Abstract: Dynamic models established from practical engineering application are non-smooth systems owing to non-smooth factors, such as impact, dry friction and switching, etc. Up to now, most studies are in terms of the non-smooth dynamic systems with a single scale or two scales. While more complex dynamic phenomena may be observed in the non-smooth dynamic systems with more scales. The main purpose of this work is to explore multiscale effect in a non-smooth electric system and the related bifurcation mechanism. Upon the traditional Chua's circuit, by introducing a periodically excited oscillator with an order gap from the natural frequency of the system and taking suitable parameter values, a coupled 4-dimensional piecewise linear dynamic system with three time-scales and two boundaries is established to study the bursting oscillations as well as the corresponding bifurcation mechanism under three time-scales. Merging the variables corresponding to the fast scale and the variables related to the intermediate scale into the fast variables, while regarding the variable corresponding to the slow scale as the slow variable, the coupled problem with three time-scales is transformed into that with two time-scales. According to the relevant Hopf bifurcation curve under two independent parameters and the stability analyses of the slow submanifold of the fast subsystem, two different bursting oscillations of the coupled dynamic system are given in the case of two different parameter values. On the basis of the fast-slow analysis method, the transformed phase portrait and the non-smooth dynamics of the slow submanifold occurring on the non-smooth boundaries, the bifurcation mechanism of the mutual transformation of different bursting oscillations is analyzed in details, in which some helpful numerical simulations are given to illustrate the validity of our study simultaneously. At the same time, a new evolution of bursting oscillations is found, i.e., the bursting oscillation induced by destructive grazing bifurcation.