Wave Concept Iterative Procedure for inhomogeneous multi-layered circuits M.Titaouine1 , N. Raveu2, H. Baudrand2 1
: Department of electronics, university center of Bordj Bou-Arréridj, El-Annasser, 34625, Algeria
[email protected] 2 : Université de Toulouse; INPT, UPS; LAPLACE; ENSEEIHT, 2 rue Charles Camichel, BP 7122, F-31071 Toulouse cedex 7, France. CNRS; LAPLACE; f-31071 Toulouse, France.
[email protected],
[email protected] Abstract - An improvement of the wave concept iterative procedure is presented to qualify multi-layered circuits with inhomogeneity. The approximation made is interesting as far as small substrate are considered, but leads to inaccuracy for large substrate. the substrate height limit is qualified in this paper.
Transform and its inverse insure conversions between the two domains. A0 is the localized source of the circuit. Relations between incoming B and outgoing A waves are defined in (1). FMT
II. METHOD IMPROVEMENT Let us consider a single interface problem, the usual WCIP scheme is very simple, see Fig. 1. The operator Γ is described in the spectral domain and represents the homogeneous media interaction on the waves. The S operator assigns boundary conditions at the interface and is expressed in the spatial domain. A Fast Fourier
~
A
I. INTRODUCTION The Wave Concept Iterative Procedure (WCIP) is an integral method based on generalized wave definitions. This method was proved to be particularly interesting for planar multilayers circuits [1]-[3] analysis and antenna coupling or radiation evaluation in unbounded media on oversized platform [4]-[5]. The WCIP method proceeds by separation of the structure under study into interfaces and homogeneous media between them. Boundary conditions on interfaces are represented by scattering operator S and homogeneous media by diffraction operator Γ, they are respectively defined in spatial and spectral domains. The resolution is therefore iterative with a uniform surface mesh so that the spatial and spectral waves are directly deduced from each other with a fast Fourier transform and its inverse. If the WCIP is well suited for homogeneous media, it is not able to characterize circuits presenting inhomogeneous media in its classical form. Some modifications in the iterative process are introduced to separate waves into homogeneous separated regions. As long as the substrate height is small compared to the wavelength, results are in good accordance with HFSS without any change in calculation time compared to the classical scheme. However results are inaccurate for large substrate. This technique is limited considering the substrate height but interesting for small inhomogeneous substrate.
Γ
Ã
B S
+
B
FMT-1
A0
Fig. 1 WCIP scheme
~ ~ ⎧B =ΓA (1) ⎨ A ⎩ =SB+A0 The iterative scheme is slightly modified to qualify inhomogeneous media: the waves emitted from the interface are separated in the spatial domain into two independent waves corresponding to the inhomogeneity of the substrate. The process is described in Fig. 2.
A1 A2 A
FMT FMT
S
Ã1 Ã2
Γ1 Γ2
~
B1
FMT-1
-1 ~ B2 FMT
+
B1
+ B2 B
A0 Fig. 2 WCIP scheme for inhomogeneous media
The substrate of surface Ω is divided into Ω1 where the substrate permittivity is ε1, and Ω2 where the substrate permittivity is ε2, (Ω1∪Ω2=Ω). Relations are defined in (2). The substrate of surface Ω is divided into Ω1 where the substrate permittivity is ε1, and Ω2 where the substrate permittivity is ε2, (Ω1∪Ω2=Ω). Relations are defined in (2).
0
-5
Transmission coefficient in (dB)
~ ~ ⎧B =Γ1 A1 1 ~ ⎪~ (2) ⎨ B2 =Γ2A2 ⎪ A=SB+A0 ⎩ Where A1 is equal to A on Ω1 and null elsewhere, A2 is equal to A on Ω2 and null elsewhere. A Fast Fourier Transform is applied to both waves independently on the whole domain ~ ~ so that A1 and A2 are generated. The modal operator is then considered as homogeneous on the whole domain ( Γ1 with a ~ ~ permittivity of ε1 and Γ2 with a permittivity of ε2). B1 and B2 ~ ~ ~ are so deduced from A1 and A2 . Incoming waves B1 and ~ B2 are then projected on their own domain to obtain B1 and B2 that are summed.
-1 0
H FS S classical W C IP
-1 5
-2 0
W C IP for inhom ogeneous m edia -2 5
- 30
- 34
III. RESULTS
7
8
9
10
11
14
13
12
f e n GH z
This formulation is successively tested on two different examples. The first one deals with homogeneous FSS that present filtering properties and comparison are made with the classical WCIP scheme. The second one is tested on an inhomogeneous FSS, the dielectric permittivity change shifts the resonant frequency, comparison is achieved with HFSS. Finally sensitivity tests are carried out on the substrate height. All the structures are depicted according to Fig. 3. 5mm
ε1 2mm ε2
air 5mm
9.53mm
ε1
ε2
ε1
h
air
Fig. 4
B. Inhomogeneous square FSS characterization
With an inhomogeneous substrate (ε1=4 and ε2=1) of height 1mm, the FSS resonant frequency is slightly shifted of 0.3GHz from 9.6GHz for the homogeneous substrate to 9.9GHz for the inhomogeneous substrate. However WCIP and HFSS results are still in good agreement as attested by the Fig. 5. As far as inhomogeneity tests are concerned, one can notice losses at the resonant frequency: the level is higher with the WCIP for inhomogeneous substrate than with HFSS or with the classical WCIP. It may be due to losses in the projection of the incoming waves. However the global behaviour of the circuit is well described. 0
periodic walls Square FSS dimensions
A. Validation on the homogeneous square ring FSS The first test consists in considering an homogeneous FR4 substrate (ε1=ε2=4) of height h=1mm (≈0.045λ0). The results obtained with the new scheme are compared to HFSS and the classical WCIP scheme of Fig. 1 results. The main difference between the classical WCIP scheme and the one presented for the inhomogeneous case appears in the projection of the incoming waves that may result in losses introduction during the calculation. For this example comparison are in good agreement as presented in Fig. 4.
-5
-10
Transmission coefficient in (dB)
Fig. 3
Validation test in the homogeneous case
- 15
- 20
W CIP HFSS
- 25
- 30
- 34
7
Fig. 5
8
9
10
11
12
13
14 f en GH z
Square inhomogeneous FSS results
C. Sensitivity to the substrate height Some tests have been made on the sensitivity to the substrate height. Results are presented in Figs. 6 and 7.
In Fig. 6, the error made on the resonant frequency is evaluated as a function of the substrate height. The substrate is the homogeneous one of Part A (ε1=ε2=4) with a variable height h. The comparison is consequently carried out between the accurate WCIP classical method, which gives the reference resonant frequency, and the new approach that may present inaccuracy but can help in inhomogeneous circuits. The mesh is 128 x 128 pixels and the convergence is stopped at 1600 iterations. The error is negligible as far as the substrate height remains under 0.01λ0. The relative error in the resonant frequency (in %)
1.2
1
0. 8
0. 6
To avoid level error evaluation, several layer of small height could be introduced to improve the precision on the circuit response. CONCLUSION The WCIP classical scheme allows only characterization of circuits on homogeneous substrate; therefore a new approach is presented in this paper. The outgoing waves are separated according to the different region of the substrate. They bring out new incoming waves in the same separated domain. This technique may introduces losses but can also allow inhomogeneous substrate qualification as far as the substrate height remains lower than 0.01λ0. Another example more sensitive to the dielectric change is under study: an open stub resonator coupled to the transmission line, the connection between the line in studied over an homogeneous and over an inhomogeneous substrate. For larger substrate, another solution is under study: use an hybrid method between the WCIP and the frequency domain transmission line matrix method (FDTLM).
0.4
REFERENCES 0. 2
0
0
0.005
0.010
0.015
0.020
0.025
0.030
0.035
h/λ0
Fig. 6
Sensitivity of the resonant frequency to the substrate height
The relative error in the transmission coefficient (in %)
Considering the transmission coefficient level at the resonance the error is also negligible for the same values of the substrate height as presented in Fig. 7. For this last parameter, as the number of iterations is locked, the observed error may also be due to convergence imprecision. Another technique should be applied for substrate larger than 0.01λ0. 60
50
40
30
20
10
0
0
0.005
0.010
0.015
0.020
0.025
0.030
0.035
h/λ0
Fig. 7
Sensitivity of the transmission level to the substrate height
[1] S. Wane H. Baudrand, "A new full-wave hybrid differentialintegral approach for the investigation of multiplayer structures including nonuniformly doped diffusion", IEEE MTT., vol.53, n°1, pp 200-213, January 2005. [2] H. Zairi, A. Gharsallah, A. Gharbi, H. Baudrand, "Analysis of Planar Circuits Using a Multigrid Iterative Method", IEE Proc. Ant. & Prop. – vol. 153-n° 3 – pp. 231-236 - June 2006. [3] H. Zairi, A. Gharsallah, A. Gharbi, H. Baudrand, "Analysis of Planar Circuits Using a Multigrid Iterative Method", IEE Proc. Ant. & Prop. – vol. 153-n° 3 – pp. 231-236 - June 2006. [4] N. Raveu, T.P. Vuong, I.Terrasse, G-P. Piau, H. Baudrand : "Wave Concept Iterative Procedure applied to cylinders", IEE Proc. Microwave Ant. and Prop., 151, (5), pp. 409-416, October 2004. [5] N. Raveu, T-P. Vuong, I. Terrasse, G-P. Piau, H. Baudrand : "Characterization of antennas over conducting cylinders by wave concept", PIERS’2004, 28-31 Mars, Pise (Italie).
