Electrodynamics of Axial-Flow Rotary Blood Pumps

In this work, electric machine of implantable rotary blood pumps (RBPs) of axial-flow type was theoretically investigated. Electromagnetic coupling of rotor and stator in axial-flow RBPs was described with the Maxwell’s equations of the classical electrodynamics given quasi-magnetostatic approximation of electromagnetic field in the simplified model consisting of permanent magnet and conductive loop rotating in free space with one spatial degree of freedom. Additionally, relative field error, created by the neglect of geometric deviations introduced through manufacturing tolerances, was estimated for a typical axial-flow RBP. Upper limit of error introduced by the simplifications was estimated less than 0.2 %, leading to the finite accuracy of the description and clearly determining the influence on the control signal. Based on the presented description and additional engineering considerations, the electric machine of axial-flow RBPs was defined as a three-phase non-salient pole synchronous machine with a permanent magnet rotor. Two key features were shown: a) unlike the conventional electric motors, signal of back electromotive force tends to be a sinusoidal waveform in any construction of axial-flow RBP with significant non-magnetic gap; b) the optimal waveform of control signal in this case is sinusoidal. Initial design and control parameters of the electric machine in axial-flow RBPs can be accurately determined with presented theoretical description. Based on the description, control system of an axial-flow RBP with the optimal waveform of control signal can be developed.


I. INTRODUCTION
According to recent reports, about 64.34 million people worldwide are suffering the heart failure, and this number is expected to further increase in the near future [1], [2]. Due to donor hearts scarcity, mechanical circulatory support (MCS) is the frequently used treatment for patients suffering the end-stage heart failure. Among other systems of MCS, continuous-flow left ventricular assist devices (LVADs) are applied in clinical practice most often [3]- [7]. LVAD acts as The associate editor coordinating the review of this manuscript and approving it for publication was Jinquan Xu . a bypass device pumping additional amount of blood from the left ventricle to the aorta. It shows high durability and can easily be implanted in the patient's thorax. Inflow and outflow cannulae are also required to ensure proper connection to the cardiovascular system.
Generally, continuous-flow LVADs are rotary blood pumps (RBPs) of axial-or radial-flow type. Basically, RBP is an electric motor with a path for blood flow generated by a rotating impeller driven through electromagnetic coupling with stator windings. Rotating impeller instead of a shaft is the essential distinction of RBP and conventional electric motor. In axial-flow RBPs, blood flow is created along the pump central axis. Autonomous operation of an RBP is implemented by a controller supplied with rechargeable batteries as a source of electric energy. Percutaneous cable provides the electric connection between RBP and controller.
General principle of operation in axial-flow RBPs is conversion of supplied electric energy into mechanical energy of rotation and, eventually, into hydraulic energy of blood flow. Therefore, structure of rotary blood pumps separates into electric machine and hydraulic machine coupled through mechanical machine [8]- [16]. The electric machine consists of electromagnetically coupled stator and rotor. Bearings of various types represent the mechanical machine. And major elements of the hydraulic machine are impeller and diffuser, producing hydraulic work determined by the pressure head across the pump H h and the generated blood flow rate Q h . The sequential chain, as shown in Fig. 1, represents energy conversion and losses occurring during operation of axialflow RBPs. Since the influence of interconnection between the structural units is negligible, each unit can be considered independently through the analysis of input and output parameters.
For conversion of electric energy into mechanical energy, an external magnetic field modulated by relative rotor position is necessary in the stator. Therefore, rotor should be a permanent magnet rigidly embedded into the impeller and driven by the electric current in stator windings, providing rotational movement with angular velocity ω (Fig. 2). This solution does not require connection of rotor circuit, as for electromagnetic rotor, greatly simplifying RBP design and increasing operational reliability. The torque T 1 created by the electric machine is not equal to the torque T 2 applied to the impeller, since it is partially compensated by the drag torque in the front and rear impeller bearings. The consistent nature of energy conversion also determines a relatively high level of losses due to their accumulation at each stage. In addition, efficient operation of the hydraulic machine with certain parameters requires a significant non-magnetic gap between the permanent magnet and the stator windings (in comparison to industrial electric motors).
A deep, fundamental description of the electromagnetic coupling between stator and rotor in terms of electrodynamics is important to determine the electric machine type and to optimize the signals of RBP control system. However, to authors' knowledge, derivation of the electrodynamic characteristics of axial-flow RBPs was not yet presented in literature.
In this paper, we describe the electromagnetic coupling of stator and rotor in axial-flow RBPs based on the Maxwell's equations given the commonly utilized simplifications. We estimate upper limit of error introduced by these simplifications, leading to the finite accuracy of presented description and determining influence on the control signal. Also, we theoretically justify the optimal control signal of axial-flow RBPs as sinusoidal waveform. Finally, we show and describe the main difference of RBPs from industrial electric motors and determine the actual type of RBP electric machine as a three-phase non-salient pole synchronous machine with a permanent magnet rotor.

II. ELECTROMAGNETIC APPROACH
The electromagnetic interaction in a medium with polarization and magnetization is described by the Maxwell's equations represented in differential form as follows (Gauss' law for electric fields is (1), Ampere's law is (2), Faraday's law is (3) and Gauss' law for magnetic fields is (4)) [17]: whereĒ andH represent the electric and magnetic field strengths, respectively;P is the electric polarization field; M is the magnetization field; ρ is the volumetric density of electric charge;J is the electric current density; and ε 0 , µ 0 are the permittivity and permeability of free space, respectively. Possibility of conversion of electromagnetic energy into mechanical energy is based on emergence of the Lorentz force described by (5) for the case of continuous charge and current distribution in the environmental volume, wherē f represents the emerging force andB is the magnetic induction:f = ρĒ +J ×B.
The magnetic induction is related generally to the magnetic field strength by the following equation: Further description of the energy conversion in axial-flow RBPs is based on the energy conservation law in differential form [18] of the Poynting's theorem: whereS is the density of energy flux [19]; w is the spatial density of energy; and P C is the spatial density of power converted from one form to another. If (2) and (3) are multiplied in dot product byĒ andH , respectively, and then added up, as described in [17], [20] and [21], the following is obtained: Terms of (8) correspond to (7) as follows: This helps to describe (8) in terms of energy conservation law, matching (7) and (8). The first and the second terms on the right side of (8) correspond to the rate of density change of the stored energy for electric and magnetic field components, respectively. The third and the fourth terms show the rate of density change of the energy stored in medium due to its polarization and magnetization. And the last term describes the electromagnetic energy conversion from one form to another at the considered point in space.

A. FEATURES AND SIMPLIFICATIONS IN AXIAL-FLOW ROTARY BLOOD PUMPS
Thus, (1)-(5) and (8) describe electromagnetic interactions in general form for an arbitrary environment, in cases the laws of classical electrodynamics are applicable. Further, these relations are applied for processes occurring in the axial-flow RBPs.
Based on the above qualitative description of the RBPs, a simplified model of electromagnetic coupling is proposed (Fig. 3) to construct analytical description. This model consists of a rigid loop made of an ohmic conductor with relatively high conductivity, powered by electric current and surrounded by a material with relatively low polarization and conductivity. The conductive loop is electromagnetically coupled with a permanent magnet and has one spatial degree of freedom, i.e. rotates about a fixed axis in a magnetic field of the magnet, with significant non-magnetic gap between the loop and the magnet.
This consideration allows to apply further simplifications. In the fully static state of the system with the loop powered with DC, energy of the magnetic field considerably prevails over the electric energy, since the voltage drop produced by high conductivity of the loop is insignificant and the capacitive coupling between the loop segments is negligible compared to the inductive coupling. This behavior of the electromagnetic system creates the prerequisites for considering it quasi-magnetostatic. Because dimensions of RBPs should not exceed L max = 10 cm and expected maximum fundamental frequency of the supply current is ν max = 10000 min −1 , we can estimate the relative error δ caused by this approximation, as described in [20]: This estimated error is negligible, allowing to use the quasimagnetostatic approximation and, therefore, to exclude the displacement currents in (2) from consideration.
Significant gap between the permanent magnet and the conductive loop influences the density of magnetic field energy in the volume of materials with weak magnetic properties, making the magnetic circuit linear. In the RBPs, gap between the magnet rotor and the stator windings is filled with flowing blood (diamagnetic), and elements of mechanical and hydraulic machines are made of paramagnetic material, e.g. titanium alloy. The absolute values of volumetric magnetic susceptibility of such materials do not exceed 10 −4 [22], [23]. This leads to the same relative error when their magnetic properties are excluded from consideration. Therefore, another simplification is that permanent magnet with residual magnetization is the only element of the system having pronounced magnetic properties.
The electric current flowing through the material of conductive loop is described by the following equation, also known as Ohm's law:Ē where σ is the conductivity of the loop material. Given the introduced simplifications and (13), (1)-(4) and (8) take the following form: Both last terms of (15) correspond to the term P C in (7) and toĒ ·J in (8). The first of them is the energy dissipation due to the heat produced by electric current flowing through the loop material. And scalar product of the torque densityτ z and the angular velocity fieldω introduces power consumption of the work performed during the relative rotation of the system elements about a fixed z axis.

B. DEVELOPMENT OF ANALYTICAL DESCRIPTION
Further, (15) is integrated over a limited volume V , since the current in the loop is limited by the finite volume of the loop itself, and a finite volume boundary, where the magnetic field strengthH becomes negligible, can always be found. Thus, difference between integral over the limited volume V and integral over infinite volume is always less than any predetermined positive number. Therefore, the integral form of the energy conservation law is: Then each integral in (16) is considered individually.

1) ELECTROMAGNETIC ENERGY FLUX
According to the divergence theorem [18], where S V is the surface bounding the volume V . The surface integral in (17) determines the rate of electromagnetic energy transfer from volume V through the surface S V into the external space. Due to quasi-static approximation, there is no electromagnetic radiation. This allows to state that the energy flux through the surface S V is nonzero only in the area S V , where surface S V intersects the terminals connecting the conductive loop to an external current source. According to [21], the integral of Ē ×H over the surface S V is opposite to the current in the loop I multiplied by the difference of potentials U s between terminals connecting the loop to the current source:

2) RATE OF CHANGE OF THE MAGNETIC FIELD ENERGY
The first term in the right side of (16) is the rate of change of the magnetic field energy that given (6) can be represented as follows: whence the following equation for the energy increment W over (t − t 0 ) is obtained: Hereinafter, (t 0 ) means integration of the corresponding integrands in terms of time, since all of them are timedependent quantities. Also, most of explicit designations of time dependence are downed for conciseness.
Since in the considered electromagnetic system the magnetic field is generated by the current flowing in the conductive loop and by the permanent magnet rotor, resulting field H is a superposition of the loop fieldH L and the permanent magnet fieldH PM :H Therefore, change of magnetic field energy in the system splits into three parts: change of the loop field energy W L , change of the permanent magnet field energy W PM , and change of the coupling energy W L,PM : Terms in (22) are represented separately given the substitution of (21) and (6) into (20) as follows: where V PM is the volume of the permanent magnet as the only material with magnetic properties. W L is the energy of magnetic field produced by the loop current in free space. The energy of the permanent magnet W PM includes the energy of magnetic field produced by the magnet in free space and the energy stored in its magnetized material. Both components of W PM have arisen before initial time instant t 0 and remain constant during operation of the system with permanent magnet driven in a fully reversible recoil region. The first term of the coupling energy W L,PM is the interaction between two sources of magnetic field in free space, and the second term is the impact of the loop field on the permanent magnet material, generally causing a displacement of the operating point on its demagnetization curve. This last term is due to the fact that magnet material is not absolutely magnetically hard.
Hence, with constant energy of permanent magnet and zero change of coupling energy in the entire volume V given one spatial degree of freedom for the conductive loop, total energy change of the system reduces to energy change produced by the loop. Therefore, we obtain Differential of magnetic induction dB can be expressed through differential of magnetic vector potential dĀ, as in [20], with the equation: Considering substitution of (27) into (26) and property of divergence ∇ · H L × dĀ , we obtain the following: The term ∇ · H L × dĀ is zero. Further, application of the Ampere's law for magnetostatics, determined by (14), gives: Since there is no current outside the conductive loop, it is possible to pass from the integral over the volume V to the integral along the loop C with current I , and then, using Stokes' theorem [18], move to the integral over the surface S C bounded by C: Next, substitution of (27) into (30) provides: is the increment of the flux linkage of the loop. Thus, it is shown that the increment of the magnetic field energy in the considered system is equal to the integral of the current in the loop I over the flux linkage . According to [21], the Ampere's law for magnetostatics (14), the vector potential definition (27), and the Coulomb gauge ∇·Ā = 0 provide the equation Ā = −µ 0J that is the Poisson's equation with the following solution: wherer 0 is the radius vector to the observation point; andr 1 is the radius vector to the element dV . Substituting the (33) into (29), we obtain: As it was shown previously, we can go to the integrals over two contours instead of both integrals over the volume V , considering zero electric current at t 0 and locating the observation point at the conductor axis of the loop: Contours C r 0 and C r 1 should not coincide, since this will lead to an uncertainty in the result. One of the contours runs along the conductor axis and another one, along the conductor surface. Indirectly, this fact confirms that it is impossible to apply the approximation using zero thickness conductors in some cases of electrodynamics.
The double integral in the result of (35) depends only on geometric parameters of the conductive loop and corresponds to the loop inductance L. The change in the magnetic field energy of the loop is expressed through the inductance L as follows: Combining (31) and (36), we obtain: As a result, we obtain the following equation for flux linkage :

3) ENERGY DISSIPATION
The following equation holds with the loop made of an ohmic conductor and at zero current outside the loop: where R is the ohmic resistance of the loop.

4) MECHANICAL ENERGY PRODUCTION
Since the conductive loop is assumed rigid, the last term in (16) can be transformed into macroscopic parameters: where T z is the torque produced by relative rotation of the loop and the permanent magnet about fixed z axis; and ω is the angular velocity of this rotation. Further, we consider the loop with the current I (Fig. 4) rotated at angle dϕ relative to the z axis from position C 0 to position C dϕ in the magnetic fieldB and apply the approach from [24].
The force exerting the segment of the loop dl in the magnetic fieldB is determined by the magnetic component of the Lorentz force in (5) as follows: The torque produced by the given force relative to the z axis is equal to: Hereinafter,ē z ,ē ρ ,ē ϕ are unit vectors directed along the corresponding axes of the cylindrical coordinate system (ρ, ϕ, z). We transform (42) to the cylindrical coordinates with the substitutionr = ρē ρ + zē z : Further, givenē z ×ē ρ =ē ϕ , we obtain: Next, we introduce a surface element ds equal to dr × dl and to ρdϕ ē ϕ × dl . Therefore, it is possible to make the substitution ρ ē ϕ × dl = ds/dϕ into (44) with the following result: Integrating (45) over the surface S bounded by C 0 and C dϕ , we obtain the equation for the loop torque T z : Taking (32) into account, one can note that the integral in the resulting part of (46) is the increment of the flux linkage d : allowing to write (40) as follows: Analysis of the terms in (16) allows to rewrite it with lumped parameters from (18), (36), (38), (39) and (48): Thence, the equation of voltage drop in the considered loop is obtained: (50) VOLUME 9, 2021 The first term describes the voltage drop due to the selfinduction phenomenon. The second term corresponds to the voltage drop at the ohmic resistance of the loop. And the third term is the back electromotive force (EMF) produced by the relative movement of the loop and the permanent magnet.
Also, (50) can be written as follows: The flux linkage included in (51) is defined as follows: where PM is the flux linkage with the magnetic field of a cylindrical permanent magnet with transverse magnetization.

III. DETERMINATION OF ELECTRIC MACHINE TYPE AND OPTIMAL CONTROL SIGNAL WAVEFORM A. BACK ELECTROMOTIVE FORCE WAVEFORM
Further, we determine how the flux linkage PM changes during relative rotation of the loop and the permanent magnet with angular velocity ω. We also determine the waveform of the back EMF by the time derivative d PM /dt. For this, we find analytical expression for spatial distribution of the permanent magnet field with transverse magnetization, shown in Fig. 5. We use scalar potential of the magnetic field, defined through the concept of magnetic charges, as proposed in [25]: where r is the volumetric density of magnetic charges;r is the radius vector of the observation point;r is the radius vector to an elementary magnetic charge. Then, we turn to the cylindrical coordinate system (ρ, ϕ, z) and to the surface density of magnetic charges ς , determined through the projection of the magnetization vectorM as follows: This transition allows to integrate over the magnet surface: where a and L are the dimensions of the magnet. The transition to the magnetic field is carried out through the definition of the scalar potential:H Equations (55) and (56) show that dependence of spatial distribution of the magnetic field strength on the coordinate ϕ is sinusoidal for a finite cylindrical magnet. Therefore, given the previous calculations, the expression for flux linkage PM , included in (52), is obtained as follows: where ψ a is the flux linkage amplitude determined by the magnetization of the permanent magnet and its geometric parameters. From (51), (52) and (57), one can obtain the following differential equation describing the processes in the considered system:

B. SUPPRESSION OF GEOMETRIC DEVIATIONS
Equation (57) is obtained for a permanent magnet of ideal cylindrical shape. However, geometric deviations occur during the RBP assembly as a result of manufacturing tolerance in various parts, such as deviations in permanent magnet shape and/or misalignment between axis of rotation and axis of symmetry of the magnet. These deviations lead to a distortion in the change of flux linkage PM , introducing high-frequency harmonics that destabilize the control and decrease overall efficiency of the system. Hereinafter, we consider that any geometric deviation introducing field error can be reduced to axis misalignment leading to the same error. Further, we evaluate the magnetic field error produced with this effect by providing the example of the asymmetric installation of the permanent magnet, illustrated in Fig. 6. For this, we again use (53) and the following expansion in a series [26]: 164706 VOLUME 9, 2021 FIGURE 6. Illustration of asymmetric installation of the permanent magnet, where G is the non-magnetic gap; a is the radius of the permanent magnet;r is the radius vector from the center of aligned magnet to the observation point;r m is the radius vector from the center of misaligned magnet to the observation point; andr err is the alignment error.
where P i r,r are the Legendre polynomials forming an orthogonal system. As a result, we obtain: The value of 0 PM (r) is zero, since there are no magnetic charges. Given P 1 cos rr = cos rr , 1 PM (r) can be expressed in terms of the dipole magnetic momentm: The dipole scalar potential 1 PM (r) corresponds to the scalar potential of an ideally cylindrical shape magnet defined in (55). The dipole magnetic momentm can be expressed for a permanent magnet through the material magnetizationM as:m For r > a r err , values of the Legendre polynomials P i cos rr tend to unity. With i increasing, the expression r i /r i+1 in (60) that determines the scalar potential of higherorder magnetic moments decreases stronger than the term 1 PM (r). Thus, prevalence of the sinusoidal field component of the rotating permanent magnet increases with distance from the surface of the permanent magnet. Given this and the monotonic change in the scalar potential with increasing distance, we conclude that influence of deviations decreases with increasing non-magnetic gap.
This suppression of deviation influence is inherent for the axial-flow RBPs, in contrast to industrial electric motors with a required minimum air gap in the magnetic circuit that causes relatively high sensitivity to deviations. Also, analysis of (60) shows the impossibility to generate arbitrary waveforms of the flux linkage in axial-flow RBPs (e.g., a waveform close to triangular shape that is necessary for producing the broadband back electromotive force of trapezoidal shape).
The exact solution of (55) and expressions for the magnetic field components of a cylindrical permanent magnet with transverse magnetization are given in [25]. These expressions are relatively complex, and we use some simplification to estimate the error produced by the asymmetric installation of the permanent magnet. Note that this magnetic field decays as a power of reciprocal distance from the magnet axis within the range of 2 (infinite cylindrical permanent magnet [27]) and 3 (point magnetic dipole [27]): whereH c is the magnetic field strength of an infinite thin cylindrical magnet; andH d is the magnetic field strength of a point magnetic dipole. Hence, for a finite cylindrical magnet, where δ am H (r, r err ) is the relative error of magnetic field value in case of axis misalignment; r err is the magnet misalignment error; and g takes the value between 2 and 3.
Therefore, the relative field error decreases as a power of reciprocal r err between square and cube. In the considered system, conductive loop is located at r = a + G, where a is the radius of permanent magnet; and G is the non-magnetic gap. In industrial electric motors, size of the gap is close to the manufacturing tolerance of the magnet radius. And in axial-flow RBPs, the gap is of the magnet radius or larger. Therefore, analytical comparison of maximum relative field error in industrial electric motors and axial-flow RBPs gives the following inequality: where δ am RBP is the relative filed error in axial-flow RBPs; and δ am EM is the relative field error of industrial electric motors.
For dimensions and tolerances in one of typical axialflow RBPs (a = 4.75 mm; G = 3.75 mm; and r err = 0.003 mm) [11], distortion of the permanent magnet field in the winding volume is up to 0.11 %. Therefore, neglect of geometric deviations in the description leads to the same value of the relative field error.
Thus, previous theoretical justification and small value of the error indicate that increased non-magnetic gap suppresses geometric deviations in the construction of axial-flow RBPs and leads to attenuation of high-order harmonics, and therefore, provides sinusoidal waveform of back EMF. Another consequence of increased non-magnetic gap and cylindrical shape of the permanent magnet rotor is non-salient pole nature of the electric machine having parameters of the magnetic circuit and inductance of the stator windings independent of the rotation angle.

C. OPTIMAL CONTROL CURRENT WAVEFORM
Conversion of the supplied electric energy into mechanical energy transferred to the mechanical machine is described by (47). To find an optimal waveform of control current, we define the optimality criteria as the maximum efficiency of electric machine. Equation (49) shows that linear losses are proportional to the squared current. Thus, optimal waveform of control current should correspond to the minimum RMS current value at the maximum produced torque. To solve this problem, we use the optimization methods from [28]- [30]. A continuous periodic current I in the winding and a continuous periodic change in the flux linked with winding d PM /dϕ expand in trigonometric Fourier series as follows: where i n q , i n d , k m q , k m d are the Fourier series coefficients given as q-and d-axis components of the rotor multiple reference frame [29], [30]; M and N are finite positive integers.
Substitution of (65) and (66) into (47) gives: Herein, we consider only the average torque T av and correspondingly use only the terms of series satisfying the conditions m − n = 0. Also, we should increase M or N to achieve H = max {M , N }, obtaining: Next, we introduce the following notations: therefore, obtaining: Rearranged (71) is:k where T * av is the desired average torque. Equation (72) contains 4H unknowns yielding infinitude of solutions. From this infinitude, it is necessary to find a solution with a minimum current value I = I min , and the maximum efficiency of electric machine. This task is performed by the method of minimum norm solution [31]: orī where k = 2T * av /(ψ akk T ).
In our case, the quantity PM is explicitly determined by (57), and then, solution to (74) is trivial: Thus, it is shown that with flux linkage varying sinusoidally in the electric machine of axial-flow RBPs, the waveform of the control current coinciding with the waveform of back EMF is optimal. This analytical conclusion is confirmed in practice by [28], [30], [32]- [34].

D. ELECTRIC MACHINE TYPE
Eventually, considered electromagnetic system consists of the permanent magnet and the conductive loop with electric current, which, if necessary, can be replaced by a multi-turn winding. However, this configuration does not provide the ability to start the machine at an arbitrary position of the rotor at rest. At least two (preferably orthogonal) windings ( Fig. 7 (a)) with the possibility of independent current regulation are necessary to start the machine. Therefore, the stator should have at least three terminals in case of combining one terminal for two windings, or four terminals in case of independent connection of windings. It is desirable to minimize the number of stator electrical leads, since larger number complicates the driveline design and increases the number of output channels, reducing the reliability of the control system. Also, it is desirable to have the complete symmetry of stator windings. This allows to unify the output channels of the control system and to optimize the load on the power source. The requirements for minimum number of leads and symmetry of windings are met by a three-phase structure, classical for alternating current circuits ( Fig. 7 (b)). Based on the foregoing, it is proposed to construct the stator of axial-flow RBP of a three-phase configuration. Similar expressions for stator voltage, mechanical torque and optimal current waveform in three-phase systems are given in detail in [28], [29], [34].
Finally, we examined the key features of axial-flow RBPs, allowing to define the device as a three-phase non-salient pole synchronous machine with a permanent magnet rotor and a sinusoidal waveform of back EMF. The device is described by (58), obtained from classical electrodynamics with a number of simplifications. These simplifications lead to errors presented in Table 1, yielding overall relative error less than 0.2 %.

IV. CONCLUSION
In summary, thorough description of electromagnetic coupling of stator and rotor in axial-flow RBPs was performed. The actual equation of stator voltage was derived from fundamental Maxwell's equations describing this coupling through quasi-magnetostatic approximation of electromagnetic field in the simplified model consisting of permanent magnet and conductive loop rotating in free space with one spatial degree of freedom. Additionally, relative field error, created by the neglect of geometric deviations introduced through manufacturing tolerances, was estimated for a typical axial-flow RBP. The upper limit of error introduced by the simplifications was estimated less than 0.2 %, allowing to determine the finite accuracy of the proposed description and its influence on the RBP control. Sinusoidal waveform was shown to be the only waveform for back electromotive force in axial-flow RBPs, in contrast to industrial electric motors. Also, it was theoretically justified as the optimal control signal waveform. Eventually, electric machine of axial-flow RBPs was formulated as a three-phase nonsalient pole synchronous machine with a permanent magnet rotor, based on the provided description and additional engineering considerations. The presented description can be used for reasonable construction of axial-flow RBP and development of control system with increased power efficiency and optimized weight, size, and reliability, crucial in MCS.
Future research is addressed to numerical and experimental validation of the proposed theoretical description and to subsequent creation of an RBP control system with the implemented optimal waveform of control signal.