When generators of a power system are subject to a large disturbance, their electromechanical oscillations (EOs) are essentially nonlinear as observed from their rotor angle waveforms due to the nonlinear power network. However, traditional modal analysis has not given enough considerations to nonlinearities in EOs. This paper analytically formulates the accurate oscillation frequency (OF) of an EO mode addressing nonlinearities. It is revealed that when the system model and condition are fixed, the OF of any EO mode is energy-dependent as a function of the oscillation amplitude (OA). That is characterized by a new tool named Frequency-Amplitude (F-A) curve regarding each mode, from which the frequency decreases from the natural frequency toward zero when the amplitude grows to a critical threshold. The paper proves that an F-A curve is actually a projection of the system trajectory between the stable equilibrium and the stability boundary onto the OF-OA plane regarding one mode. Based on the concept of F-A curve, a stability index is defined and estimated from measurements for online angle stability analysis. The proposed methodology is presented in detail using an SMIB system and then demonstrated on the IEEE 9-bus system and WECC 179-bus system having multiple EO modes.