Transient stability is an important measure in assessing a system's overall stability. It characterizes the ability of a power system to maintain stability following major disturbances. Modern power systems whose dynamics are increasingly complex and shaped by fast power electronic devices are testing the assumptions and limitations of existing transient stability simulation methods. This work presents a new method developed to improve transient stability simulation by extending its frequency bandwidth through dynamic phasors. This method constructs a continuous set of differential algebraic equations modeling a power system's dynamics using modified nodal analysis. These equations are numerically solved using a general purpose differential algebraic equations solver based on a variable step and variable order algorithm. This selection allows the method developed in this work to conduct accurate simulations while maintaining the advantages of demodulation in phasor-based analysis. Models are included for linear ac networks as well as two important components for power system simulation. The first is a new synchronous machine model, which uses a direct reference frame transformation to represent a synchronous machine as an inductance with nonlinear coupling to its rotor. The second is an HVdc transmission system model, which includes a converter component that is capable of accurately simulating low impedance faults near converter terminals. This accuracy is achieved through an ac current that is directly coupled to ac voltage transients and event monitoring to apply state changes when adverse operating conditions are detected. Simulation results are presented from a five hundred bus test system, which includes two HVdc transmission line models. The results show that the method is accurate for severe disturbances and is three hundred times faster than a detailed EMT simulation. This system was modified to also include a multimass turbine model and tuned to produce unstable subsynchronous oscillations. The results show that this method accurately simulates negatively damped modes and is still over thirty times faster than EMT simulations. Finally, results of a two thousand bus system are also presented, which are in good agreement with results from a conventional transient stability simulation program.