Why Doesn't the Current in a Transformer Diverge?
Lenz's Law states that the direction of the electric current which is induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field.
So, rather than causing a uncontrolled &quot;positive feedback loop&quot;, growing with out bound, the currents in the coils of a transformer form a &quot;negative feedback loop&quot;, which creates a stable, finite, output.
Edit:
The way transformer action is often explained by considering components of the transformer current.
First, assume that there is no load on the secondary. It is open circuit. That makes the primary a simple inductor. If the inductor were ideal, the current through the inductor would be \$90^{\circ}\$ degrees out of phase with the applied voltage. This current is called the excitation current, or magnitization current. This magnitization current creates flux in the core. The changing voltage causes changing current, which causes changing flux, which induces a back emf (voltage) across the primary, which exactly matches the applied voltage.
Even though, at this point, there is no load on the secondary, the secondary does enclose a changing flux. So, (again ignoring resistance, and leakage inductance and other complicating factors) the changing flux that induces an emf in the primary also induces an emf in the secondary. and the ratio between the two emf's is set by the turns ratio. That is, the voltage on the secondary tracks the voltage on the primary by a constant factor.
Now, when we add a load, things get more complicated. But we will consider this magnitization current as a component of whatever current may happen to exist in the primary. It will, independent of whatever else happens, cause a flux in the core, a changing flux, that will induce a voltage in both the primary and secondary coils, and these induced voltages will be instantaneously proportional to each other.
So what happens when we add a load to the secondary? Let's assume it is a resistive load. Since there is a voltage on the secondary, the resistive load will draw an instantaneous current which is proportional to the instantaneous voltage. That current, will cause a proportional replica of itself in the primary which is a separate component in the primary to the magnitization or excitation current. The load current in the secondary and the current it causes to exist in the primary above and beyond (and out of phase with) the excitation current work in opposite directions as far as creating flux goes. In combination, (again assuming ideality) they do not add any flux at all to the transformer core. The flux is completely determined by the magnitization current, rather than by the load current.
This explanation may leave you with more questions, but it will perhaps allow you to focus more clearly upon what those questions are.