# Analysis of Subharmonic Oscillation and Slope Compensation for a Differential Boost Inverter
## 1 Introduction  
DC-AC inverters find widespread usage in many residential, industrial and military applications. With the ever-increasing development of the renewable energy technology, DC-AC inverters have become one of the most attractive and viable solutions to the power conversion problem. They are extensively used and play key roles in various actual applications of power electronics technologies for renewable energy sources [1; 2; 3]. They are also used in motor drive [4; 5] and DSTATCOM applications [6] as well as in many uninterruptible power supply system applications such as plant facilities and factories, medical equipments and centers in hospitals, airline computer and communication systems in server farms and web hosting sites [7]. One of the important tasks in the design of DC-AC inverters is the control loop implementation which must ensure a system free from any kind of instabilities. However, it is well known that this aim is difficult to be achieved for all values of system parameters and that many undesired nonlinear phenomena can arise in these kinds of indispensable parts of modern and emerging energy systems. These phenomena can significantly jeopardize the system performance and can cause serious consequences on its reliability. Therefore, understanding these nonlinear phenomena, their analysis, prediction and control have increasingly become of great concern of many researchers all over the world [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. The major part of the analytical results on subharmonic oscillation in power electronics converters has been achieved for DC-DC converters [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. DC-AC inverters are more difficult to deal with, since their dynamics is governed by two vastly different frequencies, namely the high switching frequency and the low frequency of the output voltage reference sinewave. For reliable and desirable operation, the stability of the system must be guaranteed for the whole range of its parameters. In [13], the dynamics behavior of an H-bridge under a digital Current Mode Control (CMC) was investigated by using a one dimensional discrete time model. Different dynamical behaviors for the system were revealed by varying the proportional gain of the current controller. In [14] a similar approach was applied and it was demonstrated that different types of bifurcations (instabilities) can take place such as period doubling leading to Subharmonic Oscillation (SO) and border collision bifurcations leading directly to chaotic behavior. Using the quasi-static approximation, in [15] the slow-scale and fast-scale instabilities in a voltage-mode controlled H-bridge inverter are reported and analyzed using an averaged model and a discrete-time model respectively. It is well known that conventional averaged model cannot predict the fast-scale instability and for that the discrete-time model must be used. A closed form discrete time model was used in [16] to predict both the slow-scale and the fast-scale instabilities in an H-bridge inverter demonstrating that the system may undergo instability phenomenon when the proportional gain of the voltage controller is increased. In an H-bridge digital-controlled grid-connected inverter system, bifurcation behavior was investigated and loss of system stability was shown by increasing the current controller gain [19] and it was shown that in this system only slow scale instability may take place leading to low-frequency oscillation. The same system, but with double edge modulation, has been studied in [9] using an analytical closed-form expression for predicting a period doubling phenomenon. Single-stage grid-connected DC-AC conversion systems with boosting voltage capability have recently attracted the attention of many researchers. Single-stage structures of inverters not only perform DC-AC conversion but also perform voltage boosting.