A model of the dynamics of full penetration welding is presented that relates the heat input to the resulting width of the backbead. It is developed from a heat balance using the pool as a control volume, and is applicable primarily to autogenous gas-tungsten arc welding (GTAW). Two means of modulating the input are considered: 1) varying the torch current and 2) varying the torch travel velocity. The proposed model in both cases is first order, but it has non-constant) parameters (i.e., gain and time constant). Regardless of which input is used, the gain and time constant of the system are shown to depend strongly upon the thickness of the material, the preheat temperature, and the nominal torch velocity. These trends were confirmed in a series of open-loop step tests, where gain and time constant were directly measured, and in closed-lop tests, where step and frequency response methods were used. The resulting model permits rational high performance controller design, and the variable parameters of the system suggest the need for parameter adaptive control.