Transformer Zero Flux Approximation

Question

I have a question about a standard approximation made about the magnetic flux in transformer analysis. Using the magnetic circuit model, the sum of the magnetomotive forces in a transformer are:

$$ \Phi_{core} R_{core} = N_1 I_1 - N_2I_2$$

The core reluctance \$ R_{core} \$ is very small for a transformer due to \$ \mu_r \$ being very high. Therefore, the approximation is made that the term \$ \Phi_{core} R_{core} \$ can be ignored. At first glance, I would accept this argument.

However, if you think of a KVL for an electrical circuit with two voltage sources separated by a very small resistor, the approximation doesn't make sense:

$$ IR = V_1 -V_2$$

Though the resistance is small, there is still a voltage drop between the two sources due to large current. Therefore, the term on the left is non-negligible.

So is the approximation that \$\Phi_{core} R_{core} = 0\$ wrong?

EDIT: There is a good answer to my initial question. I have a follow up question about what specific math\physics let you set the term to \$\Phi_{core} R_{core}\$ term to 0. For the case of two opposing voltage sources in series, if \$V_1\$ is more stiff, with a much higher decoupling capacitance than the capacitance of \$V_2\$, then after an initial transient, the voltage of \$V_2 \$ would be the same as \$V_1\$. Is there a similar train of thought that would enable \$N_1 I_1 = N_2 I_2 \$ after an initial transient?

Answer 1

It's a simple first order model, for a transformer with high or infinite permeability.

The important thing is that for a well designed high permeability core transformer, operating within its ratings, \$ \Phi_{core} R_{core} \$ is very small compared to either \$N_1 I_1\$ or \$N_2I_2\$. It's a small difference between two big numbers, so zero is a good approximation.

It's much the same sort of approximation you would make when you say an op-amp has infinite open loop gain, so the two inputs are at the same voltage. That works very well for understanding the basic circuit. If you want a more detailed model, then you let the gain be finite, and compute the finer details.

If you want to work with the magnetising current, then you need a finite permeability, and a non-zero left hand side.

It's not appropriate for air-core or low permeability transformers, or transformers being operated into saturation which collapses the permeability.