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We introduce a new iteration procedure for calculating the periodic response of nonlinear dynamic circuits, which is based on a less known variant of the Cesari method. The successive approximations of the periodic solution of circuit equations are determined through recurrent formulas in the frequency domain starting from a low-order approximation of the solution, which is obtained by using the harmonic balance method. The use of this latter method might be thus aimed at this end only, when it is used in combination with the proposed iteration procedure. The resulting constructive process of solution is remarkably less time consuming than that of the harmonic balance method, as shown through its application to some typical nonlinear circuits.