# High Frequency Transformers for Solid-State Transformer Applications
## 3 Control System  
The implemented output voltage control establishes a phase shift angle control between the primary and secondary voltage waveforms, which is only applied to the second converter of the DAB. The first converter of the DAB assumes a null angle and is constant over time [42].  
The purpose of the phase shift angle control is to regulate the power delivered to the load so that the voltage at the load terminals assumes a constant and predefined value, regardless of the load power, considering the power limitations supported by the system. Thus, the phase shift angle is determined by measuring the voltage at the load terminals.  
Figure 8 presents the block diagram of the controller. The power converter is modulated with a gain, \(K_{d}\), a delay time, \(T_{d}\), and the load dynamics. Subtracting the reference voltage, \(V_{0_{Eq}}\), from the measured voltage at the output of the converter, \(V_{0}\), provides the value of the voltage error. The gain \(\alpha\) depends on the configuration of the instrumentation used to perform the output voltage measurement, and for simulation purposes this parameter is assumed to be a unitary.  
Figure 6: Simplified HFT equivalent circuit.  
Figure 7: Converter modeling.  
The controller \(C(s)\), has as output the phase shift angle, given by \(\sin(\delta)\), allows a zero error to be obtained, so that the output voltage, \(V_{0}\), assumes the value of the reference voltage, \(V_{0_{Ref}}\).  
Based on the simplified equivalent circuit of the HFT, represented in Figure 9, where \(\widetilde{V}_{P}\) and \(\widetilde{V}_{S}\) represent the voltage at the primary and secondary terminals, respectively, performing the mesh analysis and after algebraic simplification, it is possible to obtain Equation (32).  
\[\widetilde{I}_{P}=\frac{Z_{3}\ V_{\text{S}}\!\sin(\delta)}{A}-\left(\frac{Z_{3} \ V_{\text{S}}\!\cos(\delta)+(Z_{3}+Z_{2})\ V_{P}}{A}\right)j \tag{32}\]  
To design the controller \(C(s)\), it is necessary to establish the relationship between the input power of the DAB converter, \(P_{in}\), and the output power of the DAB converter, \(P_{0}\), which corresponds to the power delivered to the load. Assuming that the converter is ideal, to simplify the controller design, i.e., that the input power is equal to the output power, it is possible to derive the expressions exposed in Equation (33).  
\[P_{0}=P_{in}=Re\!\left(\widetilde{V}_{P}\ \widetilde{I}_{P}\right)\to V_{0}=\frac{Z_{3}\ V_{P}\ V_{\text{S}}\!\sin( \delta)}{A\ I_{0}} \tag{33}\]  
Since the controller \(C(s)\) outputs \(\sin\left(\delta\right)\), and based on the equivalent controller scheme depicted in Figure 8, it is possible to conclude that the gain \(K_{d}\) will be given by Equation (34).  
\[K_{d}=\frac{Z_{3}\ V_{P}\ V_{S}}{A\ I_{0}} \tag{34}\]  
Considering the use of a proportional integral (PI) controller, starting from the diagram represented in Figure 8, consider \(\alpha=1\) and cancelling the dominant pole with the controller zero, it is possible to obtain the closed chain transfer function, represented in Equation (35).  
\[\frac{V_{0}\ (s)}{V_{0_{ref}}\ (s)}=\frac{\frac{K_{s}}{s}\ \frac{K_{d}}{1+sT_{d}}}{ 1+\frac{K_{s}}{s}\ \frac{K_{d}}{1+s}\ \frac{K_{d}}{s}} \tag{35}\]  
By simplifying Equation (35) and comparing it with the canonical second-order equation, it is possible to determine the expressions that define the proportional and integral gain, represented in Equations (36) and (37) [44], respectively.  
\[K_{I}=\frac{{{\omega_{0}}^{2}\ T_{d}}}{K_{d}\ R} \tag{36}\]  
Figure 8: Control block diagram.  
Figure 9: Simplified HFT equivalent circuit, with the primary and secondary voltages represented.  
\[K_{P}=\frac{{{\omega_{o}}^{2}}\,T_{d}}{K_{d}\,R}\,\,\tau_{c} \tag{37}\]