Resolution of the paradox of the diamagnetic effect on the Kibble coil
Shisong Li, Stephan Schlamminger, Rafael Marangoni, Qing Wang, Darine Haddad, Frank Seifert, Leon Chao, David Newell, Wei Zhao
RResolution of the paradox of the diamagneticeffect on the Kibble coil * Shisong Li , Stephan Schlamminger , Rafael Marangoni , Qing Wang ,Darine Haddad , Frank Seifert , Leon Chao , David Newell , and Wei Zhao Department of Engineering, Durham University, Durham DH1 3LE, United Kingdom National Institute of Standards and Technology, Gaithersburg 20899, United States Department of Electrical Engineering, Tsinghua University, Beijing 100084, China * [email protected] + [email protected] ABSTRACT
Employing very simple electro-mechanical principles known from classical physics, the Kibble balanceestablishes a very precise and absolute link between quantum electrical standards and macroscopicmass or force measurements. The success of the Kibble balance, in both determining fundamentalconstants ( h , N A , e ) and realizing a quasi-quantum mass in the 2019 newly revised InternationalSystem of Units, relies on the perfection of Maxwell’s equations and the symmetry they describebetween Lorentz’s force and Faraday’s induction, a principle and a symmetry stunningly demonstratedin the weighing and velocity modes of Kibble balances to within × − , with nothing but imperfectwires and magnets. However, recent advances in the understanding of the current effect in Kibblebalances reveal a troubling paradox. A diamagnetic effect, a force that does not cancel betweenmass-on and mass-off measurement, is challenging balance maker’s assumptions of symmetry atlevels that are almost two orders of magnitude larger than the reported uncertainties. The diamagneticeffect , if it exists, shows up in weighing mode without a readily apparent reciprocal effect in the velocitymode, begging questions about systematic errors at the very foundation of the new measurementsystem. The hypothetical force is caused by the coil current changing the magnetic field, producingan unaccounted force that is systematically modulated with the weighing current. Here we showthat this diamagnetic force exists, but the additional force does not change the equivalence betweenweighing and velocity measurements. We reveal the unexpected way that symmetry is preservedand show that for typical materials and geometries the total relative effect on the measurement is ≈ × − . * Accepted for publication in
Scientific Reports . a r X i v : . [ phy s i c s . i n s - d e t ] J a n ntroduction Most human activities, especially science, industry, and trade rely on measurements. The importance ofmeasurement to global society is such that the International System of Units (SI) was created as earlyas 1875 so that all measurements might be traceable to a single compact set of common standards. Fora long historical period the SI standards were formulated by artifacts (man-made or using a property ofnature), specific objects preserved in a single location, with limited access. Undeniably inaccessible,the value that such an artifact standard realizes may also vary over time , introducing dark uncertaintiesfor precision science and high-accuracy engineering . Consequently, alternatives to artifact standardshave been sought since the beginning of the SI . The first success was measurement by counting eventsof microscopic particles (e.g., atom, electron, photon, etc), first used in time measurements based onatomic clocks, which opened the door for the quantum measurement of things . On May 20, 2019, anew International System of Units, in which all seven base units are defined by physical constants ofnature, was formally adopted
8, 9 and our daily measurement activities have entered into a quantum era.With this quantum revolution of the SI, our measurement system relies now on fundamental constantswhich are woven into the structure of our universe and are here for all times and for all people, and areno longer tied to physical objects with limited stability and availability. The new SI provides a highlyaccurate or ultra-sensitive measurement foundation to support explorations that were not possible in thepast
10, 11 . The change is most profound for mass quantities, where the quest for an atomic or quantumbased standard of mass vexed researchers for decades.To realize the unit of mass at the kilogram level from atomic or quantum standards, two com-plementing technologies were eventually found, the X-ray crystal density (XRCD) method andthe Kibble balance .The XRCD method relies on the mass of the electron, which is given by theRydberg constant and defined fundamental constants. Using mass spectroscopy and scaling that takesadvantage of a nearly perfect single crystal silicon sphere, the electron’s mass can be scaled thirtyorders of magnitude to the kilogram level with a relative uncertainty of 1 × − . The realization ofthe kilogram via the Kibble balance relies on the perfect symmetry of Maxwell’s equations and canreach a similar uncertainty
15, 16 , thanks to some Nobel prize winning quantum physics
17, 18 .In the 1980s, the discovery of the quantum Hall effect by Prof. von Klitzing provided a catalyzingpiece in the quest of a quantum mass standard. It was almost immediately recognized that the quantizedresistance standard that resulted from von Klitzing’s work could be combined with the Josephsoneffect that had been theoretically postulated in 1962 and experimentally verified a year later allowingthe measurement of electrical power solely based on quantum effects. Once electrical power could bemeasured via quantum standards, a machine that precisely compares electrical to mechanical powerwould allow the quantum realization of mechanical power given by force times velocity. Velocity iseasily measured as a unitless fraction of speed of light and the force could be, for example, the weightof a mass standard in the gravitational field of the Earth. All that is needed is a precise tool that cancompare mechanical to electrical power.Luckily, such a tool, a comparator, existed. It was proposed in 1976 by Dr. Bryan Kibble , ametrologist at the National Physical Laboratory in the United Kingdom. Kibble’s invention was initially amed a watt balance, emphasizing that it compares mechanical to electrical power, since the watt isthe unit of power, both electrical and mechanical. Kibble passed away in 2016, and the watt balancewas renamed Kibble balance to honor his contributions to metrology. The core of Kibble’s idea lies in asymmetry of electromagnetism, described by Maxwell’s equations . In a nutshell, it can be describedas follows: The energy of a current-carrying loop (a coil with one turn) in a magnetic field is given bythe product of current, I and the magnetic flux, φ threading the coil. The Lorentz force in the verticaldirection F z on the coil is the negative derivative of the energy of this loop with respect to its verticalposition, z . F z = − ∂ z φ I (1)In this text, we use the abbreviation ∂ z A : = ∂ A ∂ z for the partial derivative of a quantity A with respectto z . The current is easy to measure, but not the derivative of the flux through the loop. Here is wherethe symmetry of nature comes to the rescue: Moving the coil in the magnetic field produces an inducedelectro-motive force between both ends of the coil. By Faraday’s law of induction, the induced voltage, U is proportional to the product of the derivative of the flux times the vertical velocity, v z , of the wireloop. U = − ∂ z φ v z (2)Both equations can be combined to obtain the watt equation that shows the equivalency of mechanicalto electrical power, and conveniently the hard to measure flux derivative vanishes. F z v z = U I (3)By using the weight of a mass F z = mg for the force and the quantum measurement of the electricalpower U I = C f h , where f is the frequency that is used to drive the programmable Josephson junctionvoltage array and C is a known constant that depends, for example, on how many Josephson junctionsare used, the mass can be written as m = C f hgv z . (4)Figure 1(a) shows a typical Kibble balance. Two large components are apparent: the magnet andthe wheel. The wheel is a particular choice for a part that can be used as a moving and weighingmechanism. The wheel allows the comparison of electromagnetic force and mass weight while alsoproviding the coil’s motion needed for the velocity mode. Up to the 2000s, several different typesof magnet systems were used . Over time, the field matured, and the magnet systems’ designconverged to what is known as the air-gap type, yoke-based magnetic circuits . Figure 1(b) shows atypical construction of such a permanent magnet system. The permanent magnetic circuit’s significantadvantage is that it can supply a strong (several tenths of a tesla), uniform, and robust magnetic fieldwithout an active energy source.While the description using the derivative of the flux is accurate and was used initially by Kibble,these days, the researchers use a different description of the same numerical quantity, the so-called eometric factor, or flux integral. The geometric factor is obtained by integrating the horizontalcomponent of the magnetic flux density B that is perpendicular to the wire with a length l that forms thecoil. It is abbreviated as Bl , and by virtue of Green’s theorem it is the same as ∂ z φ . For the rest of thearticle, we consider the possible causes and consequences of inevitable imperfections in the symmetry,so that there are two different geometric factors, one for weighing mode, ( Bl ) w , and one for velocitymode, ( Bl ) v . A succinct equation for the relationship of both geometric factors was suggested byRobinson . The widely accepted equation is ( Bl ) w = ( Bl ) v ( + α I + β I ) , (5)where I is the current circulating in the coil during weighing mode. Observe that the current dependenceof ( Bl ) w is canceled to first order through a common reversal trick in the design of the weighing mode:Two measurements must be made, one without and one with mass on the mass pan of the balance. Thebalance, however, can be biased with a tare weight, m t ≈ m /
2, such that the currents in the coil havethe same absolute value but opposite signs. The forces on the balance for the two states are mass-on : I on ( Bl ) v ( + α I on + β I ) + mg = m t g mass-off : I off ( Bl ) v ( + α I off + β I ) = m t g . (6)The tare weight is adjusted such that the currents are symmetric, I on = − I off , and it is sufficient to workwith the variable I : = I off . By subtracting the mass-off equation from the mass-on equation in (6), themass can be obtained as m = ( Bl ) v g (cid:0) I + β I (cid:1) , (7)where as mentioned before ( Bl ) v is obtained from the velocity mode. By using symmetric currents,all terms containing α vanish. The only remaining systematic term, 2 β I is very small, 2 β I / ( I ) ≈ − .
39, 40
Although the term is small, it is measurable by using different mass values on the Kibblebalance, e.g., m / m , 2 m . This process is possible in the new SI, because multiple and sub-multiplesof masses can be generated without having to resort to Kibble balances using a classical scheme tosubdivide masses
15, 16 . In summary, the Kibble principle is preserved when symmetric currents areapplied during weighing mode, because the dominant term of the dependence of the magnetic field onthe weighing current drops out. The next to leading order effect is small and can be compensated forusing ancillary measurements. For the remainder of this manuscript, we assume β = M = χ H depends linearlyon the applied external field H . The proportionality factor is given by the volume susceptibility χ andis negative for diamagnetic and positive for paramagnetic materials. It is impossible to build a coilwithout using weakly magnetic materials. The magnet wire used to wind the coil is made from copper heel balanceDrive motorSuspensionMass Magnet & coil P M ( upp e r ) P M ( l o w e r ) Air gap R a R m R m E E φ A φ B z r Coil − −
10 0 10 203.63.73.83.94.0 z /mm L / H measurement L ( z ) = L − kz − −
10 0 10 20 − − z = mm z = mm z = − mm B ( I ) B ( ) − = I α ( z ) z /mm R e l a t i v e B c hange × a bcd Figure 1.
The magnet system in a Kibble balance and the coil-current effect. (a) presents the majorelements in the fourth generation Kibble balance experiment at NIST. The left subplot of (b) is thesectional view of a typical permanent magnet system with symmetry, where the color map denotes the B field distribution. The right subplot presents an equivalent electrical circuit of the air-gap typemagnet system, where R m is the magnetic reluctance of the permanent magnet, R a the magneticreluctance of the air gap, E , E respectively the magnetomotive force of upper and lower magnets. (c)shows a typical measurement of the coil inductance (frequency extrapolated to DC) as a function ofcoil vertical position z . With a up-down symmetrical magnet, it can be written as L = L − kz . (d)shows the relative magnetic field change due to the coil current in such magnet systems. The plotshows the magnetic field with a plus current I off , which produces 4.9 N magnetic force. The red curveis an average magnetic field for the coil, and this field slope has been verified at BIPM as B ( I ) B ( ) − = I α ( z ) , where α is a linear function of z . Note that the field distribution with I on is an imageof I off symmetrical to B ( ) . hich is diamagnetic with χ ≈ − − . The diamagnetic force has been impressively demonstrated bylevitating a diamagnetic object, e.g. graphite , organics , water , living cell , even a frog , in amagnetic field. The force on a very small element with volume V of weakly magnetic material in theair gap of a permanent magnet is given by F χ = χ V µ B ∂ z B . (8)There is a constant static force acting on the coil, but it is common to the mass-on and mass-offmeasurement, similar to the coil’s weight, and it will drop out in the difference of the mass-on andmass-off measurement. To be clear, the diamagnetic force on the coil is well-known but is thoughtto drop out in the reversal of the current in force mode. The systematic described below is named thediamagnetic effect in force mode. It is the diamagnetic force that does not cancel between the twomeasurements in force mode.Nevertheless, a systematic bias, the diamagnetic effect, cannot be ruled out, because B is not aconstant, rather it is a function of current in the coil according to equation (5). Consequently, F χ mustbe a function of current also. The difference between mass on and off would be ∆ F χ = χ V µ I ∂ z (cid:0) B v α (cid:1) (9)so that the quadratic nonlinearity no longer cancels and we are forced to consider α . Up until 2017, α was assumed to be constant, independent of the coil position, and dispensed with. A notable articlepublished that year associated α with the reluctance effect . The reluctance effect can be explained byconsidering the magnetic energy stored in the magnetic field surrounding the coil due to the constantcurrent during weighing, E = / LI where L is the self inductance of the coil in its surroundings.Once again, a force arises as a result and in the direction of any gradient of the magnetic energy.The vertical component of this force can be written as F z = − / I ∂ z L . The force points toward themaximum of the inductance, usually at the middle of the symmetry plane of the coil magnet system.This principle is well known from solenoid actuators, where an iron slug is retracted into a solenoidwhen it is energized with current. Here, the slug (the magnet and yoke material) is fixed, while the coilis free to move in the z direction. The inductance L ( z ) depends mostly on the symmetry of the shapeand magnetic properties of the yoke and not on the permanent magnet material. For an ideal yoke L = L − kz is a quadratic function of z with z = F z = − / I ∂ z L = ( Bl ) add I , and hence ( Bl ) add = − / I ∂ z L . As described in figure1(d), experiments at the BIPM prove that this additional magnet field does, in fact, exist
46, 47 . Hence,the parameter α introduced in equation (5) can be written as α = − kz / ( Bl ) v .The partial derivative in equation (9) can be rewritten as ∂ z B v α = α∂ z ( B v ) + B v ∂ z α . The magnetsystems for the Kibble balances are often designed such that ∂ z B v = − B v k / ( Bl ) v , and the relative size of the effect can be obtained fromequation (9) as ∆ F χ mg = − χµ A c Nl k . (10) ere, we are formulating the effect on the wire while considering multiple turns, so the volume ofthe wire, V has been replaced by the product of the wire cross sectional area A c , the length l andthe number of turns N . The derivation will also work for non-current carrying elements, like thecoil former or structures mounted on the coil, but the equations are more insightful for the wire.The relative effect consists of three factors and typical values are χ / µ = − − , A c / ( Nl ) =
200 mm / ( ·
834 m ) = . × − m, and k =
550 H / m . Multiplying the three factors togetheryields a relative force of 1 × − . An amount that is more than 100 times larger than the combinedrelative uncertainty reported by the best experiment in the world.Here we reach an impasse. The paradox. On the one hand, the above summary of current reasoning,modeling and experimentation supports the conclusion that the diamagnetic effect in force modedoes exist. On the other hand, measurements of the Planck constant using two completely differentmethodologies (XRCD and Kibble balance) agree to within 1 × − , supporting the conclusion that itdoesn’t. Where is the truth?A possibility that must be considered is that there is a common bias, or intellectual phase-lockamong the experiments. After all, the highest precision Kibble balances share similar design parameters,and the community was driven by a common goal to seek a consensus value. Perhaps the relative sizeof the effect does not vary much from balance to balance. Being common mode to all, it would notbe observed. But values of the Planck constant were compared among all Kibble balance and XRCDmethodologies. To support such a bias among the balances requires intellectual phase lock acrossthe competing methods and multiple laboratories on a global scale. This seems highly unlikely in ametrology community fiercely committed to objectivity.Another possibility that must be considered is that the diamagnetic effect in force mode doesn’texist. The deniers of this effect likened the force produced by it to the fictional force that BaronMunchausen used to pull himself out of a mire by his own hair – clearly in violation of Newton’s thirdlaw. They argue, that the current in the coil cannot exert an additional and current dependent force onitself. This force, however, is between the magnet system, altered by the current, and the coil, similarto the reluctance force that undoubtedly exists (A detailed analysis can be found in the SupplementaryInformation). Given the state of knowledge, it seems logical to suggest an experiment be performed tomeasure the effect directly. Unfortunately, this is exceedingly difficult. According to equation (10)the effect depends only on variables that are, for the most part, impossible to modify for a givenKibble balance. These are instruments designed to maintain absolutely constant physical, magnetic, andelectrical geometries save for one coordinate. Changing the mass, and hence the current in the coil, willnot change the relative contribution of the diamagnetic force. The only variable sometimes available isthe coil geometry A c / ( Nl ) , but even that is not simple. Several Kibble balances have multiple coilswound on a single former, and the Kibble experiment can be performed with different coils or differentcoil combinations. Unfortunately, the relative contribution of the diamagnetic effect does not changeas long as all coils are immersed in magnetic flux produced by the same magnetic system, regardlessif they are active (used in the experiment) or not. In summary, it is conceivable that a relative bias aslarge as 1 × − exists in all Kibble balance experiments. n this article, we will solve the paradox of the diamagnetic effect in force mode. The surprisingresult is that the diamagnetic effect exists, but we find a symmetric effect in the velocity mode. Bycombining the measurements taken in velocity mode with those made in weighing mode, the biasintroduced by the diamagnetic effect is canceled. These counteracting biases explain the paradox,restore confidence in the foundation of the new SI mass, and have never been described in the literature.The result is simple and satisfying: the symmetry of the Kibble balance experiment once again selfcorrects, and the diamagnetic effect vanishes in the combined result. This new finding will relax therequirements on the materials that the coil and components attached to it are made from. Weaklymagnetic materials can be used in these cases. Still, one has to be careful not to use ferromagneticmaterials, because materials with a nonlinear response to the external field are not covered by thissymmetry. Results
Analytical result of the diamagnetic effect in velocity measurement
In the previous section, we have argued that the diamagnetic effect exists and that it produces alarge relative bias in the weighing mode of Kibble balances. The bias is so large that Kibble balanceswould not be able to make precise measurements. Here we show that the bias in the weighing mode iscancelled by an identical bias in the velocity mode and the Kibble principle holds.We start by rewriting the self inductance of the coil L ( z ) with N turns according to the derivation inthe methods section. In a cylindrical air-gap with a mean radius r a , a radial width of w a , and a height2 h a , the inductance is given by L ( z ) = L − π µ N r a w a z h a = ⇒ k = − ∂ L ∂ z = π µ N r a A a , (11)where A a = w a h a denotes the cross-sectional area of the air gap. By employing a cross sectional areafor the coil, the relative size of the diamagnetic effect can be written compactly as ∆ F χ mg = − χµ A c Nl π r a µ N A a = − χ A c A a r a r c . (12)Next, we investigate what happens when a diamagnetic material is introduced to the air gap. Theleft plots in figure 2 show the magnetic flux density as a function of vertical position. Before thematerial is introduced, the flux density is constant throughout the gap (red line). A constant flux densityfor the air gap is assumed to keep the explanation simple, but is not necessary for the theory to work.Adding the coil, here with χ <
0, changes the flux profile. A perfectly nonmagnetic coil would have noeffect, but the vertical section occupied by the coil now restricts the flux, due to the increased magneticreluctance of the diamagnetic material in that part of the gap. The total flux produced by the permanentmagnet redistributes itself, and, as a result, the flux density in the empty space increases in directproportion to the reduction of flux through the space occupied by the coil.For χ <
0, compared to the situation without the coil, B , the value of the flux density is lower atthe coil ( B c ) and higher in the rest of the gap ( B χ ). In the physical system, there are nonlinear effects cB BχBcB BχBcB BχBcB BχBcB Bχ z z z z z z z I nn e r Y o k e O u t e r Y o k e CoilPath I, z = z χ Path II, z = z c B r H rr i r l r r r o P a t h I P a t h II P a t h I P a t h II abcde fgh Figure 2.
A qualitative illustration of the magnetic field distribution at different coil positions. (a)-(c)show the B ( r c , z ) curves at three different vertical position z , z and z . The red curves are themagnetic profile when the coil susceptibility is zero. The blue curves are the first order approximationof B field curve with a diamagnetic coil, χ <
0. The green curves are profiles of the diamagnetic coilwith higher order approaching. (d) and (e) present the B field difference under two configurations: ( z − z ) > h c and ( z − z ) < h c (coil region overlap). (f) show two paths horizontally across the airgap, respectively at z χ and z c . (g) and (h) present the B field and the H field distributions along twopaths.near the edges, shown by the green curves in figure 2. Again, these are not important for the simplifiedexplanation of the effect and can be ignored.The magnetic flux threading through the coil can be obtained as the integral from the bottom ofthe air gap to the middle of the coil, indicated by the blue shaded region for the coil in three differentvertical positions in figure 2 (a) through (c).As mentioned above, the induced voltage in velocity mode is proportional to the derivative of themagnetic flux through the coil with respect to time. The flux for the baseline position of the coil z is shown in (a), while the flux for the positions z and z are shown in (b) and (c). The difference influx with respect to the coil at baseline for these positions is depicted in (d) and (e), respectively. Weassume that the coil moves through the gap along a fixed trajectory with the same constant velocity in he z direction for the case when coil susceptibility is zero, and then again when it is χ . The relativedifference of the flux density change between these scenarios, and, hence, the induced voltage is givenby ∆ U χ U = B χ B − . (13)The flux density B χ can be calculated assuming that the total magnetic flux through the air gapremains the same. At r = r c , the flux integration vertically through the whole air gap can be written as2 h a B = h c B c + ( h a − h c ) B χ = ⇒ (cid:0) B χ − B (cid:1) ( B c − B ) = − h c h a − h c , (14)where the negative sign indicates that B χ > B and B c < B when diamagnetic material is introduced.For paramagnetic material, B χ < B and B c > B .The ratio of the change from B of B χ and B c to B is identical to the ratio of the height of theoccupied gap to the height of the empty air-gap, since 2 h a and 2 h c denote the height of the air gap andthe coil, respectively.In an actual magnet system, the magnetic height of the air gap 2 h a differs from the geometricalheight of the air gap 2 h geo as one would measure with a ruler. Due to fringe fields, h a > h geo . Weassume the magnetic height of the air gap is known.For the typical large permeabilities of the yoke materials, the metal on each side of the air gap is amagnetic equipotential surface. Hence, the magneto motive force over the air gap given by (cid:82) r o r i H ( r ) d r with r i and r o denoting the inner and outer radius of the air gap, does not change when the coil isintroduced and is independent of the vertical position z where the integration is performed. Themagnetic field H is the magnetic flux divided by the permeability, H = B / ( µ ( + χ )) and is a functionof radius and height H ( r , z ) = H ( z ) r c / r , where we have used the fact that the field drops off as 1 / r and H ( z ) is the field at the mean coil radius r c .Two paths of integration through the coil at z = z c and in the empty air gap ( z = z χ ) are shown in 2(f). Integration along these paths yield (cid:90) r o r i H ( z χ ) r c r d r = (cid:90) r o r i H ( r , z c ) d r (cid:90) r o r i B χ r c µ r d r = (cid:90) r l r i B c r c µ r d r + (cid:90) r r r l B c r c µ r ( + χ ) d r + (cid:90) r o r r B c r c µ r d r (cid:90) r o r i B χ r d r ≈ (cid:90) r o r i B c r d r − (cid:90) r r r l χ B c r d r , (15)where r l = r c − w c / r r = r c + w c / B ( r ) and H ( r ) as a function of r for both integration paths are shown in 2 (g) and (h),respectively. Integrating the terms, approximating the resulting logarithms in a Taylor series of firstorder and combining the result with equation (14) gives B = B χ (cid:18) + χ w c h c w a h a r a r c (cid:19) (16) ith equation (13) the relative change that the magnetic material has in the velocity mode can bestated as ∆ U χ U = B χ B − ≈ − χ A c A a r a r c . (17)As before, A c = h c r c and A a = h a r a denote the cross-sectional areas of the coil and the air gap,respectively.Equation (17) shows the relative change of the induced voltage in the velocity mode is identicalto the relative change in force mode, see equation (12). The robustness of Kibble’s reciprocity todeviations from the ideal experimental setup without magnetic materials, are caused by a strongsymmetry in the underlying physics. Without that robustness the Kibble balance would not be thesuccess that it has been in metrology. The relative differences of the measured force and in voltagefrom the corresponding ideal theoretical values in the absence of weakly magnetic materials are givenby U real − U ideal U ideal = F real − F ideal F ideal ≈ − χ A c A a r a r c . (18)Since r a ≈ r c , the relative effect is proportional to the magnetic susceptibility and the cross-sectionalfilling ratio of the air gap. The latter denotes how much of the cross-sectional area of the air gap istaken up by the coil. With typical values, χ = − − and A c / A a = .
1, the relative difference betweenthe real and ideal numbers is 1 × − . In conclusion, the diamagnetic force with a relative magnitudeof 1 × − about 100 times larger than the reported relative uncertainties exists. But the results of theKibble balance experiments are not affected by it, because the same relative bias will be introduced inthe velocity mode. In the combination of the measurement results from force and velocity mode, theeffect cancels perfectly.The derivation above has been made using ideal geometries to show the powerful and simple idea.But, the theory holds for more complex and realistic field situations, as is discussed in the ‘Methods’section. Numerical verification
Numerical verification of a relative force change that is as small as 10 − is impossible. Sinceengineering tasks are rarely concerned with effects that small in size, commercial finite elementprograms are not optimized for the precise prediction of these small effects. At this order of magnitudetheir results cannot be trusted. To overcome their limitations and to be able to use commercialfinite element analysis (FEA) software, we invented a new technique that we name differential FEA(dFEA). While more information on dFEA can be found in the supplemental information, the followingparagraphs explain the general idea. All effects discussed here are proportional to the magneticsusceptibility χ and it can be used as a parameter to verify the result. Setting χ to a large value amplifiesthe relative change in force and voltage. With χ ≈ × − , relative effects of 1 × are achieved.Although the theory is only weakly dependent on geometry and independent of the size of the magneticfield, typical values are used. A magnetic flux density of B = .
54 T was chosen, and it requires / − . χ value, the force on thecoil without current is calculated. Then the forces for positive and negative currents are calculated.From both the null result is subtracted. This differential approach suppresses systematic errors in thecalculation due to meshing and rounding of the small effects.The calculated force differences with positive (mass-off) and negative (mass-on) currents are shownin figure 3(a) and (b), respectively. In each subplot, the force is given as a function of coil position z with −
10 mm ≤ z ≤
10 mm for the five χ values. The clearly visible slope in each subplot, is causedby the reluctance force, in agreement with the theoretical model discussed in 48. The slope is the samefor both current directions and independent of χ . Hence, in the force difference, shown in figure 3(c),the slope vanishes and the difference is nearly independent z , exactly as described in 48. Neglecting theslopes, the observed values of F ( I off ) and F ( I on ) at a given point, for example z =
0, change with χ .The change of the force difference relative to the weight of a 1 kg mass at z = χ . Diamagnetic materials ( χ <
0) yield largerabsolute values for the forces for both current directions. Unlike the slopes, this effect does not cancelby subtracting mass-on from the mass-off measurement. The effect is clearly visible in figure 3(c)where the force differences are plotted.A linear dependence of the force differences on χ is observed, see the dashed line in figure 3(d).The slope of the line can be obtained by a numerical regression to the calculation results, and byusing the regression coefficients the effect can be scaled down to small χ values whose results wouldotherwise be in the rounding error of the numerical analysis. For χ = − × − a relative change of9 . × − is obtained, in very good agreement to the theoretically obtained result of 1 × − . Thenumerical results confirm the theoretical analysis, as well as the existence of the diamagnetic force.The same FEA calculation can be used to estimate the effect in the velocity mode. Here, wecalculate the magnetic flux density in the air gap for a coil that does not carry any current for the fivesusceptibilities discussed above. As in the text above, two symbols are used to describe the flux densityin the air gap in the presence of magnetic material. At regions that the coil occupies we use B c and allother regions B χ .Figure 3(e) and (f) show the magnetic flux densities B c and B χ as a function of z , with the coilat three different positions z c , z c = − z c = z c = χ , one of the two quantities B c and B χ is larger and the other smaller than B , which can be seenin the two middle panels with χ =
0. The curves of B χ show a transient step at the border close to B c .This is an artifact of the FEA calculation which cannot reproduce the perfect step function in B thatwould be present at the boundary in the real world, see figure 2 (b). We believe that the transient has no .9054.9104.9154.9204.925 F o ff / N − − − − − F o n / N (cid:0) F o ff − F o n (cid:1) / N − − − χ ∆ F χ m g FEA resultlinear fit χ = -0.01 -0.005 0 0.005 0.01 Weighing Measurement B c / T B χ / T − − − B c B − FEA resultlinear fit − − − χ ∆ U χ U = B χ B − FEA resultlinear fit χ = -0.01 -0.005 0 0.005 0.01 Velocity Measurement abcd efgh
Figure 3.
Results of the magnetization effect in weighing and velocity measurements. (a)-(c) Resultsof magnetic force as a function of coil position -10 mm ≤ z ≤
10 mm for five different magneticsusceptibilities. The current in the coil was equivalent to one turn with I off =- I on =11.6 A. (d) Therelative change in force difference as a function of χ . Note that mg is defined at force difference at χ =
0. (e) The B c distribution at z = ± z = χ values. No current isassigned in the calculation. (f) shows the B χ field distribution. (g) presents the relative magnetic fieldchange of B c and (h) shows the change of B χ . nfluence on the conclusion, especially since its integral evaluates to zero. For any χ (cid:54) = B χ − B isabout a seventh of B − B c , and, hence, according to equation (14), the effective gap height is abouteight times the coil height, such that h c / ( h a − h c ) = / B c and B χ with respect B areshown in figure 3(g) and (h). Both figures are plotted for z c =
0. The former shows B c ( ) the latter B χ near the end of the gap, both are relative to B at the same locations. Similar to the results in theforce mode, the results are linear with respect to the chosen magnetic susceptibility and a regression tothe calculation results is performed. From the regression coefficients, B χ / B − χ , a result that would be unobtainable directly from finite element analysis. For χ = − × − , B χ / B − . × − . For comparison, the relative effect in force mode for the same χ was9 . × − . The calculated relative effects in force mode and velocity mode agree remarkably well(the difference of 1 × − is negligible compared to the numerical uncertainty of the FEA calculation).The summary of this section is given in the last row of figure 3. The left graph shows the relativebias that is incurred in force mode as a function of the magnetic susceptibility of a weakly magneticcoil. The right graph shows the relative bias incurred in velocity mode as a function of the same χ . Theresults are identical, the relative biases depend linearly on χ . For the model discussed here the slopeof the line is approximately − /
10, which corresponds to the fraction of the cross sectional area ofthe air gap that is filled by the coil. So, a Kibble balance with this geometry and a weakly magneticcoil would produce values for both modes that differ by − χ /
10 compared to the same balance thathas a completely nonmagnetic coil. However, when the results from the force and velocity mode arecombined according to equation (3), the relative biases cancel each other and the mass measured bythe Kibble balance with a weakly magnetic coil is identical to the mass measured by a Kibble with anon-magnetic coil.
Discussion
The work that led to this article accomplished four tasks.1. We have shown that the diamagnetic effect in force mode exists and its relative magnitude canbe as large as 1 × − .2. We have discovered a corresponding effect in velocity mode that completely cancels out theeffect of the diamagnetic force in the Kibble balance experiment. Such an effect has never beendescribed before in the literature.3. We have developed a new technique to calculate very small magnetic effects caused by weaklymagnetic materials using finite element analysis.4. By using the newly developed technique we could verify the existence of (1) and (2) and showthat they have the same relative size within the numerical uncertainty.Below we summarize the most important points for these accomplishments. he force described by the diamagnetic effect exists and it is large ( ≈ × − ) compared tothe relative uncertainties that Kibble balances report ( ≈ × − ) . A Kibble balance requires a coilimmersed in a magnetic field. Often the magnet wire is made from copper that is weakly diamagneticwith χ = − × − . Without current a diamagnetic force on the coil wire exists, but it is a constantforce comparable to the weight of the coil and will not impact the result. What is understood as thediamagnetic effect is caused by the current in the coil during the weighing measurement. This currentgenerates an additional magnetic field which interacts with the magnet system in what is known asback-action. Due to the back-action, the diamagnetic force is no longer constant, but proportional tothe current in the coil, and, hence, it no longer cancels and provides a systematic bias in the weighingmeasurement of the Kibble balance experiment. The relative size of this effect can be written verycompact, see equation (12). If the coil and the air gap have the same radius the effect is proportional tothe magnetic susceptibility and the ratio of the cross-sectional areas of the coil and the air gap.Unbeknownst to the scientist and engineers working with Kibble balances, there is also an effect invelocity that arises when a weakly magnetic material is added into the gap. Introducing such a materialin the gap changes the magnetic flux density and hence the result that is obtained in the velocity mode.Adding, for example, a diamagnetic coil in the gap reduces the magnetic flux density where the coilis and increases the magnetic flux density in the remainder of the gap. This is a consequence of thechanged reluctance of part of the gap. Where the coil is the magnetic reluctance is larger leading toa smaller amount of flux. However, since the flux through the total air gap remains approximatelyconstant, the flux at the remainder of the gap increases. The increased flux causes a larger inducedvoltage when the coil is moved through the gap compared to the situation where the coil is non-magnetic.As shown in equation (17), the relative change in voltage evaluates to the same expression as for thediamagnetic force. Hence the bias introduced in the weighing mode is cancelled by an equal bias invelocity mode. Thus, the paradox of the diamagnetic force on the Kibble coil is resolvedTo prove the existence of the effect of weakly magnetic materials in force and velocity mode wehave developed a new technique that we call differential finite element analysis (dFEA). Calculatingsmall forces or field changes caused by the introduction of materials whose susceptibility is of order1 × − is impossible. The numerical uncertainties are much larger than the relative effects one desiresto calculate. In differential FEA, the susceptibility of the material to be investigated is a parameterand the model is calculated with several different large susceptibilities. Values of χ ≤ × − wereused, up to a thousand times larger than the susceptibility of the coil in the physical experiment.For differential FEA to work, it is important to keep the same mesh for all calculations. From eachcalculation result, a null-result that was obtained by setting χ equal to zero is subtracted. In the end,the quantity of interest is plotted as a function of the used χ and a smooth function is fitted to the result.The fitted parameters of the function can be used to calculate the effect for small χ that the physicalsystem has. For the cases discussed here, both effects scaled linearly with χ making the scaling simple.We used differential FEA to calculate the effect that the introduction of a weakly magnetic materialhas on measurements in force and velocity mode. We find the calculated result in agreement with asimplified analytical model that we have developed in the preceding sections. The relative sizes of he effect are of order 1 × − and would render Kibble balances useless. The effect has the samemagnitude and sign in both modes and will cancel in the combined result. We believe that this is anadditional, to date not recognized symmetry of the Kibble balance that allows it to work in the presenceof linear magnetic materials. The result of the differential FEA shows that the biases in force andvelocity agree within a difference of 1 × − , limiting the upper bound for the relative bias of thecombined measurement to 1 × − .The paradox of the diamagnetic force in Kibble balances has been solved. The ongoing discussionsin the Kibble balance community are brought to a satisfying end. The reciprocity of Kibble’s equationworks perfectly in the presence of linear magnetic materials. ethods Self-inductance L ( z ) of a coil in a symmetrical yoke Let w a , 2 h geo , and r a be the geometrically measured width, height, and mean radius of the air gap. Neglecting thefringe fields at the end, the air gap has a magnetic reluctance of R ideal = w a / ( π µ r a h geo ) . The relative correctionnecessary to account for the fringe fields scales with gap’s aspect ratio w a / ( h geo ) . The reluctances of theleakage paths at both ends of the gap are parallel to R a , lowering the total reluctance of the system. Writing thereduction factor as 1 / γ with γ >
1, yields R a = w a / ( π µ r a h geo γ ) , which can be interpreted as the reluctance ofan ideal air gap of the same dimension, but a magnetic height, 2 h a , that differs from the geometric one accordingto h a = γ h geo . In our idealized gap, the magnetic flux is purely horizontal. Neglecting the vertical flux, makes theanalysis simpler without altering the conclusion.Now, let’s investigate the flux that is produced by a coil at position z c with N turns carrying a current I . Theflux φ , generated by the magneto motive force of the coil, has to traverse the air gap above and below the coil.Hence, φ = NI ( R u + R l ) , where R u = w a π µ r a ( h a − z c ) , R l = w a π µ r a ( h a + z c ) , (19)where R u / l denote the magnetic reluctance of the partial gap that is above/below the coil. Using the definition ofthe self-inductance L = N φ / I and expanding the fractions to second order in z c yields L = π µ N r a (cid:18) h a w a − z c w a h a (cid:19) . (20)which appears in the text as equation (11). Experimentally, the inductance of the coil can be measured as afunction of z c . By fitting L = L − kz c to the data, the magnetic height of the air gap can be determined from k as2 h a = π µ N r a w a k (21)For the magnet employed by the BIPM Kibble balance, k ≈
550 H m − , see 46. Using the reported technicaldata of that magnet system, w a =13 mm, N = r a =
125 mm, a magnetic height of 2 h a =
155 mm isobtained. A comparison to the measured, geometric height, 2 h geo =
82 mm shows that γ = . h a obtained by the measurement of L ( z ) agrees with the one obtained by FEAwithin a few percent. The difference is due to the fact that magnet’s top cover was missing in the experiment. General equation of a moving cylindrical segment with finite χ Here, we investigate the effect cause by any weakly magnetic part that is co-moving with the coil. Such parts areabundant in any Kibble balance experiments and include the coil former, the supporting frame, optical elements,and fasteners. Without loss of generality, we investigate a cylindrical part with a rectangular cross sectionidentified by the subscript i , its height, width, mean radial location, and magnetic susceptibility are denoted by2 h i , w i , r i , and χ i . The symmetry axis of the part coincides with the symmetry axis of the magnet.Starting with equation (9), the diamagnetic force in weighing mode on the segment is ∆ F χ , i = χ i V i µ I ∂ z (cid:0) B v , i α (cid:1) , (22)where V i = π r i ( h i w i ) is the volume of the cylindrical segment and B v , i the magnetic flux density at the segmentposition without current in the coil. The flux produced by the permanent magnet and the coil are both horizontal, nd, hence, the flux density is proportional to 1 / r . Consequently, at the mean radial position of segment i , theflux is B v , i = ( r c / r i ) B v , yielding ∆ F χ , i ∆ F χ = χ i χ V i V (cid:18) B v , i B v (cid:19) = χ i χ r c r i . (23)Using the expression for the diamagnetic effect on the coil (12) together with (23), produces a compact expressionfor the relative diamagnetic force produced by the segment. It is ∆ F χ , i mg = − χ i A i A a r a r i , (24)where A i = h i w i the sectional area of segment i , and A a the cross sectional area of the air gap defined above.Most importantly and similarly to equation (12), the relative diamagnetic force on segment i is independentof the coil current and the magnetic field B v , and is determined only by the material property ( χ i ) and geometricalratios ( A i / A a and r a / r i ).Next, the influence of the weakly magnetic segment i on the measured value of Bl in velocity phase isinvestigated. Assuming a magnet system with perfect up-down symmetry, as shown in figure 1(a), we define thefollowing three surfaces at r = r c : A ( r ≤ r c , z = h a ) and B ( r ≤ r c , z = − h a ) present the horizontal surfacesrespectively at the upper and lower gap ends. C ( r ≤ r c , z = z c ) is the coil surface. In perfect symmetry, themagnetic flux φ A penetrating surface A , equals the flux φ B through surface B . An asymmetry can be takenaccount by introducing a flux difference ∆ φ such that φ A = φ B + ∆ φ , (25)By using an electrical circuit model following Ohm’s law of magnetism as shown in figure 1(a), φ A and φ B aredetermined as φ A = E ( R m + R a ) and φ B = E ( R m + R a ) . (26)A magnetic segment co-moving with the coil, does not contribute to the air gap reluctance R a , and, hence, R a does not depend on the vertical position of the segment. Since φ A , φ B are constant for a given magnet system,the flux difference ∆ φ must also be independent of the coil position z c .That flux that goes through surface A will then either go through the coil or through the part of the air gapthat is above the coil φ U , φ A = φ C + φ U . Similarly, all the flux penetrating B flows through the part of theair gap that is below the air gap or through the coil, φ B = − φ C + φ L . The negative sign before the coil fluxindicated the direction of the flux relative to the normal vector of the coil. It is reverse for the flux φ B .The fluxes φ U and φ L can be written as a product of the surface area and the magnetic field at the radius r i under the assumption that the field is mostly independent of z . Hence, φ U ( χ i = ) = π r i ( h a − z c ) B , i , φ L ( χ i = ) = π r i ( h a + z c ) B , i . (27)the flux through the coil can now be obtained as φ C ( χ i = ) = π r i B , i z c + ∆ φ . (28)To evaluate the induced voltage only the component that depends on z is relevant and we obtain U ( χ i = ) = π r i NB , i v z . (29) aking again advantage of the 1 / r dependence of the magnetic flux density, B , i = r c / r i B . For χ =
0, thevoltage-velocity ratio is the 2 π r c NB = B l in agreement with the conventional theory of the Kibble balance.For χ i (cid:54) =
0, the magnetic field distribution along the vertical direction at r i is given by discrete two valueswith sharp steps between them. In the coil region (from z i − h i to z i + h i ), the magnetic flux density is B c , i and themagnetic flux density of the rest air gap is B χ , i . In this case φ U and φ L are written as φ U ( χ i ) = π r i (cid:2) B c , i η i h i + B χ , i ( h a − z c − η h i ) (cid:3) , φ L ( χ i ) = π r i (cid:8) B c , i ( − η i ) h i + B χ , i [ h a + z c − ( − η i ) h i ] (cid:9) , (30)where η i defines the height fraction of the segment that is above the coil z c . For example, when segment i is fullyabove the coil, η i =
1. If segment i is coincident with the coil (like the coil itself), then η i = .
5. Note since thesegment is co-moving with the coil, η i does not change with z c . Through equation (30) and known magnetic fluxrelations, the magnetic flux through the coil is solved as φ C ( χ i ) = π r i (cid:2) B χ , i ( z c + ( η i − ) h i ) − B c , i ( η i − ) h i (cid:3) + ∆ φ . (31)Dismissing all factors that are independent of z c , the induced voltages simplifies to U ( χ i ) = N ∂ φ C ( χ i ) ∂ t = π r i NB χ , i v z . (32)Comparing equation (29) to equation (32), the relative change in the induced voltage cause by segment i is ∆ U χ , i U = B χ , i B , i − . (33)This result is the equivalent expression as given in equation (13). The result of B χ / B − ∆ U χ , i U = B χ , i B , i − = − χ i A i A a r a r i . (34)The relative effects in force and velocity mode caused by the introduction of a cylindrical segment i withfinite susceptibility are identical, compare equation (24) to equation (34).The relative changes of the induced voltage and the extraneous force produced by a single segment i dependon the ratios A i / A a and r i / r a . Hence, the scaling can be checked by comparing the relative effects produced bytwo different segments i and j with the same magnetic susceptibility. The ratio of the relative effects must scalelike ( A i / r i ) / ( A j / r j ) . The calculation was performed for both segments for both modes, velocity mode and forcemode. Again, differential finite element analysis as described in the main text was used to interpolate the effectsto χ j = χ i = − × − by using calculations that used susceptibilities ranging from -0.01 to 0.01.As segment i , we use the coil which has been already shown in the main text and, for convenience, we reiteratethe numbers here. The coils has a cross-sectional area of A i =
200 mm and a mean radius of r i =
125 mm. Itproduces a relative effect of 9 . × − .For the segment j we chose an area of A j =
50 mm . It is located at r j = . j . j is the same for force and velocity mode and it is2 . × − .The ratio of the two relative effects is 3.92 and the ratio of the corresponding geometrical factors ( A i r j ) / ( A j r i ) = .
92. The agreement between the geometrical ratios and the calculated effect validate equations (24) and (34). cknowledgments
During the last two years, many experts in the field (BIPM, NIST, NRC, NIM) have been involved in this study.We would like to thank all these colleagues for their valuable discussions with us, especially Michael Stock(BIPM) and Carlos Sanchez (NRC). Their advice was helpful to understand the effect and to finally solve theparadox out. We also would like to thank Jon Pratt (NIST) for the werb review and constructive suggestions toimprove the manuscript quality.
Author contributions statement
S. Li and S. Schlamminger developed the analytical theory. Major discussions have been ongoing among allauthors for more than two years. Numerical simulations were preformed by S. Li and R. Marangoni. Themanuscript was written by S. Li and S. Schlamminger and all authors reviewed the manuscript.
Competing interests
The authors declare no competing interests.
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IEEE Trans. Instrum. Meas. , 7752–7760 (2020). upplementary InformationThe magnetic field produced by a coil in free space and inside a yoke In this section, we compare the magnetic field of a current-carrying coil in free space and in a Kibble balancemagnet. Figure 4(a) shows the effect that the current in the coil has on the measured Bl in force and velocitymode. We present a summary of the finding that have been described in . Plotted on the vertical axis is thedifference of Bl measured in different scenarios from a velocity phase measurement ( U / v ) = B l where the coilis moving through the magnetic field without any current. The two profiles ( F / I ) off and ( F / I ) on are used inmost Kibble balances for weighing – usually the two measurements in weighing phase, mass-on and mass-off,are carried out with equal and opposite currents, I off = − I on. The profiles ( U / v ) off and ( U / v ) on are profilesdetermined in the velocity phase, respectively with plus and minus currents. The bifilar coil used in the BIPMbalance allows current to be present during the velocity mode. For conventional two-mode Kibble balances, ( U / v ) is used, while ( U / v ) off and ( U / v ) on are for one-mode measurement schemes, e.g. .The data in figure 4(a) shows that the change in Bl is a linear in coil position z , as well as in coil current I inthe weighing measurement. As discussed in the article, the field gradient ∂ z B produces a diamagnetic force biasthat can not removed by mass-on and mass-off measurements. The interesting conundrum is, can the magnetfield gradient that is produced by the coil itself produce a force on itself.To clarify the paradox we study two scenarios: 1) the coil in free space and 2) the coil inside an air gapsurrounded by iron yokes as it would in a Kibble balance . Figure 4(b) and (c) show the magnetic flux densitydistribution along z for different weighing positions and both current directions. The results shown in the figureare obtained by finite element analysis based on the BIPM magnet system with an average field in the air gapof B = .
45 T. To obtain the free-space calculation, the yoke relative permeability is simply set to one. Usingthe same geometry and mesh, results in a trustworthy comparison of both scenarios.Figure 4(b) clearly shows that the magnetic field distribution is independent of the vertical position forthe coil in free space. The size and shapes of the green, orange and violet lines are identical only horizontallydisplaced by 10 mm from each other. Hence, energetically it does not matter where the coil sits. There is nominimum and hence no force on the coil. Although magnetic gradients exist on the curve B ( z ) of a free spacesystem, any force related to these gradients is internal (the coil gets compressed), which cannot be seen by theweighing unit (balance or mass compactor).For the coil in he yoke, shown in figure 4(c), the situation is different. The additional fields are onlysymmetric for z = z (cid:54) = z as it was in the free spacescenario.The external force on the coil depends on the average field. The experimental observation, shown infigure 4(a) agrees well with the FEA calculation presented in 4(c).The comparison of average fields produced by a current-carrying coil in free space and in a yoke shows thatin the latter the broken symmetry will lead to an energy redistribution and hence a magnetic field change. Aphysical picture can be provided as follows: In the yoke B changes because the coil current magnetizes the yoke,and the magnetized yoke produces an additional magnetic field at the coil position.The diamagnetic force arises from F χ = χ V µ B ∂ B z , where the last factor is the derivative of the volume averagefield with respect to the coil position. As shown in figure 4(c), the slopes of the red and blue lines and, hencethe gradient ∂ B z is constant within reasonable ranges of z . Accordingly, the diamagnetic force is constant as a − − − − − ( F/I ) off ( F/I ) on ( U/v ) off ( U/v ) on ( U/v ) ( F/I ) off meas. ( F/I ) on meas. ( U/v ) off meas. ( U/v ) on meas. z /mm (cid:16) B ( z ) B − (cid:17) × − − − − − − average ( I off or I on) I o n , z c = − mm I o n , z c = mm I o n , z c = mm I o ff , z c = − mm I o ff , z c = mm I o ff , z c = mm z /mm (cid:16) B ( z ) B − (cid:17) × − − − − − − a v e r a g e I o n a v e r a g e I o ff I o n , z c = − mm I o n , z c = mm I o n , z c = mm I o ff , z c = − mm I o ff , z c = mm I o ff , z c = mm z /mm (cid:16) B ( z ) B − (cid:17) × abc Figure 4. (a) The magnetic field change due to the coil current. The middle thick curves are obtained byexperimental measurement in the BIPM system and the thin lines are linear extrapolations from -20 mm to20 mm. (b) presents the relative B field distribution when the yoke permeability is set to µ and (c) shows themagnetic field change with soft normal yoke. Note the reference field, B = .
45 T, is used for both cases (b) and(c). unction of z , but changes direction when the current is reversed. Because of the latter fact, the diamagnetic forcedoes not cancel in the Kibble balance experiment and can lead to a systematic bias.In conclusion, the statement that ‘a current-carrying coil can not produce a measurable force on itself’ holdsfor free space system, but not necessarily for a coil inside a yoke, unless the yoke moves with the coil. We notethat this finding does not contradict Newton’s third law as there is an equal and opposite force on the yoke. Differential FEA (dFEA) – A method to calculate small effects
The diamagnetic effects described in the main article are very small. In force mode, an additional bias of 10 µN ontop of 10 N is produced. It would be impossible with finite element analysis to detect, let alone to calculate withany uncertainty, this additional bias. We first developed the analytical equations of the diamagnetic effect in forceand velocity modes. However, we wanted to verify our analytic results with numerical calculations. In searchingfor ways to do this, we invented differential FEA (dFEA). With differential FEA, it was possible to calculate thesize of the effect with a relative uncertainty of about 0 . on a standard PC in reasonable time. The effects discussed in our work are of the order 1 × − .Hence, they are three orders of magnitude smaller than what we consider reliable results obtainable by FEA.With dFEA, small effects and their uncertainty can be calculated using commercial FEA packages withoutexponentially extending the computation time. The technique takes about five times longer than a singlecalculation because, as the reader shall see below, the same model has to be calculated about five times withdifferent χ . The idea is simple. As mentioned above, the uncertainties of the calculations are 1000 times largerthan the effect, so if one can increase the effect by a factor of 1000, the effect could be detected. In our case,where the effect scales with the susceptibility, one only needs to multiply the susceptibility of the part that isinvestigated by a factor of 1000 in question by a factor of 1000 to achieve that amplification. So, if χ nom is thesusceptibility of the part in question, here the coil, χ exag ≈ × χ nom is used in the finite element calculation.It is safe to change the susceptibility because it occurs as a linear parameter, and the underlying physics does notchange. In the end, all one has to do is scale the measured effect, say the force, by χ nom / χ exag to obtain the sizeof the previously immeasurable effect.A successful implementation of dFEA relies on three good practices:1. It is not sufficient to perform one calculation with χ exag and scale the result by χ nom / χ exag . Usually thereare small calculation biases that would be scaled and falsify the result. See, for example, the middlegraph of figure 5 , where the force calculated with zero current is not exactly zero. The best practice is tocalculate the desired effect for several χ values and then interpolate the size of the effect to the nominal χ .For example, we would like to calculate the diamagnetic effect for χ nom = × − . We use five value for χ exag , namely -0.01, -0.005, 0, 0.005, 0.01. We then plot the desired function, for example the force F , asa function of χ exag , subtracting F ( χ exag = ) . This function is linear in χ exag and the value at χ nom can beobtained from a linear fit to the data. .94.914.924.934.94 F o ff / N FEAdFEA − − F / N FEAdFEA − − − − − − − z /mm F on / N FEAdFEA abc
Figure 5.
Simulation results of the magnetic force of a coil in a magnetic field with different currents: (a) I = I off , (b) I = I = I on . Note that this example is the case with χ = − .
01. The solid lines are obtainedfrom FEA calculations, while in (a) and (c) the dashed lines are differential signals, F off − F and F on − F . Thedashed line in (b) is half the difference of the two forces, ( F off − F on ) /
2. In order to plot it in the same scale themean of 4 .
92 N has been subtracted.2. It is of utmost importance to keep all other parameters constant when using dFEA. If additional parameterswere changed for the calculation with different χ exag , they would cause a change in the calculation resultthat would then, falsely, be attributed to being driven by the susceptibility. This point applies, especiallyto the meshing. The domain should be meshed only once before the various χ exag are assigned to thecomponents.3. The third practice applies to calculations where current is involved. In this case, the best results areobtained by calculating the system twice, with and without current. The difference between the two resultsgives the desired effect. Again, other than the current, nothing can change between the calculations, mostimportantly, not the meshing.Figure 5 illustrates the importance of the three practices mentioned above. The figure shows the FEA resultsof the force on the coil at eleven z positions for three different currents, I off , 0, and I on . The forces, as a functionof z , show the same trend for all three currents. The trend is caused by numerical biases in the FEA softwareand affects all three curves in the same, nonphysical, fashion. By subtracting the data obtained with I =
0, thenumerical biases can be subtracted out. The curves labeled dFEA show relative results.The curves obtained with dFEA are not only smooth, but they are also physical. They represent the coil nductance force , a linear force curve over z . A physical result can be obtained from the original noisy result.The standard deviation of the data calculated with dFEA is 100 times smaller than standard FEA data.In summary, dFEA allows one to calculate a relative diamagnetic force of 1 × − with meaningfuluncertainties. The method relies on FEA calculations with several exaggerated susceptibilities, the largest oneabout 1000 times the size of the nominal susceptibility.with meaningfuluncertainties. The method relies on FEA calculations with several exaggerated susceptibilities, the largest oneabout 1000 times the size of the nominal susceptibility.