The electromagnetic properties of receiving and scattering structures with nonlinear components are difficult to predict, especially if more than one nonlinear load is involved. In this contribution a frame antenna with two diodes that act as nonlinear loads is analyzed. This receiving antenna structure is illuminated by a polarized plane wave, carrying a transient HPEM signal. It is then the given task to compute the electric field distribution in the vicinity of the antenna. To this end, a macromodel of the nonlinearly loaded structure that relates the transient signal to the electric field measured at an observation point is derived. As a result it is seen how recent macromodeling techniques can be applied to solve the given problem step by step. This provides the possibility to further analyze the interaction of the given nonlinear loads by a general framework.

The study of nonlinearly loaded receiving structures, such as antennas or transmission lines, has a rather
long history in Electromagnetic Compatibility. Many of the techniques used to analyze such structures are based on one of the two following concepts: The first one is the harmonic balance technique which is capable to deal with weak and strong nonlinear circuits

In a recent study, the response of nonlinearly loaded loop antennas under HPEM excitation has been investigated
where the nonlinearity was given by a single diode. This led to the observation of a rectifying effect of the diode which, in turn, led to a rather long lasting direct component which was attributed to an electric energy storage effect

If the complexity of the nonlinearly loaded structures increases by the inclusion of more than one nonlinear
element, the procedure described in the above paragraph becomes more and more impractical. In this case,
advanced methods of macromodeling turn out to be much more efficient

Having these advantages of macromodeling in mind, it is the aim of this study to further analyze the mentioned energy storage effect in the presence of more than one diode. In particular, a wire frame antenna including two diodes as nonlinear loads will be considered. The scattering behavior of this structure will be analyzed using the macromodeling and convolution techniques cited above. This allows to divide the problem in a way such that the interaction between the two nonlinear loads can be observed and understood.

The remainder of this article is organized as follows: First, a frame antenna that includes two diodes as nonlinear loads and is excited by a plane wave carrying a transient excitation is introduced in Sect.

The receiving structure introduced in Fig.

Frame antenna with two embedded diodes excited by a plane wave carrying a transient excitation.

Equivalent circuit of the diode implemented in the full wave simulation

Diode parameters.

As transient HPEM excitation signal, a double exponential pulse is used. It is carried by the incident plane wave and mathematically described by

A full wave simulation is performed with CST Microwave Studio and the electric field measured by the virtual field probe of Fig.

Electric fields measured by the virtual field probe.

The excitation introduced in Sect.

Electric fields measured by the virtual field probe for a single nonlinearly loaded receiving structure.

It seems as if in the case of two diodes there is a counteracting effect which leads to a cancellation of initially present oscillation. In the following it will be shown how macromodeling techniques can contribute to calculate and understand this cancellation.

For the following approach it is an essential idea to replace both diodes by equivalent ideal voltage sources. This is in accordance to the substitution theorem

Decomposition of the original setup

Since no nonlinear elements are involved in this constellation, the superposition principle can be used to calculate the field strength at the observation point. The field contributions of the three wire frames (b),
(c), and (d) are treated separately. The first contribution is caused by the incident field with both voltage sources

It is the aim to create a macromodel of the nonlinearly loaded structure capable to relate the voltage drops

The continuous functions

Electric circuit simulation with nonlinear elements.

In order to perform the required circuit simulation, the open circuit voltages

Computation of open circuit voltages with the diodes removed.

To numerically obtain a suitable model, a time domain full wave simulation of the setup depicted in Fig.

After having determined the open circuit voltages, the electric circuit simulation of the model depicted in Fig.

The resulting electric field at the observation point is calculated from the decomposition depicted in Fig.

Two transfer functions that relate the signal carried by the incoming plane wave to the field components

Two transfer functions that relate the voltage

Two transfer functions that relate the voltage

As a result, the macromodel of the structure introduced in Fig.

In order to verify the efficiency of the macromodel, simulation results obtained from both full wave simulations and macromodel simulations are compared. The structure introduced in Sect.

The field component in

The comparison concerning the field in

One question that was raised during the beginning of this investigation was for the reason of the absence of pronounced oscillations in the early time response for the case of two diodes in the antenna. This behavior
can now be understood by considering the separate field contributions resulting from the configurations (c) and (d). In Fig.

The particular example of a multiple nonlinearly loaded receiving structure showed that macromodeling techniques are an efficient analytical tool. The major advantage is that full wave simulations can be replaced by electric circuit simulations which are much faster and less memory consuming. It is possible to split a complex problem into several parts. Moreover, components with nonlinear characteristics are quite easy to implement in electric circuit simulations. The task to translate data from datasheets of given components correctly into lumped element parameters capable to be used in full wave simulation is not immediate.

As already discovered in former studies, nonlinear loads can lead to DC-components which may persist for a comparatively long time. The interaction of multiple nonlinear loads leads to further effects that a priori are difficult to predict. Methods to investigate these effects in depth are therefore a useful instrument with potential applications to EMC.

The data presented in this article are available from the authors upon request.

The computer simulations and data processing including all graphical representations were carried out by RM. Editorial hints as well as mathematical and physical suggestions were given by MS, SF, and FG.

At least one of the (co-)authors is a member of the editorial board of

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Kleinheubacher Berichte 2021”.

The authors are indebted to Cheng Yang and Christian Schuster for kindly introducing them to advanced macromodeling techniques.

This research work was partially funded by the Bundeswehr Research Institute for Protective Technologies and CBRN Protection, Munster (grant no. E/E590/JZ001/HF063).

This paper was edited by Alexander Kraus and reviewed by Heyno Garbe and one anonymous referee.