In this paper the current and voltage ringing of the DC/AC converter, proposed by the author in [1] , is analyzed.

It is convenient to use the circuit shown in Figure 1, which describes an alternating current with a small triangular ripple through an RL load.

The circuit can be seen as a conventional voltage source inverter when v_SA is positive and not zero and when S_B18, S_B20, S_B22, and S_B24 are switched on. On the other hand, when switches S_B17, S_B19, S_B21, and S_B23 are switched on and v_SA is negative and not zero, it can also be seen as a conventional inverter. Therefore, it can be modulated as a conventional inverter in each half period. This converter is similar to a high frequency link DC/AC converter [2] - [4] .

To carry out the ringing analysis a parasitic capacitor is placed in parallel with each switch, as shown in Figure 1, to take into account the higher-frequency oscillations in the converter. The IGBT internal capacitance value was obtained from the manufacturer’s data sheet [5] .

The circuit in Figure 1 is simulated using the PSIM software, and the results are given in Figure 2. The converter is built and tested and experimental results will be presented in the final version of this paper.

As shown in Figure 2, there are three different oscillations in v_SA during the first half-period, which are also repeated in the second half-period. Each oscillation has a different path, and the three paths are highlighted in Figure 1 [6] [7] .

The first oscillation type appears when v_SA changes from −U_SB to +U_SB. Figure 1(a) shows the current loop in this case. In this mode, the load current i_LS (shown in Figure 2) is flowing through the upper switches (S_B17, S_B18, S_B21, and S_B22); therefore, there is no current through the transformer. The resulting circuit loop consists of resistors, inductances, and capacitors, where the oscillating current passes through the capacitors in parallel with the switches S_B19 and S_B23. Since the upper switches are conducting the load current, the voltages across S_B19 and S_B23 are the same. Because the capacitor C_SB has a high value, its internal resistance R_CSB is quite low (2 mΩ); thus, the SB port can also be seen as a short circuit at high frequencies.

In order to predict the behavior of the voltage across the snubber capacitors in parallel with S_B19 and S_B23 a formulation is developed in the following. To carry out the calculations, the well-known relationship for a RLC network is applied.

The oscillation frequency of a weakly-damped parallel resonant RLC network is known to be

Using the parameters from [8] , (1) results in a frequency of 3.3 MHz. The wiring and transformer resistances are considerably larger at such a high frequency than at 10 kHz. The primary and secondary resistances of each winding of the transformer were measured at 3.3 MHz and it was found to be equal to 1.9 Ω. The cables of the prototype were also measured at this frequency. Each meter of cable presents 300 mΩ resistance. Because the complete prototype is expected to use around 2 meters of cable (see prototype in [8] ), an extra 600 mΩ must be added. Therefore, a total resistance of 4.4 Ω was inserted in series with the IGBT parasitic capacitors to study the voltage overshoot in the simulated circuit.

Using the parameters shown in [8] , the damping factor can be calculated as [9]

resulting in ζ = 0.05. A system with a damping factor lower than unity is expected to be underdamped.

The general expression for the capacitor voltage in a RLC network [10] [11] is given by

where v_f is the DC voltage component, and s₁ and s₂ are, respectively,

The coefficients A₁ and A₂ are determined by the boundary conditions. For the specific situation in Figure 1(a), A₁ and A₂ are given by, respectively,

where C is the equivalent capacitance of C_SB19 and C_SB23 in parallel, i_PA is the initial current through the transformer leakage inductance, and v_oC is the initial voltage across the capacitor C.

For the first oscillation mode, the initial current through the transformer leakage inductance and the initial voltage across C_SB19 and C_SB23 are both zero. Under these initial conditions, the voltage waveform across the switches S_B19 and S_B23 is determined by (3). The results were plotted with MATLAB software and they are similar to the simulation outcome from the PSIM software, which is shown in the Figure 3.

Independent of the current value that circulates through the load, the voltage waveform across the switches S_B19 and S_B23 never changes in the first oscillation mode because the initial conditions are always the same. The resonant frequency of the circuit was calculated and confirmed by simulation to be 3.3 MHz.

The second oscillation mode in Figure 2(a) occurs when the power source U_SB starts sending energy to the load, as shown in Figure 1(b). At that moment, the voltage across the switch S_B19 is clamped with the load voltage (v_SG). The voltage v_SG can be calculated because the current through the load at this moment is known. Therefore, the initial conditions in this case are the voltage value of v_SG (at that specific moment) and zero current through the transformer leakage inductance. Using these quantities and (3), a theoretical voltage waveform across S_B19 for the second oscillation mode can be determined. The voltage waveforms across S_B19 and S_B21 are the same in this case.

The third oscillation mode starts when the power source U_SB stops sending energy to the load, as shown in Figure 1(c). At this moment, the current through the leakage inductances (L_PA and L_SA) are equal to the load current i_LS, the voltage across S_B21 is equal to U_SB, and the voltage across S_B17 is zero. Thus, the currents coming from the leakage inductances and from the capacitor in parallel with S_B21 start charging the capacitor in parallel with S_B17. When the voltages across S_B21 and S_B17 reach the same value, oscillation starts. Under these initial conditions, the theoretical voltage waveform across the switches S_B17 and S_B21 can be calculated. When the load current is negative, the same overvoltage is generated across the switches S_B20 and S_B24. As such, the highest overvoltage will always occur on the outer switches S_B17, S_B20, S_B21, and S_B24, as shown in Figure 1.

The three oscillation modes can be summarized as follows. Once the equivalent RLC circuit parameters are determined, the first oscillation mode is dependent on the voltage U_SB, the second oscillation mode is imposed by the load current i_LS and voltage v_SG, and the third oscillation mode is based on i_LS and U_SB. For the parameters shown in Table 1 and i_LS = 40 A, the maximum overvoltage is approximately 1500 V, as shown in Figure 2(c).

Consequently, with the method described in this section it is possible to determine the maximum peak voltage across the matrix converter switches for the maximum load current (i_LS = 40 A). The results in Figure 2 show a peak voltage of approximately 1500 V over the outer switches (S_B17, S_B20, S_B21, and S_B24). The outer switches have the most stress because they are switched off when the load current is at its highest value. Thus, snubber capacitors should be connected in parallel with these switches. The design of these snubber capacitors is described in the following section. The inner switches (S_B18, S_B19, S_B22, and S_B23) do not need snubber capacitors

because in the worst conditions (i_LS = 40 A) the voltage across the switches is lower than 800 V, as can be seen in Figure 2.

To reduce the overvoltage across the matrix converter switches, snubber capacitors can be placed in parallel with the switches.

The proposed DC/AC converter requires snubber capacitors to limit the overvoltage across the outer switches (S_B17, S_B20, S_B21, and S_B24). The snubber capacitors are placed in parallel with the IGBTs to absorb the energy accumulated in the stray inductances, present in the transformer and cables of the circuit shown in Figure 1 [12] . The objective of a snubber capacitor is to reduce voltage ringings that occur when a switch is switched off, by providing an alternative path for the current flowing through stray inductances. The energy accumulated in stray inductances can be eventually dissipated directly in the switches in conduction, or in an external resistor placed in series with a snubber capacitor. The snubber design will be presented only in the final version of the paper due to the lack of space, but it results in a snubber capacitor value of 22 nF. The simulation results using these snubbers are presented in Figure 2.

For the AC/AC converter proposed in this chapter, the energy from the leakage inductances (shown in Figure 1) is dissipated directly in the IGBTs and the parasitic resistances.

The power dissipated in the IGBTs due to the snubber capacitors is determined by [13] - [15]

where C_sn is a snubber capacitor value, f_s the switching frequency, and V_sn is the voltage across C_sn just before the switch is switched on. The value of C_sn can be determined using (3), to guarantee that the overvoltage across the outer switches will not be higher than a desired maximum value.

In the implemented converter (Figure 1) a total leakage inductance of 2 μH was measured, resulnting a peak voltage of 1500 V (see Figure 2) when the maximum load current (i_LS) is 40 A. The goal is to reduce the peak voltage to 800 V to allow the use of IGBTs which can support a maximum voltage of 1000 V. Therefore, using (3) is possible to determine the peak voltage across the outer switches for different values of C_sn. However, when a different capacitor is placed in parallel with the IGBTs, the ringing frequency changes and, consequently, the resistance of the transformer and cables of the prototypes also change (see Figure 4). Therefore, in order to

facilitate the choice of the snubbers, for different values of C_sn, the corresponding transformer and cable resistances were obtained from the oscillation frequencies as given by (1) and the impedance characteristics in Figure 4. Then, the resulting voltage ringing is plotted using (3), as shown in Figure 5. According to Figure 5, with snubber capacitors between 15 nF to 22.5 nF, the voltage across S_B17 has a peak value lower than 800 V. For safety margin reasons a capacitor of 22 nF was implemented. The proposed AC/AC converter was simulated again with the chosen snubber connected in parallel to each outer switch (S_B17, S_B20, S_B21, and S_B24) and the results are shown in Figure 6. This figure confirms a maximum voltage across S_B17 of 650 V and ringing frequency of 759 kHz. The inner switches (S_B18, S_B19, S_B22, and S_B23) also have overvoltage lower than 800 V, as requested.

The extra power losses in the switches can be calculated with (8), resulting a dissipation of 17.6 W per switch. Because snubber capacitors are placed only across the outer switches (S_B17, S_B20, S_B21, and S_B24), and the proposed converter has 12 switches. As a result, the total snubber loss is 211.2 W. When these losses are added to the total losses, the efficiency of the proposed AC/AC converter reduces to 93.2%.

The converter was experimentally tested to validate the theoretical Figure 7.

The ringing and overvoltage across the switches of the proposed DC/AC converter is described. This overvoltage is generated due to the resonance between three passive components: transformer leakage inductances, switch capacitances, and wiring resistances. By applying simple RLC circuit equations, it proves possible to determine the analytic equations and reproduce the voltage across the switches to predict the overvoltage and resonance frequency. The analysis shows that the switches connected directly to the high-frequency transformer encounter higher voltage spikes compared to those connected directly to the load. Snubber capacitors are designed to decrease the peak voltage based on the theory developed in this paper.

Gierri Waltrich, (2016) Ringing and Voltage Overshoot Analysis of a Proposed DC/AC Converter. Open Access Library Journal,03,1-9. doi: 10.4236/oalib.1102821