2
Formulation and Validation of the WCIP Applied to the Analysis of Multilayer Planar Circuits
2.1. Introduction
For those working within the electromagnetic sphere, wave propagation in multilayer structures is a subject which has proven to be both somewhat significant and of great interest [SUT 03]. RF circuits and microwaves are generally produced within a single layer configuration, although multilayer circuit structures have mainly been used for low-frequency digital systems. The use of multilayer planar circuits makes it possible both to produce more compact microwaves and more flexibly-designed circuits [MIL 07]. The wave concept is a well-established procedure for dealing with electromagnetic problems [KUR 65]. It is used in the Transmission Line Matrix (TLM) method [FIC 09]. To reduce both time limits and development costs of monolithic hyper-frequency integrated circuits Monolithic Microwave Integrated Circuits (MMIC), it is of paramount importance to use a rapid and efficient tool, which may accurately predict both the electrical and electromagnetic behavior of a given device. This method is not restricted by circuit complexity and has proven to be particularly interesting in the case of planar circuits. The Wave Concept Iterative Procedure (WCIP) approach consists of separating the structure studied by an environment which is both superior and inferior, although both acting homogeneously across an interface, in the case of a planar circuit with a single interface. Figure 2.1 shows an example of a multilayer planar circuit with plating on several interfaces.
The boundary conditions operating across interfaces are shown by a diffraction operator, SΩi, which is defined within the spatial sphere. Within homogeneous environments, wave propagation between interfaces is shown by a transfer matrix T, and across the extremes of the circuit (both at the highest and the lowest parts) by a reflection operator, 
i. This propagation is defined as being within the modal sphere. The method supplies a mixed resolution within the modal and spatial sphere and takes full advantage of each sphere with a low computation time.
Figure 2.1. Example of the multilayer planar structure. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
In this Chapter, section 2.2 provides theoretical formulation with simulation results from two case studies. An original application of the concept is shown in section 2.3, together with the theoretical formulations and simulations for an ideal polarizer using the WCIP. In section 2.4, a multilayer structure as a quasi-optic compact amplifier is suggested with a detailed analysis.
2.2. WCIP formulation
2.2.1. Multilayer formulation
The method described here is based upon the complete transverse wave formulation in which the concepts of dual quantities, current density and electric field, are considered. It was brought in as a means to express the boundary conditions across the air/dielectric interface in terms of waves. The waves within both the spatial and modal spheres are directly inferred by a Fourier Modal Transform (FMT) and its converse transform the so-called Inverse Fourier Modal Transform (or FMT-1). We will now consider the general issue with a single interface, as shown in Figure 2.2 by means of its cross-section.
The source A0i generates waves, one on each side of the relevant interface Ωi.
Figure 2.2. Behavior of waves A and B, the transformation spheres within the iterative process for a single interface
The incident wave (A) and the reflective wave (B), the electric field (E) and the current density (J) may be linked by:
where i indicates the environment 1 or 2 corresponding to a given interface in Ω (resistance). Z0i is the impedance which is characteristic of the same environment i with 
being the surface current density vector.
The waves within both the spatial and modal spheres are directly inferred by the FMT and its converse transform, the so-called FMT-1. The decomposition of the electromagnetic wave in propagated modes (both evanescent and propagating) within a waveguide containing periodic walls (modes TE and TM) occur through the application of the FMT. The FMT is made up of the algorithm 2D-FFT and the transformation from the spectral sphere to the modal sphere. The conversion of the so-called modal Fourier and its converse ensures the transition between the two spheres.
The diagram normally used for the WCIP, in respect to a single layer, is very simple. Two operators relate both incident and reflective waves in the spatial sphere and the spectral sphere governs the iterative procedure. It may be shown by two equations.
When the wave concept is extended to multilayered structures, as Figure 2.3 shows. A new formulation between waves (AP-1, BP-1) and (Ap, Bp) is introduced for the transition between two adjacent interfaces [SER 10].
This formulation is obtained from the TLM impedance. The transfer matrix between two adjacent interfaces is expressed as:
or, alternatively:
where Zc is the mode impedance, Z0i the environment impedance i, γ the transmission factor and l the height between two adjacent interfaces.
Figure 2.3. Transfer matrix method, waves and the source within a multilayer structure
Finally, it is possible to state boundary conditions in terms of waves across every cell, and the tangential electric and magnetic fields can then be calculated from the following equations:
2.2.2. Simulation results
To show and evaluate the performance of the proposed tools, two case studies were completed. The cases considered made it possible to deal with both coupling problems and propagation issues.
2.2.2.1. Structure with a gap and a superimposed resonator
A simulation of a microstrip structure with a gap and a superimposed resonator was completed. Figure 2.4 shows the multilayer structure containing various interfaces.
Figure 2.4. 3-D view of the multilayer structure
On the first interface, the two microstrip lines have a width of 25.9 mm, a length of 2.3 mm with a gap of 1 mm. On the second interface, the resonator has a width of 2.3 mm and a length of 27.3 mm. The waveguide is 20 mm in width and 60 mm long, εr1 = εr2 = 2.33, εr3 = 1, L2 = L3 = 0.8382 mm. Both the simulated source for an internal effective impedance of 50 Ω and the simulated charge of the 50 Ω have dimensions of 2.3 mm × 2.3 mm.
The current density Jx upon the resonator of 4 GHz is shown in Figure 2.5. The anticipated physical behavior is clearly evident.
As may be seen in Figure 2.5, the reflection coefficient |S11| and the transmission coefficient |S21| have values which are indeed very close to the measurement results. In particular, the resonance frequency obtained by the WCIP (at 4.05 GHz) is closer to the measurement (at 4.08 GHz) than the simulation (at 4.11 GHz in [FIL 00]), as much in terms of frequency as amplitude. The calculation time for a complete simulation is approximately 20 min with a 2.53 GHz Intel Core 2 Duo processor.
Figure 2.5. Behavior of current density on the second interface at 4 GHz. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 2.6. Absolute value of the reflection coefficient, |S11| and transmission coefficient |S21| in dB according to the frequency in GHz. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
2.2.2.2. Patch antenna with slit interfaces
A patch antenna with a combined slit arrangement was simulated with the software program MatLab. Figures 2.7 and 2.8 show the multilayer structure with the various interfaces. On the first interface, the microstrip power line has a width of 1.9 mm and a length of 14.7 mm. On the second interface, the slit has a length of 4.7 mm and a length of 0.6 mm. The patch on the third interface has a width of 16 mm and a length of 6.65 mm. The wave guide is 22 mm in width and 22 mm length, εr1 = εr4 = 1, εr2 = εr3 = 3.38, l2 = l3 = 0.81 mm.
Figure 2.7. 3-D view of the patch antenna structure containing slit interfaces
Figure 2.8. Cross-section of the antenna structure
Figures 2.9 and 2.10 show the behavior of current densities upon the second and third interface with a frequency of 10 GHz. The influence of the first interface, which contains the feed line, is seen in Figure 2.9. The expected physical behavior may be observed around the dielectric opening (or slit) with an increase in current density, followed by a reduction up to the surfaces of the waveguide.
Figure 2.9. Behavior of current density across the second 10 GHz interface. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
Figure 2.10. Behavior of the current density upon the third interface at 10 GHz. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
The behavior of current density is shown in Figure 2.10, which shows that the frequency of the patch resonance is close to 10 GHz. Furthermore, the influence of the dielectric opening (or slit) in the second interface may be observed.
As can be seen in Figure 2.11 (the blue curve), the resonance frequencies of simulated results are very close to the measured frequencies. Moreover this agrees, in terms of amplitude, with the first resonance frequency which is shown in [ALI 98]. Consequently, it can be confirmed that the method suggested here gives satisfactory results, as such results are closer than the measurements of the simulations, when using the Finite-Difference-Time-Domain (FDTD) shown in [ALI 98], with a relatively low computational effort. The computational time for the completed simulation is around 45 min on a 2.53 GHz Intel Core 2 Duo processor.
Figure 2.11. Absolute value of the input reflection coefficient, |S11| in dB expressed as a function of frequency in GHz. For a color version of this figure, see www.iste.co.uk/baudrand/waves.zip
2.3. Real and ideal polarizers within planar structures using WCIP
2.3.1. Formulation
An ideal polarizer, Ox, neither disrupts nor facilitates the propagation of the electric field component Ex, while it produces a complete reflexion of the component of the electric field Ey. From [2.7], it is then straightforward to express the equation Ey:
To meet the boundary conditions, Eiy = 0, the relationship between incident and reflective waves then becomes:
Figure 2.12 shows the physical representation of a real polarizer Ox.
Figure 2.12. 3-D view of a real polarizer across the axis x, Ox
Using equation [2.10], the continuity conditions across the interface in wave terms may be written in matrix form in the spatial sphere as follows:
The linearity of the Fourier transform then leads us to write [2.11] as the following equation within the spatial domain:
The Fast Modal Transform (FMT) is made up of the algorithm 2D-FFT and the transformation from the spectral sphere to the modal sphere. This was shown in [SER 10] by using periodic walls.
If P is the FMT process and P-1 its converse, the incident and reflected waves may be written in the modal sphere as follows:
Finally, the ideal polarizer matrix may be written in the modal sphere as:
where kx = (mπ)/wx, ky = (nπ)/wy, wx and wy are the dimensions for waveguides.
Expressing the polarizers directly within the modal sphere avoids discretization within the spatial sphere, and transformation operations with the help of the FMT and its converse. As a consequence, the computation time for these operations is shorter.
2.3.2. Results
To evaluate the performance of the proposed tool, three case studies have been completed. The first structure is a patch fed by a microstrip line. The other cases are real and synthetic polarizers which are introduced between the patch and the microstrip line. The wave guide is 16 mm wide and 32 mm long in dimensions, in respect of all simulations carried out.
2.3.2.1. A multilayer structure without polarizer
A circuit made up of two interfaces was simulated using the Matlab software program. The interfaces were discretized following the procedure described for the FMT and its related inverse. Figure 2.13(a) shows the multilayer structure with the different interfaces with εr1= εr2 = 2.2, εr3 = 1 and h1 = h2 = 0.65 mm. On the first interface (Figure 2.13(b)), the microstrip line has a width of 2 mm and a length of 18 mm. On the second interface (Figure 2.13(c)), the resonator has a width of 4 mm and a length of 20 mm. Figure 2.13(d) shows the anticipated behavior of the current density at a 5 GHz resonance frequency.
Figure 2.13. a) Multilayer structure, line; b) the feed line across the first interface; c) the resonator on the second interface; d) behavior of current density across the second interface at 5 GHz
2.3.2.2. Multilayer structure with a real polarizer
In this case, a second interface corresponding to the real polarizer was placed between the power supply and the resonator (Figure 2.14(a)). The physical and electric parameters are: εr1 = εr2 = εr3 = 2.2, εr4 = 1, h1 = 0.65 mm, h2 = 0.3 mm and h3 = 0.35 mm.
Figure 2.14. a) Multilayer structure; b) behavior of the current density across the third interface at 4.1 GHz
The polarizer is made up of parallel metal strips in the direction Oy. These have a width of 2 mm and a length of 32 mm spaced at intervals of 2 mm. For this simulation, the three interfaces are discretized with all operations of the FMT and its linked converse. Figure 2.14(b) shows the behavior of current density across the resonator with the resonance frequency of 4.1 GHz. This new resonance frequency is due to the increase in capacitance and inductance of the real polarizer. In Figure 2.14(b), the influence of the metallic parallel strips upon the behavior of current density may be seen.
2.3.2.3. Multilayer structure with ideal polarizer
Figure 2.15(a) shows the same structure, where the second interface is replaced by the ideal polarizer model. In this case, only two interfaces (the power line and the resonator) are discretized together with all FMT calculations and its linked converse. The ideal polarizer interface is considered as virtual with all calculations taking place only in the modal sphere.
Figure 2.15. a) Multilayer structure; b) behavior of current density across the third interface at 3.9 GHz
Figure 2.15(b) shows the behavior of current density across a resonator with a resonance frequency at 3.9 GHz. This resonance frequency is close to that obtained for the real polarizer without the influence of parallel metallic strips. Moreover, a saving of 30% in the computation time was observed.
2.4. Amplifier structure of compact micro-waves
In this section, the analysis and simulation of a five-interface structure shown on Figure 2.16 was completed.
Figure 2.16. Block diagram of an amplifier
2.4.1. Formulation of the amplifier interface
To model the amplifier interface, the auxiliary sources method is used. This method was demonstrated by formula [SBO 01] and, more recently, through using formula [HAR 13]. It is specifically adapted to the electromagnetic analysis of active elements for the planar circuit. This method allows for electromagnetic modeling of the structure made up of passive elements (P) and an active element (A), as shown in Figure 2.17.
Figure 2.17. Example of an active circuit
The first stage of the method is electromagnetic analysis of the passive structure by replacing the active element by an auxiliary source as shown in Figure 2.18(a).
Figure 2.18. a) Analysis of the passive structure; b) analysis of the active structure
Using the WCIP, it is possible to obtain the Yij admittance matrix of the three-port circuit structure applying the formula below:
The second stage consists of substituting the auxiliary source, in this case, E1, with the electric representation of the active element. This is illustrated in Figure 2.5(b), where Zd represents the impedance of a diode. Considering that this active element is a localized element with a negative impedance [TRA 03] the relevant equation is as follows:
A new admittance matrix is computed. The hexapole becomes a two-port circuit network.
The coefficients of the equivalent diffraction matrix are then computed from the admittance matrix of this structure:
where Zc is the typical impedance of 50 Ω.
2.4.2. The simulation results
To set out and evaluate the suggested structure, a number of simulations of the amplifying structure have been completed. Figure 2.19 shows the multilayer structure with the physical interface Ω2 used for the auxiliary source method.
On the first interface, the microstrip power line has a width of 2 mm and a length of 18 mm. On the second interface the resonator in L has the dimensions 4 mm x 20 mm (l x L). On the third interface the microstrip line has the same dimensions as the first interface. Expressing the polarizers directly within the modal sphere avoids discretization within the spatial domain. Consequently, 30% of computation time is saved. The dimensions of the wave guide are 32 mm x 32 mm, with εr1 = εr2 = εr3 = 2.2, εr4 = 1. The last layer (the upper layer) is free space, and l1 = l2 = l3 = 0.65 mm.
Figure 2.19. 3-D view of the auxiliary feed structure. For a color version of this figure, www.iste.co.uk/baudrand/waves.zip
Figure 2.20. Amplification according to the impedance of the amplifying structure
The simulations of the suggested structure have been achieved by varying the negative impedance of the diode from –120 Ω to –60 Ω for a 5.6 GHz frequency. As Figure 2.20 shows, a maximum amplification of 13.4 dB is obtained with a Zd impedance of –80 Ω.
Chapter written by Alexandre Jean René SERRES and Georgina Karla DE FREITAS SERRES.