## 2 System Overview and Circuit Analysis
### Transient Analysis of Power and Information Channels  
In the previous Section, the steady-state characteristics of the SS compensated WPT system are analyzed. However, if the SWPIT system switches from the normal power transmission mode to the simultaneous power and information transmission mode, the steady-state circuit model cannot be used to analyze the behavior of the resonant tank. By using the reciprocity theorem, the characteristics of forward and backward communication are identical. Therefore, in this Section, only the backward communication is studied, and its equivalent circuit is shown in Figure 4a.  
During the backward modulation, the resonant tank departs from its steady state. For simplicity, it is assumed that the secondary compensation circuit is in the resonant state whereas the primary compensation circuit is in the non-resonant state. Then, the full-order reflected impedance circuit in Figure 4a can be simplified as the second-order equivalent circuit model shown in Figure 4b, where the reflected impedance \(Z_{\text{r}}\) seen from the primary side is given by  
\[Z_{\text{r}}=\frac{\omega^{2}M^{2}}{Z_{\text{s}}}. \tag{7}\]  
where \(Z_{\text{s}}=j\omega L_{\text{s}}+1/\left(j\omega C_{\text{s}}\right)+R_{\text {p}}+R_{\text{e}}\), and the equivalent resistance seen from secondary side is \(R_{\text{e}}\). If the WPT system works in the resonant state, the reflection impedance has pure resistive characteristic.  
Figure 4: Equivalent circuit for transient analysis. (**a**) Full-order reflected impedance circuit model. (**b**) Simplified equivalent circuit model.  
To analyze the simplified second-order circuit shown in Figure 4b, the equivalent impedance \(R_{\rm eq}\) is derived as  
\[R_{\rm eq}=R_{\rm p}+\frac{(\omega M)^{2}}{R_{\rm s}+R_{\rm e}}. \tag{8}\]  
For simplicity, the initial communication process is regarded as zero-voltage response [31]. Then, the differential equation of the simplified circuit shown in Figure 4b is expressed as  
\[L_{\rm p}C_{\rm p}\frac{d^{2}i_{\rm p}(t)}{dt^{2}}+R_{\rm eq}C_{\rm p}\frac{di _{\rm p}(t)}{dt}+i_{\rm p}(t)=0. \tag{9}\]  
Solving (9), the instantaneous current \(i_{\rm p}\) in primary side is derived as  
\[i_{\rm p}=\frac{-v_{\rm c}(0)}{\omega L_{\rm p}}e^{-t/\tau}\sin\omega t+i_{\rm p }(0)\frac{\omega}{\omega_{0}}e^{-t/\tau}\cos(\omega t+\theta). \tag{10}\]  
where \(v_{\rm c}(0)\) and \(i_{\rm p}(0)\) are initial voltage on \(C_{\rm p}\) and initial current in \(L_{\rm p}\), respectively. The relationship between attenuation angular frequency and operation angular frequency is \(\omega_{0}=\omega\sqrt{1-\xi^{2}}\), in which \(\omega=1/\sqrt{L_{\rm p}C_{\rm p}}\). In order to decrease the transient response time, the equivalent circuit is designed in an under-damped state and \(\xi<1\). Assuming that the switching point of the phase-shifting angle always occurs at zero point, then the value of \(i_{\rm p}(0)\) is zero. At the same time, \(v_{\rm c}(0)\) reaches its maximum value, then (10) can be simplified as  
\[i_{\rm p}=\frac{v_{\rm p}(t)+v_{\rm c}(0)}{\omega L_{\rm p}}e^{-t/\tau}\sin \omega t. \tag{11}\]  
where, \(\tau=2L_{\rm p}/R_{\rm eq}\). According to (8) and (11), the transient response time \(\tau\) of primary current is calculated as  
\[\tau=\frac{2L_{\rm p}}{R_{\rm p}+\frac{(\omega M)^{2}}{R_{\rm s}+R_{\rm e}}}. \tag{12}\]  
In a well-designed WPT system, the equivalent resistances of coils satisfy \(R_{\rm p}\approx R_{\rm s}\ll R_{\rm e}\). As evident from (12), in order to achieve high data rate in the communication process, the value of \(L_{\rm p}\) and \(R_{\rm e}\) should decrease, and/or the value of \(\omega\) and \(M\) should increase.