### I. Introduction

### II. Theoretical Analysis

### 1. Transmission Performance of WPT

*LC*resonant circuit, the frequency

*f*of voltage and current are not affected by the ideal resonant frequency

*f*

_{source}of the source. The difference in frequency Δ

*f*= |

*f*

_{0}−

*f*

_{source}| between the

*LC*and the source only affects the equivalent impedance

*X*

_{LC}= 2

*πf*

_{source}

*L*− 1/2

*πf*

_{source}

*C*. In WPT,

*LC*resonance, which is composed of coil inductance and compensation capacitance, exhibits the same characteristics. The simplified equivalent circuit for WPT using a series-series (SS) compensation topology is shown in Fig. 1. Notably,

*R*

_{t}=

*R*

_{S}+

*R*

_{l1}and

*R*

_{r}=

*R*

_{L}+

*R*

_{l2}refer to the total resistance of the primary and secondary sides,

*X*

_{t}=

*ωL*

_{1}− 1/

*ωC*

_{1}and

*X*

_{r}=

*ωL*

_{2}− 1/

*ωC*

_{2}denote the total reactance, and

*X*

_{M}=

*ωM*is the mutual inductance reactance. Notably, Kirchhoff’s voltage law proposes the following equation:

*İ*

_{t}and

*İ*

_{r}of the primary and second sides, respectively, can be formulated as follows:

*B*=

*R*

_{t}

*X*

_{r}+

*R*

_{r}

*X*

_{t}.

*P*

_{in}, output power

*P*

_{out}, and transmission efficiency

*η*can be expressed as follows:

*U*

_{s}and

*I*

_{r}are the modulus values of

*U̇*

_{S}and

*İ*

_{r}.

*X*

_{M}, the higher the transmission efficiency

*η*. In other words, an increasing mutual inductance

*M*or resonant frequency

*f*can improve

*η*. Furthermore, when output power

*P*

_{out}attains its maximum value,

*X*

_{M}must satisfy the following equation:

*M*that maximizes the output power is usually considered. The maximum output is

### 2. Maximum Output Power of the MCM

*L*

_{1}and

*L*

_{2}. To meet resonance conditions, it may be assumed that the compensating capacitances and inductances satisfy

*L*

_{1}

*C*

_{1}=

*L*

_{2}

*C*

_{2}. On determining the structural parameters of the transmitting coil and the resonance frequency

*f*, Eqs. (6) and (7) indicate that the maximum output power is restricted by the secondary impedance

*X*

_{r}and the mutual inductance reactance

*X*

_{M}.

*f*

_{source}of the source is equal to the ideal resonant frequency

*f*

_{0},

*X*

_{t}=

*X*

_{r}=0 and

*Z*

_{t}=

*R*

_{t},

*Z*

_{r}=

*R*

_{r}. Fig. 3 shows the impact of mutual inductance reactance

*X*

_{M}and receiver resistance

*R*

_{r}on the performance of equivalent circuit, where

*R*

_{r}is positively correlated and

*X*

_{M}is negatively correlated with input power

*P*

_{in}. Furthermore, when either

*R*

_{r}or

*X*

_{M}is kept constant, the output power

*P*

_{out}will initially increase and then decrease with an increase

*R*

_{r}or

*X*

_{M}. As for the transmission efficiency

*η*, as shown in Fig. 3(c), it increases continuously with an increase in

*X*

_{M}or a decrease in

*R*

_{r}, indicating an opposite trend as that of

*P*

_{in}.

*f*

_{source}of the source is not equal to the ideal resonant frequency

*f*

_{0},

*X*

_{t}/

*L*

_{1}=

*X*

_{r}/

*L*

_{2}≈ 0. Assuming

*f*

_{source}=

*f*

_{0}+ Δ

*f*, the receiver impedance

*Z*

_{r}cannot be considered purely resistive. Fig. 4 depicts the impact of mutual inductance reactance

*X*

_{M}and receiver resistance

*R*

_{r}on the performance of WPT under non-ideal resonant conditions. Notably, the variation trend of the output power

*P*

_{out}and transmission efficiency

*η*is the same as that shown in Fig. 3(b) and 3(c). However, the maximum output power

*P*

_{out·max}is reduced. This indicates that the power difference in

*P*

_{out·max}is related to the frequency difference Δ

*f*—the greater the Δ

*f*, the greater the reduction in

*P*

_{out·max}. Furthermore, the input power

*P*

_{in}in Fig. 4(a) is different from that in Fig. 3(a). This variation is the same as that of

*P*

_{out}under non-ideal resonant conditions—increasing initially to then decrease with an increase in

*X*

_{M}or

*R*

_{r}.

*X*

_{M}is observed, which achieves the maximum output power

*P*

_{out·max}for any receiver impedance

*Z*

_{r}, thus satisfying Eq. (6). Furthermore,

*X*

_{M}=

*ωM*is influenced by mutual inductance

*M*and angular frequency

*ω*. The primary and secondary impedances—

*Z*

_{t}=

*R*

_{t}+

*jωL*

_{t}+ 1/

*jωC*

_{t}and

*Z*

_{r}=

*R*

_{r}+

*jωL*

_{r}+ 1/

*jωC*

_{r}—are influenced by coil resistances

*R*

_{AC}, self-inductances

*L*, and angular frequency

*ω*. Moreover,

*M*,

*R*

_{AC}, and

*L*are found to be closely related to the physical parameters and the relative spatial position of the coils. Notably, during static WPT, the relative spatial position of the coils usually remains constant. This paper takes advantage of the effects of the physical parameters of coils on their electrical characteristics to optimize their structure and maximize their output power.

### III. Electrical Parameters of the MCM

*μ*. To simplify the analysis process, this study considers an MCM without electromagnetic shielding, as shown in Fig. 5(b), as an example.

### 1. Mutual Inductance

*μ*

_{0}is the permeability of vacuum,

*RC*

_{i}and

*RC*

_{j}are the radii of the two single-turn coils, and

*D*

_{ij}is the center-to-center distance of the two single coils. Furthermore,

*P*(

*α*

_{ij}) and

*Q*(

*α*

_{ij}) are the complete elliptic integrals of the first and second kinds, respectively.

*RC*is usually much larger than the wire radius

*r*, the effect of

*r*can be ignored. Therefore, the equivalent radius of the

*i*

_{th}single-turn coil from the center can be formulated as

*RC*

_{i}

*≈ RC*

_{0}+(

*i*−1)

*d*, where

*d*is the pitch. Notably, all single-turn coils have the same center. Furthermore, the distance

*D*

_{t,r}between the center of the transmitting coil and the receiving coil also remains constant. Assuming that the turns of the transmitting and receiving coils are

*N*

_{t}and

*N*

_{r}, the mutual inductance of MCM can be represented as follows:

*d*

_{t},

*d*

_{r}, turns

*N*

_{t},

*N*

_{r}, outer radii

*RC*

_{t}, and

*RC*

_{r}, and the relative spatial position parameters

*D*

_{t,r}, also known as spatial parameters of the MCM, including the vertical spacing

*b*, horizontal offset

*a*, and tilt angle

*γ*. In static WPT, spatial parameters, usually regarded as known parameters in an optimal design, can be obtained through measurement and prediction.

*ς*

_{av}represents the mean value of each parameter within the change range.

*RC*

_{r}, turns

*N*

_{r}, pitch

*d*

_{r}, and the mutual inductance

*M*were synchronously normalized based on Eq. (11), the results of which are shown in Fig. 7. The slopes of the curves in Fig. 7 represent the strength of the influence of physical parameters on mutual inductance. A positive slope indicates an active impact, while slopes less than 0 indicate a negative correlation. In other words, the greater the absolute value of the slope, the more significant the effect. According to the variations of the curves observed in Fig. 7, it is evident that

*RC*

_{r}and

*N*

_{r}are positively correlated with

*M*, while

*d*

_{r}is negatively correlated. Furthermore, the slopes corresponding to

*RC*

_{r}and

*d*

_{r}are approximately constant, indicating a linear correlation between

*RC*

_{r},

*d*

_{r}, and

*M*. It is further observed that as

*N*

_{r}increases, the rate of change of

*M*decreases continuously. This may be attributed to the fact that when

*N*

_{r}increases while

*RC*

_{r}and

*d*

_{r}are constant, the increased

*RC*gradually starts to decline, as a result of which the influence of

*N*

_{r}on

*M*will continue to weaken.

*M*can be expressed as

### 2. Self-Inductive

*RC*

_{i}is the radius of the single-turn coil and

*r*

_{i}is the radius of the wire.

*D*

_{t,r}=0. The self-inductance of a coil with

*N*

_{r}turns can be formulated as:

*N*

_{r}, radius

*RC*

_{r}, pitch distance

*d*

_{r}, and wire radius

*r*

_{r}. Therefore, during coil design,

*r*

_{r}is usually determined based on electrical parameters, such as current and voltage. Only

*N*

_{r},

*RC*

_{r}, and

*d*

_{r}are the optimal targets for the coil structure. The results processed using Eq. (11) are presented in Fig. 8, which shows that the changes in the curves are similar to those shown in Fig. 7. The radius

*RC*

_{r}and turns

*N*

_{r}are positively correlated with self-inductance

*L*

_{r}, while the pitch

*d*

_{r}is negatively correlated with the same. Furthermore,

*RC*

_{r}and

*d*

_{r}are linearly correlated with

*L*

_{r}. Notably,

*RC*

_{r}and

*N*

_{r}correspond to a larger curve slope, meaning that these parameters have a more obvious influence on

*L*

_{r}.

### 3. Resistance

*ω*denotes angular frequency, and

*σ*indicates conductivity.

*R*

_{AC}is influenced by the operating parameters (angular frequency

*ω*), wire material (conductivity

*σ*, wire radius

*r*), and structure parameters (radius

*RC*, turns

*N*, pitch

*d*). Therefore, the proposed design focused on structure parameters. The impact of physical parameters on AC resistance

*R*

_{AC}is demonstrated in Fig. 9.

*R*

_{AC}is consistent with the overall trend shown in Figs. 7 and 8. The outer radius

*RC*

_{r}and turns

*N*

_{r}are linearly positively correlated with

*R*

_{AC}, while pitch

*d*

_{r}is negatively correlated. Essentially, it is observed that an increase in

*RC*

_{r}and

*N*will lengthen the total length of wire

*l*, while the increase in

*d*

_{r}will shorten it. Furthermore, Litz wire is usually used in WPT, as it can effectively mitigate the skin effect and proximity effect. There exists a linear relationship between physical parameters and

*R*

_{AC}. It is only when

*d*

_{r}is very small that it will exhibit a reduced skin effect. In other words,

*R*

_{AC}gradually decreases and eventually stabilizes with an increases of

*d*

_{r}.

*R*

_{AC}is also restricted by radius

*RC*

_{r}, turns

*N*

_{r}, and pitch

*d*

_{r}, which can be expressed as follows:

### IV. Optimization of the MCM

### 1. Influence of Resonance Frequency

*f*is another crucial factor that affects the maximum output power

*P*

_{out·max}. Eq. (6) clearly shows that

*f*influences

*P*

_{out·max}, primary impedance

*Z*

_{t}, secondary impedance

*Z*

_{r}, and mutual inductance reactance

*X*

_{M}. The impact of

*f*and mutual inductance

*M*on transmission performance, based on Eqs. (3)–(5), are shown in Fig. 10, where the black dashed line corresponds to the ideal resonant frequency

*f*

_{0}. The area to its right represents the under-compensation area, where the primary reactance

*X*

_{t}and secondary reactance

*X*

_{r}are inductive. Moreover, input power

*P*

_{in}and output power

*P*

_{out}have the same variations. When frequency

*f*reaches close to

*f*

_{0}, the maximum output power

*P*

_{out·max}is achieved, while the corresponding mutual inductance

*M*is the smallest. Furthermore, as

*f*increases, the required

*M*for

*P*

_{out·max}increases along with it. This is because when

*f*deviates from

*f*

_{0},

*X*

_{t}/

*L*

_{1}=

*X*

_{r}/

*L*

_{2}≈ 0. This means that the larger the deviation, the larger the

*X*

_{t}and

*X*

_{r}. Since the modulus of the primary and secondary impedances |

*Z*

_{t}| and |

*Z*

_{r}| are also larger, a larger

*X*

_{M}was required. In addition, for the transmission efficiency

*η*in Fig. 10(c), the higher the

*M*, the higher the

*η*. Therefore,

*η*is also affected by

*f*. A small

*M*results in a higher

*f*and lower

*η*. At

*f*

_{0}, the required

*M*is at its minimum. This is also why both the primary and secondary sides of an MCM should work in a resonant state as much as possible.

### 2. Design of the Receiving Coil

*P*

_{out·max}is shown in Fig. 11. First, the wire radius

*r*, load

*R*

_{l}, and resonance frequency

*f*were determined based on the requirements of EV wireless charging. In a single-sided restricted MCM, the physical parameters of a transmitting coil and its relative spatial position are known conditions. Furthermore, self-inductance

*L*

_{t}and resistance

*R*

_{l1}of the transmitting coil were determined using Eqs. (15) and (17). For this purpose, the primary and secondary coils with similar or identical structures and parameters were considered the most favorable. Notably, for a restricted receiving coil, the radius

*RC*

_{r}should be as close as possible to that of the transmitting coil. Moreover, since the resistance of coils is usually small,

*R*

_{l1}is generally higher than the resistance of the source and the compensation capacitor. Therefore, it is included in

*Z*

_{t}. However, resistance

*R*

_{l2}of the receiving coil is usually ignored. Additionally, to reduce losses, the influence of the skin effect should be weakened as much as possible. The optimal pitch

*d*

_{r}was selected based on

*f*. Subsequently, based on Eq. (6), the optimal impedance

*X*

_{M}was calculated. The outer radius

*RC*

_{r}, turns

*N*

_{r}, and pitch

*d*

_{r}of the receiving coil were selected using Eqs. (12), (15), and (17).

*X*

_{t}=

*X*

_{r}=0,

*Z*

_{t}=

*R*

_{t}, and

*Z*

_{r}=

*R*

_{r}. The effect of self-inductance

*L*

_{r}of the receiving coil can be ignored, since it only needs to satisfy Eq. (6). In such a case, the turns of the receiving coil can be calculated without accounting for Eq. (15). As shown in Fig. 4, when the resonance frequency

*f*deviates from the ideal value—that is, when it is in non-ideal resonance—an optimal mutual inductance

*M*is present. Therefore, the effects of the self-inductances of both the transmitting and receiving coils must be accounted for in the design process shown in Fig. 11.

### V. Experiments

*X*of 0 in frequency

*f*= 85 kHz. Fig. 12 traces the variations in

*X*and resistance

*R*with

*f*. It is evident that

*X*increases continuously with an increase in

*f*, changing from negative (capacitive) to positive (inductive). Notably, compensation capacitor banks are composed of standard capacitors (with certain errors). Table 2 presents the actual compensation capacitances calculated. Due to the parasitic capacitances of the coils, the actual compensation capacitances were slightly lower than those calculated based on resonance. Furthermore, the resistances of the transmitter and receivers were approximately the sum of those of the coils and the corresponding capacitors.

*M*with transmission distance

*D*when the transmitting and receiving coils face each other. It is observed that as

*D*increases,

*M*continues to decrease—first fast and then slow. Furthermore, the more turns the receiving coil has, the greater the

*M*of the MCM that it forms. This result is consistent with the theoretical analysis, thus verifying that in a limited space (that is, the radius of coils remains unchanged), the adjustment of turns can ensure that the same

*M*at different

*D*.

*R*

_{L}remained constant, but the transmission distances

*D*of the MCMs changed (essentially, the change in mutual inductance

*M*in Fig. 13). Furthermore, Fig. 14 shows the variations in output power

*P*

_{out}and transmission efficiency

*η*with the transmission distance

*D*. It is evident that the trends of all three MCMs are the same. As

*D*increases,

*P*

_{out}increases at first and then declines.

*η*shows a continuous downward trend. Furthermore, the more the turns of the receiving coil, the greater the

*η*. All three MCMs achieved maximum

*P*

_{out}at

*η*of around 50%. However, the optimal

*D*corresponding to the maximum

*P*

_{out}was different—25 cm, 27 cm, and 28 cm for the three MCMs.

*P*

_{out}, Eq. (6) had to be met. At ideal resonance frequency

*f*

_{0}, the primary impedance

*Z*

_{t}refers to the sum of the resistances of the coil and compensation capacitor, which showed almost no change. Meanwhile, the secondary impedance

*Z*

_{r}includes the resistances of the coil, compensation capacitor, and load. Usually, the resistance

*R*

_{L}of a load is significantly greater than that of the receiver, with

*Z*

_{r}being approximately equal to

*R*

_{L}. Notably, the transmission distances

*D*corresponding to the three images are found to be 25 cm, 27 cm, and 28 cm, representing the optimal

*D*corresponding to the different MCMs. The three curves in Fig. 15 correspond to the

*P*

_{out}and

*η*of the three MCMs. Their variations are almost identical to Fig. 14. As

*R*

_{L}increases,

*P*

_{out}increases at first and then decreases, while

*η*shows a continuous decreasing trend. Furthermore, the more the turns of the receiving coil, the greater the corresponding optimal

*R*

_{L}. Also, the farther the

*D*, the smaller the corresponding optimal

*R*

_{L}. Moreover, the three pictures shown in Fig. 15 indicate that the MCMs corresponding to the maximum

*P*

_{out}are

*MCM1*,

*MCM2*, and

*MCM3*. This implies that the change in

*R*

_{L}does not affect the optimal

*D*corresponding to the different MCMs.

*f*tends to have an impact on mutual inductance reactance

*X*

_{M}, primary impedance

*Z*

_{t}and secondary impedance

*Z*

_{r}. In theory, as long as Eq. (6) is satisfied, WPT can extract the maximum output power

*P*

_{out}. Fig. 16 presents the variation in output power

*P*

_{out}and transmission efficiency

*η*with

*f*when the mutual inductance

*M*and load

*R*

_{L}are constant. Notably, the transmission distances

*D*corresponding to three images remains 25 cm, 27 cm, and 28 cm. However, as

*f*increases,

*P*

_{out}and

*η*first show an increasing trend that ultimately becomes a decreasing trend. The maximum values are achieved around the ideal resonant frequency

*f*=

*f*

_{0}. However, the rate of change in

*P*

_{out}is considerably higher than

*η*. Notably, similar to the influence of

*R*

_{L}on the maximum

*P*

_{out},

*f*does not influence the optimal

*D*corresponding to the different MCMs.

*D*, load

*R*

_{L}, and frequency

*f*changed, there was always an optimal MCM that was able to obtain the maximum output power

*P*

_{out·max}. The only variable for this optimal MCM was the turns

*N*

_{r}of receiving coil. This result verifies the feasibility of taking recourse to the adjustment of turns to achieve maximum output power for MCMs with different primary and secondary coils at a specific position.