# Analysis of Subharmonic Oscillation and Slope Compensation for a Differential Boost Inverter
## 3 Behavior of the Differential Boost Inverter  
The dynamical behavior of the boost inverter is explored in this section with the aim to gain insight on suitable ways of obtaining an appropriate model that can be used for its accurate stability analysis. The system is first studied through simulations using the full-order switched model of the inverter implemented using PSIM(c) software by varying suitable system parameters. The focus is first on system stability in terms of the time varying voltage reference. The fixed parameter values used for the rest of the study are reported in Table 1. Many time-domain waveforms have been computed to get a clear view of the system behavior and only representative results are shown below. The simulation is run for sufficiently long time to allow the system to reach its steady-state. The data obtained during time transient within the startup phase and during the transient regime of the regulation phase are fully eliminated. Only the last two cycles of the output voltage reference are plotted. Figure 2 shows the system waveforms when the system is stable. The figure shows the time-domain waveforms of the reference voltage \(v_{\text{ref}}\) and the output voltage \(v_{\text{o}}\), the capacitor voltages \(v_{\text{o}1}\) and \(v_{\text{o}2}\), the inductor currents \(i_{1}\) and \(i_{2}\) and the control signal \(r_{s}(i_{1}-i_{2})\) and the signal \(r_{s}i_{\text{ref}}-v_{\text{ramp}}\). It is worth noting that the output voltage cannot be distinguished from its reference signal \(v_{\text{ref}}\) due to the practically zero amplitude and phase errors. Note also that the state variables and the control signal oscillate at two main frequencies, the switching frequency (100 kHz) and the reference voltage frequency (50 Hz). From a practical point of view, the output voltage is characterized by a low value of THD as required in any application. As parameters are varied, the state variables undergo a sudden distortion by exhibiting SO at the fast switching scale as shown in Figure 3 for \(k_{p}=0.4\). This phenomenon takes place when the proportional gain \(k_{p}\) gradually increases and reaches a critical value close to 0.22. As shown in Figure 3, it can be observed that the inductor currents \(i_{1}\) and \(i_{2}\) exhibit SO leading to disrupting bubbling phenomenon of the waveforms. In particular, when \(k_{p}\approx 0.22\), the fast-scale instability develops in all the state variables but it is more visible and pronounced in the inductor current waveforms \(i_{1}\) and \(i_{2}\) and their combination \(r_{s}(i_{1}-i_{2})\). As stated before, such behavior manifests itself as a period-doubling phenomenon at the fast switching scale [9; 51]. It can also be noticed in Figure 4 that the phenomenon already becomes visible in the capacitor voltages and the output voltage hence it can deteriorate the performance of the inverter and therefore its prediction is an important task from a practical point of view. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Inductance \(L_{1}=L_{2}\) & 100 \(\mu\)H \\ Resistance \(r_{1}=r_{2}\) & 0.1 \(\Omega\) \\ Capacitance \(C_{1}=C_{2}\) & 22 \(\mu\)F \\ Input voltage \(v_{g}\) & 200 V \\ Load resistance \(R\) & 100 \(\Omega\) \\ Time constant of the voltage controller \(\tau\) & 1 ms \\ PWM switching frequency \(f_{s}\) & 100 kHz \\ RMS value of the reference voltage \(v_{\text{ref}}\) & 230 V \\ Frequency of the reference voltage \(v_{\text{ref}}\) & 50 Hz \\ Current sensor gain \(r_{s}\) & 0.1 \(\Omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The used parameters for the DC-AC differential boost inverter. Figure 2: Steady-state response of boost inverter with \(k_{p}=0.2\) and \(V_{M}=2\) V. (**a**) Capacitor voltages \(v_{o1}\) and \(v_{o2}\), output and reference voltages \(v_{o}\) and \(v_{\text{ref}}\). (**b**) Inductor currents \(i_{1}\) and \(i_{2}\) and control signals \(r_{s}(i_{1}-i_{2})\) and \(r_{s}i_{\text{ref}}-v_{\text{ramp}}\).