IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. IA-12. NO. 3, MAY/JUNE 1976 Differential Electrical Shaft Combined with a Double Kramer Cascade DANIEL SHARON, MEMBER, IEEE, ABRAHAM ALEXANDROVITZ, SENIOR MEMBER, IEEE, AND JOSEPH BEN URI Abstract-A new electrical transmission drive that insures an adjust- able constant speed difference between a pair of mechanical shafts independent of load conditions or speed level is described in the paper. This generalized electrical shaft comprises two rotor interconnected induction machines, fed from different frequencies and two dc ma- chines forming a double Kramer cascade. The use of the Kramer cas- cades offers many important advantages over the alternative simpler scheme of 'using resistors in the rotor circuit, namely, higher efflciency, much lower speed dependence on load, and speed level adjustability in the subsynchronous range. The drive performance is analyzed by means of an equivalent circuit from which the current and torque expressions are derived. Good agreement is obtained between theoreti- cal and experimental results. 1. INTRODUCTION VARIOUS electrical shafts providing exact speed corre- spondence in a pair of mechanical systems have been in- vestigated in the past [11 -[4] under different designations. Two such well-known schemes are shown in Figs. 1(a) and l(b). Features common to both are a) the pair of identical wound rotor asynchronous machines M1 and M2, supplied from a common polyphase source; b) the interconnection of the rotor windings permitting synchronizing torques to be developed and ensuring identical rotor speeds. New properties are realized, however, by connecting the induction machines to sources of different frequencies. This results in a constant speed difference between the two rotors, irrespective of load conditions or speed levels whence the designation differential electrical shaft or DES. The corresponding DES counterpart of the electrical shaft in Fig. 1, (Arbeitswelle), for example, is shown in Fig. 2, where one of the machines is connected to the regular three-phase supply and the other to an adjustable voltage and frequency SCR bridge. Various DES schemes and their general properties have been described previously by the authors [5]. These properties are essentially based on the interaction between the rotor induced voltages and currents in the asynchronous machines M1 and M2. This interaction is conditional on the equality of the rotor frequencies, that is, ( Jrl =Wr2( (1) Paper TOD-72-78, approved by the Industrial Control Committee of the IEEE Industrial Applications Society for publication in this TRANSACYIONS. Manuscript released for publication December 30, 1974. D. Sharon and J. Ben Uri are with the Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel. A. Alexandrovitz is with the University of Southern California, Los Angeles, CA, on leave from the Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel. (a) (b) Fig. 1. Electrical shafts schemes. Fig. 2. Differential electrical shaft. Denoting the corresponding stator angular frequencies by co, and W2, respectively, (1) implies l -PQI = 02 -PQ2 (2) where Q, and 22 are the angular speeds of machines Ml and M2, respectively, and P the respective number of pole-pairs assumed equal in both machines. Rearranging (2) we obtain Ql- 2 - - (1 -C2) p (3) that is, the speed difference is constant for constant frequen- cies. The objective of this paper is to analyze an improved DES (Arbeitswelle) which has important advantages over the scheme given in Fig. 2. '26 8