*2.1. Shielding Structure*

To overcome the negative impact of the addition of an active shielding coil on transmission efficiency, an improved double-coil active shielding structure is used in this paper with two half-loops instead of a single active shielding coil—and the structure diagram is presented in Figure 1.

**2. Double-Coil Active Shielding Technology**

*2.1. Shielding Structure*

gram is presented in Figure 1.

**Figure 1.** Double-coil active shielding structure. **Figure 1.** Double-coil active shielding structure.

It is apparent from Figure 1 that each half-loop is equivalent to half of a circle, with its inner radius denoted by *r*in and outer radius denoted by *r*out. Two half-loops are installed around the transmitting coil, and the radius of the transmitting coil is *r*s; thus, it is obvious that *r*in > *r*s. To facilitate the discussion at a later stage, the regions in Figure 1 are divided as follows: inside the transmitting coil, i.e., *r* < *r*s, is noted as region 1; excluding It is apparent from Figure 1 that each half-loop is equivalent to half of a circle, with its inner radius denoted by *r*in and outer radius denoted by *r*out. Two half-loops are installed around the transmitting coil, and the radius of the transmitting coil is *r*s; thus, it is obvious that *r*in > *r*s. To facilitate the discussion at a later stage, the regions in Figure 1 are divided as follows: inside the transmitting coil, i.e., *r* < *r*s, is noted as region 1; excluding the transmitting coil part, the region inside the active shielding coil, i.e., *r*in < *r* < *r*out, is denoted as region 2; outside the active shielding coil, i.e., *r* > *r*out, is noted as region 3.

To overcome the negative impact of the addition of an active shielding coil on trans-

mission efficiency, an improved double-coil active shielding structure is used in this paper—with two half-loops instead of a single active shielding coil—and the structure dia-

the transmitting coil part, the region inside the active shielding coil, i.e., *r*in < *r* < *r*out, is denoted as region 2; outside the active shielding coil, i.e., *r* > *r*out, is noted as region 3. Observing the direction of shielding coil currents in Figure 1, it can be seen that in region 3, the EMF generated by the two half-loops is in the opposite direction of the EMF generated by the transmitting coil, which creates a weakening effect on the total EMF in region 3. Moreover, in region 1, the EMF generated by the two shielding coils is consistent Observing the direction of shielding coil currents in Figure 1, it can be seen that in region 3, the EMF generated by the two half-loops is in the opposite direction of the EMF generated by the transmitting coil, which creates a weakening effect on the total EMF in region 3. Moreover, in region 1, the EMF generated by the two shielding coils is consistent with the direction of the EMF of the transmitting coil. Compared with the traditional single shielding coil, the negative impact of the two half-loops on the performance of WPT system is reduced. It achieves a dual effect, reducing the EMF leakage of the WPT system while increasing the transmission efficiency.

#### with the direction of the EMF of the transmitting coil. Compared with the traditional sin-*2.2. Magnetic Field Calculation*

gle shielding coil, the negative impact of the two half-loops on the performance of WPT system is reduced. It achieves a dual effect, reducing the EMF leakage of the WPT system while increasing the transmission efficiency. *2.2. Magnetic Field Calculation* In this work, the configuration described in Figure 1 is considered. Since the currents flowing in the radial direction of the two half-loops are opposite (as shown in Figure 1), the EMFs generated by these two current segments can be ignored. Consequently, in the process of calculating EMF, the structure shown in Figure 1 can be simplified. The simplified model is illustrated in Figure 2: neglecting the radial current segments, the two half-loops are replaced by two concentric circular coils with the same current but opposite phase. The inner radius and outer radius of the two concentric circular coils are consistent with Figure 1.

In this work, the configuration described in Figure 1 is considered. Since the currents flowing in the radial direction of the two half-loops are opposite (as shown in Figure 1), the EMFs generated by these two current segments can be ignored. Consequently, in the process of calculating EMF, the structure shown in Figure 1 can be simplified. The simpli-In Figure 2, the geometric model of the distance from the center of coil to point M is demonstrated, where the magnetic flux density at point M is a function of the coil radius, the current flowing through the coil, and the distance from the coil center to point M. The total magnetic flux density at point M is calculated by the following Equation (1) [29]:

$$\begin{split} B\_{\rm M} &\approx \sum\_{N=1}^{n} j\mu\_{o} \frac{kr\_{N}^{2}I\cos\theta\_{M}}{2d\_{M}^{2}} \bigg[1 + \frac{1}{jkd\_{M}}\Big] e^{-jkd\_{M}\cdot\theta\_{N}} \\ &- \sum\_{N=1}^{n} \mu\_{o} \frac{(kr\_{N})^{2}I\sin\theta\_{M}}{4d\_{M}} \bigg[1 + \frac{1}{jkd\_{M}} - \frac{1}{(kr\_{N})^{2}}\Big] e^{-jkd\_{M}} \cdot \theta\_{N} \end{split} \tag{1}$$

2

2

*<sup>N</sup> M M*

*kr I B j e*

total magnetic flux density at point M is calculated by the following Equation (1) [29]:

cos 1 1

+

*d jkd*

sin 1 1 <sup>1</sup>

*M M <sup>N</sup>*

*d jkd kr*

*N M jkd*

− + −

*N M jkd*

( )

2

*M*

*N*

 

(1)

*M*

*N*

−

*e*

−

In Figure 2, the geometric model of the distance from the center of coil to point M is demonstrated, where the magnetic flux density at point M is a function of the coil radius, the current flowing through the coil, and the distance from the coil center to point M. The where, *d<sup>M</sup>* = p *x* <sup>2</sup> + *y* <sup>2</sup> + *z* 2 , cos *<sup>ϕ</sup><sup>M</sup>* = √ *<sup>x</sup> x* <sup>2</sup>+*y* 2 , cos *<sup>θ</sup><sup>M</sup>* = √ *<sup>z</sup> x* <sup>2</sup>+*y* <sup>2</sup>+*z* 2 , sin *θ<sup>M</sup>* = √ *y x* <sup>2</sup>+*y* <sup>2</sup>+*z* 2 , *ϑ<sup>N</sup>* = sin *θ<sup>M</sup>* cos *ϕ<sup>M</sup>* cos *θM*.

( )

1

=

*o*

*n*

*N*

M 2 1

*o*

*kr I*

*n*

=

2

4

where,

sin *<sup>M</sup>*

*x*

2

=

2 2 2 *<sup>d</sup><sup>M</sup>* <sup>=</sup> *x y z* + + ,

2 2 2

+ +

*y*

*x y z*

**Figure 2.** Geometric model of the distance from the center of the coil to point M. **Figure 2.** Geometric model of the distance from the center of the coil to point M.

M(*x*,*y*,*z*). Therefore, Equation (1) can be simplified as (3):

Note that *μ*<sup>0</sup> is the vacuum permeability, *k* is the wave number (2*π*/*λ*), *λ* is the wavelength (*c*/*f*), *c* is the speed of light (3 × 10<sup>8</sup> m/s), *f* is the frequency, *r<sup>N</sup>* (*N* = 1, 2, ..., *n*) is the Note that *µ*<sup>0</sup> is the vacuum permeability, *k* is the wave number (2*π*/*λ*), *λ* is the wavelength (*c*/*f*), *<sup>c</sup>* is the speed of light (3 <sup>×</sup> <sup>10</sup><sup>8</sup> m/s), *<sup>f</sup>* is the frequency, *<sup>r</sup><sup>N</sup>* (*<sup>N</sup>* = 1, 2, . . . , *<sup>n</sup>*) is the coil radius, and *N* is the coil turn.

coil radius, and *N* is the coil turn. For the convenience of representation, the variable *g* is introduced here:

$$\begin{aligned} g &= \sum\_{N=1}^{n} j\mu\_o \frac{kr\_N^2 \cos\theta\_M}{2d\_M^2} \left[ 1 + \frac{1}{jkd\_M} \right] e^{-jkd\_M \cdot \theta\_N} \\ &- \sum\_{N=1}^{n} \mu\_o \frac{(kr\_N)^2 \sin\theta\_M}{4d\_M} \left[ 1 + \frac{1}{jkd\_M} - \frac{1}{(kr\_N)^2} \right] e^{-jkd\_M} \cdot \theta\_N \end{aligned} \tag{2}$$
  $\text{It is abundantly clear that g is a function that varies with the position of point M(x, y, z).}$ 

( ) 2 1 4 *M N o <sup>N</sup> M M <sup>N</sup> e d jkd kr* = It is abundantly clear that g is a function that varies with the position of point M(*x*,*y*,*z*). Therefore, Equation (1) can be simplified as (3):

$$\mathcal{B}\_M \approx \mathcal{g}(\mathbf{x}, \mathbf{y}, \mathbf{z}) \mathbf{I} \tag{3}$$

2 2

,

cos *<sup>M</sup>*

=

2 2 2

+ +

,

(2)

(3)

(4)

*z*

*x y z*

*x*

*x y*

+

cos *<sup>M</sup>*

 

*<sup>N</sup> M M M* = .

 

, sin cos cos

 

=

*B<sup>M</sup> g x y z I* ( , , ) The geometric coil array of the transmitting, receiving, and shielding coils is depicted in Figure 3. It is apparent that the transmitting, receiving, and shielding coils are all coaxial The geometric coil array of the transmitting, receiving, and shielding coils is depicted in Figure 3. It is apparent that the transmitting, receiving, and shielding coils are all coaxial coils. Thus, all coils have an identical distance for the *x*-axis and *y*-axis to point M. The distance of the *x-y* axis from point M in the Cartesian coordinate system can be calculated as √ (*x* <sup>2</sup> + *y* 2 ).

coils. Thus, all coils have an identical distance for the *x*-axis and *y*-axis to point M. The distance of the *x-y* axis from point M in the Cartesian coordinate system can be calculated as √(*x* <sup>2</sup>+ *y* ). However, in the *z*-axis direction, the distances vary from coil to coil. Therefore, the distance to point M will differ with regard to the *z*-axis. The distance from the transmitting coil to the point M *dS*<sup>1</sup> is then expressed by Equation (4).

$$d\_{S1} = \sqrt{\left(\sqrt{x^2 + y^2}\right)^2 + (z\_1 + z\_2)^2} \tag{4}$$

coil to the point M *dS*<sup>1</sup> is then expressed by Equation (4). ( ) 2 2 2 2 1 1 2 ( ) *<sup>S</sup> d y z z* = + + + *<sup>x</sup>* Since the receiving coil and shielding coil are in the same plane, the distance from the receiving coil to point M has the same expression as the distance from the shielding coil to point M, as shown in Equation (5).

$$d\_{S2} = d\_{SH\\_in} = d\_{SH\\_out} = \sqrt{\left(\sqrt{x^2 + y^2}\right)^2 + z\_2^2} \tag{5}$$

to point M, as shown in Equation (5).

**Figure 3.** Geometric coil array of transmitting, receiving, and shielding coils. **Figure 3.** Geometric coil array of transmitting, receiving, and shielding coils.

The total magnetic flux density *B*<sup>M</sup> at point M of the space is calculated by applying where *dS*<sup>2</sup> is the distance from the receiving coil to point M; *dSH\_in* and *dSH\_out* are, respectively, the distance from the internal and external shielding coils to point M.

superposition as: The total magnetic flux density *B*<sup>M</sup> at point M of the space is calculated by applying superposition as:

$$B\_{\mathbf{M}} = B\_1 + B\_2 + B\_{\mathbf{3}\dots\mathbf{i}n} + B\_{\mathbf{3}\dots\mathbf{out}} \tag{6}$$

$$\mathbf{J} = \begin{array}{ccccccccc} & \mathbf{J} & \mathbf{0} & \mathbf{J} & \mathbf{0} \\ & \ddots & \mathbf{0} & \mathbf{J} & \cdots & \mathbf{J} \\ \end{array} \tag{7}$$

( )

*<sup>S</sup>* <sup>2</sup> *SH in SH out* \_ \_ <sup>2</sup> *d d d y z* = + + <sup>=</sup> <sup>=</sup> *<sup>x</sup>*

where *dS*<sup>2</sup> is the distance from the receiving coil to point M; *dSH\_in* and *dSH\_out* are, respec-

tively, the distance from the internal and external shielding coils to point M.

2 2 2 2

(5)

(6)

(7)

(8)

(9)

where *B*1, *B*2, *B*3\_*in*, and *B*3\_*out* are the magnetic flux density of the transmitting coil, receiving coil, and the internal and external shielding coils at point M, respectively, and these are where *B*1, *B*2, *B*3\_*in*, and *B*3\_*out* are the magnetic flux density of the transmitting coil, receiving coil, and the internal and external shielding coils at point M, respectively, and these are calculated corresponding to the coil design through (1). Moreover, the shielding effectiveness (*SE*) is defined as follows:

$$SE = \left(1 - \frac{B\_{\rm M}}{B\_1 + B\_2}\right) \times 100\% \tag{7}$$

#### M 1 2 1 100% *<sup>B</sup> SE B B* = − <sup>+</sup> **3. Dynamic Shielding for WPT Systems with Double-Coil Active Shielding** *3.1. Mathematical Analysis*

**3. Dynamic Shielding for WPT Systems with Double-Coil Active Shielding**  *3.1. Mathematical Analysis* The theory of double-coil active coil shielding is here addressed using a circuit approach. A WPT system with a series–series (SS) compensation topology [33] is modeled using an equivalent circuit as shown in Figure 4, with *R*1, *R*2, and *R*<sup>3</sup> as the internal re-The theory of double-coil active coil shielding is here addressed using a circuit approach. A WPT system with a series–series (SS) compensation topology [33] is modeled using an equivalent circuit as shown in Figure 4, with *R*1, *R*2, and *R*<sup>3</sup> as the internal resistances; *L*1, *L*2, and *L*<sup>3</sup> as self-inductances; *C*1, *C*2, and *C*<sup>3</sup> as capacitances, and *R*<sup>L</sup> and *R*<sup>S</sup> as the resistive load and internal resistance of power transmitting. *M*12, *M*13, and *M*<sup>23</sup> are the mutual inductances between coils, and subscript '1', '2', '3' indicates the transmitting, receiving, and shielding coil, respectively. *V*<sup>S</sup> and *V*<sup>A</sup> are the power supply of the transmitting coil and the shielding coil, and *I*1, *I*2, and *I*<sup>3</sup> are the current of the transmitting, receiving, and shielding coil, respectively. *ω* is the angular operating frequency.

sistances; *L*1, *L*2, and *L*<sup>3</sup> as self-inductances; *C*1, *C*2, and *C*<sup>3</sup> as capacitances, and *R*<sup>L</sup> and *R*<sup>S</sup> as The equivalent circuit in Figure 4 can be calculated as follows:

$$
\begin{bmatrix} Z\_1 + \mathcal{R}\_S & Z\_{12} & Z\_{13} \\ Z\_{12} & Z\_2 + \mathcal{R}\_L & Z\_{23} \\ Z\_{13} & Z\_{23} & Z\_3 \end{bmatrix} \begin{bmatrix} I\_1 \\ I\_2 \\ I\_3 \end{bmatrix} = \begin{bmatrix} V\_S \\ 0 \\ V\_A \end{bmatrix} \tag{8}
$$

$$Z\_{i} = R\_{i} + j\omega L\_{i} + \frac{1}{j\omega \mathbb{C}\_{i}}, i \in \{1, 2, 3\} \tag{9}$$

$$Z\_{1} = \begin{array}{c} \vdots \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \ \vdots \end{array} \tag{9}$$

1 12 13 1

*Z Z R Z I*

*Z R j L i*

*S*

12 2 23 2 13 23 3 3

*L*

*Z Z Z I V*

= + +

0

*A*

*S*

$$Z\_{i\bar{j}} = \mathbf{j}\omega M\_{i\bar{j}}, i \neq \mathbf{j}, i, j \in \{1, 2, 3\} \tag{10}$$

*Z R Z Z I V*

 <sup>+</sup> + =

1 , {1,2,3} *<sup>i</sup> i i*

*j C*

*i*

injection to compensate for the EMF leakage.

**Figure 4.** Equivalent circuit of WPT systems with double-coil active shielding. **Figure 4.** Equivalent circuit of WPT systems with double-coil active shielding.

Considering, for simplicity, a three-coil configuration (1—transmitting coil, 2—re-It is worth mentioning that *C*<sup>3</sup> can be neglected since there is enough active current injection to compensate for the EMF leakage.

ceiving coil, 3—active shielding coil), the total magnetic flux density at observation point M(*x*,*y*,*z*) in (6) can be simplified as: Considering, for simplicity, a three-coil configuration (1—transmitting coil, 2—receiving coil, 3—active shielding coil), the total magnetic flux density at observation point M(*x*,*y*,*z*) in (6) can be simplified as:

$$B\_{\mathbf{M}}\left(\mathbf{x},\mathbf{y},\mathbf{z}\right) = \sum\_{k=1}^{3} B\_{k}(\mathbf{x},\mathbf{y},\mathbf{z})\tag{11}$$

, , , {1, 2,3} *ij ij Z j M i j i j* =

It is worth mentioning that *C*<sup>3</sup> can be neglected since there is enough active current

(10)

(11)

(12)

(13)

(14)

1 *k* = where *B*<sup>3</sup> is equal to the sum of *B*3\_in and *B*3\_out. Substituting Equation (3) into (11), there is:

$$B\_{\mathbf{M}}\left(\mathbf{x},\mathbf{y},\mathbf{z}\right) = \sum\_{k=1}^{3} g\_{k}(\mathbf{x},\mathbf{y},\mathbf{z}) \mathbf{l}\_{k} \tag{12}$$

M 1 ( , , ) ( , , ) *k k k <sup>B</sup> x y z g x y z I* = <sup>=</sup> It should be noted that the vectorial functions *g<sup>k</sup>* depend on the configuration of all It should be noted that the vectorial functions g<sup>k</sup> depend on the configuration of all coils, i.e., the transmitting, receiving, and active shielding coils. Therefore, the feeding of the active shielding coil depends not only on the EMF generated by the transmitting coil, but also on the EMF generated by the receiving coil.

coils, i.e., the transmitting, receiving, and active shielding coils. Therefore, the feeding of the active shielding coil depends not only on the EMF generated by the transmitting coil, Assuming that the shielding coil achieves an ideal SE, i.e., the total magnetic flux density B<sup>M</sup> measured at point M is zero, it yields:

$$B\_3(\mathbf{x}, y, z) = -\left(B\_1(\mathbf{x}, y, z) + B\_2(\mathbf{x}, y, z)\right) \tag{13}$$

Assuming that the shielding coil achieves an ideal *SE*, i.e., the total magnetic flux density *B*<sup>M</sup> measured at point M is zero, it yields: *B x y z B x y z B x y z* 3 1 2 ( , , ) ( , , ) ( , , ) = − + ( ) The above Equation (13) can only be satisfied if point M lies on the surface of the plane loop with its normal axis unit vector n<sup>a</sup> parallel to the direction of the sum of the incident fields:

$$n\_a \| (\mathbf{g}\_1(\mathbf{x}, \mathbf{y}, z)I\_1 + \mathbf{g}\_2(\mathbf{x}, \mathbf{y}, z)I\_2) \tag{14}$$

The above Equation (13) can only be satisfied if point M lies on the surface of the plane loop with its normal axis unit vector *n<sup>a</sup>* parallel to the direction of the sum of the However, in fact, for a given active coil structure, the shielding area cannot contain only one single point. Therefore, vector condition (14) cannot be completely satisfied in practice. Thus, a less restrictive but practical condition is introduced in the following.

incident fields: *n x y z I g x y z I <sup>a</sup>*‖( *<sup>g</sup>*1 1 2 2 ( , , ) ( , , ) <sup>+</sup> ) Considering the diminishing of the main component of magnetic flux density under the condition that the active coil is properly scheduled and is planar, being parallel to the *na*, the compensation for the *B*<sup>M</sup> component in the direction *n<sup>a</sup>* at a given point M is given by:

$$\begin{aligned} n\_d \cdot \mathbb{B}\_M(\mathbf{x}, y, z) &= \\ n\_d \cdot \left[ \begin{array}{cc} \mathbf{g}\_1(\mathbf{x}, y, z) & \mathbf{g}\_2(\mathbf{x}, y, z) & \mathbf{g}\_3(\mathbf{x}, y, z) \end{array} \right] \cdot \begin{bmatrix} I\_1 \\ I\_2 \\ I\_3 \end{bmatrix} &= \mathbf{0} \end{aligned} \tag{15}$$

Considering the diminishing of the main component of magnetic flux density under

the condition that the active coil is properly scheduled and is planar, being parallel to the *na*, the compensation for the *B*<sup>M</sup> component in the direction *n<sup>a</sup>* at a given point M is given

by:

Expressing the current vector in (15) into (9), it yields:

$$
\begin{array}{cc}
 n\_d \cdot \left[ \begin{array}{c} g\_1(\mathbf{x}, y, z) \\ \end{array} \begin{array}{c} g\_2(\mathbf{x}, y, z) \\ \end{array} \begin{array}{c} g\_3(\mathbf{x}, y, z) \\ \end{array} \begin{array}{c} g\_3(\mathbf{x}, y, z) \\ \end{array} \right] \cdot \\
 \begin{bmatrix} Z\_1 + R\_S & Z\_{12} & Z\_{13} \\ Z\_{12} & Z\_2 + R\_L & Z\_{23} \\ Z\_{13} & Z\_{23} & Z\_3 \\ \end{array} \end{array} \begin{array}{c} \begin{array}{c} \\ \end{array} \begin{array}{c} V\_S \\ 0 \\ V\_A \\ \end{array} \end{array} \tag{16}
$$

To make the expression more concise, a new variable t is introduced:

$$
\begin{aligned}
\begin{bmatrix} t\_1 \\ t\_2 \\ t\_3 \end{bmatrix} &= n\_d \cdot \begin{bmatrix} \ g\_1(\mathbf{x}, \mathbf{y}, z) & g\_2(\mathbf{x}, \mathbf{y}, z) & g\_3(\mathbf{x}, \mathbf{y}, z) \end{bmatrix} \cdot \\
\begin{bmatrix} Z\_1 + R\_S & Z\_{12} & Z\_{13} \\ Z\_{12} & Z\_2 + R\_L & Z\_{23} \\ Z\_{13} & Z\_{23} & Z\_3 \end{bmatrix}^{-1} \end{aligned} \tag{17}
$$

Thus, Equation (16) is transformed into:

$$t\_1 V\_S + t\_3 V\_A = 0\tag{18}$$

From this, the expression for the power supply of the active shielding coil is further derived:

$$V\_A = -V\_\mathcal{S}(t\_1/t\_\Im)\tag{19}$$

With consideration of the losses incurred by the presence of double active shielding coils, the power transfer efficiency *η* of the system can be calculated as:

$$\eta = \frac{P\_2}{P\_1 + P\_3} \tag{20}$$

$$\begin{cases} P\_1 = V\_1 I\_1 \\ \quad P\_2 = \frac{V\_2^2}{R\_L} \\ \quad P\_3 = V\_A I\_3 \end{cases} \tag{21}$$

where *P*<sup>1</sup> and *P*<sup>3</sup> are the output power of the transmitting and active shielding coils, and *P*<sup>2</sup> is the transferred power to load.

Variations in the position or current of the transmitting and receiving coils result in a corresponding change in the magnetic flux density of the WPT system. The dynamic shielding scheme proposed in this paper is to adjust the power supply *V*<sup>A</sup> in the active shielding coil according to the changes in coil position and current, allowing the excitation of the shielding coil to adapt to the changes in the EMF leakage of the WPT system.
