Bearingless PM synchronous machine with zero-sequence current driven star point-connected active magnetic thrust bearing

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 zero-sequence 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 three-phase 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 three-phase windings of the motor. The focus is on the determination of the operating area (maximum zero-sequence 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 self-bearing 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 disk-like and thereby self-stabilizing 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 AMB-suspended drives is the use as high-speed 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 so-called thrust AMB, is required. Often the thrust bearing is realized parallel to a radial AMB as combined AMB [5] on the non-drive 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 half-motors 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 star-connected 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 high-speed 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) Turbo-compressor wheel, (6) Safety bearings at drive and non-drive 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 zero-sequence 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 three-phase 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 three-phase windings fed by the same inverter.

Table 1. Motor specifications

Rated speed /  min-1	nN	60000
Rated torque / mNm	MN	105
Rated lateral force / N	FN	8.2
Rated phase voltage / V	Us,N	42
Rated phase current for torque / A	Icw,N	3.18
Rated phase current for lateral force / A	Iccw,N	2.26
Rated total phase current / A	Is,N	3.90
Rated current for axial force / A	I0,N	0.9
Stator bore outer/inner diameter / mm	ds,o/ds,i	70/35
Stator stack length / mm	lFe	40
Bandage thickness / mm	hb	1.5
Magnet height / mm	hPM	2.75
Mechanical air gap (ds,i – dr,o)/2 / mm	δ	1.0
Rotor mass / g	mr	800
Number of turns per phase (6 phases)	Ns,6	22
Number of pole pairs (torque/suspension winding)	p/psus	1/2

In this machine a combined two-layer 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 2p-pole field wave generates tangential force, whereas the 2p_sus-pole field wave generates lateral force for rotor suspension. Starting from a conventional four-pole three-phase 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 three-phase system A (B). They are both star-connected, yielding the two star-points Z_A and Z_B. If the two systems are fed in phase a counter-clockwise 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 2p-pole 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 three-phase system A and B consisting theoretically of two symmetrical three-phase 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 i₀ 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 ZERO-SEQUENCE VOLTAGE

The proposed technique relies on a conventional space vector pulse width modulation (SVPWM) [3]. That means, that for each of the elliptical three-phase current systems I_A and I_B a current controller determines a certain voltage space vector in the stator-fixed 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 counter-clockwise 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 counter-clockwise rotating voltage space vectors for torque and lateral force generation. However, since the 2p-pole rotor field does not induce into the imaginary 2p_sus-pole suspension winding, the voltage trajectory is only slightly elliptical, so that mainly the back-EMF of the 2p-pole 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 high-side 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 t₀, … t₇. For the traditional symmetrical SVPWM the calculation of t₀, … t₇ is well explained in literature [3, 22]. This technique is commonly used for three-phase inverters when operated in field-oriented control. It is important to note that only the time spans t₁, t₂ 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 zero-sequence 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 i₀ and u₀ is depicted in Fig. 2b. According to that the Ohmic-inductive voltage drop u₀ 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 stator-fixed α-β-reference frame related to the voltage limit a; and equivalent circuit for the zero-sequence 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 zero-sequence 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)

${{\overline{u}}_{\text{0}}|}_{T_{\text{sw}}}=\frac{U_{\text{DC}}\cdot k_{\text{eq}}}{2\cdot T_{\text{sw}}}\cdot (\underset{\begin{array}{l}\text{Fixed time spans by different active }\\ \text{voltage vectors in system A and B}\end{array}}{\underbrace{\frac{t_{\text{2,A}}-t_{\text{1,A}}}{3}+\frac{t_{\text{1,B}}-t_{\text{2,B}}}{3}}}+\underset{t_{\text{z,AMB}}}{\underbrace{\underset{\begin{array}{l}\text{Controllable time spans for artificial}\\ \text{zero-seqeunce voltage generation}\end{array}}{\underbrace{t_{\text{7,A}}-t_{\text{0,A}}+t_{\text{0,B}}-t_{\text{7,B}}}}}})$ (3)

$t_{\text{passive,A}}=t_{\text{0,A}}+t_{\text{7,A}}=T_{\text{sw}}-t_{\text{active,A}};t_{\text{passive,B}}=t_{\text{0,B}}+t_{\text{7,B}}=T_{\text{sw}}-t_{\text{active,B}}$ (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 under-determined system another condition has to be introduced. It is selected in a way that the overlapping time span t₇ – t₀ is proportional (t₀ → t_0,lin, t₇ → 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)

$\{\begin{array}{cc}{t}_{\text{0,A}}=\frac{1}{2}\cdot (t_{\text{passive,A}}-\frac{t_{\text{passive,B}}}{t_{\text{passive,A}}+t_{\text{passive,B}}}\cdot t_{\text{z,AMB}})& \forall t_{\text{0,A,lin}}>0\\ t_{\text{0,A}}=0\wedge t_{\text{7,A}}=t_{\text{passive,A}}& \forall t_{\text{0,A,lin}}<0\end{array}$(6a)

$\{\begin{array}{cc}{t}_{\text{7,A}}=\frac{1}{2}\cdot (t_{\text{passive,A}}+\frac{t_{\text{passive,B}}}{t_{\text{passive,A}}+t_{\text{passive,B}}}\cdot t_{\text{z,AMB}})& \forall t_{\text{7,A,lin}}>0\\ t_{\text{7,A}}=0\wedge t_{\text{0,A}}=t_{\text{passive,A}}& \forall t_{\text{7,A,lin}}<0\end{array}$ (6b)

$\{\begin{array}{cc}{t}_{\text{0,B}}=\frac{1}{2}\cdot (t_{\text{passive,B}}+\frac{t_{\text{passive,A}}}{t_{\text{passive,A}}+t_{\text{passive,B}}}\cdot t_{\text{z,AMB}})& \forall t_{\text{0,B,lin}}>0\\ t_{\text{0,B}}=0\wedge t_{\text{7,B}}=t_{\text{passive,B}}& \forall t_{\text{0,B,lin}}<0\end{array}$ (6c)

$\{\begin{array}{cc}{t}_{\text{7,B}}=\frac{1}{2}\cdot (t_{\text{passive,B}}-\frac{t_{\text{passive,A}}}{t_{\text{passive,A}}+t_{\text{passive,B}}}\cdot t_{\text{z,AMB}})& \forall t_{\text{7,B,lin}}>0\\ t_{\text{7,B}}=0\wedge t_{\text{0,B}}=t_{\text{passive,B}}& \forall t_{\text{7,B,lin}}<0\end{array}$ (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 three-phase inverter must be replaced by a novel six-phase 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 three-phase systems i_cw and i_ccw, the zero-sequence component i₀(t) introduces field harmonics of order ν = 3, 9, 15, … according to (9). This field distribution does not move but is constant if i₀ is constant and pulsates with f_ax if i₀ is sinusoidal and f_ax-frequent.

$V_{\nu }(\gamma ,t)={\sum }_{\nu =3,9,15,...}^{\infty }\frac{4}{\pi }\cdot \frac{1}{\nu }\cdot m\cdot N_{\text{c}}\cdot \frac{i_{\text{0}}\left(t\right)}{3}\cdot \cos (\nu \cdot \gamma )\text{; }{N}_{\text{c}}\text{: number of coil turns}$ (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 zero-sequence 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 zero-sequence 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 zero-sequence impedance low. However, for the given system the zero-sequence 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 six-pole 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 zero-sequence current on the Ohmic losses in the conductors is negligible and, thus, not considered here.

Torque ripple

The impact of the zero-sequence 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.

$\begin{array}{c}M\left(t\right)=\underset{0}{\overset{l_{\text{Fe}}}{\int }}\underset{0}{\overset{2\pi }{\int }}\left(f_{\text{t}}(\gamma ,t)\cdot r_{\text{s,i}}\right)\text{d}\gamma \text{d}z=\\ =M_{\text{1}}\cdot \sin ({\phi }_{\text{r,1}}-{\phi }_{\text{s,1}})+{\sum }_{i=3,9,15,...}^{\infty }{\hat{M}}_{\text{i}}\cdot \sin (i\cdot {\omega }_{\text{s}}t+i\cdot {\phi }_{\text{r,i}}-{\phi }_{\text{s,i}})\end{array}$(10)

From that, it can be concluded that apart from the well-known constant term an additional time-depending 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 zero-sequence current i₀. Here, most crucially the harmonic ν = μ = 3 produces a non-constant 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 (${\hat{B}}_{\text{r,}\mu }=0\forall i\ne \mu$).

From (10) the torque coefficients M_i can be calculated according to (11). From that, the low influence of a zero-sequence feeding can be seen. Equation (11) is valid for m = 3 phases. The analytical result (Analyt. DC-current) for the third harmonic current ripple is proved at rated operation by the FE simulation (FE DC-current) results from Fig. 4b.

$\begin{array}{l}{M}_1={\hat{i}}_{\text{cw}}\cdot k_{\text{w,1}}\cdot 2\cdot m\cdot N_{\text{s}}\cdot {\hat{B}}_{\text{r,1}}\cdot l_{\text{Fe}}\cdot r_{\text{s,i}}\text{ ; }{M}_{1,\text{N}}=124.956\text{ mNm}\\ \begin{array}{c}{\hat{M}}_i=\frac{i_{\text{0}}}{3}\cdot k_{\text{w,i}}\cdot 2\cdot m\cdot N_{\text{s}}\cdot {\hat{B}}_{\text{r,}\mu =i}\cdot l_{\text{Fe}}\cdot r_{\text{s,i}}\text{ ; }{\hat{M}}_{\text{3,N}}=0.096\frac{\text{mNm}}{\text{A}}\cdot i_{\text{0}}\\ \Rightarrow w_{\text{M}}=0.077\frac{\%}{\text{A}}\text{; }{w}_{\text{M}}=\left|\frac{M_{\max }-M_{\min }}{M_{\max }+M_{\min }}\right|\end{array}\end{array}$ (11)

So far the zero-sequence 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 i₀(t) is not constant. The calculation is not shown here, but it can be concluded that a time variant zero-sequence 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 AC-current, Fig. 4b) is smaller in any case compared to the DC feeding. Fig. 4b shows the torque ripple amplitude for different zero-sequence current feedings (î₀ = 0 … 10.5 A) at rated operation. For usual operation no more than i₀ =°1 A (i₀/3 <°0.33 A) is needed. From Fig. 4b it can be concluded that a zero-sequence 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 x-y 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).

$\begin{array}{c}{F}_{\text{x}}\left(t\right)=F_{\text{op}}\cdot \cos ({\phi }_{\text{r,1}}-{\phi }_{\text{s},2})+F_{\text{dis}}\cdot \cos ({\phi }_{\text{s},1}-{\phi }_{\text{s},2})+\\ +{\hat{F}}_{\mathrm{var},1}\cdot \cos ({\omega }_{\text{s}}t+{\phi }_{\text{s},2}-{\phi }_{\text{s},3})+{\hat{F}}_{\mathrm{var},2}\cdot \cos (2\cdot {\omega }_{\text{s}}t-{\phi }_{\text{s},2}+3\cdot {\phi }_{\text{r},3})\\ F_{\text{y}}\left(t\right)=F_{\text{op}}\cdot \sin ({\phi }_{\text{r,1}}-{\phi }_{\text{s},2})+F_{\text{dis}}\cdot \sin ({\phi }_{\text{s},1}-{\phi }_{\text{s},2})-\\ -{\hat{F}}_{\mathrm{var},1}\cdot \sin ({\omega }_{\text{s}}t+{\phi }_{\text{s},2}-{\phi }_{\text{s},3})+{\hat{F}}_{\mathrm{var},2}\cdot \sin (2\cdot {\omega }_{\text{s}}t-{\phi }_{\text{s},2}+3\cdot {\phi }_{\text{r},3})\end{array}$ (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.

$\begin{array}{l}{F}_{\text{op}}=-\frac{1}{2}\cdot {\hat{i}}_{\text{ccw}}\cdot k_{\text{w,2}}\cdot 2\cdot m\cdot N_{\text{s}}\cdot {\hat{B}}_{\text{r,1}}\cdot l_{\text{Fe}}(\frac{r_{\text{s,i}}}{p_{\text{s,2}}\cdot \delta }+1);\left|F_{\text{op,N}}\right|=8.584\text{ N}\\ {\hat{F}}_{\mathrm{var},1}={\hat{i}}_{\text{ccw}}\cdot \frac{i_0}{3}\cdot k_{\text{w,2}}\cdot k_{\text{w,3}}\cdot m^2\cdot N_{\text{s}}\cdot {\mu }_{\text{0}}\cdot l_{\text{Fe}}(\frac{r_{\text{s,i}}}{9\cdot {\delta }^2\cdot \pi }-\frac{2}{3\cdot r_{\text{s,i}}\cdot \pi }+\frac{1}{9\cdot \delta \cdot \pi });\\ \left|{\hat{F}}_{\text{var,1,N}}\right|=0,0073\frac{\text{N}}{\text{A}}\cdot i_0\to w_{\text{F,1,N}}=0,085\frac{\%}{\text{A}}\text{; }{w}_{\text{F}}=\left|\frac{F_{\max }-F_{\min }}{F_{\max }+F_{\min }}\right|\\ {\hat{F}}_{\mathrm{var},2}=-{\hat{i}}_{\text{ccw}}\cdot k_{\text{w,2}}\cdot m\cdot N_{\text{s}}\cdot {\hat{B}}_{\text{r,3}}\cdot l_{\text{Fe}}(\frac{r_{\text{s,i}}}{p_{\text{s,2}}\cdot \delta }-1);\\ \left|{\hat{F}}_{\text{var,2,N}}\right|=0,021\text{ N}\to w_{\text{F,2,N}}=0,25\%\end{array}$(13)

Fig. 4. Calculated force ripple wF a; calculated torque ripple wM b for different zero-sequence current feedings at nN = 60000 min-1, 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 six-pole field wave by i₀ and the rotating four-pole 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 i₀. From this difference the force ripple frequency is f_F,var,1 = f_s – 0 = f_s. The amplitude is proportional to the zero-sequence current i₀ (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 zero-sequence current is fed. It dominates the ripple of F_var,1 for values of i₀ < 3 A (13). The calculation with a time variant zero-sequence 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 zero-sequence 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 stator-excited 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 well-known design goal for many reasons [26]. Therefore, the following section shows the criticality of the zero-sequence 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 travelling-wave 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 travelling-wave 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 (îccw = 3,2 A)	PM	2.61 W	2.55 W
	shaft	0.79 W	0.58 W
3 (i0/3 = 1 A)	PM	0.52 W	0.51 W
	shaft	0.08 W	0.11 W
3 (i0/3 = 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 zero-sequence current per phase (i₀ < 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 i₀ < 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 zero-sequence current feedings at rated operation (nN = 60000 min-1, 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 PWM-operated 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 zero-sequence 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 design-related 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 zero-sequence 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 zero-sequence 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 zero-sequence 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 zero-sequence 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 zero-sequence 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). i₀ and u₀ 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 well-known 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 non-linear behavior which results from the zero-voltage 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 i₀ < 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 back-EMF related motor speed, on the bearing current slope has to be considered. This question is directly linked to the provided zero-voltage time spans which can be used in order to generate the zero-sequence current artificially. Therefore, Fig. 7 compares the zero-voltage 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 step-like zero-sequence current requirement occurs at 0.5 ms requiring the maximum zero-sequence voltage (positive). Therefore, d_7,A and d_0,B            (all high-side switches of system A on and all low-side 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 zero-voltage 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 zero-voltage 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 min-1);b – ûcw = 80 V (ma = 0.96, n ≈ 80000 min-1)

Inherent zero-sequence current with and without control

At high modulation degrees another problem occurs: The zero-sequence 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 zero-sequence current i_{0,non-control} for the case of a symmetrical SVM without zero-sequence 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 u₀ 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 common-mode, no voltage is active between Z_A and Z_B. Therefore, the lateral force generating common-mode voltage system does not lead to a ripple in u₀.

However, the differential-mode 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 α-β-plane. 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 u₀ = 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 u₀ = –U_DC/3. This pattern continues, so that for block commutation the voltage drop of the AMB can be given by (14), neglecting the common-mode feeding. Hence, the zero-crossings 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 differential-mode feeding, it is obvious that the condition for u₀ = 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).

$u_{\text{0}}\left(t\right)=\frac{4}{\pi }\cdot \frac{U_{\text{DC}}}{3}\cdot \cos \left(3\cdot ({\omega }_{\text{s}}\cdot t-\epsilon )\right)$  (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 (α-β-γ) diagram; red: electric potential uγ,A and uγ,B at ε = 0° for differential-mode feeding

$\begin{array}{l}{{\overline{u}}_{\gamma }|}_{T_{\text{sw}}}=\pm \frac{U_{\text{DC}}}{6}\cdot t_{\text{active}}\mp \frac{U_{\text{DC}}}{2}\cdot t_{\text{passive}}=0\\ \Rightarrow t_{\text{active}}\le \frac{3}{4}\cdot T_{\text{sw}}\wedge t_{\text{passive}}>\frac{1}{4}\cdot T_{\text{sw}}\Rightarrow {\hat{U}}_{\text{s}}\le \frac{U_{\text{DC}}}{2}\text{ ; }{m}_{\text{a}}\le \frac{\sqrt{3}}{2}\end{array}$ (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 i₀ = 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 zero-sequence current of i₀ = 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. i₀ = const., the tolerance of the zero-sequence 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.

${\hat{u}}_0=2\pi \cdot f_{\text{ax}}\cdot L_{\text{plant}}\cdot {\hat{i}}_0\Rightarrow {{\hat{i}}_0|}_{{\hat{u}}_0=U_{\text{inv,max}}=\text{ const.}}~\frac{1}{f_{\text{ax}}}$ (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 square-wave form is 4/π·U_DC the envelope of the peak current î_0,max can be estimated according to (17) (see Fig. 9).

${\hat{i}}_{0,\max }=(1-m_{\text{a}})\cdot \frac{2}{{\pi }^2}\cdot \frac{U_{\text{DC}}}{L_{\text{plant}}}\cdot \frac{1}{f_{\text{ax}}};\text{ where }{m}_{\text{a}}\text{=}\sqrt{3}\cdot \frac{{\hat{U}}_{\text{cw}}+{\hat{U}}_{\text{ccw}}}{U_{\text{DC}}}$ (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 zero-sequence 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 zero-sequence 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 i₀ < 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 zero-sequence current, since the active time spans are calculated independently. For modulation degrees m_a > 0.866 a third harmonic occurs in the zero-sequence 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 i₀ < 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 high-speed drives, taking into account the mainly non-critical constraints by the bearingless machine and the inverter. It is topic of future investigations to realize the presented technique by a prototype.