# Design of an Isolated Bidirectional Symmetric Resonant Converter
## 3 Equivalent Circuit and Mathematic Formulation  
For obtaining the voltage gain and its relationship with respect to frequency, the fundamental harmonic approximation method (FHA) [19] was used to build the equivalent circuit model of the forward charging mode as illustrated in Figure 7[20]. Here, \(V_{DC\_Grid\_FHA}\) is the fundamental component of the square voltage waveform at the primary side. \(V_{oe}\) is the fundamental component of the square voltage waveform of the resonant tank and transformer. The resonant capacitance at the primary side, resonant inductance at the primary side, magnetizing inductance, resonant capacitance at the secondary side, and resonant capacitance at the secondary side are designated as \(C_{r1}\), \(L_{r1}\), \(L_{m}\), \(C_{r2}\)', and \(L_{r2}\)', respectively. Moreover, power equivalent load resistance \(R_{oe}\) is the AC load resistance of the energy storage battery referred to the primary side and is expressed as (1). \[R_{oe}=\frac{8N^{2}}{\pi^{2}}.\frac{V_{Bat}}{I_{Bat}}=\frac{8N^{2}}{\pi^{2}}R_ {Bat} \tag{1}\]  
where \(N\) is the transformer turn ratio, and \(R_{Bat}\) is the equivalent resistance of battery pack. For obtaining the transfer function of voltage gain, the Laplace transform of the ratio between the output voltage and the input voltage is derived as follows. Figure 7: AC equivalent circuit of converter in the forward operation. First, the node voltage on the transformer is assigned as \(v_{x}\), and based on basic circuit theory, (2) is obtained. \[v_{x}(s)=v_{DC\ Grid\_FHA}(s)\cdot\frac{Z_{2}(s)}{Z_{1}(s)}v_{\alpha e}(s)=v_{x} (s)\cdot\frac{Z_{4}(s)}{Z_{3}(s)} \tag{2}\]  
In (2), \(Z_{1}(s)\), \(Z_{2}(s)\), \(Z_{3}(s)\), and \(Z_{4}(s)\) are the Laplace forms of \(Z_{1}\), \(Z_{2}\), \(Z_{3}\), and \(Z_{4}\), respectively, looking from different positions, and are as expressed by (3). \[Z_{1}(s)=sL_{r1}+\frac{1}{sC_{r1}}+Z_{2}(s)Z_{2}(s)=\frac{sL_{m}\left(sL_{r2}^ {\prime}+\frac{1}{sC_{r2}^{\prime}}+R_{\alpha e}\right)}{sL_{m}+sL_{r2}\prime +\frac{1}{sC_{r2}^{\prime}}+R_{\alpha e}}Z_{3}(s)=sL_{r2}\prime+\frac{1}{sC_{ r2}\prime}+R_{\alpha e}Z_{4}(s)=R_{\alpha e} \tag{3}\]  
By properly rearranging (2) and (3), the voltage gain function of the forward charging mode can be given as (4). \[M_{vr\_Forward}(s)=\frac{v_{\alpha e}(s)}{v_{DC\ Grid\_FHA}(s)}=\frac{sL_{m}// \left(sL_{r2}\prime+\frac{1}{sC_{r2}\prime}+R_{\alpha e}\right)}{sL_{r1}+ \frac{1}{sC_{r1}}+\left[sL_{m}//\left(sL_{r2}\prime+\frac{1}{sC_{r2}\prime}+R _{\alpha e}\right)\right]}\cdot\frac{R_{\alpha e}}{sL_{r2}\prime+\frac{1}{sC_{ r2}\prime}+R_{\alpha e}} \tag{4}\]  
In addition, for simplifying the mathematic operation and for easily observing the gain curve, some definitions are described as follows: \(f_{s}\) is the switching frequency of power switches;\(f_{r1\_F}\) is the first resonant frequency;\(f_{n\_F}\) is the ratio of \(f_{s}\) and \(f_{r1\_F}\), referred to as normalized frequency; \(Q_{1}\) represents quality factor; \(k_{1}\) is the ratio of \(L_{m}\) and \(L_{r1\_F}\); \(g_{1}\) and \(m_{1}\) represent the resonant capacitor ratio and resonant inductance ratio, respectively, with reference to the primary side. These are written as follows:  
\[f_{1\_F}=\frac{1}{2\pi\sqrt{L_{r1}C_{r1}}},f_{n}=\frac{f_{s}}{f_{r1\_F}},Q_{1 }=\frac{\sqrt{\frac{L_{r1}}{C_{r1}}}}{R_{\alpha e}},k_{1}=\frac{L_{m}}{L_{r1}},g_{1}=\frac{C_{r2}\prime}{C_{r1}},m_{1}=\frac{L_{r2}\prime}{L_{r1}} \tag{5}\]  
\[C_{r1}=\frac{1}{2\pi f_{r1\_F}Q_{1}R_{\alpha e}} \tag{6}\]  
\[L_{r1}=\frac{Q_{1}R_{\alpha e}}{2\pi f_{r1\_F}} \tag{7}\]  
By substituting the above defined quantities into (4), it yields (8), where \(C_{r1}\) and \(L_{r1}\) are expressed as (6) and (7).