2.2 Analytical Model
2.2.2 Solution of the magnetic vector potential and scalar potential
The coil system is divided into various regions to calculate magnetic field density using the pro- posed modelling method. The magnetic vector potential or scalar potential is used to calculate the magnetic field in each region. These potentials are described in the subsequent section. Sub-sub sec- tion 2.2.2.1 presents the solution of the magnetic vector potential, whereas magnetic scalar potential is explained in sub-sub section 2.2.2.2.
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2.2.2.1 Magnetic vector potential in coil
The magnetostatic Maxwell equations are used for the derivation of the equations, which is given as-
∇ ×H=J, (2.1)
∇ ·B=0, (2.2)
where H is the magnetic field strength, B is the magnetic flux density, and J is the current density vector. Considering the vector calculus identity, the divergence of the curl of any vector field is always zero as
∇ ·(∇ ×G)=0, (2.3)
where G is an arbitrary vector field. From Gauss law for magnetism (2.2) and vector identity (2.3), the magnetic field density can be written as the curl of vector potential [63], given as-
B=∇ ×A (2.4)
where A is the magnetic vector potential. The magnetic vector potential must satisfy the Coulomb Gauge condition, therefore
∇ ·A=0 (2.5)
The Magnetic flux density B and the magnetic field intensity H are related by (2.6)
B=µ0µrH (2.6)
whereµ0 is the magnetic permeability of vacuum andµrrelative magnetic permeability of the mate- rial. By using (2.1) and (2.6) the magnetic field density B can be written in terms of current density as
∇ ×B=µ0µrJ (2.7)
substituting (2.4) into (2.7)
∇ ×(∇ ×A)=µ0µrJ, (2.8)
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and using vector identity
∇(∇.A)−A(∇.∇)=µ0µrJ, (2.9) substituting (2.5) in (2.9) reduces to
∇2A=−µ0µrJ. (2.10)
The (2.10) is known as the Poisson equation. The solution of the magnetic vector potential, which satisfies (2.10), gives valid magnetic field density in the considered domain.
This region represents the current-carrying primary coil. The direction of the magnetic vector potential is along the direction of current density. Here, the current density in the primary coil flow along x and y directions, represented as Jx and Jy, respectively shown in Fig 2.2, where the current density along the z- direction is assumed to be zero. The component form of (2.10) is given as
∇2Ax = −µ0µrJx (2.11)
∇2Ay = −µ0µrJy (2.12)
∇2Az = 0. (2.13)
where Ax, Ay, and Az are the magnetic vector potentials in x-, y-, and z-directions. Here, due to the absence of the current in the z-direction, Az is considered as zero [92]. The current densities Jx(x,y) and Jy(x,y) are discussed in detail in subsection 2.2.3. The differential form of (2.11)-(2.13) is given as
∂2Ax
∂x2 + ∂2Ax
∂y2 + ∂2Ax
∂z2 = −µ0µrJx (2.14)
∂2Ay
∂x2 + ∂2Ay
∂y2 + ∂2Ay
∂z2 = −µ0µrJy (2.15)
∂2Az
∂x2 + ∂2Az
∂y2 + ∂2Az
∂z2 = 0 (2.16)
After solving(2.14)-(2.16) using the separation of variables method, we get the magnetic vector
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potential given by (2.17).
Ax = −
N
X
n=1,3,5 M
X
m=1,3,5
cII1ekzz+dII1e−kzz
cos (kxx) sin(kyy)+ µ0Jx(x,y) kz2 Ay =
XN
n=1,3,5
XM
m=1,3,5
cII2ekzz+dII2e−kzz
sin (kxx) cos(kyy)+ µ0Jy(x,y)
kz2 (2.17)
Az = 0
The magnetic vector potential may also be used to calculate the magnetic field because of the validity of Gauss law magnetism (2.2) in the air medium. But the magnetic vector potential for representing a region of coil system ( shown in Fig. 2.2 ) requires two components and, hence more unknown coefficients. These unknown coefficients need to be known to calculate the magnetic field, and they are obtained by solving the boundary condition. More unknown needs more boundary conditions which increase the complexity of the analytical model. The magnetic scalar potential can be an alternate way to calculate the magnetic field in the domain. The magnetic scalar potential and their governing equation are discussed in the next subsection.
Table 2.1: Parameters of the coil system and analytical model
Parameters Definition Value Unit
a1p Inner distance of primary coil from centre in x-direction 66.3 mm a2p Outer distance of primary coil from centre in x-direction 92.7 mm b1p Inner distance of primary coil from centre in y-direction 42.8 mm b2p Outer distance of primary coil from centre in y-direction 69.2 mm a2s Inner distance of secondary coil from centre in x-direction 92.5 mm b2s Outer distance of secondary coil from centre in y-direction 69.0 mm
w Diameter of the primary and secondary coils 2.5 mm
τx Coil pitch in x-direction 465 mm
τy Coil pitch in y-direction 353.75 mm
J Current density in the coil 0.713 A-mm−2
N Harmonics in x- direction 50
M Harmonics in y- direction 50
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2.2.2.2 Solution of magnetic scalar potential in air medium
For deriving the magnetic scalar potential from the magnetostatic equations, the current density is assumed to zero. Therefore, Ampere’s law (2.1) can be written as
∇ ×H=0. (2.18)
The vector calculus identity, the curl of the gradient of the scalar field is zero, i.e
∇ ×(∇ϕ)=0, (2.19)
whereΨis a scalar potential. Therefore, the magnetic filed strength can be expressed as the gradient of a scalar potential, given as
H= −∇ϕ. (2.20)
Here, ϕ is a magnetic scalar potential. Besides satisfying (2.18) in the region where current is absent, The Gauss law for magnetism (2.2) should also be satisfied. By using (2.6), (2.2) can be written in terms of magnetic field intensity as
∇ ·H=0. (2.21)
With the help of (2.20), (2.21) can be written as
− ∇ ·(∇ϕ)= 0. (2.22)
By using vector identity the magnetic scalar potential is given as
∇2ϕ=0. (2.23)
The (2.23) is known as the Laplace equation. The solution of the magnetic scalar potential, which satisfies (2.23), gives valid magnetic field density behaviour in the considered domain. However, this is all under the condition that no current density can exist in the considered region. Therefore, the magnetic scalar potential can be used for the formulation in the region where the current density is absent. Among all the divided regions of the coil system shown in Fig. 2.2, I and III are the regions TH-2341_126102029
2.2 Analytical Model
where the current density is absent. Therefore, magnetic scalar potential is used for the formulation in the region I and III. The Laplace equation for the region I and III are given in (2.24)-(2.25).
∇2ϕI =0 (2.24)
∇2ϕIII = 0 (2.25)
By solving (2.24) and (2.25) using separation of variables method, the expression of the magnetic scalar potential for these regions is expressed as
ϕI =
N
X
n=1,3,5 M
X
m=1,3,5
cIekzzcos (kxx) cos(kyy) (2.26)
ϕIII =
N
X
n=1,3,5 M
X
m=1,3,5
cIIIe−kzzcos (kxx) cos(kyy) (2.27)
kx = nπ
τx, ky = mπ
τy , and kz = s
nπ τx
!2
+ mπ τy
!2
(2.28) where, kx and kyare the spatial frequencies in the x and y direction with periodτx andτy respectively.
The solution of the magnetic scalar potential for region I and III which satisfied the Laplace equation has been given in this subsubsection.
In this subsection, the solution of the magnetic vector potential and scalar potential for different regions derived from the magnetostatic Maxwell equations has been discussed. From (2.17), it can be seen that for the calculation of magnetic vector potential expression, current density distribution should be known. The next subsection gives a detailed description of the current density distribution in the primary coil.