How is an inductor a lumped element?

Question

While learning the Lumped Circuit Abstraction (and the Lumped elements discipline), an inductor was considered a lumped element. But I fail to see how it fits the requirements of LCA:

1. For it, the \$\frac{d\phi}{dt}\$ is not zero.
2. In a circuit with an inductor, (in AC supply), resistances would not be the only irreversible energy loss, owing to the inductor's em waves.

Although there are various ways to put in the restrictions after which LCA is rendered valid, but these two requirements are commonly cited.

Answer 1

It is important to realize that lumped circuit element conditions must hold only outside of the circuit elements. Otherwise you would not be able to have any dynamics (current or volatge changes) at all.

I.e. the requirement for good lumped circuit element approximation is that \$\frac{d\Phi}{dt} =0\$
outside of the lumped element.

That's the case for a good inductor (e.g. a toroidal core inductor; the complete magnetic field change is kept inside).

The same is the case for \$\frac{dq}{dt}\$:
It has to be 0 outside of the circuit elements. Of course it is not true if you look at only one plate of a capacitor. It is true though for the outside of the whole capacitor.