## 2 DAB DC/DC Resonant Converter TPS Close Loop Control Scheme
### Mathematical Formulation of DAB TPS Control Method
#### 2.2.2 Resonant Circuit Load Resistance Regulation with Rectifier Phase Shift Control
 \tag{27}\]  
\[V_{m}=I_{RX}*Z_{1}+V_{c2s}=\left(\frac{Z_{1}}{Z_{c2}}+\frac{Z_{L1}}{Z_{ac}}+ 1\right)*V_{ac\_rec}=\left(\frac{Z_{1}+Z_{c2}}{Z_{c2}}+\frac{Z_{L1}}{Z_{ac}} \right)*V_{ac\_rec} \tag{28}\]  
\[I_{m}=\frac{V_{m}}{Z_{m}}=\frac{1}{Z_{m}}\left(\frac{Z_{1}+Z_{c2}}{Z_{c2}}+ \frac{Z_{L1}}{Z_{ac}}\right)V_{ac\_rec} \tag{29}\]  
\[ \begin{array}{c}I_{TX}=I_{RX}+I_{m}=\left(\frac{1}{Z_{c2}}+\frac{1}{Z_{m}} \left(\frac{Z_{1}+Z_{c2}}{Z_{c2}}+\frac{Z_{L1}}{Z_{ac}}\right)\right)V_{ac\_ rev}\\ =\left(\frac{1}{Z_{c2}}\left(\frac{Z_{m}+Z_{1}+Z_{c2}}{Z_{m}}\right)+\frac{Z_ {L1}}{Z_{m}Z_{ac}}\right)V_{ac\_rev}\\ =\left(\frac{1}{Z_{c2}}\left(\frac{Z_{m}+Z_{r}+Z_{c1}+R1+Z_{c2}}{Z_{m}}\right) +\frac{Z_{L1}}{Z_{m}Z_{ac}}\right)V_{ac\_rev}\text{Apply}:\ Z_{1}=Z_{r}+Z_{c1}+R 1\\ =\left(\frac{R1}{Z_{c2}Z_{m}}+\frac{Z_{1}}{Z_{m}Z_{ac}}\right)V_{ac\_rev} \text{Apply}:\ Z_{m}+Z_{r}+Z_{c1}+Z_{c2}=0\end{array}  \tag{30}\]  
\[ \begin{array}{c}V_{c2p}=I_{TX}*Z_{1}+V_{m}=\left(Z_{1}\left(\frac{R1}{Z_{c2 }Z_{m}}+\frac{Z_{L1}}{Z_{m}Z_{ac}}\right)+\frac{Z_{1}+Z_{c2}}{Z_{c2}}+\frac{Z _{L1}}{Z_{ac}}\right)V_{ac\_rev}\\ =\left(\frac{Z_{1}R1+Z_{m}Z_{1}+Z_{m}Z_{c2}}{Z_{c2}Z_{m}}+\left(\frac{Z_{1}+Z_ {m}}{Z_{m}}\right)\frac{Z_{L1}}{Z_{ac}}\right)V_{ac\_rev}\end{array}  \tag{31}\]  
\[ \begin{array}{c}I_{c2p}=\frac{V_{c2p}}{Z_{c2}}=\left(\frac{Z_{1}R1\ +\ Z_{m}Z_{1}\ +\ Z_{m}Z_{c2}}{(Z_{c2})^{2}Z_{m}}+ \left(\frac{Z_{1}+Z_{m}}{Z_{c2}Z_{m}}\right)\frac{Z_{L1}}{Z_{ac}}\right)V_{ac \_rev}\end{array}  \tag{32}\]\[\begin{split} I_{ac_{inv}}=I_{TX}+I_{c2p}\\ =\left(\frac{R1}{Z_{c2}Z_{m}}+\frac{Z_{11}}{Z_{m}Z_{ac}}+\frac{Z_{1 }R1+Z_{m}Z_{1}+Z_{m}Z_{2}}{(Z_{c2})^{2}Z_{m}}+\left(\frac{Z_{1}+Z_{m}}{Z_{c2}Z_ {m}}\right)\frac{Z_{11}}{Z_{ac}}\right)V_{ac\_rv}\\ =\left(\frac{(R1+Z_{m})(Z_{c2}+Z_{1})}{(Z_{c2})^{2}Z_{m}}+\left( \frac{Z_{c2}+Z_{c1}+R2+Z_{m}}{Z_{c2}Z_{m}}\right)\frac{Z_{11}}{Z_{ac}}\right)V _{ac\_rv}\text{Apply}:\ Z_{1}=Z_{r}+Z_{c1}+R1\\ =\left(\frac{(R1+Z_{m})(Z_{c2}+Z_{1})}{(Z_{c2})^{2}Z_{m}}+\left( \frac{R2}{Z_{c2}Z_{m}}\right)\frac{Z_{11}}{Z_{ac}}\right)V_{ac\_rv}\text{Apply}: \ Z_{m}+Z_{r}+Z_{c1}+Z_{c2}=0\\ V_{ac\_rv}=I_{ac\_rv}\times Z_{11}+V_{c2p}\\ =\left(\frac{Z_{11}(R1+Z_{m})(Z_{c2}+Z_{1})}{(Z_{c2})^{2}Z_{m}}+ \left(\frac{R2}{Z_{c2}Z_{m}}\right)\frac{(Z_{11})^{2}}{Z_{ac}}+\frac{Z_{1}R1+Z_ {m}Z_{1}+Z_{m}Z_{2}}{Z_{c2}Z_{m}}+\left(\frac{Z_{1}+Z_{m}}{Z_{m}}\right)\frac {Z_{11}}{Z_{ac}}\right)V_{ac\_rec}\\ =\left(\frac{Z_{11}(R1+Z_{m})(Z_{c2}+Z_{1})+Z_{c2}(Z_{1}R1+Z_{m} Z_{1}+Z_{m}Z_{c2})}{(Z_{c2})^{2}Z_{m}}+\left(\frac{R2*Z_{11}+Z_{2}(Z_{1}+Z_{m})}{Z_{ c2}}\right)\frac{Z_{11}}{Z_{m}Z_{ac}}\right)V_{ac\_rec}\\ =\left(\frac{Z_{11}(R1+Z_{m})(Z_{c2}+Z_{1})+Z_{c2}Z_{1}R1+Z_{c2} Z_{m}(Z_{1}+Z_{c2})}{(Z_{c2})^{2}Z_{m}}+\left(\frac{R2*Z_{11}+Z_{c2}Z_{1}+Z_{c2}Z _{m}}{Z_{c2}}\right)\frac{Z_{11}}{Z_{m}Z_{ac}}\right)V_{ac\_rec}\\ =\left(\frac{(Z_{11}R1+Z_{m}(Z_{11}+Z_{m}))(Z_{1}+Z_{c2})+Z_{c2} Z_{1}R1}{(Z_{c2})^{2}Z_{m}}+\left(\frac{R2*Z_{11}+Z_{c2}(R1-Z_{c2})}{Z_{c2}} \right)\frac{Z_{11}}{Z_{m}Z_{ac}}\right)V_{ac\_rec}\\ =\left(\frac{Z_{11}R1Z+Z_{11}R1Z_{c2}+Z_{c2}Z_{1}R1}{(Z_{c2})^{2 }Z_{m}}-\frac{Z_{c2}Z_{11}}{Z_{m}Z_{ac}}\right)V_{ac\_rec}\text{Apply}:\ \begin{cases}Z_{11}+Z_{c2}=0\\ Z_{m}+Z_{r}+Z_{c1}+Z_{c2}=0\\ R1=R2\end{cases} \] \[=\left(\frac{Z_{11}R1}{Z_{c2}Z_{m}}-\frac{Z_{c2}Z_{11}}{Z_{m}Z_{ ac}}\right)V_{ac\_rec}\text{Apply}: \begin{cases}Z_{11}+Z_{c2}=0\\ Z_{m}+Z_{r}+Z_{c1}+Z_{c2}=0\\ R1=R2\end{cases} \] \[=\left(\frac{Z_{11}R1}{Z_{c2}Z_{m}}-\frac{Z_{c2}Z_{11}}{Z_{m}Z_{ ac}}\right)V_{ac\_rec}\text{Apply}: \begin{cases}Z_{11}+Z_{c2}=0\\ Z_{c1}+Z_{c2}=0\end{cases} \] \[=\left(\frac{Z_{11}R1}{Z_{c2}Z_{m}}-\frac{Z_{c2}Z_{11}}{Z_{m}Z_{ ac}}\right)V_{ac\_rec}\text{Apply}: \begin{cases}Z_{11}+Z_{c2}=0\\ Z_{c1}+Z_{c2}=0\end{cases} \] \[=\left(R1*Z_{ac}-\frac{(Z_{c2})}{Z_{c2}}\right)\frac{Z_{11}}{Z_{ m}Z_{c2}Z_{ac}}V_{ac\_rec}\]  
Under the condition that the rectifier operates at unity power factor with Zac = Rac, the inverter-to-rectifier phase angle \(\delta\) is derived as (35). \[