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A mathematical model for a turbogenerator unit is presented, based on the direct- and quadrature-axis equations of the generator, and associated equations for the prime mover and governor, and for the excitation control system. The solution of the equations of the complete model by digital computer is developed. In the representation of the generator, a method is explained by which the effective inductance coefficients and time constants in the direct and quadrature operational equations may be related to, and varied in accordance with, the magnetic conditions computed at each step advance in solution, thereby accounting for the effect of the generator magnetic-circuit characteristics on its transient response. The eddy-current paths in the solid rotor are first represented by an equivalent short-circuited winding in each axis, the subtransient parameters of which are those derived from short-circuit tests in the ordinary way. An improved rotor representation is then developed from a solution of the field equations for a simplified rotor model having a rectangular section and smooth surface. Beginning with the case of a large depth of flux penetration in the rotor body, corresponding to low-frequency rotor excitation, an equivalent-circuit interpretation of the solution of the field equations is formed, the low-frequency circuit parameters of which are then modulated at each integration step throughout solution to take account of different and higher incremental rates of magnetisation. A further method of computing the effects of rotor eddy currents is given in which the eddy-current paths are represented by ladder networks in each axis. The validity of the methods developed is examined by comparing computed results with those obtained from full-scale site tests on a 30 MW unit and on a 120 MW unit, for transient conditions provoked by load rejection, 3-phase short circuit and excitation reduction leading to pole slipping and rotor overvoltages.