Constant flux from no load to full load

The core flux of a transformer is fixed by the applied primary voltage, not by the load. Therefore it stays practically constant from no load to full load.

Reason 1: voltage balance in the primary

$$V_1 = E_1 + I_1Z_1 \approx E_1 = 4.44fN_1\phi_m$$

The primary impedance drop $I_1Z_1$ is only 1 to 3% of $V_1$ at full load. Since the supply voltage $V_1$ and frequency $f$ are constant, $\phi_m = \dfrac{V_1}{4.44fN_1}$ stays practically constant.

Reason 2: mmf balance

When a load current $I_2$ flows, its mmf $N_2I_2$ tends to reduce the flux. The primary current immediately increases by $I_1'$, so that

$$N_1I_1' = N_2I_2$$

The net mmf in the core is then again $N_1I_\mu$, the no-load value, so the flux does not change.

Conclusion: the only change in flux comes from the small primary impedance drop $I_1Z_1$, which is about 1 to 3% at full load. Because the flux is constant, the core (iron) loss is constant at all loads, and the core loss measured at no load can be taken as the core loss at full load.