Membrane-supported transmission lines and circuits are excellent candidates for millimeter-wave applications where conventional substrate-supported architectures begin to suffer from parasitic effects at the dielectric/ air interface.^{1-3} Transmission lines based on the technology, known as shielded-membrane-microstrip (SMM) transmission lines, have been used to realize an asymmetric tapered coupled-line coupler operating from 10 to 60 GHz.^{3} This coupler, derived from the Klopfenstein impedance taper,^{4} has a minimum directivity of 10 dB at 60 GHz.

To build on the technology, the current work offers an analysis of a symmetric-shielded-membrane microstrip coupler (SMMC) using method of moments (MoM) electromagnetic (EM) simulations in two dimensions (2D). The modeling of this structure consists in analyzing the even- and odd-mode characteristic impedances (Z_{oe}, Z_{oo}), the coupling coefficient, k, and the primary inductive and capacitive matrices ( and ). The results of a coupler with 20- dB coupling using SMM lines show excellent directivity and isolation.

* Figure 1(a)* shows a schematic representation of the shielded membrane microstrip line. For the purposes of this analysis, the structure is assumed to be lossless. It is based on a dielectric material, silicon (Si), having a relative dielectric constant, ε

_{r3}, of 11.7. The cross-sectional view of the line shows that it has a microstrip conductor characterized by a width, w, and thickness, t, placed on a membrane comprised of SiO

_{2}/Si

_{3}N

_{4}/SiO

_{2}of thickness hm having a relative dielectric constant, ε

_{r2}, of 4.5. These types of transmission lines, as demonstrated in ref. 3, can be used to create a low effective dielectric permittivity environment in which both even and odd modes will propagate at the same velocity when used in a coupler configuration

*>. For the purposes of integration with other components in higher-level designs, SMM lines are compatible with monolithic-microwaveintegrated- circuit (MMIC) semiconductors.*

**Fig. 1(b)** Electrically, the SMMC can be described in terms of its primary parameters, the inductance and capacitance matrices, and , and its secondary parameters, the even-mode Z_{oe} and odd-mode Z_{oo} impedances and its coupling coefficient, k, where^{5}:

The inductance matrix contains the self-inductances of the two strips, with separation, s, on the diagonal, and the mutual inductances between strips in the off-diagonal terms. The capacitance matrix, , accounts for capacitance effects between the two conductive strips with conductivity of (m), characterizing energy storage in the SMM structure. In * Fig. 1*, the isolated line is described in terms of its effective dielectric constant, ε

_{eff}, and its characteristic impedance, Z

_{c}.

The numerical calculations of the EM parameters, such as and , of the analyzed SMM coupler were carried out with LINPAR for Windows (Matrix Parameters for Multiconductor Transmission Lines), a 2D MoM software program for numerical evaluation of quasistatic matrices for multiconductor transmission lines embedded in piecewise-homogeneous dielectrics.^{6} The technique used in the program is based on an electrostatic analysis. In the analysis, the dielectrics were replaced by bound charges in a vacuum, and the conducting bodies were replaced by free charges. A set of integral equations was derived for the charge distribution from the boundary conditions for the electrostatic potential and the normal component of the electric field. The MoM was applied to these equations, with a piecewiseconstant (pulse) approximation for the total charge density and the Galerkin technique.

For a chosen coupling coefficient, when and are found, it is possible to estimate the resulting scattering parameters of the SMMC using an adapted numerical model.^{7} First, the authors were interested in the analysis of the SMM line using the method of moments, with the following features: strip width, w, of 135 m; strip thickness, t, of 1 m, strip conductivity, m, of 3.9 x 10^{7} Ohms/m; groundplane separation, h_{b}, of 50 m; cover height, h_{u}, of 50 m; membrane (SiO2/ Si3N4/SiO2) with thickness, hm of 1.5 m and dielectric constant, ε_{r2}, of 4.5; silicon (Si) dielectric material with relative dielectric constant, ε_{r3}, of 11.7 and substrate geometrical parameters l_{bs}, l_{bi} and a, respectively, of 600, 750, and 900 m.

* Figure 3* shows the analyzed attenuation factor, which increases as the square root of frequency due to conductor losses. This factor obtained by MoM analysis is less than 0.072 dB/ mm at 120 GHz, whereas in ref. 3 it is less than 0.06 dB/mm at 110 GHz.

** Figure 4(a)** shows the influence of the frequency on the effective dielectric constant of the SMM line which varies between 1.08 and 1.15. The presence of the fine membrane in the structure causes a reduction in the effective dielectric constant of the SMM line, which passes very close to 1.08 at 120 GHz, whereas in ref. 3 the minimum value of the effective dielectric constant is given as 1.05 at 118 GHz.

*shows the influence of frequency on the characteristic impedance of the analyzed SMM line. Both the effective dielectric constant and the characteristic impedance vary only slightly with frequency, indicating nearly nondispersive single-mode propagation to 120 GHz.*

**Figure 4(b)** As part of the study, the authors applied the MoM-based numerical tool to the analysis and design of the SMM coupler. The MoM approach makes it possible to simulate the performance of a design and decide if a given set of constraints makes it possible to realize the coupler. * Figure 5* shows the segmentation of the charged surfaces of the shielded membrane microstrip coupler using LINPAR, with the following features: strip width, w, of 135 m; strip thickness, t, of 1 m; strip conductivity, m, of 3.9 x 10

^{7}Ohm/m; ground-plane separation, h

_{b}, of 50 m; cover height, h

_{u}, of 50 m; membrane thickness, h

_{m}, of 1.5 m; length, lbs, of 1200 m; length, l

_{bi}, of 1500 m, and substrate dimensional parameter, a, of 1800 m; and relative dielectric constants ε

_{r2}and ε

_{r3}, respectively, of 4.5 and 11.7.

* Figure 6* shows the even- and oddmode characteristic impedances for the (a) effective dielectric constants and for (b) the SMMC for w/hu= 2.7, as a function of the ratio s/hu at 40 GHz.

*shows the variation of the coupling coefficient as a function of the ratio s/h*

**Figure 7**_{u}, also at 40 GHz. It appears that for a ratio, s/h

_{u}, of 0.55, the coupling coefficient is equal to 20 dB.

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The results of the MoM analysis were used to design and build a 20-dB shielded membrane microstrip coupler. All four ports of the coupler (* Fig. 8*) were matched to 50 Ω. The fixed parameters of the 20-dB shielded membrane microstrip coupler include a characteristic impedance of (Z

_{oe}Z

_{oo})

^{0.5}= 49.5 Ω and an operating frequency of 40 GHz.

The features of the coupler obtained from the analysis results include ratios (w/h_{u}) and (s/h_{u}) of 2.7 and 0.55, respectively, coupler length of 1750 m, even- and odd- mode characteristic impedances, Z_{oe} and Z_{oo}, respectively, of 54.84 and 44.65 1, respectively, with inductance matrix of

and a capacitance matrix of

Using an adapted numerical model,^{7} the resulting scattering parameters (with respect to 50 Ω) were plotted from 2 to 80 GHz in * Fig. 9*. The results indicate that the desired 20-dB coupling occurring from 30 to 50 GHz, with minimum directivity of 10 dB and good isolation of 15 dB.

REFERENCES

1. S. V. Robertson, L. P. B. Katehi, and G. M. Rebeiz, "Micromachined Self-packaged W-Band Bandpass Filters," IEEE MTT-S Symposium Digest, Orlando, FL, May 14-19, 1995, pp. 1543-1546.

2. T. M. Weller, L. P. B. Katehi, M. I. Herman, and P. D. Wanhof, "Membrane Technology Applied to Microstrip: A 33 GHz Wilkinson Power Divider," IEEE MTT-S Symposium Digest, San Diego, CA, May 23-27, 1994, pp. 911-914.

3. S. V. Robertson, A. R. Brown, L. P. B. Katehi, and G. M. Rebeiz, "A 10-60 GHz Micromachined Directional Coupler," *IEEE Transactions on Microwave Theory and Techniques*, Vol. 46, No. 11, November 1998.

4. R. W. Klopfenstein, "A Transmission Line Taper of Improved design," Proceedings of the IRE, Vol. 44, January 1956, pp. 31-35.

5. A. Lallam, N. Benabdallah, N. Benahmed, and Y. Bekri, "Analyze EM parameters of slotted tube couplers," Microwaves and RF, Vol. 47, No. 3, March 2008, pp. 76-86.

6. A.R. Djordjevic, M.B. Bazdar and T.K. Sarkan, *LINPAR for windows: Matrix parameters of multiconductor transmission lines, Software and user's manual*, Artech House, 1999.

7. A.R. Djordjevic, M. Bazdar, G. Vitosevic, T. Sarkar, and RF. Harrington, *Scattering parameters of microwave networks with multiconductor transmission lines*, Artech House, Norwood, MA, 1990.