Boundary Conditions

3.5 Summary

4.2.4 Boundary Conditions

The boundary conditions are applied between the x and z edge of each region. Here, the divided regions are shown in Fig. 4.3, are classified as big and small regions based on the width of the region along the x-direction. The example is shown in Fig. 4.5 explains the application of boundary conditions. The big region is denoted by region p, where p can be the region I, III, V, or V II, as shown in Fig. 4.3. The small region is denoted by q_j, where q can be the region II, IV, V I, or j can be a to e (decided by Fig. 4.3). For example, if q is region II, then j is a to c, and if q is region IV, then j is a to e. The interaction of a big region with small regions occur at z^′, as shown in Figs. 4.5(a) and 4.5(b). Here, z^′can be z₂ to z₇. The interface of subregions occurs at x^′, as shown in Figs. 4.5(c) and 4.5(d). Here, the interface between a and b is shown. Although, it can be between b and c or c and d or d and e depending on the value of x^′. The value of x^′can be x2to x5.

The boundary conditions are defined by the continuity of tangential magnetic field intensity and magnetic vector potentials at x and z edge of each subregion. The tangential magnetic field strength H_xp must be equal to tangential magnetic field strength H_{xq j} of each subregion for respective width subregion at z^′, as shown in Fig. 4.5(a). The magnetic vector potential of each region q_j must be TH-2341_126102029

4.2 2-D Subdomain Model for the Coil System

equal to the magnetic vector potential of the region p at z^′, as shown in Fig. 4.5(b). Similarly, in Figs.

4.5(c) and 4.5(d), the continuity of tangential magnetic field strength and magnetic vector potential should be at x^′. The boundary conditions can be written as

j = a

Region p z^’

Region q_j

j = b j = c

H

_xqa

H

_xp

x z

H

_xqb

H

_xqc

(a)

j = a

Region p z^’

Region q_j

j = b j = c x

z

A

_qb

A

_qc

A

_p

A

_qa

A

_p

A

_p

(b)

x^’ x

z

H

_zqa

H

_zqb

(c) x x^’

z

A

_qa

A

_qb

(d)

Figure 4.5: Illustration of the interface conditions by continuity of (a) magnetic field intensity and (b) magnetic vector potential at z=z^′, and continuity of (c) magnetic field intensity and (d) magnetic vector potential at x=x^′.

H_xp x,z^′

=P

j

H_{xq j}(x,z^′) where x_{inq j} ≤ x≤ x_{inq j} +τ_{xq j} (4.15) A_{q j} x,z^′

= A_p(x,z^′) where x_{inq j} ≤ x≤ x_{inq j} +τ_{xq j} (4.16) H_zqb x^′,z

= H_zqa(x^′,z) where z_inqb ≤ z≤z_inqb +τ_zqb (4.17) Aqa x^′,z

= Aqb(x^′,z) where zinqa ≤ z≤zinqb +τ_zqb. (4.18) The boundary condition at the edge of each region is given in Appendix C. The boundary conditions, given in (4.15)-(4.18), are applied between the regions having unequal width. This unequal width implies that regions have different fundamental spatial frequencies. Equations (4.15)-(4.18) show that two waveforms of different fundamental spatial frequencies should be equal for a specific interval, and it is implemented by expressing a Fourier series of one’s region waveform into a specific interval (width of another region). Therefore, equations (4.15)-(4.18) can be implemented by (4.19)-(4.22), TH-2341_126102029

4. A Subdomain Analytical Model of Coil System with Magnetic Shields of Finite Dimensions and Finite Permeability for WPT systems

respectively.

uc^x_p = 1 τxq j

Np

X

np=1

X

j

x_{inq j}+τ_{xq j}

Z

x_{inq j}

H_{xq j} x,z^′

sin(k_xp(x−x_inp))dx (4.19)

uc_{q j}^x = 1 τ_{xq j}

Nq j

X

nq j=1

xinq j+τxq j

Z

x_{inq j}

Ap x,z^′

sin(kxq j(x−xinq j))dx (4.20)

uc^z_qb = 1 τ_zqb

Lqb

X

lqb=1

z_inqb+τzqb

Z

z_inqb

H_zqa x^′,z

sin(k_zqb(z−z_inqb))dz (4.21)

uc^z_qa = 1 τ_zqa

Lqa

X

lqa=1

z_inqa+τzzqa

Z

z_inqa

A_qb x^′,z

sin(k_zqa(z−z_inqa))dz (4.22)

where uc is an unknown coefficient of region r, and depending on z^′, it can be c_r or d_r, whereas depending on x^′, it can be e_ror f_r. By using (4.19)-(4.22), the developed simultaneous linear equations are solved by rewriting these equations into matrix form as [83, 96].

[UC]= [BC]⁻¹[S V] (4.23)

Here, the matrix [BC] and [UC] contain elements of known and unknown coefficients, respectively.

The matrix [S V] contains the elements representing the source within the equations. The expanded form of [UC] is given by (4.41).

UC=

[UC_I] [UC_IIa] [UC_IIb] [UC_IIc] [UC_III] [UC_IVa] [UC_IVb] [UC_IVc] [UC_IVd] [UC_IVe] [UC_V] [UC_{V Ia}] [UC_{V Ib}] [UC_{V Ic}] [UC_{V II}]

T

(4.24) Each element of [UC] is sub-matrix, the element of this submatrix represents unknown coefficients

TH-2341_126102029

4.2 2-D Subdomain Model for the Coil System

of magnetic vector potentials described in subsection 4.2.3, given by (4.25)-(4.39).

[UC_I] = h [d₁^x]_N

1×1

i (4.25)

[UCIIa] =

[c_IIa^x ]_N

IIa×1 [d_IIa^x ]_N

IIa×1 [ f_IIa^z ]_L

IIa×1

(4.26) [UC_IIb] =

[c0_IIb^x ]_1×1 [c^x_IIb]_N

IIb×1 [d0^x_IIb]_1×1 [c^x_IIb]_N

IIb×1 [e^z_IIb]_L

IIb×1 [ f_IIb^z ]_L

IIb×1

(4.27) [UC_IIc] =

[c_IIc^x ]_N

IIc×1 [d^x_IIc]_N

IIc×1 [e^z_IIc]_L

IIc×1

(4.28) [UCIII] =

[c_III^x ]_N

III×1 [d_III^x ]_N

III×1

(4.29) [UC_IVa] =

[c_IVa^x ]_N

IVa×1 [d_IVa^x ]_N

IVa×1 [ f_IVa^z ]_L

IVa×1

(4.30) [UC_IVb] =

[c0_IVb^x ]_1×1 [c_IVb^x ]_N

IVb×1 [d0^x_IVb]_1×1 [d_IVb^x ]_N

IVb×1 [e^z_IVb]_L

IVb×1 [ f_IVb^z ]_L

IVb×1

(4.31) [UCIVc] =

[c_IVc^x ]_N

IVc×1 [d^x_IVc]_N

IVc×1 [e^z_IVc]_L

IVc×1 [ f_IVc^z ]_L

IVc×1

(4.32) [UC_IVd] =

[c0_IVd^x ]_1×1 [c^x_IVd]_N

IVd×1 [d0^x_IVd]_1×1 [d_IVd^x ]_N

IVd×1 [e^z_IVd]_L

IVd×1 [ f_IVd^z ]_L

IVd×1

(4.33) [UC_IVe] =

[c_IVe^x ]_N

IVe×1 [d^x_IVe]_N

IVe×1 [e^z_IVe]_L

IVe×1

(4.34) [UC_V] =

[c_V^x]_N

V×1 [d_V^x]_N

V×1

(4.35) [UC_{V Ia}] =

[c_{V Ia}^x ]_N

V Ia×1 [d_{V Ia}^x ]_N

V Ia×1 [f^z_{V Ia}]_L

V Ia×1

(4.36) [UC_{V Ib}] =

[c0_{V Ib}^x ]_1×1 [c_{V Ib}^x ]_N

V Ib×1 [d0_{V Ib}^x ]_1×1 [d_{V Ib}^x ]_N

V Ib×1 [e^z_{V Ib}]_L

V Ib×1 [ f_{V Ib}^z ]_L

V Ib×1

(4.37) [UCV Ic] =

[c_{V Ic}^x ]_N

V Ic×1 [d_{V Ic}^x ]_N

V Ic×1 [e^z_{V Ic}]_L

V Ic×1

(4.38) [UC_{V II}] = h

[d_{V II}^x ]_N

V II×1

i (4.39)

The matrix [BC] with its element is expressed as

TH-2341_126102029

4.ASubdomainAnalyticalModelofCoilSystemwithMagneticShieldsofFiniteDimensionsandFinite PermeabilityforWPTsystems

[BC]=



























































































[I] h

K_I^IIai h

K_I^IIbi h K_I^IIci

[0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0]

hK_IIa^I i

[I] h K_IIa^IIbi

[0] h K_IIa^IIIi

[0] [0] [0] [0] [0] [0] [0] [0] [0] [0]

hK_IIb^I i h K_IIb^IIai

[I] h

K_IIb^IIci h K_IIb^IIIi

[0] [0] [0] [0] [0] [0] [0] [0] [0] [0]

hK_IIc^I i

[0] h K_IIc^IIbi

[I] h K_IIc^IIIi

[0] [0] [0] [0] [0] [0] [0] [0] [0] [0]

[0] h

K_III^IIai h

K_III^IIbi h K_III^IIci

[I] h

K_III^IVai h

K_III^IVbi h

K_III^IVci h

K_III^IVdi h K_III^IVei

[0] [0] [0] [0] [0]

[0] [0] [0] [0] h

K_IVa^IIIi

[I] h K_IVa^IVbi

[0] [0] [0] h

K_IVa^V i

[0] [0] [0] [0]

[0] [0] [0] [0] h

K_IVb^IIIi h K_IVb^IVai

[I] h K_IVb^IVci

[0] [0] h

K_IVb^V i

[0] [0] [0] [0]

[0] [0] [0] [0] h

K_IVc^IIIi

[0] h K_IVc^IVbi

[I] h K_IVc^IVdi

[0] h K^V_IVci

[0] [0] [0] [0]

[0] [0] [0] [0] h

K_IVd^IIIi

[0] [0] h

K_IVd^IVci

[I] h

k_IVd^IVei h K_IVd^V i

[0] [0] [0] [0]

[0] [0] [0] [0] h

K_IVe^IIIi

[0] [0] [0] h

K_IVe^IVdi

[I] h K^V_IVei

[0] [0] [0] [0]

[0] [0] [0] [0] [0] h

K_V^IVai h

K_V^IVbi h

K_V^IVci h

K_V^IVdi h K_V^IVei

[I] h

K_V^{V Ia}i h

K_V^{V Ib}i h K^{V Ic}_V i

[0]

[0] [0] [0] [0] [0] [0] [0] [0] [0] [0] h

K_{V Ia}^V i

[I] h K_IVa^{V Ib}i

[0] h K_{V Ia}^{V II}i

[0] [0] [0] [0] [0] [0] [0] [0] [0] [0] h

K_{V Ib}^V i h K_{V Ib}^{V Ia}i

[I] h

K^{V Ic}_{V Ib}i h K_{V Ib}^{V II}i

[0] [0] [0] [0] [0] [0] [0] [0] [0] [0] h

K_{V Ic}^V i

[0] h K_IVc^{V Ib}i

[I] h K^{V II}_{V Ic}i

[0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] h

K_{V II}^{V Ia}i h

K_{V II}^{V Ib}i h K^{V Ic}_{V II}i

[I]

























































































 (4.40)

78

TH-2341_126102029

4.2 2-D Subdomain Model for the Coil System

The matrix [SV] is given in (4.41).

[S V] =

[S V_I] [S V_IIa] [S V_IIb] [S V_IIc] [S V_III] [S V_IVa] [S V_IVb] [S V_IVc] [S V_IVd] [S V_IVe] [S V_V] [S V_{V Ia}] [S V_{V Ib}] [S V_{V Ic}] [S V_{V II}]

T

(4.41) Each element of vector [SV] is given in (S21)-(4.55).

[S VI] = h [0]_N1×1

i (4.42)

[S V_IIa] =

[0]_NIIa×1 [0]_NIIa×1 [0]_LIIa×1

(4.43) [S V_IIb] =

[0]_1×1 [0]_NIIb×1 [0]_1×1 [0]_NIIb×1 [0]_LIIb×1 [0]_LIIb×1

(4.44) [S VIIc] =

[0]_{NV Ic×1} [0]_{NV Ic×1} [0]_{LV Ic×1}

(4.45) [S V_III] =

[0]_NIII×1 [S VIII]_NIII×1

(4.46) [S V_IVa] =

[0]_NIVa×1 [0]_NIVa×1 [S V_IVa]_LIVa×1

(4.47) [S VIVb] =

[S V_{IVb c0}]_1×1 [0]_NIVb×1 [S V_{IVb d0}]_1×1 [0]_NIVb×1 [0]_LIVb×1 [0]_LIVb×1

(4.48) [S V_IVc] =

[0]_NIVc×1 [0]_NIVc×1 [S VeIVc]_LIVc×1 [S Vf IVc]_L

IVc×1

(4.49) [S V_IVd] =

[ES_{IVd c0}]_1×1 [0]_NIVd×1 S V_{IVd d0}]_1×1 [0]_NIVd×1 [0]_LIVd×1 [0]_LIVd×1

(4.50) [S V_IVe] =

[0]_NIVe×1 [0]_NIVa×1 [S V_IVa]_LIVa×1

(4.51) [S V_{V Ia}] =

[0]_{NV Ia}×1 [0]_{NV Ia}×1 [0]_{LV Ia}×1

(4.52) [S V_IVb] =

[0]_1×1 [0]_{NV Ib×1} [0]_1×1 [0]_{NV Ib×1} [0]_{LV Ib×1} [0]_{LV Ib×1}

(4.53) [S V_{V Ic}] =

[0]_{NV Ic×1} [0]_{NV Ic×1} [0]_{LV Ic×1}

(4.54) [S V_{V II}] = h

[0]_{NV II×1}i

(4.55) The element of matrix [BC] and [SV] is given in appendix C. The elements of matrix [BC] and [SV]

can be obtained from the simultaneous equations developed from (4.19)-(4.22). The dimension of matrix [BC] is Dmax×D_max, and the dimension of matrix [IC] and [SV] is Dmax×1. The value of D_max

TH-2341_126102029

4. A Subdomain Analytical Model of Coil System with Magnetic Shields of Finite Dimensions and Finite Permeability for WPT systems

is given by

D_max =N₁+2·N_IIa+L_IIa+2·(N_IIb+L_IIb+1)+2·N_IIc+L_IIc+2·N_III+2·N_IVa+L_IVa +2·(N_IVb+L_IVb+1)+2·(N_IVc+L_IVc)+2·(N_IVd+L_IVd+1)+2·N_IVe+L_IVe+2·N_V +2·NVIa+LVIa+2·(N_VIb+LVIb+1)

+2·N_VIc+L_VIc+N_VI1.

(4.56)

The accuracy and computational time of the subdomain model depend on the maximum number of harmonics taken into account in the solution. The maximum number of harmonics depends on the available memory of the computer [96]. After a certain number of harmonics, the linear system becomes ill-conditioned and gives inaccurate results [75]. The less number of harmonics gives inac- curate field solutions at discontinuous points in the geometry, especially at the edge of the magnetic shields. This is due to the dominance of the Gibbs phenomenon at the interface of regions. However, having different sizes of regions, the magnetic potentials series can be truncated at different points.

Here, the different number of harmonics (given in Table 4.1) for different regions has been selected to obtain a correctly converged solution. A detailed discussion on the number of the harmonic taken into account is given in [96]. The finite number of harmonics NI-NV II (x-edge) and LIIa-LV Ia(z-edge), given in Table 4.1, have been used for calculating the analytical results. With the obtained value of unknown coefficients, the magnetic field density can be calculated using (4.3). Further, the analytical results of the magnetic field are compared with 2-D FEA results in the next subsection.

4.2.5 2-D Subdomain Model Verification

For a fair comparison, the magnetic field densities obtained from subdomain model and 2-D FEA are plotted on Path 1, as shown in Fig. 4.6. Path 1 is located between the coils at z_m, which is the distance from the bottom of the primary coil to Path 1 ,and it is 15 mm. The distance between the Shield 1 and the top of the primary coil (represented as air in Fig. 4.1(a)) is 5.09 mm due to the presence of base material (acrylic) and adhesive. The same is true for the secondary coil arrangement.

The Shield 2 is at (z₆) 20.09 mm.

Fig. 4.7 depicts the magnetic field distribution for the perfect alignment of the coils. In this case, the magnetic field distribution is symmetric to the centre of the coil. Fig. 4.8 illustrates the magnetic TH-2341_126102029

4.2 2-D Subdomain Model for the Coil System

Primary coil

Shield 1 Shield2

Reduced flux lines

Rectangular bar 1 Rectangular bar 2

Center of coil Path 1

Left side Right side zm z₆

Reduced flux lines

Secondary coil

(a)

Primary coil

Shield 1

Shield2

Reduced flux lines

Rectangular bar 1 Rectangular bar 2

Center of coil Path 1

Left side Right side zm

Reduced flux lines

Secondary coil

(b)

Figure 4.6: 2-D flux distribution of the practical coil system in xz-plane (a) without shift and (b) with 40mm horizontal shift of the secondary coil arrangement.

field distribution for 40 mm horizontal displacement of the secondary coil arrangement. Here, the asymmetrical pattern in the distribution of B_xand B_zis observed because the flux distribution between the coils becomes unsymmetrical, as shown in Fig. 4.6(b). This asymmetry in flux distribution is due to an increase in flux concentration in the direction of the movement of Shield 2. Similarly, the magnetic field distribution inside the Shield 1 and Shield 2 for perfect alignment and misalignment cases is shown in Figs. 4.9-4.10. From Figs. 4.7-4.10, it can be seen that the results from the 2-D subdomain model, for the aligned as well as the misaligned cases, show an excellent agreement with 2-D FEA results. Fig. 4.11 shows the comparison in Bz obtained from the 2-D subdomain model

0.303 0.403 0.503 0.603 0.703 0.803 -6.0

-4.0 -2.0 0.0 2.0 4.0 6.0

Bx

(T)

x (m) 2-D FEA

2-D Analytical x10

-4

coil length in

x-direction

(a)

0.303 0.403 0.503 0.603 0.703 0.803 -8.0

-4.0 0.0 4.0 8.0

Bz

(T)

x (m)

2-D FEA

2-D Analytical x10

-4

coil length in

x-direction

(b) Figure 4.7: Distribution of (a) Bxand (b) Bzalong Path 1.

TH-2341_126102029

4. A Subdomain Analytical Model of Coil System with Magnetic Shields of Finite Dimensions and Finite Permeability for WPT systems

0.303 0.403 0.503 0.603 0.703 0.803 -6.0

-4.0 -2.0 0.0 2.0 4.0 6.0

Bx

(T)

x (m) 2-D FEA

2-D Analytical x10

-4

coil length in

x-direction

(a)

0.303 0.403 0.503 0.603 0.703 0.803 -8.0

-4.0 0.0 4.0 8.0

2-D FEA

2-D Analytical

Bz

(T)

x (m) x10

-4

coil length in

x-direction

(b)

Figure 4.8: Distribution of (a) B_x and (b) B_zalong Path 1 with 40mm horizontal shift of the secondary coil arrangement.

0.303 0.403 0.503 0.603 0.703 0.803 -1.5

-1.0 -0.5 0.0 0.5 1.0 1.5

Bx

(T)

x (m)

2-D FEA

2-D Analytical

coil length in

x-direction

(a)

0.303 0.403 0.503 0.603 0.703 0.803 -1.5

-1.0 -0.5 0.0 0.5 1.0 1.5

Bx

(T)

x (m) 2-D FEA

2-D Analytical

coil length in

x-direction

(b)

Figure 4.9: B_xdistribution, along the x-direction, on the center of (a) Shield 1 and (b) Shield 2.

0.303 0.403 0.503 0.603 0.703 0.803 -1.5

-1.0 -0.5 0.0 0.5 1.0 1.5

Bx

(T)

x (m)

2-D FEA

2-D Analytical

coil length in

x-direction

(a)

0.303 0.403 0.503 0.603 0.703 0.803 -1.5

-1.0 -0.5 0.0 0.5 1.0 1.5

Bx

(T)

x (m) 2-D FEA

2-D Analytical

coil length in

x-direction

(b)

Figure 4.10: B_x distribution, along the x-direction with 40mm horizontal shift of the secondary coil arrange- ment, on the center of (a) Shield 1 and (b) Shield 2.

TH-2341_126102029