**4. Modeling localized impedance in microstrip circuit**

Let us consider a microstrip line loaded by an impedance represented by a surface impedance of value *Zs*. The line is assumed to be excited by the TEM mode. The edge effects are neglected and, to simplify the numerical approach, it is assumed that the line is simply bordered by magnetic walls (**Figure 16**). A magnetic wall P also limits the functional area.

This problem can be addressed in two ways. Either the circuit is modeled globally by taking a virtual source (test function) instead of *Zs* for 0< *y*<*c*, or we take an auxiliary source for 0 <*y*<*c*. We calculate the quadrupole *Q* between this source and the main excitation, then we close this source by impedance *Zs*, following the circuit shown in **Figure 14**.

From the calculation developed in the appendix, the following conclusions can be drawn: first, if we want the same result between a direct approach and an indirect approach through auxiliary sources, it is necessary to take as many auxiliary sources as there are test functions in the first approach; it is necessary to take as many sources. This means that if one seeks precision of the results by taking into account, for example, the effects of edges to define the shape of the electric field of a single auxiliary source, other the difficulty that this way of making will present to evaluate the impedance seen of the source (as it was specified in II-2), this selected shape may be inadequate for a different impedance. If the precision is judged insufficient with the only auxiliary source, it will be necessary to introduce others and to study the action of these on the localized element, if possible. The second remark concerns the problem that arises as soon as the dimensions of the subdomain corresponding to a localized element tend to zero. In the example given in an appendix, a capacitance that tends to infinity appears parallel to an impedance whose dimension tends to zero; the capacitive nature comes from the TM modes alone present in the structure studied. For a more realistic impedance placed at the end of a microstrip line, we would find an inductance in series with the impedance tending to infinity with the width of the latter, in addition to the capacitance in parallel.

These remarks raise the difficulty of defining the behavior of a circuit of very small dimensions placed in a box. It is intuitively clear that the details of the internal functioning of this circuit are not affected by the large external circuit; only its electrical characteristics matter, the geometrical parameters do not influence in themselves. Hence, the idea of locating the impedances by admitting that their dimensions are zero. As this approach is not possible, because of the divergences of this discrepancy, we must return to the process of measuring an impedance and see how it can help to define its behavior when it is inserted in any circuit, because the circuit, because the only accessible parameters are those measured, or modeled from a fine description of the element.
