Three-Axis Coil Probe Dimensions and Uncertainties During Measurement of Magnetic Fields from Appliances

Comparisons are made between the average magnetic flux density for a three-axis circular coil probe and the flux density at the center of the probe. The results, which are determined assuming a dipole magnetic field, provide information on the uncertainty associated with measurements of magnetic fields from some electrical appliances and other electrical equipment. The present investigation extends an earlier treatment of the problem, which did not consider all orientations of the probe. A more comprehensive examination of the problem leaves unchanged the conclusions reached previously.


Introduction
This paper reconsiders a problem related to the measurement of power frequency magnetic fields from electrical appliances using three-axis circular coil probes.Specifically, it reexamines the differences between the average magnetic flux densitj' as determined using a magnetic field meter with a three-axis circular coil probe and the magnetic flux density at the center of the probe, Ba, assuming the field is produced by a small loop of alternating current, i.e., a magnetic dipole.The "average" arises as a consequence of the averaging effects of the coil probes over their cross sectional areas when placed in a nonuniform magnetic field.The differences between the average magnetic field and Bn can be regarded a.s measurement errors because the center of the probe is normally considered the measurement location.The magnetic dipole field is chosen as the relevant field because its geometry provides a good approximation of the magnetic field produced by many electrical appliances [1].
The average magnetic flux density measured by a three-axis magnetic field meter, Bjv.i, is also referred to as the resultant magnetic field and is defined as [2] B^,i^yjB} + B\ + Bl (1) where Bu Bi, and flj are average root-mean-.square(rms) magnetic field components determined by each of three orthogonally oriented coil probes.
Differences between B^jy and Ba are calculated as a function of r^a where r is the distance between the magnetic dipole and the center of the probe, and a is the radius of the three-axis probe.In addition, differences between 5".i and Ba are examined for different orientations of the magnetic dipole and rotations of the three-axis probe.Because the relative orientation of the dipole and three-axis probe is not known during most measurement situations, there is a distribution of possible differences between Bav.i and Bo, and these differences collectively represent a source of measurement uncertainty for a given ria.What will be of interest in this paper is the largest difference that occurs between B^vs and Bo as a function of rla (for all possible orientations of the dipole).This largest difference is designated ABm«j.
This investigation extends an earlier treatment of the problem which considered different orientations of the dipole, but not all possible orientations of the three-axis probe [3], The maximum difference between Fav3 and So, ABma^, is found by a numerical search during which Bavj is determined by numerical integration.The major advance over the earlier study is the development of an expression giving the average magnetic flux density for a circular coil probe for any position and orientation of the probe in the dipole magnetic field.This development allows the search for ABmaxi to consider "all" possible rotations of the three-axis probe.The extended search is shown to leave unchanged the values of ABmaxj that were determined by the earlier treatment.

Expression for Average Magnetic Field
In the derivation given below, it is assumed that the cross sectional areas of the wire in the coil probes and the opposing magnetic fields produced by currents induced in the probes are negligible.We also assume that the three orthogonally oriented coils of the three-axis probe have circular cross sections of equal area.These assumptions either can be met in practice or can be taken into account by a calibration process.
The average magnetic flux density, B^^, for a single circular coil probe with cross sectional area^^ is given by fi..=ij[«.«d4. ( where d4 is an element of probe area, « is a unit vector perpendicular to A, and B is the magnetic flux density.In spherical coordinates, the magnetic flux density for a small current loop of radius b is [4] B=i^oseu.+ i^smm,, where fM) is the permeability of vacuum, 7 is the alternating current, and Ur and ue are unit vectors in the directions of increasing r and 6, respectively.
The assumption is made that b<ir, and the sinu-soidal time dependence of the field has been suppressed.The value of Bo is given by the magnitude of fi fEq.(3)], Figure 1 shows the spherical coordinates r and 0, a small current loop at the origin of the coordinate system, and a sketch of a three-axis probe.The center of the probe coincides with the origin of the prime coordinate system jc',>-', andz'.The coil probes are labelled PI, P2, and P3, have unit normal vectors wi, n2, and m, respectively, and are shown in Figure 1 (inset) for illustrative purposes as being in the directions of prime coordinates.The orientation of the magnetic dipole with respect to the position of the probe is characterized by the angle 6. 1, Three axis magnetic field probe with its center at JT =Ji,, ^ = 0, and z =Z(,.A small current loop producing a dipole magnetic field is located at the origin of the unprimed coordinate system.The unit vectors n\, ii>, and nj arc norma! to the areas of probes PI, P2, and P3, respectively.Changes in the angle flcorrespond to varying the orientation of the dipole with respect to the probe. For our purposes, it is convenient to express B in terms of Cartesian coordinates.The magnetic flux density is then [2] B=i^+j^^+k 2f' .3^2.2f' mA (4)   where r -[x^+y^-\-z^\'^, i,j, and k are unit vectors for the Cartesian coordinates, and C is the constant The goal is to develop an expression for Bav at an arbitrary point which can be evaluated for any orientation of the coil probe.The value of B^^j can then be found by combining the rms values of B^ from three orthogonal directions according to Eq. (1).The approach described below for obtaining the desired expression for Bai-is to transform the problem into the coordinate system of the coil probe.In this coordinate system, the unit vector normal to the plane of the coil coincides with the "?-axis", B is expressed in terms of the probe coordinates, and the integration over the area of the circular coil probe is carried out numerically in polar coordinates.
We begin by considering, without loss of generality, a three-axis coil probe with its center at x ^xn, y =0, and 2=Z(i, where Xn = rsin3 and Zn = rcofiB (Fig. 1).We then focus on coil probe PI and its unit vector ni after it is rotated through angles ai and az with respect to the prime coordinate system as shown in Fig. 2. The unit vectors n^ and n^ will also change in orientation to maintain their orthogonal relationship, but are not shown for purposes of clarity.In the prime coordinate system, iii is given by Hi=i sinai cosoj +ysina;isino5 -I-Jtcosai.
The coordinate system of the probe is reached by the following transformations: (i) translation of the origin to the origin of the prime coordinates shown in Fig. 1, (ii) rotation of the prime coordinates through angle 02 about the 2'-axis, yielding the doubleprime coordinates jr'',>''', z", as shown in Fig. 2, and (iii) rotation of the double-prime coordinates through angle on about the ^"-axis yielding the triple-prime coordinatesx",>"', z'" (Fig. 2).In the triple-prime coordinate system, the normal vector, rti, given by Eq. ( 5), is along the z "'-axis as desired.oy Fig. 2. (a) Geometry of unit vector MJ and coordinates after rotation of the prime coordinates through angle aj about the z'-axis and after rotation of double-prime coordinates through angle ai about ^"-axis.

Transformation (i) is given by
The first rotation of coordinates (ii) is given by and the second rotation (iii) is given by z" =z '"cosai -X '"sinai jr"=z"'sinai+x'"cosa, y"=y'".
From Eqs. (8)-(10), we have which, when substituted into Eq.( 4), expresses B in terms of the probe coordinates.A simplification of Eq. ( 11) is had by noting that the integration over the area of the probe [Eq.( 2)] occurs in Xh&x"'-y'" plane, i.e., z'" =0.Contributions to Sav3 from each of the coil probes is found by using the appropriate norma) vector [Eiqs.( 5)-( 7)] during the integrations.As noted above, the integration is carried out numerically (using a double Simpson's Rule) in polar coordinates, i.e..
The accuracy of the numerical integrations was checked by increasing the number of divisions between the limits of integration for p and ip.The results reported below were not affected by further refinements of the intervals used during the integrations.
In the search for ASmarf, it will be necessary to perform rotations of the three-axis probe about the z"'-3xis orni direction (see Search Protocol below), i.e., the unit vectors ns and 1(3 for probes P2 and P3 are rotated about nt.This removes the constraint noted earlier that m lies in thtx'-y' plane.Because the integrand for j?" is in terms of the a angles, relationships must be found between the angle of rotation about the z^-axis, designated as tf), and the a values that appear in the integrands for P2 and P3.These relationships are found by examining the unit vectors for the probes n^ and n^ as they rotate about the z '"-axis or ni direction.
The trigonometric expressions in Eqs. ( 14) and (15) are not simplified in order to aid the reader in seeing the relationships between the three unit vectors.Following a counterclockwise rotation of <^ degrees about the z^-axis, the a;'s will increase in value and the a/s will decrease in value in the expressions for itz and tis.These changes also occur in the expression for the magnetic flux density B. After a rotation of <}> degrees, a line along the unit vector «2 will intersect the circle of rotation in thex"-^" plane at a point given by (Fig. 3) x"'=ccos<^ (16) where the radius for the rotation has been arbitrarily taken to be some constant c.From Eqs. ( 10) and ( 16), the same point in the double prime coordinate system is z''= x"'sinaio= -ccosi/>sinaio The increment to a^ for unit vector n; after rotation <f>, Ba (Fig. 3), can be found from the expre^sion for its tangent, i.e., Prior to the rotation, the angle with respect to the ^'-axis for the unit vector m is given by arto + 90° (Fig. 3).After the rotation, the corresponding angle will be 5i2 + 90° where 5|2 < an,.The value of Sn is found by noting that after the rotation <j), the line / from the origin to the projection of the circle of rotation onto the x"-}'' plane is given by i=Vixy+(yy, (19) and that the tangent for 5i2 is just [z7/| (Fig- 3).From Eqs. ( 17) and ( 19),
The a values for 03 can be determined with a similar analysis.Following a rotation of 4> degrees about the z^-axis {Fig, 4), a line along the unit vector iti will intersect the circle of rotation in the x^-y" plane at the point y'"==cc0S4^ z"'=0.21), the same point in the double prime coordinate system is z" = csin<^sinaiti Jf"--csin<^cosaiu y''=cCQs<f>. ( Following rotation <^, the angle 010 +90° for BJ [Eq.(15)] will increase by an amount Sjj as shown in Fig. 4. The increment, fe, can be found from the expression for the absolute value of its tangent, i.e.. tanfe = ^= tan <^ cosaio, 523 = tan "'(tan^cosflfio). ( Prior to the rotation, the unit vector nj makes an angle of 90° with respect to the 2'-axis (Fig. 4).After the rotation, this angle will decrease by an amount S^.The value of Sn is found by noting that after the rotation <(>, the line m from the origin to the projection of the circle of rotation onto the x"-y" plane is given by m =Vixy+ (y"y. (24) and that the tangent for Sn is justzVm (Fig. 4).

Search Protocol
The search for the largest difference between Bavj and Bu, ASmix3, for a given distance r from the dipole proceeds as follows: (i) For a fixed distance r away from the dipole, and with d = ai = aj = 0, the three-axis probe is rotated about the 7'"-axis or nj direction in 2' steps (i.e., 4> is incremented in 2° steps).Then B" for each coil probe is evaluated and combined according to Eq. (1) for each value of <^ to obtain B^^j, Bavs is compared with Bi,, and the largest difference is saved.Because of the symmetry of the problem, a total rotation of 90" is required to cover all the cases (with 2° increments).
(ii) The angle ai is advanced in 5° steps and the above comparisons are repeated as the probe is rotated about the z'"-axis or «i direction.The maximum value of ai, without duplication of results is 9Cf.
(iii) For each value of oi, 02 is incremented from 0° in steps of 5° and the above comparisons are repeated.Because of symmetry arguments, a total rotation of 180° is required to consider all the cases without duplication.
(iv) Following the above calculations, different orientations of the magnetic dipole are considered by changing the angle 8 in 15° steps and repeating steps (i) through (iii).The choices of increments indicated above were found to provide adequate sensitivity for determining ABjyj.
(v) Steps (i) through (iv) are repeated for different values oi r.
A diagram schematically indicating several pK)sitions for m, and rotations about m, as the above protocol was carried out for a fixed value of r is shown in Fig. 5.

Results and Discussion
As already noted, an earlier search for \B"aa [3] was not as comprehensive as the one described in this paper.While the ratio r/a and & could be varied without restriction, the rotations of the three-axis probe were limited to simple rotations about the x'-,y'-,OTz'-sms.ThM is, it was not possible to consider differences between flav3 and Bo for combinations of rotations about two or three axes.This problem has been overcome with a more generalized expression for B,y compared to the ones used in the earlier calculation.What is perhaps surprising, however, is that the AS^a^ values obtained with the more comprehensive search protocol are the same as previously calculated.That is, ASowi is negative and occurs for all r/a values when 6 = 90", as previously found, and correspond to the ABmaij values determined earlier following simple rotations about they'-axis (referred to as "a rotations" in Ref. [3]).Numerical values of hBma^3 are provided in Table 1 as a function of r/a.

Conclusions
The present calculations have determined the largest differences between the resultant magnetic field, Bav3, and the field value at the center of the probe Bu, assuming a dipole magnetic field.These largest differences, designated ABmnj, are reported in Table 1 as a function of normalized distance, r/a, from the center of the dipole and agree with values previously found after a far less comprehensive search [3].The quantity, ABmaxi, can be regarded as the largest error due to instrumental averaging effects.As noted earlier, because the relative orientations of the dipole and three-axis probe are not known for a given rla under typical measurement conditions, there will be a range of possible differences between Bj/.^ and Ba.Thus, ideally, it would be desirable to determine the distribution of differences between B^i and Bo and treat the problem using a statistical approach, but that has been left to a future calculation.
Because the dipole field is a good approximation of fields produced by many electrical appliances, the information in Table 1 should be taken into account when total uncertainties are being determined during measurements of magnetic fields from appliances.For example, if the resultant magnetic field is to be measured at a distance r from an appliance with a combined relative standard uncertainty [5] of less than ± 10%, magnetic field meters with three-axis probes having radii a such that rla ~3 should be considered unsuitable.Three-axis probes having radii such that r/a =5 would conservatively be considered suitable if the combined relative standard uncertainty from all other sources (e.g., calibration process, frequency response) amounted to about 3% or less, since 6.9%-t-3.0%= 9.9%.where 6.9% is taken from Table 1 for r/a =5.About the authors: M. Misakian is a physicist and C. Fenimore is a mathematician in the Electricity Division of NIST.The National Institute of Standards and Technology is an agency of the Technology Administration, U.S. Department of Commerce.

Fig. 3 .Fig. 4 .
Fig.3.Geometry of coordinates and unit vectors after Unit vector iij is rotated <j> degrees about n, or z"-axis.The rotation of nj is not shown for purposes of clarity (sec Fig.4),

Fig. 5 .
Fig. 5. Trajectories of iij during search for ikB,nM?-The unit vectors ff2 and nj are nut shown but maintain their orthogonal relationship with ni throughout the search.

Table 1 .
Values of Ai^",,3 as a function of normalized distance r/a from magnetic dipole