In an electrical panel test program an interesting thing was observed. In the panel a three phase voltage transformer with following technical characteristics was used to adapt one 110 V AC under voltage relay with main bus bar voltage.

Voltage Ratio: 380/110 V
Connection: YnYn
Rated power: 60 VA
Construction: Core type
Accuracy class: 3P

The under voltage relay was three phase and microprocessor base made by an authentic company.
The acceptance tests on the individual relay were done successfully, but the under voltage relay in combination with voltage transformer didn’t have suitable reflection against actual one phase under voltage in primary system that was simulated by removing one of three phase fuses in primary side of the voltage transformer.(Really, no under voltage was observed by the relay.) All wire interconnections and related circuits also voltage transformer was rechecked again and every thing was very well. The relevant test men couldn't explain the reason of  behavior of circuit. It seemed that interface circuits or devices didn’t work correctly.
  
How can you explain the reason of non activity of that protection system against primary under voltage?

It appears to me that the flux induced in the core from the other two - still active - phases which obviously pass through the core section of the "inactive" phase are enough to generate an almost normal voltage in both the primary and secondary coils of that phase.

This could be because the flux density of the VT is reasonably low to give an accurate transformation ratio over its entire range. The problem may be reduced by using a heavier load on the secondary so that the higher current will show more "droop" when the voltage fails. However this may mean less sensitivity to abnormally high voltage.

I must admit that we usually supply 3 separate single phase transformers for this type of application, each connected between phase and neutral thus eliminating the possibility of phase interaction. 

This is a little out of my area. I don't have a lot of practical experience with three phase power systems. But isn't this relay designed to find a phase being overloaded and therefore low voltage on that phase. Pulling the fuse for a single power phase eliminates the the supply for that phase but also eliminates the loading effect for the transformer. Of course two phases of a 3 Phase transformer are very capable of synthesizing the missing phase especially under no critical load conditions, most people don't realize that transformers are very dynamically self-correcting. The poweri nteraction of the 3 phases are inseparable and proportional to the power being pulled out of them. The resilience of the polyphase system was a major reason for its development.

I think perhaps the system under question would respond very quickly to a real overload condition, of a single phase, especially because that will pull power out of the transformer beyond its ability to self-compensate. Opening the supply side of a single phase, shouldn't present a critical condition especially under no power draw conditions.

Having read a lot about Tesla's development of polyphase power distribution systems this is my take on it, I may be completely wrong, but it is also a little difficult to understand the configuration of the system being described, and the relay being decribed. Of course if one really wants to make sure all supply side phases are present, one would have to measure them separately, no single relay could perform the task, but the testing relay circuit sounds like it would trip under most real world overloading conditions since it measures a missing phase under load.

 

As the three pairs of windings share the same magnetic core the secondary without an input to its primary will have an ouput and this could be high enough to be seen as "good" by the monitoring relay.

Also there could be a voltage on the phase without fuse due to other three phase equipment feeding back onto the open circuit phase 

If 1 Link is open say R then windings R-S & R-T will have no input. Only Windings S -N- T in series get full voltage. So only S & T wrt N  each other have voltage. So you should confirm the voltage across each windings.

Nothing can be derived out of the picture you placed untill at least a working block-diagram is uploaded.
 

The core type three phase voltage transformer with neutral grounding in primary side consist three induced voltage in each phase even one of primary three phase voltage is disconnected; because the symmetrical magnetizing path of this type of transformer leads residual magnetic flux of two active limbs to other dead limb, therefore residual induced voltage due to vector summation of two voltage with 120 degree difference phase can cause third secondary induced voltage without existing corresponded phase in primary side. This condition can be cause non operation of under voltage relay in secondary side of same voltage transformer. In this case removing of primary grounding can help to prevention of relay error operation.