Transformer, how the primary current is adjusted to be equal to the secondary load (without considering magnetizing current)?
It seems to me that you have to be helped to recognize this: -



Picture from here.

In other words, if you apply 1 volt to the blue winding then 1 volt will appear on the red winding. This happens when the turns on both windings are in-phase and very closely coupled but I haven't given a proper reason why this happens yet.

Consider ONLY the blue winding that has 1 volt applied to it. The current into that winding is the magnetization current and, it ramps upwards governed by: -

$$\dfrac{di}{dt} = \dfrac{V}{L}$$

In that blue winding there is a back-emf that is equal to the applied voltage. This is an induced voltage and, if you introduced the red winding (the secondary) you would see the same induced voltage. It would be the same because the magnetic flux produced by the magnetization current in the blue winding is fully coupled to the red secondary winding.

Now what would happen if we attached a load resistor of 1 ohm to the secondary winding?

Current (1 amp) would flow into that 1 ohm resistor like so: -



That current (on the face of it) could create an extra magnetic flux in the transformer windings. The extra flux might (possibly) do two things. It might: -


cause the output voltage to rise (due to \$V = L\dfrac{di}{dt}\$)
cause the output voltage to fall (due to \$V = -L\dfrac{di}{dt}\$)


If it caused the output voltage to rise then we would have an unstable situation because, due to that voltage increase, there has to be more current flowing into the 1 ohm resistor and this means more flux and more voltage and this ends in disaster.

If it caused the output voltage to fall then the back emf in the primary would also fall and we would have a situation where the primary took an unholy amount of current.

But, the reality is that neither of the above happens. If either of the above happened we end with contradictions - if the secondary voltage rose driving more current through the 1 ohm load, the primary voltage also has to rise but it can't because, it is limited by the applied 1 volt hence, current has to simultaneously flow back into the 1 volt source on the blue winding whilst also flowing into the secondary load i.e. we have uncontrolled perpetual energy that self-destructs.

If the secondary voltage fell we have the contradiction of primary (blue) current rising uncontrollably which means flux rises uncontrollably which means secondary (red) voltage has to rise - a contradiction.

The sensible equilibrium is found when we examine the 3rd scenario: -


The rate of change of flux in the windings remains constant


This can only happen when the load current flowing from the secondary (red) is matched by a load current flowing into the primary (blue). These load currents produce equal and opposite magnetic fluxes and hence they cancel leaving the original magnetization flux and, that magnetization flux carries on doing what it did all along - setting the output voltage to be equal to the input voltage (1:1).

For a transformer with a non unity turns ratio, the secondary load ampere-turns are equal and opposite with the primary current ampere-turns that flow due to the load.