Given a real turbine characteristic, equations giving discharge and torque are derived in terms of certain characteristic slopes, speed changes, and pressure deviations. It is shown that for the majority of reaction turbines the flow through the machines is speed dependent and the effect of this phenomenon on the pressure dependence of flow is examined and found to be significant. Similarly the conventional representation of the pressure dependence of torque is shown to be correct only in the case of one particular type of turbine. A formula allowing the establishment of the correct relationship for any characteristic is given. The nonlinearities of the quantity-gate and torque-gate relationships are examined, and the concept of effective gate is introduced in order to improve the accuracy of any predictions. The modified relationships are used in establishing the bounds of the region in which stable operation of the unit can be achieved, and these are shown to vary both from turbine type to type and with gate on any particular turbine, depending on the slope of the appropriate unit quantity-unit speed curve. Similarly, the governor parameters leading to critical damping are shown to be a function of this characteristic slope. The analysis leads to the conclusion that the neglect of the speed dependence of the turbine discharge can result in significant errors, especially when dealing with both low and high head machines.

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