2.2 Inductance and mutual inductance of DH coil

As shown in Figure 5, for simplicity, the DH coil is separated into several cells and each cell consists of an inner loop (Loop 1) and an outer loop (Loop 2).

By superposition, the inductance of the DH coil is equal to the sum of the cell inductance and the mutual inductance between each cell (Figure 5a), namely:

$$L\_{DH} = \sum\_{i=1}^{n} \sum\_{j=1 \atop j \neq i}^{n} M\_{i,j} + \sum\_{i=1}^{n} L\_i = \sum\_{i=1}^{n} \sum\_{j=1 \atop j \neq i}^{n} M\_{i,j} + nL\_{cell}.\tag{1}$$

where Mi,j denotes the mutual inductance between cell i and cell j, and Lcell denotes the inductance of a single cell.

#### Figure 4.

(a) Schematic representation of the DH coil and current directions indicated by arrows. Conductors are printed on both sides of the flexible PCB; (b) the PCB is wrapped to form a DH coil.

Figure 5. (a) The DH coil can be separated into n cells, having mutual inductance with each other; (b) cell model.

According to the cell model (Figure 5b), the inductance of a single cell is the sum of the inductances of Loops 1 and 2 and their mutual inductance. Since the loops are perpendicular to each other, the mutual inductance between the loops is zero so that the inductance of a single cell is given by

$$L\_{cell} = L\_1 + L\_2 + 2M\_{12} = L\_1 + L\_2 \tag{2}$$

where,

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

Mi�<sup>1</sup>,j�<sup>2</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup>

Mi�<sup>1</sup>,j�<sup>1</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup>

Mi�<sup>2</sup>,j�<sup>2</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup>

4π ð0 2π ð<sup>2</sup><sup>π</sup> 0

4π ð<sup>2</sup><sup>π</sup> 0

4π ð<sup>2</sup><sup>π</sup> 0

MTotal <sup>¼</sup> <sup>X</sup><sup>n</sup>

<sup>M</sup>Ti‐<sup>1</sup> <sup>¼</sup> <sup>μ</sup>0<sup>m</sup> 4π

8 >>>>>><

>>>>>>:

Figure 7.

75

<sup>M</sup>Ti‐<sup>2</sup> <sup>¼</sup> <sup>μ</sup>0<sup>m</sup> 4π

i¼1

ð<sup>2</sup><sup>π</sup> 0 ð<sup>2</sup><sup>π</sup> 0

ð<sup>2</sup><sup>π</sup> 0 ð<sup>2</sup><sup>π</sup> 0

with only one turn as shown in Figure 7b. Then, MTi-1 and MTi-2 are given by

ð<sup>2</sup><sup>π</sup> 0

DOI: http://dx.doi.org/10.5772/intechopen.89120

ð<sup>2</sup><sup>π</sup> 0

of the calculated inductances by Eq. (2).

tances between Tx and each loop in the DH coil, namely,

i¼1

MTi�<sup>1</sup> <sup>þ</sup>X<sup>n</sup>

R1R<sup>2</sup> cos θ cos ϕ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup>ð Þ 2 sin <sup>θ</sup> sin <sup>ϕ</sup> <sup>þ</sup> cos <sup>θ</sup> cos <sup>ϕ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup>ð Þ 2 sin <sup>θ</sup> sin <sup>ϕ</sup> <sup>þ</sup> cos <sup>θ</sup> cos <sup>ϕ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>q</sup> <sup>d</sup>θd<sup>ϕ</sup>

<sup>q</sup> <sup>d</sup>θd<sup>ϕ</sup>

<sup>q</sup> <sup>d</sup>θd<sup>ϕ</sup>

<sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup>

<sup>1</sup> � <sup>2</sup>R<sup>2</sup>

<sup>2</sup> � <sup>2</sup>R<sup>2</sup>

<sup>ð</sup>MTi�<sup>1</sup> <sup>þ</sup> MTi�<sup>2</sup>Þ ¼ <sup>X</sup><sup>n</sup>

<sup>2</sup> þ 2R1R<sup>2</sup> cosð Þ θ þ ϕ

:

(5)

(7)

<sup>1</sup> cosð Þ θ � ϕ

<sup>2</sup> cosð Þ θ � ϕ

i¼1

MTi: (6)

<sup>R</sup><sup>1</sup> cos <sup>θ</sup> � <sup>R</sup><sup>2</sup> cos <sup>ϕ</sup> <sup>þ</sup> <sup>d</sup>ij � �<sup>2</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup>

<sup>R</sup><sup>1</sup> cos <sup>θ</sup> � <sup>R</sup><sup>1</sup> cos <sup>ϕ</sup> <sup>þ</sup> <sup>d</sup>ij � �<sup>2</sup> <sup>þ</sup> <sup>2</sup>R<sup>2</sup>

<sup>R</sup><sup>2</sup> cos <sup>θ</sup> � <sup>R</sup><sup>2</sup> cos <sup>ϕ</sup> <sup>þ</sup> <sup>d</sup>ij � �<sup>2</sup> <sup>þ</sup> <sup>2</sup>R<sup>2</sup>

Therefore, the total inductance of the DH is derived based on the superpositions

As shown in Figure 7a, the mutual inductance between the DH coil and the transmitter (Tx) can also be regarded as the sum of all individual mutual induc-

i¼1

(a) The total mutual inductance between Tx and the DH coil is calculated by the superpositions of the mutual inductances between Tx and all cells; (b) Modeling of the mutual inductance between Tx and one cell.

where MTi-1 and MTi-2 are the mutual inductances between Tx and the ith inner and outer loops, respectively, MTi is the mutual inductance between Tx and the ith cell, and n denotes the cell number. For simplicity, Tx is modeled as a circular coil

> <sup>R</sup>1RTX cosð Þ <sup>θ</sup> � <sup>γ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>R</sup><sup>1</sup> cos <sup>θ</sup> � <sup>R</sup>TX cos <sup>γ</sup> <sup>2</sup> þ �ð Þ <sup>R</sup><sup>1</sup> sin <sup>θ</sup> � <sup>R</sup>TX sin <sup>γ</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>R</sup><sup>1</sup> cos <sup>θ</sup> � <sup>d</sup> <sup>2</sup> <sup>q</sup> <sup>d</sup>θd<sup>γ</sup>

<sup>R</sup>2RTX cosð Þ <sup>ϕ</sup> � <sup>γ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>R</sup><sup>2</sup> cos <sup>ϕ</sup> � <sup>R</sup>TX cos <sup>γ</sup> <sup>2</sup> þ �ð Þ <sup>R</sup><sup>2</sup> sin <sup>ϕ</sup> � <sup>R</sup>TX sin <sup>γ</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>R</sup><sup>2</sup> cos <sup>ϕ</sup> <sup>þ</sup> <sup>d</sup> <sup>2</sup> <sup>q</sup> <sup>d</sup>ϕd<sup>γ</sup>

MTi�<sup>2</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>

R2

Wireless Power Transfer for Miniature Implantable Biomedical Devices

R2

where L<sup>1</sup> and L<sup>2</sup> are the inductances of Loops 1 and 2, respectively, given by

$$\begin{cases} L\_1 = \frac{\mu\_0}{4\pi} \Big|\_{0}^{2\pi} \int\_0^{2\pi} \frac{R\_1(R\_1 \cdot a) \left(2\sin\theta\sin\phi + \cos\theta\cos\phi\right)}{\sqrt{2(R\_1 \cos\theta - (R\_1 \cdot a)\cos\phi)^2 + (R\_1 \sin\theta - (R\_1 \cdot a)\sin\phi)^2}} d\theta d\phi \\\ L\_2 = \frac{\mu\_0}{4\pi} \Big|\_{0}^{2\pi} \int\_0^{2\pi} \frac{R\_2(R\_2 \cdot a) \left(2\sin\theta\sin\theta' + \cos\theta\cos\theta'\right)}{\sqrt{2(R\_2 \cos\theta - (R\_2 \cdot a)\cos\phi)^2 + (R\_2 \sin\theta - (R\_2 \cdot a)\sin\phi)^2}} d\theta d\phi \end{cases} \tag{3}$$

where R<sup>1</sup> and R<sup>2</sup> are the projected radii of Loops 1 and 2 along the z axis (Figure 5a).

Figure 6 shows a model containing two cells. The mutual inductance between the two cells is calculated by

$$\begin{split} M\_{i,j} &= M\_{i \cdot \mathbf{1}\_j, j \cdot \mathbf{1}} + M\_{i \cdot \mathbf{1}\_j, j \cdot \mathbf{2}} + M\_{i \cdot \mathbf{2}\_j, j \cdot \mathbf{1}} + M\_{i \cdot \mathbf{2}\_j, j \cdot \mathbf{2}} \\ &= M\_{i \cdot \mathbf{1}\_j, j \cdot \mathbf{1}} + 2M\_{i \cdot \mathbf{1}\_j, j \cdot \mathbf{2}} + M\_{i \cdot \mathbf{2}\_j, j \cdot \mathbf{2}} \end{split} \tag{4}$$

Figure 6. Modeling of the mutual inductance between two cells.

Wireless Power Transfer for Miniature Implantable Biomedical Devices DOI: http://dx.doi.org/10.5772/intechopen.89120

where,

According to the cell model (Figure 5b), the inductance of a single cell is the sum of the inductances of Loops 1 and 2 and their mutual inductance. Since the loops are perpendicular to each other, the mutual inductance between the loops is

(a) The DH coil can be separated into n cells, having mutual inductance with each other; (b) cell model.

where L<sup>1</sup> and L<sup>2</sup> are the inductances of Loops 1 and 2, respectively, given by

where R<sup>1</sup> and R<sup>2</sup> are the projected radii of Loops 1 and 2 along the z axis

Figure 6 shows a model containing two cells. The mutual inductance between

Mi,j <sup>¼</sup> Mi‐<sup>1</sup>,j‐<sup>1</sup> <sup>þ</sup> Mi‐<sup>1</sup>,j‐<sup>2</sup> <sup>þ</sup> Mi‐<sup>2</sup>,j‐<sup>1</sup> <sup>þ</sup> Mi‐<sup>2</sup>,j‐<sup>2</sup> <sup>¼</sup> Mi‐<sup>1</sup>,j‐<sup>1</sup> <sup>þ</sup> <sup>2</sup>Mi‐<sup>1</sup>,j‐<sup>2</sup> <sup>þ</sup> Mi‐<sup>2</sup>,j‐<sup>2</sup>

Lcell ¼ L<sup>1</sup> þ L<sup>2</sup> þ 2M<sup>12</sup> ¼ L<sup>1</sup> þ L<sup>2</sup> (2)

(3)

(4)

<sup>R</sup>1ð Þ <sup>R</sup><sup>1</sup>‐<sup>a</sup> ð Þ 2 sin <sup>θ</sup> sin <sup>ϕ</sup> <sup>þ</sup> cos <sup>θ</sup> cos <sup>ϕ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>ð Þ <sup>R</sup><sup>1</sup> cos <sup>θ</sup> � ð Þ <sup>R</sup><sup>1</sup>‐<sup>a</sup> cos <sup>ϕ</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>R</sup><sup>1</sup> sin <sup>θ</sup> � ð Þ <sup>R</sup><sup>1</sup>‐<sup>a</sup> sin <sup>ϕ</sup> <sup>2</sup> <sup>q</sup> <sup>d</sup>θd<sup>ϕ</sup>

<sup>R</sup>2ð Þ <sup>R</sup><sup>2</sup>‐<sup>a</sup> 2 sin <sup>θ</sup> sin <sup>θ</sup><sup>0</sup> <sup>þ</sup> cos <sup>θ</sup> cos <sup>θ</sup><sup>0</sup> ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>ð Þ <sup>R</sup><sup>2</sup> cos <sup>θ</sup> � ð Þ <sup>R</sup><sup>2</sup>‐<sup>a</sup> cos <sup>ϕ</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>R</sup><sup>2</sup> sin <sup>θ</sup> � ð Þ <sup>R</sup><sup>2</sup>‐<sup>a</sup> sin <sup>ϕ</sup> <sup>2</sup> <sup>q</sup> <sup>d</sup>θd<sup>ϕ</sup>

zero so that the inductance of a single cell is given by

<sup>L</sup><sup>1</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup> 4π

8 >>>>>><

Figure 5.

>>>>>>:

(Figure 5a).

Figure 6.

74

<sup>L</sup><sup>2</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup> 4π ð<sup>2</sup><sup>π</sup> 0

ð<sup>2</sup><sup>π</sup> 0

the two cells is calculated by

Modeling of the mutual inductance between two cells.

ð<sup>2</sup><sup>π</sup> 0

Recent Wireless Power Transfer Technologies

ð<sup>2</sup><sup>π</sup> 0

$$\begin{cases} M\_{i-1,j-2} = \frac{\mu\_0}{4\pi} \Big|\_{2\pi}^0 \int\_0^{2\pi} \frac{R\_1 R\_2 \cos\theta \cos\phi}{\sqrt{\left(R\_1 \cos\theta - R\_2 \cos\phi + d\_{\vec{\text{u}}\right)^2 + R\_1^2 + R\_2^2 + 2R\_1 R\_2 \cos\left(\theta + \phi\right)}} d\theta d\phi \\\ M\_{i-1,j-1} = \frac{\mu\_0}{4\pi} \Big|\_0^{2\pi} \int\_0^{2\pi} \frac{R\_1^2 (2\sin\theta \sin\phi + \cos\theta \cos\phi)}{\sqrt{\left(R\_1 \cos\theta - R\_1 \cos\phi + d\_{\vec{\text{u}}\right)^2 + 2R\_1^2 - 2R\_1^2 \cos\left(\theta - \phi\right)}} d\theta d\phi \\\ M\_{i-2,j-2} = \frac{\mu\_0}{4\pi} \Big|\_0^{2\pi} \int\_0^{2\pi} \frac{R\_2^2 (2\sin\theta \sin\phi + \cos\theta \cos\phi)}{\sqrt{\left(R\_2 \cos\theta - R\_2 \cos\phi + d\_{\vec{\text{u}}\right)^2 + 2R\_2^2 - 2R\_2^2 \cos\left(\theta - \phi\right)}} d\theta d\phi \end{cases} \tag{5}$$

Therefore, the total inductance of the DH is derived based on the superpositions of the calculated inductances by Eq. (2).

As shown in Figure 7a, the mutual inductance between the DH coil and the transmitter (Tx) can also be regarded as the sum of all individual mutual inductances between Tx and each loop in the DH coil, namely,

$$M\_{Total} = \sum\_{i=1}^{n} M\_{Ti-1} + \sum\_{i=1}^{n} M\_{Ti-2} = \sum\_{i=1}^{n} (M\_{Ti-1} + M\_{Ti-2}) = \sum\_{i=1}^{n} M\_{Ti}.\tag{6}$$

where MTi-1 and MTi-2 are the mutual inductances between Tx and the ith inner and outer loops, respectively, MTi is the mutual inductance between Tx and the ith cell, and n denotes the cell number. For simplicity, Tx is modeled as a circular coil with only one turn as shown in Figure 7b.

Then, MTi-1 and MTi-2 are given by

$$\begin{cases} M\_{\text{Ti}:1} = \frac{\mu\_0 m}{4\pi} \int\_0^{2\pi} \int\_0^{2\pi} \frac{R\_1 R\_{\text{TX}} \cos\left(\theta - \eta\right)}{\sqrt{\left(R\_1 \cos\theta - R\_{\text{TX}} \cos\eta\right)^2 + \left(-R\_1 \sin\theta - R\_{\text{TX}} \sin\eta\right)^2 + \left(R\_1 \cos\theta - d\right)^2}} d\theta d\eta\\ M\_{\text{Ti}:2} = \frac{\mu\_0 m}{4\pi} \int\_0^{2\pi} \int\_0^{2\pi} \frac{R\_2 R\_{\text{TX}} \cos\left(\phi - \eta\right)}{\sqrt{\left(R\_2 \cos\phi - R\_{\text{TX}} \cos\eta\right)^2 + \left(-R\_2 \sin\phi - R\_{\text{TX}} \sin\eta\right)^2 + \left(R\_2 \cos\phi + d\right)^2}} d\phi d\eta \end{cases} \tag{7}$$

Figure 7.

(a) The total mutual inductance between Tx and the DH coil is calculated by the superpositions of the mutual inductances between Tx and all cells; (b) Modeling of the mutual inductance between Tx and one cell.

where RTX is the radius of Tx, μ<sup>0</sup> is the magnetic permeability of free space, and d is the vertical distance between the cell and Tx. Accordingly, we have

$$M\_{\rm Ti} = M\_{\rm Ti\cdot 1} + M\_{\rm Ti\cdot 2}.\tag{8}$$

The coupling factor k between Tx and Rx is given by

$$k = \frac{M\_{\text{Total}}}{\sqrt{L\_{\text{DH}}L\_{\text{TX}}}} \tag{9}$$

where

$$L\_{\rm TX} = \mu\_0 R\_{\rm TX} \left( \ln \left( \frac{R\_{\rm TX}}{a\_{\rm TX}} \right) - 1.75 \right). \tag{10}$$

#### 2.3 Coupling factor simulations

According to the magnetic resonant WPT theory, the coupling factor k largely influences system performance [20]. In order to evaluate the coupling factor in the system with the misalignment between the transmitter and receiver coils, the proposed DH coil was compared to a conventional double-layer solenoid by simulation. The transmitter was modeled as a PSC with the outer and inner radii being 30 and 15 mm, respectively. For the DH coil and conventional solenoid, R<sup>1</sup> and R<sup>2</sup> were 5 and 6 mm, respectively. The variation of k was investigated with respect to the lateral and angular misalignments.

Figure 8 shows the simulation models with lateral or angular misalignments. X0 and Y0 in Figure 8a and b indicate the displacements along the x-direction and y-direction, respectively. α and β indicate the rotating angles around the x-axis and y-axis, respectively. The vertical distance d in Figure 8c and d is 20 mm in all cases. The simulation was used to calculate the coupling factor in different scenarios. The results are shown in Figure 9.

It is seen that offers the maximum coupling factor k of the conventional solenoid much smaller than that of the DH coil, comparing Figure 9a and b. Additionally, the coupling factor k of the DH coil is larger than that of the conventional solenoid in most of the measurement range. Figure 9c showed that k is essentially invariable as the conventional solenoid rotates around the x-axis. However, the DH coil has an optimal angle as which the largest k is achieved. With β decreases, the central axes of both the DH coil and conventional solenoid change from being perpendicular to being parallel with the plane of Tx as shown in Figure 9d. Within such a process, the coupling factor of the solenoid decreases, while the coupling factor increases for the DH coil. In addition, when β is 0, the coupling factor of the DH coil is much larger than that of the conventional solenoid. Accordingly, the orthogonal-coil structure enhances the mutual inductance. This phenomenon verifies the superiority of the DH coil over the traditional solenoid.

#### 2.4 Experimental results

We constructed several prototypes of DH coils with variable turns, gaps and widths. Then, the coil with the largest quality factor was chosen and tuned to a resonant frequency of 5.2 MHz using capacitors. This DH coil is presented in Figure 10.

Figure 9.

77

Figure 8.

around (c) the x-axis and (d) the y-axis, respectively.

The coupling factor as a function of the relative position between the receiver and transmitter coils in different cases: (a) DH coil with lateral misalignment, (b) traditional coil with lateral misalignment, (c) the angular

Simulation models of (a) DH coil and (b) solenoid coil for lateral misalignment, and for angular shifting

Wireless Power Transfer for Miniature Implantable Biomedical Devices

DOI: http://dx.doi.org/10.5772/intechopen.89120

shifting around the x-axis, and (d) the angular shifting around the y-axis.

To study WPT performances in different misalignment scenarios, the PTE was chosen as an evaluation index. The PTE was measured based on the scatter

Wireless Power Transfer for Miniature Implantable Biomedical Devices DOI: http://dx.doi.org/10.5772/intechopen.89120

#### Figure 8.

where RTX is the radius of Tx, μ<sup>0</sup> is the magnetic permeability of free space, and

<sup>k</sup> <sup>¼</sup> <sup>M</sup>Total ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LDHLTX

> aTX � �

According to the magnetic resonant WPT theory, the coupling factor k largely influences system performance [20]. In order to evaluate the coupling factor in the system with the misalignment between the transmitter and receiver coils, the proposed DH coil was compared to a conventional double-layer solenoid by simulation. The transmitter was modeled as a PSC with the outer and inner radii being 30 and 15 mm, respectively. For the DH coil and conventional solenoid, R<sup>1</sup> and R<sup>2</sup> were 5 and 6 mm, respectively. The variation of k was investigated with respect to the

Figure 8 shows the simulation models with lateral or angular misalignments. X0 and Y0 in Figure 8a and b indicate the displacements along the x-direction and y-direction, respectively. α and β indicate the rotating angles around the x-axis and y-axis, respectively. The vertical distance d in Figure 8c and d is 20 mm in all cases. The simulation was used to calculate the coupling factor in different scenar-

It is seen that offers the maximum coupling factor k of the conventional solenoid much smaller than that of the DH coil, comparing Figure 9a and b. Additionally, the coupling factor k of the DH coil is larger than that of the conventional solenoid in most of the measurement range. Figure 9c showed that k is essentially invariable as the conventional solenoid rotates around the x-axis. However, the DH coil has an optimal angle as which the largest k is achieved. With β decreases, the central axes of both the DH coil and conventional solenoid change from being perpendicular to being parallel with the plane of Tx as shown in Figure 9d. Within such a process, the coupling factor of the solenoid decreases, while the coupling factor increases for the DH coil. In addition, when β is 0, the coupling factor of the DH coil is much larger than that of the conventional solenoid. Accordingly, the orthogonal-coil structure enhances the mutual inductance. This phenomenon verifies the superior-

We constructed several prototypes of DH coils with variable turns, gaps and widths.

To study WPT performances in different misalignment scenarios, the PTE was

Then, the coil with the largest quality factor was chosen and tuned to a resonant frequency of 5.2 MHz using capacitors. This DH coil is presented in Figure 10.

chosen as an evaluation index. The PTE was measured based on the scatter

� �

� 1:75

<sup>M</sup>T<sup>i</sup> <sup>¼</sup> <sup>M</sup>Ti‐<sup>1</sup> <sup>þ</sup> <sup>M</sup>Ti‐2: (8)

p (9)

: (10)

d is the vertical distance between the cell and Tx. Accordingly, we have

<sup>L</sup>TX <sup>¼</sup> <sup>μ</sup>0RTX In <sup>R</sup>TX

The coupling factor k between Tx and Rx is given by

where

2.3 Coupling factor simulations

Recent Wireless Power Transfer Technologies

lateral and angular misalignments.

ios. The results are shown in Figure 9.

ity of the DH coil over the traditional solenoid.

2.4 Experimental results

76

Simulation models of (a) DH coil and (b) solenoid coil for lateral misalignment, and for angular shifting around (c) the x-axis and (d) the y-axis, respectively.

#### Figure 9.

The coupling factor as a function of the relative position between the receiver and transmitter coils in different cases: (a) DH coil with lateral misalignment, (b) traditional coil with lateral misalignment, (c) the angular shifting around the x-axis, and (d) the angular shifting around the y-axis.

frequency. However, the PTE decreases as the misalignment increases, similar to

more efficient power delivery than the traditional solenoid.

Wireless Power Transfer for Miniature Implantable Biomedical Devices

resonant at the operating frequency in the RF range.

3. Mat-based wireless power transfer to moving targets

It can be observed that the PTE is strongly dependent on the inductive coupling, which was discussed in the previous section. Therefore, the proposed DH coil offers

Neural stimulation and recording provide emerging prosthetic and treatment options for spinal cord injury, stroke, sensory dysfunction, and other neurological diseases and disorders. Neural recording from awake animals with observable behavior has greatly enhanced our understanding of central and peripheral nervous systems. Although there has been substantial studies on miniaturized, implantable electronic circuits that record neural data and stimulate neuronal networks in freely moving laboratory animals, the mobility of the animal subject is often limited, and the experimental results obtained under restricted conditions may not reflect the full repertoire of brain activity corresponding to their natural behaviors [16]. There are similar problems in the study of new drugs which often requires monitoring a number of variables from the inside of the animal body and observation of their

Traditionally, magnetic induction was used for WPT using a similar form to a transformer [21]. This form of the magnetic induction method is highly efficient (>90%) in the near-field range, but much less efficient as the transmission distance increases. In 2007, an efficient mid-range WPT via strongly coupled magnetic resonance was reported [22]. This system consists of four coils (Figure 12), namely, driver, primary (or transmitter), secondary (or receiver), and load coils. Inductive coupling is used between the driver and primary coils as well as between the secondary and load coils. The primary and secondary coils with the same resonant frequencies tend to exchange energy efficiently. This mechanism is valuable in the application to medical implants because biological tissues are generally non-

Because our special WPT system involves moving targets (animals, e.g., rodents), the vertical component of the magnetic field generated by the transmitter is required to be distributed as even as possible over the entire area of interest (e.g., floor of the rodent cage). When this condition is satisfied, the device carried by or implanted within each rodent can receive steady power at any location of the floor. Previously, we designed a WPT system in which multiple circular spiral coils were printed on hexagonal PCBs [23]. These PCBs were then tiled hexagonally forming a "power mat" shown in Figure 13. Note that the use of hexagons in the pack of coils is not an arbitrary choice, rather it has been proven that this design will leave the

A resonance based WPT system including four coils, namely driver, primary, secondary, and load coils.

the variation of the coupling factor.

DOI: http://dx.doi.org/10.5772/intechopen.89120

mobility and behaviors.

Figure 12.

79

Figure 10. Experimental setups for efficiency measurement.

#### Figure 11.

PTE measurements vs. frequency with variable misalignments: (a) lateral misalignment in the x-direction, (b) lateral misalignment in the y-direction, (c) angular misalignment around x-axis, and (d) angular misalignment around y-axis.

parameters measured by a network analyzer. PTE measurements were also performed for both lateral and angular misalignments. The results shown in Figure 11 indicate that the WPT system achieves the maximum PTE at the resonant

#### Wireless Power Transfer for Miniature Implantable Biomedical Devices DOI: http://dx.doi.org/10.5772/intechopen.89120

frequency. However, the PTE decreases as the misalignment increases, similar to the variation of the coupling factor.

It can be observed that the PTE is strongly dependent on the inductive coupling, which was discussed in the previous section. Therefore, the proposed DH coil offers more efficient power delivery than the traditional solenoid.

## 3. Mat-based wireless power transfer to moving targets

Neural stimulation and recording provide emerging prosthetic and treatment options for spinal cord injury, stroke, sensory dysfunction, and other neurological diseases and disorders. Neural recording from awake animals with observable behavior has greatly enhanced our understanding of central and peripheral nervous systems. Although there has been substantial studies on miniaturized, implantable electronic circuits that record neural data and stimulate neuronal networks in freely moving laboratory animals, the mobility of the animal subject is often limited, and the experimental results obtained under restricted conditions may not reflect the full repertoire of brain activity corresponding to their natural behaviors [16]. There are similar problems in the study of new drugs which often requires monitoring a number of variables from the inside of the animal body and observation of their mobility and behaviors.

Traditionally, magnetic induction was used for WPT using a similar form to a transformer [21]. This form of the magnetic induction method is highly efficient (>90%) in the near-field range, but much less efficient as the transmission distance increases. In 2007, an efficient mid-range WPT via strongly coupled magnetic resonance was reported [22]. This system consists of four coils (Figure 12), namely, driver, primary (or transmitter), secondary (or receiver), and load coils. Inductive coupling is used between the driver and primary coils as well as between the secondary and load coils. The primary and secondary coils with the same resonant frequencies tend to exchange energy efficiently. This mechanism is valuable in the application to medical implants because biological tissues are generally nonresonant at the operating frequency in the RF range.

Because our special WPT system involves moving targets (animals, e.g., rodents), the vertical component of the magnetic field generated by the transmitter is required to be distributed as even as possible over the entire area of interest (e.g., floor of the rodent cage). When this condition is satisfied, the device carried by or implanted within each rodent can receive steady power at any location of the floor. Previously, we designed a WPT system in which multiple circular spiral coils were printed on hexagonal PCBs [23]. These PCBs were then tiled hexagonally forming a "power mat" shown in Figure 13. Note that the use of hexagons in the pack of coils is not an arbitrary choice, rather it has been proven that this design will leave the

#### Figure 12.

A resonance based WPT system including four coils, namely driver, primary, secondary, and load coils.

parameters measured by a network analyzer. PTE measurements were also performed for both lateral and angular misalignments. The results shown in

Figure 10.

Figure 11.

78

misalignment around y-axis.

Experimental setups for efficiency measurement.

Recent Wireless Power Transfer Technologies

Figure 11 indicate that the WPT system achieves the maximum PTE at the resonant

PTE measurements vs. frequency with variable misalignments: (a) lateral misalignment in the x-direction, (b) lateral misalignment in the y-direction, (c) angular misalignment around x-axis, and (d) angular

Figure 13. WPT system in which the floor of the animal cage is located over a hexagonally packed power mat.

smallest gap between circular resonator coils [24]. The power mat is able to deliver wireless power to the implants and "carry-on" devices to multiple rodents which move freely on the floor above the mat.

> An adjustable RF oscillator produces a sinusoidal signal, which is amplified by a power amplifier. The output of the amplifier is connected to an array of driver coils which are inductively coupled with primary coils [23]. At the power receiving site (a laboratory animal), the receiver coil is inductively coupled with a load coil to

> In the coupled mode theory (CMT), the first eigen-mode is used to analyze a resonant system. The approximation by the first eigen-mode is quite accurate under the condition of a strong coupling in the WPT system. In practical applications, it is not possible or necessary to directly work on an arbitrary large number of resonators. Rather, for a large hexagonally packed transmitter (HPT) mat (Figure 15), every one of the resonators can be treated as in the middle of the mat, except for those ones at the edge, and fortunately the edge effect can be solved simply by making the mat larger than the animal cage floor. For this reason, we may simplify

Let us index the seven transmitters from 1 to 7 and the single receiver have an index of 8. In order to describe the strongly coupled system, a set of differential

8

j ¼ 1 j 6¼ i

ð Þ jω<sup>0</sup> � Γ<sup>1</sup> jκ<sup>12</sup> ⋯ jκ<sup>17</sup> jκ<sup>18</sup> jκ<sup>21</sup> ð Þ jω<sup>0</sup> � Γ<sup>2</sup> ⋯ jκ<sup>27</sup> jκ<sup>28</sup> ⋮ ⋮⋱⋮ ⋮ jκ<sup>71</sup> jκ<sup>72</sup> ⋯ ð Þ jω<sup>0</sup> � Γ<sup>7</sup> jκ<sup>78</sup> jκ<sup>81</sup> jκ<sup>82</sup> ⋯ jκ<sup>87</sup> jω<sup>0</sup> � Γ<sup>8</sup> � Γ<sup>L</sup>

7

j¼1

jκijajð Þþt fi

jκijajð Þt

ð Þt , i ¼ 1, …, 7

a1ð Þt a2ð Þt ⋮ a7ð Þt a8ð Þt

� �

(11)

f tð Þ f tð Þ ⋮ f tð Þ 0

(12)

supply the power to the electronic components within the implant.

Wireless Power Transfer for Miniature Implantable Biomedical Devices

DOI: http://dx.doi.org/10.5772/intechopen.89120

<sup>8</sup>ðÞ¼ <sup>t</sup> ð Þ <sup>j</sup>ω<sup>0</sup> � <sup>Γ</sup><sup>8</sup> � <sup>Γ</sup><sup>L</sup> <sup>a</sup>8ðÞþ<sup>t</sup> <sup>X</sup>

<sup>a</sup>\_iðÞ¼ <sup>t</sup> ð Þ <sup>j</sup>ω<sup>0</sup> � <sup>Γ</sup><sup>i</sup> aið Þþ<sup>t</sup> <sup>X</sup>

analysis by examining the seven-resonator case.

equations based on CMT is given by [22].

a\_

Or, if written in matrix form

a\_1ð Þt a\_2ð Þt ⋮ a\_7ð Þt a\_8ð Þt

Figure 15.

A large hexagonally packed power mat.

81

It has been found from computer simulation that the packing of a circle by identical disks demonstrates interesting patterns [25]. Denser packings consist of specific numbers of disks. The first several numbers are: 1, 7, 19, 37, 61, and 91. Although the packing density increases as more disks are packed within the enclosing circle, the density is upper-bounded by π<sup>2</sup>=12 ≈ 0:822. Although, in general, using more disks produces a higher density, the implementation complexity increases as the number of disks increases. Additionally, it is difficult to connect the RF signal to numerous transmitter (Tx) coils, and the cost involved is high. On the other hand, we have previously reported that the RF signal can be easily connected to a pair of open concentric rings to power seven Tx coils simultaneously [23]. Therefore, if more than one coil is used in the Tx, the seven-coil design is often the best choice for most practical applications to power free-position devices although its packing density is not the highest.

## 3.1 Theoretical analysis of mat-based WPT system

As shown in Figure 14, the mat-based WPT system enables magnetically coupled resonance between an array of transmitter coils and a single receiver coil.

Figure 14. Mat-based WPT system design.

Wireless Power Transfer for Miniature Implantable Biomedical Devices DOI: http://dx.doi.org/10.5772/intechopen.89120

Figure 15. A large hexagonally packed power mat.

smallest gap between circular resonator coils [24]. The power mat is able to deliver wireless power to the implants and "carry-on" devices to multiple rodents which

WPT system in which the floor of the animal cage is located over a hexagonally packed power mat.

It has been found from computer simulation that the packing of a circle by identical disks demonstrates interesting patterns [25]. Denser packings consist of specific numbers of disks. The first several numbers are: 1, 7, 19, 37, 61, and 91. Although the packing density increases as more disks are packed within the enclosing circle, the density is upper-bounded by π<sup>2</sup>=12 ≈ 0:822. Although, in general, using more disks produces a higher density, the implementation complexity increases as the number of disks increases. Additionally, it is difficult to connect the RF signal to numerous transmitter (Tx) coils, and the cost involved is high. On the other hand, we have previously reported that the RF signal can be easily connected to a pair of open concentric rings to power seven Tx coils simultaneously [23]. Therefore, if more than one coil is used in the Tx, the seven-coil design is often the best choice for most practical applications to power free-position devices although

As shown in Figure 14, the mat-based WPT system enables magnetically coupled resonance between an array of transmitter coils and a single receiver coil.

move freely on the floor above the mat.

Recent Wireless Power Transfer Technologies

Figure 13.

Figure 14.

80

Mat-based WPT system design.

its packing density is not the highest.

3.1 Theoretical analysis of mat-based WPT system

An adjustable RF oscillator produces a sinusoidal signal, which is amplified by a power amplifier. The output of the amplifier is connected to an array of driver coils which are inductively coupled with primary coils [23]. At the power receiving site (a laboratory animal), the receiver coil is inductively coupled with a load coil to supply the power to the electronic components within the implant.

In the coupled mode theory (CMT), the first eigen-mode is used to analyze a resonant system. The approximation by the first eigen-mode is quite accurate under the condition of a strong coupling in the WPT system. In practical applications, it is not possible or necessary to directly work on an arbitrary large number of resonators. Rather, for a large hexagonally packed transmitter (HPT) mat (Figure 15), every one of the resonators can be treated as in the middle of the mat, except for those ones at the edge, and fortunately the edge effect can be solved simply by making the mat larger than the animal cage floor. For this reason, we may simplify analysis by examining the seven-resonator case.

Let us index the seven transmitters from 1 to 7 and the single receiver have an index of 8. In order to describe the strongly coupled system, a set of differential equations based on CMT is given by [22].

$$\begin{aligned} \dot{a}\_8(t) &= (j a\_0 - \Gamma\_8 - \Gamma\_L) a\_8(t) + \sum\_{j=1}^7 j \kappa\_{ij} a\_j(t) \\\\ \dot{a}\_i(t) &= (j a\_0 - \Gamma\_i) a\_i(t) + \sum\_{j=1}^8 j \kappa\_{ij} a\_j(t) + f\_i(t), \quad i = 1, \dots, 7 \end{aligned} \tag{11}$$

#### Or, if written in matrix form

$$
\begin{bmatrix}
\dot{a}\_{1}(t) \\
\dot{a}\_{2}(t) \\
\vdots \\
\dot{a}\_{7}(t) \\
\dot{a}\_{8}(t)
\end{bmatrix} = \begin{bmatrix}
(j\omega\_{0} - \Gamma\_{1}) & j\kappa\_{22} & \cdots & j\kappa\_{7} & j\kappa\_{8} \\
j\kappa\_{21} & (j\omega\_{0} - \Gamma\_{2}) & \cdots & j\kappa\_{7} & j\kappa\_{8} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
j\kappa\_{71} & j\kappa\_{72} & \cdots & (j\omega\_{0} - \Gamma\_{7}) & j\kappa\_{8} \\
j\kappa\_{81} & j\kappa\_{82} & \cdots & j\kappa\_{87} & (j\omega\_{0} - \Gamma\_{8} - \Gamma\_{L})
\end{bmatrix} \begin{bmatrix}
a\_{1}(t) \\
a\_{2}(t) \\
\vdots \\
a\_{7}(t) \\
a\_{8}(t)
\end{bmatrix} + \begin{bmatrix}
f(t) \\
f(t) \\
\vdots \\
f(t) \\
f(t) \\
\vdots
\end{bmatrix} \tag{12}
$$

where aið Þt , i ¼ 1, 2, ⋯, 7, and a8ð Þt are, respectively, the first eigenmodes of the transmitter and receiver resonators corresponding to the natural frequency ω0, Γis are the intrinsic loss rates of resonators due to absorption and radiation, Γ<sup>L</sup> represents the rate of energy going into the load, κs are pairwise coupling coefficients between resonators, and fi s are the inputs to the transmitter resonators. In our case, all fi s are the same, i.e., f <sup>1</sup> ¼ f <sup>2</sup> ¼ ⋯ ¼ f <sup>7</sup> ¼ f. Note that ais are also known as positive frequency components in terms of CMT. Although ai (generally complexvalued) does not represent a voltage or current directly, the energy contained in each resonator can be represented as ai j j<sup>2</sup> , and the power output of the system is 2ΓLj j a<sup>8</sup> 2 . Using the CMT concept, the goal of obtaining a uniform power output becomes finding a constantj j a<sup>8</sup> within the WPT space.

To make Eq. (12) more concise, we write it into the following form:

$$
\dot{\mathbf{a}} = \mathbf{A}\mathbf{a} + \mathbf{f} \tag{13}
$$

results of multiple cells can be obtained simply by superposition of single cell results. In most cases, changes in the position of a device lead to a variation in mutual inductance which results from a change in the magnetic field distribution. Although some unevenness in the distribution is unavoidable, we expect this distribution to be nearly uniform with enhanced misalignment tolerability for WPT

We utilize the concentric model to approximate the coil where the total magnetic field is a superposition of the fields of individual loops in the coil. Assuming that a loop with a radius of a is centered at the origin carrying a current I. Based on the Biot-Savart law, the x-, y- and z-components of the magnetic field at point

> <sup>2</sup> E k<sup>2</sup> � <sup>α</sup><sup>2</sup> K k<sup>2</sup>

<sup>2</sup> E k<sup>2</sup> � <sup>α</sup><sup>2</sup>

where <sup>ρ</sup><sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup>2, <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>z</sup>2, <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> � <sup>2</sup>aρ, <sup>β</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>aρ,

With these simplifications, we can calculate the magnetic field of the seventransmitter powering platform. Figure 16 shows a magnetic field with different distances between the transmitter and the surface (evaluating plane) where the field was evaluated. It can be seen that the separation between the evaluating plane and the transmitter enables the variation of the z-component of magnetic field. When the distance is increasing, the flatness of the magnetic field also improves at first. However, when the height is too large, the magnetic field distribution becomes similar to that of a larger spiral coil, which affects both the flatness and magnitude of the magnetic field. On the other hand, when the height is too small, although the peak magnitude is larger, the z-component is inversed at the gap between resona-

In order to investigate the effect of mutual inductance between turns in the spiral coil, we also performed a simulation study on the 7-coil mat using a commercial finite element (FE) software HFSS (Ansys Corp., Pittsburgh, PA). Figure 17 shows the 3D model of the HPT mat used in the simulation, where each PSC was 20 cm in outer diameter, 1 cm in conductive trace width, and 1 cm in trace spacing. The input power was set at 1 W. As stated previously, the goal of the power mat design was to obtain a nearly uniform magnetic field within an extended region to support WPT for moving targets, rather than optimizing PTE (the animal cage is

We excited the seven PSCs simultaneously using a common RF power source. Energy was injected into the driver coil array to maintain resonance in the presence of losses and energy drawn from the magnetic field by the receiver coil. Figure 18 shows the z-component distribution of the magnetic field at 8 and 20 cm distances,

<sup>2</sup> E k<sup>2</sup> � <sup>α</sup><sup>2</sup> K k<sup>2</sup>

the first and second kinds, respectively. For easier calculation, without loss of generality, the current is chosen so that C ¼ 1. With the magnetic field of a single loop, we can apply the superposition rule to get the magnetic field generated by a multi-loop circular coil. Since the receiver coil is always in parallel with the transmitter coil (which is installed below the animal cage floor), the fluxes in the receiver coil are contributed by the z-component of the magnetic field. Therefore, we will

K k<sup>2</sup> <sup>¼</sup> <sup>y</sup>

, C ¼ u0I=π, and K(.) and E(.) are the complete elliptic integrals of

x Bx (15)

applications involving moving targets.

DOI: http://dx.doi.org/10.5772/intechopen.89120

Bx <sup>¼</sup> Cxz

By <sup>¼</sup> Cyz

Bz <sup>¼</sup> <sup>C</sup>

<sup>2</sup>α2βρ<sup>2</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup>

Wireless Power Transfer for Miniature Implantable Biomedical Devices

<sup>2</sup>α<sup>2</sup>βρ<sup>2</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup>

<sup>2</sup>α<sup>2</sup><sup>β</sup> <sup>a</sup><sup>2</sup> � <sup>r</sup>

focus on the z-component of the magnetic field in our analysis.

tors, which implies a large fluctuation of the magnetic field.

powered from a regular AC socket).

83

r (x, y, z) are given by [28].

<sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>α</sup><sup>2</sup>=β<sup>2</sup>

where the vectors and matrices are in correspondence with Eq. (12).

If the WPT system is driven by a sinusoidal input, e.g., <sup>f</sup>ðÞ¼ <sup>t</sup> Fej<sup>ω</sup>0<sup>t</sup> ½ � <sup>1</sup>, <sup>1</sup>, <sup>⋯</sup>1, <sup>0</sup> <sup>T</sup>, the positive frequency components have the form ofaðÞ¼ <sup>t</sup> <sup>a</sup>e<sup>j</sup>ω0<sup>t</sup> at the steady state. Substituting this into Eq. (13), we can solve for að Þt

$$\mathbf{a}(t) = -\mathbf{B}^{-1}\mathbf{f}(t) \tag{14}$$

where

$$\mathbf{B} = \begin{bmatrix} -\Gamma\_1 & j\kappa\_{12} & \cdots & j\kappa\_{17} & j\kappa\_{18} \\ j\kappa\_{21} & -\Gamma\_2 & \cdots & j\kappa\_{27} & j\kappa\_{28} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ j\kappa\_{71} & j\kappa\_{72} & \cdots & -\Gamma\_7 & j\kappa\_{78} \\ j\kappa\_{81} & j\kappa\_{82} & \cdots & j\kappa\_{87} & -\Gamma\_8 - \Gamma\_L \end{bmatrix}.$$

In case where B is not invertible, its pseudo inverse can be used instead. Thus, given Γi, κij, and fi , we can compute aið Þt analytically by Eq. (14). The CMT approach provides a powerful analytical tool for the multi-resonator WPT system. For example, it has been utilized to maximize the efficiency of power transfer and investigate the relay effect by inserting one or more resonators between the transmitter and receiver [26]. Using CMT, we have studied the dynamics of the system involving an array of resonators [27]. Although the previous studies have shown that CMT well characterizes the temporal behavior of the WPT system, it has clear limitations when the system parameter changes. For example, when the receiving resonator moves over the HPT mat, the coupling coefficients κ8<sup>i</sup>ð Þ i ¼ 1, 2, ⋯, 7 change, and the variations of the system behavior are difficult to determine analytically. In order to study the motion effect of the receiving resonator and answer the critical question whether the receiver resonator can harvest sufficient amount of power at different locations over the HPT mat, we performed numerical simulation and conducted an experimental test.

#### 3.2 Simulation of mat-based WPT system

For a clear illustration of the design principle of the mat-based WPT system, we simulated a single HPT cell consisting of seven PSCs, as limited by the computational complexity. This simulation does not cause a loss of generality because the

Wireless Power Transfer for Miniature Implantable Biomedical Devices DOI: http://dx.doi.org/10.5772/intechopen.89120

results of multiple cells can be obtained simply by superposition of single cell results. In most cases, changes in the position of a device lead to a variation in mutual inductance which results from a change in the magnetic field distribution. Although some unevenness in the distribution is unavoidable, we expect this distribution to be nearly uniform with enhanced misalignment tolerability for WPT applications involving moving targets.

We utilize the concentric model to approximate the coil where the total magnetic field is a superposition of the fields of individual loops in the coil. Assuming that a loop with a radius of a is centered at the origin carrying a current I. Based on the Biot-Savart law, the x-, y- and z-components of the magnetic field at point r (x, y, z) are given by [28].

$$\begin{aligned} B\_{\mathbf{x}} &= \frac{\mathbf{C}\mathbf{x}\mathbf{z}}{2\alpha^{2}\beta\rho^{2}} \left[ (a^{2} + r^{2})E(\mathbf{k}^{2}) - \alpha^{2}K(\mathbf{k}^{2}) \right] \\ B\_{\mathbf{y}} &= \frac{\mathbf{C}\mathbf{y}\mathbf{z}}{2\alpha^{2}\beta\rho^{2}} \left[ (a^{2} + r^{2})E(\mathbf{k}^{2}) - \alpha^{2}K(\mathbf{k}^{2}) \right] = \frac{\mathbf{y}}{\mathbf{x}}B\_{\mathbf{x}} \\ B\_{\mathbf{z}} &= \frac{\mathbf{C}}{2\alpha^{2}\beta} \left[ (a^{2} - r^{2})E(\mathbf{k}^{2}) - \alpha^{2}K(\mathbf{k}^{2}) \right] \end{aligned} \tag{15}$$

where <sup>ρ</sup><sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup>2, <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>z</sup>2, <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> � <sup>2</sup>aρ, <sup>β</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>aρ, <sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>α</sup><sup>2</sup>=β<sup>2</sup> , C ¼ u0I=π, and K(.) and E(.) are the complete elliptic integrals of the first and second kinds, respectively. For easier calculation, without loss of generality, the current is chosen so that C ¼ 1. With the magnetic field of a single loop, we can apply the superposition rule to get the magnetic field generated by a multi-loop circular coil. Since the receiver coil is always in parallel with the transmitter coil (which is installed below the animal cage floor), the fluxes in the receiver coil are contributed by the z-component of the magnetic field. Therefore, we will focus on the z-component of the magnetic field in our analysis.

With these simplifications, we can calculate the magnetic field of the seventransmitter powering platform. Figure 16 shows a magnetic field with different distances between the transmitter and the surface (evaluating plane) where the field was evaluated. It can be seen that the separation between the evaluating plane and the transmitter enables the variation of the z-component of magnetic field. When the distance is increasing, the flatness of the magnetic field also improves at first. However, when the height is too large, the magnetic field distribution becomes similar to that of a larger spiral coil, which affects both the flatness and magnitude of the magnetic field. On the other hand, when the height is too small, although the peak magnitude is larger, the z-component is inversed at the gap between resonators, which implies a large fluctuation of the magnetic field.

In order to investigate the effect of mutual inductance between turns in the spiral coil, we also performed a simulation study on the 7-coil mat using a commercial finite element (FE) software HFSS (Ansys Corp., Pittsburgh, PA). Figure 17 shows the 3D model of the HPT mat used in the simulation, where each PSC was 20 cm in outer diameter, 1 cm in conductive trace width, and 1 cm in trace spacing. The input power was set at 1 W. As stated previously, the goal of the power mat design was to obtain a nearly uniform magnetic field within an extended region to support WPT for moving targets, rather than optimizing PTE (the animal cage is powered from a regular AC socket).

We excited the seven PSCs simultaneously using a common RF power source. Energy was injected into the driver coil array to maintain resonance in the presence of losses and energy drawn from the magnetic field by the receiver coil. Figure 18 shows the z-component distribution of the magnetic field at 8 and 20 cm distances,

where aið Þt , i ¼ 1, 2, ⋯, 7, and a8ð Þt are, respectively, the first eigenmodes of the transmitter and receiver resonators corresponding to the natural frequency ω0, Γis are the intrinsic loss rates of resonators due to absorption and radiation, Γ<sup>L</sup> represents the rate of energy going into the load, κs are pairwise coupling coefficients

s are the same, i.e., f <sup>1</sup> ¼ f <sup>2</sup> ¼ ⋯ ¼ f <sup>7</sup> ¼ f. Note that ais are also known as positive frequency components in terms of CMT. Although ai (generally complexvalued) does not represent a voltage or current directly, the energy contained in

To make Eq. (12) more concise, we write it into the following form:

where the vectors and matrices are in correspondence with Eq. (12). If the WPT system is driven by a sinusoidal input, e.g., <sup>f</sup>ðÞ¼ <sup>t</sup> Fej<sup>ω</sup>0<sup>t</sup>

the positive frequency components have the form ofaðÞ¼ <sup>t</sup> <sup>a</sup>e<sup>j</sup>ω0<sup>t</sup> at the steady state.

�Γ<sup>1</sup> jκ<sup>12</sup> ⋯ jκ<sup>17</sup> jκ<sup>18</sup> jκ<sup>21</sup> �Γ<sup>2</sup> ⋯ jκ<sup>27</sup> jκ<sup>28</sup> ⋮ ⋮⋱⋮ ⋮ jκ<sup>71</sup> jκ<sup>72</sup> ⋯ �Γ<sup>7</sup> jκ<sup>78</sup> jκ<sup>81</sup> jκ<sup>82</sup> ⋯ jκ<sup>87</sup> �Γ<sup>8</sup> � Γ<sup>L</sup>

In case where B is not invertible, its pseudo inverse can be used instead. Thus,

For a clear illustration of the design principle of the mat-based WPT system, we simulated a single HPT cell consisting of seven PSCs, as limited by the computational complexity. This simulation does not cause a loss of generality because the

approach provides a powerful analytical tool for the multi-resonator WPT system. For example, it has been utilized to maximize the efficiency of power transfer and investigate the relay effect by inserting one or more resonators between the transmitter and receiver [26]. Using CMT, we have studied the dynamics of the system involving an array of resonators [27]. Although the previous studies have shown that CMT well characterizes the temporal behavior of the WPT system, it has clear limitations when the system parameter changes. For example, when the receiving resonator moves over the HPT mat, the coupling coefficients κ8<sup>i</sup>ð Þ i ¼ 1, 2, ⋯, 7 change, and the variations of the system behavior are difficult to determine analytically. In order to study the motion effect of the receiving resonator and answer the critical question whether the receiver resonator can harvest sufficient amount of power at different locations over the HPT mat, we performed numerical simulation

, we can compute aið Þt analytically by Eq. (14). The CMT

<sup>a</sup>ðÞ¼� <sup>t</sup> <sup>B</sup>�<sup>1</sup>

. Using the CMT concept, the goal of obtaining a uniform power output

s are the inputs to the transmitter resonators. In our case,

, and the power output of the system is

a\_ ¼ Aa þ f (13)

fð Þt (14)

½ � <sup>1</sup>, <sup>1</sup>, <sup>⋯</sup>1, <sup>0</sup> <sup>T</sup>,

between resonators, and fi

each resonator can be represented as ai j j<sup>2</sup>

Recent Wireless Power Transfer Technologies

becomes finding a constantj j a<sup>8</sup> within the WPT space.

Substituting this into Eq. (13), we can solve for að Þt

B ¼

and conducted an experimental test.

3.2 Simulation of mat-based WPT system

all fi

2ΓLj j a<sup>8</sup> 2

where

given Γi, κij, and fi

82

respectively, above the power mat (i.e., the X-Y plane). Color indicates the magnitude of the magnetic field in the z-direction. It can be seen that, at z = 8 cm (Figure 18a), the magnitude of the magnetic field was the highest (peak) at the center of each coil, and the lowest (valley) at the junction of three coils. When the distance to the HPT mat increased to 20 cm, a smoother magnetic field distribution was observed, but approaching to the field generated by a large spiral coil (Figure 18b). In order to evaluate the evenness of distribution quantitatively, the coefficient of variation (COV) was utilized which was defined as the standard deviation of the field values divided by the mean. Thus, a smaller value of the COV indicates a more uniform distribution. Figure 19 shows the COVs of the magnetic field in the z-direction above the HPT mat at distances from 5 to 40 cm. It can be observed that the COV achieves a value <10% when the distance is larger than the size of the transmitter coil.

Figure 17.

Figure 18.

Figure 19.

85

mat at the resonant frequency of 85.2 MHz.

at the resonant frequency of 85.2 MHz.

HFSS simulation. (a) 3D model of the transmitter mat; (b) Dimensions of each PSC.

Wireless Power Transfer for Miniature Implantable Biomedical Devices

DOI: http://dx.doi.org/10.5772/intechopen.89120

Distribution of the z-component of the magnetic field in a plane at (a) 8 cm and (b) 20 cm above the HPT mat

Variation in coefficient of variation (COV) of vertical field distribution as a function of distance above the HPT
