# Transformer Zero Flux Approximation

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?

• It's not a generally accepted approximation - what makes you say that? Maybe some mickey mouse website? Oct 1, 2021 at 7:21
• For an ideal transformer, the current is related by the turns ratio via $I_2 = I_1 \frac{N_1}{N_2}$. This is because the assumption is made that $\phi_{core} R_{core} = 0$
Oct 1, 2021 at 18:35

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.

• Thanks for your response; it makes perfect sense. My follow up question is what math/physics enables you to set the $\Phi R$ term to 0? that For a voltage buffer amplifier, the error in input voltage is $\frac{1}{A+1}$, and as the open loop gain approaches infinity, the inputs become the same voltage. Is there some similar math argument that explains the approximation for a transformer?