## 3 Optimization of the Field-Null Configuration
### Typical Examples of Field-Null Optimization  
Section 3.1 presents a scheme to obtain the PF coil current waveforms in order to obtain an optimum field-null configuration, which is valid for both the circumstances of static and evolving coil currents with a self-consistent eddy current in passive conductors. This provides a useful tool for designing not only static but also evolving field-null configurations, which is particularly useful when the eddy current is high and requires careful compensation. In addition, by choosing different regions **A** in Equations (7)-(11), one can conveniently design field-null configurations for various purposes.  
For example, if the field-null area is desired to be as large as possible to ensure robust breakdown, one can choose **A** to be a big circular region as shown by the red dashed circle line in Figure 5a. The CS current is fixed to be the control target waveform used in HL-3 experiments, which is treated as known, and PF6 + PF8 coils are used to form the field-null configuration for simplicity. By following the workflow shown in Figure 3, a set of PF6 and PF8 current waveforms is obtained for minimizing the averaged \(|B_{\perp}|\) inside region **A**, as shown in Figure 5b. These waveforms, together with the preset CS current waveform, can be used as control targets in actual experiments. In another case, the PF4 and PF7 coils are used additionally as unknowns together with PF6 and PF8, a much better field-null configuration is obtained, as shown in Figure 5c. The corresponding CS and PF current waveforms are presented in Figure 5d.  
Figure 4: A typical example of the simultaneous field-null optimization and eddy current calculation. (**a**), (**b**) and (**c**) refer to \(|B_{\perp}|\) contour plots in iteration cycles 1, 29 and 30, respectively, and the red dashed circle lines refer to the desired field-null area **A** in Equations (7)–(11). In (**d**), the blue, red and magneton lines show the calculated total eddy currents inside the vacuum vessel in iteration cycles 1, 29 and 30, respectively. The coils filled with orange and red colors in (**a**)–(**c**) refer to the “unknown” and “known” coils, and those filled with white refer to coils that are not energized, respectively.  
In other situations, there might be requirements to prevent the plasma from losing vertical stability at the early stage of current ramp-up, and the oblate elliptic region as shown in Figure 6a could be chosen to make the plasma breakdown primarily near the midplane, avoiding the formation of an early plasma shape that is too elongated and could be vertically unstable. For some harsh circumstances, like the wall condition being bad and the impurity level being high, a field-null region shown in Figure 6c could be selected to align with the high-loop-voltage region on the high field side (HFS) of a tokamak. The CS and PF currents corresponding to Figure 6a,c are shown in Figure 6b,d, respectively. These waveforms can provide the basis for feedback-control simulations and could greatly reduce the time cost of tuning the field-null configuration by modifying coil currents only experimentally.  
Figure 5: Typical examples of field-null optimization using the scheme presented in Section 3.1. (**a**) The large-area case, (**b**) the large-area case using PF4 coils. (**c**) and (**d**) Show the CS and PF currents corresponding to the \(|B_{\perp}|\) contour plots in (**a**) and (**b**), respectively. The coils filled with orange and red colors in (**a**) and (**c**) refer to the “unknown” and “known” coils, and those filled with white refer to coils that are not energized, respectively. Some of the current values in (**b**,**d**) has been multiplied by a constant for a clearer view, e.g., “10”IP6” in the legend means the curve referring to IP6 in (**b**,**d**) is multiplied by 10.