Representation of synchronous machines using constant-parameter voltage-behind-reactance (VBR) formulations improves accuracy and numerical efficiency of power systems transient simulation programs. This paper extends the VBR representation to the rotor circuit and presents two new formulations that achieve direct constant-parameter interfacing of the rotor and stator terminals with arbitrary external networks. In the first model, the entire machine is represented by constant RL branches that have algebraic coupling among the circuit variables. In the second model, all damper windings are implemented in state-space form to increase the numerical efficiency, while the stator and field windings are provided as constant-parameter circuits. The proposed models are validated against the commonly used and some state-of-the-art alternative models using a single machine with a simplified ac excitation system. Computer studies demonstrate the improved accuracy and efficiency of the proposed models when external rotor circuitry is considered.