**3. Use of auxiliary sources in planar circuits**

The auxiliary sources allow the passage from a large scale at which the calculation is purely electromagnetic, to a smaller scale that corresponds to the connection of purely electromagnetic, to a smaller one which corresponds to the connection of circuit elements, most often characterized elsewhere.

The main and auxiliary sources are of such dimensions that the definition of voltage and current are intensity are unambiguously defined, as mentioned in the previous paragraph. Taking the general scheme of a planar circuit comprising one or more main sources and auxiliary sources, we can represent the different interactions in the following way (**Figure 14**): *Q* is the multipole of electromagnetic interactions.

**Figure 14.** *Interactions between main and auxiliary sources.*

At the junction level, surfaces between sources and the surrounding medium, *S* and *S*<sup>0</sup> , it must be possible to define at the same time an electric field, a voltage, a current, and a magnetic field; this is why the condition of Eq. (12) has been imposed.

However, this is not without problems because a global calculation (without auxiliary sources) gives a form of the field which does not verify this condition. This is shown by the simple example treated in the following.

This raises the problem of the validity of introducing the auxiliary sources; the analysis of this example gives an order of magnitude of the maximum dimensions to take for these last ones (**Figure 15**).
