# 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
As \({G_{13}}^{*}\) continues to increase, the magnitude of \(v_{\text{ABf}}\) remains at \(v_{\text{ABf,I,max}}\) to minimize the phase difference between \(i_{\text{r1}}\) and \(v_{\text{EF}}\). On this basis, the transfer power is only controlled by \(\varphi_{13}\). (2) Case II (\(k_{1}\geq 1\)): The magnitude of \(v_{\text{ABf}}\) is always smaller than that of \(v_{\text{EF}i}\) in this case, thus there must be a phase difference between \(i_{\text{r1}}\) and \(v_{\text{EF}}\). In order to minimize the phase difference between \(i_{\text{r1}}\) and \(v_{\text{EF}}\), the magnitude of \(v_{\text{ABf}}\) remains at the maximum value \(v_{\text{ABf,II,max}}\) in the entire power range. In both cases, the modulation of port 1 is simplified to SPS modulation. The corresponding soft-switching operation conditions have been analyzed in detail in [15]. The ZV-on condition of port 1 is expressed as  
\[\cos(\varphi_{13}\pi)>k_{1} \tag{33}\]  
The operation state of port 2 is the same as that of state 1, hence, the ZV-on condition of port 2 is the same as (31). For port 3, since the transformer currents of port 1 and port 2 are not in phase, it is difficult to get the analytical solution of \(i_{\text{r3}}\). Combing the analysis in [15], the ZV-on condition is \(\theta_{3}>\varphi_{13}\). \(\theta_{3}\) is the lagging angle between \(i_{\text{r3}}\) and \(v_{\text{AB}}\). **State 3:** when \(k_{1}<1\), \(k_{2}<1\), \({G_{13}}^{*}\leq\sqrt{1-{k_{1}}^{2}}\) and \({G_{23}}^{*}>\sqrt{1-{k_{2}}^{2}}\) or \(k_{1}<1\), \(k_{2}\geq 1\) and \({G_{13}}^{*}\leq\sqrt{1-{k_{1}}^{2}}\), the optimization result is expressed as  
\[\left\{ \begin{array}{l}{{D_{1}}^{*}=\frac{2}{\pi}\arccos\left(\sqrt{{k_{1}} ^{2}+{G_{1}}^{*2}}\right)}\\ {{D_{2}}^{*}=0}\\ {{\varphi_{13}}^{*}=\frac{1}{\pi}\arctan(\frac{{G_{1}}^{*}}{{k_{1}}})}\\ {{\varphi_{23}}^{*}=\frac{1}{\pi}\arcsin({G_{2}}^{*})}\\ \end{array} \right. \tag{34}\]  
The phasor diagram is shown in Figure 5c. In this state, the range of the voltage gain and transfer power between port 1 and port 3 is the same as that of state 1. According to the range of \(k_{2}\), there are two cases of state 3: \(k_{2}<1\) and \(k_{2}\geq 1\). They are represented by \"1\" and \"1\" in Figure 5c. The physical interpretation of the two cases is similar to that of state 2, thus no detailed description is given here. The soft-switching performance of this state is similar to that of state 2. The ZV-on condition of port 1 is the same as (30). The ZV-on condition of port 3 is \(\theta_{3}>\varphi_{13}\). The ZV-on condition of port 2 is expressed as  
\[\cos(\varphi_{23}\pi)>k_{2} \tag{35}\]  
**State 4:** when \(k_{1}<1\), \(k_{2}<1\), \({G_{13}}^{*}>\sqrt{1-{k_{1}}^{2}}\) and \({G_{23}}^{*}>\sqrt{1-{k_{2}}^{2}}\) or \(k_{1}\geq 1\) and \(k_{2}\geq 1\), the optimization result is expressed as  
\[\left\{ \begin{array}{l}{{D_{1}}^{*}=0}\\ {{D_{2}}^{*}=0}\\ {{\varphi_{13}}^{*}=\frac{1}{\pi}\arcsin({G_{1}}^{*})}\\ {{\varphi_{23}}^{*}=\frac{1}{\pi}\arcsin({G_{2}}^{*})}\\ \end{array} \right. \tag{36}\]  
The phasor diagram is shown in Figure 5d. In this state, according to the range of \(k_{1}\) and \(k_{2}\), there are two cases of state 4. In both cases, the modulation strategies of port 1 and port 2 are simplified to SPS modulation. (1) Case I (\(k_{1}<1\) and \(k_{2}<1\)): As \({G_{13}}^{*}\) and \({G_{23}}^{*}\) increase, both \(D_{1}\) and \(D_{2}\) reduce to 0. The magnitude of \(v_{\text{ABf}}\) and \(v_{\text{CDf}}\) increases to the maximum value \(v_{\text{ABf,I,max}}\) and \(v_{\text{CDf,I,max}}\). In order to reduce the RMS value of the transformer current, the magnitude of \(v_{\text{ABf}}\) remains at \(v_{\text{ABf,l,max}}\), and the magnitude of \(v_{\text{CDf}}\) remains at \(v_{\text{CDf,l,max}}\), as \(G_{\text{13}}\)' continues to increase. (2) Case II (\(k_{1}\geq 1\) and \(k_{2}\geq 1\)): The magnitude of \(v_{\text{ABf}}\) and \(v_{\text{CDf}}\) are smaller than that of \(v_{\text{EFf}}\).