Transient stability and frequency stability of virtual synchronous generator (VSG)-controlled converters during grid faults are critical for the stable operation of high-share renewable energy power systems. However, the existing transient stability methods often overlook the consideration of frequency stability. This paper presents a method to improve both frequency stability and transient stability of VSGs during grid faults. The influence mechanism of inertia and damping on transient stability and frequency stability is comprehensively analyzed under two types of faults, with or without stable equilibrium points. To address the conflict between frequency stability and transient stability of VSGs with large inertia under the first type of fault, an improved VSG control with angular frequency deviation feedback is proposed. The transient response of this control method is analyzed using small-signal and large-signal models. Furthermore, to achieve optimal dynamic performance, this improved VSG control introduces the optimal damping ratio and presents a parameter design method. Finally, the correctness of the proposed method and its theoretical analysis is verified through experimental results.