Analysis and design of excitation system in homopolar inductor machine

Fu Xinghe Fu Xiangda Xu Yutian He Hang

(School of Electrical Engineering, Southeast University, Nanjing 210096, China)

Abstract：The excitation system of the homopolar inductor machine (HIM) is analyzed and designed to establish the design approach and evaluation criteria of the excitation system. The finite element method is used to calculate the three-dimensional electromagnetic field in the HIM, and the distribution and characteristics of the magnetic field are described. The analytical method is applied to investigate the design process of the excitation winding. The ratio of the axial length of the armature winding to the excitation winding and the ratio of the axial length of the working air-gap to the non-working air-gap are investigated by the numerical calculation method. A prototype HIM is designed and manufactured, and some experiments are implemented to verify the correctness of the theoretical analysis and numerical calculation results. The research results show that the established design method of the excitation winding is practical and feasible. Under the conditions of constant excitation magnetomotive force or constant excitation power, the optimum range of the axial length ratio of the armature winding and the excitation winding is 0.45 to 0.5. The optimal axial length of the non-working air-gap can be determined by the ratio of the stator inner diameter and pole-pairs.

Key words：homopolar inductor machine;excitation system;exciting coil;axial length ratio

The homopolar inductor machine (HIM) has the favorable advantages of an adjustable air-gap magnetic field, robust rotor structure, high reliability, and low idling losses. As a result, the HIM has renewed interest as an appealing candidate in flywheel energy storage[1]and capacitor charge power supply[2]. The power or torque density of the HIM is not high enough due to the homopolar magnetic field. Therefore, a multi-unit out-rotor HIM is proposed to boost the output power and stored energy[3]. Another investigated four-stator-cores HIM, which consists of an alternator and a motor, enjoys the merits of a high-voltage and high-frequency discharge performance[4]. A novel excitation compensation method is proposed to improve the power density as a capacitor charge power supply by installing a compensation winding[5]. Furthermore, the novel HIM with multistage stator cores is developed to solve the low power density owing to the length limitation and homopolar air-gap flux density[6].

The homopolar air-gap flux density in the HIM raises another question whether the analytical methods applied to the conventional machine can be used on the HIM? The controversy whether the two-reaction theory is applicable or not to the HIM is settled and it is applicable to the HIM[7]. Moreover, it is feasible to regard the HIM as a wound field synchronous machine (WFSM) or a combination of a WFSM and synchronous reluctance machine, which helps to simplify the HIM analysis[8]. An essential 3D flux distribution in the HIM brings more challenges to predicting machine behavior[9]since 3D finite element analysis (FEA) is a quite tough and time-consuming task when optimization design is implemented[10-11]. A nonlinear equivalent magnetic circuit and an air-gap permeance function are introduced to avoid the 3D magnetic field calculation[12]. Furthermore, a 2D analytical model is proposed to calculate the magnetic field and back-EMF (electromotive force)[13].

The relationship between the flux density in the working air-gap with the arced slot and the salient in the HIM is analyzed by using the finite element method (FEM)[14]. The optimal geometrical parameters of the tooth are obtained to achieve the maximum back-EMF in the armature winding[15]. The non-overlapping coils are used to obtain the maximal fundamental winding factor and back-EMF in a 32 kW HIM[16]. The core loss in the HIM with DC-biased magnetic induction is different from that in the conventional electrical machine with sinusoidal induction[17]. The inductance mathematic model and characteristics are presented and analyzed[18], and the operation mode of the HIM is discussed[19]. The performances of the HIM with a diode-bridge rectifier and capacitive load are analyzed, and the simplified average value mathematical model is established[20]. Narayanan et al.[21]developed an interesting HIM with three sections of the rotor, in which two sections work as alternators, and another section works as a motor.

Some investigations described the characteristics of the HIM in detail. However, except for the peripheral space competitions between the stator teeth and armature conductors, the axial space competitions between the armature windings and the excitation coils must be considered when designing the HIM. How to resolve this conflict and how to properly allocate the limited axial length are vital to the performance improvement of the HIM. At present, they are far from being adequately addressed.

This paper investigates the excitation system of the HIM by analytical and numerical methods. The magnetic field distribution of the HIM is revealed by finite element calculation. The design principle of the excitation coil in the HIM is discussed in detail, and an example is presented to illustrate the design process. The axial length ratios are investigated. The optimal parameters under different constraint conditions are predicted. Finally, a prototype HIM is manufactured, and some experiments are implemented to verify the correctness of the finite element calculation and theoretical analysis.

1 Structure and Magnetic Field

The schematic structure of the HIM is shown in Fig.1. The HIM comprises of a 9-slots stator core with three-phase concentrated windings, a rotor with 4-teeth and 4-slots, excitation coils, and a frame. The excitation coils are fixed by the coil brackets and placed between the end windings and the end cap. Since there is no brush for the excitation coils, less maintenance is required for the HIM. In Fig.1, the lines with arrows denote the flux path. The flux generated by the excitation current goes axially through the rotor yoke, then radially through the working air-gap into the stator teeth, then axially along the frame, again radially inward passing the non-working air-gap, and finally returns to the rotor yoke. The magnetic field of the HIM is a complex 3D field. Therefore, 3D finite element computation is required to illustrate the machine performance. Fig.2 shows the flux distribution in the HIM under no-load, which is consistent with the theory analysis above.

Fig.1 Section view of the HIM

Fig.2 shows that both the working and non-working air-gap magnetic fields are homopolar fields. For the non-working air-gap, the magnetic field is almost constant.

Fig.2 Three-dimensional magnetic field of the HIM

But to the working air-gap, the magnetic field looks like a rectangle waveform with low values under the rotor slots and high values under the rotor teeth. Besides, the magnetic field in the working air-gap changes along the axial direction, which is quite different from the traditional electrical machine, such as the induction motor, PMSM, and WFSM. The homopolar and uneven magnetic field makes it challenging to design the HIM. The calculated line-to-line back-EMF in the HIM under no-load and rated speed is shown in Fig.3. The waveforms are close to sinusoidal ones, although there are specific harmonic components.

Fig.3 Waveforms of the back-EMF in the HIM

2 Excitation Coil Design

The magnetomotive forceFgenerated by the excitation current is defined as

F=NIf

(1)

whereNis the number of turns of the excitation coil, and the exciting currentIfis expressed as

(2)

whereUfis the voltage of the field power supply;Rfis the resistance of the exciting coil andRfis given as

(3)

whereρis the resistivity of the copper;ktis the temperature coefficient of the resistivity, andkt=1 at normal temperature;lsumis the total length of the excitation coil;Scuis the cross-section area of the excitation wire;dcuis the diameter of the excitation wire;laveis the average length per turns of the excitation coil andlaveis given as

(4)

whereDaveis the average diameter of the virtual cylinder annulus filled by the excitation coil;Doutis the outer diameter of the cylinder annulus; andDinis the inner diameter of the cylinder annulus.

According to Eqs.(1) to (4), the magnetomotive forceFcan be rewritten as

(5)

Eq.(5) shows that diameterdcuof the excitation wire is determined when the excitation magnetic potential, excitation voltage, and outer and inner diameters of the cylinder filled by the excitation coil are constant. The excitation magnetic potential has nothing to do with the number of turns of the field winding.

The exciting powerPfcan be expressed as

(6)

Eq.(6) shows that the excitation power is inversely proportional to the number of turns of the field winding.

The axial cross-section areaSof the virtual cylinder annulus can be defined as

(7)

wherekslotis the copper space factor of the excitation coil;Jis the electric current density;Lefis the axial length of the virtual cylinder annulus;ksis a comprehensive coefficient and it is given as

The above analysis shows that the relationships between the magnetomotive force, the exciting power, the exciting current, and the number of turns of the field winding are relatively complicated. Fig.4 describes the complicated relationships when the magnetomotive forceFand the exciting voltageUfare constant.

Suppose that the axial cross-section areaSlocates at pointA. According to Eq.(7), an appropriate pointCon the vertical coordinates can be determined. Then, according to Eqs.(5) and (6), pointsD,E, andFcan be calculated, respectively.

An example is given to verify the effectiveness of the proposed method. Given thatF=1 000 A,Uf=31 V,ρ=1.678×10-8Ω·m,kt=1,Dout=68 mm,Din=47

Fig.4 Relationships between the variables in the process of the excitation coil design

mm,Lef=46 mm, andkslot=0.8, we can obtain the calculation results as follows:S=4.83×10-4m2,N=2 395,If=0.42 A,Pf=13 W,dcu=0.35 mm, andJ=4.3 A/mm2.

3 Numerical Analysis of Axial Length Ratio

3.1 Axial length ratio of the armature winding to the excitation coil

Ignoring the axial length of the end winding and the end cap, the whole axial lengthLeof the HIM mainly depends on the axial lengthLefof the virtual cylinder annulus filled by the excitation coil and the axial lengthLeaof the armature winding, which means that

Le=Lea+Lef

(8)

The ratio of the two axial lengths may affect the operating characteristic of the HIM whenLeis constant. Therefore, coefficientγLis defined as

(9)

The influences of the magnetomotive force and the exciting power on the electromagnetic design are considered. Therefore, the following two cases are discussed based on the finite element method.

Suppose that the total magnetomotive force generated by the excitation current is constant. The calculated flux linkage coefficientφof the armature winding and the calculated exciting powerPfwith differentγLare shown in Fig.5. Here, coefficientφis defined as the ratio of the flux linkage to the number of turns of the armature winding. Fig.5 shows that coefficientγLmay affect the values of the output voltage and the exciting power. It is helpful to increase the axial length of the armature winding to produce a greater back-EMF. However, a longer axial length of the armature winding can lead to more excitation power. Moreover, when coefficientγLexceeds a critical value, the change rate of the flux linkage coefficient decreases, and the change rate of the exciting power increases. For the curves in Fig.5, the utilization rate of the exciting power reaches a maximum when the two curves have the longest vertical distance. For the physical parameters of the test HIM, the proper coefficientγLshould be in the range of 0.45 to 0.47. Therefore, according to the defined parameters (Lef=46 mm) above,Leashould be close to 40 mm.

Fig.5 Flux linkage and exciting power with constant magnetomotive force

Suppose that the exciting power is constant. With the increase in coefficientγL, the calculated flux linkage coefficientφof the armature winding, and the calculated magnetomotive forceFare presented in Fig.6. When coefficientγLis small, the magnetic saturation appears in the stator core of the HIM owing to a shorter axial length of the armature winding and larger field ampere-turn. As a result, the flux linkage and the back-EMF with a smallerγLare lower than those with a largerγL. With the increase inγL, the flux linkage increases. When coefficientγLexceeds a critical value, both the flux linkage and the back-EMF decrease owing to a smaller axial length ofLefand less magnetomotive force. Therefore, the proper coefficientγLis in the range of 0.45 to 0.5, which is almost consistent with the above results.

Fig.6 Flux linkage and field ampere-turn with constant exciting power

3.2 Axial length ratio of the working air-gap to the non-working air-gap

There are two air-gaps in the HIM: the working air-gap and the non-working air-gap. The radial lengths of the two air-gaps are the same. However, the axial length of the non-working air-gap is less than that of the working air-gap, and the circumferential length of the non-working air-gap is greater than that of the working air-gap. Therefore, the magnetic reluctances of the two air-gaps are different. The effective air-gap flux density in the HIM may be affected by the magnetic reluctance of the two air-gaps. Under the action of 1 000 A field ampere-turn, the flux density in the two air-gaps with specific parameters (Lea=40 mm,Lgap2=8 mm) are calculated by the finite element method as shown in Fig.7. The flux density in the non-working air-gap reaches 1.7 T, while in the working air-gap, it is only 0.6 T. Therefore, the axial length ratio of the two air-gaps is quite vital and should be investigated to improve the density of the working air-gap.

Fig.7 Flux density in the working air-gap and the non-working air-gap

A coefficientksgis defined as

(10)

whereSshaftis the cross-section area of the shaft;Sgap2is the circumferential area of the non-working air-gap;Dis the inner diameter of the stator iron;Lδis the radial length of the two air-gaps.

Eq.(10) can be substituted by Eq.(11) sinceLδis much smaller thanD.

(11)

To avoid saturation,suppose that coefficientksgis equal to 1. Therefore,Lgap2should be equal toD/4.

To consider the magnetic saturation in the HIM, a magnetic calculation based on the FEM is implemented to validate the analysis process above. The inner diameterDof the stator iron is 45 mm. Also, the total axial length of the two air-gap is 52 mm. With different axial lengths of the non-working air-gap, the flux density in the two air-gaps is computed and shown in Fig.8.

Fig.8 shows that the flux density in the working air-gap increases with the increase in the axial length of the non-working air-gap. However, the axial length of the armature winding decreases while the axial length of the non-working air-gap increases. The flux linkage of the ar-mature winding is relative to both the flux density in the working air-gap and the axial length of the armature winding. Therefore, the flux linkage of the armature winding does not increase with the increase in the axial length of the non-working air-gap. The computed results of the flux linkage are shown in Fig.9. From Fig.9, it can be seen that the optimalLgap2is 12 mm, which is slightly higher than that of the analytical calculation (45/4=11.25 mm).

Fig.8 Flux density in the two air-gaps

Fig.9 Flux linkage of the armature winding

The numerical calculation results verify the correctness of the theoretical analysis. The figure 4 in the denominator of Eqs.(10) and (11) presents the pole-pairspof the HIM. Therefore, the general formula to determine the axial length of the non-woking air-gap and ensure low magnetic saturation can be expressed as

(12)

Given that both the outer diameter and the total axial length of the HIM are constant, the volume of the HIM does not change. The power density of the HIM is defined as the output electrical power divided by the volume. With the variety in the axial lengths of the non-working air-gap, the numerical computing results of the power density are described in Fig.10. In Fig.10, the maximum value of the power density is selected as the reference of the per-unit value. The maximum value of the power density is reached when the axial length of the non-working air-gap is 14 mm, which deviates slightly from the calculation results. The reason is that the resist-ance of the armature winding decreases with the increase in the axial length of the non-working air-gap when the total axial length remains unchanged.

Fig.10 Power density of the HIM

The optimal axial length of the non-working air-gap is 12 mm when considering the weight and manufacture of the HIM.

4 Experiment

A prototype HIM with 8-poles and 9-slots is designed and tested to verify the result of the theoretical analysis and numerical calculation. The specifications are listed in Tab.1. The product photos are shown in Fig.11.

Tab.1 Primary specifications of the prototype HIM

Fig.12 presents the experimental result of the line-to-line back-EMF of the HIM under the no-load and rated speed. The experimental waveform has 9 small ripples within one electrical cycle owing to the influence of the stator slot-open. The amplitude of the measured back-EMF under no-load is 46 V. Compared with the finite element computation, there is little error, which is mainly caused by manufacturing tolerances and some variations in the magnetic properties.

Fig.12 Waveform of measured line-to-line back-EMF of the HIM under no-load

Some experiments are designed and implemented to validate the axial length allocation of the HIM. First, the line-to-line back-EMF of the HIM is recorded when the axial length of the armature windingLeais equal to 50 mm. The HIM prototype is disassembled, and the salient poles of the rotor are cut short until the axial lengthLeais equal to 45 mm. Then, the output voltage of the HIM prototype is measured again. After that, the measurement experiments are repeated when the axial lengthLeais equal to 42.5 and 40 mm. Fig.13 describes the variations of the magnitude of the line-to-line back-EMF under different conditions. It can be seen that the amplitudes of the line-to-line back-EMF are almost equal under different situations when the excitation current is over 0.4 A. Therefore, considering material utilization, the optimal axial length of the armature winding is equal to 40 mm, which is consistent with the theoretical analysis and calculation.

Fig.13 Magnitude of the line-to-line back-EMF with different axial lengths Lea

5 Conclusions

1) The design approach of the excitation winding proposed in this paper is practical and instrumental in determining the optimal parameters of the field winding.

2) The proper coefficientγLis in the range of 0.45 to 0.5, which applies both to the constant exciting magnetomotive force and to the constant exciting power.

3) The axial length of the non-working air-gap in the HIM should be equal to the ratio of the stator inner diameter and pole-pairs to ensure higher material utilization and lower magnetic saturation.