I have described the condition in the transformer picture I have attached. Will the transformer still function in this condition?
2 Answers
If the primary and secondary turns are identical (i.e. a 1:1 turns ratio) then the transformer will continue to function but of course it won't isolate any more because you've bypassed the isolation.
Consider taking two long wires and wrapping them round a transformer core many times. As a pair they can be shorted together at the ends and this just forms a single winding. If they are open circuited the AC on one winding produces exactly the same AC voltage on the other and so clearly they can be connected together as per your diagram.
Any other turns ratio and the thing will smoke.

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3A 1:1 turns ratio and the windings connected in opposition will also smoke. – Chu Dec 09 '15 at 14:04
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@Chu this will indeed - just like connecting batteries in parallel but one the wrong way! – Andy aka Dec 09 '15 at 14:05
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Yes. Zero core flux, so short-circuit. – Chu Dec 09 '15 at 14:05
Consider a isolation transformer. If the primary winding having 230 VAC supply and its also connected with secondary winding of the transformer. We know that the primary and secondary winding having not a electrical contact . So the isolation transformer is create the another 230vac in the secondary of the transformer . We connect the series connection for secondary of the transformer and the input 230vac supply.
Therefore we got 460vac in 2 wire single phase voltage. Its used for drives and etc... I am panel wiring engineering . I now used this type of connection for driving VFD. Thanking you..