Transformer: loaded vs open-circuited vs short-circuited
Why do these equations contradict each other?


First of all, you're using ideal transformer equations and that's fine as long as you apply them properly.

But you haven't applied them properly in this case.

Assuming an ideal transformer, if the secondary is loaded with a short-circuit, the voltage across the secondary is zero volts and thus, the primary voltage must be zero.

The fact is that the equations you provide must be satisfied simultaneously.

So, assuming the secondary is loaded with impedance \$Z\$, the transformer equation becomes

$$V_s = I_s \cdot Z = kV_p $$

but

$$I_s = \frac{I_p}{k} $$

thus

$$V_p = \frac{I_p}{k^2}\cdot Z$$

And there you have it, \$V_p\$ and \$I_p\$ are not independent of the secondary load \$Z\$.

In fact, you see that when \$Z = 0 \mathrm{\Omega}\$, the primary voltage must be zero for any finite primary current.

So, there's no contradiction.


  I want to know one equation that describes all parametrs of
  transformer where is possible to see how primary and secondary
  voltages and currents change depending on impedences in transformer.


Then start with this model of a physical transformer



and solve for the primary and secondary voltages and currents.

If I recall correctly,


R1 models the primary winding resistance
X1 models the primary leakage inductance
Rm models core loss due to hysteresis
Xm models the finite permeability of the core
X2 models the secondary leakage inductance
R2 models the secondary winding resistance