Analysis of a Radial Flux Air-Cored Permanent Magnet Machine with a Double-sided Rotor and Non-overlapping Windings

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Double-sided Rotor and

Non-overlapping Windings

PhD – Oral Examination

Peter Jan Randewijk

Stellenbosch University, Faculty of Engineering,

Dep. of Electrical & Electronic Engineering, Electrical Machines Research Group http://research.ee.sun.ac.za/emr/

Supervisor – Prof. M.J. Kamper 17 February 2012

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1 Introduction

2 Subdomain analysis method

3 Winding Configuration

4 Back-EMF Calculation

5 Torque Calculation

6 _Conclusion

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The Radial Flux Air-cored Permanent Magnet (RFAPM) machine is a new type of machine The RFAPM machine is a dual of the Axial Flux Air-cored Permanent Magnet (AFAPM) machine with a double-sided rotor and double layer, non-overlapping, air-cored windings, [Kamper et al, 2008]

The RFAPM machine was first presented by [Randewijk et al, 2007]

The salient features of the RFAPM machine:

cylindrically shaped double-sided rotor air-cored stator

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Rotor Outer Yoke

Rotor Inner Magnets Rotor Inner Yoke

Rotor Backplate Stator Yoke & Backplate

Stator Coils

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The advantages of RFAPM machines over AFAPM machines:

structural integrity of cylindrically shaped rotor yokes is much higher than that of disc shaped rotor yokes

+ [Stegmann & Kamper, 2009] – more than 30 %

lower mass for a similar 6.75 kW machine

The advantages of an air-cored machine:

no stator iron losses

+ higher efficiency

no cogging torque

+ lower wind cut-in speeds

The advantages of non-overlapping windings:

lower end-winding length lower copper losses

+ higher efficiency

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Application fields for RFAPM machines:

large diameter, short “stack” or stator lengths multiple pole machines

ideal for direct drive application

+ small to medium direct drive, wind turbine

generators

Focus of Dissertation:

thorough analytical analysis of the RFAPM machine

reduce blind reliance on FEM

simplify calculation of back-EMF & torque solve the ripple torque riddle

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In order to analytically find the solution to the magnetic field inside the machine, the machine is divided into different 2D regions or subdomains

π p 2πp φ r rn h ℓg ℓg hm hm hy hy kmπp I II III IV V

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+ The RFAPM of [Stegmann & Kamper, 2011]

where used as a Test Case.

Description Symbol Value

Number of pole-pairs p 16

Number of coils per phase q 8

Nominal stator radius rn 232 mm

Active stator/copper length ` 76 mm

Stator coil thickness/height h 10 mm

N48 NdFeB PM thickness/height hm 8,2 mm

Rotor yoke thickness/height hy 8 mm

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The governing equations for the different regions or subdomain:

Regions µ_r Governing equation

I µ_r|_y ∇2A = 0~ II 1 ∇2A = −µ~ ₀∇ × ~M₀ III 1 ∇2A = 0~ IV 1 ∇2A = −µ~ ₀∇ × ~M₀ V µ_r|_y ∇2A = 0~ Assumptions:

+ in the rotor yokes, µr|y is finite

+ in the rotor yokes, µr|y is constant

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With the annulus shaped subdomains and a periodic forcing function of the radially magnetised PMs π p 2πp 3πp 4πp 5πp φ r M0 kmπ_p + where _M~₀₌ ~Brem µ0

the solution to the partial differential equations has a known Fourier series expansion form.

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+ General solution: Az,gen(r, φ) = ∞ X m=1,3,5,... Cmrmp+Dmr−mpcos mpφ + Particular solution: Az,part(r, φ) = ∞ X m=1,3,5,... Gmcos mpφ with Gm =−4rcmBrem_m₂_pπcos mpβ and β = 1 − km 2 π p

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The relationship between the magnetic vector potential, the flux-density and the magnetic field intensity, is given by ~ B = ∇ × ~A and ~ H = B~ µ

Using the boundary condition, A(v)_z =0

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B_r(v)=B(v+1)_r and H_φ(v)=H_φ(v+1)

on the boundaries between two subdomains

The solution for C_m(v) & D_m(v) for each subdomain,

can be obtained from the following boundary matrix,           r_imp r−mp i · · · r_iimp r−mp ii −riimp · · · r_iimp−1 −r_ii−mp−1 −r_iimp−1 · · · r_iiimp · · · r_iiimp−1 · · · ... ... ... ...           ·          CI m DI m CII m DII m CIII m ...          =          0 GII m 0 −GIIm 0 ...         

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with, ri =rn−h₂ − `g− hm− hy rii =rn−h₂ − `g− hm riii =rn−h₂ − `g riv =rn+h₂ +`g rv =rn+h₂ +`g+hm rvi =rn+h₂ +`g+hm+hy

the radii of the boundaries of the different subdomains

C_m(v) & D_m(v)are only solved for, m = 1, 3, 5, . . . 27

This results in 140 values, completely describing the magnetic fields inside the RFAPM machine

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The magnetic vector potential, Az, inside the RFAPM machine 0.21 0.22 0.23 0.24 0.25 x (m) −0.04 −0.02 0.00 0.02 0.04 y (m) -14.4 -14.4 -13.6 -13.6 -13.6 -12.8 -12.8 -12 -12 -11.2 -10.4 -9.6 -8.8-8 -7.2 -6.4 -5.6 -4.8 -4 -3.2 -2.4 -1.6_-0.8 0 0 0 0.8 0.8 1.6 1.6 2.4 2.4 3.2 3.2 4 4 4.8 4.8 5.6 5.6 6.4 6.4 7.2 7.2 8 8 8.8 8.8 9.6 9.6 10.4 10.4 11.2 11.2 12 12 12 12 12.8 12.8 12.8 12.8 13.6 13.6 13.6 13.6 14.4 14.4 -12.8 -9.6 -6.4 -3.2 +0.0 +3.2 +6.4 +9.6 +12.8 Magnetic vector potential, Az| PM (mWb/m)

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And from ~B = ∇ × ~A, the magnitude of magnetic flux-density inside the RFAPM machine

0.21 0.22 0.23 0.24 0.25 x (m) −0.04 −0.02 0.00 0.02 0.04 y (m) 0.00 0.32 0.64 0.96 1.28 1.60 1.92 2.24 2.56 Flux-density ,B (T)

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The radial flux-density distribution in the centre of the PM – & stator regions

+ The analytical solution is only calculated up to

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The azimuthal flux-density distribution in the centre of the PM – & stator regions

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We are however only interested in the magnetic vector potential & the radial flux-density in the stator region, i.e. region III, where

AIII_z (r, φ) = ∞ X

m=1,3,5,...

C_mIIIrmp+D_mIIIr−mpcos mpφ and BIII_r (r, φ) = −1 r · ∞ X m=1,3,5,...

mp(C_mIIIrmp+D_mIIIr−mp)sin mpφ

with CIII

m and DIIIm the coefficients obtained from

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The magnetic vector potential allows us to calculate:

the flux-linkage from the stator’s winding distribution

and hence the back-EMF induced

The radial flux-density allows us to calculate:

the developed torque from the stator’s current density distribution

To try and simplify the calculation of the back-EMF & torque further

and due to the fast solution provided by the analytical solution, 6 s vs. 300 s for the FEM solution,

we varied the pole arc-width of the PMs to see if we could obtain a near sinusoidal radial

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The peak, fundamental and %THD of the radial

flux-density distribution as a function of km, in the

centre of the stator winding:

0.5 0.6 0.7 0.8 0.9 1.0

Pole embracing factor, km

0.0 0.2 0.4 0.6 0.8 1.0 Radial flux-density ,B r (T) Br Br,1 %THD(Br) 0 4 8 12 16 20 % THD (B r )

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The classical overlapping winding configuration with the number of coils, q, equal to the number of pole pares, p ωmech Br,1cos(pφ − pωmecht − γ) dr ds φ N S S N N S S N 3 2 1 1’2’3’ a a′ 3 2 1 1’2’3’ a a′ 3 2 1 1’2’3’ b b′ 3 2 1 1’2’3’ b b′ 3 3’2’1’ 1 2 c′ c 3 3’2’1’ 1 2 c′ c 2∆ q π p γ p π q πq

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The winding distribution for phase a, n φ qN 2∆ a a′ a a′ a a′ 2∆ q π q 2πq

and can be given by the follow Fourier series,

n_a|O(φ) =

∞ X

n=1

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with

b_n|n,O =−2qN

π · kw,n|O

where kw,n, the winding factor, is defined as

k_w,n|O =k_w,pitch,n|O_{· k}_w,slot,n|O with the pitch factor

k_w,pitch,n|O =1,0

and the “virtual” slot – or coil-side width factor,

k_w,slot,n|O = sin(n∆)

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Uses single layer non-overlapping windings with the number of coils, q, half the number of pole pares, p ωmech Br,1cos(pφ − pωmecht − γ) dr ds φ N S S N N S S N 4 3 2 1 1’2’3’4’ a a′ 4 3 2 1 1’2’3’4’ c c′ 4 3 2 1 1’2’3’4’ b b′ 2∆ q π p γ p π Q πQ

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φ qN 2∆ a a′ a a′ 2∆ q π Q 2πq

Surprisingly, the only difference in die winding distribution for the single layer non-overlapping windings, is in the pitch factor

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Uses double layer non-overlapping windings to maximise the flux-linkage per coil, but with the same number of coils. . .

ωmech Br,1cos(pφ − pωmecht − γ) dr ds φ N S S N N S S N 4 3 2 1 1’2’3’4’ a a′ 4 3 2 1 1’2’3’4’ c c′ 4 3 2 1 1’2’3’4’ b b′ 2∆ q π p γ p 2π Q

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φ qN 2∆ a a′ a a′ 2∆ q 2(π−3∆) Q 2πq

Again the only difference in die winding

distribution for the double layer non-overlapping windings, is in the pitch factor

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Using Stokes’ integral theorem and the winding distribution function,

the amplitude of the flux-linkage in the stator windings, for the overlapping winding

configuration, can be calculated as follows,

Λ_a,b,c|O =₋1 h Z r_n+h₂ rn−h2 2q`N a ∞ X m=1

k_w,m|OaIII_m|A_zdr with

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Assuming that the average flux-linkage is the

same as that occurring at r=rnand

Using only the fundamental component, i.e. m=1, the amplitude of the flux-linkage can be approximated as,

Λ_a,b,c|O ≈ 2qr_apn`Nk_w,1|OBr,1

This will result in the amplitude for the back-EMF, E_a,b,c|O ≈ −2qr_an`Nkw,1|OBr,1ωmech

For the single – & double layer, non-overlapping windings,

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due to the fact that number of coils, q is half the number of pole pairs, p,

only the 2nd order space harmonics of the winding distribution, “n = 2m”, will rotate at the same speed as the space harmonics of the magnetic vector potential (and that of the flux-density)

Only considering “m = 1” for the flux-density distribution,

it implies we also only need to consider “n = 2” for the winding distribution,

so that for the non-overlapping winding configurations:

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Λ_a,b,c|I,II _≈ 2qrn`N

ap kw,2|I,IIBr,1 and

E_a,b,c|I,II ≈ −2qr_an`Nkw,2|I,IIBr,1ωmech

This allows us to define a voltage constant, kE

similar to a “traditiona” brushless DC machine ˆ

E_f =k_Eωmech

with

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The back-EMF comparison for the double layer, non-overlapping winding configuration

0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00 11.25 12.50 Time, t (ms) −800 −600 −400 −200 0 200 400 600 800 Bac k EMF (V) ea(t) eb(t) ec(t) ea(t) (Approx.) eb(t) (Approx.) ec(t) (Approx.) ea(t) (Maxwell® 2D) eb(t) (Maxwell® 2D) ec(t) (Maxwell® 2D)

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Average Torque

From the equivalent circuit diagram of the RFAPM machine, − Eaf + Rs Ia Ls ωmech Tmech Pmech + − Va

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and with the back-EMF and the phase current regulated to be in phase, Tmech= 3E_ω afIa mech = 3 ˆEafˆIa 2ωmech

where ˆE_af and ˆIaare the peak values of the

back-EMF and phase current respectively, the steady state torque developed by the machine can be written as

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with the torque constant, kT, equal to the voltage constant,

kT =kE = 2qr_an`NkwBˆr,1 and the stator current space vector

ˆIs = 3₂ˆIa

Thus with the stator current space vector current regulated to be in-phase with the back-EMF space vector,

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the torque will be equal to the magnitude of the stator current space vector and

the RFAPM, regardless of the type of winding configuration used,

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Ripple Torque

With the RFAPM machine being an air-cored machine, i.e. no cogging torque

The Lorentz method provides a quick and easy way in which the torque can be calculated From Lorentz’s law, the volumetric force density can be calculated as

~fv = ~J × ~B

=_−Jz(φ)BφIII(r, φ)~ar +Jz(φ)BIIIr (r, φ)~aφ =fr~ar +fφ~aφ

Only the azimuthal component of the volumetric

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From the winding distribution, assuming balanced sinusoidal currents,

the three-phase current density distribution, for the overlapping winding configuration,

J_z|O(φ) =            −3qI_ar_np_hπN ∞ X n=1 k_w,n|Osin nqφ + ωt, n = 3k−2 −3qI_ar_np_hπN ∞ X n=2 k_w,n|Osin nqφ − ωt, n = 3k−1 with k ∈ N1

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For the overlapping winding configuration,

The 3k−2 harmonics, 1, 4, 7, . . . “moves right” The 3k−1 harmonics, 2, 5, 8, . . . “moves left” ~J φ qNi 2arnh∆ a a′ a a′ a a′ b′ b b′ b b′ b c′ c c′ c c′ c 2∆ q π q 2πq

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The three-phase current density distribution, for both the non-overlapping winding configuration, will be almost similar

J_z|I,II(φ) =            −3qI_ar_np_hπN ∞ X n=1 k_w,n|I,IIsin nqφ + ωt, n = 3k−1 −3qI_ar_np_hπN ∞ X n=2 k_w,n|I,IIsin nqφ − ωt, n = 3k−2 with k ∈ N1

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For the single layer, non-overlapping winding configuration,

The 3k−1 harmonics, 2, 5, 8, . . . “moves right” The 3k−2 harmonics, 1, 4, 7, . . . “moves left” ~J φ qNi 2arnh∆ a a′ c c′ b b′ a a′ 2∆ q π Q Qπ

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Similar for the double layer, non-overlapping winding configuration,

The 3k−1 harmonics, 2, 5, 8, . . . “moves right” The 3k−2 harmonics, 1, 4, 7, . . . “moves left” ~J φ qNi 2arnh∆ a a′ c c′ b b′ a a′ 2∆ q 2(π−3∆) Q 2π q

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The netto electromagnetic or mechanical torque, can then be calculated as

~Tmech = Z v ~_{r ×~f}_v_dv = Z vrJz(φ)B III r (r, φ)~azdv

with ~r the radial vector from the centre of the machine, resulting in Tmech=` Z r_n+h₂ rn−h2 Z 2π 0 r 2_J z(φ)BrIII(r, φ) dφ dr

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Radial flux-density distribution in the centre – as well as on the inner – and outer edge of the stator −11.2500−8.4375 −5.6250 −2.8125 0.0000 2.8125 5.6250 8.4375 11.2500 Angle, φ (◦₎ −1.0 −0.5 0.0 0.5 1.0 Radial flux-density ,B r (T) Br|rn−h/2 Br|rn Br|rn+h/2 Br|rn−h/2(Maxwell® 2D) Br|rn(Maxwell® 2D) Br|rn+h/2(Maxwell® 2D)

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Tmech =−3q`NI_ar p nh ∞ X m=1,5,7,... kw,2mRmSm with Rm = Z rn+h₂ rn−h₂ r2_·mp(CmIIIrmp+DmIIIr−mp) r dr =mp" C III mrmp+2 mp + 2 − DIII mr−mp+2 mp − 2 #rn+h₂ rn−h₂ and

Sm =(cos (m−1)pωmecht for m=6k+1, k ∈ N0

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A comparison between the different torque calculation methods 0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00 11.25 12.50 Time, t (ms) 200 205 210 215 220 225 Mechanical Torque ,Tmech (Nm)

Using the Lorentz Method Using Maxwell® 2D

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A comparison between the different torque

calculation methods with hy =20 mm to

eliminate rotor yoke saturation

0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00 11.25 12.50 Time, t (ms) 200 205 210 215 220 225 Mechanical Torque ,Tmech (Nm)

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A comparison between the different torque

calculation methods, also with Hc = _µBrem

rrecoil 0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00 11.25 12.50 Time, t (ms) 200 205 210 215 220 225 Mechanical Torque ,Tmech (Nm)

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A comparison between the Maxwell_{2D and}R SEMFEM, [Gerber et al, 2010]

0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00 11.25 12.50 Time, t [ms] 200 205 210 215 220 225 Tmech [Nm] Using Maxwell® 2D Using the Lorentz Method Using SEMFEM

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The harmonic spectrum for the different torque calculation methods: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Harmonic Number 0.00 0.05 0.10 0.15 0.20 0.25 Tmech |h Tmech |a ve . × 100 % Analytical Maxwell® 2D Maxwell® 2D (Tweaked) SEMFEM

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A comparison between the different torque calculation methods for [Groenewald 2011]

0.000 0.104 0.208 0.312 0.417 0.521 0.625 0.729 0.833 0.938 1.042 Time, t (ms) 54.0 54.5 55.0 55.5 56.0 56.5 57.0 57.5 58.0 Mechanical Torque ,Tmech (Nm)

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Sub-domain analysis method provides quick and fairly accurate results for the electromagnetic fields inside a RFAPM machine

The calculated value of the radial flux density is 3 % higher than that using FEM

due to the fact that the recoil permeability of the PMs used is not unity and

and to a lesser extent, due to the saturation of the iron core yokes

The analytical solution provided a practical way in which to optimise the pole arc-width of the PMs that would produce a quasi sinusoidal radial flux-density distribution in the centre of the stator windings

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From the analytical solution for the magnetic vector potential, the flux-linkage and back-EMF could be calculated

It was shown that with sinusoidal radial

flux-density distribution in the centre of the stator

windings a voltage constant, kE, for the RFAPM

machine could be defined, similar to a “normal” brushless DC machine

From the voltage constant and with the phase current regulated to be in phase with the back-EMF of the RFAPM, a torque constant,

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the Lorenz method provides a quick and accurate way in which to calculate the ripple torque component of a RFAPM machine – 6 s. vs. ≈15 min.

The 3,3 % error in the torque calculation, compared to FEM, could largely be contributed to the difference in the radial flux-density

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M.J. Kamper, R-J. Wang and F.G. Rossouw,

Analysis and Performance of Axial Flux Permanent-Magnet Machine With Air-Cored Non-overlapping Concentrated Stator Windings,

IEEE Transactions on Industry Applications, May. 2008, p 1495-1504

P.J. Randewijk, M.J. Kamper and R-J. Wang,

Analysis and Performance Evaluation of Radial Flux Air-Cored Permanent Magnet Machines with Concentrated Coils,

Power Electronics and Drives Systems Conference, PEDS 2007

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J.A. Stegmann

Design and Analysis Aspects of Radial Flux Air-cored Permanent Magnet Wind Generator System for Direct Battery Charging Applications

Masters Thesis, Stellenbosch University 2010

J.A. Stegmann and M.J. Kamper

Design aspects of medium power double rotor radial flux air-cored PM wind generators

Energy Conversion Congress and Exposition, ECCE 2009

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J.A. Stegmann and M.J. Kamper

Design Aspects of Double-Sided Rotor Radial Flux Air-Cored Permanent-Magnet Wind Generator,

IEEE Transactions on Industry Applications, Feb. 2011, p 767–778

S. Gerber, J.M. Strauss and P.J. Randewijk

Evaluation of a hybrid finite element analysis package featuring dual air-gap elements,

Electrical Machines, International Conference on,, ICEM-2010

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D. Groenewald

Evaluation of a radial flux air-cored permanent magnet machine drive with manual transmission drivetrain for electric vehicles,