Lorentz force sigmometry: A contactless method for electrical conductivity measurements Robert P. Uhlig, Mladen Zec, Marek Ziolkowski, Hartmut Brauer, and André Thess Citation: J. Appl. Phys. 111, 094914 (2012); doi: 10.1063/1.4716005 View online: http://dx.doi.org/10.1063/1.4716005 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i9 Published by the AIP Publishing LLC.
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JOURNAL OF APPLIED PHYSICS 111, 094914 (2012)
Lorentz force sigmometry: A contactless method for electrical conductivity measurements Robert P. Uhlig,1,a) Mladen Zec,1 Marek Ziolkowski,1,2 Hartmut Brauer,1 and Andre´ Thess3 1
Department of Advanced Electromagnetics, Ilmenau University of Technology, P. O. Box 100565, D-98684 Ilmenau, Germany 2 West Pomeranian University of Technology, KETiI, Sikorskiego 37, PL-70313 Szczecin, Poland 3 Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology, P. O. Box 100565, D-98684 Ilmenau, Germany
(Received 17 February 2012; accepted 8 April 2012; published online 15 May 2012) The present communication reports a new technique for the contactless measurement of the specific electrical conductivity of a solid body or an electrically conducting fluid. We term the technique “Lorentz force sigmometry” where the neologism “sigmometry” is derived from the Greek letter sigma, often used to denote the electrical conductivity. Lorentz force sigmometry (LoFoS) is based on similar principles as the traditional eddy current testing but allows a larger penetration depth and is less sensitive to variations in the distance between the sensor and the sample. We formulate the theory of LoFoS and compute the calibration function which is necessary for determining the unknown electrical conductivity from measurements of the Lorentz force. We conduct a series of experiments which demonstrate that the measured Lorentz forces are in excellent agreement with the numerical predictions. Applying this technique to an aluminum sample with a known electrical conductivity of rAl ¼ 20:4 MS=m and to a copper sample with rCu ¼ 57:92 MS=m we obtain rAl ¼ 21:59 MS=m and rCu ¼ 60:08 MS=m, respectively. This demonstrates that LoFoS is a convenient and accurate technique that may find application in process control and thermo-physical C 2012 American Institute of Physics. property measurements for solid and liquid conductors. V [http://dx.doi.org/10.1063/1.4716005]
I. INTRODUCTION AND MOTIVATION
The braking effect of the magnetic field caused by a permanent magnetic system (DC magnetic field) on a translating, electrically conducting solid body or on the flow of a liquid metal is well-known.1 It is also well-known that the braking force is a Lorentz force, whose magnitude depends both on the velocity and the electrical conductivity of the material in the immediate vicinity of the magnet system. By contrast, it is less widely known that a Lorentz force with the same magnitude as in the solid or liquid but with opposite direction acts by virtue of Newton’s third law upon the magnetic-field-generating system, for instance, a permanent magnet. This phenomenon has recently found practical application for the contactless velocity measurement in metallurgy in the form of “Lorentz force velocimetry” (LFV)2 and for the detection of defects in solid metals in the form of “Lorentz force eddy current testing” (LET).3,4 The goal of the present work is to demonstrate that the same Lorentz force that is used in LFV and LET can be exploited for the contactless measurement of electrical conductivities. Both measurement techniques—LFV and LET—have in common that they require the material properties of the investigated materials to be known. The characteristic material property for any measurement technique using eddy currents and/or electro-magnetic fields is the electrical conductivity of the specimen or fluid under test. Nowadays, there are only a few established measurement techniques for the determination a)
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of conductivity.5–7 Fluids are usually treated with amperometric and potentiometric measurements which are basically of the same kind, whereas solid state bodies are treated with impedance spectroscopy and the four-point-method. A big disadvantage is—dependent on the application—the contact with the specimen, that might be not possible, e.g. in hot metals or if the specimen is moving fast. Contactless methods, as e.g., the eddy current method,6 suffer strongly from deviations in lift-off distance and cover only the subsurface region and cannot provide conductivity measurements deeper within the material due to the skin effect. The conductivity measurement technique presented in the current work overcomes the above-mentioned disadvantages since it provides a contactless measurement deep inside the material, no matter whether it is a fluid or a solid body. Since the internationally widely used Greek symbol for the conductivity is r and the exploited physical effect is the Lorentz force we call the method “Lorentz force sigmometry” (LoFoS). The paper is organized as follows. In Sec. II, we explain the basic idea of the technique and discuss scaling relations which connect the measured Lorentz force with the electrical conductivity to be determined. Section III is devoted to numerical computations of the calibration function necessary to implement LoFoS. In Sec. IV, we demonstrate by means of laboratory experiments that the predicted calibration functions accurately describe the reality and can therefore be used to perform contactless measurements of electrical conductivity.
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II. BASIC PRINCIPLE
The problem to be investigated is shown in Fig. 1. An infinite plate of arbitrary thickness is moved below a magnetic field source that can be a set of coils or a permanent magnet (PM). The source is fixed at a lift-off distance dz above the conductor with the magnetization direction transversal to the moving direction. The Lorentz force is generated due to the interaction of ~ The the eddy currents ~ j and the external magnetic field B. eddy currents are induced due to the relative movement between the electrical conductor and the primary magnetic field. The physical relationship can be drawn from Ohm’s law for moving conductors ~ ~ þ~ ~ j ¼ rðE v BÞ;
(1)
~ is the electric field that is zero in the present case. where E The Lorentz force can be calculated from ððð ~ ~ ~ j BdV: (2) F¼ V
The drag component of the Lorentz force, Fx, i.e. along the movement direction, depends linearly on the magnetic Reynolds number Rm, whereas the lift force component Fz is proportional to the square of Rm for small Rm: Fx / Rm ;
(3)
Fz / R2m :
(4)
The magnetic Reynolds number Rm is a nondimensional number that describes the ratio between the convection and the diffusion of the magnetic field.8 It can be calculated according to Rm ¼ l0 rvL;
(5)
where l0 is the vacuum permeability, r is the electrical conductivity, v is the relative velocity between the conductor and the magnetic field source, and L is the characteristic length scale.
An obvious idea is to calculate the conductivity using only the drag force component of the Lorentz force. Since the Reynolds number is linearly dependent on the conductivity, one can find an coefficient b that is fulfilling r ¼ bFx :
(6)
The coefficient b can be determined experimentally and numerically. It depends on the geometry of the magnet system, on the velocity and strongly on the lift-off distance, and ~ 9 The strong dependency on the the magnetic flux density B. lift-off distance leads to a very high uncertainty in conductivity measurement. Manufacturing errors and mechanical oscillations can lead easily to deviations of up to 100 lm resulting in force deviations of some millinewton depending on the liftoff distance. The magnetic field strength of the PM is usually not given and it is difficult to determine since magnetization direction and mounting errors have to be taken into account. To overcome the disadvantages of using only the drag component a modified approach is applied. As a consequence of Eqs. (3) and (4), the ratio of Fz =Fx depends linearly on Rm Fz / Rm : Fx
(7)
For thin plates of infinite extension, Reitz10 found that for any shape of the magnetic field source the lift-to-drag ratio can be written as Fz v ¼ ; Fx w
(8)
where w is the characteristic velocity of the conductor under test determined by w ¼ 2=ðl0 rDÞ, where D is the thickness of the sheet. However, Eq. (8) can be used for plates and sheets as long as their thickness does not exceed the motional skin depth d
1 d¼D 2Rm
12 (9)
and velocities are kept in the low Rm-range which is the case up to moderate velocities (v 5 m=s).11 Considering L ¼ D=2 to be the characteristic length scale of the present problem, Eq. (8) can be rewritten as Fz ¼ Rm : Fx
(10)
The result implies that the lift-to-drag ratio neither depends on ~ of the PM nor on the lift-off distance dz. the magnetic field B Under the assumption that the conductivity is homogeneous and isotropic we can state that for a constant velocity the lift-to-drag ratio can be described as r¼a
FIG. 1. LoFoS sample problem—conducting plate moving with a constant velocity below a permanent magnet.
Fz ; Fx
(11)
where the calibration coefficient a depends on the geometry of the magnet system, on the translational velocity, and
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TABLE I. Geometrical and material properties used for comparison of the analytical with the numerical model and experiments. Parameter Specimen length L Specimen width W Specimen depth D Magnet length l/Diameter Ø Magnet width w Magnet height d Remanent flux density Br
Fig. 2
Experiments
0.25 m 0.25 m 0.05 m 0.015 m 0.015 m 0.025 m 1.17 T
0.25 m 0.05 m 0.05 m Ø 0:015m 0.025 m 1.17 T
(weakly) on the distance between the plate and the magnet. Since a is a priori unknown, we have to use a calibration with two specimens. The conductivity of these specimens can be determined by means of certificated methods, such as the Van-der-Pauw-method5,12 and should be approved by the National Metrology Institute (e.g. for Germany: Physikalisch Technische Bundesanstalt).
III. ANALYTICAL AND NUMERICAL MODEL
To illustrate the presented technique, we analyze a typical LoFoS problem shown in Fig. 1. We use two methods to calculate the drag force Fx and the lift force Fz acting on the moving parallelepipedial magnet: a semi-analytical one, which can be applied to configurations where the size of the magnet is small compared with the tested specimen (L; W w; l; d; D) and a fully numerical one, without these restrictions. The semi-analytical method is based on Lee’s approach13 where forces acting on a current coil, which is moving above a conducting sheet of arbitrary thickness, are calculated using a Fourier transform approach. The permanent magnet is substituted by a solenoid consisting of n parallel coils carrying the current I0 equal to nI 0 ¼ Br h=l0 , where Br and h are the remanence and the height of the used permanent magnet, respectively. The semi-analytical method is implemented numerically and enables fast force calculations. In order to overcome the restrictions of the analytical model and to test the limits of its applicability, we additionally use a numerical model. We apply a finite element method (FEM) to solve the governing partial differential
equations taking into account the real geometry of the problem (Table I). According to the basic principle of the LoFoS technique, the magnet system and the material under test are set into relative motion. We fix the coordinate system to the magnet as a reference frame, while the bar is moving with constant velocity. Additionally, if the cross-section of the plate, normal to the direction of the motion, remains constant, the quasi-static approximation (QS) can be used. This considerably reduces the computational costs since stationary analysis can be used in order to obtain an accurate steady state solution. Following the assumptions mentioned above, the following equations for electric scalar potential (V) and mag~ are applied: netic vector potential (A) 1 ~ ~ þ rrV ¼ 0; (12) ~ r A M r~ v ðr AÞ r l0 ~ rrV ¼ 0; r½r~ v ðr AÞ
(13)
~ is the magnetization vector of the permanent where M magnet. If not stated otherwise, we used in both models the geometrical and material properties collected in Table I. The magnetization of the permanent magnet is considered to be coaxial with the z-axis of the applied coordinate system. As already stated before, in the semi-analytical model, an infinite plate of finite thickness (D) is used. Therefore, in order to verify the results of the numerical model, the plate is considered to be long and wide enough compared with the size of the PM so that the plate can be considered as infinite. We analyze the lift-to-drag ratio of the Lorentz force acting on the magnet as a function of the plate’s conductivity to compare the applied methods (see Eq. (11)). Figure 2 shows the characteristic calibration curves of the LoFoS technique calculated for different magnet positions (dz) and plate velocities (v). The numerical results show a very good agreement with the analytical expressions in the whole range of the considered conductivities. This enables us to use the fast semi-analytical model to analyze the simplified LoFoS configurations, whereas the numerical model should be used if the conducting plate has finite dimensions or when a more complex magnet system is used.
FIG. 2. Calibration curves for plates obtained analytically (anl.) and numerically (num.) (anl.—infinite plate with thickness 50 mm, num.—W D L ¼ 250 50 250 mm) for various velocities (a) lift-off distance dz ¼ 3 mm (b) lift-off distance dz ¼ 5 mm.
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TABLE II. Basic linear fitting coefficients for plate calibration. dz ¼ 3 mm
Coeff. v (m/s) a (MS/m) roff (MS/m)
0.5 299.95 1.44
1.0 171.51 4.11
dz ¼ 5 mm 1.5 129.89 6.56
0.5 266.35 1.50
1.0 153.42 4.34
1.5 117.06 6.96
The calibration curves shown in Fig. 2 represent the basis for the implementation of the LoFoS technique. Using the assumption of a linear dependency between conductivity and lift-to-drag ratio (cf. Eq. (11)) we use basic linear fitting to obtain the function according to Table II in the form Fz r ¼ a þ roff ; Fx
(14)
where roff is the offset that results from the linear fit at the measurement range between 20 MS/m and 60 MS/m. IV. EXPERIMENTAL VALIDATION
In order to test the validity of the numerical predictions, we set up an experiment shown in Fig. 3. Due to effects of inertia, the relative movement between the permanent magnet and the specimen is realized by a linear drive carrying the object under test rather than the permanent magnet. The range of velocity is between 0 m/s and 3.75 m/s which is sufficient to reach intermediate magnetic Reynolds numbers for highly conductive specimen. We use a three-component force sensor which has a measurement range of Fx;y ¼ 3 N longitudinal and transversal to the direction of movement and Fz ¼ 10 N in the vertical direction. The difference in the ranges is chosen due to the weight of the used magnet. The force component transversal to the movement direction Fy is used as a quality indicator for the alignment of the magnet with respect to the object under test. At the symmetry line in the direction of motion it should obey Fy ¼ 0. This statement is valid for plates as well as for bars.
FIG. 3. Sketch of experimental measurement setup containing (1) a threecomponent force sensor, (2) a permanent magnet, and (3) the specimen mounted on a linear belt-driven drive.
A. Characterization of the solid state bar
In order to validate the basic principle of the proposed measurement technique LoFoS, two solid bars of known conductivity have been considered. Two parameters are studied: (i) the velocity v and (ii) the lift-off distance dz. The investigations help to understand the necessity of using the lift-to-drag ratio in terms of accuracy enhancement. For low magnetic Reynolds numbers, the force can be calculated applying the model of Reitz.10 The drag component of the Lorentz force is rising linearly with the velocity, whereas the lift component is rising with the square of velocity (see Fig. 4). The range of low Reynolds numbers has been obtained by using an aluminum bar with a conductivity of rAl ¼ 20:4 MS=m. For higher Reynolds number, i.e. a copper bar with rCu ¼ 57:9 MS=m, the drag force is rising non-linearly and the lift component linearly with increasing velocities. Nevertheless, the ratio between drag force and lift force is almost linear for both materials for low and intermediate velocities (see Fig. 5). The experiments with copper show that the linear dependency on velocity is still valid on the bounds of the existing conductivity range.
FIG. 4. Dependency of force on velocity for Al-alloy at a lift-off distance of dz ¼ 3 mm and Cu at dz ¼ 5 mm (a) drag force (b) lift force.
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FIG. 5. Dependency of the ratio between lift force and drag force on velocity for Al-alloy and Cu (lift-off distance dz ¼ 3 mm and 5 mm, respectively).
J. Appl. Phys. 111, 094914 (2012)
FIG. 7. Dependency of the ratio between lift force and drag force on lift-off distance for Al-alloy, where the permanent magnet is PM ø15 mm 25 mm.
Another important issue is the dependency of the Lorentz force on the lift-off distance. In order to increase the total force a smaller lift-off distance dz is preferable. The disadvantage of a decreasing lift-off distance is the increase of the sensitivity to surface dependent lift-off distance changes, e.g. surface roughness. In Fig. 6 is shown that a moving PM of finite thickness cannot be substituted by an equivalent magnetic dipole14 because the Lorentz force is not scaled with dz3 but approximately dz4.7. The high sensitivity with respect to the lift-off distance leads to uncertain conductivity measurements when only the drag or the lift force is applied. The use of the lift-to-drag ratio reduces the sensitivity to the lift-off distance drastically as can be seen from Fig. 7. The investigation of both effects, the velocity and the liftoff distance dependency on the lift-to-drag ratio, has shown the advantages while using the ratio instead of a single Lorentz force component. The overall measurement error can be calculated by the total derivative of the lift-to-drag ratio Fz Fx dFz Fz dFx ¼ ; (15) d Fx F2x where the values denoted by a d are the measurement uncertainties. In the case of the presented experiment, the overall uncertainty can be read from Fig. 8 for fixed velocity and lift-off distance. As a result, we propose to use medium velocities and small lift-off distances in order to minimize the measurement uncertainty.
FIG. 8. Dependency of measurement uncertainty on fixed parameters of (a) lift-off distance dz ¼ 3 mm and (b) velocity v ¼ 2 m=s (PM ø15 mm 25 mm).
FIG. 6. Dependency of force on lift-off distance for Al-alloy (different magnet sizes, v ¼ 2 m=s), (a) drag force, (b) lift force.
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FIG. 9. Numerically obtained calibration curves for the used bar geometry (W D L ¼ 50 50 250 mm) for different velocities (a) lift-off distance dz ¼ 3 mm, the particular example from Table IV is visualized colorfully (b) lift-off distance dz ¼ 5 mm.
B. Conductivity measurement
In this section, we discuss how a measurement procedure might work for LoFoS. Due to the fact that the expected Lorentz force produced by the PM is unknown, a calibration with standard conductor targets is necessary. Since the force sensors can be calibrated by national mass standards, the results of the measurement can be traced back to SI units. The calibration can be done with two points, i.e. two materials (e.g. aluminum and copper) or more points (using titanium, copper, and other materials of intermediate conductivity). Through the obtained ratios can be drawn a line or, in case of more points, a polynomial fit can be made. The calibration should be done for different velocities. We propose measurement velocities of v1 ¼ 0:5 m=s, v2 ¼ 1:0 m=s, and v3 ¼ 1:5 m=s for our particular setup ensuring an intermediate magnetic Reynolds number Rm. The specimen has to be measured with the same velocities. The obtained lift-to-drag ratios have to be marked in the calibration graphs. If three velocities are used, it is possible to average the obtained conductivity. Furthermore, due to the very good agreement between measurements and numerical results, we propose a numerical calibration as well, when the calibration curves can be generated by changing the conductivity of the specimen. These calibration graphs can be used to estimate a measured, unknown conductivity (cf. Fig. 9). The resulting calibration coefficients a (cf. Eq. (11)) are summarized in Table III. We measured the acting Lorentz force on two solid bars of known conductivity, namely, aluminum (rAl ¼ 20:4 MS=m) and copper (rCu ¼ 57:92 MS=m). The given conductivities have been measured with the eddy current device Sigmatest 2.069 (Institut Dr. Foerster GmbH & Co. KG). The measured lift-to-drag ratio has been used to TABLE III. Basic linear fitting coefficients for numerical bar calibration. dz ¼ 3 mm
Coeff. v (m/s) a (MS/m) roff (MS/m)
0.5 417.91 0.26
1.0 216.26 1.04
dz ¼ 5 mm 1.5 151.43 2.18
0.5 389.46 0.25
1.0 201.54 1.01
1.5 140.89 2.10
determine the conductivity of these bars. Therefore, the purely numerically obtained calibration curves in Fig. 9 have been used. The measurement uncertainties and the intermediate results are given in Table IV. Applying the linear fitting coefficients from Table III and the uncertainty calculation according to Eq. (15) we have obtained the conductivities of rAl ¼ 21:59 MS=m and rCu ¼ 60:08 MS=m, respectively. The combined measurement uncertainties of uAl ¼ 9:82 MS=m and uCu ¼ 3:46 MS=m, respectively, are based on the uncertainty of the force sensor (dFx , dFz ) and can be reduced when the measurements are repeated several times or by applying a more appropriate force sensor. C. Limitations
The proposed measurement technique for the determination of conductivity benefits strongly from a wide range force sensor with high resolution and low uncertainty. In order to obtain high values of the force components a small lift-off distance and a high velocity are recommended. Without any appropriate protection, the force sensor could be damaged. During the production process, the lift-off distance might fluctuate by a few millimeters, and thus an active control can be necessary. This is more challenging for short process times and high velocities of moving media. Another challenge in the case of high velocities is the limited sampling frequency of the force sensor. For short specimen, there might be not enough measuring points to obtain a precise lift-to-drag ratio. TABLE IV. Conductivity calculation using LoFoS. Parameter
Unit
Material v dz Fx dFx Fz dFz Fz Fz Fx 6d Fx r6u
m/s mm N N N N
Al 1.5 3 0.735
MS/m
0:15760:065 21:5969:82
Cu 1.5 3 1.92 0.015
0.115
0.789 0.050 0:41160:022 60:0863:46
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V. CONCLUSIONS
The proposed technique called LoFoS is able to provide the electrical conductivity of a specimen. In a nutshell, LoFoS has the following advantages: (i) LoFoS is contactless, (ii) LoFoS can be applied continuously during production processes, (iii) LoFoS is a method that enables the user to measure conductivity beyond the surface, and (iv) the proposed data processing makes LoFoS resistant to changes in lift-off distance, velocity, and strength of the magnetic field source. LoFoS is suitable for specimen of any kind of physical condition. The only limitation is given by the minimal measurable Lorentz force components of a commercial multi-component force sensor. Due to the fact that LoFoS is a contactless method for conductivity measurements it can be implemented in the production process of any nonmagnetic material (e.g. aluminum and brass). It is possible to measure the conductivity of hot or aggressive materials. Due to the use of DC magnetic fields, LoFoS is not limited by source frequencies like conventional eddy current based techniques. The penetration depth during test is bigger than the one reached by classical eddy current methods which is usually limited to the order of a few micrometers. LoFoS is providing a robust measurement technique because of the use of the lift-to-drag ratio of the components of the Lorentz force. For a wide range of conductivities, the ratio is linearly dependent on the conductivity which makes it easy to calibrate. Additional feature is its easy combination with Lorentz force eddy current testing using the same mea-
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surement setup. The conductivity is directly traceable to SI units if the measurement has been calibrated appropriately. ACKNOWLEDGMENTS
The current work has been supported by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Research Training Group “Lorentz force velocimetry and Lorentz force eddy current testing” (GK 1567) at Ilmenau University of Technology. 1
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