**2. Analysis of the LCC-S Compensation Network**

The circuit diagram of the proposed system using LCC-S compensation with the buck converter and UC is shown in Figure 3.

**Figure 3.** Circuit diagram of the proposed WPT system.

*Lf* <sup>1</sup> and *Cf* <sup>1</sup> are the compensation inductor and compensation capacitors for the transmitting side, respectively. *L*<sup>1</sup> and *L*<sup>2</sup> are the self-inductances of the primary and secondary coils, and *C*<sup>1</sup> and *C*<sup>2</sup> are the series capacitors for the transmitting and receiving side, respectively. *VAB* is the inverted AC voltage by the high-frequency inverters, and *Vab* is the output voltage of the receiving coil. *M* is the mutual inductance between the two coils, and *ω*<sup>0</sup> is the operating resonant frequency. *Uin* is the input DC voltage, and *Uout* is the output DC voltage. *Rf* 1, *R*1, and *R*<sup>2</sup> are the self-resistances of the compensation inductor, primary coil, and secondary coil, respectively. *L*, *C*, and *S*<sup>5</sup> are the inductor, capacitor, and IGBT switch of the buck converter, respectively. *RL* is the equivalent resistance of the UC, which varies with its state of charge (SOC).

The equivalent circuit diagram of the proposed system is shown in Figure 4. The *Req* is the equivalent resistance of the buck converter, which is equal to:

$$R\_{eq} = \frac{1}{\mu^2} R\_L \tag{1}$$

**Figure 4.** Equivalent diagram of proposed system.

The system can be analyzed using the two-port network theory. To convert the system into a two-port network, the secondary side parameters are transferred to the primary side. The two-port network of the system is shown in Figure 5. *Zr* is the transferred impedance of the secondary side, and *V*∗ *ab* is the voltage across *Zr*. *Zr* can be calculated as follows,

$$Z\_I = \frac{\omega^2 M^2}{R\_{eq} + R\_2} \tag{2}$$

The system's two-port network can be expressed using the following equations:

$$\begin{cases} V\_{AB} = & Z\_{11}I\_1 + Z\_{12}I\_2 \\ V\_{ab}^\* = & Z\_{21}I\_1 + Z\_{22}I\_2 \end{cases} \tag{3}$$

Converting Equation (3) into matrix form:

$$
\begin{bmatrix} V\_{AB} \\ V\_{ab}^\* \end{bmatrix} = \begin{bmatrix} Z\_{11} & Z\_{12} \\ Z\_{21} & Z\_{22} \end{bmatrix} \begin{bmatrix} I\_1 \\ I\_2 \end{bmatrix} \tag{4}
$$

**Figure 5.** Two port network of the proposed system.

To find the system impedance matrix, two modes are used. In the first mode, shown in Figure 6a, the load is disconnected, which makes *I*<sup>2</sup> = 0. In the second mode, shown in Figure 6b, the input source is disconnected, which makes *I*<sup>1</sup> = 0. Using the two cases, the system impedances are calculated as:

$$Z\_{11} = \frac{V\_{AB}}{I\_1} \mid\_{I\_2=0} = R\_{f1} + j\left(\omega L\_{f1} - \frac{1}{\omega C\_{f1}}\right) \tag{5}$$

$$Z\_{21} = \frac{V\_{ab}^\*}{I\_1} \mid\_{I\_2=0} = \frac{1}{j\omega \mathcal{C}\_{f1}} \tag{6}$$

$$Z\_{22} = \frac{V\_{ab}^\*}{I\_2} \mid\_{I\_1=0} = R\_1 + j\left(\omega L\_1 - \frac{1}{\omega \mathbb{C}\_{f1}} - \frac{1}{\omega \mathbb{C}\_1}\right) \tag{7}$$

$$Z\_{12} = \frac{V\_{AB}}{I\_2} \mid\_{I\_1=0} = \frac{1}{j\omega \mathcal{C}\_{f1}} \tag{8}$$

**Figure 6.** (**a**) Mode 1: Load is disconnected. (**b**) Mode 2: Input source is disconnected.

Using the following equations, the system's parameters are tuned in such a way that the system input voltage and current have zero-phase difference.

$$\mathbb{C}\_{f1} = \frac{1}{\omega^2 L\_{f1}}\tag{9}$$

$$\mathcal{C}\_1 = \frac{1}{\omega^2 (L\_1 - L\_{f1})} \tag{10}$$

$$\mathcal{C}\_2 = \frac{1}{\omega^2 L\_2} \tag{11}$$

When the system's parameters satisfy Equations (9)–(11), then Equations (5) to (8) become:

$$\begin{cases} Z\_{11} = & R\_{f1} \\ Z\_{12} = Z\_{21} = & j\omega L\_{f1} \\ Z\_{22} = & R\_1 \end{cases} \tag{12}$$

The voltage gain from inverter output voltage to the secondary coil output voltage can be calculated as follows,

$$G\_V = \left| \frac{V\_{ab}}{V\_{AB}} \right| = \left| \frac{V\_{ab}}{V\_{ab}^\*} \right| \left| \frac{V\_{ab}^\*}{V\_{AB}} \right| \tag{13}$$

According to the two-port network theory, *V*∗ *ab VAB* can be calculated using the following formula,

$$\left|\frac{V\_{ab}^\*}{V\_{AB}}\right| = \frac{Z\_{21}Z\_r}{Z\_{11}\left(Z\_r + Z\_{22}\right) - Z\_{12}Z\_{21}} = \frac{\omega^3 L\_{f1}M^2}{\left(R\_2 + R\_{eq}\right)\left(R\_{f1}\left(\frac{\omega^2 M^2}{R\_2 + R\_{eq}} + R\_1\right) + \left(\omega L\_{f1}\right)^2\right)}\tag{14}$$

From the circuit diagrams, shown in Figures 5 and 6, *Vab* and *V*<sup>∗</sup> *ab* can be derived as,

$$V\_{ab} = \left(\frac{j\omega M I\_2}{R\_2 + R\_{eq}}\right) R\_{eq} \tag{15}$$

$$V\_{ab}^{\*} = \frac{\omega^2 M^2 I\_2}{R\_2 + R\_{eq}} \tag{16}$$

Substituting Equations (14)–(16) into Equation (13), the system's voltage gain is derived as,

$$G\_V = \frac{\omega^2 L\_{f1} M R\_{eq}}{\left(R\_2 + R\_{eq}\right) \left(R\_{f1} \left(\frac{\omega^2 M^2}{R\_2 + R\_{eq}} + R\_1\right) + \left(\omega L\_{f1}\right)^2\right)}\tag{17}$$

Similarly, according to the characteristics of the two-port network theory, the inverter output current *I*<sup>1</sup> and primary coil current *I*<sup>2</sup> and the current gain *GI* can be calculated as:

$$I\_{1} = \frac{V\_{AB} \left(Z\_{22} + Z\_{r}\right)}{Z\_{11} \left(Z\_{22} + Z\_{r}\right) - Z\_{12}Z\_{21}} = \frac{V\_{AB} \left(R\_{1}R\_{2} + R\_{1}R\_{eq} + \omega^{2}M^{2}\right)}{R\_{f1} \left(R\_{1}R\_{2} + R\_{1}R\_{eq} + \omega^{2}M^{2}\right) + \left(\omega L\_{f1}\right)^{2} \left(R\_{2} + R\_{eq}\right)}\tag{18}$$

$$H\_{2} = \frac{V\_{AB}Z\_{21}}{Z\_{11}\left(Z\_{22} + Z\_{r}\right) - Z\_{12}Z\_{21}} = \frac{V\_{AB}\omega L\_{f1}\left(R\_{2} + R\_{eq}\right)}{R\_{f1}\left(R\_{1}R\_{2} + R\_{1}R\_{eq} + \omega^{2}M^{2}\right) + \left(\omega L\_{f1}\right)^{2}\left(R\_{2} + R\_{eq}\right)}\tag{19}$$

$$\mathbf{G}\_{I} = \left| \frac{I\_{2}}{I\_{1}} \right| = \frac{Z\_{21}}{Z\_{I} + Z\_{22}} = \frac{\omega L\_{f1} \left( R\_{2} + R\_{eq} \right)}{\omega^{2} M^{2} + R\_{1} \left( R\_{2} + R\_{eq} \right)} \tag{20}$$

Using Equation (19), the secondary coil current can be derived as,

$$I\_3 = \frac{\omega^2 M L\_{f1} V\_{AB}}{R\_{f1} \left(R\_1 R\_2 + R\_1 R\_{eq} + \omega^2 M^2\right) + \left(\omega L\_{f1}\right)^2 \left(R\_2 + R\_{eq}\right)}\tag{21}$$

Under the assumption that the system is under resonance condition and there are no conduction losses in the inverter, rectifier, and buck converter, the system efficiency can be calculated as,

$$\eta = \frac{V\_{ab}^\* I\_2}{V\_{AB} I\_1} \frac{R\_{\varepsilon q}}{R\_2 + R\_{\varepsilon q}} = G\_V G\_I \frac{R\_{\varepsilon q}}{R\_2 + R\_{\varepsilon q}} \tag{22}$$

Substituting Equations (17) and (20) into Equation (22), the system's efficiency can be obtained as,

$$\eta = \frac{\omega^4 L\_{f1}^2 M^2 R\_{eq}}{\left(R\_{f1}\left(\frac{\omega^2 M^2}{R\_2 + R\_{eq}} + R\_1\right) + \left(\omega L\_{f1}\right)^2\right)\left(\frac{\omega^2 M^2}{R\_2 + R\_{eq}} + R\_1\right)\left(R\_2 + R\_{eq}\right)^2} \tag{23}$$

For the system parameters listed in Table 1 and 2, the relationship between the system's efficiency *η* and load resistance *Req* is shown in Figure 7. It can be seen that at a particular resistance *Rop*, i.e., 11 Ω, the system can operate at maximum efficiency. *Rop* can be derived by differentiating Equation (23) with respect to *Req*, and *Rop* can be derived as follows,

$$\frac{\partial \eta}{\partial R\_{eq}} = 0 \Rightarrow R\_{op} = \sqrt{\frac{\left\{R\_1 R\_2 R\_{f1} + R\_2 \left(\omega L\_{f1}\right)^2 + R\_{Lf1} \omega^2 M^2\right\} \left(R\_1 R\_2 + \omega^2 M^2\right)}{R\_1 \left\{R\_1 R\_{Lf1} + \left(\omega L\_{f1}\right)^2\right\}}}\tag{24}$$

According to Equation (1), for varying *RL*, the buck converter duty cycle *u* can be used to regulate *Req*=*Rop* to make sure that the system operates at maximum efficiency. For the ease of control designing, the optimal resistance *Rop* can be translated into optimal power *Pop*. When the output power of the system is *Pop*, the system will operate at maximum efficiency. Using Equation (17), *Pop* can be obtained as,

$$P\_{op} = \frac{V\_{ab}^2}{R\_{op}} = \left(\frac{1}{R\_{op}}\right) \left[\frac{\omega^2 L\_{f1} M R\_{cq} V\_{AB}}{\left(R\_2 + R\_{cq}\right) \left(R\_{f1} \left(\frac{\omega^2 M^2}{R\_2 + R\_{cq}} + R\_1\right) + \left(\omega L\_{f1}\right)^2\right)}\right]^2 \tag{25}$$

Using Equation (25), the relationship between the efficiency and the output power is shown in Figure 8. It can be seen that at *Pop*, i.e., 120 Watts, the system efficiency is highest. To track the system output power *Pout* to this maximum efficiency point, the DFTSMC controller is designed in the next section to control the duty cycle *u* of the secondary side buck converter.

**Figure 8.** Efficiency, *η* vs. *Pout*.
