## 3 Continuous-Time Average State-Space Model
### Isolation Stage (DC-DC)  
In this stage, the DAB converter is considered for DC-DC conversion. A high-frequency transformer is placed between DAB, as depicted in Figure 4a, and the equivalent diagram is depicted in Figure 4b. The DAB converter deals with three main parameters, namely, phase shift between two bridges (\(\mathcal{O}\)), duty cycle ratios (\(\mathbb{D}\)), and switching frequency (fs). The other parameters are as follows: \(\mathrm{R_{tf}}\) represents the equivalent of winding resistance and switching on resistance; \(\mathrm{L_{tf}}\) is the leakage inductance of the transformer. The leakage inductance is calculated using Equation (20). Refer to Equation (15), the variation in transformer output with different frequencies and leakage inductance is shown in Figure 5.  
Figure 4: (a) DC-DC stage architecture, (b) Equivalent diagram of DC-DC stage. Figure 3: Control block representation of input stage. Moreover, the duty cycle ratio of the proposed scheme is fixed to 0.5. Subsequently, the phase shift modulation technique is used here to control the operation of DAB. The transferred power from one stage of DAB to the other stage can be expressed in Equation (16). Further, the state-space model of the DAB can be formulated as Equation (17) by assuming the two-state variables of the proposed systems, namely, the current through the inductor (\(i_{L}\)) and the voltage across the capacitor (U\({}_{0}\)). Equations (17) and (18) are time-variant and nonlinear. To ensure that the system is time-invariant and linear, the traditional state-space averaging procedure is applied to characterize a state variable x(t) by utilizing its Fourier series expansion as given in Equation (19). To make a simple analysis, we have considered the simple full bridge converter circuit in the output side of the transformer with a single DC link. When the Fourier analysis is applied in Equations (17) and (18), Equations (20)-(25) are obtained. In this Fourier analysis, only the 0 th and 1 st-order terms are considered, while other higher-order terms of state variables are neglected. \[\mathrm{P}=\frac{\mathrm{U}^{2}}{2\times\pi\times\mathrm{f}\times\mathrm{L}} \varnothing\bigg{(}1-\frac{\varnothing}{\pi}\bigg{)} \tag{15}\]  
\[\mathrm{P}_{\mathrm{DAB}}=\frac{\mathrm{U}_{\mathrm{dc}}\mathrm{U}_{0}}{2 \mathrm{N}\mathrm{f}_{\mathrm{S}}\mathrm{L}_{\mathrm{tf}}}\mathbb{D}(1- \mathbb{D}) \tag{16}\]  
\[\mathrm{here},\;\mathrm{L}_{\mathrm{tf}}=\frac{\mathrm{L}_{1}}{\mathrm{N}^{2 }}+\mathrm{L}_{2}\]  
\[\frac{\mathrm{d}\mathrm{i}_{\mathrm{L}}(\mathrm{t})}{\mathrm{dt}}=-\frac{ \mathrm{R}_{\mathrm{tf}}}{\mathrm{L}_{\mathrm{tf}}}\mathrm{i}_{\mathrm{L}}( \mathrm{t})+\frac{\mathrm{T}_{1}(\mathrm{t})}{\mathrm{L}_{\mathrm{tf}}} \mathrm{U}_{\mathrm{dc}}(\mathrm{t})-\frac{\mathrm{T}_{3}(\mathrm{t})}{ \mathrm{L}_{\mathrm{tf}}}\mathrm{U}_{0}(\mathrm{t}) \tag{17}\]  
\[\frac{\mathrm{d}\mathrm{U}(\mathrm{t})}{\mathrm{dt}}=-\frac{1}{\mathrm{RC}_{0} }\mathrm{U}(\mathrm{t})+\frac{\mathrm{T}_{2}(\mathrm{t})}{\mathrm{C}_{0}} \mathrm{i}_{\mathrm{L}}(\mathrm{t})-\frac{\mathrm{I}_{0}}{\mathrm{C}_{0}} \tag{18}\]  
\[\mathrm{X}(\mathrm{t})=\sum_{k=\infty}^{\infty}\mathrm{X}_{k}(\mathrm{t}) \mathrm{e}^{\mathrm{i}k\omega t} \tag{19}\]  
Figure 5: Inductance versus (**a**) frequency with a phase shift of 400 V, (**b**) power with a phase shift of 400 V (**c**) frequency with a phase shift of 200 V, (**d**) power with a phase shift of 200 V.  
where \(\omega=\)2\(\pi\)f and X\({}_{\rm k}\)(t) is the k\({}^{\rm th}\) coefficient in the Fourier series analysis. Figure 4:0  
Figure 4:1  
Figure 4:2  
Figure 4:3  
Figure 4:4  
Figure 4:5  
The transient behavior of the DAB converter in terms of voltage variations is considerably slower in the input than in the output. Therefore, the 0 th term of input voltage is \(U_{dc}^{0}=\rm U_{dc}\) and \(i_{\rm o}^{0}=\rm i_{\rm o}\). The transient behavior of the first term of input voltage and load current is neglected, which are \(\rm U_{\rm 1 real}\) and \(\rm i_{\rm 1 real}\).