For an Ideal transformer where permeability of the iron is infinity, why is the excitation current 0 if the secondary winding is open circuit?

Question

I have read this question Why is the Exciting Current zero in an ideal transformer?. However I don't quite understand where his initial equation for the excitation current came from. Can someone guide me though the process step by step from a lower level, starting from the moment a voltage is applied?

My intuition tells me that when a voltage is applied, a current will begin to flow, causing an induced magnetic field in the coil. However, since the permeability is infinity, that will result in an infinite B field, and an infinite flux.(this is apparently not true, and I am quite confused)

Answer 1

My intuition tells me that when a voltage is applied, a current will begin to flow, causing an induced magnetic field in the coil. However, since the permeability is infinity, that will result in an infinite B field, and an infinite flux.(this is apparently not true, and I am quite confused)

Intuition and infinity don't play nicely together.

Field = inductance x current.

If the inductance is infinite, then you can get any field with zero current, as \$0 \times \infty \$ is undefined, it can be any real number.

I'm an engineer, so I don't know what mathematicians mean by infinity, but whatever it is, it's not really relevant to real life. Engineers generally mean it in the terms of a limit, as something becomes so big that its reciprocal becomes negligible.

Going back to field = inductance x current, and ignoring any constants of proportionality, let's say we get unit field with one 1A and 1H. If we had 10H, then we'd need only 100mA. With 100H, we need only 10mA. As we let the inductance get larger and larger, the required current gets smaller and smaller. It gets far smaller than any load current, it gets smaller than anything we can measure.

This is what engineers mean by infinite inductance, that the current required to create any finite field in it is negligible for all practical purposes.

Answer 2

Infinite quantities are only a writing habit. We have many physical relations where A/B=C. One of them is Ohm's law. In some physical constructions quantities A and C can be both meaningful, but C happen to be always zero, no matter how we set A. Obviously something is different than in cases where C=A/B where B has a certain numeric value. In these cases we generally write that B=infinite. It has no other meaning. It starts from the basic assumption C=0.

In math infinite is not a number. It cannot obey normal calculation rules. It's proven that insert even a single new number to the set of normal finite real and complex numbers, then calculation rules cannot be any more valid.

An example: Obviously you can agree that infinite+1 = infinite. If infinite were a valid number, then 1 would be equal to zero.

Infinity has been adapted to valid math in two ways:

1) as the size (=cardinality) of sets

2) as a limit process where a variable in a calculation is let to grow, but it always still stays a number. If the result of the calculation approaches to some certain value we call it the limit value of the result. The growing variable is said to be approaching infinite.

One can also manipulate a full set of equations which describe how a circuit behaves. The equations can approach to some simpler set of "limit" equations if some quantities grow with no limits or shrink towards zero.

Ideal transformer is such limit case of ordinary transformers. It's behaviour is the limit of ordinary behaviours when inductances grow and losses get reduced towards zero. But all calculation attempts with actual infinite inductances are nonsense except one: Current is and stays zero.