# Stability and Heat Input Controllability of Two Different Modulations for Double-Pulse MIG Welding
## 2 Design of Current Waveform Parameters for Double-Pulse MIG Welding with Two Different Modulations
The peak of the high-energy pulse train is the same as that of the low-frequency modulation signal, whereas its base value is \(I_{bs}\). The duty cycle is \(D_{s}\). Based on these parameters, the corresponding time of the high-frequency signal in [0, \(T_{hs}\)] is:  
\[\widetilde{t}_{hs}=\widetilde{t}_{l}-INT\bigg{(}\frac{\widetilde{t}_{l}}{T_{ hs}}\bigg{)}\times T_{hs}. \tag{3}\]  
Figure 5: High-frequency modulation signal energy pulse group graph: (**a**) High-energy pulse train of the high-frequency modulation signal; (**b**) Low-energy pulse train of the high-frequency modulation signal. The current values corresponding to this period can be expressed as:  
\[I_{hs}=\bigg{(}\frac{I_{p}+I_{bs}}{2}\bigg{)}-\bigg{(}\frac{I_{p}-I_{bs}}{2} \bigg{)}sign\bigg{\{}\widetilde{t}_{l}-INT\bigg{(}\frac{\widetilde{t}_{l}}{T_{ hs}}\bigg{)}\times T_{hs}-D_{s}T_{hs}\bigg{\}}. \tag{4}\]  
According to the same principles, the period of the low-energy pulse train is \(T_{hw}\), and its interval is \([T_{s}+T_{d},T_{s}+T_{d}+T_{hw}]\). In this interval, the time and current values corresponding to the high-frequency signal are:  
\[\widetilde{t}_{hw}=\widetilde{t}_{l}-T_{s}-T_{d}-INT\bigg{(}\frac{\widetilde{ t}_{l}-T_{s}-T_{d}}{T_{hw}}\bigg{)}\times T_{hw} \tag{5}\]  
\[I_{hs}=\bigg{(}\frac{I_{p}+I_{bs}}{2}\bigg{)}-\bigg{(}\frac{I_{p}-I_{bs}}{2} \bigg{)}sign\bigg{\{}\widetilde{t}_{l}-T_{s}-T_{d}-INT\bigg{(}\frac{\widetilde {t}_{l}-T_{s}-T_{d}}{T_{hs}}\bigg{)}\times T_{hs}-D_{s}T_{hw}\bigg{\}}. \tag{6}\]  
It should be noted that the following relation must be satisfied to ensure the transfer of one droplet for each pulse:  
\[D_{s}T_{hs}=D_{w}T_{hw}. \tag{7}\]  
This can be rewritten as:  
\[\frac{D_{s}}{D_{w}}=\frac{T_{hw}}{T_{hs}}=\frac{f_{hs}}{f_{hw}}. \tag{8}\]  
The high- to low-energy and low- to high-energy transitions have a fixed time span, and the corresponding energy variation plots are shown in Figure 6. In the interval \([T_{s},T_{s}+T_{d}]\) of the high- to low-energy transition, the base current can be expressed as:  
\[I_{b}=I_{bs}-\frac{I_{bs}-I_{bw}}{T_{d}}(\widetilde{t}_{1}-T_{s}). \tag{9}\]  
The period in Figure 6(a) that corresponds to the interval \([T_{s},T_{s}+T_{hs}]\) is:  
\[\widetilde{t}_{hd}=\widetilde{t}_{l}-INT\bigg{(}\frac{\widetilde{t}_{l}-T_{s} }{T_{hs}}\bigg{)}\times T_{hd}-T_{s}. \tag{10}\]  
The corresponding currents are given as:  
Figure 5:0  
Figure 5:1 Transition pulse group of high-frequency modulation signal: (**a**) High- and low-energy transition pulse group; (**b**) Low-energy transition pulse group. Similarly, the base current corresponding to the interval [\(T_{s}+T_{d}+T_{w}\), \(T\)] of the transitional low-to high-energy pulse train can be determined. In Figure 6b, the period corresponding to the interval [\(T_{s}+T_{d}+T_{w}\), \(T\)] is:  
Figure 5:2  
The corresponding current values can be expressed as:  
\[ \begin{split}& I_{hl}=\left(\frac{I_{p}+I_{bw}+\frac{I_{bs}-I_{bw}}{ T_{d}}(\widetilde{t}_{1}-T_{s}-T_{d}-T_{w})}{2}\right)-\left(\frac{I_{p}-I_{ bw}-\frac{I_{bs}-I_{bw}}{T_{d}}(\widetilde{t}_{1}-T_{s}-T_{d}-T_{w})}{2} \right)\\ &\times sign\bigg{\{}\widetilde{t}_{1}-INT\bigg{(}\frac{ \widetilde{t}_{1}-T_{s}-T_{d}-T_{w}}{T_{hs}}\bigg{)}\times T_{hs}-T_{s}-D_{s }T_{hs}\bigg{\}}\end{split}  \tag{13}\]  
The calculations show that the total number of high- (low-) energy pulse trains will decrease with increases in \(t_{bw}\) and \(t_{bs}\), if \(T_{w}\) and \(T_{s}\) are fixed. Therefore, if the current is to be limited to lower values, it is necessary to reduce \(n\) and \(m\) while appropriately increasing \(t_{bw}\) and \(t_{bs}\). In addition, the dependence of current on the number of pulses must be closely monitored to minimize the latter's impact on current. In conventional DP-MIG welding, the energy of the welding processes undergoes rapid switching, which is detrimental to welding stability. In addition, this increases the probability of screw-like wires being formed when the arc is started in \"cold plate\" conditions.