IS TRANSFORMER A CONSTANT FLUX MACHINE ?

There may be two different kind of discussion regarding this. 

First let me consider:

Ideal Transformer

For an ideal Transformer, to be precise, the internal drops of across resistance and reactances(r1 & x1) will be zero. So primary voltage be the voltage across the shunt branch. Hence in that case we could consider the Transformer as a constant flux machine.

Practical Transformer

For practical Transformer we could say that it is not a constant flux machine. It is because in Transformer no load current will be 2% to 6% of Full load current. So drop must be considered across the series resistance and reactances (r1 & x1). Due to this drop the voltage across shunt branch in equivalent circuit decreases with decrease in terminal voltage. 

Conclusion

Though in practical Transformer, the drop across series branch will be considered but that drop will be very less, so in general we could say that Transformer is a constant flux machine but in practice flux is not constant, rather it is not considered due to very less value.