CHEN Lei, HE Xiaoxia, LI Dongmei, LI Haixia, ZHANG Rong

(Department of Precision Instrument, Tsinghua University, Beijing 100084, China)

Abstract: A suspending metal rotor is the core component of electromechanical gyroscope, reaction sphere and other instruments, and the temperature of the rotor has an impact on the performance of the instrument. It takes a long time for the instrument to enter thermal equilibrium state from a cold start. A method to contactless heat a spinning metal sphere that suspended inside a vacuum housing without disturbing its motion is studied. When a spinning metal sphere is placed in an alternating magnetic field, eddy current and induction heat is generated on metal sphere as expected, but simultaneously additional disturbing torque is also produced. To inductively heat the metal sphere without applying disturbing torque to change its momentum, an elliptical rotating magnetic field is proposed to heat the sphere with no torque applied during the heating process. Firstly, the existence of the elliptical rotating magnetic field with zero moment applied induction heating is proved by theoretical calculation. Then, a FEA model with Helmholtz coils is modeled and the simulation result perfectly verified the theory. To coincide with the engineering prototype, a compact two-phase two-pole winding with high magnetic permeability is modeled and the simulation results show that the compact design also has null-torque potential by adjusting phase difference between the two alternating magnetic field. Finally, a prototype is designed and built. Null-torque heating is achieved by using dichotomy method to fine-tune the phase angle. The phase angle differs from the finite element simulation result by 1.4 °, and thus verified the theory.

Key words: contactless heating, alternating magnetic field, elliptical rotating magnetic field, finite element analysis

Temperature stability has an impact on the performance of inertia instrument[1,2], and temperature control system is usually employed[3]. A suspending metal rotor is the core component of electromechanical gyroscope, attitude control reaction sphere[4-6]and other similar instruments[7-10]. It takes a long time for the rotor to enter thermal equilibrium state from a cold start. Pre-heating is usually used to speed up the warm-up process of the key component. The current situation is that a sphere is contactless suspended in a vacuum housing and only exchanges heat through heat radiation and it takes more than ten hours to warm up the sphere to a thermal equilibrium state. In many cases, the temperature difference between the vacuum housing and sphere is still significant after half a day since starting up which is not allowed in certain application.

Other than thermal radiation, heating a metal through inducted eddy current can be used in practice. In general, it is much more efficient to heat the sphere to a certain temperature using inducted eddy current than heat radiation. However, eddy current interacting with source magnetic field usually induces torque. It is difficult to apply inductance eddy by magnetic field while no disturbing torque is introduced.

In this paper, we propose an elliptical rotating magnetic field to apply heat to the sphere with null-torque. Firstly, we prove the existence for such a solution, i.e., there must be a type of elliptical magnetic field to introduce null-torque while inductance eddy occurs. Then, we developed a 3-D Finite Element Analysis (FEA) model and engineering prototype to verify the theoretical results by simulation and experimental study.

We start in Section 1 from determining the torque of a sphere in an elliptical rotating magnetic field and condition for null-torque. In Section 2, we use FEA to verify the null-torque theory. Furthermore, a compact two-phase two-pole winding model for null-torque is designed and verified its null-torque capability using FEA. In Section 3, an experiment device is developed and tested. We conclude the paper in Section 4.

1 Theoretical calculation

1.1 Geometry and Torque equations

Consider a conducting thin shell in a rotating magnetic field. The shell is contactlessly suspended and rotating with an angular velocity about the→ axis. Fig.1 illustrates the geometry for determining the torques by a rotating spherical shell placed in a magnetic field.

Fig.1 A rotating thin shell in a rotating frame.

The geometry parameters and physics constants are given in Tab.1.

For simplicity, we denote the following coordinate and symbols.

ω— sphere’s angular velocity.

Equations for the torqueexperienced for an isotropic body is as follows[7].

Where A is the magnetic vector potential, E the electric field,the unit radius vector, r the shell’s radius,ρthe charge per unit volume,τthe resistivity of the sphere,ψ? the electrical potential which is the electrostatic potential for static fieldsψ.

In the absence of static electric fields and charge distributions on the sphere,ψandρboth vanish. The torque equation under these conditions then becomes

1.2 Torque in uniform magnetic field

Firstly, we list some known results of sphere’s torque in situation of uniform magnetic field, then the solution in elliptical rotating magnetic field will be carried out in detailed.

When alternating magnetic field isthe theoretical accelerating torquezTalong the sphere-rotating axis is depicted[7]as follows.

Whereμis permeability of vacuum andis surface resistivity of the thin shell,aandbis the outer and inner radius of the thin shell. For a given sphere, spin speedω, to achieve null torque mean a certain field rotation speedΩ. This is not convenient in engineering.

In uniform magnetic field ofThe theoretical torqueTz2along the sphere rotating axis is as follows[8],

The uniform field produces retarding torque for a spinning metal sphere[8].

To find a magnetic field suitable for generate null torque. Now, we will deduce the torque equation in elliptical rotating magnetic field.

1.3 Torque in elliptical rotating magnetic field

field in

Firstly, we define the elliptical rotating magneticcoordination system using parametric ellipse equation as follows.

The relationship between the magnetic vector follows,

The magnetic vector potential is specified by the differential equation

Take Eq.(3) into Eq.(4), we get

The equation above suggests the solution to Eq.(6) and Eq.(7) is of the form.

Where

According to

Eq.(11) can be expressed in spherical coordination as

Substitute Eq.(12) into Eq.(10) and from Eq.(8),, the torque expression can be reduced to

Substituting the above equation into Eq.(13), we get

Transforming back to the stationary coordinate

The transforming matrixCis

After integrating of Eq.(13), we get the expression for the torque as follows.

According to Halverson and Cohen’s conclusion[8], for a spherical shell,

Substitute Eq.(15) and Eq.(16) into Eq.(14), we get the torque of a spherical metal shell in a rotating elliptical magnetic field.

Averaging in one magnetic field rotating period,

For convenience, elliptical rotating magnetic field Eq.(3) can also be expressed using phase equation as follows.

Where

Substitute Eq.(19) into Eq.(18), we get torque for phase equation

Where

1.4 Solution for null-torque

LetTz= 0and solve the semi-major and semi-minor axis magnitude of the elliptical magnetic field, we get null-torque solution for standard ellipse form.

Where

And for phase form of the solution Eq.(20),Tz= 0 when phaseΦsatisfies

Furthermore, we can easily deduct from Eq.(20)

For a spherical metal shell with parameters shown in Tab.1, the null-torque field condition can be calculated and the result is

If magnetic field meets the condition above, the metal shell will maintain a constant spin speed while being heated.

Tab.1 Parameters and Constants

2 Simulation

2.1 FEA model with Helmholtz coils

To verify the theoretical calculations, a 3D-FEA model of a spherical metal shell in an elliptical rotating magnetic field is developed.

Helmholtz coils is generally used to generate a uniform magnetic field. Helmholtz coils composes a pair of circular coils separated by distance equal to the coils’ radius. When both coils are applied with same direct current, a uniform magnetic field is generated in a small vicinity between the two coils.

To generate a rotating magnetic field, two sets of Helmholtz coils are placed perpendicular to each other. An elliptical rotating magnetic field is generated when two sinusoidal current of the same amplitude but with a phase difference is applied to each pair of Helmholtz coils. The elliptical field is adjusted by changing the phase.

When a spherical metal shell is place in rotating magnetic field, eddy current induced in the shell interacting with magnetic field produces torque. Fig.2(a) shows the magnetic field and Fig.2(b) shows the eddy current of a metal shell rotating at 400 rps placed in magnetic field rotating at 1000rps. Inducted eddy current suggests that the metal shell will generate heat.

Fig.2 FEA result

We simulated torque of the shell with respect to phase ranging from 0 ° to 360 ° at a step value of 1 °. Other simulation parameters is shown in Tab.1.

Fig.3 shows both theoretical and FEA analysis torque with respect to phase of the elliptical rotating magnetic field. Theoretical torque is calculated by Eq.(18). FEA analysis shows that the torque on sphere is null at phase of 2.2°. The phase is very close to that of theoretical method of 2.29° as calculated in Eq.(25).

Fig.3 FEM and Theoretical Torque with respect to phase.

2.2 FEA model with high magnetic permeability core

The above design is ideal to generating uniform magnetic field, but there are two major disadvantages of this design. The device is bulky and the magnetic field it generates is relatively weak due to absence of stator.

To overcome the above disadvantage, core with high magnetic permeability is adopted and a two-phase two-pole design is developed as shown in Fig.4. It can generate a strong rotating magnetic field with compact size shown in Fig.5, the disadvantage of this design is the magnetic field is not as uniform as the field generated by Helmholtz coils.

Fig.4 winding design

Fig.5 magnetic field generated

The magnetic field generated by the device is simulated with similar process as by using Helmholtz coils. Comparing with Fig.2(b), the magnetic field is not as homogenous as the magnetic field generated by Helmholtz coils. However, the relation between phase and torque obtained by FEA simulation shows the compact coil design can also achieve null-torque with a phase of 68.11° as shown in Fig.6.

Fig.6 FEM and Experiment Torque with respect to phase.

3 Experiment

A prototype with structure of Fig.4 is designed to carry out the experimental test. It consists of a two-phase two-pole winding, a stator made of soft magnetic alloy and a spherical metal shell suspended in high vacuum to eliminate air friction. The winding module and experiment setup is shown in Fig.7. A driver circuit to spin up the metal shell and calibrate phase of null-torque is developed. The driver is composed of two parts; a two-channel wave generator followed by an amplifying circuit that produce 200 mA current in the coils to generate the elliptical magnetic field.

Fig.7 pictures of winding module and prototype

We follow the below steps to achieve null-torque.

Step1. Accelerate the rotor to 400 rps with two-phase alternating current.

Step2. Dichotomy method is adopted to adjust the phase to find the null-torque. By synthesizing Eq.(20), Eq.(24) and all the simulating result, the phase with null-torque point between - 90°＜Φ＜90 °is suggested, so we set the phase searching range of （α，β）=（-9 0 °,90° ）.

Step3. To implement dichotomy, firstly, set the phase to midpoint of the interval（α，β）. Then observe the spin speed of the metal shell. If the sphere is accelerated which means the torque is positive, then the null-torque phase lies inIf the sphere is decelerated, the null-torque phase is inIf the metal shell holds its rotation speed, null-torque is achieved. Repeat Step.3 and narrow the range until the null-torque is found.

The experiment process is recorded in Tab.2.

Tab.2 Finding null-torque phase using dichotomy

After implementing dichotomy 8 times, the null-torque phase interval is narrowed down to less than 1° between （-69.3°, 70°）. At this point, it is more suitable to use the dial on the function generator to fine-tune the phase. When the phase is dialed to 69.51°, the spin speed holding stable(< 0.001 Hz/s) and null-torque phase is found. Comparing with the null-torque phase of the FEA result shown in FIG.6, there is an error of 1.4° between the two results. This is due to numerical accuracy of FEM, errors induced by drive circuit.

4 Conclusions

In this paper, we analytically proved that by adopting an elliptical rotating magnetic field that satisfies condition (17), heating a contactless suspended sphere without disturbing its movement could be accomplished. A 3D FEA analysis is conducted, and the condition of null-torque coincides with the theory. Furthermore, we developed a compact two-phase two-pole winding system and verified its null-torque capability using FEA. Finally, an experiment device is developed and null-torque phase is found by using dichotomy method. The experiment result coincides with FEA result with error of 1.4°

Online adjustment of the null-torque phase and the engineering applications will be further studied in the future.