Large-amplitude self-excited thermoacoustic oscillations are unwanted in many premixed propulsion systems, such as rocket motors and gas turbines. However, they are desirable in some other practical applications, such as thermoacoustic prime movers or cooling systems. Thus, predicting the onset of such oscillations and a better understanding of its generation mechanism are important. Previous studies have confirmed that the time delay between unsteady combustion and flow perturbations plays a critical role in generating such thermoacoustic oscillations. In this work, the stability of a 1D thermoacoustic system is investigated theoretically and numerically. A premixed flame is confined in the combustor. The coupling between the unsteady heat release and oncoming flow disturbance is characterized by using interaction index-delayli-Z model. H denotes the interaction index between unsteady combustion and flow. To make the present studies more generalized, acoustic damping 1 is considered. The effects of H and l on the stability of the thermoacoustic combustor are investigated. Two interesting stability regions associated with the heating power H and the acoustic damping are found: (a) delay-independent region and (b) stability switches region. When the thermoacoustic system is 'locked' in the delay-independent region, it is found that changing the time delay will not affect the stability of the system. However, if the thermoacoustic system is in the stability switches region, then a continuous increase in the time delay can lead to stability switches between stability and instability. Moreover, such switches become more frequent, when H and g are coupled and near the boundary between the stability switches region and delay-independent region. The present work opens up an applicable approach to design a stable combustor. (C) 2017 Elsevier Ltd. All rights reserved.