## 2 System Overview and Circuit Analysis
### Power Transmission Characteristics of SS Compensated Circuit  
As mentioned in the previous Section, the information and power channels share the same modulator, so it is necessary to evaluate the power transmission characteristics. In order to simplify the analysis, a common serial-serial (SS) compensation circuit is selected and studied. As shown in Figure 3, the equivalent circuit based on the mutual inductance model is established.  
The operating frequency studied in this paper covers the Qi standard and SAE J2954 recommended practice, which ranges from 80 kHz to 200 kHz. Ignoring the stray capacitance of coils and semiconductors and supposing the WPT system in the resonant state [28], the operating angular frequency \(\omega\) of the system is given by  
\[\omega=\frac{1}{\sqrt{L_{\mathrm{P}}C_{\mathrm{P}}}}=\frac{1}{\sqrt{L_{\mathrm{ s}}C_{\mathrm{s}}}} \tag{1}\]The subscripts \"p\" and \"s\" represent the primary side and the secondary side, respectively. The symbols \(L\), \(C\) and \(R\) are the corresponding self-inductance, capacitance, and equivalent resistance, respectively. \(M\) is coupling inductance which is equal to \(k\sqrt{L_{\rm P}L_{\rm s}}\). \(V_{\rm p}\) is the voltage generated by power inverter, and \(R_{\rm e}\) is the equivalent resistance seen from rectifier.  
By using the fundamental harmonic approximation (FHA) method and [PERSON]'s voltage low (KVL), the following matrix equation can be obtained [29].  
\[\left[\begin{array}{cc}Z_{\rm p}&-j\omega M\\ -j\omega M&Z_{\rm s}\end{array}\right]\left[\begin{array}{c}I_{\rm p}\\ I_{\rm s}\end{array}\right]=\left[\begin{array}{c}V_{\rm p}\\ 0\end{array}\right] \tag{2}\]  
where, \(Z_{\rm p}=j\omega L_{\rm p}+1/(j\omega C_{\rm p})+R_{\rm p}\) and \(Z_{\rm s}=j\omega L_{\rm s}+1/(j\omega C_{\rm s})+R_{\rm s}+R_{\rm e}\). According to (1) and (2), the root-mean-square (RMS) value of primary resonant current \(I_{\rm p}\) and secondary resonant current \(I_{\rm s}\) are given by  
\[\left\{\begin{array}{c}I_{\rm p}=\frac{\left(R_{\rm s}+R_{\rm e}\right)V_{ \rm p}}{\omega^{2}M^{2}+R_{\rm p}\left(R_{\rm s}+R_{\rm e}\right)}\\ I_{\rm s}=\frac{j\omega MV_{\rm p}}{\omega^{2}M^{2}+R_{\rm p}\left(R_{\rm s}+R_ {\rm e}\right)}.\end{array}\right. \tag{3}\]  
Figure 3: Equivalent mutual inductance model of serial–serial (SS) compensated circuit.  
Figure 2: Schematic diagram of the proposed simultaneous wireless power and information transmission (SWPIT) system.  
According to (3), the input power \(P_{\text{in}}\) from the full-bridge inverter and output power \(P_{\text{out}}\) of the secondary compensation circuit are calculated as (4) and (5), respectively.  
\[P_{\text{in}}=\Re\left[V_{\text{P}}(I_{\text{p}})^{*}\right]=\frac{(R_{\text{s} }+R_{\text{e}})\,V_{\text{p}}^{2}}{\omega^{2}M^{2}+R_{\text{p}}\left(R_{\text{s }}+R_{\text{e}}\right)} \tag{4}\]  
\[P_{\text{out}}=\Re\left(R_{\text{e}}|I_{\text{s}}|^{2}\right)=\frac{R_{\text{e }}(\omega M)^{2}V_{\text{p}}^{2}}{\left[\omega^{2}M^{2}+R_{\text{p}}\left(R_{ \text{s}}+R_{\text{e}}\right)\right]^{2}} \tag{5}\]  
where \(\Re(\cdot)\), \((\cdot)^{*}\) and \(|\cdot|\) represent the real, conjugate and absolute variables, respectively. Taking the ratio of (5) and (4), the system efficiency \(\eta\) is calculated as  
\[\eta=\frac{P_{\text{out}}}{P_{\text{in}}}=\frac{\omega^{2}M^{2}R_{\text{e}}}{ \omega^{2}M^{2}(R_{\text{p}}+R_{\text{e}})+R_{\text{s}}(R_{\text{p}}+R_{ \text{e}})^{2}}. \tag{6}\]  
For a well-designed WPT system, the values of \(\omega\), \(M\), \(R_{\text{p}}\) and \(R_{\text{s}}\) are usually fixed [30], and the input power and output power are mainly determined by \(V_{\text{p}}\) and \(R_{\text{e}}\), respectively. It can be seen from (6) that the system efficiency is related to \(V_{\text{p}}\), but it is determined by \(R_{\text{e}}\). Thus, the modulation of input voltage \(V_{\text{p}}\) has the potential to realize forward communication while the modulation of the equivalent resistance \(R_{\text{e}}\) can realize backward communication.