The inter-turn short circuit of field windings in synchronous machines will cause the field current to increase and reactive power output to decrease. It will also introduce the aggravation of vibration. To achieve online fault monitoring and protection， electrical quantities such as fault currents of stator and rotor must be calculated accurately and efficiently. In this paper， based on the multi-loop theory and the steady-state characteristics of stator and rotor fault currents， the steady-state expression of each loop current is obtained first. Then， differential equations in the steady-state system are reduced into linear equations to acquire the steady-state mathematical model on inter-turn short circuit of field windings. This is based on the principle that components of the same frequency are equal. By solving the linear equations using Gaussian elimination， the steady-state solution of the AC and DC components of each loop current can be obtained and the steady-state calculation of the fault can be achieved rapidly. The field winding inter-turn short circuit experiments have been carried out on a 12 kW synchronous machine with three pairs of non-salient poles on no-load. The results show very good match with that of steady-state simulations. The steady calculation method proposed can save nearly ninety percent of the computational time compared with the transient simulation approach. It provides a useful foundation for the inter-turn short circuit fault detection and protection of the field windings in synchronous machines.