# Research on a Novel Nanosecond Marx Generator and Its Efficiency Analysis
## 2 The Proposed Nanosecond Pulse Generator
### The Series-Resonant Charging Power Supply  
Compared with the traditional DC voltage source, the full-bridge series-resonant source has small volume and high-efficiency advantages. \(L_{\tau}\) is the resonant inductance and \(C_{\tau}\) is the resonant capacitance. The Field Programmable Gate Array (FPGA) controls the switches S1-S4 through the driver chips TLP5702. The switches S1-S4 are turned on sequentially through timing control and a resonant current \(i_{\tau}\) is generated. The resonant current passes through the transformer to charge the energy-storage capacitors \(C_{1-5}\). According to the relationship between the full-bridge switching frequency \(f_{s}\) and the resonant frequency \(f_{\tau}\), the circuit can work in three modes [11; 12], one discontinuous current mode (DCM, where \(f_{s}<0.5f_{\tau}\)) and two continuous current modes (CCM, where \(0.5f_{\tau}<f_{s}<f_{\tau}\) or \(f_{s}>f_{\tau}\)). In the DCM mode, the switches can realize zero-current turn-on and turn-off and reduce the switching loss. Therefore, the discontinuous current mode was chosen in this research. The waveform diagram of the main operating points of the series resonant capacitor charging circuit in DCM is shown in Figure 3[13]. C\({}^{\prime}\) is the equivalent capacitance of the energy-storage capacitor \(C_{1-5}\). It can be seen that the voltage on the energy-storage capacitor increases evenly. Figure 3: The waveform of series resonant circuit in DCM. Within a switching cycle \(T_{\rm s}\), the operation of the series-resonant charging source can be divided into four phases [13; 14]. In phase 1, the switches S1 and S4 are turned on, and the DC source charges \(C_{\rm r}\) through \(L_{\rm r}\), \(i_{\rm r}>0\). In phase 2, the switches S1 and S4 are turned off, and \(C_{\rm r}\) charges \(L_{\rm r}\) through continuous diodes D1 and D4, \(i_{\rm r}<0\). Phase 3 is the same as phase 1, except that the switches S2 and S3 are turned on, \(i_{\rm r}<0\). Phase 4 is the same as phase 2, except that \(C_{\rm r}\) charges \(L_{\rm r}\) through continuous diode D2 and D4, \(i_{\rm r}>0\). The resonant current \(I\)(t) can be derived during the charging period, as shown in Equation (1). \(I\)(t) represents the resonant current in a switching cycle. At this point, the effect of isolation resistance \(R_{\rm c}\) can be ignored. \[I(\rm t)= \begin{cases}\frac{V_{\rm dc}-V_{\rm N1}-V_{\rm N1}}{\omega L_{\rm r }}\sin\omega(t-t_{0}),&t_{0}<t<t_{1}\\ \frac{V_{\rm dc}-V_{\rm N1}+V_{\rm N2}}{\omega L_{\rm r}}\sin\omega(t-t_{1}),& t_{1}<t<t_{2}\\ \frac{-V_{\rm dc}-V_{\rm N1}(1+1)^{\prime}+V_{\rm 2(N+1)}}{\omega L_{\rm r}} \sin\omega(t-t_{3}),&t_{3}<t<t_{4}\\ \frac{-V_{\rm dc}-V_{\rm N1}(1+1)^{\prime}-V_{\rm 2(N+1)}t^{\prime}}{\omega L_{\rm r }}\sin\omega(t-t_{4}),&t_{4}<t<t_{5}\\ 0,&t_{2}<t<t_{3},\,t_{5}<t<t_{6}\end{cases}  \tag{1}\]  
Here,  
\[\omega= \sqrt{\frac{1}{L_{\rm r}C_{\rm r}}+\frac{1}{L_{\rm r}C^{\prime}}} \tag{2}\]  
where \(C^{\prime}\) is the equivalent capacitance of the energy-storage capacitor \(C_{\rm 1-5}\), subscript \(N\) represents the \(N\)th resonant period, and \(\omega\) is the angular frequency of the resonant loop. The average charging current \(I_{\rm avg}\), which is constant in a resonant period \(T_{\rm r}\), can be expressed by Equation (3). \(I_{\rm peak}\) is the peak current of each phase. \[I_{\rm avg}=\frac{1}{2}\ \times\ \frac{2}{\pi}\left(I_{\rm peak1,3}+I_{\rm peak2,4}\right)\ =\ \frac{2}{\pi}\ \times\ \frac{C^{\prime}-C_{\rm r}}{(C^{\prime}+C_{\rm r})\omega L_{\rm r}}V_{\rm dc} \tag{3}\]  
The average charging current \(I_{\rm avg}^{*}\) in a switching period \(T_{\rm s}\) is given by Equation (4) [15]. It can be seen that the average value of resonant current in each cycle is the same and has nothing to do with the number of cycles, which provides a basis for the stabilization of repeated frequency discharge in subsequent experiments.