TRITA-EE 2007:048
Numerical Modeling and Evaluation of
the Small Magnetometer in Low-Mass
Experiment (SMILE)
Space and Plasma Physics Royal Institue of Technology
Israel Alejandro Arriaga Trejo
ABSTRACT
Fluxgate magnetometers have played a major role in space missions due to their stability, range of operation and low energy consump-tion. Their principle of operation is relatively simple and easy to implement, a nonlinear magnetic material is driven into saturation by an alternating excitation current inducing a voltage that is mod-ulated by the external field intended to be measured. With the in-creasing use of nanosatellites the instruments and payload on board have been reduced considerably in size and weight.
The Small Magnetometer in Low-Mass Experiment, SMILE, is a miniaturised triaxial fluxgate magnetometer with volume compen-sation incorporating efficient signal processing algorithms within a field programmable gate array (FPGA). SMILE was designed in col-laboration between the Lviv Centre of Institute of Space Research in Ukraine where the sensor was developed and the Royal Institute of Technology (KTH) in Stockholm, Sweden where the electronics used to operate the instrument were designed and programmed. The characteristic dimensions of the SMILE magnetometer and geome-try of its parts make impractical the task to find an analytical ex-pression for the voltages induced in the pick-up coils to evaluate its performance. In this report, the results of numerical simulations of the SMILE magnetometer using a commercial finite element method (FEM) based software are presented. The results obtained are com-pared with the experimental data available and will serve as a first step to understand the behaviour of the nonlinear components that could lead to improvements of its design in a future.
CONTENTS 1. Theoretical Background . . . 8 1.1 Maxwell’s Equations . . . 8 1.1.1 Magnetostatics . . . 9 1.1.2 Induced currents . . . 12 1.1.3 Inductance . . . 13 1.1.4 Magnetism in matter . . . 14 1.1.5 Boundary conditions . . . 17
1.2 A glimpse into the history of magnetism . . . 18
2. Fluxgate Magnetometers . . . 22
2.1 Principle of operation . . . 22
2.2 Single core sensor . . . 23
2.2.1 Induced voltage . . . 26
2.3 Double core sensor . . . 28
2.3.1 Induced voltage . . . 28
2.4 Volume Compensation . . . 29
3. Simulation software . . . 32
3.1 The COMSOL Multiphysics software . . . 32
3.1.1 3D Electromagnetics Module . . . 32
3.1.2 Mesh generation . . . 33
3.1.3 Solvers . . . 34
3.2 Finite Element Method . . . 35
3.2.1 Distinctive Features . . . 39
4. Accuracy estimation . . . 40
4.1 Sphere of permeable material inmersed in a uniform magnetic flux density . . . 40
4.2 Theoretical solution . . . 40
Contents 2
5. The SMILE Magnetometer. . . 47
5.1 Sensor . . . 47 5.2 Electronics . . . 48 6. Results. . . 51 6.1 Fluxgate element . . . 51 6.2 Double core . . . 59 6.3 Excitation circuit . . . 63 6.4 Compensation coils . . . 68 7. Conclusions . . . 75 Bibliography . . . 76 Appendix 79 Appendices . . . 80
A. Code for sphere of permeable material . . . 80
LIST OF FIGURES
1.1 The upper figure shows the dependance of the relative permeability µr with the magnetic field for a
nonlin-ear material. In the bottom figure, a characteristic B − H curve is shown. . . 18 2.1 In the single core configuration a nonlinear magnetic
material is surrounded by excitation coils where the current that saturates the core periodically flows. . . 24 2.2 B − H curve of the core, the saturation is attained
for Hmax = 10A/m with Bsat= 0.5T. . . 26
2.3 (Top) Excitation field with amplitude H0 = 15A/m
and period τ = 125µs. (Centre) Magnetic flux den-sity inside the magnetic material. No external mag-netic field is present. (Bottom) Induced voltage in the sensing coil for N = 1 and Asns= 1 cm2. . . 27
2.4 (Top) Excitation field with amplitude H0 = 15A/m
and period τ = 125µs. (Centre) Magnetic flux den-sity inside the magnetic material. An external mag-netic field Hext= 3A/m is present. (Bottom) Induced
voltage in the sensing coil for N = 1 and Asns = 1
cm2. . . 28
2.5 Double core sensor used in the SMILE magnetome-ter. The cores are two tapes of an amorphous alloy with dimension 16x1x0.02mm. The excitation coils are wound around each core and are connected in se-ries (800turns). The pick-up coils surround the whole configuration. . . 29 2.6 Excitation field driving the cores. Both cores are
wound by the same coil in opposite direction and are consdiered to be identical, the excitation fields sat-isfy the relation Hexc1 = −Hexc2. The period of the
List of Figures 4
2.7 Magnetic flux density inside the cores when an exter-nal magnetic field Hext = 3A/m is present. . . 30
2.8 Induced voltage in the sensing coil for N = 1, Asns=
2 cm2 and τ = 125µs for the double core sensor. . . 31
3.1 Mesh of the SMILE model in COMSOL. The geome-try has been decomposed in a total of 68873 elements. 34 3.2 Base function and its derivative for k = 2, when the
geometry has been meshed with n = 5 elements. . . . 36 3.3 Solution of Poisson equation in the unit interval [0, 1]
using the FEM for ρ = 1 and ǫ = 1. The approxi-mate solutions were calculated using 5 and 10 mesh elements of the unit interval. . . 38 4.1 Sphere of permeable material inmersed in an
uni-form magnetic flux density of 1mT. The radius of the sphere is 1cm and µr= 4 × 104. . . 41
4.2 Comparison of the numerical and theoretical solution at z = -45mm. The length of the simulation box is 30cm and the SPOOLES solver together with a fine mesh were used to obtain the numerical solution. . . 43 4.3 Comparison of the numerical and theoretical solution
at z = -20mm. The length of the simulation box is 30cm and the SPOOLES solver together with a fine mesh were used to obtain the numerical solution. . . 44 4.4 Comparison of the numerical and theoretical solution
at z = 0mm. The length of the simulation box is 30cm and the SPOOLES solver together with a fine mesh were used to obtain the numerical solution. . . 45 4.5 Estimated error for the numerical solutions obtained
using different meshing modes. . . 46 5.1 SMILE magnetometer. The sensor has a mass of 21g
and dimensions of 2cm per side. . . 48 5.2 Electronic board containing the FPGA,
microcon-troller and additional components used to operate the instrument. The board was designed at the de-prtament of Space and Plasma Physics in The Royal Institute of Technology (KTH). . . 49
List of Figures 5
5.3 Block diagram of the different modules implemented in the electronic board. . . 49 6.1 COMSOL model of the fluxgate element with
cor-responding pick-up coils used in the SMILE magne-tometer. . . 52 6.2 Measured inductance of the three pick-up coils of
Sen-sor 6. . . 52 6.3 Magnetic flux density produced when a current of
10µA flows through the pick-up coils. The number of elements used to mesh the geometry is 43088. . . 54 6.4 Inductance of the pick-up coils as a function of the
relative permeability of the cores. . . 55 6.5 Magnetic flux through a circular region with radius
b = 8cm situated at z = 0cm. The radius of the sphere is a = 1cm and the magnitude of the induction field B0 = 1mT. . . 56
6.6 B-H curves for different values of the α parameter. The maximum induction field is 0.36T. . . 57 6.7 Inductance of the pick-up coils for different values of
the α parameter in model (6.6). . . 57 6.8 Measured and simulated inductance of the pick-up
coils for Sensor 6. . . 60 6.9 Measured and simulated inductance for Sensor 6 for
different turns in the pick-up coils. . . 60 6.10 Measured and simulated inductance for Sensor 6 for
different saturation values. . . 61 6.11 Nonlinear cores with respective excitation coils. Each
excitation coils consists of 800 turns. . . 62 6.12 Magnetic flux through the pick-up coils for different
values of the excitation current and external induc-tion field. . . 62 6.13 Distribution of the magnetic flux density [T] along
the axis of one core when no external magnetic field is present. . . 63 6.14 Magnetic flux density (norm) [T] inside the cores for
different excitation currents and external fields. . . . 64 6.15 Magnetic field (norm) [A/m] inside the cores for
List of Figures 6
6.16 Relative permeability of the cores for different exci-tation currents and external fields. . . 66 6.17 Excitation current used to saturate the cores. . . 67 6.18 Magnetic flux and induced voltage in the pick-up coils
when different external induction fields are present. . 67 6.19 Resonant circuit used to generate the excitation
cir-cuit. The circuit elements r and Lc represent the
in-put resistance and inductance of the excitation coils respectively. . . 68 6.20 Inductance of the excitation coils. . . 69 6.21 Block diagram model of the resonant circuit shown
in Figure 6.19 used in Simulink to determine the ex-citation current. . . 69 6.22 (Top). A periodic rectangular pulse modulated with
a ramp is used to drive the resonant circuit. (Bottom) Excitation current generated. . . 70 6.23 Excitation current generated for different amplitudes
of the voltage source. . . 70 6.24 Compensation coils. . . 71 6.25 Compensation field [T] produced when a 1mA
cur-rent is used to bias the coils. The geometry was par-titioned with a total of 29257 elements. . . 72 6.26 Compensation field [T] along the axis of axis of
sym-metry of the fluxgate element. . . 72 6.27 Linear model used to fit the data in Table 6.2. . . 74
LIST OF TABLES
6.1 Inductance of the pick-up coils. . . 53 6.2 Compensating field for different currents driving the
1. THEORETICAL BACKGROUND
The present chapter is intended to settle the theory upon which the SMILE magnetometer bases its principle of operation. An overview of magnetism of steady currents is adresssed at the beginning fol-lowed by a description of magnetic materials. The scope used to discuss the properties of magnetism in matter is a macroscopic one (phenomenological) without going in details of quantum properties of atoms that constitute magnetic materials. The material intro-duced in this chapter is based on references [15], [8], [23], [14] and [17]. For the historical approach references [26] and [27] were con-sulted.
1.1 Maxwell’s Equations
It is an experimental fact that electromagnetic interactions in a physical system can be described by Maxwell’s equations
∇ · E = ρ ǫ0 (1.1) ∇ · B = 0 (1.2) ∇ × E + ∂B ∂t = 0 (1.3) ∇ × B − 1 c2 ∂E ∂t = µ0J (1.4)
with appropiate boundary conditions according to the geometry analysed and the so called constitutive relations which provide in-formation about the medium.
1. Theoretical Background 9
B = µH (1.6)
J = σE (1.7)
The E and B fields completely charaterise the electric and mag-netic fields in the system.
In the systems which are the object of our study the fields do not vary rapidly in time and the effects of the finite velocity of prop-agation of electric and magnetic fields can be considered to be in-stantaneous at two distinct points, known as the quasistatic approx-imation. With these assumptions is possible to simplify Maxwell’s equations: ∇ · E = ρ ǫ0 (1.8) ∇ · B = 0 (1.9) ∇ × E + ∂B ∂t = 0 (1.10) ∇ × B = µ0J (1.11)
The set of equations (1.8)-(1.11) together with (1.5) - (1.7) will be used to describe quantitatively the principle of operation of fluxgate magnetometers.
1.1.1 Magnetostatics
Although it is completely valid to assert that a magnetic field is just a relativistic effect of an electric field described in a moving reference frame with respect to an inertial frame where no magnetic interaction is detected, a more conventional treatment is followed here. During the year of 1820 Andre-Marie Amp`ere, after having attended the presentation of Fran¸cois Arago at the Paris Academy of Sciences, completed the work initiated by Hans Christian Ørsted by conducting experiments with current carrying wires[27]. In one of his experiments Amp`ere discovered that two rectilinear wires attract or repell each other depending on the direction of the currents that
1. Theoretical Background 10
flow through them. This effect is not restricted to straight wires only, indeed any two closed circuits carrying steady currents interact with each other as Amp`ere realised. The force excerted by a circuit C1
carrying a current i1 on a circuit C2 with current i2 is given by the
Biot-Savart law FC1→C2 = µ0 4πi1i2 Z C1 Z C2 dl2× (dl1× r) r3 (1.12)
the constant µ0 is the permeability of free space and is defined as
µ0 = 4π × 10−7newton/ampere2 (1.13) It is possible to rewrite (1.12) as FC1→C2 = i2 Z C2 dl2× µ0 4πi1 Z C1 dl1× r r3 (1.14)
The integral on the right hand side of equation (1.14) defines a vector valued function B : S ⊂ R3 → R3, the magnetic flux density,
B = µ0 4πi1 Z C1 dl1× r r3 (1.15)
which is specified by the geometry and current flowing through C1. By symmetry, the circuit C2 also produces a magnetic flux
density as a result of the current circulating in it. The magnetic flux density is measured in Teslas [T] in the International System of Units.
In general, for a distribution of steady currents in a region V in space, the magnetic flux density is given by
B = µ0 4π Z V J × r r3 d 3r (1.16)
where J is the current density distribution.
As can be shown by direct substitution, the magnetic flux den-sity defined by (1.16) is consistent with (1.9). The fact that the divergence of B is always zero has important physical consequences. First of all, the non existence of free magnetic monopoles1 and
sec-1Several experiments have been conducted to detect magnetic monoples. On February 14,
1982 Blass Cabrera at Stanford University recorded an event that would have indicated the pass of a monopole through his laboratory with his instruments [3].
1. Theoretical Background 11
ond, the field lines of the magnetic flux density always form closed trajectories, i.e., they can not originate from a point and diverge.
Since every vector valued function with zero divergence can be expressed as the curl of a vector field it is possible to write
B = ∇ × (A + ∇ψ) (1.17)
for some vector field A and scalar field ψ. The vector valued function A : V ⊂ R3 → R3 proposed in (1.17) is known in the
liter-ature as the magnetic vector potential and it has been shown that its usage does not change the physics of the problem[8]. The equation that should satisfy the magnetic vector potential A is obtained by substituting (1.17) in (1.11):
∇2A = ∇(∇ · A) − µ0J (1.18)
If the Coulomb gauge is used, ∇·A = 0, Poisson’s equation results and the magnetic potential can be calculated as
A = µ0 4π Z V J rd 3r (1.19)
the unit of A is the Ampere [A].
When the vector field B of a distribution of currents in space is calculated at points situated far from the sources, the denominator in (1.16) or equivalently in (1.19) can be expanded in series leading to an expansion of B (or A). For the magnetic vector potential we have,
A = AM+ AD+ AO+ . . . (1.20)
The first term of the expansion in A always vanishes meanwhile the second term depends on the distribution of the currents in space. This second term is referred as the magnetic dipole moment of the configuration AM = µ0 4π m × r r3 (1.21)
where the vector m is the dipole moment and is defined as m = 1
2 Z
V
1. Theoretical Background 12
The magnetic dipole moment plays an important role in physics as the dominant term in expression (1.20). It is the term that has the major contribution to the magnetic field at distances far from the sources since the remaining ones decrease considerably fast. Earth’s magnetic field is well described with a dipole moment m = 8 × 1022Am2, tilted about 11 degrees with respect to the rotation axis.
The magnetic field at the surface of the Earth is about 30µT at the equator and up to 60µT near the poles. It is on the top of this geomagnetic background that small disturbances due to auroral currents (space physics) or crustal mineral deposits (in exploration geophysics) need to be measured. Accurate vector measurements including stable and precise orientation of the vector magnetometer axes are required since the disturbances created by auroral currents are in the order of 50−500nT. The Earth is not the only planet with a characteristic dipole field, in other planets the dipole field dictates the dynamics of charged particles in vast region of space (that can be much larger than the planet itself), called magnetosphere. For instance, the magnetosphere of Mercury has been also approximated with a dipole displaced from its centre[1].
1.1.2 Induced currents
When experiments showed that charge in motion produced mag-netic fields that influenced the orientation of permanent magnets and magnetised needles, physicists started to wonder if the inverse effect could be possible, this is, that a magnetic field originated the motion of charges. During the year 1820, the galvanometer was al-ready invented and measure the current through a wire was possible. Experiments where a conductor was placed near a permanent mag-net were performed expecting that the field produced by the magmag-net induced a current in the conductor. If the configuration remained static no induced current was detected. Michael Faraday realised that is possible to induce a current but in order to achieve it the configuration should not remain static. Indeed he noticed that when the magnetic flux through the circuit composed by the conductor and galvanometer changed with time, a deflection in the needle of the instrument was observed.
Let Γ represent a closed path that delimits a surface S, the flux through S is defined as
1. Theoretical Background 13
Φ = Z
S
B · da (1.23)
Faraday induction law states that the line integral of E around Γ equals the rate of change of the flux Φ through S
Z
Γ
E · dl = −dΦ
dt (1.24)
The integral in the left side of (1.24) is called the induced elec-tromotance and is measured in volts [V].
1.1.3 Inductance
When two electrical circuits C1 and C2 are placed near one to the
other, the magnetic flux density produced by the current flowing through C1 will contribute to the magnetic flux generated in C2
by the current flowing through it. The symmetric effect is also observed, the magnetic flux desity due to the current in C2 affects
the flux produced by the induction field in C1. It is possible to
characterise quantitatively this magnetic coupling of circuits with the mutual inductance,
M12 = µ0 4π Z C1 Z C2 dl1· dl2 r (1.25)
and the magnetic flux produced by circuit C1 on C2 can be
de-termined by
Φ12 = M12i1 (1.26)
The integral in (1.25) is known in literature as Neumann integral and it is symmetric with respect to the subscripts as one would expect from the physics of the problem.
Since every circuit is coupled to itself by its own magnetic flux it is natural to introduce the concept of self-inductance,
Φ = Li (1.27)
where i is the current flowing through the circuit. The mutual and self-inductance depend only on the geometry of the circuits and are measured in henrys [H].
1. Theoretical Background 14
1.1.4 Magnetism in matter
Experiments have shown that three kinds of magnetic materials exist[23]. In two of these groups the interaction with magnetic fields is feeble even if they are in presence of strong magnetic fields, on the other hand in the remaining group the interaction is quite strong and can be perceived macroscopically without any special instru-ment. Paramagnetic and diamagnetic materials constitute the first two groups mentioned and the other one is constituted by ferro-magnetic materials. Magnetism is a quantum effect and henceforth quantum mechanics gives complete account of magnetism in matter, nonetheless classical models can be used to model macroscopically magnetised bodies.
Ferromagnetic materials can be modeled as a distribution of mag-netic dipole moments. The vector valued function M : S ⊂ R3 →
R3 which gives the density of these dipoles at every point within
the magnetised body is referred in literature as magnetisation. Us-ing (1.21) it is possible to calculate the magnetic vector potential at any point outside the magnetised body. The magnetic potential of a body with magnetisation M is given by
A = µ0 4π Z S M × n r d 2r + µ0 4π Z V ∇ × M r d 3r (1.28)
where V denotes the volume occupied by the body and S the surface delimiting this volume. The potential given by (1.28) is composed of two terms and it is equivalent to the potential that a surface current density distributed on S with value
λm = M × n (1.29)
and a volume current density in V given by
Jm = ∇ × M (1.30)
would produce. A letter m was added as a subscript in expres-sions (1.29) and (1.30) to emphasise that these are not conduction currents as the ones present in conductors. The current given by (1.30) is the result of the magnetisation of the body. The magnetic flux density produced by this current (and by the surface current density) is obtained by taking the curl of (1.28). For a physical sys-tem where conduction currents and currents due to magnetisation
1. Theoretical Background 15
are present, the magnetic flux density is calculated as the superpo-sition of the respective fields produced by each current.
If the nature of the different kind of currents is distinguished, (1.11) can be written as
∇× B = µ0(Je+ Jm) (1.31) where Je denotes the conduction currents. Using (1.30) results
in
∇× (B µ0
− M) = Je (1.32)
The term in parenthesis in (1.32) is the vector valued function H : R3 → R3 defined as the magnetic field
H = B µ0
− M (1.33)
The units of the magnetic field vector are ampere/meter. By rewriting (1.32) in terms of the magnetic field H one arrives to
∇× H = Je (1.34)
When there are no free currents and only currents resulting from magnetisation are present (1.34) reduces to
∇× H = 0 (1.35)
and the scalar magnetic potential φm : R3 → R proposed by
Pierre Simon de Laplace can be introduced. The scalar magnetic potential is not uniquely determined just as the potential used in electrostatics. The magnetic field can be obtained by taking the gradient of the scalar potential.
H = −∇φm (1.36)
Equation (1.36) is mathematically identical to the electrostatic situation where the electric field can be derived from a scalar po-tential by taking its gradient. Indeed if the analogy is continued the magnetic charge density ρm, can be determined by taking the
diver-gence of H. It should be stressed that this is only a mathematical artifice that does not have physical reality.
1. Theoretical Background 16
Even though the magnetic charge density is a mathematical arti-fice, it can serve to introduce and visualise a physical field of major importance in the design of fluxgate sensors, the demagnetising field HD: R3 → R3. It is well known that a uniformly magnetised sphere
can be modeled as a distribution of magnetic charge on its surface with one hemisphere containing charge of one sign and the remaing of opposite sign. The magnetic field in the interior of the sphere as a result of this charge distribution is uniform and it is directed in opposite direction to the magnetisation field. The field is said to try to oppose the magnetisation field and henceforth its name. In gen-eral the demagnetising field is not antiparallel to the magnetisation field, only for a uniformly ellipsoid (the sphere being a particular case) this is true. The relation between HD and M is given by
HD = −DM (1.38)
where D is the demagnetising tensor. Linear Isotropic materials
In general there is no physical law deduced from basic principles that relates the B and H fields within a magnetised sample, nonetheless for linear isotropic materials it is possible to introduce the magnetic susceptibility as
M = χmH (1.39)
From (1.33) one can write B in terms of the magnetisation and the magnetic field
B = µ0(H + M) (1.40)
by substituting (1.39) in (1.40) it is possible to relate the netic flux density and the magnetic field within an isotropic mag-netic material
B = µ0µrH (1.41)
where
1. Theoretical Background 17
is the relative permeability. It should be noticed from (1.39) and (1.42) that χm and µr are dimensionless constants.
Ferromagnetic materials
For ferromagnetic substances like iron, the constitutive relation among B and H is nonlinear and non single valued[14],
B = B(H) (1.43)
A typical curve showing the relationship given by (1.43) is de-picted in Figure 1.1 and it is called hysteresis loop. The most im-portant feature of the curve is the induction saturation Bsat, which
plays a major role in the principle of operation of fluxgate magne-tometers. The width of the hysteresis loop determines the magnetic hardness of the material (its sensitivity for past magnetisation his-tory). It is common to use soft magnetic materials as cores in the fluxgate magnetometer for a robust operation and low losses. In specialised literature it is common to find an additional definition for the permeability of a material[19]. The differential permeability is defined as the slope at every point in the B − H curve,
µd = 1 µ0 dB dH (1.44) 1.1.5 Boundary conditions
In order to find a unique solution to Maxwell’s equations that de-scribe the phenomena observed in a given region of study where two different media are present, a set of restrictions that the magnetic field (or the magnetic flux density) should satisfy in the boundary that delimits the interface must be imposed. These set of restrictions are referred as boundary conditions,
(B2− B1) · n = 0 (1.45)
B2× n =
µ2
µ1
B1× n (1.46)
where the subscripts denote the value of the magnetic flux density and permeability for each medium. Equation (1.45) states that the
1. Theoretical Background 18 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1 −0.5 0 0.5 1x 10 −3
Magnetic Field H[A/m]
Magnetic Flux Density B[T]
−1 −0.5 0 0.5 1 0 1 2 3 4x 10 4
Magnetic Field H[A/m]
Relative permeability
µr
Fig. 1.1:The upper figure shows the dependance of the relative permeability µr with
the magnetic field for a nonlinear material. In the bottom figure, a charac-teristic B − H curve is shown.
normal component of the magnetic flux density at the interface is continuous (in the mathematical sense) meanwhile the tangential component is discontinuous.
1.2 A glimpse into the history of magnetism Those who cannot remember the past are condemned to repeat it George Santayana
The following section is intended to serve as a brief reference to the history of magnetism. The history of magnetism is vast and in-teresting and is the result of the contribution of many illustrated as well as affortunated persons that were eager to unravel the misteries that Nature hides. Nowadays, most of the technological achieve-ments that we enjoy in society are the result of the discoveries of those men that were able to understand the basic principles of an unified aspect of nature, electromagnetism.
Magnets and their nature have been known for long time to the humankind. The latin poet Titus Lucretius Carus, author of De rerum natura poem, wrote about the inherent attraction of iron
1. Theoretical Background 19
by magnets whose name derived from the northern part of Greece, Magnesia[7]. The repulsion and attraction of magnets was an empir-ical fact also observed as well as the intrinsic aligment of lodestones with the geographical north-south direction. Besides their physical properties, magnets and lodestones were endowed with additional properties and powers with the pass of time. It was said that their usage restored husbands to wives, heal poissoned women, expelled demons and detected gold among other things. During the year of 1269, in the city of Lucera in Italy, Petrus Peregrinus de Maharn-curia, wrote the Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucort, miltem, de magnete (Letter on the Magnet of Peter Peregrinus of Maricourt to Sygerius of Foucaucourt, Soldier) where he summarised all the knowldege about lodestones and instruments using them till then. It is important to mention that apparentely Petrus Peregrinus was the firts to shape a piece of lodestone into a sphere, since in the letter he explained how an iron needle interacted with it. Influenced by the dogmas of his epoch, he attributed the orientation of hanging lodestones to heaven forces in the celestial sphere.
Almost 300 years had to pass for the human kind until another step was made toward the deciphering of the real nature of mag-netism. In the year 1600, William Gilbert (1544-1603) the royal physician of Queen Elizabeth I published his De Magnete, Magneti-cisque Corporibus, et de Magno Magnete Tellure (On the Magnet: Magnetic Bodies Also, and On the Great Magnet the Earth). The importance of Gilbert’s treatise is that he confronted the supersti-tions in which magnetism was held captive. Gilbert correctly pro-posed in his work that the Earth itself behaves as a big magnet. Besides that, Gilbert carried on a series of experiments to study in a objective manner the magnetic field. Nonetheless Gilbert also believed that inside magnets there was a kind of essence that per-turbated the dormant soul of iron with their presence in order to explain the attraction observed.
During the eighteenth century admirable progress was done with electricity (phenomenon considered to be completely independent of magnetism). In this century Peter van Musschenbrok (1692-1761) discovered by accident that electricity can be stored, constructing the first capacitor in the Netherlands (Leyden jars). In France, Charles Augustus Coulumb (1736-1806) quantitatively proved the inverse
1. Theoretical Background 20
square nature of electric attraction with his torsion balance. But the major achievement reached during this epoch is without hesitation the voltaic cell developed in Italy by Alessandro Volta (1745-1827) after having heard of the experiments conducted by Luigi Galvani (1737-1798) professor of anatomy at the University of Bologna and his assistant Giovanni Aldini.
Advances in magnetism would have to wait until the nineteenth century when experiments with electricty were done in laboratories in Europe and in part of America. In Denmark, a man influenced by the famous german philosopher Immanuel Kant (1724-1804) started to wonder about the possibility of a relation between electricity and magnetism. His name was Hans Christian Ørsted (1777-1851) who during a public lecture placed a compass near a thin platinum wire that conducted electricity and noticed together with the audience the deflection of the compass. The results of this public demonstra-tion were presented in Paris on September 4, 1820 by Fran¸cois Arago (1786-1853) to the Academie des Sciences. They were really aston-ishing to the scientific community of that time since electricty and magnetism were considered two unrelated phenomena. Among the attendants to the meeting in Paris was Andre Marie Amp`ere (1775-1836). It is said that within a few weeks Amp`ere had reproduced Ørsted’s experiments and discovered that a short solenoid behaved in the same way as a permanent magnet near a current carrying wire; this encouraged him to affirm that magnetism was the result of circular currents flowing inside magnets. A major breakthrough in the progress of magnetism was acheived in England by Michael Faraday (1791-1867). Faraday’s life can be considered a Cinderella science tale, where the courage of a man to discover the basic prin-ciples behind nature made him succeed in life. Faraday lacked of a formal education in science, indeed it was by reading newspapers that he started to get in touch with it. He worked in a binding shop and one of his clients invited him to attend a series of lectures given by Sir Humphry Davy (1778-1829). Faraday took notes of the lectures and bind them. Davy was impressed by the work done by Faraday and this opened him the doors to work as an assistant in his the laboratory. The lack of a mathematical training in Faraday’s formation was more than compensated with his ability to perform experiments and explain in a neat way the basic principles behind them. In the beginning Faraday was not attracted by magnetism; he
1. Theoretical Background 21
started to work in it by an invitation of his friend Richard Phillips, editor of the Annals of Philosophy, who persuaded him to write an article about the knowldege by then of electricty and magnetism. It is well known the results of this work, Faraday performed a consid-erable number of experiments to undertsand the action at distance observed in interactions with magnets. One of the most remarkable contributions was the concept of field, a corner stone concept in the study of physics. Faraday also discovered the law of induction, giv-ing the basis for a enormous stride in technological achievements (generation of electricity). The man who integrated all the knowl-edge in a single elegant frame was the Scotish scientist James Clerk Maxwell(1831-1879). Maxwell was the opposite to Farday, he was an excellent theoretical physicist. From his youth Maxwell started to show his mathematical skills when he correctly stated in an essay contest that the rings of Saturn could not be solid bodies. He gave solid mathematical fundation to Faraday’s ideas. When he formu-lated the equations of electromagnetism in mathematical terms he was able to predict theoretically a missing term in Amp`ere’s law, the displacemet current. This was one of the milestones in human history, when the basic laws of electromagnetism were started to be understood.
The understanding of the electromagnetic field has lead us to enjoy the benefits of the generation of electicity, mobile communications, medical applications, exploration of space, among others achieve-ments.
2. FLUXGATE MAGNETOMETERS
Fluxgate magnetometers are sensors designed to measure magnetic fields and date back to the beginning of the 1900’s. They were used during the Second World War to detect submarines and since the 1950’s, with the start of the space era, became an essential payload in space missions. Even though different configurations have been tested and reported in literature their principle of operation is basi-cally the same and relatively simple. A nonlinear magnetic material is driven periodically into saturation by an alternating current in-ducing a voltage rich in even harmonics of the excitation current frequency when an external induction field is present. Using differ-ent signal processing techniques in the time or frequency domain, the information contained in the induced voltage can be used to de-termine the magnitude of the external field. The performance of the fluxgate magnetometers is considerably improved when volume com-pensation is used. With volume comcom-pensation a set of coils nullify the external magnetic field in the region where the fluxgate element is situated, eliminating nonlinearities in the sensor and allowing it to work as a zero level indicator.
In this chapter the theory of operation of the single and dou-ble core geometries are described. The analysis is restricted to these geometries since the Small Magnetometer in Low-Mass Experi-ment (SMILE) sensor uses these configurations to measure magnetic fields.
2.1 Principle of operation
The basic components that constitute an elementary fluxgate mag-netometer are a nonlinear magnetic material surrounded by a pri-mary winding and a secondary set of coils used to measure the in-duced voltage. Through the winding flows a current that saturates periodically the magnetic core in both directions, inducing a
volt-2. Fluxgate Magnetometers 23
age in the sensing coils as a result of the change in magnetic flux with time. The external magnetic flux through the sensing coils is gated with changes of the relative permeability due to the saturation of the core by the excitation current. In the absence of an external magnetic field the voltage induced will contain the odd harmonics of the excitation current frequency. If the sensor is placed in a region where a magnetic field is present the symmetry found previously will be broken and this will be manifested in the voltage induced which will contain harmonics not found in the driving signal. All the information about the external magnetic field can be deduced by analysing the harmonics added to the sensing coils signal. Differ-ent configurations have been proposed to measure magnetic fields. In specialised literature two groups are distinguished[20]: parallel and orthogonal sensors. With parallel sensors, the magnetic field produced by the excitation current is parallel to the external field component to be measured meanwhile with orthogonal geometries the direction of the aforementioned fields are perpendicular to each other. In this report the analysis is focused on a specific pair of parallel sensors: the single and double core sensor (also known as Vacquier sensor) since the SMILE magnetometer bases its principle of operation on them.
2.2 Single core sensor
The most basic fluxgate parallel detector is the single core sensor which consist of a nonlinear magnetic core (a soft magnetic material) surrounded by excitation and sensing coils. The geometry of the configuration is shown in Figure 2.1.
The excitation coils are driven by an alternating current that pro-duces a magnetic field which varies the permeability of the core an henceby saturates it. The core is saturated equally in both direction by the positive and negative excursions of the excitation current in the absence of an external magnetic field. With an external field the core remains saturated for a longer time in one direction than the other; unbalance shown in the harmonic content of the induced voltage.
It is possible to determine the waveform of the induced voltage at the pick-up coils for the single core geometry using Maxwell’s equations and further assumptions as it is shown in refereces [20]
2. Fluxgate Magnetometers 24
Fig. 2.1:In the single core configuration a nonlinear magnetic material is surrounded by excitation coils where the current that saturates the core periodically flows.
and [21]. The magnetic flux density inside the magnetic core is given by (1.40) which is writen again for convenience
B = µ0(H + M) (2.1)
When the sensor is operating in the linear region (1.39) can be used to relate the magnetizing field and the magnetic field in the interior of the core. A demagnetising field will be present due to the geometry of the core (open ends),
M = χmH (2.2)
HD = −DM (2.3)
H = Hext− DM (2.4)
In general the demagnetising and the magnetisation fields are not collinear vectors as was pointed out in Section 1.1.4. For the simplicity of the analysis it is found in the literature that the de-magnetising factor is considered to be a scalar. From (2.1), (2.2), (2.3) and (2.4) it is possible to obtain
2. Fluxgate Magnetometers 25 M = µr− 1 1 + D(µr− 1) Hext (2.5) H = Hext 1 + D(µr− 1) (2.6) B = µ0 µr 1 + D(µr− 1) Hext (2.7)
where the expression relating the magnetic susceptibility and the relative permeability has been used. From (2.6) it can be seen that the magnetic field inside the core differs from the external magnetic field, the demagnetising factor being responsible of this departure. Equation (2.7) is commonly rewritten as,
B = µr
1 + D(µr− 1)
Bext (2.8)
with the apparent permeability µa: R → R defined by[20],
µa=
µr
1 + D(µr− 1)
(2.9) The voltage induced in the sensing coils is obtained using Fara-day’s law and can be approximated by
Vsns = −NsnsAsns
dB
dt (2.10)
where the subscript sns was added to indicate parameters associ-ated with the geometry of the pick-up (or sensing) coils; N denotes the number of turns and A the area. The vector notation can be abandoned since all the vector fields considered are colinear. Using (2.8), (2.9) and (2.10) results in,
Vsns= −NsnsAsns " (1 − D)dµr dt [1 + D(µr− 1)]2 Bext+ µr 1 + D(µr− 1) dBext dt # (2.11) Since fluxgate magnetometers measure slowly varying magnetic fields, the second term in (2.11) can be disregarded leading to
Vsns= −NsnsAsns
(1 − D)dµr
dt
[1 + D(µr− 1)]2
2. Fluxgate Magnetometers 26 −10 −5 0 5 10 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 H[A/m] B[T]
Fig. 2.2: B − H curve of the core, the saturation is attained for Hmax= 10A/m with
Bsat= 0.5T.
known in literature as the fluxgate equation [24]. The dynamics of the systems under consideration is completely specified by this equation.
2.2.1 Induced voltage
In order to determine the induced voltage in the pick-up coils it is necessary to solve (2.12) nonetheless, it is possible to estimate the shape of the induced signal by using simplified waveforms for the excitation field and further assumptions such as zero demagnetising factors. In Figure 2.2 an idealised B − H curve for a nonlinear magnetic core is shown; the behaviour of the magnetic material is described by a piecewise linear model and the demagnetising field ignored to simplify even more the analysis,
B(H) = −Bsat if H ≤ −Hmax Bsat
HmaxH if −Hmax < H ≤ Hmax
Bsat if H > Hmax
(2.13)
Let us assume for the moment that no external field Hext is
2. Fluxgate Magnetometers 27 0 20 40 60 80 100 120 −15 −10 −5 0 5 10 15 Time[µs] Excitation field[A/m] 0 20 40 60 80 100 120 −1 −0.5 0 0.5 1 Time[µs]
Magnetic flux density[T]
0 20 40 60 80 100 120 −2 −1 0 1 2 Time[µs] Induced voltage(V)
Fig. 2.3:(Top) Excitation field with amplitude H0 = 15A/m and period τ = 125µs.
(Centre) Magnetic flux density inside the magnetic material. No external magnetic field is present. (Bottom) Induced voltage in the sensing coil for N = 1 and Asns= 1 cm2.
a field of the form,
Hexc(t) = 4H0 τ t if t ≤ τ 4 −4H0 τ (t − τ 4) + H0 if τ 4 < t≤ 3τ 4 4H0 τ (t − 3τ 4 ) − H0 if t > τ (2.14)
where H0 and τ represent the maximum amplitude of the
excita-tion field and the period of the driving signal respectively. Due to the nonlinear characteristics of the core, the external flux is gated out in the pick-up coils in a symmetric way since saturation is achieved in both directions during equal time intervals (see Figure 2.3).
When an external magnetic field Hext is present, the field
pro-duced by the excitation current will have an offset which will unbal-ance the time intervals during which the core is saturated not being equal any more (see Figure 2.4).
The induced voltage can be calculated from Farday’s law and using signal processing techniques in the time domain (correlation filter) or in the frequency domain (harmonic content analysis) the external field can determined.
2. Fluxgate Magnetometers 28 0 20 40 60 80 100 120 −15 −10 −5 0 5 10 15 Time[µs] Excitation field[A/m] 0 20 40 60 80 100 120 −1 −0.5 0 0.5 1 Time[µs]
Magnetic flux density[T]
0 20 40 60 80 100 120 −2 −1 0 1 2 Time[µs] Induced voltage(V)
Fig. 2.4:(Top) Excitation field with amplitude H0 = 15A/m and period τ = 125µs.
(Centre) Magnetic flux density inside the magnetic material. An external magnetic field Hext = 3A/m is present. (Bottom) Induced voltage in the
sensing coil for N = 1 and Asns= 1 cm2.
2.3 Double core sensor
The double core configuration or Vacquier sensor is composed of two cores wound by the same wire in opposite directions to ensure that the current flowing through the excitation coils will drive the cores into saturation in opposite directions in a symmetric way when no external field is present. The pick-up coils may surround each core independently or both of them. In the first case the sensing coils are connected in series. If the double core sensor is situated in a region where no external induction field is present the voltage measured in the pick-up coils will be zero, since the magnetic flux on each core cancels mutually. When an external magnetic field is present the flux will not cancel any more and a signal with twice the frequency of the excitation current is induced in the pick-up coils. Figure 2.5 shows one of the double core sensor that forms part of the SMILE sensor.
2.3.1 Induced voltage
Like in the previous section, it is possible to illustrate the waveform induced in the pick-up coils by using piecewise linear models for the
2. Fluxgate Magnetometers 29
Fig. 2.5:Double core sensor used in the SMILE magnetometer. The cores are two tapes of an amorphous alloy with dimension 16x1x0.02mm. The excitation coils are wound around each core and are connected in series (800turns). The pick-up coils surround the whole configuration.
excitation fields; for an analytical approach of the Vacquier sensor refer to [19]. For the analysis of this configuration, identical cores are considered and are saturated in opposite directions by the same current. Assuming the same B −H characteristics for the single core sensor, the magnetic flux density inside the cores when no external field is present is plotted in Figure 2.6. It is necessary to break the symmetry with an external field to measure a non zero voltage in the pick-up coils (Figure 2.7).
2.4 Volume Compensation
The compensation technique is used to nullify the external mag-netic field in the region where the fluxgate sensor is located. Two approaches have been proposed in literature to accomplish this[20]: component compensation and volume compensation. By using feed-back coils, the external magnetic field component parallel to the axis of the fluxgate sensor can be cancelled. With this arrangement, each fluxgate element is equipped with its own set of feedback coils and the current used to drive them is extracted from the pick-up
2. Fluxgate Magnetometers 30 0 20 40 60 80 100 120 −10 0 10 Time[µs] H1 [A/m] 0 20 40 60 80 100 120 −10 0 10 Time[µs] H2 [A/m] 0 20 40 60 80 100 120 −1 0 1 Time[µs] B1 [T] 0 20 40 60 80 100 120 −1 0 1 Time[µs] B2 [T]
Fig. 2.6:Excitation field driving the cores. Both cores are wound by the same coil in opposite direction and are consdiered to be identical, the excitation fields satisfy the relation Hexc1 = −Hexc2. The period of the excitation field is
τ = 125µs. 0 20 40 60 80 100 120 −10 0 10 Time[µs] H1 [A/m] 0 20 40 60 80 100 120 −10 0 10 Time[µs] H2 [A/m] 0 20 40 60 80 100 120 −1 0 1 Time[µs] B1 [T] 0 20 40 60 80 100 120 −1 0 1 Time[µs] B2 [T]
Fig. 2.7:Magnetic flux density inside the cores when an external magnetic field Hext=
2. Fluxgate Magnetometers 31 0 20 40 60 80 100 120 −1 0 1 Time[µs] B1 [T] 0 20 40 60 80 100 120 −1 0 1 Time[µs] B2 [T] 0 20 40 60 80 100 120 −0.5 0 0.5 Time[µs] B1 + B 2 0 20 40 60 80 100 120 −10 0 10 Time[µs] Induced voltage(V)
Fig. 2.8:Induced voltage in the sensing coil for N = 1, Asns= 2 cm2 and τ = 125µs
for the double core sensor.
coils by means of a feedback loop. The main disadvantage of com-ponent compensation is that the fluxgate elements are sensitive to transverse magnetic fields, both present externally and those gen-erated by compensating coils of other elements. This places strict requirements on mutual mechanical alignment and stability of the fluxgate elements. The effect of the external transverse fields can-not be removed, however. It is possible to avoid these drawbacks if the fluxgate elements are located in a common null field region with vector feedback. With this approach a set of compensation coils in three orthogonal directions create a field that will cancel completely an exterior field in the region where the fluxgate sensors are located. The advantage of this configuration is that the axes of of the sen-sor are defined by the orientation of the compensating coils only. For tri-axial sensors it has been found experimentally that a volume compensated sensor offers major advantages than independent coils for each axis [22].
3. SIMULATION SOFTWARE
The software used to simulate the sensor elements that form part of the SMILE magnetometer was COMSOL Multiphysics. COMSOL Multiphysics is a commercial software for simulation of scientific and engineering models described by a system of partial differential equations using the Finite Element Method (FEM). In this chapter a brief introduction to COMSOL Multiphysics and the libraries used to model the components is given. For a detailed description of the software refer to [4].
3.1 The COMSOL Multiphysics software
COMSOL Multiphysics is an interactive software for modeling and simulation of scientific and engineering problems based on a partial differential equation formulation. The software contains a Graphic User Interface (GUI) that allows the user to create the geometry of the problem under study and additional tools to mesh the model and solve it. A set of predefined libraries are available and cover wide areas of physics such as: fluid mechanics, chemistry, photonics, quantum mechanics, electromagnetism among others. Once a model has been solved it is possible to visualise the solution obtained and if necessary use the post processing tools for further analysis of the data. It is also possible to create the geometry of the model with COMSOL Script, a programming language where the models can be saved in .M files and can be accesed with the interface to Matlab.
3.1.1 3D Electromagnetics Module
All the models presented in this document were simulated with the Electromagnetics module of COMSOL v3.2 (AC/DC module in COMSOL v3.3). This is a specialised module containing a diversity of libraries designed to solve electromagnetic related problems. Since the purpose of the project is to model the components of the SMILE
3. Simulation software 33
magnetometer for a static regime, the 3D Magnetostatic class was selected. This library solves simultaneoulsy the system of equations given by[5],
−∇ · (−σv × (∇ × A) + σ∇V − Je
) = 0 (3.1)
∇ × (µ−10 µ−1r ∇ × A − M) − σv × (∇ × A) + σ∇V = Je (3.2) where σ, v and V represent the electrical conductivity, the ve-locity of the medium and electric potential respectively. Since only magnetostatics simulations were performed, the system of equations is simplified to
∇ × (µ−10 µ−1r ∇ × A − M) = Je
(3.3) Boundary conditions
Different boudary conditions can be specified in the surfaces that de-limit the geometry of the models specified to be consistent with the physical situations simulated. The boundary conditions available in the quasistatic library are: magnetic field, surface current, electric insulation, magnetic potential, magnetic insulation and continuity among others. For a detailed description of the different boundary conditions in COMSOL, refer to [5]. The magnetic insulation,
n × A = 0 (3.4)
and the continuity condition,
n × (H1− H2) = 0 (3.5)
were the most used ones in the simulations performed. 3.1.2 Mesh generation
In order to find a solution for a given model in COMSOL it is necessary to mesh the geometry before applying the finite element method. Different parameters can be specified independently by the user such as the maximum element size of the elements, their growth rate, and resolution of curved regions. A set of mesh modes
3. Simulation software 34
Fig. 3.1:Mesh of the SMILE model in COMSOL. The geometry has been decomposed in a total of 68873 elements.
are included in COMSOL v3.2 and v3.3 with predefined values for these parameters which in most of the cases sucessfully mesh the given geometry.
The majority of the models used to simulate parts of the SMILE magnetometer were meshed with the extremely coarse, specially those where the cores are included. With this mode the number of elements produced was below 50000, for which the memory of the computer was sufficient to obtain a solution. Figure 3.1 shows a meshed model of the SMILE magnetometer including all its com-ponents. Due to the relative dimensions of the cores compared with other parts of the sensor, the number of elements increases consid-erably even though the coarsest mesh mode is used.
3.1.3 Solvers
COMSOL Multiphysics includes a variety of solvers appropiate to the nature of the problem analysed. The results reported in this document were obtained with time independent linear and nonlinear solvers. The solver used was SPOOLES1. COMSOL v3.3 includes a
3. Simulation software 35
solver that uses efficiently double processor architectures and saves considerable time of computation, the PARDISO solver.
3.2 Finite Element Method
The finite element method is a numerical method used to solve par-tial differenpar-tial equations. It started to be applied in electromagnetic problems during the 1960s and it is well suited to analyse complex geometries[25]. The basic ideas behind courtaines are[25]: (i) dis-cretise the analysed geometry in elements, (ii) define a set of basis functions over each element, (iii) construct an approximate solution using the basis functions, (iv) apply Garlekin’s method to obtain a system of equations and (v) solve the resultant system to determine the contribution of each basis function to the approximation[18]. The essence of the method is exemplified by applying it to a 1D problem. For this purpose, the Poisson equation will be solved in the unit interval [0, 1],
−∇2Φ = ρ
ǫ (3.6)
with boundary conditions
Φ(0) = Φ(1) = 0 (3.7)
Mesh
It is necessary to discretise first the geometry where the solution is computed. For the case considered the simplest discretization is a uniform partition of the unit interval in n segments, being the norm of the partition h,
h= 1
n (3.8)
and each node in the mesh labeled by xk, k = 0, 1, . . . , n.
Basis Functions
The set of basis functions used in the FEM are well localised2. The
most common are polynomial approximations of the form,
3. Simulation software 36 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x φ2 (x) 0 0.2 0.4 0.6 0.8 1 −6 −4 −2 0 2 4 6 x d φ2 /dx
Fig. 3.2:Base function and its derivative for k = 2, when the geometry has been meshed with n = 5 elements.
ϕk(x) = 0 if 0 ≤ x < xk−1 x−xk −1 h if xk−1 ≤ x < xk xk+1−x h if xk≤ x < xk+1 0 if xk+1 ≤ x ≤ 1 (3.9)
Equation (3.9) defines a set of well localised functions in the unit interval. Figure 3.2 shows the second base function and its derivative when the geometry is divided in 5 subintervals (elements).
Approximate solution
The approximate solution to (3.6) is constructed with the basis func-tions {ϕk(x)} as
˜
Φ(x) =Xn−1k=1akϕk(x) (3.10)
3. Simulation software 37
Garlekin method
The system of equations that the ak coefficients should satisfy is
obtained by substituting (3.10) in (3.11) Z 1 0 ǫd ˜Φ dx dϕk dx dx= Z 1 0 ρϕkdx (3.11)
for k = 1, 2, . . . , n − 1. Equation (3.11) is a particular case of the most general integral used in Garlekin’s method. The Garlekin method is a technique developed by the russian mathematican Boris Grigoryevich Garlekin to determine the weighting coefficients of an expansion of the form
˜
Φ(x) = a1ϕ1(x) + a2ϕ2(x) + . . . + anϕn(x) (3.12)
for an approximate solution to the differential equation
LΦ = f (3.13)
such that the error of the approximation is orthogonal to the basis functions, where L is a differential operator and f is the inho-mogeneous term. The system of equations resulting from (3.11) is of the form Ma = f (3.14) with M = 1 h 2 −1 0 . . . 0 −1 2 −1 . . . 0 0 −1 2 . . . 0 0 0 −1 . . . 0 .. . ... ... . .. ... 0 0 0 . . . −1 (3.15) a = a1 a2 .. . an−1 (3.16)
3. Simulation software 38 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 x Φ (x) N = 5 N = 10 Theoretical
Fig. 3.3:Solution of Poisson equation in the unit interval [0, 1] using the FEM for ρ= 1 and ǫ = 1. The approximate solutions were calculated using 5 and 10 mesh elements of the unit interval.
and f = hρ ǫ 1 2 ... n− 1 (3.17)
It can be seen that M is a sparse matrix and the elements in the diagonal are non vanishing. This is a characteristic feature of the FEM, the matrices obtained are tridiagonal. The advantage of sparse matrices is that efficient algorithms such as Gauss elimination can be implemented in computer programs to obtain an inverse.
In Figure 3.3 the approximate and analytical solutions for (3.6) are shown. For the numerical approximation 5 and 10 elements were used to mesh the unit interval. As the number of elements increases the discrepancy between the real and the approximated solution is reduced.
In general, the method previously outlined is extended to two and three dimensions. For 2D problems the mesh elements consist of triangular or quadrilateral regions and for 3D cases the geometry is
3. Simulation software 39
discretised with tetrahedral elements.
It is important to mention that for the 1D situation described above, the FEM reduced to the method of Finite Differences (FDM). In general these methods are not equal, the FEM is used to find ap-proximations to solutions of differential equations meanwhile the FDM is an approximation to the differential equation itself [9].
3.2.1 Distinctive Features
The FEM has proven to be a powerful technique to solve not only EM problems. Among its most distinctive features are that it can be used to analyse complex geometries taking into account inho-mogeneties of the medium. Besides that, the resultant system of equations obtained is characterised by sparse matrices with most of their entries equal to zero allowing the use efficient computer meth-ods to solve them.
Like every numerical method the FEM has also disadvantages. The most critical probably being the mesh of the geometry even though considerable advances have been achieved nowadays with computer programs in this area. A direct consequence of the discretization of the geometry is the dimensions of the matrices used to calculate the coeffiecients of the basis functions which can surpass the memory required to find a solution.
4. ACCURACY ESTIMATION
In order to have an estimate of the accuracy of the solutions obtained with the numerical approximation, different simulations were per-formed with a model whose theoretical solution is well known. The model used is a sphere of permeable material immersed in a region pervaded by a uniform induction field. The results obtained indicate that the accuracy of the numerical approximations depends on the number of elements used in the mesh as well as the extent of region where the solution is computed.
4.1 Sphere of permeable material inmersed in a uniform magnetic flux density
The model used as reference to compare the theoretical solution with the numerical approximations was a sphere of relative perme-ability µr = 4 × 104, radius of 1cm situated in a region where a
uniform magnetic flux density of 1mT is present. The geometry of the problem is depicted in Figure 4.1.
4.2 Theoretical solution
It is possible to find a closed expression that characterises the mag-netic flux density in the interior and exterior of the sphere shown in Figure 4.1. As can be noticed from the description of the problem, no free currents are present, therefore it is possible to express the solution via the scalar potential introduced in section 1.1.4. The symmetry of the configuration allows to use spherical coordinates (r, θ, φ). The magnetic scalar potential should satisfy Laplace’s equation and be of the form
φim =
∞
X
n=0
4. Accuracy estimation 41
Fig. 4.1:Sphere of permeable material inmersed in an uniform magnetic flux density of 1mT. The radius of the sphere is 1cm and µr= 4 × 104.
for points situated in the interior of the sphere and
φem = −H0rcosθ+ ∞
X
n=0
βnr−(n+1)Pn(cosθ) (4.2)
for those situated in the exterior; where a and H0 denote the
radius of the sphere and the external magnetic field respectively. The coefficients αnand βncan be determined by applying boundary
conditions at r = a, ∂φe m ∂θ (r = a) = ∂φi m ∂θ (r = a) (4.3) µ0 ∂φe m ∂r (r = a) = µ0µr ∂φi m ∂r (r = a) (4.4)
which lead to the system of equations for the only two non-vanishing coefficients α1 and β1
β1
4. Accuracy estimation 42
µrα1+
2β1
a3 = −H0 (4.6)
By solving the linear system given by (4.5) and (4.6) the magnetic scalar potential can be expressed in a closed form as
φim= − 3H0 µr+ 2 rcosθ (4.7) φem= −H0rcosθ+ a3H0 µr− 1 µr+ 2 cosθ r2 (4.8)
The components of the magnetic flux density and the magnetic field can be derived from (4.7) and (4.8). For the magnetic flux density, its components in cartesian coordinates are,
Bxi = 0 (4.9) Byi = 0 (4.10) Bzi = 3 µr µr+ 2 B0 (4.11) Bxe = 3a3B0 µr− 1 µr+ 2 xz (x2+ y2+ z2)52 (4.12) Bye = 3a3B0 µr− 1 µr+ 2 yz (x2+ y2+ z2)52 (4.13) Bze = " 1 − a3µr− 1 µr+ 2 x2 + y2− 2z2 (x2+ y2+ z2)52 # B0 (4.14) 4.3 Numerical Solution
The numerical simulations were implemented using COMSOL Mul-tiphysics v.3.2. With the results obtained it was realised that the accuracy of the solution depends on the number of mesh elements and the dimensions of the region where the model is meshed which we shall refer henceforward as the simulation box. The results im-proved as the dimensions of the simulation box were considerably greater than the radius of the sphere and a finer mesh was used,
4. Accuracy estimation 43 −0.05 0 0.05 −0.05 0 0.05 1 1.01 1.02 1.03 1.04 x 10−3 x[m] Slice at z =−0.045m y[m] |BTheory |[T] −0.05 0 0.05 −0.05 0 0.05 1 1.01 1.02 1.03 x 10−3 x[m] Slice at z =−0.045m y[m] |BComsol |[T] −0.05 0 0.05 −0.05 0 0.05 0 1 2 3 4 x 10−6 x[m] Slice at z =−0.045m y[m] ||B Theory |−|B Comsol ||[T] −0.05 0 0.05 −0.05 0 0.05 0 0.1 0.2 0.3 0.4 x[m] Slice at z =−0.045m y[m] |1−|B Comsol |/|B Theory ||*100%
Fig. 4.2:Comparison of the numerical and theoretical solution at z = -45mm. The length of the simulation box is 30cm and the SPOOLES solver together with a fine mesh were used to obtain the numerical solution.
however this increased the time for computation and memory used. Figure 4.2 to 4.4 show the solution obtained with the simulation software and the theoretical one in a 30x30x30cm region. The mag-nitude of the magnetic flux density is plotted at slices parallel to the xy plane at three different heights. The plots in the upper part of the figures show the magnitude of the magnetic flux density, the calculated and the one obtained numerically. In the lower part, the absolute and percentual error are plotted. As can be seen from the aforementioned figures, with these parameters the absolute error is at least two orders of magnitude less than that of the solutions. It can be noticed also, that discrepancy of the solutions tends to increase as the slice used to compare them approaches a height of z = 0cm and decreasess for planes situated far from the sphere.
4. Accuracy estimation 44 −0.05 0 0.05 −0.05 0 0.05 0.95 1 1.05 1.1 1.15 1.2 1.25 x 10−3 x[m] Slice at z =−0.02m y[m] |BTheory |[T] −0.05 0 0.05 −0.05 0 0.05 0.95 1 1.05 1.1 1.15 1.2 1.25 x 10−3 x[m] Slice at z =−0.02m y[m] |BComsol |[T] −0.05 0 0.05 −0.05 0 0.05 0 0.5 1 1.5 2 x 10−5 x[m] Slice at z =−0.02m y[m] ||B Theory |−|B Comsol ||[T] −0.05 0 0.05 −0.05 0 0.05 0 0.5 1 1.5 2 x[m] Slice at z =−0.02m y[m] |1−|B Comsol |/|B Theory ||*100%
Fig. 4.3:Comparison of the numerical and theoretical solution at z = -20mm. The length of the simulation box is 30cm and the SPOOLES solver together with a fine mesh were used to obtain the numerical solution.
4. Accuracy estimation 45 −0.05 0 0.05 −0.05 0 0.05 0 0.5 1 1.5 2 2.5 3 x 10−3 x[m] Slice at z =0m y[m] |BTheory |[T] −0.05 0 0.05 −0.05 0 0.05 0 1 2 3 4 x 10−3 x[m] Slice at z =0m y[m] |BComsol |[T] −0.05 0 0.05 −0.05 0 0.05 0 1 2 3 4 5 6 x 10−5 x[m] Slice at z =0m y[m] ||B Theory |−|B Comsol ||[T] −0.05 0 0.05 −0.05 0 0.05 0 2 4 6 8 10 12 x[m] Slice at z =0m y[m] |1−|B Comsol |/|B Theory ||*100%
Fig. 4.4:Comparison of the numerical and theoretical solution at z = 0mm. The length of the simulation box is 30cm and the SPOOLES solver together with a fine mesh were used to obtain the numerical solution.
4. Accuracy estimation 46 10 15 20 25 30 35 40 45 50 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Length of simulation box [cm]
Relative error normal coarse coarser extra coarse extremely coarse
Fig. 4.5:Estimated error for the numerical solutions obtained using different meshing modes.
In order to quantify the deviation of the numerical solution with the theoretical one, an estimated error using the norm of the mag-netic flux density was calculated in a fixed region τ for different mesh modes and simulation boxes. With |BC|, |BT| : τ ⊂ R3 → R
denoting the norm of the magnetic flux density of the numerical and theoretical solutions respectively in the τ region, the estimated error ε: τ ⊂ R3 → R was calculated as ε= R R R τ 1 − |BC| |BT| dxdydz R R R τdxdydz (4.15) Figure 4.5 summarises the results of the simulations performed with different lengths of simulation boxes and mesh modes. The dimensions of the τ region were kept constant for all the cases shown to 5x5x5cm. As can be seen from the figure the results improve as the simulation box increases its size and a finer mesh is used.
5. THE SMILE MAGNETOMETER
The Small Magnetometer in Low-Mass Experiment, SMILE, is a miniaturised state of the art fluxgate magnetometer that combines the use of a miniature triaxial sensor with volume compensation and digital signal processing routines implemented in a field pro-grammable gate array (FPGA) to measure low varying magnetic fields. SMILE is the result of the collaboration bewtween the Lviv Centre of the Institute of Space Research in Ukraine, where the sen-sor was designed, and the Royal Institute of Technology in Stock-holm, Sweden where the electronics used to operate the sensor were implemented and programmed. In this chapter a brief description of the SMILE magnetometer and the electronics used to operate the instrument is given. For a deep technical description of the sensor please refer to [11]. The visual material printed in this chapter is reproduced with permission of Dr. Nickolay Ivchenko from the De-partment of Space and Plasma Physics, KTH in Stockholm, Sweden.
5.1 Sensor
The sensor, developed at the Lviv Centre of the Institute of Space Research in Lviv, Ukraine uses the fluxgate principle with volume compensation, described in Chapter 2, to measure magnetic fields. The fluxgate elements (three in total, one for each direction) consist of two tapes of an amorphous alloy with dimensions 16x1x0.02mm, adhered onto fiber glass substrates and surrounded by the excitation coils. Three cylindrial spooles, around which the pick-up coils are wound, are used to contain the cores and fix their relative positions within the cubic geometry of the instrument (see Figure 5.1).
The compensation coils are wound around the cubic frame con-taining the fluxgate elements. Each compensation coil consists of three rectangular coils connected in series. A set of compensation coil is provided for each double core arrangement. The
distinguish-5. The SMILE Magnetometer 48
ing features of the SMILE magnetometer are its dimensions, reduced mass and materials employed in its design. The sensor has dimen-sions of 20x20x20mm, a mass of 21g and most of its parts are made of Macor.
Fig. 5.1:SMILE magnetometer. The sensor has a mass of 21g and dimensions of 2cm per side.
5.2 Electronics
The electronics used to operate the sensor were designed at the Royal Institute of Technology (KTH) in Stockholm, Sweden at the departement of Space and Plasma Physics. The circuit board con-tains a field programmable gate array (FPGA), a microcontroller, and additional analog components. It is in the FPGA where the functionality of the instrument relies since it contains the digital signal processing (DSP) core, clock generator and digital to analog conversion (DAC) logic. In Figure 5.2 the electronic board with the components used to operate the sensor are shown and in Figure 5.3 a block diagram showing the different electronic subsystems.
Excitation current
In order to saturate the cores an alternating current must flow through the excitation coils. This current is generated by a reso-nant circuit driven by the FPGA. The frequency of the excitation current is 8kHz.
5. The SMILE Magnetometer 49
Fig. 5.2:Electronic board containing the FPGA, microcontroller and additional com-ponents used to operate the instrument. The board was designed at the deprtament of Space and Plasma Physics in The Royal Institute of Technol-ogy (KTH). ADC DSP Communications Filters Microcontroller Filters Exc Pickup coils Compensation coils Excitation coils
DAC DAC DAC
Sensor
