**1. Introduction**

Since the concept of magnetic coupled resonant wireless power transfer (MCR-WPT) was proposed by MIT in 2007, the technology has developed rapidly and has been widely applied in implantable medical devices, home appliances, mobile devices and electric vehicles [1–8]. Increasing transmission power is particularly important for wireless charging of electric vehicles. The main ways to improve power levels are: (1) power supply parallel topology, which uses multiple parallel power supplies for power distribution, but the parallel topology is too redundant [9]; (2) multi-phase parallel, there are shortcomings of current imbalance between phases and serious coil loss. In addition, the compensation topology also has a grea<sup>t</sup> impact on the output power level.

At present, the basic compensation topologies are series-series (SS), series-parallel (SP), parallel-series (PS), and parallel-parallel (PP) [10,11]. The most widely used is the SS topology, but this topology has inherent disadvantages: once coil parameters are determined, the rated output power can be regulated only by the input voltage. In addition to the four basic resonance topologies described above, hybrid-series-parallel (LCC, composed by an inductor and two capacitors) compensation topology has been extensively studied for its excellent performance. It can implement zero phase angle (ZPA) and zero voltage switching (ZVS), and its output current is independent of the load [12–17].

Most of the current studies on WPT compensation topology focus on improving the output performance by adopting complex control strategies. However, the load resistance to achieve optimal

efficiency of SS topology WPT system is unchangeable after the coil-parameters are fixed whereas the one of LCC topology can be adjusted according to the parameters of compensation topology. In this paper, the WPT system is modeled and the transfer performance is analyzed. Methods are proposed to increase output power level while maintaining optimal transmission efficiency without redesign of the coupled coils by optimizing the parameters of LCC compensation topology. Both theorical and experimental results indicate that the proposed parameter optimization strategy is effective in improving transfer efficiency and adjusting output power under different load resistances. It avoids the disadvantages of redesigning the coil due to power level changes to maintain high transmission efficiency. At the same time, it has obvious advantages in safety under the condition of output short circuit.

## **2. Theoretical Analysis**

### *2.1. System Analysis*

Figure 1 shows the schematic diagram of the magnetic coupled wireless power transfer system, where AC power on the grid side is rectified to form DC power, the high-frequency inverter is composed of H-bridge, and the direct current is converted by inverter circuit into alternating current with a system rated frequency, which flows through the transmitter end and transmits the power to the receiver coil. The primary and secondary compensation circuit is composed of SS or LCC compensation topology. The transmitter and receiver coils are planar spiral coils designed in Section 3, and the receiver end is connected to electric vehicle battery through a rectifier circuit.

**Figure 1.** The structure diagram of wireless power transfer system.

Because the resonance part of wireless power transfer system has the characteristics of band-pass filtering, only the fundamental wave component is considered later, and the square wave generated by the full-bridge inverter can be equivalent to an AC voltage source for theoretical analysis. The square wave is expanded according to the Fourier series, the duty cycle is set to *D*, and the fundamental wave of the square wave whose amplitude is *Ud* is as follow:

$$f\_l = \frac{4lI\_d}{\pi} \sin(\pi D) \cos(\omega t - \pi D) \tag{1}$$

The battery is a typical non-linear load. With the use of battery, the battery charging current and the state of charge are different, and different external load characteristics are displayed. In order to simplify the difficulty of system analysis, the battery is generally equivalent to a resistive load according to the ratio of charging voltage and current of battery and then analyzed. Therefore, the battery and its managemen<sup>t</sup> system are equivalent to a load in this paper.

#### *2.2. Theoretical Analysis of SS and LCC Compensation Topology*

Modeling and analysis of SS-type and LCC-type compensation topologies are shown below. Figures 2 and 3 show simplified topologies of SS-type and LCC-type resonant circuits, respectively. The parasitic resistance of inductors and capacitors are much smaller than the internal resistance of coil, so it can be ignored. The subsequent experimental results show that this approximate analysis does not affect the accuracy of experiment.

#### 2.2.1. Analysis of SS-type Resonance Circuit

The SS topology is the most basic resonance structure. Both primary and secondary side use series resonance. The equivalent circuit is shown in Figure 2, where *US* is a voltage-stabilized source, the source internal resistance is ignored, *M* is the mutual inductance between two coils.

**Figure 2.** The equivalent circuit model of SS-type.

According to Figure 2 and Kirchhoff voltage law (KVL), we can obtain:

$$\begin{cases} (R\_1 + j\omega L\_1 + \frac{1}{j\omega \mathbb{C}\_1})I\_1 - j\omega MI\_2 = \mathcal{U}\_s\\ (R\_2 + R\_L + j\omega L\_2 + \frac{1}{j\omega \mathbb{C}\_2})I\_2 - j\omega MI\_1 = 0 \end{cases} \tag{2}$$

When the system meets the resonance conditions:

$$f\_0 = \frac{1}{2\pi\sqrt{L\_1C\_1}} = \frac{1}{2\pi\sqrt{L\_2C\_2}}\tag{3}$$

When *RL* is much greater than *R*1 and *R*2, *R*1 and *R*2 can be ignored, the currents are as follows:

$$\begin{cases} I\_1 = \frac{\mathcal{U}I\_sR\_L}{\left(\omega \mathcal{M}\right)^2} \\ I\_2 = \frac{\mathcal{U}I\_s}{\omega \mathcal{M}} \end{cases} \tag{4}$$

The transmission power and efficiency of the system are:

$$P\_{\rm ss} = \frac{\mathcal{U}\_s^{\cdot 2} \mathcal{R}\_L}{\left(\omega \mathcal{M}\right)^2} \tag{5}$$

$$\eta\_{\mathbb{M}} = 1 - \frac{R\_L R\_1}{\left(\omega M\right)^2} - \frac{R\_2}{R\_L} \tag{6}$$

The maximum efficiency and corresponding load value are:

$$
\eta\_{\text{sc\\_max}} = 1 - \frac{2\sqrt{R\_1 R\_2}}{aM} \tag{7}
$$

$$R\_{\infty} = \sqrt{\frac{R\_2}{R\_1} (\omega M)^2} \tag{8}$$

Due to the inverse relationship between distance and mutual inductance, in practical applications, it is common for charging object to be far away from primary coil. It can be seen from (4) and (5) that when the secondary side is open, the primary current will increase sharply; when a short circuit occurs on the secondary side, the primary current and transmission power will decrease. On the other hand, in the case of the SS structure with constant coil parameters, the load resistance to achieve optimal efficiency is unchangeable.

#### 2.2.2. Analysis of LCC-type Resonance Circuit

The LCC structure is a new type of composite resonant structure. As shown in Figure 3, *Lf* 1, *C*1, *Cp*1 and *Lf* 2, *C*2, *Cp*2 are the corresponding resonant circuit units of primary and secondary coils, respectively.

$$U\_s \overbrace{\begin{matrix} U\_1 \\ U\_1 \\ \end{matrix}}^{L\_{f1}} \overbrace{\begin{matrix} \overbrace{I\_1}^{\alpha\_1} \\ \end{matrix}}^{C\_1} \overbrace{\begin{matrix} \overbrace{I\_1}^{\alpha\_1} \\ \end{matrix}}^{C\_1} \overbrace{\begin{matrix} \overbrace{I\_1}^{\alpha\_1} \\ \end{matrix}}^{L\_{f2}} \overbrace{\begin{matrix} \overbrace{I\_2}^{\alpha\_2} \\ \end{matrix}}^{L\_{f3}} \overbrace{\begin{matrix} \overbrace{I\_3}^{\alpha\_3} \\ \end{matrix}}^{C\_2} \overbrace{\begin{matrix} \overbrace{I\_4}^{\alpha\_4} \\ \end{matrix}}^{L\_{f4}} \overbrace{\begin{matrix} \overbrace{I\_5}^{\alpha\_5} \\ \end{matrix}}^{R\_{f1}} \overbrace{\begin{matrix} \overbrace{I\_6}^{\alpha\_6} \\ \end{matrix}}^{R\_{f2}}$$

**Figure 3.** The equivalent circuit model of LCC-type.

Similarly, we can formulate KVL equation according to Figure 3:

$$\begin{cases} (j\omega L\_{f1} + \frac{1}{j\omega \mathbb{C}\_1})I\_1 - \frac{1}{j\omega \mathbb{C}\_1}I\_2 = \mathbb{U}\_s\\ (R\_1 + j\omega L\_1 + \frac{1}{j\omega \mathbb{C}\_1} + \frac{1}{j\omega \mathbb{C}\_{p1}})I\_2 - \frac{1}{j\omega \mathbb{C}\_1}I\_1 - j\omega M I\_3 = 0\\ (R\_2 + j\omega L\_2 + \frac{1}{j\omega \mathbb{C}\_2} + \frac{1}{j\omega \mathbb{C}\_{p2}})I\_3 - \frac{1}{j\omega \mathbb{C}\_2}I\_4 - j\omega M I\_2 = 0\\ (R\_L + j\omega L\_{f2} + \frac{1}{j\omega \mathbb{C}\_2})I\_4 - \frac{1}{j\omega \mathbb{C}\_2}I\_3 = 0 \end{cases} \tag{9}$$

The resonance conditions are as follows:

$$\begin{aligned} \mathbb{C}\_1 &= \frac{1}{\alpha^2 L\_{f1}}, \mathbb{C}\_{p1} = \frac{1}{\omega^2 L\_1 - \frac{1}{\mathbb{C}\_1}},\\ \mathbb{C}\_2 &= \frac{1}{\alpha^2 L\_{f2}}, \mathbb{C}\_{p2} = \frac{1}{\omega^2 L\_2 - \frac{1}{\mathbb{C}\_2}} \end{aligned} \tag{10}$$

The currents of the loops are:

$$\begin{cases} I\_1 = \alpha^6 \mathbf{C}\_1 \mathbf{^2C\_2} \mathbf{^2M} \mathbf{^2R\_L} \mathbf{U\_s} \\\ I\_2 = -\alpha \mathbf{C}\_1 \mathbf{U\_s} \\\ I\_3 = \alpha^4 \mathbf{C}\_1 \mathbf{C}\_2 \mathbf{^2MR\_L} \mathbf{U\_s} \\\ I\_4 = -\alpha^3 \mathbf{C}\_1 \mathbf{C}\_2 \mathbf{M} \mathbf{U\_s} \end{cases} \tag{11}$$

The transmission power and e fficiency of the system are:

$$P\_{\rm LCC} = \frac{M^2 \mathcal{U}\_s^2 \mathcal{R}\_L}{a^2 L\_{f1}^{-2} L\_{f2} ^2} \tag{12}$$

$$\eta\_{L\text{CC}} = 1 - \frac{R\_1 L\_{f2}}{R\_L M^2} - \frac{R\_2 R\_L}{\omega^2 L\_{f2}^2} \tag{13}$$

The maximum e fficiency and corresponding load values are:

$$
\eta\_{LCC\\_max} = 1 - \frac{2\sqrt{R\_1 R\_2}}{aM} \tag{14}
$$

$$R\_{L\text{CC}} = \sqrt{\frac{R\_1 \mu^2 L\_{f2}{}^4}{R\_2 M^2}} = \frac{R\_1 L\_{f2}}{R\_2 M^2} R\_{SS} \tag{15}$$

It can be known from (11) that when the mutual inductance decreases, the currents of primary and secondary coils decrease, which is a safe working condition. And when the system coil-parameters are fixed, the load resistance to achieve optimal e fficiency can be adjusted by the inductance of the secondary resonance circuit *Lf* 2. *C*2 also needs to be adjusted to keep resonant according to (10). Compared with the SS structure, it avoids the disadvantages of redesigning the coil parameters to maintain e fficient operation when the system's rated parameters are changed, which makes the system design more flexible and reduces the manufacturing cost.

#### **3. Parameter Design**

#### *3.1. E*ff*ect of Coil Design on Optimal E*ffi*ciency Load*

When a WPT system is actually designed, some parameters are fixed due to the limitation of frequency or coil size. At this time, choosing a specific resonance mode to obtain better transmission efficiency according to the existing parameters becomes the key to the design.

This paper presents a design scheme of planar spiral coil [18] that optimizes the system transmission e fficiency. The core idea of designing a high-e fficiency MCR-WPT system is to maintain the system's high-e fficiency operating state by adjusting the other coil parameters (mainly turn number, wire diameter, turn pitch, etc.) when the rated load is the optimal load to achieve maximum e fficiency.

Figure 4 is a schematic diagram of planar spiral coil. The internal resistance *R*, self-inductance *L* and mutual inductance *M* between the coils [19–21] can be obtained by:

$$\begin{cases} R = \sqrt{\frac{\rho \mu\_0 a \nu}{2}} \frac{N r\_{\text{avg}}}{a} \\ L = \mu\_0 N^2 r\_{\text{avg}} c\_1 \left[ \ln(c\_2/\lambda) + c\_3 \lambda + c\_4 \lambda^2 \right] \\ M = \frac{\mu\_0 \pi N\_1 N\_2 \left(\frac{r\_{1\text{-avg}}}{2}\right)^2 \left(\frac{r\_{2\text{-avg}}}{2}\right)^2}{2 \left[ h^2 + \left( \frac{r\_{1\text{-avg}}}{2} \right)^2 \right]^{1.5}} \end{cases} \tag{16}$$

where ρ is the conductor resistivity, μ0 is the vacuum permeability, ω is the current angular frequency, *N* is the number of coil turns, *rmin* is the inner radius of coil, *d* is the turn spacing of coil, *ravg* = *rmin* + (*N* − 1)*d*/2 is the average radius of coil, *a* is the conductor radius, *c*1, *c*2, *c*3 and *c*4 are fitting coe fficients, for circular coils they are 1, 2.46, 0 and 0.2, respectively, λ = (*N* − 1)*d*/(2\**ravg*), *h* is the transmission distance.

**Figure 4.** Schematic diagram of planar spiral coil.

According to the actual application scenario, some parameters of the system can be determined. In this paper, the transmission distance *h* = 20cm, the inner radius of the transmitter and receiver coil *<sup>r</sup>*1\_*min* = *<sup>r</sup>*2\_*min* = 2 cm. In the case of tightly wound, the conductor diameter and turn spacing are not considered, the influence of coil turns on transmission e fficiency occupies the main factor. Where the

load resistance *RL* is optimal to realize maximum transmission efficiency, the coupling coefficient *k* = *M*/ √*<sup>L</sup>*1 ∗ *L*2, self-inductance *L*1 and *L*2, the coil internal resistance *R*1 and *R*2, and the optimal load *RL* are all related to the turn number of primary coil *N*1 and secondary coil *N*2:

$$\begin{cases} \begin{array}{c} P\_{\rm L} = \zeta(k(N\_{1}, N\_{2}), L\_{1}(N\_{1}), L\_{2}(N\_{2}), R\_{1}(N\_{1}), R\_{2}(N\_{2}), R\_{L}(N\_{1}, N\_{2})) \\ \eta = \xi(k(N\_{1}, N\_{2}), L\_{1}(N\_{1}), L\_{2}(N\_{2}), R\_{1}(N\_{1}), R\_{2}(N\_{2}), R\_{L}(N\_{1}, N\_{2})) \end{array} \end{cases} \tag{17}$$

(17) shows that the transmission power *PL* and efficiency η are quantities related to the coupling coefficient, load value, self-inductance, and coil internal resistance. Under the tightly wounding condition, the independent variable is mainly affected by the turn number of the coils. Therefore, the ultimate optimization goal of this paper can be expressed as finding the optimal coil turn number while meeting the rated power value of the system design, that is, to find max(η(*N*1,*N*2)).

Figure 5 can be made by Equation (16). It can be seen from Figure 5a that the transmission power decreases with increasing turns; from Figure 5b, the efficiency increases quickly first, then slowly increases as the turn number increases, only when the turn number of primary and secondary coils reaches a critical point can the efficiency be maintained at a higher level. Finally, the iterative method is used to obtain the number of turns, and the coil parameters are reasonably designed to achieve a high-efficiency transmission system that meets the power requirements. The specific design process is shown in Figure 6.

**Figure 5.** (**a**) Transmission power and (**b**) efficiency with respect to the turn number of coil.

**Figure 6.** Flow chart of system design.

The following Table 1 shows the optimized resonant parameters of SS compensation topology designed to the load (50 Ω) corresponding to optimal efficiency point.


**Table 1.** The resonant circuit parameters of SS-type.

#### *3.2. Selection of LCC Resonance Parameters*

By optimizing the coil design, for the SS resonance structure, the system is in an optimal efficiency condition when the load is 50 ohms. According to (15), after the coil parameters of LCC resonance structure are determined, the optimal efficiency load value is only related to the parameter *Lf* 2. The optimal efficiency load of LCC resonance structure can be determined by designing the inductance *Lf* 2. From (5) and (12), we have:

$$P\_{LCC} = \frac{M^4}{L\_{f1}\,^2L\_{f2}\,^2}P\_{SS} \tag{18}$$

For the LCC resonance structure, since the load and coil parameters have been determined, it can be known that by adjusting the value of *Lf* 1, the level of rated output power can be changed. The parameter configuration method is as follows: (1) Firstly, according to the design parameters of SS compensation structure system, the value of *Lf* 2 is obtained by Equation (15); (2) Secondly, according to the rated power of system, determine the value *Lf* 1 through Equation (18); (3) Finally, determine the remaining parameter values according to Equation (10).

As the rated output power of SS resonant structure is 2 kW, a margin of 0.85 is taken, and the rated output power of LCC resonant structure is 2.45 kW, so the values of two inductors *Lf* 1 and *Lf* 2 can be determined. The specific parameter values are as follows in Table 2.


**Table 2.** The resonant circuit parameters of LCC-type.

#### **4. Simulation and Experimental Verification**

#### *4.1. Magnetic Simulation*

For the coil designed in Section 3, the finite element simulation software ANSYS EM was used for simulation, and the vertical distance between two coils was 200 mm, the remaining parameters are shown in Tables 1 and 2. The 3D finite element model is shown in Figure 7a, the boundary condition is radiation, the excitation currents *Ip* = 7A and *Is* = 7A, the maximum mesh element lengths of solution region, transmitter and receiver coil are 75 mm, 40 mm and 40 mm, respectively. Figure 7b is the distribution diagram of magnetic field strength around the system. It can be seen that the magnetic field distribution around the system is evenly distributed, mainly concentrated near the coils, and has little effect on the surrounding environment.

**Figure 7.** (**a**) Simulation setup and (**b**) magnetic field strength distribution.

Another purpose of simulation is to obtain the self-inductance, mutual inductance and coupling coefficient of the coil. There are many theoretical calculation methods for calculating these parameters, which will not be repeated here. They can also be obtained by actual measurement. According to the SAEJ2954 standard [22], the actual measured coupling coefficient formula is:

$$k = \sqrt{V\_{\text{oc}} I\_{\text{sc}} / V\_1 I\_1} \tag{19}$$

where, *V*1 and *I*1 are the voltage and current of primary coil respectively, *Voc* and *Isc* are the open circuit voltage and short circuit current of secondary coil, respectively.

The actual mutual inductance value can be obtained by "open circuit and short circuit test". This method needs to measure three quantities, *Lp*1 is the measured primary inductance when the secondary circuit is open, *Ls*1 is the measured secondary inductance when the primary circuit is open, and *Lp*2 is the measured primary inductance when the secondary circuit is shorted, as follow:

$$M = \sqrt{(L\_{p1} - L\_{p2})L\_{s1}}\tag{20}$$

Table 3 shows the theoretical, simulated and actual measured parameter values of the system under rated parameters. It can be seen from Table 3 that the deviation of the theoretical, simulated and actual value of the primary and secondary coil self-inductance can be ignored.


**Table 3.** System parameter values.
