# Optimization Design of Planar Circle Coil for Limited-Size Wireless Power Transfer System
## 2 Design Variables and Objectives of Planar Circular Coil
Therefore, the design variables are inverter frequency \(f\), circuit natural angular frequency \(\omega_{0}\), the mutual inductance of the coils \(M\), coils resistances \(R\), and coils self-inductance \(L\). Generally, the mutual inductance and self-inductance of two coils are the inherent parameters of the coupler, only related to the design of the coupler. However, the AC resistance is not only associated with the coupler design, but also affected by the inverter frequency. The inverter frequency of underwater WPT systems is usually between kHz-MHz. When the inner diameter of the copper wire is 2 mm, the coil adopts the traditional design (Coil inner diameter/outer diameter = 0.4, Turn pitch = 3 * Wire inner diameter) [12; 13], and the distance between the primary and secondary side is 5 cm (the design distance between the primary side and the secondary side when the underwater vehicle docks with the base station. ), the relationship between coil's self-inductance \(L\), mutual inductance \(M\), and AC resistance \(R\) and inverter frequency \(f\) is shown in Figure 3.  
The data show that the mutual inductance and self-inductance do not change much with frequency, while the AC resistance changes greatly. Therefore, we take \(L=84.61\)\(\mu\)H as an example, when the load resistance \(R_{L}=50\)\(\Omega\), the circuit natural angular frequency \(\omega_{0}=2\pi f_{0}=2\pi\cdot 500\) kHz, then substitute AC resistance data to solve \(\eta\) and \(K\). The relationship between voltage modulus gain \(K\), inverter frequency \(f\), and mutual inductance \(M\) is shown in Figure 4a. Figure 4b illustrates the relationship between efficiency \(\eta\), inverter frequency \(f\), and mutual inductance \(M\). Figure 2: Coupler structure of the WPT system. As shown from Figure 4, the voltage modulus gain \(K\) increases first and then sharply decreases with the increase of mutual inductance \(M\) and reaches its maximum value at the natural resonant frequency (500 kHz). With the further increase of mutual inductance \(M\), the frequency splitting phenomenon occurs. That is, there are two frequencies to maximize the load voltage. The greater the \(M\), the more fragmentation. Figure 4 illustrates that the efficiency \(\eta\) continues to increase with the increase of mutual inductance \(M\). Meanwhile, when the coupling coefficient is the same, the efficiency at the resonant frequency is the highest. Obviously, if the coil is optimized for PTE only, the TP will be insufficient when the input voltage is restricted. Similarly, while the TP is ideal, the TPE is only approximately 50%; this lower efficiency is undesirable in engineering applications. To ensure that \(K\) and \(\eta\) have good values simultaneously, it is necessary to study the influence of the same frequency and coils parameters for \(K\) and \(\eta\) of the WPT system at the same time. Therefore, the partial contour diagram of voltage modulus gain \(K\) and system efficiency \(\eta\) is drawn in a coordinate system, as shown in \[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]0  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]1 (**a**) The relationship between voltage modulus gain \(K\), inverter frequency \(f\), and mutual inductance \(M\); and (**b**) the relationship between efficiency \(\eta\), inverter frequency \(f\), and mutual inductance \(M\). \[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]2 The relationship between coil’s self-inductance \(L\), mutual inductance \(M\), AC resistance \(R\), and frequency \(f\).