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This paper proposes a generalization of the MoM-GEC method [1] needed for studying planar structures excited with a source located at perpendicular plan relative to circuit plan. A general formulation is detailed to allow modeling excitation of a planar structure with one or more sources located in plans other than the circuit plan. The numerical approach elaborated is based on the definition of new admittance operators and rotational transformations describing the transition from one plan to another. To validate this approach, we consider the case of a single source located in the perpendicular plan to the circuit.

Studying of microwave planar circuits is based on EM characterization of the structure via determination of electrical (E) and magnetic (H) fields and the electric current density J using Maxwell equations. To solve these equations, iterative methods are used (FDTD [

As the source is the knowledge of an electromagnetic field distribution on a circuit surface independently of the load, one distinguishes two source models: localized source modal [

Localized source remains a theoretical and relatively simple model to describe studied structure. Such model, is not fully consistent with actual excitations located in another plan than the circuit which favors the extended source, hence the importance of our study.

Our goal is to develop an exact source method based on MoM-GEC method for studying planar structures excited by sources located in any other plan. We first establish a general formulation for the case of N sources. We verify, then the accuracy of the hypothesis of sources simplification: localized planar sources.

3D extension of MoM-GEC method has not been really focused on in previous work. Related work mainly includes Hamdi et al. work in [

In this paper, input impedance, current density and electric field distribution are evaluated and discussed in the case of a single vertical source. Results are compared to commercial software HFSS and CST. A couple of structures are studied: microstrip short-circuited line and microstrip open circuit line as an application.

This paper is organized as follows: we start presenting the new approach in the case of N sources by exposing the general formulation of integral equations to determine the admittance matrix. Then, we detail the formulation and the determination of input admittance (

In this section, we introduce and explain the general formulation for 3D structures by considering the case of N-ports (N sources) located at perpendicular plans to the circuit.

voltage sources (

where L is an integro-differential operator, g is the excitation source and f is the unknown to be determined.

In this work, L is an admittance operator, f is the electric field E tangential to the circuit plan and g represents the excitation sources. The method consists in solving this equation by the Galerkin method (a variety of the MoM method) to determine the electric field E and deduce the input admittance

The circuit is printed on a dielectric substrate of relative dielectric constant

To characterize this discontinuity, we used the formalism of admittance operator and assume that excitation sources are totally independent and completely decoupled (electromagnetic coupling) for each other [

The electromagnetic quantities in source plans (from P_{1} to P_{N}) and planar circuit (c) are expressed in following relations (Equation (2)):

where:

^{th} plan is the circuit plan.

E and J are the electric field and the current density defined in the circuit plan.

The admittance operators ^{th} plan and the current density generated by these modes on the i^{t}^{h} plan provided that all sources k # j are switched off and the circuit plan is metallized. These

If

For modeling the electric field E of the circuit plan, we choose test functions of electric field type

With

Test functions satisfy the boundary conditions of the circuit plan. They are zero on the metal and non-zero on the dielectric. And conversely, the current density J defined on the circuit plan is zero on the dielectric and non- zero on the metal. Therefore, the test function

The application of the Galerkin method to the Equation (2) involves projecting the first N equations respectively on the unitary sources functions from ^{th} equation on the various test functions which gives us a second sub system.

We suppose that:

With

Considering the orthonormalization relationships verified by these unitary sources: (where

With:

The determination of the admittance matrix

In this section, we presented a general formulation of the problem by taking the case of N vertical sources. By applying the Galerkin method, we determined the admittance matrix binding current density and E field. This matrix is characterized by new admittance operators describing the transition from one plan to another. To validate our approach, we develop in this paper the case of a single vertical source that is the subject of the next section.

The vertical source located at perpendicular plan to the circuit is the most used excitation source in real conditions. We focus now on the study of a single vertical source located at the perpendicular plan to the circuit. We start presenting the studied structures. Then, we detail the determination of the admittance operators.

To validate our method, we consider two structures: a micro-strip short-circuited line

second structure allows us to verify the boundary conditions and to ensure the validity of the numerical approach.

With:

l_{p}: length of the microstrip line,

l_{s}: length of the dielectric substrate,

l_{b}: length of the box,

Length | Width | Height | ||
---|---|---|---|---|

Line | l_{p} = λ/2 | - | ||

Dielectric substrate | l_{s} = λ/2 | |||

Box | l_{b} = λ/2 | |||

Length | Width | Height | |
---|---|---|---|

Line | l_{p} = λ/2 | - | |

Dielectric substrate | l_{s} = λ | ||

Box | l_{b} = λ |

w_{p}: width of the microstrip,

a: width of the box/dielectric substrate,

h_{0}: thickness of the dielectric substrate,

h: thickness of the box.

By using the generalized equivalent circuit method, we can model each of the two structures of the

The circuit is excited by a single source of electric field type. This source is defined by a unitary function

With:

To determine the electric field

Using the formulation of the source method developed in the Section 2, the current densities of the source and the circuit are associated to the corresponding electric fields by the admittance operators:

Applying the same procedure in (2), the Galerkin method is used to solve Equation (16) while taking into account the boundary conditions of electromagnetic fields on the circuit plan. The first step in the Galerkin method is to define test functions

The test functions

plan (xoz) (Equation (18)).

where:

The second step is to project the Equation (1) of the System (16) on the unitary function

With:

To calculate the input admittance, we must determine the different admittance operators:

To calculate the different operators, we need to impose some conditions, namely:

The

Similarly, the

This procedure is ensured after establishing in each plan (xoy) and (xoz) a basis of TE and TM mode functions satisfying the boundary conditions and allowing the decomposition of operators

We explain in the next two paragraphs the determination method of the operators

To have an equation system containing only the two operators

Hence the circuit plan split the structure into two homogeneous areas separated by an electric wall. In the straight section of each guide, basis functions

The mode functions TE and TM in the plan (xoy) are given by the Equation (25) and Equation (26):

With:

The propagation constant of the mode

The decomposition of the

With:

The operator

In fact, knowing the electric field in the source plan (xoy) which is decomposed on the mode functions

axes (ox) and (oy), we use the following relationship:

The third component of

Equation (32) presents the relationship binding the operator _{2} (xoz)) and the electric field E_{1} (of the plan P_{1} (xoy)).

To describe the operator

Similarly, we can describe the field E_{1} on the basis of mode functions

Using the Equation (34), we can express the current

By identification (Equation (33) and Equation (35)), the current

Using Equation (33) and the fact that

From the Expression (37), we deduce the expression of the

With:

We define _{1} to the plan P_{2}. To deduce this new operator, we establish an expression for the current density in the plan (xoz) while using Maxwell equations. Applying the Maxwell-Faraday equation, the current density has the following expression (Equation (40)):

By substituting the expression of E_{1} (Equation (34)) in the expression of

By identification between Equations (36) and (41), we can deduce the expression of the new operator

After expressing the different operators and transformations, the next section will be dedicated to numerical results for two chosen structures and make some comparisons to validate our new approach.

The new numerical approach is based on the definition of several admittance operators used to describe the passage from one plan to another. The implementation of these operators require several large-sized matrices manipulation and cpu-consuming integral calculations. In our case, using development environments dedicated to numerical calculations such as MATLAB is not suitable, lacks of fast hybrid symbolic/numeric calculation and has no built-in cache support neither save-points concept (we cannot resume calculation when needed).

There are several alternatives, namely programming languages: C, C++ and Java. In literature, several researchers recommended JAVA for scientific treatment [

In our research laboratory SYS’COM, DrTaha Ben Salah has developed during his research work a TMWLib library (for Tiny MicroWave Library) [

We applied our modelling approach to both structures: microstrip short-circuited line and microstrip open circuit line. We used the microstrip short-circuited line as a reference structure to compare obtained input impedance with theoretical input impedance of this structure. We also deduce for these structures some electromagnetic characteristics (current density J and electric field E) to verify the boundary conditions.

The chosen studied structure to validate the obtained input impedance is a microstrip short-circuited line. This structure must respect two approximations. First, the structure is considered as a transmission line submitted to the line’s fundamental mode (characterized by its propagation constant β_{g}). Then, the line length L should be large enough to assume that higher order modes reflected at the short circuit are attenuated before reaching excitation source. The expected value of the theoretical input admittance is given by the Equation (43).

With

functions (trigonometric type), 94,000 TE and TM mode functions

We observe that the two curves of the input impedance are very close with a relative error lower than 1%. This confirms the validity of our numerical approach and the perfect adaptation between source and circuit. In fact, among the parameters affecting the consistency of results precision of the fundamental mode taken as excitation source has the higher effect.

With X is the length l of the microstrip line.

In this section, we present some electromagnetic characteristics (current density and electric field) for a microstrip short-circuited line. We also compare the obtained results to the results found with two commercial software HFSS and CST.

_{y} along the propagation direction (oy). We note that the electric field satisfies the boundary conditions. It is maximum at the source and presents a fast attenuation at source/line discontinuity. The result with the new approach MOM-GEC’3D contains small attenuation that tend to cancel due to the Gibbs effect. HFSS gives a less step attenuation at source/line discontinuity; whereas CST’s result has the best consistency

_{x}. The two figures obtained with CST and MoM-GEC’3D confirms results alignment.

In this section, we present results of current density and electric field for a microstrip open circuit line to validate our numerical approach.

wavelength, variations along (x) should not be relevant, which is the case of (a). Still CST have some important variation (at the center of the line). Moreover, a better attenuation (

_{y} away from source and magnetic wall (line edge). Again, MoM-GEC’3D gives better results but still very similar to

CST while HFSS gives a little more fuzzy results. All of three results still consistent with boundary conditions though.

Similarly, the electric field component E_{x} verifies the boundary conditions for the result obtained with MOM-GEC’3D and CST while CST, for this case, presents a better boundary conditions. This may be explained with the forced usage of (y) based test functions (in order to validate more generic approach) whereas line width is too small relatively to wavelength, so that Gibbs effect remains a little substantial (

The different simulations made for short circuit and open circuit demonstrates the accuracy of our new approach. This was approved by the verification of the boundary conditions and comparison with two commercial simulation software HFSS and CST.

In this paper, we present a new formulation of the source method to characterize discontinuities in planar circuits. A new definition of the excitation source is introduced to overcome the discontinuity problem at the source/circuit transition. We expose a general formulation of the source method, by determining the Input admittance matrix of N-port discontinuity in a planar circuit. To validate our approach, we considered the case of a single vertical source. We detailed the determination of the various operators and rotational transformations required to calculate the input impedance. In the last part of our work, we presented and interpreted some results in the case of a microstrip short-circuited line and microstrip open circuit line.

The numerical results obtained using this approach were compared to results obtained by both commercial software HFSS and CST. Our results show a concordance and consistency with those obtained using HFSS and CST, with even better results in most cases.

We also demonstrated that considering the fundamental mode of the access line to circuit as the excitation source gives us a perfect adaptation between the source and the circuit.

This new approach can be applied to any type and number of excitation sources (coaxial cable at perpendicular plan [

definition of the source in the integral analysis and the determination of admittance operators to link the electromagnetic quantities of sources and magnitudes of the circuit, which are defined in vertical plans.

Oueslati Basma,Ben Salah Taha,Larbi Chiraz,Aguili Taoufik, (2016) New 3D Aware Formulation of MoM-GEC Method for Studying Planar Structures with Vertical Sources. Journal of Electromagnetic Analysis and Applications,08,79-94. doi: 10.4236/jemaa.2016.84009