GB2586504A

GB2586504A - Method and apparatus for estimating closed-circuit hysteresis curves for magnetic materials

Info

Publication number: GB2586504A
Application number: GB1912101.1A
Authority: GB
Inventors: Paul Douglas Clewett James; Wade Jack; Nathan Cornelius Robin; Dudding John
Original assignee: Hirst Magnetic Instruments Ltd
Current assignee: Hirst Magnetic Instruments Ltd
Priority date: 2019-08-22
Filing date: 2019-08-22
Publication date: 2021-02-24
Also published as: WO2021032987A1; WO2021032987A4; GB201912101D0

Abstract

A method for calculating the closed circuit hysteresis curve of e.g. a permanent magnet sample combines a mathematical model of the sample with open circuit measurements e.g. from a Pulsed Field Magnetometer (PFM). The model is used to create a hysteresis curve defined by variables which can be adjusted to best fit the measurements. From the best fit variables, the model can be adjusted to simulate the closed circuit hysteresis curve. The model may be a lattice of nodes. The minimum energy configuration of dipoles within the sample may be modelled. Periodic boundary conditions may be applied to model the closed-circuit environment. The local field of the model may be the sum of the external field and the self-demagnetisation field of the dipoles. Hamiltonian or Monte Carlo methods may be used to optimize the model. The invention may particularly be used for testing of rare earth magnets.

Description

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Hirst Magnetic Instruments Limited

METHOD AND APPARATUS FOR ESTIMATING CLOSED-CIRCUIT HYSTERESIS CURVES FOR MAGNETIC MATERIALS

TECHNICAL FIELD OF THE INVENTION

This invention relates to a method and apparatus for more accurately estimating closed-circuit hysteresis curves for magnetic materials based on open-circuit readings.

BACKGROUND

There are many circumstances in which it is necessary to know the magnetisation of a sample of magnetic material in response to an externally applied magnetic field. Important examples include the design and manufacture of components for electric vehicle motors, wind/wave turbine generators, and quality control in permanent magnet production.

Permanent magnets are characterised by their full-loop hysteresis curve (hysteresis loop). This curve is measured by driving a positive to negative oscillating external magnetic field across a sample, such that the sample achieves full saturation in both -2 -

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polarisations. The second quadrant of the hysteresis loop (see Fig. 2), also known as the demagnetisation curve, is particularly important because magnets usually work in opposition to an applied field. Historically, the demagnetisation curve has been measured in a permeameter by placing a sample magnet in a closed magnetic circuit, as illustrated in Fig. 1. Iron pole pieces p1 and p2 which form part of a low-reluctance iron magnetic circuit a are used to deliver magnetic flux to the poles of the sample c provided with a pick-up coil b. Magnetisation coils d generate the magnetic flux in the magnetic circuit a producing a flux path e. The design has evolved as a practical way of producing a large, steady-state and uniform magnetic flux across the sample magnet. There are two important limitations with this traditional permeameter design: (i) Modern rare-earth magnets are so powerful that the required saturation field also saturates the iron magnetic circuit. Such rare earth magnets cannot therefore be accurately characterised using pernneameters.

(ii) In order to take a quasi-static measurement the variation of the external magnetic field is deliberately slow in order to minimise the impact of eddy currents on the measured hysteresis loop. As a result, traditional permeameters are unsuitable for high volume testing in production or quality control environments.

Manufacturers of rare-earth magnets have adopted Pulsed-Field Magnetometers (PFM) in order to characterise their products. PFM measurements can be thousands of times faster than permeameter measurements. However, there are two important -3 -

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factors which must be taken into consideration with PFM measurements: (i) The high-speed measurements induce eddy currents in the magnet sample that are known to distort the hysteresis loop. (GB 2 532 702-A of Hirst Magnetic Instruments Ltd. comprehensively resolves this issue.) (ii) The sample magnet is no longer in a closed magnetic circuit. As such, the sample tends to develop a strong self-demagnetisation field in addition to its magnetisation in response to the externally applied field. To address this issue a self-demagnetisation factor (SDF) is used to skew the open-circuit measurement such that it more closely matches the closed-circuit pernneameter measurement. However, it is important to recognise that, even in the best cases, the self-demagnetisation factor correction is a relatively poor approximation. Firstly it is necessary to assume that the magnetisation of the sample is homogeneous in order to calculate the inhomogeneous magnetisation in the sample, that is to say the approach is not self-consistent. Secondly, the approximation gets worse for high external fields because the method ignores the dependence of the self-demagnetisation field on the applied field. Together these limitations result in a rounding of the knee in the open-circuit SDF-corrected hysteresis loop with respect to the corresponding closed-circuit measurement, as illustrated in Fig. 2. This figure shows the second quadrant of the full-loop hysteresis curve, also known as the demagnetisation curve, which plots magnetisation M against the externally-applied magnetic field H. Curve I is the open-circuit demagnetisation curve measured using a Hirst -4 -

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Magnetic Instruments PFM. Curve II is the SDF-corrected open-circuit demagnetisation curve. Curve III (solid line) is the closed-circuit demagnetisation curve measured using a typical pernneameter. Curve IV (dashed line superimposed on curve III) is the closed-circuit-like curve obtained using the method proposed herein, based on the open-circuit data obtained in curve I. It will be noted that the relative error between the SDF-corrected open-circuit demagnetisation curve measured using a conventional PFM (curve II) and the closed-circuit demagnetisation curve measured using a permeanneter (curve III) can be as large as 10% in the knee region. These errors are directly responsible for inefficiencies in motor and generator design when using rare-earth magnets, and subsequently the production of tens of gigatons per year of excess carbon dioxide gas.

The present invention seeks to provide a new method and apparatus for using readings obtained using open-circuit PFM techniques to more accurately match the closed-circuit hysteresis curve measured using a pernneanneter.

SUMMARY OF THE INVENTION

When viewed from one aspect the present invention proposes a method of estimating a closed-circuit hysteresis curve for a -5 -mag netic sample: - obtain open-circuit measurements from the magnetic sample, e.g. using a pulsed-field magnetometer (PFM); - create a mathematical model of the sample comprising a lattice of nodes; - using the model, create a hysteresis curve defined by variables; - adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.

The invention also provides apparatus for performing the method.

BRIEF DESCRIPTION OF THE DRAWINGS

The following description and the accompanying drawings referred to therein are included by way of non-limiting example in order to illustrate how the invention may be put into practice. In the drawings: Figure 1 is a diagrammatic representation of a permeameter as used for measuring the closed-circuit hysteresis curve of a magnetic sample; Figure 2 is a demagnetisation curve for a magnetic sample, showing a comparison of the curves obtained by -6 -various methods; Figure 3 is a flow chart illustrative of the method proposed herein; Figure 4 is a second flow chart illustrating one way of minimising the internal energy of the sample which uses the Monte-Carlo method; Figure 5 is an illustrative example of a model of a typical cylindrical magnet comprising a tetrahedral mesh; Figure 6 is a schematic diagram of apparatus for performing the method.

DETAILED DESCRIPTION OF THE DRAWINGS

The iterative method described herein uses a numerical model of a sample of the magnetic material under investigation to find the minimum energy configuration of the dipoles using measured open-circuit data. When the variable parameters of the model have thus been established the model is re-run under conditions which simulate an infinitely long sample in which the magnetic flux at opposite north and south poles are substantially identical. This enables a hysteresis loop to be produced from the open-circuit data which closely matches closed-circuit conditions. -7 -

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The method is based on a significant observation. In a pernneameter the magnetic circuit provides a low-reluctance path for the externally applied field flux. Secondary to that, the magnetic circuit also provides a return path for the sample magnet flux. As such, to a good approximation, all flux leaving the north face of the sample magnet is returned identically to the south face of the sample magnet. In a numerical context, matching the north pole flux to the south pole flux is referred to as "periodic boundary conditions". By measuring the sample magnet with periodic boundaries the sample responds as if it were infinitely long, and crucially, infinitely long magnets exhibit no self-dennag netisation. Therefore, in effect, the closed-circuit measurement is related to the open-circuit measurement, not by an SDF correction, but by pseudo-infinite extrusion. It follows that if the parameters for a hysteresis loop produced by an open-circuit model can be fitted to an open-circuit hysteresis loop measured by a PFM or similar equipment, then it is possible to precisely reproduce the closed-circuit-like hysteresis loop by adding periodic boundary conditions to the model.

The method has been verified using an Ising model of a magnetic sample under investigation. The model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or -1). The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors. The model allows the identification of phase transitions, as a simplified model of reality. -8 -

The proposed method for producing a closed-circuit full-loop hysteresis curve for a magnetic sample broadly comprises the following steps.

(a) Measure the open-circuit hysteresis loop using a PFM or other open magnetic circuit technique.

(b) Correct for eddy currents if required. (GB 2 532 702-A).

(c) Generate a mathematical model of the sample based on an array of magnetic dipoles.

(d) Set the local field in the model to the sum of the externally applied field and the calculated self-demagnetisation field due to the dipoles.

(e) Find the minimum internal energy configuration of the dipoles in the sample volume. Energy minimisation is an established physical principle based on the observation that a system will not remain in any given state if there is a lower-energy state into which it can move.

(f) Repeat (d) and (e) for all externally applied fields.

(g) Fit the appropriate parameters of the model (e.g. Hamiltonian parameters) to match the open-circuit-like model to the hysteresis loop measured in open-circuit.

(h) Re-run the model in a closed-circuit-like configuration with substantially identical magnetic flux at the opposite north and south poles, for example using periodic boundaries, using the best-fit parameters to calculate a full-loop hysteresis curve which closely matches closed-circuit conditions.

It may not be necessary to plot the entire hysteresis loop if only part of the curve is of interest, e.g. the demagnetisation curve, -9 -but the same principles would apply.

In order to better illustrate the proposed method one embodiment of the method will now be described in greater detail, being further illustrated in the flow charts of Fig.s 3 and 4. The figures in parentheses appearing in the flow charts refer to the steps described below.

(1) Measure the open-circuit hysteresis loop. A sample of magnetic material is analysed under open-circuit conditions using a PFM or other open-circuit measurement technique, to record data representing the full hysteresis loop, or at least a relevant part of the open-circuit hysteresis curve. Different externally-applied fields are used to create the points on the hysteresis loop.

(2) Create a lattice. A cubic, tetrahedral or other lattice is created mathematically, preferably, but not necessarily, a numerical computer model. The lattice represents the three-dimensional volume of the sample magnet. The lattice is preferably based on an Ising model, but this is just one of a family of Monte-Carlo methods that could be used, which all use random number generation to evolve the system. If the sample is concave, or it is not simply-connected, then the region of air surrounding the magnet would also be modeled. A typical tetrahedral lattice of a cylinder magnet is shown in Fig. 5. A magnetic dipole is assigned to each node i of the magnet volume. The moment of each dipole m, is set in proportion to the volume of the element, AV -10 -

QV A

m m. z

V

Equation 1 where v is the total volume, m is the total magnetic moment of the sample magnet, and Z is the direction of the externally applied field.

(3) Calculate the self-demagnetisation field. After first setting hard boundary conditions and initialising the dipoles parallel to a uniform magnetisation the externally applied field is set. The far-field magnetic potential, 1), , is calculated for each dipole, i.e. at each point in the lattice. This is the self-demagnetisation potential.

(4) Calculate the externally applied magnetic field for each dipole. The local magnetic field, H, , is calculated as the gradient of the local magnetic potential: 11 VC Equation 2 (5) Sum the self-demagnetisation magnetic field and the externally applied field for each dipole. At each point in the lattice the total local magnetic field is calculated as the sum of the self demagnetisation field and the externally applied field.

(6) Calculate the internal energy of the magnet. The internal energy, U, of the sample magnet may be defined by a Hamiltonian. The form of the Hamiltonian depends upon the nature of the material. For example, in an Ising-like material the Hamiltonian would be Equation 3 where j is the dipole-dipole coupling constant for nearest neighbours, and h is the dipole-field coupling constant for the far-field, as calculated in steps (3) -(5) above. The self-demagnetisation field is kept fixed. The dipoles may be evolved using the Monte-Carlo method as illustrated in the flow chart of Fig 4. The Monte-Carlo method may be summarised as (a) Select a random dipole.

(b) Rotate the dipole by a random amount in three dimensions, maintaining a constant magnitude.

(c) Re-calculate the internal energy. If the internal energy is reduced, then accept the rotation, otherwise accept the rotation with a probability proportional to the Boltzmann factor: -expi -AU/kBT) Equation 4 The Monte-Carlo method iterates as in the flow chart until the -12 -sample has reached a steady-state of minimum energy. In the end state the configuration of the dipoles represents the configuration that minimises the total energy of the system.

(7) The self-demagnetisation field calculation and Monte-Carlo method, steps (3) -(6), are repeated until magnetisation has reached a steady-state.

(8) The magnetisation is recalculated, steps (3) -(7), for each externally applied field. This creates a numerical hysteresis curve defined by variables.

(9) The Hamiltonian parameters j and h are calculated by using a least-squares fit to the open-circuit full-loop hysteresis curve.

(10) The model is then re-calculated, steps (3) -(8), in a closed-circuit configuration, using the best-fit parameters, in order to obtain the corresponding closed-circuit-like hysteresis loop. The closed-circuit configuration may, for example, be simulated by applying periodic boundary conditions in a direction parallel to the driving field.

Although the resultant closed-circuit-like hysteresis loop is a calculation based on measured data and not a direct measurement, investigations confirm that the calculated loop matches with permeameter measurements with considerably greater accuracy than using an SDF correction, being a near-perfect reproduction of the closed-circuit measurement as -13 -

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demonstrated by Fig. 2.

The method may be embodied in apparatus adapted to perform the necessary calculations, preferably (but not confined to) a Pulsed-Field Magnetometer. Referring to Fig. 6, the apparatus may comprise a probe 10 to obtain open-circuit measurements from a magnetic sample MS. Readings from the probe are sent to a processing unit 11 for carrying out the method which stores the open circuit data in a first memory location 12. The numerical model is created in a second memory location 13. The resulting pseudo-closed-circuit data allows the processing unit 11 to create a corrected hysteresis curve which may for example be produced on a visual display 14.

Whilst the above description places emphasis on the areas which are believed to be new and addresses specific problems which have been identified, it is intended that the features disclosed herein may be used in any combination which is capable of providing a new and useful advance in the art.

Claims

-14 -CLAIMS1. A method of estimating a closed-circuit hysteresis curve for a magnetic sample: - obtain open-circuit measurements from the magnetic sample, e.g. using a pulsed-field magnetometer (PFM); - create a mathematical model of the sample comprising a lattice of nodes; - using the model, create a hysteresis curve defined by variables; - adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.
2. A method of estimating a closed-circuit hysteresis curve for a magnetic sample: - obtain open-circuit measurements from the magnetic sample, e.g. using a pulsed-field magnetometer (PFM); - create a mathematical model of the sample comprising a lattice of nodes; - using the model, create a hysteresis curve defined by variables; - adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - apply periodic boundary conditions to the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.
-15 - 3. A method of estimating a closed-circuit hysteresis curve for a magnetic sample: - obtain open-circuit measurements from the magnetic sample for a range of applied fields; - create a mathematical model of the sample comprising a lattice of nodes; - using the model, create a hysteresis curve defined by variables; - adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.
4. A method of estimating a closed-circuit hysteresis curve for a magnetic sample: - obtain open-circuit measurements from the magnetic sample, e.g. using a pulsed-field magnetometer (PFM); - create a mathematical model of the sample comprising a lattice of nodes; - find the minimum-energy configuration of the nodes; - using the model in the minimum energy configuration, create a hysteresis curve defined by variables; - adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.
-16 - 5. A method of estimating a closed-circuit hysteresis curve for a magnetic sample: - obtain open-circuit measurements from the magnetic sample, e.g. using a pulsed-field magnetometer (PFM); - create a mathematical model of the sample comprising an array of magnetic dipoles; - (A) set the local field in the model to the sum of the externally applied field and the calculated self-demagnetisation field due to the dipoles; - (B) find the minimum internal energy configuration of the dipoles; - repeat (A) and (B) for all externally applied fields creating a hysteresis curve defined by variables; - adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.
6. A method of estimating a closed-circuit hysteresis curve for a magnetic sample: - obtain open-circuit measurements from the magnetic sample, e.g. using a pulsed-field magnetometer (PFM); - create an Ising model of the sample comprising an array of magnetic dipoles; - (A) set the local field in the model to the sum of the externally applied field and the calculated self-demagnetisation field due to -17 -the dipoles; - (B) find the minimum internal energy configuration of the dipoles; - repeat (A) and (B) for all externally applied fields creating a hysteresis curve defined by variables; - adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.
7. A method of estimating a closed-circuit hysteresis curve for a magnetic sample: - obtain open-circuit measurements from the magnetic sample, e.g. using a pulsed-field magnetometer (PFM); - create a mathematical model of the sample comprising an array of magnetic dipoles; - (A) set the local field in the model to the sum of the externally applied field and the calculated self-demagnetisation field due to the dipoles; - (B) find the minimum internal energy configuration of the dipoles using the Monte-Carlo method; - repeat (A) and (B) for all externally applied fields creating a hysteresis curve defined by variables; - adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated -18 -closed-circuit hysteresis curve.
8. A method of estimating a closed-circuit hysteresis curve for a magnetic sample: - obtain open-circuit measurements from the magnetic sample, e.g. using a pulsed-field magnetometer (PFM); - create a mathematical model of the sample comprising an array of magnetic dipoles; - (A) set the local field in the model to the sum of the externally applied field and the calculated self-demagnetisation field due to the dipoles; - (B) find the minimum internal energy configuration of the dipoles; - repeat (A) and (B) for all externally applied fields creating a hysteresis curve defined by Hamiltonian parameters; - adjust the Hamiltonian Parameters to best-fit the hysteresis curve of the model to the open-circuit measurements; - adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.
9. Apparatus for estimating a closed-circuit hysteresis curve for a magnetic sample: - means to obtain open-circuit measurements from the magnetic sample, e.g. a pulsed-field magnetometer (PFM); - means to create a mathematical model of the sample comprising a lattice of nodes; - means for using the model to create a hysteresis curve defined -19 -by variables; - means to adjust the variables to best-fit the hysteresis curve of the model to the open-circuit measurements; - means to adjust the model to simulate closed circuit conditions and re-run the model using the best-fit variables to produce an estimated closed-circuit hysteresis curve.