We develop an analytical model for the Direct-Current (DC) and capacitance-voltage (CV) characteristics of AlGaN/GaN High Electron Mobility Transistors (HEMTs), providing accurate predictions of the transconductance and the gate capacitance in both linear and velocity saturation regions. The models provide an accurate and smooth connection in the area near the knee point for the DC characteristics, which is attributed to precise descriptions of the channel charge. An accurate model for the low-field mobility of the two dimensional electron gases (2DEG) has been developed, considering the nonmonotonic dependence of the carrier velocity on the electric field perpendicular to the channel for the first time. The calculated transconductance and output conductance are proved accurate and to have high order continuity, especially in large voltage biases, which is hard for some other analytical or numeral models. The gate capacitance has been obtained analytically and verified by experimental data. The slight decrease in the measured gate-to-source capacitance over the velocity saturation region which indicates the results of neutralization of donors and the contribution of the free electron in the AlGaN layer has been modeled accurately and smoothly for the first time. The predicted cut-off frequency is in excellent agreements with measured data over the full range of applied biases. The models are implemented into the HSPICE simulator for the DC, AC and transient simulations, with good speed and convergence characteristics.