COMPARISON BETWEEN DIFFERENTMODELS FOR THE THIN METALLICSUBSTRATE IN MULTILAYERMICROSTRIP STRUCTURES

Lu Zhang and Jiming SongDepartment of Electrical and Computer EngineeringIowa State University2215 Coover HallAmes, IA 50011

Received 30 November 2005

ABSTRACT: The dispersive characteristics of multilayer microstripstructures with one thin metallic substrate are studied using the spec-tral-domain approach. The numerical results of several approximatemodels are compared with that of the immittance approach as one exactmodel. The comparison shows that the widely used resistive sheet (R-card) model is not appropriate to handle such structures over a broadfrequency range. © 2006 Wiley Periodicals, Inc. Microwave OptTechnol Lett 48: 1113–1117, 2006; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.21556

Key words: multilayer microstrip line; sheet impedance; skin depth;spectral domain method; dispersion

1. INTRODUCTION

Since high-speed semiconductor chips and VLSI circuits usuallyoperate at a frequency of up to 100 GHz, the electromagnetic (EM)effects from the interconnections become considerably critical [1].Although many technologies, such as the deep-submicron technol-ogy and multilevel interconnect structure among others, promote ahigher integration density, they also make circuits more vulnerableto the harsh EM effects (substrate coupling of noise, ringing andcrosstalk, and so on). To depress such crosstalk and distortions,one feasible solution is to add properly-designed neighboringmetallization or high-conductive substrates into the planar multi-layer circuits. Such inner metallic substrates are usually fabricatedwith a thickness of only a small fraction of one micron, and theywork as the ground or signal returns in silicon wafers. But at somelower frequencies the skin depth of metal becomes much largerthan the thickness of the metallic substrate. This may cause prob-lems, such as EM power leakage, coupling, and limitation of thecircuit bandwidth. Some of the previous studies of planar multi-layer microstrip lines mostly assumed that the ground and signalstrips were perfect electric conductor (PEC) with zero thickness[2], while others discussed the effects coming only from thefinite-thickness signal strips [3, 4]. Less attention was paid to thismultilayer structure with very thin metallic substrates sandwichedin the middle.

For such finite-thickness metallic substrates, some widely usedboundary conditions, such as the PEC and impedance boundarycondition (IBC), are not proper because they are impenetrableboundary conditions and cannot describe the leakage phenomenon.In the literature, several approximation models were proposed forthe thin metallic film, such as resistive sheet (R-card) [5, 6],transverse electric/longitudinal section electric (TE/LSE) mode,transverse magnetic/longitudinal section magnetic (TM/LSM)mode sheet impedance [7], and so forth. However, little compar-ison has been made among these models for some complex appli-cation structures by far. In this paper, the canonical immittanceapproach, as a spectral-domain approach, is used. Because alllayers are treated as a cascaded transmission line in this approach,it is pretty straightforward to substitute different approximate

models for the characteristic impedance of this thin metallic sub-strate. The goal of this paper is to study the influence from this thinmetallic substrate and perform a comparison among different ap-proximate models based on a case study of multilayer microstripstructure. The conclusions from this case study can be used toguide the application of these approximate models. Section 2briefly reviews the immittance approach and Galerkin’s methodfor the eigenvalue problem of transmission modes. In section 3, adescription of the different models for the thin metallic film ispresented. Then some numerical results are given in section 4 tomake a comparison, whereas section 5 summarizes the conclu-sions.

2. SPECTRAL DOMAIN APPROACH

Figure 1 illustrates the geometry of a multilayer microstrip linesandwiched by a thin metallic substrate (shadowed region). As-sume all substrates are uniform and infinite in both x and zdirections. The strip width is w, and t is the thickness of themetallic substrate. In this paper, the signal strip and the lowestboundary are treated as infinitesimal and perfectly conductive, soas to study the influence attributed only from the thin metallicsubstrate.

The basic idea of the spectral-domain approach is briefly re-viewed as follows. By applying the spatial Fourier transformationin the x direction, the relation between the electrical field on theinterface hi�1 and the current on the signal strip is expressed in thealgebraic Green’s functions as [8, 9] (hs � hi�1):

� Ex(�, hs)Ez(�, hs)

� � � Zxx(�, hs) Zxz(�, hs)Zzx(�, hs) Zzz(�, hs)

�� Jx(�)Jz(�)� , (1)

where Zxx, Zxz, Zzx, and Zzz are the four elements of the dyadicGreen’s functions in the spectral domain. � denotes the spectral-domain variable in the x direction. Jx and Jy are the current densityon the surface of the stripline. The immittance approach [8] de-couples the field into two independent configurations as TEy (LSE)and TMy (LSM) modes. Using transmission-line modelling, theGreen’s functions are the parallel combination of the impedances(Zup and Zlow) seen above and below the interface where thecurrent source is located, as illustrated in Figure 1. Galerkin’smethod is used to solve for the propagation constant. First, thecurrent density is expanded in terms of series of current-basisfunctions with unknown coefficients. According to the Parsevaltheorem, this leads consequently to a homogeneous system oflinear equations for the coefficients of the current basis as follows:

Figure 1 Geometry and equivalent transmission-line model (TLM) for amultilayer microstrip structure

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 1113

�m�1

M

amKi,mxx � �

n�1

N

bnKi,nxz � 0 �i � 1, . . . , M�,

�m�1

M

amKj,mzx � �

n�1

N

bnKj,nzz � 0 � j � 1, . . . , N�, (2)

where am and bn are the coefficients of the current basis. K is theintegral related to the Green’s functions and the current basis. Thepropagation constant is directly correspondent to the eigenvalue ofsuch a linear system.

3. MODELS FOR PLANAR CONDUCTIVE SHEET

In this section, different models for the impedance of the thin-filmconductor are listed in order to present a general background. Forthe immittance approach, each substrate is strictly modelled as theTEy (LSE) and TMy (LSM) admittance modes. Even for the highlyconductive substrate with finite thickness, the thin-film conductorstill can be modelled as one lossy dielectric layer with the complexpermittivity defined as [9]:

� � �� � j�� � �0�r � j�

�, (3)

where � is the conductivity of the metal and � is the angularfrequency. By applying the above assumption, the equivalentcascaded impedance Zin seen in the negative y direction is givenby [8]:

ZinTE/TM � ZC

TE/TMZL

TE/TM � ZCTE/TMtanh��ct�

ZCTE/TM � ZL

TE/TMtanh��ct�,

ZCTE � j�/�c, ZC

TM � �c/� j���,

�c2 � �2 � 2 � �2�, (4)

where ZLTE/TM is the equivalent wave impedance downward from

the metallic substrate. �c is the wave number of the conductor inthe y direction, while the guiding wave propagates along the zdirection with wave number . This model can be regarded as oneexact solution to calibrate the other approximate models in thefollowing.

The IBC describes the relationship between the electric andmagnetic field on the boundary when the metal thickness is muchlarger than the skin depth. The metal substrate attenuates incidentwave and becomes impenetrable [5]. The impedance can be ap-proximated as

ZS �1 � j

��, � � � 2

��, (5)

where ZS is also defined as the surface impedance or the intrinsicimpedance for a good conductor. As the thickness t of the con-ducting substrate decreases and is much smaller than the skindepth �, the IBC model becomes inappropriate. The cascade trans-mission-line model [10] extends the concept of the surface imped-ance ZS by using it as the characteristic impedance ZC

TE/TM forthe metallic slab and approximating the �c by (1 � j)/� in theformula (4).

Another widely used approximation is called the electricallyresistive sheet (R-card) or thin dielectric sheet model. For a goodconductor, it has the form [3, 5]:

RSH �1

�t, t���. (6)

For a thin-film conductor, if considering the situation of highconductivity and small t, this sheet resistance turns out to be ashunt resistance in the cascade of TLM [6].

However, the above IBC and sheet resistance models neglectthe dependency of the surface impedance on the field distribution(TE/TM mode). Considering this, Amari et al. represented a TE/TM-mode sheet impedance model [7] of the form:

ZSHTE/TM � ZS

ZLTE/TM � ZStanh��ct�

ZLTE/TMtanh��ct� � ZS�1 � 1/cosh��ct�

, (7)

where ZLTE/TM is the same as the equivalent input TE/TM mode

impedance found in (4). But ZSHTE/TM is derived from the definition

as tangential electric field over current density (Etan/J), which hasa different physical meaning when compared with the equivalentcascaded impedance Zin

TE/TM in (4). In some extreme conditions,such as t goes to zero, the ZSH

TE/TM will converge to the RSH [7] andbecome a shunt impedance. When very thick, this sheet impedance(7) will converge to the surface impedance ZS in (5).

4. MODEL COMPARISON FOR THE DISPERSION OFMULTILAYER MICROSTRIP

In this section, the models mentioned previously are first comparedon the basis of a two-layer planar cascaded structure, as shown inFigure 2. The structure consists of a thin conductive slab withthickness t and conductivity � backed by the infinite free space.The conductivity is 5.88(107) S/m, and the frequency is 10 GHz.The equivalent TE/TM input impedances are calculated by usingthe TLM formula (4), IBC (5), R-card (6), and TE/TM mode sheetimpedance (7) models, respectively.

The dispersive characteristics of the impedance for this struc-ture are studied. Figures 2 and 3 show the input impedances forboth TE mode and TM mode as functions of thickness t. The inputimpedance is described respectively by its real part and imaginarypart. The thickness of metallic substrate t is normalized by the skindepth as t/�, and the impedance is normalized by the surfaceresistance RS (�1/(��)). A medium value of �c is adopted (set�2 � 2 � 10�2�00 in (4)). The surface resistance RS is directlyrelated to the IBC. Thus, as shown in Figures 2 and 3, theimpedances of the exact immittance approach and the TE/TMmode sheet impedance model are found to converge to the IBCsurface impedance ZS when the thickness t becomes large enough,say two to three times of the skin depth � (refer to the horizontalline with unit value on the y axis). The R-card model only agreeswith them closely in the region where the t is small enough. For thereal part of impedance, the R-card model requires that the thick-ness be at least less than one skin depth, and for the imaginary partof impedance, it requests the thickness be less than one-tenth of theskin depth. Otherwise, the R-card model diverges significantly.When t increases, the input impedance Zin of the R-card modelgoes to zero; hence, the thin conductive slab can be regarded as aperfect conductor. This phenomenon is related to the physicalprospect that the resistive sheet is modelled simply in shunt withthe impedance of free space on the backside. As the thickness tincreases, the RSH (�1/(�t)) becomes small enough to be thedominant contribution to the equivalent Zin of two parallel-con-nected impedances. Thus physically the R-card model tends tomake the equivalent circuit be a perfect conductor when thethickness t is relatively large. But under this condition, the skindepth � is on the same order with the thickness t, which invalidates

1114 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 DOI 10.1002/mop

the assumption of the R-card model that the thickness t is negli-gible. As a result, the R-card model exceeds its formal validity.

Furthermore, the different models are implemented in a morecomplex structure to study their impact on the overall transmissionperformance. In this structure, a thin metallic layer is inserted intoone open lossless single layer microstrip line to form a metal-insulator–metal-insulator (MIMI) structure, as illustrated in Figure4. Our program is validated by the numerical results of onemetal-insulator–silicon structure [11].

The dispersive characteristics, such as the effective permittivity�eff and the attenuation constant, as functions of frequency areplotted in Figures 4(a) and 4(b). As illustrated in Figure 4(a), atlower frequencies all models describe the slow-wave effect (higheffective permittivity and small phase velocity). The slow-wavemode physically corresponds to the situation when the electricfield is shielded from the thin metal substrate, whereas the mag-netic field penetrates the metallic substrate to interact with the

lower structures. At higher frequencies, the dominant mode is thequasi-transverse electric and magnetic (TEM) mode. The curves ofimmittance model and TE/TM mode sheet impedance model con-verge into the dashed line that represents the IBC or infinite-thickness metal boundary condition when the thin substrate be-comes completely opaque in order to block fields from reachinglower substrates. In addition, with frequency increasing, all thecurves approach the lowest dot line (� � , PEC) that stands forthe situation where the thin metal substrate becomes a PEC and themicrostrip line becomes lossless. Between the slow-wave modeand quasi-TEM mode is the transition region. Compared with theexact immittance model, the IBC is appropriate for the �eff merelywhen the metallic substrate is much thicker than the skin depth �(at least two skin depths thick). Similarly, the TE/TM mode sheetimpedance is also more accurate at higher frequencies than lowerones. On the contrary, the R-card model behaves very close to the

Figure 2 TE-mode equivalent input impedance Zin (normalized to thesurface resistance RS � 1/(��)) as the function of t/�. Models: Z(IMM)(exact immittance approach) (4), Z(R-card) (6), Z(TE mode sheet imped-ance) (7); conductive slab �: 5.88(107) S/m; f: 10 GHz; �2 � 2 �10�2�00: (a) real parts; (b) imaginary parts. [Color figure can be viewedin the online issue, which is available at www.interscience.wiley.com]

Figure 3 TM-mode equivalent input impedance Zin (normalized to thesurface resistance RS � 1/(��)) as the function of t/� for the samestructure shown in Fig. 2(a). Models: Z(IMM) (exact immittance approach)(4), Z(R-card) (6), Z(TM mode sheet impedance) (7); conductive slab �:5.88(107) S/m; f: 10 GHz; �2 � 2 � 10�2�00: (a) real parts; (b)imaginary parts. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 1115

TE/TM-mode sheet impedance model at lower frequencies, but itdiffers from others in the transition region and high-frequencyrange. With frequency increasing, the R-card converges to thelowest boundary (� � , PEC) faster than the exact immittanceand other models. This is due to the tendency of the R-card modelto degrade the equivalent circuit into a PEC, which is also shownin Figures 2 and 3. Figure 4(b) shows the attenuation/loss changesas a function of frequency. When the thickness t is equal to therelevant skin depth �, the curves of the IBC and immittanceapproach merge together. In this high-frequency region, the classicsquare root of the frequency behavior of the loss can be observed.This is attributed to the fact that electric current mainly flowsthrough only a part of the whole metallic region, so the surfaceimpedance ZS becomes the source of loss. At the low-frequencyregion, the attenuation is proportional to the square of the fre-quency. This attenuation is due to the ohmic loss from the wholemetallic substrate, which can be modelled as a shunt conductance.In addition, if compared with the R-card model, the TE/TM mode

sheet impedance model shows better agreement with the exactresult at higher frequencies, whereas the R-card model fails toexhibit this square root of frequency behavior. The same explana-tion as before can account for this.

Finally, the accuracy of the R-card model and TE/TM modesheet impedance model is compared. Figure 5 illustrates howrelative errors of the effective permittivity and attenuation constantvary with frequency and the thickness t. The result from theimmittance approach is used as the calibration. Both Figures 5(a)and 5(b) show that the thinner the substrate, the better the accu-racy. But one pronounced phenomenon is that, for a fixed thick-ness t, the relative errors of the permittivity and attenuation do notdecrease but are almost constant as frequency drops. This meansthese two models still have noticeable errors, even when themetallic thickness is negligible compared to the skin depth. Theinvalidation comes from the violation of the basic assumptions forthese two models. The R-card model was proposed originally tothe scattering problem of thin conductive film when the tangentialelectric field is dominant. The TE/TM mode sheet impedance,

Figure 4 Dispersive characteristics as functions of frequency for theMIMI-structured microstrip under different models (lower dielectric: 80m, �r: 10.2; �: 5.8(107) S/m; upper dielectric: 20 m, �r: 10.2; w: 200m; metal thickness: 5 m): (a) effective permittivity; (b) attenuationconstant. [Color figure can be viewed in the online issue, which is availableat www.interscience.wiley.com]

Figure 5 Relative errors as functions of frequency and thickness t (sameMIMI microstrip structure as in Fig. 4): (a) effective permittivity; (b)attenuation constant. [Color figure can be viewed in the online issue, whichis available at www.interscience.wiley.com]

1116 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 DOI 10.1002/mop

similarly, is derived as the definition of the tangential electric fieldover current density. As a result, the normal component of thepolarization current is regarded as insignificant in the sheet [12].Although this assumption is fairly reasonable for the scatteringproblem, it is not very suitable for the multilayer microstrip lines.This is because, for lower frequencies or quasi-transverse electricmodes, the normal component of the electric field (Ey) is toodominant to be ignored. The zigzag shape of the curves is due tothe sign changes on the absolute error. Furthermore, at higherfrequencies, the TE/TM mode sheet impedance model shows bet-ter accuracy, whereas in the lower frequencies the R-card modelhas less error for the attenuation estimation. In the transition regionof the figures, the error in the R-card model increases and becomeseven worse for the attenuation. The reason is that the R-card modeloverdoes the approximation to make the substrate become a PEC.On the whole, Figure 5 shows that the R-card cannot maintain aconstant error over a broad frequency range, while the TE/TMmode sheet impedance model poses a modestly improved accuracywith increasing frequency.

5. CONCLUSION

To study the very thin metallic substrate sandwiched in a multi-layer microstrip structure, this paper has compared the differentmodels for this thin-film conductor. Then these models wereimplemented to analyze an open metal-insulator–metal-insulator(MIMI) structure by using the rigorous spectral-domain approach.The numerical results show that all these models can exhibit theslow wave effect. The IBC is applicable when the thickness t islarger than two skin depths thick for the estimation of the effectivepermittivity or at least 1 skin depth thick for the attenuationseparately. For low-frequency ranges and fixed thickness, theapproximation error of the R-card model and TE/TM mode sheetimpedance model is almost constant, due to ignoring the normalcomponents of the electric field. Although the TE/TM mode sheetimpedance has better high-frequency performance, there is nonumerical saving (CPU time and efficiency) compared with theexact immittance approach. The performance of the R-card athigher frequencies becomes rather unacceptable, which suggests itis not appropriate to implement the R-card model to analyze thismultilayer microstrip transmission line over a wide frequencyrange. Furthermore, the results remind us to consider the normalcomponent of the electric field inside the sheet when doing moreaccurate modelling and analysis of the finite-thickness signal strip.

REFERENCES

1. T.C. Edwards and M.B. Steer, Foundations of interconnect and mi-crostrip design, Wiley, New York, 2000.

2. T. Itoh and R. Mittra, Spectral-domain approach for calculating thedispersion characteristics of microstrip lines, IEEE Trans MicrowaveTheory Tech 21 (1973), 496–499.

3. J.D. Morsey, V.I. Okhmatovski, and A.C. Cangellaris, Finite-thicknessconductor models for full-wave analysis of interconnects with a fastintegral equation method, IEEE Trans Adv Packaging 27 (2004),24–33.

4. J.C. Rautio and V. Demir, Microstrip conductor loss models forelectromagnetic analysis, IEEE Trans Microwave Theory Tech 51(2003), 915–921.

5. T. Senior, Approximate boundary conditions, IEEE Trans AntennasPropagat 29 (1981), 826–829.

6. D.C. Jenn, A. Prata Jr., W.V.T. Rusch, and M.R. Barclay, A resistivesheet approximation for mesh reflector antennas, IEEE Trans AntennasPropagat 37 (1989), 1484–1486.

7. S. Amari and J. Bornemann, LSE- and LSM-mode sheet impedancesof thin conductors, IEEE Trans Microwave Theory Tech 44 (1996),967–970.

8. T. Itoh, Spectral domain immittance approach for dispersion charac-teristics of generalized printed transmission lines, IEEE Trans Micro-wave Theory Tech 28 (1980), 733–736.

9. J.P. Gilb and C.A. Balanis, Pulse MIS slow-wave structures over awide range of parameters, IEEE Trans Microwave Theory Tech 40(1992), 2148–2154.

10. M. Konno, Conductor loss in thin-film transmission line, ElectronCommun Japan 82 (1999), 83–91.

11. L. Zhang and J.M. Song, Dispersion characteristics of multilayermicrostrip lines with thin metal ground, Proc IEEE AP-S InternationalSymp Dig 2 (2005).

12. R.F. Harrington and J.R. Mautz, An impedance sheet approximationfor thin dielectric shells, IEEE Trans Antennas Propagat 35 (1975),531–534.

© 2006 Wiley Periodicals, Inc.

A PATTERN-RECONFIGURABLEMICROSTRIP ANTENNA ELEMENT

K. W. Lee,1 R. G. Rojas,2 and N. Surittikul21 Texas Instruments IncorporatedDallas, TX 752432 The Ohio State UniversityColumbus, OH 43212

Received 30 November 2005

ABSTRACT: A novel radiation-pattern-reconfigurable circularly polarizedmicrostrip antenna structure is presented. The measured results show thatthe antenna is able to provide a magnitude difference of more than 6 dB at90° from broadside (in two principal planes) for the tangential (tothe antenna surface) field component between the on and off statesof the switches. Nonetheless, the vertical field component remainsrelatively unchanged. © 2006 Wiley Periodicals, Inc. Microwave OptTechnol Lett 48: 1117–1119, 2006; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.21555

Key words: microstrip antenna; reconfigurable aperture; GPS

1. INTRODUCTION

Reconfigurable antennas have gained considerable interest becauseof their potential to improve the overall performance of wirelesscommunication systems. A frequency-reconfigurable antenna canoperate at different frequency bands, depending on the operation ofthe switches. Only one element is required for a multistandardsystem. Thus, frequency-reconfigurable antennas can reduce thecomplexity and improve the size form factor of the overall system[1, 2]. On the other hand, the ability to change the radiation patternin real time is important. Radiation-pattern-reconfigurable anten-nas can help to avoid noise source or intentional jamming signalsand hence improve isolation between neighboring systems oper-ating in the same frequency band [3].

This paper presents a radiation-pattern-reconfigurable circu-larly polarized microstrip antenna element that consists of a mi-crostrip patch surrounded by a parasitic metallic ring loaded withelectronic switches. The antenna is capable of changing its radia-tion pattern in real time, depending on the state of the switches(on/off) while maintaining other antenna characteristics, namely,input impedance and axial ratio (AR), relatively unchanged.

2. DESIGN APPROACH

The proposed reconfigurable antenna consists of a circularly po-larized patch element and a parasitic metallic ring loaded with

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 1117