Transformer: loaded vs open-circuited vs short-circuited

I am trying to understand the working principle of current and voltage transformers.

I understand that V.T. and C.T. are basically just transformers, but I can't understand how secondary winding voltage and current are defined by the load on the secondary winding. Hence, for current transformer, the secondary winding is short circuted (very low impedence), and if roughly speaking, voltage on secondary is determined by the equation:

$$ U_{secondary} = U_{primary} * k; where\\k = \frac{Ns}{Np} $$

and current

$$ I_{secondary} = \frac{I_{primary}}k $$

Then the current transformer works as voltage step-up transformer and voltage should be some value, but instead, because of the low load impedence Z = 0. It follows from Ohm's Law

$$U = I * Z $$

that U_secondary is 0. Why do these equations contradict each other?

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.

Answer

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