Electromagnet: Design and Numerical Simulation
3.2.2 Magnetic field Simulation
Figure 3.8: Final geometry with material properties and definition. The axis scale is presented incm.
Figure 3.9: Magnetic field and Coil definition.
associated error function. For the creation of the mesh two important parameters need to be defined:
Maximum element size, and minimum element size. Two different meshes will be created, one for the air box in the sample site, and another for the rest of the geometry. The reason for this arises from the fact that a more detailed resolution in the magnetic field of the sample site is desired, hence a more detailed mesh.
The sizes of the parameters of each mesh can be seen in the Figure 3.10. The problem is now fully defined.
Parameter sweeping
The problem is defined and the variables require definition. Four parameters matter in order to define the simulation: Maximum current applied to the coils, number of coils, number of turns per coil and the E-shaped plates size defined byb. They all influence each other. A certain maximum current requires a certain coil wire cross-section area, which will limit the number of coils since of size limitation imposed byb. The number of coils is also limited bybwhich influences the number of turns in the electromagnet.
All the length quantities will be expressed inmm. The first parameter to consider is the number of coils in the electromagnet. In order to obtain an expression that allows to calculate this parameter the following considerations should be taken into account:
• The available space in each feet isb/2−7.5mm(given the gap is15mm)
• 5mmmust be left between the sample site and the first coil (homogeneity purposes)
• Each coil has a final height of around14mm(coil + casing)
Figure 3.10: Mesh parameters of the sample site air box and the rest of the geometry (left) Visual detail of the mesh (right) .
• In between coils a2.5mmgap is required for air circulation which allows cooling
• 5mmare necessary for the auxiliary coil
Having this in consideration the resulting expression is the following:
NC(b) = (b−3.5)/1.65 (3.1)
The number of coils corresponds to the result rounded to the even lower integer,Ncoils. As we’ve seen in Table 3.1 the maximum current is between ]0−5]A.
Applying 5 A to the main coils results in a higher maximum magnetic field but also higher power consumption and Joule losses since the heating power is proportional to the product of the resistance and the square of the current as mentioned in section 2.9. Being this the case two possible currents are considered,3Aand5A.
According to the manufacturer a current of3Arequires a0.9mmdiameter circular wire (in order to avoid damage by overheating) and a current of5 Arequires a1.1mm diameter wire. Some technical aspects of the coil construction were also acquired that must be accounted for: the wire is coated with a special insulator in order to avoid short circuit and the winding technique has some properties. A circular wire of0.9mmdiameter ends up with a1.02mm(D3A) diameter with the coat, and the1.1mmwire with 1.28mm(D5A) diameter. This reduces the available space and hence the number of turns.
The coils are composed by several layers. The winding technique consists in odd layers havingn turns, the even layersn−1turns while the last layer always hasn−1turns. The coil height is fixed to14 mm, in order to fit properly inside the gap and allow some extra space for looseness. The width of each coil is defined by the following expression:
w=b/6−3[mm] (3.2)
Figure 3.11: Coil winding technique used in the current project.
Some looseness is desired as well, and 3mmare considered for each side . Given the details of the winding technique and the fact that 1mmof thermal tape are required on each side of the coils the number of turns per layer(NT L)alternates between1.2/Dand(1.2/D)−1.
The number of layers(NL)is simply given byw/D(coil coating considered) and the number of turns per coil is given by the expression:
NT =
NT L×(NL−1)/2 + (NT L−1)×(NL+ 1)/2, ifNL is odd NT L×(NL)/2 + (NT L−1)×(NL)/2, ifNL is even
(3.3)
With this data the number of coils, maximum number of turns of each coil can be calculated for a given current andb.
All this information is compiled in Table 3.2 for different values ofb.
It is desired to reduced the size of the electromagnet compared to the previous versions and for this reason theb= 18cmwasn’t considered.
bNcoilsCoilWidthcm(w)Max.currentIcoil[A]WirediameterD[mm]NturnsperlayersNTLNlayersNLNturnspercoilNT 31.021217114 1241.7 51.28913110 31.021219218 13.561.9 51.28915127 31.021217161 1562.2 51.28917144 Table3.2:Parametersweeping
Magnetic field - Results
The parameters are now defined for each case of Table 3.2 and a ”Stationary Study” can be computed.
Different plots are created to evaluate the magnetic field. A standard example of the observations can be made in Figure 3.12.
Figure 3.12: Magnetic field plots: Volume and Arrow line.
The magnetic flux vector can be observed as well as the density plot. Higher density can be observed in the inner corners, and the opposite in the outer corners. The arrow surface shows the path and direction of the magnetic flux lines along the electromagnet and the fringing effect is seen in the outer area of sample site. The differences in the magnetic flux density, and its direction are all expected effects that confirm the accuracy of the simulation interpreting the problem.
A more detailed view of the density of the magnetic flux can be seen in Figure 3.13.
Figure 3.13: Magnetic field plots: Volume, Contour and Arrow line.
The magnetic flux is almost negligible in the outer corners since this areas would represent a longer path for the flux lines. A plot of the fields line can be observed in Figure 3.14 for a more detailed view of the fringing effect and field lines path. The fringing effect, although significant doesn’t seem to affect the field uniformity in the middle of the sample site.
Further evaluation is necessary and a plot of the magnetic field in the sample site is desired to evaluate the maximum magnetic field available to the NMR studies. This is achieved by creating a parallel plane to the section surface.
The simulation is performed for all cases of Table 3.2 in order to evaluate the magnetic field obtained in the sample site.
In Table 3.3 the results are presented as well as a comparison with the previously built electromagnet [1].
The electromagnet volume is by the following equation:
V(b) = b2
3 ×(b−1.5)cm3 (3.4)
Figure 3.14: Magnetic field plots: Volume and Streamline.
WhereVocorresponds to the volume of ab= 18cmelectromagnet,Vo=V(18)cm3. The weight is calculated by:
W(V) =V(b)×10−6×ρiron (3.5)
Where ρiron stands for the iron volumetric density, ρiron = 7870 [m3/Kg]. The wire length (l) is obtained by considering the average length of a turn equal tob/2×4. The coil resistance is calculated with the expression:
R=ρl
A (3.6)
Whereρrepresents the electrical resistivity andAthe cross section area.
By evaluating all the available cases it was decided to pursue the electromagnet with the character- istics: b= 13.5cmandIcoil = 3Awhich possesses a maximum sample site magnetic field of0.329T.
The decision of pursuing this electromagnet configuration was based on the volume ratio (significant im- provement in size and portability compared to the previous electromagnets), maximum applied current (less power consumption) and the cooling requirements being similar to the available FFC equipment.
bCoreWeight[Kg]VolumeRatioV(b)/VoIcoil[A]TotalnofturnsWirelength[m]CoilResistance[Ω]JouleLosses[W]Maximum
~ Bo
[T] 3780187.04.944.50.196 123.9660.283 5440105.61.435.00.185 31308353.29.384.00.329 13.55.7370.409 5762205.72.767.20.310 31518455.412.0108.30.386 157.970.568 5864259.23.485.20.365 1814.02415640230.43.278.90.208 Table3.3:Coreweight,volumeratioandmaximummagneticfieldforeachconfiguration
Equivalent magnetic circuit for sample site magnetic field calculation
The magnetic field of the sample size can also be estimated by establishing the corresponding magnetic circuit. This is performed in order to verify the conformity of the simulation.
Figure 3.15: a) Top 2D electromagnet view with path length (mm) and b) Magnetic circuit.
In Figure 3.15 the electromagnet ”average” path length is indicated along with the equivalent mag- netic circuit . By simplifying the circuit an equivalent circuit is obtained:
Figure 3.16: Equivalent magnetic circuit.
The equations to consider are:
Tt=N I=<eqφt (3.7)
φt=BgapSgap (3.8)
N I =<eqBgapSgap (3.9)
Bgap= N
<eqSgap
Icoil (3.10)
The magnetic field magnitude , according to equation 3.10 is a function of the number of turn (6×218), the applied current (Icoil), and sample site cross sectional-area (4.5cm×4.5cm) and<eq.
<stands for the reluctance of the circuit which is calculated through the following expression:
<= l
µoµrS (3.11)
Wherel is the length of the circuit, S the cross-sectional area,µoandµr the vacuum permeability and relative permeability of the material, respectively . The equivalent reluctance <eq of the magnetic circuit is calculated by:
<eq =<eq1
2 +<eq3+<B1E1 = 6049010H−1 (3.12)
Where:
<eq1=<AB+<AF+<F E
<eq2=<BC+<CD+<DE
<eq1=<eq2
<eq3=<BB1+<EE1
The magnetic field in the sample site is proportional to the applied current of the coils:
Bgap≈0.107Icoil (3.13)
For a3Acurrent a0.3217T magnetic field is expected. This can be considered similar to the result obtained through the simulation (deviation of0.3%).
A more detailed study of the magnetic field is desired for better analysis of the field homogeneity.