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There are two types of induction motors which have double rotor slots. In the first type, both slots, upper and inner, are filled with the same conducting material (aluminum for example) and as a consequence the motor has a single cage finally, despite the double rotor slots. In the second type, usually different materials are to fill the upper and inner rotor slots (for example aluminum for the upper slots and copper for the inner slots). So, in this category the motor has two electrically independent rotor cages.The only way for the two windings of any type (coils OR bars) to be electrically independent is: 1) insulate each winding with some form of groundwall, and 2) leave the ends of the windings unconnected from each other.
In the case of most double-cage windings, there is some form of electrical pathway between the upper cage material (which is in direct contact with the lamination steel) and the lower cage (which is ALSO in direct contact with the lamination steel).
Usually, the choice for a double-cage arrangement boils down to optimization of the overall cage performance between two "standard" materials available to a manufacturer: one is often a high resistance material and the other is a low resistance material. When combined, the result is a somewhat-balanced "medium" resistance material.
Two cages are connected considerably via the core and the direct contact between the laminations and the inner and outer cage bars. The (inter-bar) currents that flow between the two cages are - in my limited knowledge and experience - likely to be considerably lower than the inter-bar currents that flow between bars in the same cage. Remember that the cages will experience very similar end-to-end bar voltages because their lengths (inside the core / slots) are equal, and apart from leakage effects - the flux linked by them is also equal.
Bearing all this in mind - the only reason to combine the end-rings into one on each side, is for simplicity or cost-reduction. However - in my experience (and this is certainly not universal!) - for larger machines - the end-rings are usually separate. And I believe the main reason for this is to reduce stresses on the bar extensions (outside the slots, near the end-rings) due to differential expansion of the inner and outer cages. It is not my main area of experience but I would imagine that Die-cast rotors with double cages will of course have single end-rings per side.
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