial-harmonic orders can be mademore strict. Bianchi et al. [29] do not consider higher time-harmonic content.

As recent as 2014 and 2015, Wei et al. [31],Wang et al. [32], and Zhang et al. [33] have presented inter-esting insights on the harmonic content of electric machines.Wei et al. [31] do consider a preliminary study of the time- andspatial-harmonic content. However, his discussion is limitedto machines with an integer number of slots per pole and perphase.

As can be seen from the above, the existing literature doesnot give a complete overview of the harmonic content inelectric machines. Moreover, in most articles, the focus isnot on the harmonic content as such but on describing the

MMF or analyzing the magnetic field. Therefore, this paperaims at complementing the existing literature by discussing theentire harmonic spectrum of machines with fractional slots andby differentiating between machines with an even number ofslots per period and machines with an odd number of slotsper period. Moreover, these equations have to be as generallyvalid as possible. Specifically, all synchronous machines thatare wound using the star-of-slots (SOS) technique, i.e., bothmachines with integer winding and FSW, are considered. Thispaper is limited to unsaturated synchronous machines thatoperate in steady state, but can be extended to account forsaturation and/or asynchronous operation. Apart from provid-ing equations that are very generally valid, this paper also aimsat giving a clear physical explanation for those equations.

B. Assumptions

In order to perform the following study, a number ofassumptions are made. As one of the main goals of thispaper is to reduce the computational time of FB models, thefollowing assumptions are limited to those in the majority ofpublications on analytical models [5]–[16].

A first and very important assumption is that the effect ofthe end windings can be neglected. This is of course only truefor machines with a relatively high length to diameter ratio.It is, however, an assumption that is used by most authorswhen constructing analytical models.

A second assumption is made with regard to the stator-current density. Only balanced current systems with an oddnumber of phases are considered. Note that some systemswith an even number of phases are eligible as well. Indeed, ifthe reduced version of such a system is radially symmetrical,the phases can be grouped in an odd number of neighboringphases [23]. The system is then similar to a system with anodd number of phases.

Third, it is assumed that the winding distribution is definedby the SOS. Although some machines are wound usingother techniques, the large majority of winding topologies isconstructed using this technique.

The effect of saturation on the harmonic content is notregarded in this paper. Although saturation occurs in a lotof machines, it is disregarded in most analytical modelingtechniques. Finally, it was assumed that the rotor of the studiedmachine is rotating at synchronous speed. This reduces thevalidity of this paper to synchronous machines that are in asteady state.

The above assumptions are listed as follows:

1) no end-effects;2) balanced system with an odd number of phases;3) winding distribution according to SOS;4) no saturation;5) synchronous operation.

C. Paper Outline

The discussion is structured as follows. In Section II, theapplied reference system is introduced, and a basic introduc-tion on the time and spatial dependence of the magnetic fieldis presented. Sections III–V discuss the three aspects that

HANNON et al.: TIME- AND SPATIAL-HARMONIC CONTENT IN SYNCHRONOUS ELECTRICAL MACHINES 8101011

Fig. 1. Geometry of a machine with 12 slots and five pole pairs.

determine the harmonic content of the magnetic field in apermanent-magnet synchronous machine (PMSM): the per-manent magnets, the stator-current density, and the machine’sgeometry. The findings from these sections are summarized inSection VI, and general rules for the harmonic combinationsunder no-load, armature reaction, and load conditions arepresented. Section VI also discusses a simple example ofhow the obtained knowledge can be used. The validity of thepresented study for general synchronous machines is discussedin Section VII. Section VIII concludes this paper. Finally, anintroduction to the SOS technique and its slot groups is givenin the appendixes.

Note that Sections III and IV-A present some basic informa-tion. However, this information is required for the discussionin Sections IV-B – VI.

II. TIME- AND SPATIAL-HARMONIC ORDERS

To study the time- and spatial-harmonic content of synchro-nous electrical machines, radial-flux synchronous machineswith surface-mounted permanent magnets are considered.In Section VII, the presented study is extended to synchronousmachines in general.

Fig. 1 shows an example of the studied machines. Thetopology of these machines lends itself to using a cylindricalcoordinate system (r, φ, z) to describe their magnetic field.In the following, the spatial coordinate system is fixed to thestator, and (r, φ, z) is thus a stator reference system. However,apart from a spatial dependence, the magnetic field in themachine also has a time (t) dependence.

Obviously, the machine’s geometry implies a spatial peri-odicity over 2π mechanical radians in the φ-direction. Thisperiod is called the basic spatial period. If the machine isoperated in steady state, it also has a periodicity over time.This period is called the mechanical time period or the basictime period (Tm), it is the time the rotor needs to performone revolution. Usually, the magnetic field is expressed usingauxiliary quantities, such as the magnetic scalar potential (ψ)[5]–[7] or the magnetic vector potential (A) [8]–[16]. If theend effects are neglected, ψ and A are independent of z.Moreover, the magnetic vector potential will then only have a

z-component and thus

A = Az · ez = A · ez. (2)

Because of the above mentioned periodicities, ψ and A canbe expressed using the following Fourier series over time andspace:

ψ(r, φ, t) =∞∑

n=0

∞∑

k=0

ψcosn,k (r) cos (kφ − nωmt)

+ψsinn,k(r) sin (kφ − nωmt) (3a)

A(r, φ, t) =∞∑

n=0

∞∑

k=0

Acosn,k(r) cos (kφ − nωmt)

+ Asinn,k(r) sin (kφ − nωmt) (3b)

or in the exponential notation

ψ(r, φ, t) =∞∑

n=−∞

∞∑

k=−∞ψn,k(r)e

j (kφ−nωmt) (4a)

A(r, φ, t) =∞∑

n=−∞

∞∑

k=−∞An,k(r)e

j (kφ−nωmt). (4b)

In (3) and (4), n is the time-harmonic order and k is the spatial-harmonic order. The machine’s mechanical rotational speed isdenoted as ωm

ωm = 2π

Tm. (5)

In the rest of this paper, the magnetic vector potential willbe regarded, although a completely similar approach couldbe used for the magnetic scalar potential. The exponen-tial notation (4b) will be used to reduce the length of theequations.

Since the magnetic field can be written as a Fourier seriesover time and space, every Fourier coefficient

(An,k(r)

)

depends on both the time-harmonic order n and the spatial-harmonic order k. Therefore, n and k should always beregarded together, such time- and spatial-harmonic combina-tion is referred to as (n, k).

The part of the field related to harmonic combination (n, k)is referred to as the (n, k)-component of the magnetic field.If kφ− nωmt is assumed constant, the rotational speed of thiscomponent can be calculated as

dφ

dt= n

kωm . (6)

This means that the rotational speed depends on the n to kratio. Considering (4), it can easily be seen that both positiveand negative rotational speeds are possible.

Although (4) regards every possible harmonic combination(n, k), the magnetic field in a PMSM does not necessarilycontain all these combinations. The machine’s field, andthereby its harmonic content, is defined by three aspects: thepermanent magnets, the distribution of the current density,and the machine’s geometry. In order to predict the harmoniccontent of the magnetic field, a good understanding of theseaspects is required. Therefore, Sections III–V discuss each ofthe aforementioned aspects and their impact on the harmoniccontent of the machine.


III. PERMANENT MAGNETS

The remanent magnetic induction of the permanent magnets(Brem) has a r - and a φ-components. If expressed in a statorreference system, Brem depends on both space and time andcan, therefore, be written as an exponential Fourier series overspace and time

Brem = Brem,r (r, φ, t) · er + Brem,φ(r, φ, t) · eφ

=∞∑

n=−∞

∞∑

k=−∞Brem,r,n,k(r)e

j (kφ−nωmt) · er

+∞∑

n=−∞

∞∑

k=−∞Brem,φ,n,k(r)e

j (kφ−nωmt) · eφ. (7)

As a source of magnetic flux, the magnets can, evidently,introduce time- and spatial-harmonic combinations in themachine’s magnetic field. They can, however, only introducecombinations that are present in Brem, i.e., Brem,r,n,k �= 0 orBrem,φ,n,k �= 0. For that reason, it is important to understandwhich harmonic combinations are available in the distributionof the remanent magnetic induction. Note that the fact thatthe magnets can only introduce harmonic combinations thatare present in Brem does not imply that only these combi-nations will be present in a no-load situation. As discussedlater, the machine’s geometry can also introduce harmoniccombinations.

As mentioned above, it is assumed that the magnets rotateat the mechanical speed (ωm), evidently the magnets as suchdo not change in time. The remanent magnetic induction willthus only contain harmonic combinations that rotate at ωm

mechanical radians per second. Referring to (6), this impliesthat only harmonic combinations with n = k are eligible.

Second, as can be seen in Fig. 1, the magnet distributioncontains p identical parts along the φ-direction, where p isthe number of pole pairs. These repetitions imply that thespatial period in the φ-direction is p times smaller than thebasic spatial period. This smaller period is referred to as thefundamental spatial period and it equals 2π/p mechanicalradians, which in turn equals 2π electrical radians. To complywith the fundamental spatial period, Brem can only containharmonic orders that are a multiple of p. Note that this demandrequires p identical repetitions of the magnet distribution.If, for example, one of the magnets is demagnetized, it is nolonger valid.

Finally, in most machines, the spatial distribution of theremanent magnetic induction is symmetrical in the φ-directionover half a fundamental period. This symmetry can only bemaintained if, when referred to the fundamental spatial period,there are no even harmonic orders, i.e., k/p is odd. Indeed, asshown in Fig. 2, even harmonic orders do not show symmetrywith respect to the middle of half a period. Note that thedemand for odd spatial-harmonic orders is common but notabsolute, and one could build a machine with asymmetricalmagnets.

As a conclusion, it can be stated that the magnets will onlyintroduce time-harmonic orders, n, wherefore n ∈ hm . Withhm , the set of time-harmonic orders for which Brem,r,n,k �= 0or Brem,φ,n,k �= 0. In a healthy machine, hm will only contain

Fig. 2. Illustration of symmetry in the magnet distribution.

Fig. 3. Balanced five-phase current system.

multiples of p. Due to the assumption of synchronous opera-tion, the magnets will only introduce harmonic combinationsfor which k = n. If the magnets are symmetrical over half aperiod, an extra constraint can be imposed: k/p should thenbe odd.

IV. STATOR-CURRENT DENSITY

Like the magnets, the current density introduces harmoniccombinations. But again, restrictions can be imposed as towhich harmonic combinations are induced. In order to studythese restrictions, the applied current and the spatial distrib-ution of the windings through which these currents flow areregarded separately.

The time-dependent current will determine which time-harmonic orders are introduced by the current density. Thedistribution of the windings, in contrast, has a spatial depen-dence and will determine which spatial-harmonic orders arepresent.

A. Current SystemThe applied current system is a balanced system with an

odd number of evenly distributed phases (m). This means thatthe rotation between neighboring phases is 2π/m electricalradians, as shown in Fig. 3 for a five-phase system.

The current, related to an arbitrary phase with number im ∈[1,m], can be written as a Fourier series over time

I (im ) =∞∑

n=−∞Ine

− j n(ωmt−(im−1) 2π

m

)

. (8)

The current density in the machine can only introduce com-ponents of the magnetic field whose time-harmonic order n


corresponds to a nonzero I (i)n . The following general consid-eration on the time-harmonic content of I (i) can be made.

For the sake of uniformity, the mechanical pulsation hasbeen used in (8). However, the base pulsation of current I (i)

is the electrical pulsation ωe = pωm . This implies that the onlymultiples of p are eligible for n. Similar as the fundamentalspatial period, the fundamental time period (Te) can be definedas the base time period Tm divided by p.

The assumption that the applied system is balanced impliesthat the sum of the current phasors should always equal zero.When referred to the fundamental time period, this means thatthe current does not contain time-harmonic orders that are amultiple of m. This consideration results in n/p �= cm, withc being an integer.

Often the current waveform is assumed to be symmetricalover half a fundamental time period. If that is the case noeven harmonic orders, when referred to the fundamental timeperiod, are present, i.e., n/p is odd.

The above considerations allow stating that the currentdensity will only introduce harmonic combinations with time-harmonic orders that are multiples of p and wherefore n/pis no multiple of m. If the current waveform is symmetricalover half a fundamental time period, n/p has to be odd. Moregenerally it can be stated that, if hc contains the time-harmonicorders that are present in I (i), the current density will onlyintroduce time-harmonic orders, wherefore n ∈ hc.

B. Winding Distribution

As mentioned, the distribution of the windings will deter-mine which spatial-harmonic orders are present in themachine’s magnetic field. There are a great number of pos-sibilities to distribute the windings around the stator surface.However, mostly the so-called SOS technique is used to assignthe slots to one or more phases [23], [30], [34], this techniqueis introduced in Appendix A. Both integer winding and FSWcan be constructed using this technique. In an attempt tobe as general as possible, this paper regards all the windingtopologies that are feasible using the SOS.

As mentioned, the basics of the SOS theory are discussedin Appendix A. An important parameter related to the SOS isthe machine’s period (τ ), calculated as the greatest commondivider of the amount of pole pairs (p) and the number ofslots (Ns ).

In Appendix B, the term slot group has been defined as a setof adjacent stator slots so that, at synchronous operation, themechanical shift between similar slots of different slot groupscorresponds to the time shift of the current densities linked tothese slots. These slot groups are shown in Fig. 4.

It was shown in Appendix B that, depending on whetherNs/τ is odd or even, the magnetic field will be identical butrotated over 2π/mτ or π/mτ mechanical radians after Tm/mτor Tm/2mτ seconds. This can be written mathematically forthe magnetic vector potential

A(r, φ, t0) = A

(r, φ + 2π

ςmτ, t0 + Tm

ςmτ

)(9)

where ς is 1 if Ns/τ is odd and 2 if Ns/τ is even.

Fig. 4. Slot groups in electrical machines. (a) Machine with Ns/τ odd(Ns = 9, p = 4, and τ = 1). (b) Machine with Ns/τ even (Ns = 12, p = 5,and τ = 1).

This time periodicity is not only valid for the completefunction, it is also valid for every separate (n, k)-component ofthe magnetic field. Indeed, according to (6), another harmoniccomponent of the magnetic field can only have the samerotational speed if it has both a different time and a differentspatial-harmonic order. This, in turn, would imply a differentsource term, I (i)n′ .

It can thus be written that

An,k(r)ej (kφ−nωmt0) = An,k(r)e

j(

k(φ+ 2π

ςmτ

)−nωm

(t0+ Tm

ςmτ

))

.

(10)

Knowing that ωm Tm = 2π , the above can be simplified

1 = e j (k−n) 2πςmτ . (11)

With c an integer, this results in

k − n = cςmτ. (12)

Equation (12) imposes a relation between the time- and spatial-harmonic orders. Note that (1) indeed corresponds with (12).More specifically, (1) is the special case where m = 3, and


the machine has an integer amount of slots per pole and perphase, which implies ς = 2 and τ = p. Equation (1) onlyconsiders the fundamental time-harmonic order, i.e., n = p.

As a conclusion, it can be stated that the current densitywill only introduce time-harmonic orders that are present inthe applied currents (n ∈ hc). Due to the distribution ofthe windings, the induced spatial-harmonic orders have tosatisfy (12).

V. MACHINE GEOMETRY

The third aspect that determines the magnetic field’s har-monic spectrum is the geometry. At no-load for example,harmonic combinations, different from the synchronous onesfound in Section III, are present in the magnetic field.

The source of these extra harmonic orders is a variationalong φ of the magnetic permeance. This effect is best knownas the slotting effect and is mostly associated with slottedmachine topologies. The latter is because the amplitude ofthe induced harmonic components depends on the differencein magnetic permeance, the greater this difference the greaterthe amplitude. In slotless machines, where the highly perme-able teeth is replaced with non-magnetic teeth, the differencebetween the permeability of the copper windings and that ofthe non-magnetic teeth is very small. The slotting effect isthen so small that most authors neglect it.

As mentioned, the source of the induced spatial-harmonicorders is a difference in magnetic permeance. The reasonis that such differences introduce a time periodicity, similarto the one in (9). Indeed, under synchronous operation, themachine’s rotor will have rotated over one slot pitch after atime of Tm/Ns seconds. The rotor will then experience thesame stator topology. Under no-load conditions, the magneticfield will then be equal but shifted over one slot pitch. This canbe expressed mathematically in terms of the magnetic vectorpotential

A(r, φ, t0) = A

(r, φ + 2π

Ns, t0 + Tm

Ns

). (13)

As explained in Section IV, this has to be true for every time-and spatial-harmonic combination separately

An,k(r)ej (kφ−nωmt0) = An,k(r)e

j(

k(φ+ 2π

Ns

)−nωm

(t0+ Tm

Ns

))

. (14)

And, again a relation between the spatial- and time-harmonicorders is found

k − n = cNs (15)

where c is an integer. The above mentioned time periodicity isonly introduced due to different magnetic permeances in theφ-direction, consequently changes in the r -direction do notintroduce harmonic combinations.

If the studies machine has teeth with different widths, therewill always be a repetition in the shape of the teeth. The right-hand side of (15) should then be divided by the number ofsubsequent teeth after which this set of teeth is repeated. If thisnumber is defined as Nt , (15) can be rewritten as

k − n = cNs

Nt= cNs,eq. (16)

With Ns,eq the number of repetitions in the shape of the slots,if the teeth are all equal, Nt = 1 and Ns,eq = Ns .

Note that, concerning the armature reaction, the effect of thegeometry is embedded in the winding distribution. Therefore,the geometry will have no further effect on the harmoniccombinations introduced due to the current distribution.

Finally, it should also be noted that differences in themagnetic permeance can also occur on the rotor. However,due to the synchronous rotation of the rotor, these differencesdo not affect the periodicity found in (13).

As a conclusion, it can be stated that, under no-loadconditions, the machine’s geometry will introduce harmoniccombinations, wherefore k−n = cNs,eq. However, in a slotlessmachine, the field components related to combinations forwhich c �= 0 may be considered negligible.

VI. HARMONIC COMBINATIONS

In Sections III–V, the harmonic orders introduced due to themagnets, the current density, and the geometry were discussed.Based on that discussion, it can be concluded that, on the onehand, the source terms, being the permanent magnets and theapplied current density, determine which time-harmonic orderswill be present. On the other hand, the distribution of thewindings and the machine’s geometry determine the presentspatial-harmonic orders.

Practically, no-load, armature reaction, and load conditionsare considered. This section discusses the present harmoniccombinations for each of these conditions based on the find-ings in Sections III-V. This section is concluded with a simpleexample of how the presented results may be used.

A. No Load

Under no-load conditions, the currents in the slots equalzero. This means that the harmonic combinations found inSection IV will not be present. The magnets will introducesynchronous harmonic combinations (n = k) that are presentin the magnet distribution. For every present time-harmonicorder n, the geometry will introduce spatial-harmonic ordersthat satisfy (16). The restriction on the harmonic combinationsunder no-load conditions can thus be summarized as

{n ∈ hm

k − n = cNs,eq.(17)

With c an integer and hm , the time-harmonic orders for whichBrem,r,n,k �= 0 or Brem,φ,n,k �= 0. In a healthy machine,hm can only contain time harmonics that are multiples of p.If the magnets are symmetrical over half a fundamental period,no time-harmonic orders, wherefore n/p is even, are present.

B. Armature Reaction

When Brem = 0, the permanent magnets will not intro-duce any harmonic combinations. The current density willonly introduce the time-harmonic orders that are present inthe applied current. The introduced spatial-harmonic ordersare defined by the distribution of the current density (12).The restrictions on the harmonic combinations can then be


TABLE I

HARMONIC CONTENT OF SYNCHRONOUS ELECTRIC MACHINES

summarized as{

n ∈ hc

k − n = cςmτ.(18)

With c an integer and hc, the time-harmonic orders for whichIn �= 0. hc can only contain multiples of p and will not containany time-harmonic orders for which n/p is a multiple of m.If the currents are symmetrical over half a period, no n valueswherefore n/p is even are present.

C. Load Conditions

The load situation is a superposition of the no-load and thearmature-reaction situations. This implies that all the harmoniccombinations that satisfy either (17) or (18) will be present inthe magnetic field.

The above is summarized in Table I.

D. Examples

To illustrate the applicability of the above theory, twoexamples will be presented in this section. The first, verysimple, example shows how Table I can be used to get abetter understanding of the machine’s physics. Second, theabove theory was used to construct an FB analytical modelof a more complex machine geometry. This model was thenvalidated with finite-element (FE) software to confirm thetheory’s validity. It is shown that applying the above theoryresults in a huge reduction of the FB model’s computationaltime.

The first example evaluates the effect of an increasingnumber of slots per pole and per phase on the harmonic contentof the no-load field using (17).

If q is increased, the number of slots (Ns = 2mpq)increases as well. According to (17), this implies fewer spatial-harmonic content. This was indeed expected, increasing thenumber of slots per pole and per phase is a well-knowntechnique to reduce the harmonic content. Usually only thefundamental time-harmonic order (n = p) is considered. Thisresults in the commonly known spatial-harmonic contents fora machine with three phases and one pole pair

{k = · · · ,−17,−11,−5, 1, 7, 13, 19, . . . if q = 1

k = · · · ,−11, 1, 13, . . . if q = 2.(19)

However, (17) also allows prediction of the spatial-harmoniccontent at higher time-harmonic orders. For example if n = 5,the spatial-harmonic content will be{

k = · · · ,−19,−13,−7,−1, 5, 11, 17, . . . if q = 1

k = · · · ,−19,−7, 5, 17, . . . if q = 2.(20)

Fig. 5. Geometry of a machine with 5 phases, 15 slots and 7 pole pairs.

TABLE II

PARAMETERS OF THE VALIDATED FIVE-PHASE MACHINE

The above shows that the results presented in this section canlead to a better understanding of the machine’s performance.Not only for the fundamental time-harmonic order but alsofor higher harmonic orders. The presented example is a verysimple one, but knowledge of the magnetic field’s harmoniccombinations may also contribute to the understanding of morecomplicated physical phenomena like the production of torqueand torque ripple.

The second example considers a five-phase, outer-rotormachine with 15 slots and 7 pole pairs, as shown in Fig. 5.Note that the yellow cylinder in Fig. 5 is a shielding cylinder,i.e., a conductive sleeve wrapped around the magnets toprotect them from higher harmonics in the magnetic field. Themachine is equipped with a four-layer winding and has a pitchfactor of 1. Its parameters are listed in Table II. By means ofexample, the spatial-harmonics corresponding to n = p = 7,i.e., the fundamental time-harmonic order, and to n = 3 p = 21can be calculated from Table I. In no-load conditions, thefollowing spatial-harmonic orders are obtained:{

k = · · · ,−53,−38,−23,−8, 7, 22, 37, 52, . . . if n = 7

k = · · · ,−54,−39,−24,−9, 6, 21, 36, 51, . . . if n = 21.

(21)


Fig. 6. Tangential component of the magnetic flux density in the center ofthe air gap at armature-reaction conditions.

A similar prediction can be made for armature-reactionconditions{

k = · · · ,−13,−8,−3, 2, 7, 12, 17, 22, · · · if n = 7

k = · · · ,−19,−14,−9,−4, 1, 6, 11, 16, · · · if n = 21.

(22)

The above theory was used to construct a computational-time efficient, i.e., only the time- and spatial-harmonic combi-nations that satisfy Table I are considered, time-dependent FBanalytical model for the five-phase machine of Fig. 5. Assum-ing a sinusoidal current, the resulting tangential componentof the armature-reaction field is compared with the resultsfrom an FE model in Fig. 6. To ensure a good accordance,in the periodic subdomains, 130 spatial-harmonic orders wereconsidered, in the slots, the amount of spatial-harmonic orderwas set to 15 and the amount of time-harmonic orders wasset to 130 as well. The very good accordance betweenthe FE model and the FB model confirms the validity ofTable I. Moreover, compared with a traditional time-dependentFB model, the computational time was reduced from636.02 to 191.27 s. Similar results are obtained for the radialcomponent and for machines in no-load operation. Note thatthe FB model has a higher computational time than onemight expect from a semi-analytical model, this is due tothe fact that it has to consider the field’s time dependence.Indeed, that is the only way to account for the eddy-currentreaction-field of the shielding cylinder [13]. Nevertheless, thecomputational time is still low when compared with that ofthe time-dependent FE model (2734 s) seconds, which has42133 degrees of freedom and was run for eight periods toensure convergence.

Note that the FE model was constructed using the sameassumptions as those in Section I-B.

VII. VALIDITY

In Sections III–VI, a synchronous machine with surface-mounted magnets was considered. However, the results, pre-sented in Section VI, may be applicable more generally.Therefore, this section regards the validity of the performedstudy.

A. Rotor TopologiesAs already mentioned in Section V, the differences in

magnetic permeance between the magnets and the magnetgaps have no effect on the present harmonic orders. This istrue for every difference in magnetic permeance on the rotor.Indeed, such differences do not affect the time periodicityfound in (13). This implies that the results from Section VIare also valid for machines with interior magnets.

Despite an increasing popularity of permanent-magnetexcited machines, the vast majority of synchronous machinesare still excited using electromagnets. Although the excita-tion source is different in such machines, it has the samecharacteristics as regards periodicity. If the excitation currentis constant, the excitation as such does not vary in time.Combined with the previous consideration that differencesin magnetic permeance have no effect, this implies that theobtained results are also valid for machines with a classicalexcitation.

B. Winding TopologiesThe winding topologies considered in Appendixes A and B

are double-layer topologies. However, the SOS tech-nique can also be used to construct single or multilayertopologies [30], [34]. Nevertheless, the findings fromSection VI are still valid. Indeed, switching to a single ormultilayer topology does not change the fact that a number ofslot groups can be defined, resulting in a time periodicity asin (9).

Note that for single-layer topologies, only half of the slotsis considered to construct the SOS [30]. This may affect thenumber of phase zones.

VIII. CONCLUSION

In this paper, the study of the harmonic content of syn-chronous electrical machines has been extended to harmoniccombinations, where both the time- and spatial-harmonics areregarded. This has resulted in general rules to determine,which harmonic combinations are present in the magneticfield, see Section VI. Sections III–V clearly indicate, wherethe different harmonic combinations originate from, this infor-mation can be used to get a better insight when study-ing harmonic-related phenomena in synchronous electricalmachines. The latter was also illustrated in Section VI-D witha simple example.

This paper hereby provides an answer to the need for abetter understanding of the harmonic content in the magneticfield of synchronous machines. Moreover, some very simpleequations to determine the harmonic content of a very broadrange of electric machines were presented.

Although the range of machines to which this paper appliesis very large, induction machines and machines that arenot wound according to the SOS theory may contain otherharmonic combinations. It would, therefore, be interesting toperform a similar study for these machines. Another interestingtopic for future research is the application of the abovefindings in FB models. It is the authors’ conviction that thecomputational time of such models can be drastically reducedby considering the results of this paper.


Fig. 7. SOS for a machine with Ns = 12 and p = 5.

APPENDIX ASTAR-OF-SLOTS

The so-called SOS is a technique that is used to assignthe phases of the applied current system to the slots ofan electrical machine. It is described by a large number ofauthors [23], [30], [34] and can be used for both integralwinding and FSW. Bianchi and Pré [30] have extended thetechnique to determine the winding layout of single layerand multilayer [34] topologies. An extensive description ofthe SOS technique is beyond the scope of this appendix.However, an understanding of the SOS’s basics is requiredfor the discussion in Section IV.

In the first step, the machine’s periodicity (τ ) is calculatedas the greatest common divisor of the number of slots and thenumber of pole pairs

τ = gcd(Ns , p). (23)

Second, a system of Ns/τ phasors, called spokes, with amutual shift of p2π/Ns radians is drawn. This is shownin Fig. 7 for a machine with 12 slots and five pole pairs.An example of such a machine is shown in Fig. 1. Everyspoke now corresponds to a slot in the electrical machine.Logically, the spoke with number i corresponds to the slotwith number i .

The third step is drawing the phase zones. Every phaseis assigned two zones, a positive zone and a negativezone. Each phase zone spans π/m radians. The shiftbetween two phase zones equals the shift between theircorresponding phases, as shown in Fig. 7 for a three-phase system. Consequently, the time shift between the cur-rents linked to the phases of subsequent phase zones is1/ωeπ/m seconds.

The resulting diagram defines one conductor of each coilby linking the spokes, and thereby the slots, to the phases ofthe applied current system. The slot corresponding to the other

conductor of the same coil is defined by the coil throw, whichis calculated as

yq = round

(Ns

2 p

). (24)

The obtained distribution is repeated after Ns/τ slots.The winding distribution obtained from the SOS in Fig. 7

is shown in Fig. 1.

APPENDIX BSLOT GROUPS

In this section, the term slot group is introduced as a numberof subsequent slots so that, under synchronous operation, themechanical shift between two slot groups equals the time shiftof their corresponding current densities. In other words, thetime the rotor needs to rotate from one slot group to the nextequals the time shift of the current densities related to thoseslot groups.

For simplicity reasons, the slot groups are chosen sothat each group is dominated by one phase, this is shownin Fig. 4.

In the following, the mechanical shift and the time shift ofdifferent slot groups are calculated to prove that they indeedcorrespond.

A. Mechanical ShiftTwo cases are regarded: the cases where Ns/τ is odd and

the case where Ns/τ is even.If the number of slots per machine period is odd, every

phase will dominate one slot group per machine period.Indeed, every phase should dominate an equal number ofsimilar slot groups, otherwise the winding distribution cannever be balanced. A phase dominating more than oneslot group, on the other hand, would imply two coincidingspokes in the SOS. This in turn would imply that all ofthe following spokes also coincide with another spoke, thiscan only happen if these spokes belong to another machineperiod. Therefore, a machine with Ns/τ odd contains mτ slotgroups. The number of slots per slot group (Ng) can then becalculated as

Ng = Ns

mτ. (25)

The above is illustrated in Fig. 4(a).If Ns/τ is even, spoke i+Ns/2τ will be opposite to spoke i .

Indeed, the rotation between these spokes is

Ns

2τp

2π

Ns= p

τπ. (26)

Since Ns/τ is even and τ is the greatest common divisor ofNs and p, p/τ has to be odd. This means that spokes i andi + Ns/2τ are indeed opposite.

Because of the fact that this is true for every spoke, everyphase will dominate two similar groups of slots per machineperiod. One due to the spokes in its positive phase zone andone due to the spokes in its negative phase zone, this canbe seen in Fig. 4(b). This implies that the number of slot


groups is now 2mτ , the number of slots in every slot group isthen

Ng = Ns

2mτ. (27)

The mechanical shift between similar slots of subsequent slotgroups can now be calculated as

Ns

ςmτ

2π

Ns= 2π

ςmτ. (28)

The mechanical shift, found in (28), translates to a time shiftwhen divided by the synchronous pulsation

1

ωm

2π

ςmτ= Tm

ςmτ. (29)

B. Time ShiftIn the SOS, the angle between two subsequent slots is

p2π/Ns electrical radians. This implies that in the SOS,the angle between similar slots of subsequent slot groupsis

Ns

ςmτp

2π

Ns= p

2π

ςmτ. (30)

The SOS consists of 2m phase zones with a mutual shift ofπ/m radians, see Appendix A. Knowing this, the number ofphase zones between similar slots of subsequent slot groupscan be calculated as

p 2πςmτπm

= 2 p

ςτ. (31)

The time shift between currents linked to consecutive phasezones in the SOS is π/m electrical radians, see Appendix A.Consequently, the shift in electrical radians between thecurrents linked to similar slots of subsequent slot groupsis

2 p

ςτ

π

m. (32)

Which results in a time shift when divided by the electricalpulsation

2 p

ς

π

mτ

1

ωe= Tm

ςmτ. (33)

This indeed equals (29).Practically, this implies that the rotor will experience

the same current density after Tm/mτ or Tm/2mτ seconds,depending on whether Ns/τ is odd or even. From the statorpoint of view, it means that after Tm/mτ or Tm/2mτ seconds,the armature-reaction field will be identical but shifted over2π/mτ or π/mτ mechanical radians, respectively.

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