Due to various controls in voltage-source converters, their impacts and mutual influences on transient stability of new-generation power systems are still obscure. Different from the mature different-order transient equations for synchronous generator (SG), fully accepted transient equations of converters still lack. In this article, a novel methodology is proposed to uniformly evaluate the controller effects on the transient stability and then establish the hierarchical transient equations of converters based on the properties of controlling-unstable-equilibrium point (CUEP). The participation factors on unstable eigenvalue of CUEP are applied to measure the contribution of state variables in transient dynamics, and dominant variables and key loops of grid-following and grid-forming converters are determined and compared. It is found that the synchronous loop, e.g., phase-locked loop in grid-following converters and virtual-synchronous loop in grid-forming converters, plays a primary role, and the power balance loop on the DC capacitor, including DC-voltage control and DC capacitor dynamics, plays a secondary role. Additionally, the impacts of synchronization and power balance are compared with the rotor swing of the SG. Analytical results are demonstrated by time-domain simulations and transient stability assessments. The dominant transient models and the CUEP-based participation factor analysis provide an improved physical insight on our understanding of transient dynamics in the new-generation power systems dominated by not only grid-following but also grid-forming converters.