## 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  
Using Fourier series expansion on rectifier AC pulsing voltage generated by phase shift PWM modulation, the relationship of rectifier DC voltage and AC RMS voltage can be derived as (6). Applying rectifier input AC and output DC power are balanced with power device loss ignored, the rectifier AC input RMS current and averaged output DC current is derived as (7), where \(\varphi(PF)_{rec}\) is rectifier AC input power factor angle. \[\mathrm{v_{ac\_rec}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{\mathrm{s}})*\mathrm{ V_{dc\_rec}} \tag{6}\]\[\mathrm{I_{dc_{\text{-rec}}}}=\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{I_{ ac_{\text{-rec}}}}*\cos(\varphi(PF)_{rec}) \tag{7}\]  
If the rectifier PWM gating signals are disabled, the body diodes of the rectifier H-bridge devices will work as a diode rectifier. In this case, \(\beta\)s = 0.5 (\(\beta\)s (\(\text{deg}\)) = 180\({}^{\circ}\)). The relationship of DC vs. AC RMS voltage/current of the diode bridge rectifier is further simplified as shown in (8) and (9). \[\mathrm{v_{ac\text{-rec}}}=\frac{2\sqrt{2}}{\pi}*\mathrm{V_{dc\text{-rec}}} \tag{8}\]  
\[\mathrm{I_{dc\text{-rec}}}=\frac{2\sqrt{2}}{\pi}*\mathrm{I_{ac\text{-rec}}} \tag{9}\]  
For DAB resonant converter with rectifier phase shift control, the AC load resistance of the resonant circuit is the rectifier AC resistance (Rac), which can be derived as (10) from (6) and (7). \[\mathrm{R_{ac}}=\frac{\mathrm{v_{ac\text{-rec}}}}{\mathrm{I_{ac\text{-rec}}}} =\frac{\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{s})*\mathrm{v_{ac\text{-rec}}}}{ \frac{\mathrm{I_{dc\text{-rec}}}}{\left(\frac{2\sqrt{2}}{\pi}\sin(\pi\beta_{ s})*\cos(\varphi(PF)_{rec})\right)}}=\frac{8}{\pi^{2}}(\sin(\pi\beta_{s}))^{2}\cos( \varphi(PF)_{rec})\frac{\mathrm{V_{dc\text{-rec}}}}{\mathrm{I_{ac\text{-rec}}}} \tag{10}\]  
For rectifier unity power factor operation, the relationship of rectifier AC and DC equivalent resistance is further simplified as (11), where \(\mathrm{R_{dc}}\) = \(\mathrm{V_{dc\text{-rec}}/\mathrm{I_{dc\text{-rec}}}}\) is the equivalent battery resistance. \[\mathrm{R_{ac}}=\frac{8}{\pi^{2}}(\sin(\pi\beta_{s}))^{2}\frac{\mathrm{V_{dc \text{-rec}}}}{\mathrm{I_{ac\text{-rec}}}}=\frac{8}{\pi^{2}}(\sin(\pi\beta_{s} ))^{2}R_{dc} \tag{11}\]  
Figure 3 illustrates significant equivalent battery load resistance variation in a typical CC-CV mode battery charging operation. The battery CC or CV charging mode is decided by the battery state of charge (SOC) and the battery voltage level. When the battery voltage is much smaller than its nominal value, and the battery SOC is low, the maximum charging current is applied for CC mode charging. When the battery voltage is approaching its nominal value, and the battery SOC is approaching 100%, the charging current is greatly reduced, and the battery works in CV mode charging. The load resistance of the DAB resonant converter is the equivalent battery resistance which varies significantly from CC to CV mode charging operation. For DAB resonant converter, according to Equation (11), it is possible to modify the AC load resistance of the resonant circuit by regulating the rectifier phase shift angle \(\beta\)s to improve power efficiency in battery CV mode charging operation. Figure 3: Battery equivalent resistance variation in CC-CV charging operation. 2.3 Inverter-to-Rectifier Phase Shift Angle Setting for Unity Power Factor Operation of DAB LCC Resonant Converter  
In TPS PWM modulation signal waveform shown in Figure 2, the inverter-to-rectifier phase shift angle \(\delta\) is defined as the phase angle difference between inverter voltage and rectifier voltage as expressed in (12). \[\delta=phase\left(\frac{V_{ac,inv}}{V_{ac,rec}}\right) \tag{12}\]  
The inverter-to-rectifier phase shift angle \(\delta\) setting for unity power operation of DAB LCC resonant converter bridges can be derived from the equivalent LCC resonant circuit model with inverter ZPA constraint for compensation circuit design applied.