A zigzag transformer is a special purpose transformer. It has primary windings but no secondary winding. One application is to derive an earth reference point for an ungrounded electrical system. Another is to control harmonic currents. Consider a three-phaseY (wye) transformer with an earth connection on the neutral point. Cut each winding in the middle so that the winding splits into two. Turn the outer winding around and rejoin the outer winding to the next phase in the sequence (i.e. outer A phase connects to inner B phase, outer B phase connects to inner C phase, and outer C phase connects to inner A phase). This device is the zigzag transformer. 
When a zigzag transformer is connected to a healthy electrical system, negligible current flows in the connection between the zigzag's neutral point and earth. By design, the magnetic fluxes in the split windings cancel each other out. With negligible current flows in the neutral under normal conditions, engineers typically elect to undersize the transformer. A short time rating is applied (i.e. the transformer can only carry full rated current for, say, 60 seconds). If one phase(s) of the delta faults to earth, the voltage applied to each phase of the zigzag is no longer in balance. Fluxes in the windings no longer oppose. Zero sequence (earth fault) current flows between the zigzag’s neutral to the phase affected by the fault. Hence, the purpose of a zigzag transformer, to provide a return path for earth faults on delta-connected systems. An application example: Occasionally engineers use a combination of Y (wye or star), delta, and zigzag windings to achieve a vector phase shift. For example, an electrical network may have a transmission network of 110/33 kV star/star transformers, with 33/11 delta/star for the high voltage distribution network. If a transformation is required directly between the 110/11 kV network the most obvious option is to use 110/11 kV star/delta. The problem is that the 11 kV delta no longer has an earth reference point. Installing a zigzag transformer near the secondary side of the 110/11 kV transformer provides the required earth reference point.
