**1. Introduction**

Microwave circuits are now most often made in planar technology on various substrates: Silicon for medium frequencies of mobile telephony, Alumina for hybrid circuits, gallium arsenide for millimeter waves. The active parts have dimensions in the micron range for the lines; we pass to the ten of microns as far as transverse dimensions are concerned; as, for the other dimensions (lines or selves), they can be of the order of a few millimeters. For many years, it appeared that it was inappropriate to simulate such an ensemble with a single software that, by a very tightly meshed, would describe the details of the multiple heterogeneities of a localized element, then, with a looser mesh, would take into account the excitation lines and the box (waveguide).

The first idea to avoid the unnecessary approach of simulating small elements that can be isolated from the surrounding circuit to measure them has resided the introduction of localized elements ("localized elements") in electromagnetic calculation software and electromagnetic calculations: Finite Elements, finite differences, method of moments [1, 2].

The second idea is more rigorous. It simulates the different parts of the circuit by software adapted to each function (method of moments for the homogeneous parts and finite differences for the strongly heterogeneous elements, for example). In a second step, the values of the fields are equalized to the limits of the domains [3, 4].

In the many studies published on the subject, one can note that the coupling problem is always expressed in a matrix form that Maxwell's equations take after a spatial or spectral truncation operation. The "compression method" [5] is, in this sense, clear and efficient. It applied to planar circuits modeled by a method of moments. The "basis functions" or test functions are triangular functions (or "rooftop" functions). The problem is then discretized, and the relations between voltage and current in a localized element are then introduced as relations between neighboring roof-tops. We use the edges as impedance ports to introduce localized elements in the finite element method [6, 7]. The approach is similar for the finite difference time domain methods and transmission line method (FDTD and TLM).

Coupling two sub-domains of a circuit, one being the "external" circuit and the other, a "localized" or "interior" part, is an objective that must be achieved independently of the mesh or the chosen numerical method. This objective is achieved in the context of integral methods by applying the "equivalence principle": An electric field on a given surface is replaced by two opposing magnetic current sources [8]. We can also reason by duality with sources of electric currents. Therefore, the model exists but must be redefined in the sense of a change of scale until it can integrate localized elements. The difficulties appear then because the passage to the limit of the zero dimensions for an element or a source is not possible in general electromagnetism under the penalty of divergences in the series. However, it is easy to admit that the meticulous description of the internal functioning of a transistor is not indispensable to predict its behavior in a circuit. The only thing that counts is its extrinsic electrical characteristics, i.e., those accessible to measurement. Then, we can affirm that the electromagnetic effects in the external domain depend only on certain parameters. No matter how the active element is manufactured, only a few average values of the electromagnetic fields count. Rather than approaching the active element by going to the zero-dimensional limit, which is impossible, we should approach the problem by defining an equivalence between elements by identifying their electrical characteristics.

To clarify these notions and evaluate the limits of validity of the hypotheses, it was judged simpler to detail the calculation on an analytical example after recalling the definition of the different types of sources used in the electromagnetic simulation. In the context of integral methods for studying planar circuits, we place them in the spectral domain (or simply in "transverse resonance").
