# Analysis of Subharmonic Oscillation and Slope Compensation for a Differential Boost Inverter
## 3 Behavior of the Differential Boost Inverter
For each subplot, traces correspond to the shown voltages in [V] and currents in [A]. Figure 3: Steady-state response of boost inverter with \(k_{p}=0.4\) 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 \(r_{s}i_{\text{ref}}-v_{\text{ramp}}\). For each subplot, traces correspond to the shown voltages in [V] and currents in [A]. By carefully examining the waveforms, the following statements can be made:  
Figure 4: Steady-state response of boost inverter with \(k_{p}=0.8\) 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 \(r_{s}i_{\text{ref}}-v_{\text{ramp}}\). For each subplot, traces correspond to the shown voltages in [V] and currents in [A]. * By increasing progressively the proportional gain and when this parameter reaches the critical value, SO oscillation starts first occurring in a very limited number of switching cycles during the first half cycle of the sinewave signal eventually in the neighborhood to the quarter of the cycle where the sinewave signal is maximum. * The number of the switching cycles, during which SO is exhibited, gets larger and the fast-scale SO is more pronounced when the proportional gain \(k_{p}\) is increased. * At the left and at the right of the maximum values during the same half cycle, one has the same values of quasi-steady-state duty cycles and therefore, theoretically, a perfect symmetry is expected in the critical phase angles at which SO takes place. However, an asymmetry can take place because the slope of the reference sinewave signal at the left of the peak point is positive while it is negative at the right side. * The SO interval is repetitive from a sinewave cycle to the next one and the study of SO phenomenon can be restricted to one sinewave cycle in terms of the phase angle \(\varphi\) as a slowly varying parameter in the range \(\varphi\in(0,2\pi)\). * Apparently, if SO is avoided for the first half cycle of the sinewave signal, it will also be avoided for the second half cycle. Therefore, the numerical and the analytical studies to be presented later will be restricted to the first half cycle of the sinewave signal for \(\varphi\in(0,\pi)\), i.e., only within the duty cycle range \(D\in(0.5,1)\). A powerful tool for clearly illustrating the SO phenomenon is by using the sampled waveforms. In order to clearly appreciate the change in the behavior of the system, sampled steady-state values of the state variables at time instants \(t=nT\) (\(n\in\mathbb{N}\)) are obtained. Therefore, the state variables are sampled at every clock instant and then plotted in the time domain. A priori, any one of the state variables can be used for illustrating the behavior of the system. However, as observed in the previous time domain numerical simulations, SO is more pronounced in some state variables than others. An interesting and naturally sampled variable for which SO is well noticed is the duty cycle of the binary signal \(u\). Figure 5 shows the waveforms of the duty cycle \(d(nT)\) (\(n\in\mathbb{N}\)) during one complete sinewave cycle for four different values of the proportional gain \(k_{p}\). The duty cycle waveforms are plotted in terms of the phase angle within the interval (0, 2\(\pi\)). For \(k_{p}=0.2\), the system exhibits a stable periodic regime in steady-state, the duty cycle does not present any disruption and its samples represent a clean and smooth waveform. When the SO regime starts taking place, one gets a different picture. For instance, for \(k_{p}=0.4\), it can be clearly seen that there is a certain phase interval within the first half of the sinewave cycle during which the duty cycle waveforms is disrupted.