In designing machines and apparatus with solid or partly solid magnetic circuits, such as eddy-current brakes, clutches, impact exciters, and rolling-mill motors, flux/time curves have often to be predetermined to secure a desired rapidity of response. The prediction has hitherto been uncertain, especially in the absence of test results on similar machines, and expensive changes of design, introducing laminated poles and yokes have sometimes been resorted to where the standard design or a simple modification of solid sections would have sufficed.The paper considers the rise of the total flux as retarded by eddy currents and shows that for this purpose all eddy-current paths can be represented by a single fictitious damping-ring whose resistance and leakage reactance are calculated from first principles. Simple expressions for the usual shapes of iron section are developed, and the test results which are given provide agreement between theory and experiment.For magnetic circuits which consist of solid iron in parts only and with fairly large air-gaps as in d.c. machines and alternators, it will generally suffice to take account only of the resistance of the fictitious damping-ring and to ignore its leakage reactance. A graphical procedure may then be used for obtaining the flux/time curve very rapidly. The error thus introduced makes the response appear rather less rapid than it is in reality.Where greater accuracy is needed, the leakage reactance of the substitute damping-ring has to be introduced and the solution found by solving transformer equations for successive parts of the characteristic.The substitute damping-ring, as a theoretical device, should prove helpful in solving other problems involving eddy currents by converting the system into a transformer with short-circuited secondary.