# 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
\(I_{\rm r1f,rms}\), \(I_{\rm r2f,rms}\), and \(I_{\rm r3f,rms}\) are the magnitude of the phasors \(\stackrel{{\rightarrow}}{{I}}_{\rm r1f}\), \(\stackrel{{\rightarrow}}{{I}}_{\rm r2f}\) and \(\stackrel{{\rightarrow}}{{I}}_{\rm r3f}\), respectively. Hence, \(I_{\rm r3f,rms}\) is calculated as  
\[I_{\rm r3f,rms}=\sqrt{\left(\frac{n_{1}}{n_{3}}I_{\rm r1f,rms}\right)^{2}+ \left(\frac{n_{2}}{n_{3}}I_{\rm r2f,rms}\right)^{2}+\frac{2n_{1}n_{2}}{n_{3} ^{2}}I_{\rm r1f,rms}I_{\rm z2f,rms}\cos(\varepsilon\pi)} \tag{18}\]  
From (18), under the same transfer power, when \(I_{\rm r1f,rms}\), and \(I_{\rm r2f,rms}\) are both the minimum, \(I_{\rm r3f,rms}\) is also the minimum. Therefore, the functions of the optimization objective are expressed as  
\[ \begin{array}{l}Minimized\ I_{\rm r1f,\ rms}(D_{1},\ D_{3},\ \varphi_{13})\\ Subject\ to\left\{\begin{array}{l}G_{\rm 13}(D_{1},\ D_{3},\ \varphi_{13})-G_{\rm 13} {}^{*}(D_{1}{}^{*},\ D_{3}{}^{*},\ \varphi_{13}{}^{*})=0\\ D_{1}\in[0,\ 1),\ D_{3}\in[0,\ 1),\ \varphi_{13}\in[-0.5,\ 0.5]\end{array}\right.\end{array}  \tag{19}\]  
\[ \begin{array}{l}Minimized\ I_{\rm r2f,\ rms}(D_{2},\ D_{3},\ \varphi_{23})\\ Subject\ to\left\{\begin{array}{l}G_{\rm 23}(D_{2},\ D_{3},\ \varphi_{23})-G_{\rm 23} {}^{*}(D_{2}{}^{*},\ D_{3}{}^{*},\ \varphi_{23}{}^{*})=0\\ D_{2}\in[0,\ 1),\ D_{3}\in[0,\ 1),\ \varphi_{23}\in[-0.5,\ 0.5]\end{array}\right.\end{array}  \tag{20}\]  
where \(G_{\rm 13}\) and \(G_{\rm 13}{}^{*}\), \(G_{\rm 23}\) and \(G_{\rm 23}{}^{*}\) are the index of the actual and desired transfer power of port 1 and port 2, respectively. \(G_{\rm 13}\) and \(G_{\rm 23}\) are respectively expressed as  
\[G_{\rm 13}=\cos(\frac{D_{1}\pi}{2})\cos(\frac{D_{3}\pi}{2})\sin(\varphi_{13}\pi) \tag{21}\]  
\[G_{\rm 23}=\cos(\frac{D_{2}\pi}{2})\cos(\frac{D_{3}\pi}{2})\sin(\varphi_{23}\pi)\] (22\(D_{3}\) is analyzed at first. Combining (3), (16), and (17), the average transfer power of port 1 and port 2 are respectively calculated and expressed as  
\[P_{13}=\frac{n_{1}}{n_{3}}V_{\text{EFF, rms}}I_{\text{r1f,rms}}\cos(\theta_{1}\pi)=\frac{\sqrt{8}n_{1}v_{3}}{\pi n _{3}}\cos(\frac{D_{3}\pi}{2})I_{\text{r1f,rms}}\cos(\theta_{1}\pi) \tag{23}\]  
\[P_{23}=\frac{n_{2}}{n_{3}}V_{\text{EFF, rms}}I_{\text{r2f,rms}}\cos(\theta_{2}\pi)=\frac{\sqrt{8}n_{2}v_{3}}{ \pi n_{3}}\cos(\frac{D_{3}\pi}{2})I_{\text{r2f,rms}}\cos(\theta_{2}\pi) \tag{24}\]  
where \(V_{\text{EFF,rms}}\) is the RMS value of \(v_{\text{EFF}}\). \(\theta_{1}\) is the phase difference between \(v_{\text{EFF}}\) and \(i_{\text{r1f}}\). \(\theta_{2}\) is the phase difference between \(v_{\text{EFF}}\) and \(i_{\text{r2f}}\). From (23) and (24), under the same transfer power, the smaller \(D_{3}\) is, the smaller \(I_{\text{r1f,rms}}\), and \(I_{\text{r2f,rms}}\) will be. Therefore, \(D_{3}\) is always set to 0 under the proposed modulation to minimize the transformer current. On this basis, the KKT conditions based on the Lagrange multiplier method are used to solve the nonlinear optimization problem shown in (19) and (20). According to the range of voltage gain and the transfer power, the optimization results include the following four states and are expressed as (25), (32), (34), and (36). Furthermore, it is useful to construct the phasor diagram of the fundamental equivalent converter to verify the principle of the optimization strategy in the time domain. The optimization results of the four states and the corresponding phasor diagrams are described as follows. **State 1:** when \(k_{1}<1\), \(k_{2}<1\), \({G_{13}}^{*}\leq\sqrt{1-{k_{1}}^{2}}\) and \({G_{23}}^{*}\leq\sqrt{1-{k_{2}}^{2}}\), the optimization result is expressed as  
\[\left\{ \begin{array}{l}{D_{1}}^{*}=\frac{2}{\pi}\arccos\left(\sqrt{{k_{1}}^ {2}+{G_{13}}^{*2}}\right)\\ {D_{2}}^{*}=\frac{2}{\pi}\arccos\left(\sqrt{{k_{2}}^{2}+{G_{23}}^{*2}}\right) \\ {\varphi_{13}}^{*}=\frac{1}{\pi}\arctan(\frac{{G_{13}}^{*}}{k_{1}})\\ {\varphi_{23}}^{*}=\frac{1}{\pi}\arctan(\frac{{G_{23}}^{*}}{k_{2}})\end{array} \right.