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Development of the theoretical basis  

of magnetic measurement uncertainty evaluation
O.V. Degtiarov, R.S. Alrawashdeh 

Measurement and Technical Expertise department 
Kharkiv National University of Radio Electronics  

Kharkiv, Ukraine  
oleksandr.degtiarov@nure.ua 

 

Abstract — The theoretical foundations of measurement 

uncertainty evaluation magnetic moment of radio electronic 

means using point methods are considered. A sequence and 

implementation of the uncertainty assessment procedure is 

proposed. A model equation based on the multipole description 

of the object’s magnetic field is obtained. An analysis of the 

equation is made. The uncertainty budget has been drawn up. 

Found standard and extended uncertainty. The results are in 

demand for control procedures and tests in the area of 

technical magnetism. The results contribute to the 

standardization of the quality assessment of magnetic 
measurements. 

Keywords — external magnetic field, dipole magnetic 

moment, measurement uncertainty, induction sensor 

I. INTRODUCTION 

The area of application of magnetic measurements is 
constantly expanding. These measurements are in demand in 
many areas of science, technology and industry, such as 
electric power, electromagnetic compatibility, etc. 

Electrical machines and radio devices are the main 
sources of electromagnetic interference. This interference 
interferes with the normal functioning of various objects of 
radio electronics, and has a negative effect on people in the 
zone of influence. This situation requires control of the 
electromagnetic environment in the areas of industrial and 
energy facilities. To solve the problem of electromagnetic 
compatibility, international organizations develop 
recommendations and standards for the application in 
European countries of permissible and limiting levels of 
external magnetic field strength in production and non-
production conditions. Based on these recommendations, 
national standards are being developed. The external 
magnetic field levels used in different countries differ in 
magnitude. 

At distances comparable to the overall dimensions of the 
source of an external magnetic field, its external magnetic 
field in the surrounding space is of a dipole nature. Such a 
field is proportional to the equivalent dipole magnetic 

moment. The dipole magnetic moment ( 2,M A m ) is a 

constant value characterizing the magnetic field of an object. 
Its value is governed by standards for certain types of 
electrical and electronic equipment. In contrast to the 
magnetic field strength, this value does not depend on the 
coordinates of the observation points. This allows you to 
determine the magnetic field strength at specified points in 
external space from measured values of the components of 
the dipole magnetic moment of the source, to get an idea of 
the spatial configuration of the external magnetic field [1, 2]. 

Currently, point and integral methods are used to 
measure the dipole magnetic moment of electrical, radio-
electronic objects. Point methods involve performing 

measurements at certain points in the external space using - 
the number of induction sensors. Induction method involves 
the use of contour measuring windings. These methods have 
their advantages, disadvantages, spheres of application. 

Modern tasks in the field of magnetic measurements 
require the improvement of metrological assurance of 
measurements of the dipole magnetic moment. Improving 
the metrological assurance of measuring the dipole magnetic 
moment includes the development of more accurate 
measurement methods, the development of methods for 
estimating the uncertainty of measurements, the elaboration 
of the questions of validation of measurement methods. 

Currently, the regulatory framework for assessing the 
uncertainty of measurements in the field of measuring 
magnetic quantities needs to be improved. In scientific 
publications [3, 4] the general algorithm for estimating 
uncertainty is described, the general algorithm for carrying 
out and processing the results of indirect measurements is 
described. There is no method for estimating the uncertainty 
of measurements of the dipole magnetic moment. A 
scientific publication [5] proposes a method for estimating 
the methodological error of point measurement methods. 
There are no estimates of uncertainty. The research [6] 
considers actual questions of magnetic measurements, but do 
not contains recommendations on estimation of measurement 
uncertainty. The scientific publication [7] do not takes into 
account the categories of measurement uncertainty. 

Based on the above, work aimed at improving the 
metrological assurance of measurements of magnetic 
quantities is relevant. This involves the development of the 
theoretical foundations of the model and empirical 
approaches to the assessment of the uncertainty of magnetic 
measurements and the elaboration of the questions of 
validation of measurement methods. 

II. METHODOLOGY DEVELOPMENT  

OF MEASUREMENT UNCERTAINTY EVALUATION OF  

THE DIPOLE MAGNETIC MOMENT  

The processes of international standardization of quality 
assessment of measurements and the presentation of their 
results involves the introduction of the concept of 
uncertainty in the practice of measurement, control and 
testing. 

Processing the uncertainty of the results of any 
measurements performed by the general algorithm. 
However, the structure of this algorithm and the 
implementation of its blocks are specific for a particular 
type of measurement. This applies to measurements of 
magnetic quantities and measurements of the dipole 
magnetic moment in particular. 

978-1-7281-1814-7/19/31.00 c© 2019 IEEE

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The main point in estimating the uncertainty of any 
measurements is the compilation of a model equation and its 
analysis. Therefore, we consider the theoretical aspects of 
the compilation of the model equation for measuring the 
dipole magnetic moment using point-based measurement 
methods. 

Any source of the external magnetic field can be 
represented as a collection of a finite number of elementary 
magnetic dipoles, each of which is in the centre of a small 
volume element. According to the principle of 
superposition, the sum of elements of elementary dipoles is 
equal to the equivalent dipole moment [2]. In external space, 
the magnetic field is characterized by a scalar magnetic 
potential ,U A . By expanding the function U  in a power 

series with respect to the radius of the observation point, we 
obtain a multipole model of the external magnetic field. 
Such a model corresponds to partial solutions of the Laplace 
equation. In the spherical coordinate system, this model has 
the form 

1

0

nn

m

R
U R

r





 
  

 
  

   cos sin cos ,nm nm nmg m h m P              (1) 

where , ,r    – the coordinates of the observation point; 

nmg , nmh , – coefficients of spatial harmonics; R - the 

radius of the base surface on which the coefficients are 

determined nmg , nmh ; (cos )m
nP   – Legendre polynomials. 

The resulting series corresponds to the provisions of the 
multipole theory of the external magnetic field proposed by 
Gauss. 

To find the magnetic field, it is necessary to differentiate 
the function U  according to the corresponding coordinate. 

For the radial component, the expression is 

2
1

1 1

4
R n

n

U n
H

R R






 
   

 
  

 
0

cos sin (cos ).
n

m
nm nm n

m

g m h m P


           (2) 

The magnetic field strength of the first spatial harmonic 
is described by the expression (3) 

1 3

1

4
RH

R
 


 

10 11 11[ cos ( cos sin )sin ] .g g h m           (3) 

The coefficients nmg , nmh  in expressions (1-3) have a 

certain physical meaning. They are equal to the magnetic 
moments of the elementary multipoles of the n  order of 

the spatial m  harmonic. The coefficients 10g , 11 11,g h , 

are respectively equal to the components of the dipole 

magnetic moment 11 xg M , 11 10, .y zh M g M   

In accordance with the developed point method, 

magnetic field strength RiH  measurements are performed 

in eight external spaces. Coordinates of points 

( 1)90i i   , ( 5)90k k   , where 1...4, 5...8i k  . 

As a result of a series of transformations, the resulting signal 

,xE V  is described by the following expression 

11

3

12
x

f

g
E

k R
   

 
3

2 1

2 2 2
5 02 1

/ 11 1 ( 1)

/ 1

nn n

xnmn
f n m

R R
M

k R R R




 

 
 


  . 

Then the magnitude of the radial dipole moment with the 
methodological error is described by the expression (4) 

3
11 /12.x x fM g E k R                       (4) 

This expression (4) is a model equation for further 
estimating uncertainty. 

Perform assessment of the measured value 2,xM Am  

and the associated standard and extended uncertainty. The 
source of measurements is the source of an external 
magnetic field of the “black box” type. To find the 
measured value, direct repeated observations 20n   of the 

resulting signal ,xE V  and distance were made (the object 

of measurement is the sensor) ,R m . Rough errors were 

excluded from the number of observations. Corrected for 
systematic errors. 

Specification of measurement data: measurements were 
performed under normal laboratory conditions. The 

resulting signal ,xE V  was recorded with a universal digital 

voltmeter. The distance was measured with a steel ruler. An 

inductive sensor 36 10 /fk A m V    was used as a 

converter of the magnetic field strength into an electrical 
signal. Input values do not have a correlation. 

Calculate the arithmetic average of multiple observations 
of the resulting signal 

20

1

1 ˆ 10.65 .
20

ind ind i
i

E E mV


   

The standard deviation of the observation of the 
resulting signal: 

 ˆ 0.05indS E mV . 

Standard uncertainty of evaluation of the resultant 
measurement result 

0,05
( ) 0.011

20
indS E mV  . 

Calculate the arithmetic average of multiple observations 
of the distance object – sensor 

20

1

1 ˆ 500
20

ind ind i
i

R R mm


  . 

The standard deviation of the observation object distance 
- sensor: 

ˆ( ) 0.01indS R mm . 

Standard uncertainty of evaluation of the result of 
measuring the distance object – sensor 

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0.01
( ) 0.002

20
indS R mm  . 

Standard Type B uncertainty estimates of input 
quantities that have an uniform law: 

40,001
( ) 5.7 10

3
f

A
u k

m V

  


. 

Find the sensitivity coefficients using the model 
equation (4) 

1 2

;
, ,...

i
mi i

f f
c

x x xx X

 
 
 

 

2
3ˆ( ) 0.083 0.066ind f

Am
c E k R

V
   ; 

2ˆ( ) 0.25 0.004ind fc R k E R Am   ; 

3 5 31ˆ( ) 1.8 10
12

fc k R V Vm   . 

Further make the uncertainty budget (Table 1) 

The model (4) is nonlinear. It is necessary to use the 
Taylor formula to find the total measurement uncertainty of 
the radial magnetic moment. 

3
2 2

1

( ) ( )c x i i
i

u M c u x



 


  

2
2 33

2
, 1

1/2
2

2 2

1

2

( ) ( )

i j ii j i j

i j

f f f

x x x x x

u x u x



        
         


 
 



2 2 2 2 2 2ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )ind ind ind ind f fc V u V c R u R c k u k   


 

1/2
22 2

2 2 2

2 2 2

1
...

2

x x x

f

M M M

U R k

                                 

 

7 21.1 10 Аm  . 

The evaluation of measurement result of the radial magnetic 
moment is determined by the expression  

20

1, 1, 1, 1,
1 1

1 1
( , ) ( , )

20

n

x k k k k
k k

M f E R f E R
n  

     

4 27 10 A m   .                                 (5) 

According to [8] coverage ratio 

0,95( )effk t  . 

Find effective number of freedom degrees 

4

4 4

1

( )

c
eff m

i

ii

u

u y



  




 

     

4

44 4 ˆ ˆ( ) ( )( ) ( ) ( ) ( )

c

f f

i i

u

u k c ku U c U u R c R

 
   
  
    
 

 

 
   

4
7

4 4

1,1 10
23

( ) ( ) ( ) ( )u U c U u R c R


 

  

. 

Then get the extended uncertainty. 

7
0,95

7 7 2

ˆ( ) (23) 1.1 10

2.07 1.1 10 2.27 10 .

c xU k u M t

Am



 

    

    

 

As a result, the assessment of the measured value, taking 
into account the expanded uncertainty, is 

2(0.700 0.002)xM mA m   , 0.95p  . 

In order to study the measurements carried out by the 
developed point magnetometric methods and measuring 
instruments, for repeatability and reproducibility, 
experimental studies were carried out using an exemplary 
source of magnetic field. 

Five observation groups were obtained, each of which 
contained eight, nine, ten, eight, and seven replications, 
respectively. 

  

TABLE I.  THE UNCERTAINTY BUDGET OF MAGNETIC MOMENT MEASUREMENTS 

Input value 
Estimated input 

value 
Standard uncertainty 

Number of 

degrees of 

freedom 

Probability 

distribution 

function of  

the input values 

Sensitivity 

сoefficient 

Contribution of 

uncertainty, 

2A m  

indE  10.65 mV 0,011 mV 19 
Normal 

distribution 
0,066 

2A m

V


 

77,3 10  

indR  500 mm 0,002 mm 19 
Normal 

distribution 
0.004 A m  50,8 10  

fk  6 
A

mV
 

45.7 10
A

mV

    
uniform 

distribution 
5 31.8 10 Vm  

70.1 10  

xM  4 27 10 A m    
7 21.1 10 Am  – – – – 

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Observations were obtained by measuring the dipole 
magnetic moment with a device with sensors located in the 
equatorial plane. 

To check the homogeneity of the observations, the 
following mathematical methods were used: 

1) the Bartlett criterion to test the admissibility of 
differences between estimates of variances; 

2) Fisher's method for checking the admissibility of 
differences between arithmetic averages. 

These methods are used in the analysis of several groups 
of observations. 

For groups of observations 5L   and repeated 

observations in  in each group, we obtain 

5 5
2 2

1 1

1
37ln ( ) ln )

37
i i i i

i i

M k S k S
 

   , 
2
1

37

54
M  . 

The level of significance is assumed to be 0.05, which 
corresponds to the probability 0.95P  . Find the tabulated 

value 2 2
0,05 3,325q     and the values of the unbiased 

estimates of the variances of each of the five groups, 
2 2( )Am : 2 4

1 5.7 10S   ; 2 4
2 11.5 10S   ; 2 4

3 8.2 10S   ; 

2 4
4 17.3 10S   ; 2 4

5 23.1 10S   ; 2
5S =23,1·10-4. 

From the known values of the dispersion estimates 
2
1S ,..., 2

5S , we find the value 2
1 2.687  . 

Since the value 2
1  determined from the experimental 

data is less than the tabulated value 2
q  (condition 2

1 < 2
q  

is satisfied), the differences between the estimates of the 
variances are valid. 

Thus, as a result of a study on the convergence of 
measurements, it was established that the measurements are 
stable and the dispersions are homogeneous. Now, to answer 
the question about the reproducibility of measurements, we 
find the variance of reproducibility. Since the number of 
repeated observations in groups is different, the dispersion 
of reproducibility is determined by the formula: 

2 2

1 1

L L

i i i
i i

S k S k
 

  . 

Value 2 4 2 211.1 10 ( )S Am  , which confirms the 

reproducibility of measurements. 

Similar studies have been conducted for devices with 
sensors located on spherical and cylindrical surfaces. For 
them, the values of the estimates of the variances of 

reproducibility 4 48.3 10 ; 7,6 10   2 2( )Am  are obtained.  

Using the Fisher criterion, we perform an analysis of the 
admissibility of differences between arithmetic averages of 

groups whose values are equal, 
2Am : 1M =5,467; 

2M =5,464; 3M =5,466; 4M =5,469; 5M =5,462. Then the 

values of the variance estimates 2
LS , 2S  are: 

2 55.7 10LS   2 2( )Am , 2 56.6 10S  
2 2( )Am . 

Thus, the analysis showed that measurements carried out 
using the developed point magnetometric methods and 
measuring instruments have the qualities of convergence 
and reproducibility. 

CONCLUSION 

The proposed approach can be used in estimating the 
uncertainty of all point methods for measuring the magnetic 
moment. This can be used for appropriate control and test 
procedures to provide and compare results. 

Based on the multipole model for describing an external 
magnetic field of an object, a model equation is obtained for 
estimating the uncertainty of measurements of the radial 
magnetic moment by the point method. A budget of 
uncertainty has been drawn up. Standard and extended 
uncertainties were obtained. 

Weaknesses of this study occur when the object of 
measurement of large longitudinal form. It is also important 
to note the high requirements for the accuracy of the 
installation of sensors according to the proposed scheme of 
their location, which may affect the accuracy of 
measurements. This circumstance imposes special 
requirements for the qualification of operators. 

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