# TAB Series-Resonant DC-DC Converter and Multi-Phase-Shift Based Global Optimization Modulation
## 3 Global Optimization of Multiple Phase Shifts to Minimize Ohmic Loss
\tag{25}\]  
Substituting (25) into (1)-(3), the voltage phasors of the fundamental equivalent converter satisfy  
\[\left\{ \begin{array}{l}v_{\text{AB}}\cos({\varphi_{13}}^{*}\pi)=v_{\text{ EF}}\\ v_{\text{CD}}\cos({\varphi_{23}}^{*}\pi)=v_{\text{EF}}\end{array} \right. \tag{26}\]  
Thus, the phasor diagram is shown in Figure 5a. It can be seen that \(\text{ir1f}\) and \(\text{ir2f}\) are always in a phase of vEF in this state. Both the equivalent two-port converters operate at the unity power factor. Hence, the RMS value of the transformer current is the smallest. As the transfer power increases, \(D_{1}\), \(D_{2}\), \(\varphi_{13}\) and \(\varphi_{23}\) vary accordingly. It ensures \(i_{\text{r1f}}\) and \(i_{\text{r2f}}\) are always in a phase of \(v_{\text{EFF}}\), that is, the converter operates at the minimum RMS value of the transformer current. In this state, the per-unit transfer power of port 1 and 2 are simplified and expressed as  
\[G_{13}=\cos(\frac{D_{1}\pi}{2})\sin(\varphi_{13}\pi) \tag{27}\]  
\[G_{23}=\cos(\frac{D_{2}\pi}{2})\sin(\varphi_{23}\pi) \tag{28}\]The soft-switching conditions of this state are analyzed as follows. Combining the analysis of the soft-switching principle in [25], the ZV off is realized naturally. The ZV on is achieved by gating on the in-coming power switch while the anti-parallel diode is conducting. Hence, take port 1 as an example, the ZV-on conditions of the switches are expressed as  
\[\left\{ \begin{array}{ll}i_{\rm r1}(T_{\rm h}-D_{1}T_{\rm h}/2)>0&\quad\text{ for S}_{1},\;\text{S}_{3}\\ i_{\rm r1}(D_{1}T_{\rm h}/2)<0&\quad\text{for S}_{2},\;\text{S}_{4}\end{array} \right. \tag{29}\]  
Substituting (5) into (29), the ZV-on conditions of the switches on port 1 are expressed as  
\[\left\{ \begin{array}{ll}\cos(\varphi_{13}\pi-D_{1}\pi/2)>k_{1}&\quad\text{ for S}_{1},\;\text{S}_{3}\\ \cos(\varphi_{13}\pi+D_{1}\pi/2)>k_{1}&\quad\text{for S}_{2},\;\text{S}_{4} \end{array} \right. \tag{30}\]  
Similarly, substituting (6) into (29), the ZV-on conditions of the switches on port 2 are expressed as  
\[\left\{ \begin{array}{ll}\cos(\varphi_{23}\pi-D_{2}\pi/2)>k_{2}&\quad\text{ for S}_{5},\;\text{S}_{7}\\ \cos(\varphi_{23}\pi+D_{2}\pi/2)>k_{2}&\quad\text{for S}_{6},\;\text{S}_{8} \end{array} \right. \tag{31}\]  
Meanwhile, since \(i_{\rm r3}\) is always in a phase of \(v_{\rm EF}\) in this state, the ZC on is always achieved and independent of the topology parameters for all the switches on port 3. **State 2:** when \(k_{1}<1\), \(k_{2}>1\),\({G_{13}}^{*}>\sqrt{1-{k_{1}}^{2}}\) and \({G_{23}}^{*}\leq\sqrt{1-{k_{2}}^{2}}\) or \(k_{1}\geq 1\), \(k_{2}<1\) and \({G_{23}}^{*}\leq\sqrt{1-{k_{2}}^{2}}\). The optimization result is expressed as  
\[\left\{ \begin{array}{ll}{D_{1}}^{*}=0\\ {D_{2}}^{*}=\frac{2}{\pi}\arccos\left(\sqrt{{k_{2}}^{2}+{G_{2}}^{*2}}\right) \\ \varphi_{13}^{*}=\frac{1}{\pi}\arcsin({G_{1}}^{*})\\ \varphi_{23}^{*}=\frac{1}{\pi}\arctan(\frac{{G_{2}}^{*}}{k_{2}})\end{array} \right. \tag{32}\]  
The phasor diagram is shown in Figure 5b. In this state, the range of the voltage gain and transfer power between port 2 and port 3 is the same as that of state 1. According to the range of \(k_{1}\), there are two cases of state 2: \(k_{1}<1\) and \(k_{1}\geq 1\). They are represented by \"I\" and \"II\" in Figure 5b. In both cases, the relationship between the phase shifts of port 1 and \({G_{13}}^{*}\) is the same. The modulation of port 1 is simplified to SPS modulation, while there are some differences in the physical interpretation of the two cases. Figure 5: Phasor diagram under proposed modulation. (**a**) State 1, (**b**) State 2, (**c**) State 3, (**d**) State 4. (1) Case I (\(k_{1}<1\)): As \({G_{13}}^{*}\) increases, \(D_{1}\) reduces to 0, and the magnitude of \(v_{\text{ABf}}\) increases to the maximum value \(v_{\text{ABf,I,max}}\). Since the smaller the phase difference between \(i_{\text{r1}}\) and \(v_{\text{EF}}\), the smaller the reactive power component and the RMS value of the transformer current.