## 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
Figure 4 shows the equivalent LCC AC resonant circuit connected to the wireless charging transformer. The parameters for Lm, Lrp, Lrs, R1, and R2 are mutual inductance, primary winding leakage inductance, secondary winding leakage inductance, primary winding resistance, and secondary winding resistance. Zac is the resonant circuit load impedance generated by rectifier control. Zac_in is the inverter AC load impedance. In the LCC resonant circuit design, the assumption is made that unity power factor operation can be achieved in rectifier control. The rectifier voltage and current are in phase with Zac = Rac. The LCC resonant circuit parameters are designed to meet the resonant requirement (13) and (14) and inverter ZPA operation constraints (15) and (16) as explained in [2,9]. The LCC resonant circuit compensation design ensures that with the unity power factor operation of the rectifier, the LCC resonant circuit compensation parameters are designed to achieve inverter ZPA operation with Zac_in = Rac_in. \[\omega_{sw}L_{1p}-\frac{1}{\omega_{sw}C_{2p}}=0 \tag{13}\]  
\[\omega_{sw}L_{1s}-\frac{1}{\omega_{sw}C_{2s}}=0 \tag{14}\]  
\[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]0  
\[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]1  
Assuming the wireless charging transformer parameters for the transmitter and receiver coils are identical (\(L_{rp}=L_{rs}=L_{r}\), \(R_{1}=R_{2}=R\)). The LCC resonant compensation circuit is symmetrical with \(L_{1p}=L_{1s}=L_{1}\), \(C_{1p}=C_{1s}=C_{1}\), and \(C_{2p}=C_{2s}=C_{2}\). The impedances of LCC AC resonant circuit components can be expressed as (17) to (22). \[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]2  
\[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]3  
\[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]4 Equivalent LCC AC resonant circuit with detailed wireless charging transformer parameters. \[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]5  
\[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]6  
\[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]7  
\[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]8  
The frequency constraint (13) and (14) is re-written as (23) in component impedances. The inverter ZPA constraint (15) and (16) is re-written as (24) in component impedances. \[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]9  
\[\mathrm{v_{ac\text{-rec}}}=\frac{2\sqrt{2}}{\pi}*\mathrm{V_{dc\text{-rec}}} \tag{8}\]0  
Utilizing KVL and KCL rule, with constraints (23) and (24) applied, the AC currents and AC voltages of the LCC resonant circuit branches are derived as a function of rectifier AC voltage \(V_{ac\_rec}\) as shown from (25) to (34). \[\mathrm{v_{ac\text{-rec}}}=\frac{2\sqrt{2}}{\pi}*\mathrm{V_{dc\text{-rec}}} \tag{8}\]1  
\[\mathrm{v_{ac\text{-rec}}}=\frac{2\sqrt{2}}{\pi}*\mathrm{V_{dc\text{-rec}}} \tag{8}\]2  
\[ \begin{array}{c}I_{RX}=I_{c2s}+\frac{V_{ac\_rec}}{Z_{ac}}\\ =\left(\frac{1}{Z_{c2}}\left(\frac{Z_{L1}}{Z_{ac}}+1\right)+\frac{1}{Z_{ac}} \right)*V_{ac\_rec}\\ =\left(\frac{1}{Z_{c2}}+\frac{1}{Z_{ac}}\left(\frac{Z_{L1}+Z_{c2}}{Z_{c2}} \right)\right)*V_{ac\_rec}=\frac{1}{Z_{c2}}*V_{ac\_rec}\text{Apply}:\ Z_{L1}+Z_{c2}=0\end{array}