The equivalence theorem and inverse source reconstruction methods are powerful tools in electromagnetics and can be used to extract equivalent current sources from measurements of the radiated fields. However, when only partial measurements are available over a part of the closed surface enclosing the radiating source, the extraction of equivalent sources becomes more challenging, and there are certain conditions and considerations that need to be taken into account.
First, let's briefly discuss the equivalence theorem. The equivalence theorem is a fundamental concept in electromagnetics that states that the fields produced by an electric current distribution on a closed surface are equivalent to the fields radiated by a localized electric current source located inside the closed surface. The equivalence theorem allows you to replace the complex structure and interactions inside the closed surface with an equivalent source that generates the same fields outside the surface.
When you have access to measurements of all tangential fields over the entire closed surface, you have complete information to reconstruct the underlying equivalent current sources accurately. However, if you only have partial measurements of the fields over a part of the surface, you need to consider the following factors:
Spatial Sampling: The spatial sampling of the partial measurements should be carefully chosen. Irregular or sparse sampling can lead to inaccuracies in the extraction process. Ideally, the measurements should be well-distributed over the region of interest to provide sufficient information for source reconstruction.
Boundary Conditions: The partial measurements should be taken under appropriate boundary conditions. These conditions may depend on the specific radiating source and the closed surface geometry. Inaccurate boundary conditions can lead to errors in source extraction.
Source Localization: When you have partial measurements, it becomes crucial to accurately localize the position of the radiating source inside the closed surface. Incorrect localization can result in significant errors in the extracted equivalent current sources.
Regularization: Inverse source reconstruction methods typically involve solving an ill-posed problem, which can be sensitive to noise and measurement errors. Regularization techniques are often used to stabilize the inversion process and improve the accuracy of the extracted sources.
Modeling Assumptions: The success of inverse source reconstruction also depends on the accuracy of the underlying models used to describe the electromagnetic behavior of the system. Assumptions made during the modeling process can introduce uncertainties in the extraction results.
In summary, while it is possible to extract equivalent current sources from partial measurements using the equivalence theorem or inverse source reconstruction methods, there are several challenges and conditions that need to be addressed to ensure accurate results. Proper spatial sampling, appropriate boundary conditions, accurate source localization, regularization techniques, and reliable modeling assumptions are all important factors that need to be carefully considered in the extraction process. Additionally, the complexity of the problem may increase with the amount of partial measurements and the characteristics of the radiating source and closed surface.
