## 3 Results
### Ringing in the Charging Path  
After the end of the resonant charging process, the voltage at the generator's terminal for charging exhibits ringing. Thereby, voltage and current, as shown in Figures 13 and 14, are shifted in phase by 90\({}^{\circ}\). These oscillations have also been seen in measurements [10].  
In order to explain these oscillations, the capacitance between the two windings of one current-compensated choke has been measured. The measurement was performed using an LCR bridge [15] at a frequency of 10 kHz. The measurement result of 50 nF has been considered in the circuit model according to Figure 3 with the parameter C_STR = 25 nF, i.e., half of the measured value, which has been added to both terminals of each current-compensated choke to form a pi-shaped equivalent circuit. The high value of 50 nF for this capacitance is reasonable due to an interleaved winding construction of the choke in order to keep the leakage inductance small and the use of a wire with a rectangular cross-section of 10 mm \(\times\) 1 mm [16].  
The reason for this oscillation is the capacitance between the two windings of the current-compensated chokes. During the resonant charging process, the resonant circuit is formed by the inductance L15 and the sum of all stage capacitors C_ST plus the sum of the capacitances between the windings of the current-compensated chokes C_STR. After the charging process, the stage capacitors are disconnected by the blocking diodes and the remaining part of the resonant circuit formed by L15 and the sum of the winding capacitances C_STR causes the oscillations.  
In order to investigate this lemma in more detail, the differential equation describing the oscillations has been considered:  
\[v_{C\_STR}(t)+R\cdot C\cdot\dot{v}_{C\_STR}(t)+L15\cdot C\cdot\dot{v}_{C\_STR} (t)=V_{0} \tag{15}\]  
Thereby, the capacitance \(C\) represents the sum of the winding capacitances \(C\_STR\) and \(R\) the resistive losses of the resonance circuit.  
For the solution of the differential equation, the following initial conditions just after charging have been assumed: \(v_{C}(0)=2V_{0}\) and \(\dot{v}_{C}(0)=0\)\(\frac{V}{s}\).  
The observed oscillation is described by the damped oscillatory case of the three possible solutions of the differential equation.  
Figure 17: Charging voltage V\({}_{\rm ch}\) and the charging current I\({}_{\rm ch}\) at the charging input terminal of the generator, on-time of the solid-state switches: t\({}_{\rm on}\) = 10 \(\mu\)s.  
\[v_{\text{C\_STR}}(t)=V_{0}\left(1+e^{-\frac{R}{2}\cdot t}\left(\cos(\sqrt{\frac{1}{ LC}-\frac{R^{2}}{4L^{2}}}\cdot t)+\frac{1}{\sqrt{\frac{4L}{R^{2}C}-1}}\sin(\sqrt{ \frac{1}{LC}-\frac{R^{2}}{4L^{2}}}\cdot t)\right)\right) \tag{16}\]  
with \(L\) being equal to L15.  
The resonance frequency is described by  
\[f_{\text{res}}=\frac{1}{2\pi}\sqrt{\frac{1}{LC}-\frac{R^{2}}{4L^{2}}} \tag{17}\]  
For an only weakly damped system the approximation of \(R=0\) Ohm can be applied. The resonance frequency for L15 = 1760 uH and C = 8:50 nF = 400 nF is \(f_{\text{res}}=5998\) Hz. This value fits well with the measured frequency of the oscillations of 5963 Hz according to [10], and, hence, supports the above-stated lemma. The measurements are also shown in the section \"Discussion\".  
The charging circuit has been improved by adding a diode in series to L15. This diode avoids a discharge of the winding capacitances through L15 and, hence, the oscillations.