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.

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    \$\begingroup\$ All lumped elements are unrealistic perfect models. \$\endgroup\$ – PlasmaHH Jan 27 '17 at 10:56
  • \$\begingroup\$ But even for a perfect inductor, would not the rate of change of flux be non-zero? \$\endgroup\$ – Satwik Pasani Jan 27 '17 at 11:03
  • \$\begingroup\$ @PlasmaHH: that's not answering the questions. Although it is a model it is satisfied to a very very large extent in very very many practical cases; namely in all cases where it makes sense to work with a circuit diagram (with extensive quantities contained in ODEs) instead of Maxwells Equations (with intensive quantities contained in PDEs). The latter one is much more difficult. \$\endgroup\$ – Curd Jan 27 '17 at 11:57

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.

  • \$\begingroup\$ What about the power dissipation? \$\endgroup\$ – Satwik Pasani Jan 27 '17 at 11:54
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    \$\begingroup\$ Do you mean power dissipation because of resistance in an inductor? You can model this simply by an equivalent series resistor (ESR) which would still statisfy lumped circuit approximation. Power loss by E/M field radiation would, however, violate the lumped circuit approximation. \$\endgroup\$ – Curd Jan 27 '17 at 12:04

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