Scope for Future Research

quantity, non-linearity and hysteretic property of magnetic shield have been neglected to simplify the analytical model. These assumptions limit the applicability of proposed models. Moreover, The presented subdomain model in chapter 4 calculates mutual inductance only for horizontal and vertical misalignments.

6.3 Scope for Future Research

The following recommendations are made to improve the range of applicability of the proposed model:

• Much research has been carried out to improve the coupling between the coils in the WPT system. For this, different coil structures such as DD, DDQ, multiple coil couplers, etc. [106–

108], have been designed. By changing the definition of the current density in the proposed analytical models, one can develop the analytical model for the WPT system having these coils.

• Include the current in the z-direction.

• By re-defining magnetic potential in the proposed 3-D analytical model, the eddy currents can be included in the modelling. .

• The analytical model using the subdomain technique discussed in chapter 4 predicts mutual inductance does not consider angular and planar misalignments. The magnetic vector potentials can be further modified to include these misalignments.

• The applicability of the proposed electromagnetic model can be increased with the inclusion of non-linear magnetic property of the applied magnetic shield.

• The transferred power in the WPT system depends on the resistance of the coils. The resistance of coils varies with the temperature of the coils. A thermal model can be included in the proposed models to predict the parameter variation with temperature in the WPT system.

• The research has been focussed on the magnetic design of the WPT system. Moreover, the vari- ation in the electrical parameter for the SP compensated WPT system also has been studied. For the SP compensated WPT system, resonance frequency depends on the mutual inductance and TH-2341_126102029

6. Conclusion and Future Works

the self inductances of the coils. Due to the misalignment between the coils, the value of M, Lp, and L_schanges, and the WPT system is no more in resonance (it gets mistuned), this mistuning impacts the electrical parameters of the SP compensated WPT. The dc-ac converter with the ability of self-tuning can overcome the variation in electrical parameters during misalignments.

TH-2341_126102029

A

Calculation of Expression of Mutual Inductance

TH-2341_126102029

A. Calculation of Expression of Mutual Inductance

A.1 Calculation of Expression

To get a generalized expression a case is considered where the coil is shifted and tilted by angle θand shifted in∆x in x-direction as shown in Fig. A.1, a generalized expression can be derived as follows:

In (3.26), the integration has been done from the coordinate point P(x_i,y_i) to Q(x_f,y_i) and P(x_i,y_i) to Q(x_f,y_i) indicated in Fig.(A.1).

Figure A.1: Generalized coil position.

The value of x_i,x_f,y_iand y_f can be written in terms ofθand∆x as-

x_i = (n_s×w−a_2s)×cos(θ)+∆x (A.1)

xf = (ns×w−a_2s)×cos(θ)+∆x (A.2)

y_i = n_s×sw−b_2s (A.3)

yf = ns×sw−b_2s (A.4)

The coil parameters such as a_2s, b_2s, etc. are given in Table 2.1. Since the integration limit is taken in x and y coordinate, writing z in terms of x as-

z= −tan(θ)×(x−∆x)+ ∆z (A.5)

TH-2341_126102029

A.1 Calculation of Expression

Now From (3.26)

M(x,y,z)= Z yf

yi

Z xf

xi













 (B_III).

sin(θ)^∧i+cos(θ)k^∧

cos(θ)















dxdy (A.6)

TH-2341_126102029

A. Calculation of Expression of Mutual Inductance

TH-2341_126102029

B

Magnetic Vector Potential of Different Regions

TH-2341_126102029

B. Magnetic Vector Potential of Different Regions

B.1 Applied Boundary Conditions Between Different Regions

In this section the boundary condition between each regions of coil system shown in Fig. 4.3 is presented. The boundary condition is applied at the interface between the each regions.// At z₃

I II

_a

II

_b

II

_c

A = 0

x

₁

x

₂

x

₅

x

₆

z

₁

z

₂

II

_b

H_xIIa H_xIIb H_xIIc H_xI

A_I

A_IIa A_IIa A_I

A_IIa A_I

Figure B.1: Illustration of boundary condition at z=z₂.

H_xI(x,z₂) = H_xIIa(x,z₂)|_x

1≤x≤x₂ + H_xIIb(x,z₂)|_x

2≤x≤x₅ + H_xIIc(x,z₂)|_x

5≤x≤x₆ (B.1) AIIa(x,z2) = AI(x,z2) where x₁≤ x≤ x2 (B.2) A_IIb(x,z₂) = A_I(x,z₂) where x₂≤ x≤ x₅ (B.3) A_IIc(x,z₂) = A_I(x,z₂) where x₅≤ x≤ x₆ (B.4) At z₃

III

II

_a

II

_b

II

_c

A = 0

z

₃

z

₆

II

_b

A = 0

H_xIIa H_xIIb H

xIIc

H_xIII

A_III

A_IIa A_IIb

A_III A_III

A_IIc

x

₁

x

₂

x

₅

x

₆

Figure B.2: Illustration of boundary condition at z=z₃.

TH-2341_126102029

B.1 Applied Boundary Conditions Between Different Regions

HxIII(x,z3) = HxIIa(x,z3) _x

1≤x≤x₂ + HxIIb(x,z3) _x

2≤x≤x₅ + HxIIc(x,z3) _x

5≤x≤x₆ (B.5) A_IIa(x,z₃) = A_III(x,z₃) where x₁ ≤ x≤ x₂ (B.6) A_IIb(x,z₃) = A_III(x,z₃) where x₂ ≤ x≤ x₅ (B.7) AIIc(x,z3) = AIII(x,z3) where x₅ ≤ x≤ x6 (B.8) At z₄

III ^HxIII

H_xIVa H_xIVb H_xIVc H_xIVd H_xIVe A_III

A_IVb A_III

A_IVa

A_III A_IVc

A_III A_IVd

A_III A_IVe

x

₁

x

₂

x

₃

x

₄

x

₅

x

₆

A = 0 A = 0

IV_a IV_b IV_c IV_d IV_e

z

₄

Figure B.3: Illustration of boundary condition at z=z4.

H_xIII(x,z₄) = H_xIVa(x,z₄) _x

1≤x≤x₂ + H_xIVb(x,z₄) _x

2≤x≤x₃ + H_xIVc(x,z₄) _x

3≤x≤x₄ (B.9) + H_xIVd(x,z₄)

_x

4≤x≤x₅ + H_xIVe(x,z₄) _x

5≤x≤x₆ (B.10)

A_IVa(x,z₄) = A_III(x,z₄) where x₁ ≤ x≤ x₂ (B.11) A_IVb(x,z₄) = A_III(x,z₄) where x₂ ≤ x≤ x₃ (B.12) A_IVc(x,z₄) = A_III(x,z₄) where x₃ ≤ x≤ x₄ (B.13) A_IVd (x,z₄) = A_III(x,z₄) where x₄ ≤ x≤ x₅ (B.14) A_IVe(x,z₄) = A_III(x,z₄) where x₅ ≤ x≤ x₆ (B.15) Region z₅

TH-2341_126102029

B. Magnetic Vector Potential of Different Regions

V

H_xV

H_xIVa H_xIVb H_xIVc H

xIVd H

xIVe

A_III A_V A_V

A_IVa A_IVc

AV

A_IVd A_IVe A_V

x

₁

x

₂

x

₃

x

₄

x

₅

x

₆

A_V

IV

_a

IV

_b

IV

_c

IV

_d

IV

_e

z

₅

Figure B.4: Illustration of boundary condition at z=z₅.

H_xV (x,z₅) = H_xIVa(x,z₅) _x

1≤x≤x2 + H_xIVb(x,z₅) _x

2≤x≤x3 + H_xIVc(x,z₅) _x

3≤x≤x4 (B.16) + H_xIVd (x,z₅)

_x

4≤x≤x₅ + H_xIVe(x,z₅) _x

5≤x≤x₆ (B.17)

AIVa(x,z5) = AV(x,z5) ; where x₁ ≤ x≤ x2 (B.18) A_IVb(x,z₅) = A_V(x,z₅) ; where x₂ ≤ x≤ x₃ (B.19) A_IVc(x,z₅) = A_V(x,z₅) ; where x₃ ≤ x≤ x₄ (B.20) AIVd(x,z5) = AIII(x,z5) ; where x4 ≤ x≤ x5 (B.21) A_IVe(x,z₅) = A_V(x,z₅) ; where x₅ ≤ x≤ x₆ (B.22) At z₆

V

^HxV

H_xIVa H_xIVb

H_xIVc A_V

A_VIb A_V

A_VIa

A_V A_VIc

x

₂

x

₅

x

₆

x

₁

z

₆

A = 0 A = 0

VI

_a

VI

_b

VI

_c

Figure B.5: Illustration of boundary condition at z=z6.

TH-2341_126102029

B.1 Applied Boundary Conditions Between Different Regions

H_xV (x,z₆) = H_{xV Ia}(x,z₆) _x

1≤x≤x2 + H_{xV Ib}(x,z₆) _x

2≤x≤x₅ + H_{xV Ic}(x,z₆) _x

5≤x≤x₆ (B.23) AV Ia(x,z6) = AV(x,z6) ; where x1 ≤ x≤ x2 (B.24) A_{V Ib}(x,z₆) = A_V(x,z₆) ; where x₂ ≤ x≤ x₅ (B.25) A_{V Ic}(x,z₆) = A_III(x,z₆) ; where x₅≤ x≤ x₆ (B.26) At z₇

VII

H_xVII

H_xIVa H_xIVb H_xIVc

A_VII A_VIb A_VII

A_VIa

A_VII AVIc

x

₂

x

₅

x

₆

x

₁

z

₇

A = 0 A = 0

A = 0

VI

_a

VI

_b

VI

_c

Figure B.6: Illustration of boundary condition at z=z₇.

H_{xV II}(x,z₇) = H_{xV Ia}(x,z₇)|_x

1≤x≤x₂ + H_{xV Ib}(x,z₇)|_x

2≤x≤x₅ + H_{xV Ic}(x,z₇)|_x

5≤x≤x₆ (B.27) AV Ia(x,z7) = AV II(x,z7) ; where x1 ≤ x≤ x2 (B.28) A_{V Ib}(x,z₇) = A_{V II}(x,z₇) ; where x₂ ≤ x≤ x₅ (B.29) AV Ic(x,z7) = AV II(x,z7) ; where x₅ ≤ x≤ x6 (B.30) At x₂

TH-2341_126102029

B. Magnetic Vector Potential of Different Regions

H_zIIa H

zIIb

A_IIb A_IIa

x

₂

z

₂

A_IVa A

IVb

H_zVIa

H_zIVb A_VIa A_VIb H_zVIb

H_zIVa

z z

₃₄

z

₅

z

₇

z

₆

II

_a

II

_b

VI

_a

VI

_b

IV

_a

IV

_b

Figure B.7: Illustration of boundary condition at x=x₂.

AIIa(x₂,z) = AIIb(x₂,z) ; where z2≤ z≤z3 (B.31) H_zIIb(x₂,z) = H_zIIa(x₂,z) ; where z₂ ≤z≤ z₃ (B.32) A_IVa(x₂,z) = A_IVb(x₂,z) ; where z₄ ≤z≤ z₅ (B.33) H_zIVb(x₂,z) = H_zIVa(x₂,z) ; where z₄≤ z≤z₅ (B.34) A_{V Ia}(x₂,z) = A_IVb(x₂,z) ; where z₆ ≤z≤ z₇ (B.35) H_{zV Ib}(x₅,z) = H_{zV Ia}(x₅,z) ; where z₆≤ z≤z₇ (B.36) At x₃

x

₃

A_IVb A_IVc

H_zIVc H_zIVb

z

₄

z

₅

IV

_c

IV

_b

Figure B.8: Illustration of boundary condition at x=x₃.

TH-2341_126102029

B.1 Applied Boundary Conditions Between Different Regions

H_zIVb(x₃,z) = H_zIVc(x₃,z) ; where z₄ ≤z≤z₅ (B.37) AIVc(x3,z) = AIVb(x3,z) ; where z4≤ z≤z5 (B.38) At x4

x

₄

A_IVc A_IVd

H_zIVd H_zIVc

z

₄

z

₅

IV

_c

IV

_d

Figure B.9: Illustration of boundary condition at x=x4.

A_IVc(x₄,z) = A_IVd(x₄,z) ; where z₄≤ z≤z₅ (B.39) H_zIVd(x₄,z) = H_zIVc(x₄,z) ; where z₄ ≤z≤z₅ (B.40) At x₅

H_zIIb(x₅,z) = H_zIIc(x₅,z) ; where z₂≤ z≤z₃ (B.41) A_IIc(x₅,z) = A_IIb(x₅,z) ; where z₂ ≤ z≤ z₃ (B.42) HzIVd(x₅,z) = HzIVe(x₅,z) ; where z4 ≤z≤z5 (B.43) A_IVe(x₅,z) = A_IVd(x₅,z) ; where z₄≤ z≤z₅ (B.44) H_{zV Ib}(x₅,z) = H_{zV Ic}(x₅,z) ; where z₆≤ z≤z₇ (B.45) AV Ic(x₅,z) = AIVd(x₅,z) ; where z6≤ z≤z7 (B.46)

TH-2341_126102029

B. Magnetic Vector Potential of Different Regions

HzIIb H_zIIc A_IIc A_IIb

x

₅

z

₂

A_IVd A_IVe H_zVIb

H_zIVe A_VIb A_VIc H_zVIc

H_zIVd

z

₃

z

₄

z

₅

z

₇

z

₆

VI

_b

VI

_c

IV

_d

IV

_e

II

_b

II

_c

Figure B.10: Illustration of boundary condition at x=x₅.