Bearingless PM synchronous machine with zerosequence current driven star pointconnected active magnetic thrust bearing
 Authors: Dietz D.^{1}, Binder A.^{1}
 Affiliations:
 Institute of Electrical Energy Conversion – Darmstadt, Technical University
 Issue: Vol 4, No 3 (2018)
 Pages: 525
 Section: Review
 URL: https://transsyst.ru/transsyst/article/view/10422
 DOI: https://doi.org/10.17816/transsyst201843525
 Cite item
Abstract
Common cylindrical bearingless drives require a separate thrust bearing, which is fed by a DC supply. Here, a technique is presented, which enables the feeding of the thrust bearing by an artificially generated zerosequence current between the two star points of the two parallel windings in the bearingless PM synchronous machine. This way, no additional DC supply for an axial active magnetic bearing is needed. It is replaced by two threephase inverters as stator winding supply, which are needed in any case to generate torque and lateral rotor force in the motor. This examination explains the technique of adapting the electric potential of the star points in two threephase windings of the motor. The focus is on the determination of the operating area (maximum zerosequence current and band width). It is constrained by the bearingless motor due to torque and lateral force ripple as well as additional eddy current losses. On the other hand, the DC link voltage and the modulation degree of the inverter for simultaneous motor operation as well as the bearing inductance limit the system dynamic. It is shown that the proposed technique is applicable for a modulation degree < 0.866, taking into account that other constraints by the bearingless machine and the inverter are mainly noncritical.
Full Text
INTRODUCTION
Active magnetic selfbearing motors, often referred to as bearingless motors (BM), combine two functions in a single device: the torque generation by an electric machine and the suspension force generation by an active magnetic bearing (AMB) [1, 2, 3]. BMs can be categorized mainly into two groups: First and most prominent, into motors, which generate lateral rotor forces but require a separate axial AMB [4–7]. Second, into motors which do not require an additional axial AMB. Among these, there are BMs which do not need an axial position control due to a disklike and thereby selfstabilizing rotor [8–12]. Other topologies can actively generate axial rotor force by a conical rotor [13, 14] or a chessboard structure on the rotor surface [15]. However, the complexity in terms of manufacturing effort and control disable these interesting solutions from industrial use. For lower power classes (< 500 W) also topologies are available, which passively stabilize the axial rotor position by means of several permanent magnet (PM) layers [16].
The predominant field of application for AMBsuspended drives is the use as highspeed drive because AMBs inherently mitigate the friction losses and enable the rotor rotation around its inertia axis to suppress vibration forces [1, 2, 17]. As such, small rotor diameters are necessary to keep the mechanical stress in the rotor at a suitable level. However, for these rotors, commonly cylindrical, a separate AMB for axial position control, a socalled thrust AMB, is required. Often the thrust bearing is realized parallel to a radial AMB as combined AMB [5] on the nondrive end (NDE) of the shaft, whereas the BM is mounted on the drive end (DE) in order to achieve a short axial length. But even if two BMs as two halfmotors are used [18], an additional axial AMB is required. In any case, this axial AMB is usually fed by a DC chopper, which is costly.
Here, a technique is presented to avoid the additional DC supply. Therefore, it is made use of the fact that BMs typically are equipped with two starconnected winding systems. That means all the six motor terminals are used to generate torque, radial and axial force at the same time. In [19] it is shown that the electric scalar potential between the two star points Z_{A} and Z_{B} can be artificially adapted in order to generate a controllable current between Z_{A} and Z_{B}. Altogether, the proposed technique is beneficial for highspeed bearingless motors with cylindrical rotors. A schematic overview of the drive system is given in Fig. 1a.
Fig. 1. Drive components (schematic) (1) Position and rotor angle sensors, (2) Axial part of combined AMB, (3) Radial part of combined AMB, (4) PM with magnetization direction, (5) Turbocompressor wheel, (6) Safety bearings at drive and nondrive end (DE / NDE) a; simplified electric circuit of the bearingless machine and the axial AMB b
In the section “The bearingless PM synchronous machine” the bearingless machine is presented which the novel technique will be applied to. This is followed by the explanation of generating artificially a zerosequence current system. The focus of the article is on the last two sections where the constraints inherently given by the motor and by the inverter are explained.
THE BEARINGLESS PM SYNCHRONOUS MACHINE
A prototype machine with a double threephase winding was built which is similar to that in Fig. 1. In this prototype the thrust bearing is conventionally fed by a DC supply. It is not possible to apply the presented technique since the star points are insulated and not accessible. The function of this machine is, however, introduced shortly because the demonstrated principle refers to this machine topology. In [4, 20, 21] the winding topology as well as measuring results are presented in detail. Its main parameters are listed in Table 1. It is important to note that the technique is applicable for every bearingless machine which exhibits two threephase windings fed by the same inverter.
Table 1. Motor specifications
Rated speed / min^{1}  n_{N}  60000 
Rated torque / mNm  M_{N}  105 
Rated lateral force / N  F_{N}  8.2 
Rated phase voltage / V  U_{s,N}  42 
Rated phase current for torque / A  I_{cw,N}  3.18 
Rated phase current for lateral force / A  I_{ccw,N}  2.26 
Rated total phase current / A  I_{s,N}  3.90 
Rated current for axial force / A  I_{0,N}  0.9 
Stator bore outer/inner diameter / mm  d_{s,o}/d_{s,i}  70/35 
Stator stack length / mm  l_{Fe}  40 
Bandage thickness / mm  h_{b}  1.5 
Magnet height / mm  h_{PM}  2.75 
Mechanical air gap (d_{s,i} – d_{r,o})/2 / mm  δ  1.0 
Rotor mass / g  m_{r}  800 
Number of turns per phase (6 phases)  N_{s,6}  22 
Number of pole pairs (torque/suspension winding)  p/p_{sus}  1/2 
In this machine a combined twolayer winding replaces the common drive and suspension windings. When accordingly fed it is possible to generate a field wave of pole count 2p = 2, equal to the rotor field pole count, and a field wave of pole count 2p_{sus} = 2p + 2 = 4 simultaneously. In interaction with the rotor field, the 2ppole field wave generates tangential force, whereas the 2p_{sus}pole field wave generates lateral force for rotor suspension. Starting from a conventional fourpole threephase winding, this is possible if the winding per phase (e.g. phase U) is separated into two spatially opposed coil groups (e.g. U_{A} and U_{B}). Each of these coil groups is now fed by a separate phase yielding six phases consisting of the phase belt sequence +U_{A}, +W_{B}, +V_{A}, +U_{B}, +W_{A}, +V_{B}. The phase belts indexed by A (B) are referred to as threephase system A (B). They are both starconnected, yielding the two starpoints Z_{A} and Z_{B}. If the two systems are fed in phase a counterclockwise rotating 2p_{sus}pole field wave occurs so that the phase sequence is reversed (i_{ccw}) for clockwise field wave rotation. If they are fed in phase opposition a clockwise rotating 2ppole field wave occurs by i_{cw}. In practice, both field waves are needed with clockwise rotation. So a superposition yields an elliptical current space vector orbit per threephase system A and B consisting theoretically of two symmetrical threephase current space vectors with reversed rotation [4, 20, 21]. This principle is used to realize the supply of the separate thrust bearing (inductance L_{AMB}) by an artificially generated current i_{0} between the star points Z_{A} and Z_{B}. This is shown in Fig. 1b where U_{DC} is the DC link voltage of the inverter.
GENERATION OF THE REQUIRED ZEROSEQUENCE VOLTAGE
The proposed technique relies on a conventional space vector pulse width modulation (SVPWM) [3]. That means, that for each of the elliptical threephase current systems I_{A} and I_{B} a current controller determines a certain voltage space vector in the statorfixed coordinate systems (α_{A}β_{A} and α_{B}β_{B}). This voltage requirement depends solely on the torque and radial suspension force requirement, whereas the axial force requirement is introduced at a later state. The amplitude of the applied phase voltages ${\hat{u}}_{A,s}$ and ${\hat{u}}_{B,s}$ which in the case of a symmetrical voltage system is equal to the length of the voltage space vectors u_{A} and u_{B} is limited by the hexagon in Fig. 2a. In this case, due to the superposition of the clockwise and counterclockwise voltage system, the main axes of the ellipses determine the inverter voltage rating. These main axes are determined by the algebraic sum of the required clockwise and counterclockwise rotating voltage space vectors for torque and lateral force generation. However, since the 2ppole rotor field does not induce into the imaginary 2p_{sus}pole suspension winding, the voltage trajectory is only slightly elliptical, so that mainly the backEMF of the 2ppole rotor field determines the voltage requirement as in common rotating field machines.
According to Fig. 2a the inverter states 0, …, 7 are related to the phase terminal electric potentials φ_{U}, φ_{V} and φ_{W}. They can take the discrete values U_{DC}/2 (U_{DC}/2) when the related highside switches are turned on (off). As the voltage space vector moves through the sectors I, …, VI (Fig. 2a) the adjoining inverter states are realized for a calculated time t_{0}, … t_{7}. For the traditional symmetrical SVPWM the calculation of t_{0}, … t_{7} is well explained in literature [3, 22]. This technique is commonly used for threephase inverters when operated in fieldoriented control. It is important to note that only the time spans t_{1}, t_{2} and t_{passive} are mandatory for the torque and lateral force generation. The position of the time spans as well as the composition of t_{passive} within one switching period, however, can be arbitrarily chosen. This is made use of for the here presented technique. The independent control of the zerosequence voltage and the phase voltages by means of extended Park and Clarke transformations is discussed in in detail in [19]. The equivalent circuit of the current loop including the motor windings A and B and the axial AMB and defining the orientation of i_{0} and u_{0} is depicted in Fig. 2b. According to that the Ohmicinductive voltage drop u_{0} over the thrust AMB is dependent on the difference in electric potential between phase terminals φ_{U,A}, φ_{V,A}, φ_{W,A} and φ_{U,B}, φ_{V,B}, φ_{W,B}. That is, every time a difference in potential between the star points occurs, a current is flowing if the star points are connected. This is inherently the case if the active voltage vectors in the two systems A and B are different (t_{1,A} ≠ t_{1,B} and t_{2,A} ≠ t_{2,B}).
Fig. 2. Switching states (circled) and voltage space vector trajectories in the statorfixed αβreference frame related to the voltage limit a; and equivalent circuit for the zerosequence current component i0 b
However, the overlapping time spans where a difference in potential is present can be influenced by variation of the time spans t_{0,A}, t_{7,A}, t_{0,B} and t_{7,B}. Aside from the dependency of the difference in the star point potential also the relation between motor and AMB impedance as voltage divider k_{eq} (1) affects the maximum voltage over the terminals of the thrust AMB. Obviously the AMB impedance should be high in relation to the motor zerosequence impedance for a dynamic current response. On the other hand, the overall impedance should be small for the same reason. The voltage at the AMB can be calculated according to (2) and can be written as (3), where ${\overline{){\overline{)u}}_{0}}}_{{T}_{SW}}$ is the average voltage over one switching period T_{sw}.
${u}_{0}~{k}_{eq}=\frac{3*(\omega {L}_{AMB,h}+\omega {L}_{AMB,\sigma}+{R}_{AMB})}{3*(\omega {L}_{AMB,h}+\omega {L}_{AMB,\sigma}+{R}_{AMB})+2*(\omega {L}_{BM,\sigma ,0}+{R}_{BM,s}}$ (1)
${\overline{){\overline{)u}}_{0}}}_{{T}_{SW}}=\frac{{k}_{eq}}{{T}_{sw}}\underset{0}{\overset{{T}_{sw}}{\int}}(\frac{{\phi}_{U,A}\left(t\right)+{\phi}_{V,A}\left(t\right)+{\phi}_{W,A}\left(t\right)}{3}\frac{{\phi}_{U,B}\left(t\right)+{\phi}_{V,B}\left(t\right)+{\phi}_{W,B}\left(t\right)}{3}dt$ (2)
${{\overline{u}}_{\text{0}}}_{{T}_{\text{sw}}}=\frac{{U}_{\text{DC}}\cdot {k}_{\text{eq}}}{2\cdot {T}_{\text{sw}}}\cdot (\underset{\begin{array}{l}\text{Fixedtimespansbydifferentactive}\\ \text{voltagevectorsinsystemAandB}\end{array}}{\underset{\u23df}{\frac{{t}_{\text{2,A}}{t}_{\text{1,A}}}{3}+\frac{{t}_{\text{1,B}}{t}_{\text{2,B}}}{3}}}+\underset{{t}_{\text{z,AMB}}}{\underset{\u23df}{\underset{\begin{array}{l}\text{Controllabletimespansforartificial}\\ \text{zeroseqeuncevoltagegeneration}\end{array}}{\underset{\u23df}{{t}_{\text{7,A}}{t}_{\text{0,A}}+{t}_{\text{0,B}}{t}_{\text{7,B}}}}}})$ (3)
${t}_{\text{passive,A}}={t}_{\text{0,A}}+{t}_{\text{7,A}}={T}_{\text{sw}}{t}_{\text{active,A}};\text{}{t}_{\text{passive,B}}={t}_{\text{0,B}}+{t}_{\text{7,B}}={T}_{\text{sw}}{t}_{\text{active,B}}$ (4)
The time spans t_{1,A}, t_{2,A}, t_{1,B} and t_{2,B} are known since they are previously calculated. Therefore, up to now the parameters {t_{z,AMB}, t_{passive,A}, t_{passive,B}} are known (3, 4) whereas the unknown parameters are {t_{0,A}, t_{7,A}, t_{0,B}, t_{7,B}}. To solve this underdetermined system another condition has to be introduced. It is selected in a way that the overlapping time span t_{7} – t_{0} is proportional (t_{0} → t_{0,lin}, t_{7} → t_{7,lin}) to the relation between the time span for the passive voltage vectors of one system and the total time span for passive voltage vectors. From that system (5) results, which can be solved by applying Cramer’s rule. However, (5) yields solutions which can be both positive and negative. Therefore, the solution space must be limited to (6), resulting in a nonlinear switching behavior.
$\left(\begin{array}{cccc}1& 1& 0& 0\\ 0& 0& 1& 1\\ 1& 1& 1& 1\\ {t}_{\text{passive,B}}& {t}_{\text{passive,B}}& {t}_{\text{passive,A}}& {t}_{\text{passive,A}}\end{array}\right)\cdot \left(\begin{array}{c}{t}_{\text{0,A,lin}}\\ {t}_{\text{7,A,lin}}\\ {t}_{\text{0,B,lin}}\\ {t}_{\text{7,B,lin}}\end{array}\right)=\left(\begin{array}{c}{t}_{\text{passive,A}}\\ {t}_{\text{passive,B}}\\ {t}_{\text{z,AMB}}\\ 0\end{array}\right)$ (5)
$\{\begin{array}{cc}{t}_{\text{0,A}}=\frac{1}{2}\cdot ({t}_{\text{passive,A}}\frac{{t}_{\text{passive,B}}}{{t}_{\text{passive,A}}+{t}_{\text{passive,B}}}\cdot {t}_{\text{z,AMB}})& \forall \text{}{t}_{\text{0,A,lin}}0\\ {t}_{\text{0,A}}=0\text{}\wedge \text{}{t}_{\text{7,A}}={t}_{\text{passive,A}}& \forall \text{}{t}_{\text{0,A,lin}}0\end{array}$(6a)
$\{\begin{array}{cc}{t}_{\text{7,A}}=\frac{1}{2}\cdot ({t}_{\text{passive,A}}+\frac{{t}_{\text{passive,B}}}{{t}_{\text{passive,A}}+{t}_{\text{passive,B}}}\cdot {t}_{\text{z,AMB}})& \forall \text{}{t}_{\text{7,A,lin}}0\\ {t}_{\text{7,A}}=0\text{}\wedge \text{}{t}_{\text{0,A}}={t}_{\text{passive,A}}& \forall \text{}{t}_{\text{7,A,lin}}0\end{array}$ (6b)
$\{\begin{array}{cc}{t}_{\text{0,B}}=\frac{1}{2}\cdot ({t}_{\text{passive,B}}+\frac{{t}_{\text{passive,A}}}{{t}_{\text{passive,A}}+{t}_{\text{passive,B}}}\cdot {t}_{\text{z,AMB}})& \forall \text{}{t}_{\text{0,B,lin}}0\\ {t}_{\text{0,B}}=0\text{}\wedge \text{}{t}_{\text{7,B}}={t}_{\text{passive,B}}& \forall \text{}{t}_{\text{0,B,lin}}0\end{array}$ (6c)
$\{\begin{array}{cc}{t}_{\text{7,B}}=\frac{1}{2}\cdot ({t}_{\text{passive,B}}\frac{{t}_{\text{passive,A}}}{{t}_{\text{passive,A}}+{t}_{\text{passive,B}}}\cdot {t}_{\text{z,AMB}})& \forall \text{}{t}_{\text{7,B,lin}}0\\ {t}_{\text{7,B}}=0\text{}\wedge \text{}{t}_{\text{0,B}}={t}_{\text{passive,B}}& \forall \text{}{t}_{\text{7,B,lin}}0\end{array}$ (6d)
Altogether, the active inverter switching instants can be calculated according to [3], whereas the passive inverter states are given by (6). However, the time spans of the two systems are coupled by (6) so that a common module for a threephase inverter must be replaced by a novel sixphase module.
CONSTRAINTS BY THE BEARINGLESS MOTOR
If the two windings A and B are fed by a DC current as described above the M.M.F. distribution V(γ,t) normal to the stator surface for the considered PM synchronous machine results as to see in Fig. 3. It is important to note that the current through the axial AMB is only one third in each phase, following Kirchhoff’s law (see Fig. 2 b)). From that it can be concluded that, aside from the symmetrical threephase systems i_{cw} and i_{ccw}, the zerosequence component i_{0}(t) introduces field harmonics of order ν = 3, 9, 15, … according to (9). This field distribution does not move but is constant if i_{0} is constant and pulsates with f_{ax} if i_{0} is sinusoidal and f_{ax}frequent.
${V}_{\nu}(\gamma ,t)={\sum}_{\nu =3,9,15,...}^{\infty}\frac{4}{\pi}\cdot \frac{1}{\nu}\cdot m\cdot {N}_{\text{c}}\cdot \frac{{i}_{\text{0}}\left(t\right)}{3}\cdot \mathrm{cos}(\nu \cdot \gamma )\text{;}{N}_{\text{c}}\text{:numberofcoilturns}$ (9)
Fig. 3. M.M.F.distribution V(γ,t) normal to the stator surface at the stator inner bore (circumferential angle γ) for a pure zerosequence current feeding i0(t), where coil current ic(t) = i0(t)/3 and coil turn count is Nc
Especially the third harmonic (ν = 3) is harmful since higher harmonics only account for 2 % of the total field normal to the stator surface. The resulting field has to be taken care of for two reasons. First, like in a transformer, it induces a voltage in the phase winding due to a certain zerosequence inductance L_{BM,σ,0}. This inductance together with the phase resistance forms a voltage divider according to (Fig. 2b). Therefore, it is desirable to keep the motor zerosequence impedance low. However, for the given system the zerosequence motor inductance L_{BM,σ,0} = 57 μH is negligible compared to the AMB inductance L_{AMB} = 15 mH (both from 2D finite element (FE) simulation by means of the software JMAG, winding overhang inductance analytically calculated). The second harmful influence of the sixpole field distribution is on the motor operation and treated hereafter. It yields torque ripple, force ripple and eddy current losses in the PM.
For all these fields of interest the worst case is active for a pure DC current feeding as shown in Fig. 4, 5. That means, the field distribution in Fig. 3 does not pulsate but is constant. The influence of the zerosequence current on the Ohmic losses in the conductors is negligible and, thus, not considered here.
Torque ripple
The impact of the zerosequence current on the torque ripple w_{M} is estimated analytically and via 2D FE simulation (Fig. 4). First, the analytical calculation process is shown in order to explain the origin of the torque ripple. It relies on the 2D Maxwell stress tensor (f_{r}, f_{t})^{T} (r: radial, t: tangential) in cylindrical coordinates and uses the assumptions of infinite iron permeability as well as neglection of the slotting, curvature and end effect.
The field wave of the νth stator and the μth rotor harmonic with respect to the circumferential angle γ of the stator field B_{s,ν}(γ,t) as well as of the rotor B_{r,μ}(γ,t) are superimposed at each time and are integrated over a closed surface in the air gap (stack length l_{Fe}). This yields the time depending torque M(t) (10, 11); N_{s}: Number of turns per phase, k_{w,ν}: winding factor of νth harmonic, m: phase count, r_{s,i}: bore radius, i: torque harmonic order.
$\begin{array}{c}M\left(t\right)=\underset{0}{\overset{{l}_{\text{Fe}}}{\int}}\underset{0}{\overset{2\pi}{\int}}\left({f}_{\text{t}}(\gamma ,t)\cdot {r}_{\text{s,i}}\right)\text{d}\gamma \text{d}z=\\ =\text{}{M}_{\text{1}}\cdot \mathrm{sin}({\phi}_{\text{r,1}}{\phi}_{\text{s,1}})+{\sum}_{i=3,9,15,...}^{\infty}{\hat{M}}_{\text{i}}\cdot \mathrm{sin}(i\cdot {\omega}_{\text{s}}t+i\cdot {\phi}_{\text{r,i}}{\phi}_{\text{s,i}})\end{array}\text{}$(10)
From that, it can be concluded that apart from the wellknown constant term an additional timedepending component is present. This component is of sinusoidal character, oscillating with frequencies i·ω_{s} (i = ν = μ) according to the product of synchronous frequency ω_{s} and order of magnetic field space harmonics of equal pole count ν = μ = 3, 9, 15, … excited by the zerosequence current i_{0}. Here, most crucially the harmonic ν = μ = 3 produces a nonconstant torque, oscillating with frequency 3·f_{s}. One option to get rid of any torque ripple of this nature is to improve the magnetization pattern so that the PM excites a field distribution which is purely sinusoidal (${\hat{B}}_{\text{r,}\mu}=0\text{}\forall \text{}i\ne \mu $).
From (10) the torque coefficients M_{i} can be calculated according to (11). From that, the low influence of a zerosequence feeding can be seen. Equation (11) is valid for m = 3 phases. The analytical result (Analyt. DCcurrent) for the third harmonic current ripple is proved at rated operation by the FE simulation (FE DCcurrent) results from Fig. 4b.
$\begin{array}{l}{M}_{1}={\hat{i}}_{\text{cw}}\cdot {k}_{\text{w,1}}\cdot 2\cdot m\cdot {N}_{\text{s}}\cdot {\hat{B}}_{\text{r,1}}\cdot {l}_{\text{Fe}}\cdot {r}_{\text{s,i}}\text{;}{M}_{1,\text{N}}=124.956\text{mNm}\\ \begin{array}{c}{\hat{M}}_{i}=\frac{{i}_{\text{0}}}{3}\cdot {k}_{\text{w,i}}\cdot 2\cdot m\cdot {N}_{\text{s}}\cdot {\hat{B}}_{\text{r,}\mu =i}\cdot {l}_{\text{Fe}}\cdot {r}_{\text{s,i}}\text{;}{\hat{M}}_{\text{3,N}}=0.096\text{}\frac{\text{mNm}}{\text{A}}\cdot {i}_{\text{0}}\\ \Rightarrow {w}_{\text{M}}=0.077\text{}\frac{\%}{\text{A}}\text{;}{w}_{\text{M}}=\left\frac{{M}_{\mathrm{max}}{M}_{\mathrm{min}}}{{M}_{\mathrm{max}}+{M}_{\mathrm{min}}}\right\end{array}\end{array}$ (11)
So far the zerosequence current was assumed to be constant. However, sometimes due to mechanical imbalance and external force disturbances the axial force requirement as well as the linked current i_{0}(t) is not constant. The calculation is not shown here, but it can be concluded that a time variant zerosequence current introduces new frequencies in the torque harmonic spectrum. This can be harmful if certain resonances are excited by these frequencies. However, the force ripple (FE ACcurrent, Fig. 4b) is smaller in any case compared to the DC feeding. Fig. 4b shows the torque ripple amplitude for different zerosequence current feedings (î_{0} = 0 … 10.5 A) at rated operation. For usual operation no more than i_{0} =°1 A (i_{0}/3 <°0.33 A) is needed. From Fig. 4b it can be concluded that a zerosequence feeding to this extent has no crucial influence on the torque ripple, staying below 0.2 %.
Force ripple
The origin of the ripple w_{F} in the lateral force is explained equivalent to the torque ripple calculation. To do so the Maxwell stress tensor (f_{r}, f_{t})^{T} has to be evaluated, taking into account that the force in the xy coordinate system is needed for control purpose. In the calculation, stator and rotor field harmonics of order ν,μ > 3 are neglected for clarity since their influence is very small. The composition of the active lateral force is depicted in (12).
$\begin{array}{c}{F}_{\text{x}}\left(t\right)={F}_{\text{op}}\cdot \mathrm{cos}({\phi}_{\text{r,1}}{\phi}_{\text{s},2})+{F}_{\text{dis}}\cdot \mathrm{cos}({\phi}_{\text{s},1}{\phi}_{\text{s},2})+\\ +{\hat{F}}_{\mathrm{var},1}\cdot \mathrm{cos}({\omega}_{\text{s}}t+{\phi}_{\text{s},2}{\phi}_{\text{s},3})+{\hat{F}}_{\mathrm{var},2}\cdot \mathrm{cos}(2\cdot {\omega}_{\text{s}}t{\phi}_{\text{s},2}+3\cdot {\phi}_{\text{r},3})\\ {F}_{\text{y}}\left(t\right)={F}_{\text{op}}\cdot \mathrm{sin}({\phi}_{\text{r,1}}{\phi}_{\text{s},2})+{F}_{\text{dis}}\cdot \mathrm{sin}({\phi}_{\text{s},1}{\phi}_{\text{s},2})\\ {\hat{F}}_{\mathrm{var},1}\cdot \mathrm{sin}({\omega}_{\text{s}}t+{\phi}_{\text{s},2}{\phi}_{\text{s},3})+{\hat{F}}_{\mathrm{var},2}\cdot \mathrm{sin}(2\cdot {\omega}_{\text{s}}t{\phi}_{\text{s},2}+3\cdot {\phi}_{\text{r},3})\end{array}$ (12)
From (12) it can be seen that the lateral force in one distinct direction consists of four components whereof two are constant. The force coefficients can be calculated according to (13). Equation (13) is valid for m = 3 phases. The result is proved by FE simulations (see Fig. 4a). The two constant components cover the force component F_{op}, necessary for operation, and the component F_{dis}, representing the disturbing influence of the stator drive field by i_{cw} on the stator suspension field by i_{ccw}. The latter is not further discussed here.
$\begin{array}{l}{F}_{\text{op}}=\frac{1}{2}\cdot {\hat{i}}_{\text{ccw}}\cdot {k}_{\text{w,2}}\cdot 2\cdot m\cdot {N}_{\text{s}}\cdot {\hat{B}}_{\text{r,1}}\cdot {l}_{\text{Fe}}(\frac{{r}_{\text{s,i}}}{{p}_{\text{s,2}}\cdot \delta}+1);\text{}\left{F}_{\text{op,N}}\right=8.584\text{N}\\ {\hat{F}}_{\mathrm{var},1}={\hat{i}}_{\text{ccw}}\cdot \frac{{i}_{0}}{3}\cdot {k}_{\text{w,2}}\cdot {k}_{\text{w,3}}\cdot {m}^{2}\cdot {N}_{\text{s}}\cdot {\mu}_{\text{0}}\cdot {l}_{\text{Fe}}(\frac{{r}_{\text{s,i}}}{9\cdot {\delta}^{2}\cdot \pi}\frac{2}{3\cdot {r}_{\text{s,i}}\cdot \pi}+\frac{1}{9\cdot \delta \cdot \pi});\text{}\\ \left{\hat{F}}_{\text{var,1,N}}\right=0,0073\text{}\frac{\text{N}}{\text{A}}\cdot {i}_{0}\to {w}_{\text{F,1,N}}=0,085\text{}\frac{\%}{\text{A}}\text{;}{w}_{\text{F}}=\left\frac{{F}_{\mathrm{max}}{F}_{\mathrm{min}}}{{F}_{\mathrm{max}}+{F}_{\mathrm{min}}}\right\\ {\hat{F}}_{\mathrm{var},2}={\hat{i}}_{\text{ccw}}\cdot {k}_{\text{w,2}}\cdot m\cdot {N}_{\text{s}}\cdot {\hat{B}}_{\text{r,3}}\cdot {l}_{\text{Fe}}(\frac{{r}_{\text{s,i}}}{{p}_{\text{s,2}}\cdot \delta}1);\\ \left{\hat{F}}_{\text{var,2,N}}\right=0,021\text{N}\to {w}_{\text{F,2,N}}=0,25\text{}\%\end{array}$(13)
Fig. 4. Calculated force ripple wF a; calculated torque ripple wM b for different zerosequence current feedings at nN = 60000 min1, MN = 105 mNm, FE: results from FE simulation (JMAG), AC feeding at ωax = ωs; Other operating points behave similarly
Of importance are the two time variant components F_{var,1} and F_{var,2}. F_{var,1} is excited by the interaction between the constant sixpole field wave by i_{0} and the rotating fourpole field wave by i_{ccw}(t). The force generation results from the difference in pole count by ± 2. However, one field wave is caused by a f_{s}frequent current i_{ccw}(t), whereas the other results from a DC current i_{0}. From this difference the force ripple frequency is f_{F,var,1} = f_{s} – 0 = f_{s}. The amplitude is proportional to the zerosequence current i_{0} (13). For rated operation this force ripple is w_{F,1,N} < 0.1 % (13) and not crucial compared to other disturbing influences. The second time variant component F_{var,2} results from the third rotor field harmonic, rotating with synchronous velocity v_{syn} = f_{r,μ}/μ and, therefore, f_{r,3} = 3·f_{s} in case of a not purely sinusoidal magnetization in interaction with the four pole stator field wave, excited by the f_{s}frequent current i_{ccw}(t). From this difference the force ripple frequency is f_{F,var,2} = 3·f_{s} – f_{s} = 2·f_{s}. This force ripple is in effect even if no zerosequence current is fed. It dominates the ripple of F_{var,1} for values of i_{0} < 3 A (13). The calculation with a time variant zerosequence current is not shown here. It can introduce new frequencies in the lateral force spectrum that may be harmful. However, for the often considered case (n f_{s} ω_{ax}) it can be seen that no force ripple harmonic is introduced.
From Fig. 4a it can be concluded that a zerosequence feeding to this extent has no crucial influence on the force ripple, staying below 0.5 %. That means, the force ripple is mostly dominated by the 2·ω_{s}frequent part F_{var,2}, which is caused by the 3^{rd} rotor field harmonic (μ = 3).
Eddy current losses in the PM
Generally, every statorexcited magnetic field wave differing from the operating wave induces voltages in eddy current loops formed by the conductive permanent magnet material due to the difference in rotational speed. Keeping the losses caused by these eddy currents low yields low rotor losses. This is a wellknown design goal for many reasons [26]. Therefore, the following section shows the criticality of the zerosequence current with respect to its induced eddy current losses in the PM. To understand the process a 2D analytical eddy current calculation in the form of a multilayer travellingwave problem was carried out first. For simplicity it was made for a planar geometry. The analytical calculation of eddy current losses in the form of multilayer travellingwave problems has been extensively discussed in literature [26–28]. The calculation process is elaborate and not shown here. The results from this calculation are shown in Table 2 and compared with the 2D FE simulation results under the same assumptions.
Table 2. Calculated eddy current losses in the rotor for the simplified 2D planar geometry (κ_{PM} = 1.25 Ms·m^{1}, κ_{shaft} = 1.92 Ms·m^{1}, μ_{r,PM} = 1, μ_{r,shaft}= 100)
Harmonic ν  Machine part  Analytical  FE 
2 (î_{ccw} = 3,2 A)  PM  2.61 W  2.55 W 
shaft  0.79 W  0.58 W  
3 (i_{0}/3 = 1 A)  PM  0.52 W  0.51 W 
shaft  0.08 W  0.11 W  
3 (i_{0}/3 = 3,5 A)  PM  6.33 W  6.32 W 
shaft  1.04 W  1.08 W 
The results from the simplified 2D Cartesian geometry show the same tendency as the results from the 2D cylindrical geometry (Fig. 5). If less than 1 A zerosequence current per phase (i_{0} < 3 A) is injected, there are no relevant additional losses in the rotor of the machine. However, from 1 A the losses increase quickly according to I²·R. Consequently a bearing current of i_{0} < 3 A is not crucial and is seen to be the upper limit for the operating current, i.e. i_{0,max} = 3 A (see Fig. 15).
Fig. 5. Calculated eddy current losses Pd,Ft,PM in the permanent magnets for different zerosequence current feedings at rated operation (nN = 60000 min1, MN = 105 mNm) from 2D FE simulations (JMAG); other operating points show similar characteristics
CONSTRAINTS BY THE INVERTER
This section focuses on the dependency between the bearing performance and the operation point of the PWMoperated MOSFET inverter. Generally, one important design goal of the magnetic suspension control to ensure stability is the sufficiently fast change rate of the bearing current which is influenced by the required current amplitude, the frequency and voltage rating of the inverter as well as of the bearing inductance. In this case the amplitude of the current is limited by the motor tolerance for a zerosequence current (i_{0,max} = 3 A). The frequency range and voltage amplitude of the current is limited by the inverter (U_{DC,max} = 150 V, f_{sw,max} = 60 kHz). Apart from these given parameters, the system designrelated time constant of the plant (τ_{plant} = 10.64 ms, see Fig. 2b) plays an important role. The choice of the desired parameters for a stiff control is limited in reality. Given the time constant of the plant and staying within the tolerance range of the zerosequence current for the motor it is mainly dependent of the inverter.
In addition to this general statement, the difficulty has to be treated that the inverter capacity is constrained by the generation of the torque and suspension field in the bearingless motor. As can be seen from the switching instant calculation (section “Generation of the required zerosequence voltage”), the generation of the two voltage systems is treated with priority. In accordance with that, it can be shown by simulation that neither in steady state nor in transient condition the zerosequence voltage changes its time harmonic spectrum of the torque and lateral force generating motor currents. On the other hand, the operation point of the motor, determining the inverter operation point, influences the zerosequence current slope and the axial bearing performance. Different points of operation that show this crucial impact are investigated by means of the software Simulink assuming ideal switching behavior in the following order: variable bearing current step response at rated motor operation, fixed bearing current step response at variable motor speed, inherent zerosequence current with and without control, operating area for variable sinusoidal bearing current.
Variable bearing current step response at rated motor operation
Fig. 6 gives insight into the transient behavior of the bearing current for rated motor operation. In this and the following time plots i_{0,ideal} gives the theoretical current in case of a pure DC voltage feeding u_{0,ideal} (ideal inverter behavior, no PWM). i_{0} and u_{0} are the real current and real voltage, resulting from the pulse width modulated inverter.
In Fig. 6a it can be seen that current follows within 0.3 ms in comparison with the ideal case where the current follows within 0.1 ms. In this scenario the inverter shows the wellknown lack element behavior. For a reference step of i_{0,ref} = 3 A (Fig. 6b) the current follows within 0.9 ms which is again three times the time span of the pure DC voltage feeding. Further, the nonlinear behavior which results from the zerovoltage time span calculation can be seen. However, it is already can be shown that the disturbing effect of the inverter on the controller circuit is acceptable for current requirements i_{0} < 1 A.
Fig. 6. Simulated step response response for a reference current step of i0,ref =1 A a; and i0,ref =3 A b; 0.5 ms at rated motor operation (fs,N = 1000 Hz, Ûcw = 60 V, Ûccw = 3.2 V, fsw = 60 kHz)
Fixed bearing current step response at variable motor speed
The influence of the inverter utilization by the motor operation, i.e. the modulation degree given by the backEMF related motor speed, on the bearing current slope has to be considered. This question is directly linked to the provided zerovoltage time spans which can be used in order to generate the zerosequence current artificially. Therefore, Fig. 7 compares the zerovoltage duty states d_{0,A}, d_{7,A}, d_{0,B}, d_{7,B} for Fig. 6a n/n_{N} = 2/3 and Fig. 6b n/n_{N} = 4/3.
A steplike zerosequence current requirement occurs at 0.5 ms requiring the maximum zerosequence voltage (positive). Therefore, d_{7,A} and d_{0,B} (all highside switches of system A on and all lowside switches of system B on) are as big as possible whereas d_{0,A} and d_{7,B} are kept zero. This clearly shows that the remaining zerovoltage time span declines from approximately T_{sw}/2 to T_{sw}/8. Reciprocally to that, these time spans are kept constantly at their maximum level four times longer at n/n_{N} = 4/3 than at n/n_{N} = 2/3. Consequently, the current rise time is longer for higher modulation degrees. It can be shown approximately that twice the modulation degree leads to a four times longer rise time.
Fig. 7. Simulated zerovoltage duty states d0,A, d7,A, d0,B, d7,B, for a reference current step of i0,ref =1 A at 0.5 ms at rated suspension operation (ûccw = 3.2 V, îccw = 3.2 A) and different modulation degrees: a – ûcw = 40 V (ma = 0.5, n ≈ 40000 min1);b – ûcw = 80 V (ma = 0.96, n ≈ 80000 min1)
Inherent zerosequence current with and without control
At high modulation degrees another problem occurs: The zerosequence current exhibits a 3^{rd} harmonic with regards to the synchronous frequency disturbing the bearing operation which cannot be counteracted by the current controller since the DC link voltage is too small to realize fast current changes within the short passive voltage instant (compare Fig. 9). Therefore, a third harmonic occurs for modulation degrees m_{a} > √3/2 which is equal to a voltage space vector of length U_{DC}/2. For comparison the zerosequence current i_{0,noncontrol} for the case of a symmetrical SVM without zerosequence current control is given. In order to explain the existence of the third harmonic, Fig. 8 can be consulted. It shows the electric potential of the two star points Z_{A}, i.e. u_{γ,A} and Z_{B}, i.e. u_{γ,B}, for the discrete switching states. For the case of block commutation, i.e. t_{active} = T_{sw} and t_{passive} = 0, the potential of one star point pulsates between U_{DC}/6 and –U_{DC}/6 three times per electrical period. The magnitude of the u_{0} between the two star points depends on both electric potentials u_{γ,A} and u_{γ,B}. This means, if the two voltage systems are fed in commonmode, no voltage is active between Z_{A} and Z_{B}. Therefore, the lateral force generating commonmode voltage system does not lead to a ripple in u_{0}.
However, the differentialmode feeding of u_{cw} leads to a difference in potential, since the two clockwise systems u_{cw,A} and u_{cw,B} are geometrically opposed in the αβplane. That is, while system A is e.g. in state 1 (u_{γ,A} = U_{DC}/6), system B is in instant 4 (u_{γ,B} = –U_{DC}/6), leading to an amplitude of u_{0} = U_{DC}/3. A sixth of an electric period later system A is in instant 2 (u_{γ,A} = –U_{DC}/6), system B is in instant 5 (u_{γ,B} = U_{DC}/6), leading to an amplitude of u_{0} = –U_{DC}/3. This pattern continues, so that for block commutation the voltage drop of the AMB can be given by (14), neglecting the commonmode feeding. Hence, the zerocrossings of this function are at the switching states when the voltage space vector is between two discrete switching states at odd multiples of ε = π/6. The maxima of (14) are located at even multiples of ε = π/6 (Fig. 8). In order to explain, from which modulation degree this third harmonic occurs, a condition must be found, so that the amplitude of (14) is zero. As explained, only the instants, when the active voltage vector is composed of only one active switching state, must be considered, e.g. t_{active,A} = t_{1,A} and t_{active,B} = t_{4,B}. Since t_{active,A} = t_{active,B} due to symmetrical differentialmode feeding, it is obvious that the condition for u_{0} = 0 must be fulfilled over one switching period by ${{\overline{u}}_{\gamma ,\text{A}}}_{{T}_{\text{sw}}}=0$ and ${{\overline{u}}_{\gamma ,\text{B}}}_{{T}_{\text{sw}}}=0$ yielding (15).
${u}_{\text{0}}\left(t\right)=\frac{4}{\pi}\cdot \frac{{U}_{\text{DC}}}{3}\cdot \mathrm{cos}\left(3\cdot ({\omega}_{\text{s}}\cdot t\epsilon )\right)$ (14)
By applying (3) and (15), it can be shown that the maximum space vector amplitude is Û_{s} = U_{DC}/2, i.e. m_{a} = √3/2, which is in accordance with (15). Consequently, it can be said that the system is applicable for n < 1.25 n_{N} which is related to a maximum axial force frequency f_{ax,max}. This gives an accurate estimation of the maximum applicable modulation degree regardless the exact system parameters. Moreover, the given system meets the shown behavior despite its slightly elliptical voltage space vector orbit, since û_{ccw} ≈ 0.05·û_{cw}.
Fig. 8. Switching instants 0, 1, , …, 7 and related electric potentials at the star points ZA (uγ,A) and ZB (uγ,B) in a 3D (αβγ) diagram; red: electric potential uγ,A and uγ,B at ε = 0° for differentialmode feeding
$\begin{array}{l}{{\overline{u}}_{\gamma}}_{{T}_{\text{sw}}}=\pm \frac{{U}_{\text{DC}}}{6}\cdot {t}_{\text{active}}\mp \frac{{U}_{\text{DC}}}{2}\cdot {t}_{\text{passive}}=0\\ \Rightarrow \text{}{t}_{\text{active}}\le \frac{3}{4}\cdot {T}_{\text{sw}}\wedge {t}_{\text{passive}}\frac{1}{4}\cdot {T}_{\text{sw}}\text{}\Rightarrow \text{}{\hat{U}}_{\text{s}}\le \frac{{U}_{\text{DC}}}{2}\text{;}{m}_{\text{a}}\le \frac{\sqrt{3}}{2}\end{array}\text{}$ (15)
Operating area for variable sinusoidal bearing current
In order to describe the operating area of the axial AMB system it is necessary to point out which requirements the position controller sets for the current controller. The controller design of the thrust magnetic bearing is not in the focus here. However, from the system it is known that an average current of i_{0} = 0.9 A is needed in order to levitate the rotor if it is operated vertically so that the thrust bearing carries the rotor. From that it is estimated that the maximum zerosequence current of i_{0} = 3 A is sufficient even for vertical operation.
In the previous section, it was shown that the slew rate of the current is limited by the inverter voltage rating. This results in a limited operating frequency range for the system: At low frequency, e.g. i_{0} = const., the tolerance of the zerosequence current in the bearingless motor limits the operation in terms of maximum axial force generation (solid lines in Fig. 9). At high frequency, however, the inverter voltage rating (together with the given time constant of the plant) limits the operation (dashed line in Fig. 9). That is, the current cannot follow the reference signal anymore. If the voltage drop over the plant resistance R_{plant} is neglected the relation between bearing current and required voltage is given by (16) according to [1]. Here, f_{ax} is the bearing current frequency and L_{plant} is the bearing and motor inductance of the circuit.
${\hat{u}}_{0}=2\pi \cdot {f}_{\text{ax}}\cdot {L}_{\text{plant}}\cdot {\hat{i}}_{0}\text{}\Rightarrow \text{}{{\hat{i}}_{0}}_{{\hat{u}}_{0}=\text{}{U}_{\text{inv,max}}=\text{const.}}~\frac{1}{{f}_{\text{ax}}}$ (16)
This shows that the maximum possible bearing current amplitude is inversely proportional to its applied frequency, since the inverter voltage is limited. Usually it is limited by the inverter voltage rating. In this case additionally the modulation degree m_{a} for the required voltage space vectors of the torque and force generating voltage systems in the bearingless motor has to be considered. Taking into account that the fundamental amplitude of the applied voltage of a squarewave form is 4/π·U_{DC} the envelope of the peak current î_{0,max} can be estimated according to (17) (see Fig. 9).
${\hat{i}}_{0,\mathrm{max}}=(1{m}_{\text{a}})\cdot \frac{2}{{\pi}^{2}}\cdot \frac{{U}_{\text{DC}}}{{L}_{\text{plant}}}\cdot \frac{1}{{f}_{\text{ax}}};\text{where}{m}_{\text{a}}\text{=}\sqrt{3}\cdot \frac{{\hat{U}}_{\text{cw}}+{\hat{U}}_{\text{ccw}}}{{U}_{\text{DC}}}$ (17)
Fig. 9. Simulated operating area of the axial magnetic bearing system
The black line in Fig. 9 shows the envelope of the operating area at the given conditions (U_{DC} = 150 V, L_{plant} = 15 mH, m_{a} = 0.73). i_{0,N} = 0,9 A and refers to the current necessary for rotor levitation at vertical operation. f_{s,N} refers to the rated speed of 60000 min^{1}. It can be seen that for vertical as well as for horizontal operation the system can be operated without any problems as long as any imbalance force excitation requires less than half the rated bearing current. However, this is not to expect since imbalance forces mainly act radially. Apart from that, there are several options to enlarge the operating area by adapting the system boundary conditions. The most effective is to increase the DC link voltage to increase the current slope and to decrease the modulation degree (blue line in Fig. 9). Another is to operate the drive at lower speeds which requires less modulation degree (green line in Fig. 9) Moreover a decrease in plant inductance yields higher current slopes. However, since L ~ N² but F ~ N·I a reduction in inductance by means of coil turn reduction is on the costs of moderately higher currents in order to achieve the same bearing force.
CONCLUSIONS
A new method of operating the axial AMB for a cylindrical bearingless machine was introduced which relies on the feeding by the zerosequence current between the two star points of the machine. Exemplary a 1 kW / 60000 min^{1} PM synchronous machine is considered. It was shown that it is possible to manipulate artificially the electric potential of the star point of a three phase winding system by the adaption of the time spans for the passive voltage instants in a SVPWM. However, the voltage drop between the two star points depends on both potentials of the star points, leading to a coupling between the two current systems. It is explained, how this problem of five unknown phase currents can be separated into smaller problems of two and three dimensions. The operation of the system is dependent on the constraints, given by the BM as well as by the inverter capacity.
Firstly, it is shown that a zerosequence current leads to torque and force ripple and to an increase in eddy current losses in the PM of the rotor. However, all these fields of interest are not crucial for currents i_{0} < 3 A. For the given AMB the rated current to compensate for the rotor weight is i_{0,N} = 0.9 A. So it can be concluded that the constraints by the motor do not determine significantly the applicability of the method. Finally, it is demonstrated that the more crucial constraints are given by the inverter. For pure DC current requirements at steady state and in transient conditions it is shown that the motor currents are not influenced by the zerosequence current, since the active time spans are calculated independently. For modulation degrees m_{a} > 0.866 a third harmonic occurs in the zerosequence current, prohibiting operations at speeds n > 1.25·n_{N} which is caused by the influence of the active voltage time spans on the star point potential. Finally, the step response and the sinusoidal current requirements yield suitable results for currents i_{0} < 1 A. Certainly the operating area can be enlarged by a higher DC link voltage, a smaller AMB inductance or if the modulation degree at rated operation is reduced.
Altogether it can be stated that the presented technique is applicable for active magnetic suspensions of highspeed drives, taking into account the mainly noncritical constraints by the bearingless machine and the inverter. It is topic of future investigations to realize the presented technique by a prototype.
About the authors
Daniel Dietz
Institute of Electrical Energy Conversion – Darmstadt, Technical University
Author for correspondence.
Email: ddietz@ew.tudarmstadt.de
Germany, LandgrafGeorgStr. 4, 64283 Darmstadt
M.Sc.
Andreas Binder
Institute of Electrical Energy Conversion – Darmstadt, Technical University
Email: abinder@ew.tudarmstadt.de
Germany, LandgrafGeorgStr. 4, 64283 Darmstadt
Prof., Dr.Ing. habil. Dr. h.c.
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Supplementary files
Supplementary Files  Action  
1. 
Fig. 1. Drive components (schematic) (1) Position and rotor angle sensors, (2) Axial part of combined AMB, (3) Radial part of combined AMB, (4) PM with magnetization direction, (5) Turbocompressor wheel, (6) Safety bearings at drive and nondrive end (DE / NDE) a; simplified electric circuit of the bearingless machine and the axial AMB b

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Fig. 7. Simulated zerovoltage duty states d0,A, d7,A, d0,B, d7,B, for a reference current step of i0,ref =1 A at 0.5 ms at rated suspension operation (ûccw = 3.2 V, îccw = 3.2 A) and different modulation degrees: a – ûcw = 40 V (ma = 0.5, n ≈ 40000 min1);b – ûcw = 80 V (ma = 0.96, n ≈ 80000 min1)

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