# Development of Wireless Power Transmission System for Transfer Cart with Shortened Track
## 3 Thermal Analysis and Pickup Module Fabrication  
### Thermal Analysis  
Litz wire is made up of multiple strands of multiple wires where a piece of string of copper has a diameter of 0.32 mm, and there are 45 pcs in 12 strands (see Figure 1a). As discussed in [26], serious power loss such as proximity effects can occur as turns of wiring increase in the system. Therefore, a single-turn cable is employed in this study, and as a result, we can neglect the power loss safely and propose a simple approach described below to estimate the skin depth caused by the skin effects in the cable. To estimate the heat generation owing to an electric field in a wire, the skin depth [16,25] should be estimated. It is evident that the heat generation increases when the coil resistance increases, and therefore, the coil resistance has to be minimized by employing conductors of appropriate shapes and cross-sections. The conductor resistance is increased due to the skin effect, which distributes the current primarily on the surface of the conductor instead of uniformly over the cross-section [15]. As a piece of string of copper has a diameter of \(d=0.32\) mm and there are 540 pcs (45 pcs \(\times\) 12 strands; see Figure 1a), the overall area of the cable (\(A\)) is given by \(A=\pi d^{2}/4\times 540\) pcs. Therefore, the effective radius (\(r_{\text{e}}\)) was obtained to be \(r_{\text{e}}=3.7181\) mm from \(A=\pi r_{\text{e}}\)2. The skin depth (\(\delta\)) in the cable is caused by the skin effect. Here, \(\delta\) depends on the frequency of the current, as well as the electrical and magnetic properties of the conductor, as follows [15],  
Footnote 2: The power transfer efficiency is defined as \(\delta=\frac{\pi\delta}{\pi\delta}\). \[\delta=\sqrt{\frac{2}{\omega\sigma\mu}} \tag{1}\]  
where \(\omega\) is the angular frequency of current (\(\omega=2\pi f\) with frequency \(f=20\) kHz), \(\sigma\) is the conductivity (\(\sigma=5.8\times 10^{7}\) S/m for Cu), and \(\mu\) is the permeability (\(\mu=1.2566\times 10^{-6}\) H/m for Cu). From the above relation, the skin depth was obtained as \(\delta=0.4673\) mm. In order to verify the estimated skin depth, thermal analysis was conducted by establishing a model for the heat generation including the skin depth. The heat generation through a wire section made of copper in a duct is described in Figure 1a. The heat generation per length (\(q^{\prime}\)) can be expressed as follows [27],  
\[q^{\prime}=l^{2}\frac{R}{L} \tag{2}\]  
where \(R/L\) can be written as \(R/L=\rho/A_{\text{eff}}\) from the relationship of \(R=\rho L/A_{\text{eff}}\). Here, \(I\) is the current (in this study, \(I=125\) A with an RMS value of a 20 kHz sine wave), \(R\) is the resistance, \(L\) is the duct length (\(L=0.3\) m), \(\rho\) is the copper resistivity (\(\rho=1.68\times 10^{-8}\)\(\Omega\cdot\)m), and \(A_{\text{eff}}\) is the effective area of the copper wire (refer to Figure 2b). To determine \(A_{\text{eff}}\) of the copper cable, effective links between \(\delta\) and \(r_{\text{e}}\) need to be established, as depicted in Figure 2b. \(A_{\text{eff}}\) is easily calculated directly by \(A_{\text{eff}}=\pi\) [\(r_{\text{e}}{}^{2}-(r_{\text{e}}-\delta)^{2}\)]. Using previously obtained \(\delta\) and \(r_{\text{e}}\) and from Equation (2), the volumetric heat generation (\(q^{\prime\prime}\)) is written as follows,  
\[q^{\prime\prime}=\frac{q^{\prime}}{A_{\text{duct}}} \tag{3}\]  
where \(A_{\text{duct}}\) is the duct cross-sectional area (i.e., \(A_{\text{duct}}=H\times W\); the rectangular cavity has internal dimensions of height H = 18.8 mm and width \(W=35.5\) mm, as represented in [23]). As a result, we obtained \(q^{\prime\prime\prime}=38{,}674\) W/m\({}^{3}\) for \(I=125\) A.  
Simplified geometry of the WPT system was employed to minimize the calculation time of the system to the extent that the accuracy is not affected in the thermal analysis calculation. As mentioned previously, particular emphasis was placed on quantifying the skin depth caused by skin effects in the cable.