02-04-2012, 12:00 PM
Concept of sinusoidal distributed windings
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Electrical machines are designed in such a manner that the flux density distribution in the
airgap due to a single phase winding is approximately sinusoidal. This appendix aims to make
plausible the reason for this and the way in which this is realized. In this context the so-called
sinusoidally distributed winding concept will be discussed.
Figure A.1 represents an ITF based transformer or IRTF based electrical machine with a
finite airgap g. A two-phase representation is shown with two n1 turn stator phase windings.
The windings which carry the currents i1α, i1β respectively, are shown symbolically. This
implies that the winding symbol shown on the airgap circumference represents the locations of
the majority windings in each case, not the actual distribution, as will be discussed shortly. If
we consider the α winding initially, i.e. we only excite this winding with a current i1α, then the
aim is to arrange the winding distribution of this phase in such a manner that the flux density in
the airgap can be represented as B1α = ˆBα cos ξ. Similarly, if we only excite the β winding
with a current i1β, a sinusoidal variation of the flux density should appear which is of the form
B1β = ˆBβ sin ξ. The relationship between phase currents and peak flux density values is of the
form B1α = Ci1α, B1β = Ci1β where C is a constant to be defined shortly. In space vector
terms the following relationships hold
i1 = i1α + ji1β (A.1a)
B
1 = ˆB1α + j ˆB1β (A.1b)
Given that the current and flux density components are linked by a constant C, it is important to
ensure that the following relationship holds, namely
B
1 = Ci1 (A.2)
If for example the current is of the formi1 = ˆi1ejρ then the flux density should be of the
form B1 = Cˆi1ejρ for any value of ρ and values ofˆi1 which fall within the linear operating
range of the machine. The space vector components are in this case of the form i1α =ˆi1 cos ρ,
i1β =ˆi1 sin ρ. If we assume that the flux density distributions are indeed sinusoidal then the
resultant flux density Bres in the airgap will be the sum of the contributions of both phases
namely
Bres (ξ) = Cˆi 1cos ρ
ˆB
1α
cos ξ + Cˆi 1sin ρ
ˆB
1β
sin ξ (A.3)
328 FUNDAMENTALS OF ELECTRICAL DRIVES
Figure A.1. Simplified ITF model, with finite airgap, no secondary winding shown
Expression (A.3) can also be written as Bres = Cˆi1 cos (ξ − ρ) which means that the resultant
airgap flux density is again a sinusoidal waveform with its peak amplitude (for this example)
at ξ = ρ, which is precisely the value which should appear in the event that expression (A.2)
is used directly. It is instructive to consider the case where ρ = ωst, which implies that the
currents iα, iβ are sinusoidal waveforms with a frequency of ωs. Under these circumstances the
location within the airgap where the resultant flux density is at its maximum is equal to ξ = ωst.
A traveling wave exists in the airgap in this case, which has a rotational speed of ωsrad/s.
Having established the importance of realizing a sinusoidal flux distribution in the airgap for
each phase we will now examine how the distribution of the windings affects this goal.
For this purpose it is instructive to consider the relationship between the flux density in the
airgap at locations ξ, ξ +Δξ with the aid of figure A.2. If we consider a loop formed by the
two ‘contour’ sections and the flux density values at locations ξ, ξ +Δξ, then it is instructive
to examine the sum of the magnetic potentials along the loop and the corresponding MMF
enclosed by this loop. The MMF enclosed by the loop is taken to be of the form Nξi, where Nξ
represents all or part of the α phase winding and i the phase current. The magnetic potentials
in the ‘red’ contour part of the loop are zero because the magnetic material is assumed to be
magnetically ideal (zero magnetic potential). The remaining magnetic potential contributions
when we traverse the loop in the anti-clockwise direction must be equal to the enclosed MMF
which leads to
g
μo
B (ξ) − g
μo
B (ξ +Δξ) = Nξi (A.4)
Expression (A.4) can also be rewritten in a more convenient form by introducing the variable
n(ξ) = Nξ
Δξ which represents the phase winding distribution per radian. Use of this variable
Appendix A: Concept of sinusoidal distributed windings 329
Figure A.2. Sectional view of phase winding and enlarged airgap
with equation (A.4) gives
B (ξ +Δξ) − B (ξ)
Δξ
= − g
μo
n(ξ)i (A.5)
which can be further developed by imposing the conditionΔξ → 0 which allows equation (A.5)
to be written as
dB (ξ)
dξ
= − g
μo
n (ξ) i (A.6)
The left hand side of equation (A.6) represents the gradient of the flux density with respect to
ξ. An important observation of equation (A.6) is that a change in flux density in the airgap is
linked to the presence of a non-zero n(ξ)i term, hence we are able to construct the flux density
in the airgap if we know (or choose) the winding distribution n(ξ) and phase current. Vice versa
we can determine the required winding distribution needed to arrive at for example a sinusoidal
flux density distribution.
A second condition must also be considered when constructing the flux density plot around
the entire airgap namely
π
−π
B (ξ) dξ = 0 (A.7)
Equation (A.7) basically states that the flux density versus angle ξ distribution along the entire
airgap of the machine cannot contain an non-zero average component. Two examples are
considered below which demonstrate the use of equations (A.6) and (A.7). The first example
as shown in figure A.3 shows the winding distribution n(ξ) which corresponds to a so-called
‘concentrated’ winding. This means that the entire number of N turns of the phase winding
are concentrated in a single slot (per winding half) with width Δξ, hence Nξ = N. The corresponding
flux density distribution is in this case trapezoidal and not sinusoidal as required.
The second example given by figure A.4 shows a distributed phase winding as often used in
practical three-phase machines. In this case the phase winding is split into three parts (and three
slots (per winding half), spaced λ rad apart) hence, Nξ = N
3 . The total number of windings
of the phase is again equal to N. The flux density plot which corresponds with the distributed
330 FUNDAMENTALS OF ELECTRICAL DRIVES
Figure A.3. Example: concentrated winding, Nξ = N
Figure A.4. Example: distributed winding,Nξ = N
3
winding is a step forward in terms of representing a sinusoidal function. The ideal case would
according to equation (A.6) require a n(ξ)i representation of the form
n (ξ) i =
g
μo
ˆB
sin (ξ) (A.8)
in which ˆB represents the peak value of the desired flux density function B (ξ) = ˆB cos (ξ).
Equation (A.8) shows that the winding distribution needs to be sinusoidal. The practical implementation
of equation (A.8) would require a large number of slots with varying number of turns
placed in each slot. This is not realistic given the need to typically house three phase windings,
hence in practice the three slot distribution shown in figure A.4 is normally used and provides a
flux density versus angle distribution which is sufficiently sinusoidal.
Appendix A: Concept of sinusoidal distributed windings 331
In conclusion it is important to consider the relationship between phase flux-linkage and
circuit flux values. The phase circuit flux (for the α phase) is of the form
φmα =
π2
−π2
B (ξ) dξ (A.9)
which for a concentrated winding corresponds to a flux-linkage value ψ1α = Nφmα. If a
distributed winding is used then not all the circuit flux is linked with all the distributed winding
components in which case the flux-linkage is given as ψ1α = Neff φmα, where Neff represents
the ‘effective’ number of turns.