## 3 Results and Discussion
### Results of Response Surface Method  
The synergistic effect of any two factors on target results is analyzed by the RSM. The regression equations of the three factors on these results are formulated based on Equation (14), thus obtaining Equations (15) and (16):  
\[\begin{array}{ll}Y_{\text{Max. mean temperature}}&=14.06+10.15A+0.02234B-0.17 18C-0.4485A^{2}\\ &+0.000078B^{2}+0.001302C^{2}-0.007136 AB+0.01805 AC\\ Y_{\text{Temperature uniformity}}&=22.36-6.551A-0.01150B+0.5264C+0.5630A^{2}\\ &+0.000013B^{2}-0.003771C^{2}-0.001142 AB-0.02700 AC\end{array} \tag{15}\]  
In the above equations, A, B, and C are heating depth, current, and frequency, respectively. Figure 6 presents the response surface diagrams for the synergistic effect of any two parameters on target results. Therefore, (a), (b) and (c) illustrate the synergistic effects on maximum mean temperature and (d), (e) and (f) exhibit the synergistic effects on temperature uniformity. As mentioned above, the current has a much greater impact on the maximum mean temperature than the other two factors. It can also be observed from Figure 6, compared to increasing frequency, increasing current and heating depth can cause a greater temperature rise, with the current having a greater impact on the maximum mean temperature. This can also be observed from Figure 6a-c. Because frequency has less impact, it is the current or heating depth that exert more impact when they are present with frequency, which is consistent with the conclusions reached through ANOVA. Figure 6d indicates that the temperature uniformity first rises and then declines with the increase of heating depth within a certain range of current, while it decreases slightly with increased currents. Therefore, the change of this uniformity is not just a linear variation of increased or decreased heating depths and currents but the result from the interaction between them. When the heating depth is about 7 mm and the current fluctuates around 850 A, the temperature uniformity is optimal, which is in line with the conclusions drawn by Taguchi method. In Figure 6e,f is similar to (b) and (c). That is, frequency has little impact on temperature uniformity and, consequently, current and heating depth play a leading role.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & **DF** & **Seq SS** & **Adj MS** & **F-Value** & **\(p\)-Value** & **Contribution** \\ \hline
**Max. mean temperature (\(\overline{\text{x}}\))** & & & & & & \\ Depth/mm & 2 & 0.9878 & 0.4939 & 30.77 & 0.031 & 4.17\% \\ Current/A & 2 & 22.6452 & 11.3226 & 705.43 & 0.01 & 95.54\% \\ Frequency/KHz & 2 & 0.0688 & 0.0344 & 2.14 & 0.318 & 0.29\% \\ Error & 2 & 0.0321 & 0.0161 & & & \\ Totle & 8 & 23.7339 & & & & \\
**Temperature uniformity (s)** & & & & & & \\ Depth/mm & 2 & 136.303 & 68.1514 & 1343.91 & 0.001 & 59.59\% \\ Current/A & 2 & 91.894 & 45.9469 & 906.05 & 0.001 & 40.18\% \\ Frequency/KHz & 2 & 0.529 & 0.2647 & 5.22 & 0.161 & 0.23\% \\ Error & 2 & 0.101 & 0.0507 & & & \\ Totle & 8 & 228.827 & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 6: ANOVA for max. mean temperature and temperature uniformity.  
Figure 6: Response surface diagrams for the synergistic effect of different parameters on target results: (a) interaction of depth and current on max. mean temperature; (b) interaction of frequency and current on max. mean temperature; (c) interaction of depth and frequency on max. mean temperature; (d) interaction of depth and current on temperature uniformity; (e) interaction of frequency and current on temperature uniformity; (f) interaction of depth and frequency on temperature uniformity.