Metrological characterisation of rotating-coil magnetometer systems


ACTA IMEKO 
ISSN: 2221-870X 
June 2021, Volume 10, Number 2, 30 - 36 

 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 30 

Metrological characterisation of rotating-coil magnetometer 
systems 

Stefano Sorti1,2, Carlo Petrone2, Stephan Russenschuck2, Francesco Braghin1 

1 Politecnico di Milano, Department of Mechanical Engineering, Via La Masa, 1, 20156, Milano 
2 European Organization for Nuclear Research 1205 Geneva, Switzerland, 1211 Meyrin 

 

 

Section: RESEARCH PAPER  

Keywords: Magnetic measurements; magnetic field; rotating-coil magnetometers; rotating shaft 

Citation: Stefano Sorti, Carlo Petrone, Stephan Russenschuck, Francesco Braghin, Metrological characterisation of rotating-coil magnetometer systems, Acta 
IMEKO, vol. 10, no. 2, article 6, June 2021, identifier: IMEKO-ACTA-10 (2021)-02-06 

Section Editor: Giuseppe Caravello, Università degli Studi di Palermo, Italy 

Received January 7, 2021; In final form April 13, 2021; Published June 2021 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Stefano Sorti, e-mail: stefano.sorti@polimi.it  

 

1. INTRODUCTION 

Rotating-coil systems are a special type of induction-coil 
magnetometers, based on Faraday's law of induction. They are 
used to measure integral field harmonics in the magnet bore. 
Rotating-coil systems consist of arrays of coils mounted on a 
rotating shaft, aligned with the magnet axis. The varying flux 
linkage with the coil induces a voltage. Typically, one long coil 
(or a chain of shorter coils) spans the entire magnet, including 
the fringe-field areas, because the transversal, integrated field is 
typically sufficient for beam tracking in particle accelerators [1]. 

Even if local measurements are required, through point-wise 
magnetometers [2], integral field is still a primary requirement. 

Precise measurements of magnetic fields require a careful 
evaluation of the mechanical properties of the rotating coils, 
subject to static and dynamic forces [3]. This implies the need for 
the evaluation of vibrations [4]. Precautions for reducing the 
impact of mechanical deformations should also be taken at the 

design stage. A properly designed system should have natural 
frequencies higher than the operating frequencies [5]. 

Compensation schemes for the main field harmonic, 
commonly referred to as bucking, provide effective mitigation of 
spurious field harmonics caused by these vibrations [6]. 
Nevertheless, the learned design methodologies are not always 
sufficient to match the requirements. Analytical formulas exist 
only for static phenomena such as misalignments and gravity [6]. 
Vibrations are instead modelled directly as spurious field 
harmonics in the measured field, and therefore they can only be 
evaluated in a prescribed field, as in [4]. The approach adopted 
in literature is typically to design the instrument aiming at a 
reasonably high stiffness, evaluating its compliance with 
controlled input like motor torque, as in [3]. Therefore, all the 
approaches proposed in the literature are limited in describing all 
the relevant mechanical phenomena, particularly shaft flexibility 
and propagation of mechanical vibration from motor and 
supports. 

An analytical model for the mechanical description of 
rotating-coils was proposed in [7]. This model can predict the 

ABSTRACT 
Rotating-coil magnetometers are among the most common and most accurate transducers for measuring the integral magnetic-field 
harmonics in accelerator magnets. The measurement uncertainty depends on the mechanical properties of the shafts, bearings, drive 
systems, and supports. Therefore, rotating coils require a careful analysis of the mechanical phenomena (static and dynamic) affecting 
the measurements, both in the design and in operation phases. The design phase involves the estimation of worst-case scenarios in 
terms of mechanical disturbances, while the operation phase reveals the actual mechanical characteristics of the system. In previous 
publications, we focused on modelling the rotating-coil mechanics for the design of novel devices. In this paper, we characterise a 
complete system in operation. First, the mechanical model is employed for estimating the forces arising during shaft rotation. Then, the 
effect of the estimated disturbances is evaluated in a simulated measurement. This measurement is then performed in the laboratory 
and the two results are compared. In order to characterise the robustness of the system against mechanical vibrations, different 
revolution speeds are evaluated. This work thus presents a complete procedure for characterising a rotating-coil magnetometer system.  

mailto:stefano.sorti@polimi.it


 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 31 

effect of static coil deformations, coil-axis to magnet alignment 
tolerances, and vibration modes on the measured field 
harmonics. The model was then expanded in a finite-element 
formulation (FEM) and applied to the design of a rotating-coil 
bench in [8]. This paper proposes a further refinement of the 
model, applied to the metrological characterisation of the 
designed rotating-coil system. 

The shaft is an anisotropic rotor described in a non-inertial 
frame, thus including rotor-dynamics effects, coupled with 
steady space-frame support. Vibrations of the real system during 
operation are measured and introduced in simulations. A sample 
dipole magnet is measured both with the real system and through 
a simulation of the device in order to validate the robustness and 
the reliability of the system. 

Therefore, the conclusion of this study is a complete 
magneto-mechanical description of a rotating-coil system in 
operation. 

2. THE MAGNETO-MECHANICAL MODEL 

The 3D finite-element method (FEM) [8] is adopted to 
describe rotating-coil shafts and their supporting structures. The 
two parts are modelled in different frames and coupled as 
described in the next section. The resulting mechanical 

deformation field of the shaft 𝒖(𝒓) (where 𝒖 is the displacement 
vector and 𝒓 the position) is applied to the coil geometry to 
evaluate its effects on the magnetic measurements. The FEM 
model is shown in Figure 1. 

2.1. The mechanical model  

The model is based on Timoshenko beams with 12 degrees 
of freedom (DOF) per node, with the possibility of adding 
lumped masses, springs, and dampers. It is possible to include 
also non-ideal boundary conditions: clamped or hinged ends can 
be replaced by elastic foundations with a suitable stiffness.  

The equations for the degrees of freedom (DOFs) 𝒑 of the 
fixed support structure are: 

𝑀�̈� + 𝐶�̇� +𝐾𝒑 = 𝒇, (1) 

where 𝑀, 𝐾 and 𝐶 are the mass, stiffness, and damping matrices 
of the system. Damping is characterised experimentally as modal 

damping [7], [8], and 𝒇 accounts for external forces acting on the 
system. These include gravity, unbalancing, and vibration of 
moving parts. 

The same equation holds for the rotating shaft but with the 
additional effects of not being in an inertial frame. The main 
advantage of adopting a rotating frame for the shaft is to have 
constant mechanical properties when reducing the shaft 
subsystem (shown below). This is a common approach for 

anisotropic shafts [9]. For the sake of simplicity, the rotating 

speed Ω is assumed to be constant, and therefore the angular 
displacement can be expressed as a function of time: 𝜃 = Ω𝑡. 

Let us consider the transformation expressing the rotation of 

the shaft 𝒒 = 𝑅𝒒f, where 𝑅 =  𝑅(𝜃), and 𝒒f, 𝒒 are the shaft 
DOFs expressed in the fixed and rotating frames, respectively. 
Applying the transformation to the shaft equation in a fixed 

frame, following Eq. (1), and multiplying by 𝑅, the shaft equation 
in the rotating frame results is 

𝑀�̈� + (𝐶 + 2𝑅𝑀�̇�T)�̇� + (𝐾 − Ω2𝑀)𝒒 = 𝑅T𝒇. (2) 

In this equation, the term added to 𝐶 denotes the Coriolis 
force, while the extra stiffness term is the centrifugal force. 

Before coupling the subsystems, each one is reduced with the 
Craig-Bampton (CB) method [10]. The CB method is a popular 
technique that allows for an independent reduction of 
subdomains before the matrix assembly. It is based on splitting 

the set of DOFs of each subsystem into internal DOFs 𝒒i and 
boundary DOFs 𝒒b (or 𝒑i and 𝒑b, respectively, for the support 
structure). Boundary DOFs are the ones at the interface between 
the subsystems and thus directly involved in the assembly of the 
full structure. The reduced set of DOF from the CB method is a 

combination of constraint modes 𝛹b and internal vibration 
modes 𝛹i . They are expressed by the coordinate transformation 
 

𝒒 = [
𝒒b
𝒒i
] = [

𝐼 𝟎
𝛹i 𝛹b

][
𝒒b
𝝔i
], (3) 

 

where 𝐼 is the identity matrix, and 𝝔i are modal coordinates of 
the shaft. The modal coordinates of supporting structure are 

denoted by 𝝅i. 
The computation of 𝛹i for the rotating shaft may be affected 

by the presence of non-symmetric matrices in the equation. To 
avoid a more complicated approach, typically involving left and 

right eigenvectors, it is proposed to compute 𝛹i as the free 
vibration modes of the shaft, neglecting the �̇� terms [9]. 
The computation of 𝛹b (for both subsystems) is performed by 
imposing fictitious unitary displacements to each boundary DOF 
and computing the static response: 

𝛹b = 𝐾i,i
−1𝐾𝑖,𝑏, (4) 

where 𝐾i,i and 𝐾i,b are the submatrices of the stiffness matrix 𝐾 
accounting for internal-internal and internal-boundary 
interactions, respectively.  

The reduction of each subsystem can be performed by 

truncating the 𝛹i set of modes. Different criteria can be 
enforced, mainly based on the maximum frequency component 
of the expected external inputs [11]. 

Finally, primal assembly [10] is adopted for the coupling 
between the support structure and rotating shaft. Only the subset 

𝒒b must be transformed before coupling, resulting in 𝒒b,f. Modal 

coordinates 𝝔i are not affected, as they are not in the physical 
space.  

A generalised set of coordinates is introduced to account for 
both subsystems as 

𝒖 = [
𝐼 0 0 0
0 𝐼 𝐼 0
0 0 0 𝐼

]
⏟        

𝐿T

[

𝝔i
𝒒b,f
𝒑b
𝝅i

], (5) 

where 𝐼 is the identity matrix and 0 is the zero matrix. 
 

 

Figure 1. Layout of the mechanical model. The equations for the support 
structure and for the shaft are expressed in the fixed frame (𝒙,𝒚,𝒛) and in 
the rotating frame (𝒙𝒔,𝒚𝒔,𝒛𝒔), respectively.  



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 32 

The assembled equations finally result in 

𝐿T [
𝑀s 0
0 𝑀r

]𝐿�̈� + 𝐿T [
𝐶s 0
0 𝐶r

]𝐿�̇� + 𝐿T [
𝐾s 0
0 𝐾r

]𝐿𝒖

= 𝐿T [
𝒇s
𝒇r
], 

(6) 

where the subscripts s and r identify the matrices for the support 
and the rotating shaft, respectively. Employing the FEM shape 
functions, the deformation field of the shaft can be computed 
and applied to the geometry of the coil. 

2.2. The magnetic model  

The mechanical model is coupled with the magnetic model by 
computing the magnetic flux linkage with the induction coil. 

In order to calculate the system response to a given field 
distribution, the magnetic flux density is expressed analytically. 
The typical description for integral flux density is the 2D 
multipoles expansion [1]. However, for correct modelling, the 
full 3D field is required because long integral coils also intercept 
the magnet fringing field. Therefore, 3D pseudo-multipoles are 
adopted [12]: 

𝐵𝑟(𝑟,𝜑,𝑧) = −𝜇0 ∑𝑟
𝑛−1

∞

𝑛=1

(𝒞𝑛(𝑟,𝑧)sin𝑛𝜑

+ 𝒟𝑛(𝑟,𝑧)cos𝑛𝜑), 

(7) 

 

𝐵𝜑(𝑟,𝜑,𝑧) = −𝜇0 ∑𝑛𝑟
𝑛−1

∞

𝑛=1

(�̃�𝑛(𝑟,𝑧)cos𝑛𝜑

− �̃�𝑛(𝑟,𝑧)sin𝑛𝜑), 

(8) 

 

𝐵𝑧(𝑟,𝜑,𝑧) = −𝜇0 ∑𝑟
𝑛

∞

𝑛=1

(
∂�̃�𝑛(𝑟,𝑧)

∂𝑧
sin𝑛𝜑

+
∂�̃�𝑛(𝑟,𝑧)

∂𝑧
cos𝑛𝜑), 

(9) 

where 

𝒞𝑛(𝑟,𝑧) = 𝑛𝒞𝑛,𝑛(𝑧) −
(𝑛 + 2)𝒞𝑛,𝑛

(2)
(𝑧)

4(𝑛 +1)
𝑟2 + ⋯ 

 

(10) 

�̃�𝑛(𝑟,𝑧) = 𝒞𝑛,𝑛(𝑧) −
𝒞𝑛,𝑛
(2)
(𝑧)

4(𝑛 + 1)
𝑟2 + ⋯ (11) 

In the interest of brevity, the similar expressions for the skew 

components �̃�𝑛, 𝒟𝑛 have been omitted here. It is, therefore, 
possible to compute the magnetic flux density at any point in 
space. The flux linkage in the coils is then calculated numerically 
with 

Φ = ∫ 𝐁
𝒜

⋅ d𝐚. (12) 

In a discrete setting, the flux increments 𝜙𝑚 for any angular 
positions 𝜃𝑚 can be developed into a discrete Fourier series: 

Ψ𝑛 = ∑ 𝜙𝑚

𝑀−1

𝑚=0

⋅ 𝑒
−𝑖2𝜋𝑚𝑛

𝑁 . (13) 

This yields the harmonic content of the response: 

𝐶𝑛(𝑟0) = 𝑟0
𝑛−1

Ψ𝑛
𝑘𝑛
, (14) 

where 𝐶𝑛
a(𝑟0) = 𝐵𝑛

a(𝑟0) + 𝑖𝐴𝑛
a(𝑟0) are the measured (apparent) 

field harmonics and 𝑘𝑛 are the coil sensitivity factors for the 𝑛-
th field harmonic: 

𝑘𝑛 =
𝑁𝑇𝐿𝑐
𝑛

(𝑟2
𝑛 − 𝑟1

𝑛), (15) 

where 𝑁𝑇 is the number of coil turns, 𝐿𝑐 the total length of the 
coil and the two radii are the position of the go and return wires 

of the coil. The 𝑘𝑛 express the integral sensitivity of the coil and 
are therefore not functions of 𝑧. The differences between the 
imposed 𝐴𝑛, 𝐵𝑛 and the apparent 𝐴𝑛

a , 𝐵𝑛
a coefficients at the 

reference radius of measurement are the main figure of merit to 
evaluate the effects of mechanical defects. 

3. THE MEASUREMENT SYSTEM REVIEW 

The proposed model was used to design the measurement 
system, as described in [8]. A brief recap of the design 
considerations is given before presenting the model of the final 
system. 

3.1. The design process  

The design process was based on separate studies of the two 
subsystems. The shaft was designed considering an 
approximated (non-optimal) support. Then the supporting 
structure was designed for the expected worst-case scenario (i.e. 
for the least stiff shaft expected to be mounted on the support). 
A recall of the design procedure is presented in Figure 2. On the 
left, the probability density function of errors in the field 

multipole 𝐴3 for different design iterations of the shaft, from 
bulk resin (1) to carbon-only shaft (4). On the right, the standard 
deviation of the error in measuring the main component of a 

quadrupole 𝐵2 for different designs of the supporting structure. 
The shaft was designed for measuring the field harmonics in 

dipoles and quadrupole magnets. Compensation schemes were 
applied in simulating the measurement procedure. In order to 
calculate the performance of the compensation in the design, coil 
imperfections were introduced as Gaussian distributions. The 
external forces considered were gravity, support vibrations (in 
the form of harmonic forces on the bearings), bearing-friction-
related torque, and shaft unbalancing due to eccentricity and sag. 
The boundary conditions for the shaft were elastic supports 
(lumped springs and dampers), accounting for the equivalent 
stiffness of the supporting structure.  

 

Figure 2. Overview of the design phase for the shaft, on the left, and for the 
support, on the right (adapted from [8]). 

]

1

3

2

4

Variants

0 0.5 1 1.5 2 2.5
A3[10

-4

0

2

4

6

8

10

12

P
ro

b
a

b
il

it
y

 f
u

n
ct

io
n

 [
%

]

Variants

s
o

f 
B

2
e

rr
o

r 
[1

0
-4

]

1

2

3

4



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 33 

Regarding the design of the support, the error of the main 
field component was the most relevant figure of merit. The 
design process was thus an iteration of different space-frame 
layouts for the support structure. 

3.2. The model of the measurement system 

The complete model of the measurement system is shown in 
Figure 3 (top image). It consists of a 1.5-m-long shaft made of 
carbon-fiber composite profiles, supporting a printed circuit 
board (PCB) coil array. The array is made of five equal radial coils 
of 1.48 m length, 11.2 mm width, and 60 turns. The diameter of 
the shaft is 72 mm. The supporting structure is a two-meter-wide 
aluminium frame. The boundary conditions for the support 
structure are two clamped and two pinned nodes. This comes 
from the different layouts of the joints and aims at being 
conservative in terms of mechanical stiffness. 

Figure 3 (bottom image) also shows the modelled magnetic 
field for the characterisation of the system, which is the main 
topic of the next section. The shaft model consists of two parallel 
beams, representing the PCB and the carbon profiles, 
respectively. Bolt joints are modelled as rigid links between mesh 
nodes. Bearing stiffness is included in the interface between the 
shaft and the structure as a set of lumped linear springs in the 
fixed frame. The flexible joint, which links the shaft with the 
motor, is included in the bearing stiffness matrix as a torsional 
spring. The motor-drive unit is introduced as a rigid body on one 
end of the structure. Forces and torques generated by the 
rotation are modelled as external loads.  

The support structure weights 65 kg, it guarantees a stiffness 

at the tip of a minimum 1.5 × 105N/m in all three directions 
and has its first natural frequency at 28 Hz. The shaft is 
approximately 2.5 kg and has the first natural frequency (a 
bending mode) at 74 Hz. 

3.3. The expanded model for input estimation 

The effects of the motor on the mechanical system are 
modelled as equivalent lumped inputs. In particular, we 

introduce a force 𝒇M (with unknown magnitude and direction) 
and a torque 𝑡M about the shaft rotation axis. The force and 
torque are applied in the node supporting the shaft on the motor 
side. Due to the linking between the structure and the shaft 

(through stiffness 𝑘B), their effects are visible on the coils and 
therefore have an effect on the magnetic measurements. 

Due to the simplification in the modelling of the motor 
effects, it is necessary to provide the model with the vector of 

forces and torque 𝒈 = [𝒇M, 𝑡M]. 
In order to minimise the impact of the mechanical 

measurements on the magnetic measurements, an indirect 

measurement of 𝒈 is performed. Instead of mounting force 
sensors at the joints between the parts, like torque transducers 
[14], accelerometers are mounted at the shaft supports. 
Therefore, accelerations are measured, and forces are estimated. 
To perform this operation, a Kalman Filter is introduced. 

The mechanical model of Eq. (6) is expanded to include the 
inputs in Kalman Filter's estimation [13]. It is required to write 
the model in state-space form, collecting the variables in the 

vector 𝒙 = [�̇�,𝒖]. Before assembling the estimator, it is 
suggested to convert the dynamic system in a discrete-time 

domain (with time step ∆t). It is, therefore, possible to write: 

In Eq. (16), the first term stems from Eq. (6), while the second 
term is added for the sake of state estimation. It contains the 

mechanical disturbances 𝐿𝒛 and the fictitious disturbances for 
the input ∆t 𝐼𝒘. This term is required to consider time-varying 
inputs in the estimation, otherwise, the inputs would obey 

𝒈𝑘+1 = 𝒈𝑘. More specifically, 𝐿 can be assumed to be the 
identity matrix in case no external disturbances are expected, 

while 𝒛 and 𝒘 are zero-mean, normally-distributed disturbances 
with covariance matrices 𝑄𝑧 = cov(𝒛,𝒛) and 𝑄𝑤 = cov(𝒘,𝒘). 
The Kalman filter also requires an output equation, which is in 
our case the subset from Eq. (6) accounting for the accelerations 
of the nodes at which position the accelerometers are mounted. 
The output is expected to suffer from measurement noise, 

assumed to be a zero-mean, normally-distributed variable 𝒏 with 
covariance matrix 𝑅 = cov(𝒏,𝒏). 

4. EXPERIMENTAL VALIDATION 

The experimental validation campaign aims at two objectives: 
to evaluate the accuracy of the model in predicting the outcome 
of a magnetic measurement, and to characterise metrologically 
the effect of vibrations in the real system. 

Regarding the model-prediction capabilities, the model is the 
same as the one in the design phase. It is based on conservative 
hypotheses for boundary condition of the structure (Figure 3 and 
Section 3.2) and without updates based on vibration 
measurements. The comparison between the model and the 
measurements is therefore expected to be consistent up to a 
safety factor. Approaches to update the system model in order 
to replicate the real system more accurately are discussed in the 
next section. 
  

 

 

Figure 3. The rotating-coil system without the magnet (top) and its magneto-
mechanical model (bottom). A few elements are highlighted: the force 𝒇𝐌 
and the torque 𝒕𝐌 generated by the motor, the bearing stiffness matrix 𝒌𝐁, 
the array of coils (𝒄𝟏 to 𝒄𝟓) and the magnetic flux density 𝑩 to be measured. 
The rigid bodies accounting for motor and shaft ends are omitted.  

[
𝒙𝑘+1
𝒈𝑘+1

] = [
𝐴 𝐵
0 𝐼

][
𝒙𝑘
𝒈𝑘
] + [

𝐿 0
0 ∆t 𝐼

][
𝒛
𝒘
]. (16) 

 M
𝐵𝐶5

𝑡M

𝑘B

𝑘B

𝐶1



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 34 

4.1. Experimental setup 

As a first case study, a dipole magnet is selected. It is a 0.4 m 
long dipole from the CERN East Area, with a nominal integral 
field of 0.36 Tm at 240 A [15]. Future studies will also cover the 
case of quadrupoles and higher-order magnets. 

The magnetic flux density distribution generated by the 
magnet was already measured with a short rotating-coil, and 

pseudo-multipoles were computed up to the 5𝑡ℎcomponent. 
Pseudo-multipoles coefficients are therefore available for the 
measurement model. 

The magnetic measurement performed with the system under 
investigation is a rotating-coil measurement assessing both the 
integral absolute field and the integral multipole coefficients. 

Therefore, the fluxes from the main coil 𝑐1 and from the 
compensation scheme 𝑐1 − 𝑐3 are acquired. 

The 𝑘1 sensitivity factor for each of the coil of the array is 
calibrated in a reference magnet. In order to match the calibrated 
area, coil widths in the model are adjusted. This is also to 
decrease the performance of bucking compensation in the 

model, aiming at more realistic values. Coefficients 𝑘𝑛 for 𝑛 > 1 
are computed directly from the nominal geometry of the model.  

A single measurement consists in a revolution at constant 
speed. Different speeds are considered for both the mechanical 
and the magnetic measurement campaigns so that, adopting the 
units of revolutions-per-minute, Ω ∈ [30, 45, 60, 75, 90, 105, 120]. 
In order to prevent unpredicted phenomena (like thermal 
transients) from correlating with the speed wrongly, the different 

values for Ω are evaluated in a randomised order. 20 repetitions 
are performed per each speed. 

For the mechanical evaluation of the measuring system, a 
Leica geosystem is employed for the alignment with respect to 
the magnet. In addition, two tri-axial accelerometers (6g range, 
0-200 Hz bandwidth) are mounted on the support structure next 
to the coil bearings to assess the vibrations of the system. Given 
their weight of 50 g, they have been neglected in the mechanical 
model. 

Due to the low values expected for accelerations, the 
accelerometers have been calibrated [16] in the range of ±0.5 g 
(or 0.5 g - 1.5 g for the axes parallel to gravity): each device was 
mounted on a support that allows a controlled angular 
displacement with respect to gravity (accuracy 0.02 mrad). DC 
acceleration is then measured in a series of known angular 
positions, so that the gravity component per each axis could be 
computed. The overall accuracy is estimated to be 0.0025 m/s2. 
This value includes the uncertainty due to the sensor noise, 
averaged over 20 repetitions. 

The impact of accelerometer accuracy on the force estimation 
is estimated to be about 0.3% in amplitude; it has thus negligible 
effects on the measurement. The main source of uncertainty for 
the estimation is the model itself, which is in fact, the focus of 
the validation procedure. 

4.2. Mechanical measurements results 

The most relevant figure of merit for the mechanical 
alignment of the system is the misalignment between magnet axis 
and coil rotation axis. After the alignment procedure, the final 
angle between the two axes resulted in 0.052 mrad. 

The positioning error is estimated to be less than 20 μm, 
which is the overall accuracy of the geosystem, while the linear 
stages can perform smaller motions relying on a sub-micrometer-
resolution encoder. 

However, these positioning errors are negligible for the 
proposed magnetic measurements (estimated to result in errors 
of the order of 10-8). 

The vibrations are measured for different rotation speeds of 
the shaft. For each of the speeds considered, the estimation 
procedure is performed on a series of 20 revolutions. 

Figure 4 shows the RMS of the measured vibrations and the 
estimated force and torque as the rotating speed function.  

4.3. Magnetic measurements results 

The main figure of merit for the performance of the 
measurement system is the result of the magnetic measurement. 
Consistent with the mechanical measurements, different values 
are considered for the rotation speed of the shaft. The twofold 
scope of the campaign is to validate the robustness of the 
measurement results against the mechanical vibrations and the 
capability of the model in reproducing the system behaviour.  

Figure 5 shows the measurements of the absolute integral 
field as a function of the speed. Each rotation is processed 
independently to evaluate the dispersion caused by random 

 

Figure 4. RMS values for the measured accelerations (averaging all the 
accelerometers) and for the estimated force magnitude and torque. The RMS 
are computed over the full bandwidth (0-200 Hz) of the accelerometers.  

 

Figure 5. Integral absolute field resulting from measurements (top) and 
simulations (bottom). The measured results are presented by the distribution 
of 20 rotations (blue line is the average). 

30 40 50 60 70 80 90 100 110 120
Speed [rpm]

0.1

0.15

0.2

0.25

0.3

A
cc

e
le

ra
ti

o
n

 [
m

/s
2
]

RMS of the measured accelerations

30 40 50 60 70 80 90 100 110 120
Speed [rpm]

0.1

0.15

0.2

0.25

F
o

rc
e

 [
N

]

0

0.2

0.4

T
o

rq
u

e
 [

N
 m

]

RMS of the estimated inputs

Real measurement

Speed [rpm]

0.36588

0.365885

0.36589

0.365895

∫
𝐵
d
𝑧

[T
m

]

30 40 50 60 70 80 90 100 110 120
Speed [rpm]

0.365116

0.365118

0.36512

0.365122

∫
𝐵
d
𝑧

[T
m

]

Simulated measurement

30 40 50 60 70 80 90 100 110 120



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 35 

disturbances and noise. The distribution shown is a probability 
density estimate based on a normal kernel function.  

The overall spread concerning all the measured values is 
0.4 units (10-4 of the average field), while averaging the turns, the 
spread is 0.2 units. Moreover, the trend of the mean values as a 
function of the speed is not monotonic, as is the RMS of the 
accelerations. The reason is that other phenomena are speed-
related such as the signal-to-noise ratio for the acquisition 
system. Therefore, it is plausible to identify a relative accuracy of 
±0.1 units from the mechanical vibrations on the absolute 
integral field. 

The second plot in Figure 5 shows the simulated 
measurement of the integral field. The difference between the 
measured and simulated values is on average 21 units, while the 
spread between measurements is 0.15 units. Comparing the 
trend, these results confirm that the model correctly predicted 
the amount of disturbances from mechanical vibration in the 
magnetic measurement.  

The error in the absolute field can be explained by the missing 
high-order harmonics in the simulation. As the absolute field is 
constant with respect to vibration amplitude, this is not 
investigated further. 

Figure 6 shows the measurement results of the integral 
multipoles. The multipoles coefficients are measured with both 

the main coil 𝑐1 and the compensation scheme 𝑐1 − 𝑐3. It is 
therefore possible to appreciate the performance enhancement 
of the compensation in the measurement of high-order 
multipoles (between 10th and 15th). Nevertheless, the effects of 
vibrations on the multipoles measured by the main coil are at 
most 0.1 units for the speeds considered. Thus, the rotating-coil 
system yields measurement results better than one unit, even 
without a compensation scheme. As far as the simulated 
measurement is concerned, the multipoles from the main coil are 
given. Thus, it is possible to appreciate the matching of the 
behaviour for both the multipoles values and their trends as a 

function of the speed. In particular, the model is slightly 
overestimating the effect of the mechanical vibrations on high-
order multipoles, but this is consistent with the model's expected 
behaviour. The field harmonic of order 9 was not included in the 
pseudo-multipoles provided to the model. 

5. CONCLUSIONS 

The proposed magneto-mechanical model, already applied to 
the design of a rotating-coil system, is now validated against the 
actual device in a real magnetic measurement campaign. The 
vibrations of the system during the measurement have been 
recorded, and the corresponding input was estimated. The 3D 
magnetic flux density of the magnet was provided, and the 
measurement procedure was simulated. 

The results agree with the trends and the magnitudes of the 
real measurement. If a closer matching is required, the model 
should be updated accordingly. This updating would require a 
proper mechanical measurement campaign to experimentally 
characterise all the system features (for instance, by modal 
analysis, with an array of accelerometers and a controlled input). 
Nevertheless, the real system magnetic measurements confirmed 
that the performance required in the design phase had been 
exceeded, with an overall error by mechanical vibrations 
estimated to a few units of 10-5. 

Therefore, we can conclude that the proposed procedures 
have led to the design, construction, and commissioning of a 
novel rotating-coil magnetometer system with unprecedented 
control over some mechanical properties typically critical for the 
magnetic measurement result. 

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Figure 6. Field multipoles (modulus) as measured with the compensated 
scheme 𝒄𝟏 −𝒄𝟑 (top), by the main coil 𝒄𝟏 (middle), and as simulated for 𝒄𝟏 
(bottom). For the measured quantities, each rotation is processed 
independently and pictured by a dot. 

0

50

100

150

200

|
𝐶
𝑛
 (

r 0
)|

 [
u

n
it

s]

 𝒏, 𝒄𝟏, simulation

2.8

164.1

0.7
26.2

0.1 0.1 0.1

2 3 4 5 6 7 8
n []

0

0.1

0.2

0.3

0.4

9 10 11 12 13 14 15
n []

0

50

100

150

200

|
𝐶
𝑛
 (

r 0
)|

 [
u

n
it

s]

 𝒏, 𝒄𝟏, measurement

4.0

166.3

0.8
26.1

0.1 0.1 0.0

2 3 4 5 6 7 8
n []

0

0.1

0.2

0.3

0.4

9 10 11 12 13 14 15
n []

0

50

100

150

200

|
𝐶
𝑛
 (

r 0
)|

 [
u

n
it

s]

 𝒏, 𝒄𝟏 − 𝒄𝟑, measurement

3.6

165.0

0.8
25.9

0.1 0.1 0.0

2 3 4 5 6 7 8
n []

0

0.1

0.2

0.3

0.4

9 10 11 12 13 14 15
n []

30
45
60
75
90

105
120

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ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 36 

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