The WIM concept based on iterative method is to calculate amplitude and direction of incident wave, reflected wave and transmitted wave in the multi-layers planar structure. The electric field, magnetic field and network parameters of equivalent circuit are results that we want to solve and display.
Transmission line is represented by equivalent circuit, as shown in
Considering the input port, as shown in
The relation equation base on the incident wave (A) and reflected wave (B) is presented by
and
where
Then, the input voltage and current equation can be written as
Rewrite the equations in the form of an electric field and current density that are as
Equation (9) and (10) are the electric field and current density (or magnetic field) in following the wave equation. The variable A (incident wave) and B (reflected wave) are the key parameters used in the WIM algorithm.
Considering, the scattering parameter (S) of a two ports network as shown in
The S parameters defined by the incident and reflected wave are expressed as
where An, Bn are the wave variables and
The parameters variable Sii is called the reflection coefficients at port
Wave propagation described by incident, reflected and transmitted waves is represented in the planar structure. We see that the waves will be reflected continuously, as shown in
In iterative procedure, the excited wave
dielectric region (i) 1 and 2, the wave
The WIM procedure, as shown in
1) Define the excited wave
2) Convert the waves in the real domain to the spectrum domain by the FFT:
3) Apply the reflection coefficient
4) Transform the waves in the spectrum domain to the real domain by the IFFT:
5) Calculate the reflected waves using the scattering parameters of planar circuit:
6) Repeat step 2 to step 5 until the convergence of the network parameters are obtained.
After testing the convergence at the k iterations, the tangential electric field and current density in the discontinuity using Equations (9) and (10), can be written as
Thus, the admittance parameter of two ports network are obtained as
also, the impedance parameter can be written as
Finally, the scattering parameter of planar circuit is given by
The detail of mathematical operator in the WIM procedure, as shown in
The excited wave
where
For simplify the calculation of the generalized TEm,n, TMm,n mode wave description, the Modal FFT pair permits movement the transverse filed components from the real domain to the spectrum domain, the modal wave equation in x direction can be defined as
And also, the equation in y direction is defined as
Thus, the modal transform matrix using WIM algorithm can be represented as
Similar, the Modal IFFT pair permits movement the modal filed components from the spectrum domain comeback to the real domain, the spatial wave equation in x direction can be defined as
And also, the wave equation in y direction is defined as
Thus, the spatial wave matrix using WIM algorithm can be represented as
where
The expression of reflection coefficient at the upper and bottom side of box in the spectrum domain is given by
where the TEm,n, TMm,n mode admittances in the metallic box are
At the printed surface of the discontinuity, the boundary conditions of fields, as shown in
Case 1, on the metal regions (M), we have the condition;
Case 2, on the dielectric regions (D), we have the conditions;
Case 3, on the planar source regions (P), we have the condition;
where E0 refers the excited electric field and the Z0 refers the source internal impedance, and
Finally, at the planar circuit in the real domain, the scattering parameters of wave equation are summarized on each printed surface region using Equations (23)-(25). The wave relation equation can be expressed as
where
When considering the condition of each region, on the dielectric region: