Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
The harmonic analysis of machine excitation
C G Hodge MSc FREng CEng FIMarEST1
F Eastham DSc FRSE FREng CEng FIEE2
A C Smith DSc CEng FIET3
1
BMT Defence Services Ltd, UK
2
University of Bath, UK
3
University of Manchester, UK
Synopsis
Electrical rotating machines operate through the interaction of electrical currents with
magnetic flux that may itself be created through other electrical currents. In both cases the
magnetic flux present in the air gap of the machine is a crucial component of its overall
performance dictating, among other things, torque (and hence machine size) and noise and
vibration. Both these particular aspects are of prime importance to designers of electrical
propulsion motors for naval applications. The prime aspect of the magnetic flux in the air gap
is its shape or profile which has a direct bearing on both torque density and torque pulsations.
This paper describes a method to analyse a machine winding through application of Fourier
Series and the Discrete Fourier Transform to determine the efficacy of the winding with regard
to machine excitation.
INTRODUCTION
The increasingly common application of electrical propulsion to naval warships is creating a focus on machine
performance in terms of both torque density and noise and vibration; the former for reasons of minimising the
impact on volume and mass (and thereby easing the naval architecture problem) and the latter in order to
improve, or at least preserve, the military effectiveness of the platform. However these two aspects: noise and
vibration and torque density, compete within the design because the straight forward route to increased torque
density is a combination of both increased flux levels and adoption of squarer, more trapezoidal, air gap flux
profiles, both of which increase the presence of non-uniform torque (and hence noise and vibration).
As a result the need to carefully design the magnetic performance of the machine is a crucial aspect which
requires to be carefully balanced in order to optimise the overall machine performance.
There are four main classes of electrical machines:
1.
2.
3.
4.
Commutated DC Machines
Synchronous AC Machines
Permanent Magnet AC Machines (with a sub-set of brushless DC machines acknowledged).
Induction Machines.
The first of these has unique, and interesting, aspects of its noise and vibration performance which would of
itself offer a useful avenue of investigation and could provide a worthwhile paper in the future. The rotor
structure of the next two (Synchronous and Permanent Magnet AC machines) both experience a magnetic field
with two key aspects:
1.
2.
It is largely steady (sometimes described as DC).
It has a fixed pole number.
This simplifies the magnetic design of the machine in general and of the stator windings in particular because
only fields with the same pole number as the rotor can interact to create torque. The situation with regard to the
fourth type, induction machines, where the rotor windings are most commonly short circuited as a standard
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
squirrel cage is quite different. In these machines currents of any stator pole number will reflect into the rotor
and will interact to create torque which for non-fundamental currents may create noise and vibration. The
analysis in this paper will now concentrate on induction machines with squirrel cage windings.
THE IDEAL AIR GAP FLUX PROFILE
One aspect needs to be clearly understood with respect to the field in an electrical machine. There is only one
field and it is comprised of two components, considering an induction machine:
1.
2.
The direct field – resulting from the rotor currents which form the machine excitation.
The quadrature or reaction field resulting from the load currents flowing in the stator windings of the
machine.
Whilst these form a single unified field it is still valuable to consider them separately when assessing torque
production. Currents cannot interact with the field they themselves create to produce torque. (This would be the
electrical equivalent of a person pulling themselves up by their own boot straps.) Rather it is the cross products
that develop torque:
1.
2.
The stator currents with the direct field arising from the rotor currents.
The rotor currents with the reaction field arising from the stator currents.
These by necessity are equal and opposite. If it is desired to derive the torque in the machine from the single
uniform air gap flux then the methods of Maxwell Stresses or Co-Energy can be applied.
It is a common, and inherently true, assertion that a sinusoidal flux density distribution across the pole face will
provide a quiet machine. What is not clear from this statement – at least to the non-specialist – is what is meant
by the pole face and especially so in the case of an induction machine. The pole face in question is most clearly
seen in a DC machine: it is stationary, fixed to the stator and is clearly a pole face. A synchronous machine
(permanent magnet or separately excited) is in many aspects an inside out DC machine and again the pole face
is visible but now it is in motion being attached to the rotor. In the case of the induction machine the pole face
does not exists as a physical object in either the stator or the rotor. It is rather a “pole” of flux rotating around
the air gap at synchronous speed relative to the stator and at slip speed relative to the rotor. In this case however
the situation is greatly simplified by the fact that stator currents lead to rotor currents and therefore a sinusoidal
distribution of flux in the air gap must derive from a sinusoidal stator flux profile. The presence of a sinusoidal
variation of flux density across the pole face prevents sharp changes in force production and avoids oscillatory
torques.
PRACTICAL WINDINGS
If a sinusoidal variation of air gap flux is required and if this is created by the sum of the fluxes arising from
separate phase windings then, when of the same frequency, sinusoids add to sinusoids to create new sinusoids,
the air gap flux density distribution from each phase alone must also be sinusoidal. One method where this could
be closely approximated is to use a set of windings in each phase where the number of turns (coils) in each pair
of slots varies sinusoidally around the air gap of the machine. This can be done and does produce a quiet
machine but it is a difficult winding to create, and install, and is inefficient with respect to machine volume as
the number of conductors in each slot (after aggregating all the phases) is not uniform. An alternative, method,
is to wind the machine in phase bands.
Single Layer Windings
An example of a 24 slot single layer three phase winding is shown in Figure 1. It can be seen that the pole pitch
and the coil pitch is 12 slots.
Figure 1: Three Phase Single Layer 24 Slot Winding Distribution
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
Consider just the red phase as is shown in Figure 2:
Figure 2: Extracted Red Phase Winding Distribution from Figure 1.
The red phase conductor count in full, over 360 electrical degrees, for this phase is listed in Table 1:
0 to
Conductors
1
1
1
1
0
0
0
0
0
0
0
0
180 O
Slot No
1
2
3
4
5
6
7
8
9
10
11
12
Conductors
-1
-1
-1
-1
0
0
0
0
0
0
0
0
Slot No
13
14
15
26
17
18
19
20
21
22
23
24
180 O to
360 O
Table 1: Conductor Distribution for the Single Layer Red Phase Winding in Figure 2
It is clear from this conductor distribution alone that the winding does not approach the ideal of a sinusoidal
conductor distribution; rather the distribution is simple patches of current – the phase bands. One way that this
can be improved is by moving to a two layer winding and allowing some overlap between the two layers so that
the number of phase conductors in each slot is stepped around the stator circumference. An example of a 24 slot
two layer three phase winding is shown in Figure 3.
Figure 3: Three Phase Two Layer 5/6 Short Chorded 24 Slot Winding Distribution
It can be seen that the pole pitch is 12 slots but, now, the coil pitch 10 slots hence it is called a 5/6 short chorded
winding or occasionally it is said to be short pitched by 2 slots or 1/6. Consider just the red phase as shown in
Figure 4:
Figure 4: Extracted Red Phase Winding Distribution from Figure 3
Which is equivalent to the distribution shown in Figure 5.
Figure 5: Re-Drawn Extracted Red Phase Winding Distribution from Figure 3
The distribution of conductors shown in Figure 5 illustrates the main advantage of a two layer winding – the
distribution is a good deal closer to a sinusoidal distribution than that in a single layer. Indeed the two layer
short pitched winding, of which Figure 3 is just one example, is now all but universally used for “commodity”
induction motors. The designer still has some flexibility in terms of how much overlap is used and the impact of
varying overlap can be considered by using the method of harmonic analysis of winding distributions and air
gap flux profiles that will now be described.
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
The red phase conductor count in full, over 360 electrical degrees, for the red phase two layer distribnution of
Figure 5 is listed in Table 2:
0 to
Conductors
1
1
2
2
1
1
0
0
0
0
0
0
180 O
Slot No
1
2
3
4
5
6
7
8
9
10
11
12
Conductors
-1
-1
-2
-2
-1
-1
0
0
0
0
0
0
Slot No
13
14
15
26
17
18
19
20
21
22
23
24
180 O to
360 O
Table 2:Conductor Distribution for the Two Layer Red Phase Winding of Figure 5
HARMONIC ANALYSIS OF AIR GAP FLUX
Given that a sinusoidal variation of air gap flux density is required then harmonic analysis becomes of immense
value and the harmonic flux density that it extracts become a useful vantage point for assessing the efficacy of a
machine design. But harmonic analysis can provide more information of equal value, it can indicate the
presence of negative sequence (reverse rotating), and less commonly zero sequence fields, and can also indicate
how well each harmonic couples into the machine (harmonic winding factors).
This paper will now compare the single and two layer windings, however to do so two aspects need adjustment.
To compare the harmonics that will be extracted, the total number of conductors needs to be the same between
the two distributions (so that harmonic amplitudes are comparable) and the magnetic axes of the two coils need
to be aligned (so that harmonic phases are comparable). As previously defined the single layer winding has a
total of four coils and eight conductors, whereas the two layer winding has a total eight coils and sixteen
conductors. In addition, by starting the red phase winding in both cases in slot one, the magnetic axes are
displaced by one slot between them. The Table, and Figures, that follow define conductor distribution for the
single layer and double layer windings that have the same number of turns and conductors and a matched coil
angular alignment:
Single Layer
0
2
2
2
2
0
0
0
0
0
0
0
0
2
2
2
2
0
0
0
0
0
0
0
Double Layer
1
1
2
2
1
1
0
0
0
0
0
0
1
1
2
2
1
1
0
0
0
0
0
0
Table 3: Revised Winding Distribution for Single and Double Layer Harmonic Analysis
Figure 6 : Single Layer Red Phase Winding Distribution Used For Harmonic Analysis
Figure 7: Double Layer Red Phase Winding Distribution Used For Harmonic Analysis
Winding Distribution Harmonics
The conductor distributions in Table 3 can be plotted and this is shown in Figure 8 and Figure 9 each of which,
in addition to the winding distribution (in red), also show the fundamental of that winding distribution (in blue),
the third harmonic (in green), the fifth (in cyan), the seventh (in mauve) and the sum of these four harmonics (in
black).
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
Figure 8: Plot of Winding Distribution of the Single Layer Winding in Table 1 including Fundamental, Third,
Fifth and Seventh Harmonic
Figure 9: Plot of Winding Distribution of the Double Layer Winding in Table 1 including Fundamental, Third,
Fifth and Seventh Harmonic
At this stage only the “shape” of the winding has been considered, nevertheless it is even now, clear that the
distribution of a two layer short chorded winding is significantly better than that for a single layer winding; it
approaches the ideal of a sinusoidal conductor distribution more closely containing lower amplitude harmonic
spatial components. The next consideration is how a coil winding distribution relates to the air gap flux resulting
from its excitation alone, this condition corresponds with zero rotor current.
Coil Winding Distribution to Coil Loading to Flux
For the red phase, as currently being considered alone, the coil loading is simply the winding (in this analysis
defined as a distribution as illustrated in Figure 9) multiplied by the current it carries. For other phases to the one
so far considered their will be a time phase between them (that ideally would match the spatial phase between
the phases). It is not possible to incorporate this time phase information together with the spatial phase in the
expression of the winding distribution in a simple or indeed meaningful way. The spatial angle between phases
can be incorporated as a simple complex multiplier of the form 𝑒 𝑗𝜓 where 𝜓 is the physical angular
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
displacement between them (in electrical degrees). But this is not possible for the time dependant excitation,
hence the analysis is best retained on a single phase basis until there is a need to move away from coil loading.
The coil loading once calculated can be considered as a surface current density and this can be used to develop
the air gap flux through integration:
Assume that the fundamental of the surface current density 𝐽 𝜃 varies sinusoidally with stator angle. I.e. it has
the form:
𝐽 𝜃 = 𝐽𝑠𝑖𝑛 𝜃
Where:
𝐽
is the peak current density.
𝜃
is the electrical angle measured across the pole (a pole pair would occupy an arc of 2𝜋).
This is clearly not the case for the largely square profile of winding as shown in Figure 9 but it is true for each
of the harmonics once extracted.
Each harmonic will create more pole pairs than the fundamental. Referring to Figure 8 and Figure 9it is clear
that across the winding distributions as shown (which covers 2𝜋 radians electrically) the fundamental creates
two poles (1 pole pair) and the third harmonic creates six poles (three pole pairs) and so on. The effective pole
pitch is important for the generation of flux, the longer the pole pitch the greater the resultant flux, thus
harmonics become gradually less important as their order increases due to the consequent reduction in harmonic
pole pitch. (And a smaller pole pitch also leads to a greater proportion of leakage flux, again reducing their
importance in excitation).
Then for each harmonic of number 𝑚 across the stator segment covered by the harmonic pole:
𝐽𝑚 𝑠 = 𝐽𝑚 𝑠𝑖𝑛
2𝑚𝜋𝑠
𝑝
Where:
𝑚
is harmonic number
𝐽𝑚
is the peak harmonic current of order m.
𝑠
is displacement measured across the pole (in mechanical radians).
𝑝
is the displacement at the extent of a full pole-pair measured around the air gap (in mechanical measure).
Then the peak harmonic air gap flux, 𝐵𝑔𝑚 , is given by:
𝜇0 𝐽𝑚
𝑔
𝐵𝑔𝑚 =
𝑝
4𝑚
𝑠𝑖𝑛
0
2𝜋𝑠
𝑝𝜇0 𝐽𝑚
𝑑𝑠 =
𝑝
2𝑚𝜋𝑔
Where:
𝑔
is the size of the air gap
And then:
𝐵 𝑠 = 𝐵𝑔𝑚 𝑐𝑜𝑠
2𝜋𝑚𝑠
𝑝
And it can be seen that the flux is inversely proportion to the harmonic number, m. Having worked out the air
gap flux due to a sinusoidal current distribution (as implied by the harmonics of any given winding distribution)
it is useful to return to consider how the air gap flux builds up in a single coil (i.e. one not sinusoidally
distributed). The resultant flux from a single coil is trapezoidal; this is because the iron in the stator and rotor
has a very high relative permeability and so the flux travels across the low permeability air gap in straight lines
perpendicular to the air gap, i.e along radii. This is a useful result as it allows the effect of additional conductors
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
to be added into the winding – as has already been done at Figure 8 and Figure 9 without losing the fidelity of
their final air gap flux. The shape of the air gap flux then becomes the integral of the winding distribution
around the air gap over the pole pair, with the constant of integration chosen such that the flux distribution has a
zero mean, and this has been done for the same windings and is shown in Figure 10 and Figure 11.
Figure 10: Air Gap Flux for Winding Distribution of Single Layer Winding Distribution in Table 3
Figure 11: Air Gap Flux for Winding Distribution of Double Layer Winding Distribution in Table 3Figure 9
Once more, in addition to the winding’s flux profile (in red) Figure 10 and Figure 11 also show the fundamental
of that winding distribution (in blue), the third harmonic (in green), the fifth (in cyan), the seventh (in mauve)
and the sum of these four harmonics (in black). It should be noted however that if the DFT is to be used to
extract the harmonics then the flux profile must be defined by more than one point per slot in order to avoid
aliasing in the DFT output, albeit this error is small if uncorrected. 1
1.
The harmonic amplitudes quoted in this paper are actually derived by integrating the winding
harmonics and are identical with those derived from a traditional analysis using harmonic winding factors.
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
Note that there is an inherent assumption here that the conductors form point currents, this simplification is
relatively unimportant when it is the shape of the flux profile that is desired, and in any case it is capable of
straight forward extension to physical conductors if thought necessary.
It is clear that all the flux harmonics are reduced between the single and double layer windings. Indeed the fifth
and seventh are barely distinguishable in Figure 11 and so Figure 12 shows an expanded section of Figure 11 to
enable their inspection.
Figure 12: Expanded Section of Figure 11 Showing Detail of Fifth and Seventh Harmonics of Double Layer
Winding in Table 3
THE IMPACT OF EXCITATION HARMONICS
To create torque the excitation field must link with the reaction field that has the same pole number. For the
fundamental this is relatively straight forward and in the case of a synchronous or permanent magnet machine is
an inherent par of the design. The excitation harmonics, as discussed above, in this case will not create torque as
they do not match the pole number of the excitation. They will however have the capacity to induce internal
vibration (due to the forces they create which whilst cancelling around the full stator circumference will have
local presence) and also contribute to the creation of stray losses.
In the case of the induction machine the situation is more complicated, here the rotor is capable of supporting
any number of pole pairs compatible with its multi-phase nature (a rotor of “m” short circuited bars will have
“m” independent phase currents and “m” current loops).
Induction Machines with Pure Sinusoidal Excitation
Even with a pure sinusoidal excitation voltage the presence of winding (and hence air gap flux) harmonics are
important and can create torque pulsations. In the example used previously, the “standard” three phase two layer
chorded winding, it is clear that on a single phase basis there are significant third, fifth and seventh harmonics.
Although the third harmonic cancels between the full set of the three phases, the fifth and seventh do not. The
fifth harmonic produces a ten pole (five pole pair) winding with a strong negative sequence (i.e. reverse
rotating) while the seventh produces a fourteen pole (seven pole pair) winding with a strong positive sequence
(i.e. forward rotating). The Annex gives a mathematical derivation of rotating flux fields from coil distributions
and current loading.
When the machine is rotating in its nominal forward direction the fifth harmonic of the winding is essentially
being plugged and it will create vibration as the machine runs up to speed but this will reduce significantly as it
nears synchronous speed due to the dependence of the effective rotor impedance on slip frequency. The seventh
harmonic however is forward rotating and it is possible for the machine to “stall” during run up and end up
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
running stably at one seventh of the expected speed. It is major challenge for an induction machine designer to
minimise the impact of the third and fifth spatial winding harmonics and indeed this is an additional reason for
the use of short pitching – or chording – in addition to minimising noise and vibration by more nearly
approximating to a sinusoidal distribution of excitation conductors.
For example the magnitude of the first four harmonics for the single and two layer windings already considered
are given in Table 4. Together with the same harmonic comparisons for the air gap flux in Table 5.
Winding
Fundamental
Third
Fifth
Seventh
Ninth
Eleventh
Single Layer Winding
1.277
0.871
0.274
0.210
0.361
0.168
2 Layer Winding
1.233
0.616
0.071
0.054
0.255
0.162
Table 4: Comparison of Winding Harmonics for the Single and Double Layer Windings of Table 3
Flux
Fundamental
Third
Fifth
Seventh
Ninth
Eleventh
Single Layer Winding
4.891
1.112
0.21
0.115
0.154
0.059
2 Layer Winding
4.725
0.787
0.053
0.03
0.109
0.057
Table 5: Comparison of Air Gap Flux Harmonics for Single and Double Layer Windings of Table 3
The windings used for the comparisons in Table 4 and Table 5 are the simple three phase single layer 24 slot
winding and the three phase two layer 5/6 short chorded 24 slot winding of Figure 6 and Figure 7 for which the
total conductors and coils of these two windings are the same. It should also be noted that both the third and the
ninth harmonics cancel in these balanced three phase windings. Note that a traditional analysis by winding
factors would yield negative values for the harmonic flux amplitudes at the 7th, 9th and 11th harmonics for the
single layer harmonic fluxes in Table 5 this is present in the data as extracted via the DFT but as a phase shift of
180o.
Hence for normal design-speed operation the rotor is running super-synchronously (generating) for all forward
positive sequence harmonics (7, 13 and 17) and backwards (plugging) for the remainder (5,11 and 19).
Considering these fields in isolation, would show that they would all produce negative torque. But due to the
differing rotational speed of the space harmonic MMFs these will actually cyclically modulate the fundamental
torque, at one point adding and another subtracting. Therefore they are a source of torque pulsations.
Induction Machines with Harmonic Excitation
Induction Machines supplied from power electronic converters will have a complex frequency spectrum of
supply harmonics. The derivation of harmonic torques is, as with the process described above for the case of
sinusoidal excitation, relatively straight forward, if somewhat laborious, a full treatment is beyond the scope of
this paper but a summary follows.
Once a non sinusoidal excitation supply is present then the result is a mixture of winding spatial and converter
time harmonics and the rotor, being a squirrel cage, will respond to all of these. In similar fashion to the space
harmonics. the important time harmonics in the supply are of numbers 5,7,11,13 (again similar to the space
harmonics triplens cancel for a three-phase winding).
But unlike the case for the space harmonics the airgap synchronous speed of the MMFs associated with the time
harmonics is also is also scaled. Importantly; whereas the scaling of the synchronous speed was a reduction for
the space harmonics (because the pole number increased); now with the time harmonics the scaling is the exact
inverse (the supply frequency is increased). Hence whereas the interaction of the fundamental excitation with
the space harmonics produced differing rotational speeds of the harmonic flux waves, now the time harmonic of
the supply can recover the inherent speed reduction implicit in the space harmonic. This can produce a constant
torque that will either add or subtract to that of the fundamental rather than modulating it. For example: consider
an Nth time harmonic exciting a machine with a two pole fundamental winding (as has been so far considered):
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
The Nth space harmonic of the winding has N pole pairs and, for a balanced three phase supply, will produce
either a positive or a negative sequence field.
The speed of the rotating MMF associated with this space harmonic for fundamental excitation f1 is:
𝜔𝑠1 =
2𝜋𝑓1
𝑁
The frequency of the Nth harmonic is simply:
𝑓𝑁 = 𝑁𝑓1
From which the rotational speed of the Nth MMF space harmonic excited by an Nth time harmonic is:
𝜔𝑠𝑁 =
2𝜋𝑁𝑓1
= 2𝜋𝑓1
𝑁
Which is the same as the fundamental field from the fundamental frequency; hence this will add to the torque
developed by the machine.
Other interactions are also possible, and where the space harmonic and the time harmonic do not share the same
harmonic number then the situation is similar to that described for pure sinusoidal excitation. Due to the
differing rotational speed of the space harmonic MMFs these will modulate the fundamental torque and they are
a further source of torque pulsations.
CONCLUSION
A method of analysing a machine’s performance through a harmonic analysis of its excitation has been
described. The method as so far described in this paper is capable of significant extension and this is being
investigated. In particular its application to machines being fed by power electronic converters and the influence
of harmonics due to slotting (permeance harmonics) is being considered.
Nevertheless the method, even as described here, offers useful insights to machine performance and the
fundamental causes of machine torque pulsations and noise and vibration.
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
ANNEX
ROTATING FIELDS FROM MACHINE WINDINGS
Single Phase
Let a sinusoidal winding distribution be defined as:
𝑁 𝜃 = 𝑁𝑐𝑜𝑠 𝜃
And alternatively:
𝑁 𝑗𝜃
𝑒 + 𝑒 −𝑗𝜃
2
𝑁 𝜃 =
Let a time varying current exciting this winding be defined as:
𝐼 𝑡 = 𝐼𝑐𝑜𝑠 𝜔𝑡
And alternatively:
𝐼 𝑡 =
𝐼 𝑗𝜔𝑡
𝑒
+ 𝑒 −𝑗𝜔𝑡
2
Then, using the first definitions, the coil loading is:
𝑁𝐼 𝑡, 𝜃 = 𝑁𝐼𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠 𝜔𝑡
Which is sinusoidally alternating standing wave of flux sinusoidally distributed around the stator air gap.
Returning to the second definitions, the coil loading is alternatively:
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼 𝑗𝜔𝑡
𝑒
+ 𝑒 −𝑗𝜔𝑡
4
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼 𝑗𝜔𝑡 𝑗𝜃
𝑒 𝑒 + 𝑒 𝑗𝜔𝑡 𝑒 −𝑗𝜃 + 𝑒 −𝑗𝜔𝑡 𝑒 𝑗𝜃 + 𝑒 −𝑗𝜔𝑡 𝑒 −𝑗𝜃
4
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼 𝑗
𝑒
4
𝜔𝑡 +𝜃
+ 𝑒𝑗
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼 𝑗
𝑒
4
𝜔𝑡 +𝜃
+ 𝑒 −𝑗
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼
𝑐𝑜𝑠 𝜔𝑡 + 𝜃 + 𝑐𝑜𝑠 𝜔𝑡 − 𝜃
2
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼
𝑁𝐼
𝑐𝑜𝑠 𝜔𝑡 + 𝜃 +
𝑐𝑜𝑠 𝜔𝑡 − 𝜃
2
2
𝑒 𝑗𝜃 + 𝑒 −𝑗𝜃
𝜔𝑡 −𝜃
+ 𝑒 −𝑗
𝜔𝑡 −𝜃
+ 𝑒 −𝑗
𝜔𝑡 +𝜃
+ 𝑒𝑗
𝜔𝑡 −𝜃
+ 𝑒 −𝑗
𝜔𝑡 −𝜃
𝜔𝑡 +𝜃
Which is a pair of constant half amplitude travelling waves: 𝑁𝐼𝑐𝑜𝑠 𝜔𝑡 − 𝜃 is a forward travelling wave and
𝑁𝐼𝑐𝑜𝑠 𝜔𝑡 + 𝜃 is a reverse travelling wave.
Three Phase
Let a balanced three phase set of sinusoidal winding distributions be defined as:
𝑁𝑅 𝜃 =
𝑁 𝑗𝜃
𝑒 + 𝑒 −𝑗𝜃
2
𝑁𝑌 𝜃 =
𝑁 𝑗
𝑒
2
𝜃 −2𝜋 3
+ 𝑒 −𝑗
𝜃 −2𝜋 3
𝑁𝐵 𝜃 =
𝑁 𝑗
𝑒
2
𝜃 +2𝜋 3
+ 𝑒 −𝑗
𝜃 +2𝜋 3
Let a set of time varying current exciting this winding be defined as:
𝐼𝑅 𝑡 =
𝐼 𝑗𝜔𝑡
𝑒
+ 𝑒 −𝑗𝜔𝑡
2
Paper presented at the INEC 2012 conference held 15-17 May in Edinburgh.
𝐼𝑌 𝑡 =
𝐼 𝑗
𝑒
2
𝜃 −2𝜋 3
+ 𝑒 −𝑗
𝜃 −2𝜋 3
𝐼𝐵 𝑡 =
𝐼 𝑗
𝑒
2
𝜃 +2𝜋 3
+ 𝑒 −𝑗
𝜃+2𝜋 3
Then the coil loading is:
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼
4
𝑒 𝑗𝜔𝑡 + 𝑒 −𝑗𝜔𝑡
+ 𝑒𝑗
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼 𝑡, 𝜃 =
𝑁𝐼 𝑡, 𝜃 =
𝜔𝑡 +2𝜋 3
𝜔𝑡 −2𝜋 3
𝑒 𝑗𝜃 + 𝑒 −𝑗𝜃 + 𝑒 𝑗
+ 𝑒 −𝑗
𝜔𝑡 +2𝜋 3
𝑒𝑗
𝑁𝐼
𝑐𝑜𝑠 𝜔𝑡 − 𝜃 + 𝑐𝑜𝑠 𝜔𝑡 + 𝜃
2
+ 𝑐𝑜𝑠 𝜔𝑡 − 2𝜋 3 − 𝜃 − 2𝜋 3
+ 𝑐𝑜𝑠 𝜔𝑡 + 2𝜋 3 − 𝜃 + 2𝜋 3
+ 𝑒 −𝑗
𝜃 +2𝜋 3
𝜔𝑡 −2𝜋 3
+ 𝑒 −𝑗
𝑒𝑗
𝜃 −2𝜋 3
+ 𝑒 −𝑗
𝜃 −2𝜋 3
𝜃 +2𝜋 3
+ 𝑐𝑜𝑠 𝜔𝑡 − 2𝜋 3 + 𝜃 − 2𝜋 3
+ 𝑐𝑜𝑠 𝜔𝑡 + 2𝜋 3 + 𝜃 + 2𝜋 3
𝑁𝐼
𝑐𝑜𝑠 𝜔𝑡 − 𝜃 + 𝑐𝑜𝑠 𝜔𝑡 + 𝜃 + 𝑐𝑜𝑠 𝜔𝑡 − 𝜃 + 𝑐𝑜𝑠 𝜔𝑡 + 𝜃 − 4𝜋 3
2
+ 𝑐𝑜𝑠 𝜔𝑡 − 𝜃 + 𝑐𝑜𝑠 𝜔𝑡 + 𝜃 + 4𝜋 3
𝑁𝐼
3𝑐𝑜𝑠 𝜔𝑡 − 𝜃
2
+ 𝑐𝑜𝑠 𝜔𝑡 + 𝜃 + 𝑐𝑜𝑠 𝜔𝑡 + 𝜃 − 4𝜋 3 + 𝑐𝑜𝑠 𝜔𝑡 + 𝜃 + 4𝜋 3
3
𝑁𝐼 𝑡, 𝜃 = 𝑁𝐼 𝑐𝑜𝑠 𝜔𝑡 − 𝜃
2
Which is a constant 150% amplitude forward travelling wave.