System Overview

3.5 Summary

4.2.1 System Overview

Fig. 4.1(a) shows a 3-D illustration of the considered coil system. The rectangular coils in Fig.

4.1(a) are named as primary coil and secondary coil, and the magnetic shields below and above them are called Shield 1 and Shield 2, respectively. In this work, the primary coil together with Shield 1, is called the primary coil arrangement, whereas the secondary coil together with Shield 2 is called the secondary coil arrangement. Here, two geometrically identical rectangular coils and magnetic shields are taken. However, the presented model is general and can be used for different dimensions of magnetic shields and coils.

Fig. 4.3 shows the detailed cross-sectional view (xz-plane) of the coil system (Fig. 4.1(a)), used for formulating the 2-D subdomain model, and Table 4.1 gives the geometrical parameters for the coil TH-2341_126102029

4.2 2-D Subdomain Model for the Coil System

I

II_a II_b II_c

IV_a IV_b IV_c IV_d IV_e

V VI_a

VII

VI_c VI_b

III

A = 0

x

₁

x

₂

x

₃

x

₄

x

₅

x

₆

z

₅

z

₁

z

₂

z

₃

z

₆

z

₇

z

₈

z

₄

Figure 4.3: xz-face of the coil system.

z

_inr

z

x z

_fr

x

_inr

x

_fr

τ

x

= x

_fr

- x

_inr

τ r

z

= z

_fr

- z

_inr

wherercan be any region from I to VII.

Figure 4.4: Co-ordinate representation of different regions demonstrated in Fig. 4.3.

system. These parameter has been chosen as an example to verify the subdomain model; however, one can use this subdomain model for different dimensions of the coil system shown in Fig. 4.1(a).

The whole coil system is divided into several regions in the horizontal (x-direction) as well as in the vertical (z-direction) directions, as shown in Fig. 4.3. These regions are divided based on the interaction of different material properties. Region I represents air medium below the Shield 1, whereas Region V II represents the air medium above the Shield 2. Regions II_b and V I_b represent Shield 1 and Shield 2, respectively. The air medium to the left of the Shield 1 is called Region II_a, whereas the one to the right, is called Region II_c. Similarly, the air medium to the left and right side of Shield 2 is represented as Regions V I_a and V I_c, respectively. Region III is the air gap between the primary coil and the Shield 1, and Region V represents the air gap between the primary coil and Shield 2. Since the secondary coil is open-circuited hence carries no current, it is included in the Region V. Regions IV_aand IV_e represent the air medium to the left and the right side of the primary TH-2341_126102029

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

coil, respectively. Regions IVb and IVd, respectively, represent the two coil sides (carrying current inward and outward of the xz-plane , i.e., in the y-direction) of the primary coil. The Region IV_c represents the air medium between the two coil sides of the primary coil.

The proposed analytical model considers the following assumptions:

• The eddy-current induced in magnetic shields are neglected;

• The magnetic shields used in coil system are considered as isotropic;

• The coil system is surrounded by a vacuum through a rectangular box, as shown in Fig. 4.3.

The magnetic vector potential to at the edge of this rectangular box is assumed zero;

• The temperature-dependent quantity such as resistance of the primary coil is neglected in mag- netic modelling;

• No current exists in the z-direction.

Table 4.1: Parameters of the Coil System and Subdomain Model

Parameters, symbols Value

Diameter of the primary and secondary coils (w) 2.4 mm Number of turn in primary and secondary coils (turn) 11 Relative permeability of Shield 1 and Shield 2 (µ_IIb,µ_{V Ib}) 20000

Geometrical specification in the x-direction (x₁, x₂, x₃, x₄, x₅, x₆) (0, 45, 47.75, 60.85, 63.60, 108.6) cm

Coil length in the x-direction x5-x2 18.6 cm

Coil length in the y-direction y5-y2 14.11 cm

Geometrical specification in the x-direction (z1, z2, z3, z4, z5) (-100.511, -0.511, -0.509, 0, 0.25) cm Current density in the primary coil (at 3.5 A) 0.56 A-mm⁻² Number of Harmonics in x- direction for Big region (N_I, N_III, N_V, N_{V II}) 210

Harmonics in x- direction for Regions IIa,IIb,IIc(NIIa, NIIb, NIIc) 100 Harmonics in z- direction for Regions II_a,II_b,II_c(L_IIa, L_IIb, L_IIc) 30 Harmonics in x- direction for Regions IVa,IVc,IVe(NIV_a, NIVc, NIVe) 50 Harmonics in z- direction for Regions IV_a,IV_c,IV_e(L_IVa, L_IVc, L_IVe) 30 Harmonics in x- direction for Regions IVb,IVe(NIV_a, NIVc, NIVe) 60 Harmonics in z- direction for Regions IV_b,IV_e(L_IVa, L_IVc, L_IVe) 60 Harmonics in x- direction for Regions V Ia, V Ib, V Ic(NIIa, NIIb, NIIc) 100 Harmonics in z- direction for Regions V I_a, V I_b, V I_c(L_IIa, L_IIb, L_IIc) 30

TH-2341_126102029

4.2 2-D Subdomain Model for the Coil System

The magnetic vector potential formulation has been used in subdomain modelling. A region r has been defined to represent the magnetic vector potential for each subdomain, as shown in Fig. 4.4. The inial and final co-ordinates in the x-direction of region r are denoted by x_inr and x_fr, whereas in the z-direction, they are denoted by z_inr and z_fr, respectively. The region r can be any of the regions, from I to V II, as shown in Fig. 4.3. The exact values of x_inr, x_fr, z_inr, and z_fr for a particular region can be taken from Fig. 4.3. For example, the values of x_inr, x_fr, z_inr, and z_fr for the region IV_c (r = IV_c) are x3, x4, z4, and z5, respectively.

The magnetic field for each described region is calculated by magnetic vector potential, which is obtained using the Poisson and Laplace equations, as discussed in the next subsection.