efficient finite element solution of low-frequency scattering problems via anisotropic metamaterial...

EFFICIENT FINITE ELEMENT SOLUTIONOF LOW-FREQUENCY SCATTERINGPROBLEMS VIA ANISOTROPICMETAMATERIAL LAYERS

Ozlem Ozgun and Mustafa KuzuogluDepartment of Electrical and Electronics Engineering, Middle EastTechnical University, 06531 Ankara, Turkey; Corresponding author:[email protected]

Received 4 August 2007

ABSTRACT: We introduce a new technique which remedies the draw-backs in the Finite Element solution of low-frequency electromagneticscattering problems, through the usage of an anisotropic metamateriallayer which is designed by employing the coordinate transformation ap-proach. The usual finite element method should utilize a “challenging”mesh generation scheme to accurately simulate the “small” objects inscattering problems; on the contrary, the proposed technique provides aconsiderable reduction in the number of unknowns, and requires a moreconvenient and simpler mesh structure inside the computational domain.The most interesting feature of the proposed method is its capability tohandle arbitrarily shaped “small” scatterers by using a “single” meshand by modifying only the constitutive parameters inside the matamate-rial layer. We report some numerical results for two-dimensional elec-tromagnetic scattering problems. © 2008 Wiley Periodicals, Inc.Microwave Opt Technol Lett 50: 639–646, 2008; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23167

Key words: low-frequency electromagnetic scattering; anisotropic meta-materials; coordinate transformation; finite element method (FEM)

1. INTRODUCTION

The scattering of electromagnetic waves from electrically smallobjects (viz., objects whose dimensions are small compared to thewavelength) has been investigated in various fields of science andengineering (such as physics, electrical engineering, geophysics,astrophysics, and biology) for more than a century. The low-frequency scattering problem was pioneered by Lord Rayleigh [1]in 1881. The term “low-frequency scattering” is also named as“Rayleigh scattering” in many studies based on Rayleigh’s contri-bution. In the Rayleigh scattering problem, the unknown fieldquantities are expressed as a convergent series in positive integralpowers of the propagation constant k, and then the unknownexpansion coefficients are determined from Maxwell’s equationsand boundary conditions [2]. These expansion coefficients arefunctions of the direction of incidence and observation, as well asthe geometry of the scatterer. Because of the dependence on thegeometry of the scatterer, the calculation of the unknown coeffi-cients in the Rayleigh series is usually a difficult task especially forobjects of arbitrary shape. The Rayleigh technique has been uti-lized in the literature to solve low-frequency scattering problemsfor scatterers of some specific shapes [3-7].

Apart from the above-mentioned analytical approaches, someapproximate computational methods have been devised to solvelow-frequency scattering problems on account of the advances inthe computer technology. However, accurate solution of low-frequency scattering problems is still a challenging task in thecontext of numerical approaches [such as the method of moments(MoM) and the finite element method (FEM)]. In the MoM ap-proach employing the electric field integral equation (EFIE), thematrix is known to be ill-conditioned at low frequencies, and somemethods have been proposed to overcome this problem [8, 9]. Inaddition, the MoM approximation of the magnetic field integral

equation (MFIE) has also some drawbacks at low frequencies interms of the stability of the solution, as reported in [10], whichstates that small errors in the real part of the current distributioncauses significant errors in the far-field profile, and which proposesa method to resolve this issue.

In the usual (FEM), accurate simulation of the low-frequencyscattering problems may not be manageable especially due to thehigh memory requirement to take into account the fine sections ofthe “small” objects with high numerical precision. That is, al-though the FEM is a powerful method to handle any type ofgeometry and material inhomogeneity, the FEM mesh may requirea large number of unknowns to define properly the geometry of theobject whose size is only a fraction of wavelength inside thecomputational domain. Furthermore, it is known that to employ theFEM to the solution of scattering problems involving spatiallyunbounded domains, the physical domain must be truncated by anartificial boundary or layer to achieve a bounded computationaldomain. The most common approaches in mesh truncation areabsorbing boundary conditions (ABCs) [11] and perfectly matchedlayers (PMLs) [12]. However, to employ the ABC, the truncationboundary must be located sufficiently far away from the scatterer(at least in the order of wavelength) to reduce spurious reflectionsof the propagating waves at the outer boundary. The consequenceof this requirement results in an increase in the computationaldomain, especially at low frequencies, due to a large number ofelements inside the white-space (i.e., usually free-space in scatter-ing problems) which is not occupied by the objects. On thecontrary to the ABC, the PML region may be designed very closeto the scatterer, thus is capable of minimizing the white-spaceespecially in high frequencies [13]. However, at low frequencies,the distance between the scatterer and the PML boundary shouldalso be sufficiently large so that the evanescent waves becomenegligible at the outer boundary [14]. In addition, the PML mayyield ill-conditioning in the FEM matrix equation at low frequen-cies [15] due to the large propagation constant k. Therefore, theFEM mesh size increases enormously at low frequencies for themesh truncation techniques to work well, and to achieve a finediscretization of small objects inside a relatively large computa-tional domain.

In this article, we propose a new technique to solve efficientlylow-frequency electromagnetic scattering problems via FEM usingan anisotropic metamaterial (AMM) layer, as illustrated in Figure1. The original scattering problem for a two-dimensional (2D)electrically small, perfect electric conductor (PEC) cylinder ofcircular crosssection is shown in Figure 1(a) together with itspossible mesh structure employing triangular elements and PMLregion inside a square computational domain. As mentioned pre-viously, since the free-space region (�FS) should be large enough,and since the mesh around the boundary of the scatterer (��S)should be refined to achieve accurate results, the computationaldomain requires a large number of unknowns even if a nonuniformmesh generation scheme is employed (i.e., the element size isgradually increased from the boundary of the scatterer toward theoutermost boundary). Proper development of such a nonuniformmesh algorithm may be a challenging task if the scatterer is ofarbitrary shape, especially in three-dimensional (3D) problems.However, in Figure 1(b), we design an arbitrarily shaped AMMlayer (�A) which is located at an arbitrary distance from thescatterer, and we solve this problem which turns out to be “equiv-alent” to the original problem in Figure 1(a). It should be noted thatthe “empty” region in Figure 1(b) is no longer included in thecomputational domain. The two configurations in Figure 1(a,b) areequivalent in the sense that they yield identical field values in theircommon free-space regions, and that the field values calculated in

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 50, No. 3, March 2008 639

Figure 1(b) are inherently related to those in the original problemin Figure 1(a). In other words, the equivalent problem in Figure1(b) provides a complete information about the field distribution(both near- and far-field) corresponding to the “small” scatterer,even though the mesh of the equivalent problem does not containelements at the close vicinity of the scatterer which is shown by adotted-curve in Figure 1(b). The problem in Figure 1(b) may beinterpreted as a scattering problem involving a “larger” PECscatterer (shown by ��A,in) coated by an AMM layer. Therefore,the equivalent problem in Figure 1(b) transforms the original

low-frequency problem into a relatively high-frequency problem,and makes the solution of the problem more feasible by decreasingthe number of unknowns considerably and by employing a moreconvenient, even uniform, mesh generation scheme inside thecomputational domain. Another special feature of the proposedtechnique is that low-frequency scattering of waves from arbi-trarily shaped scatterers can be solved using a single mesh if theconstitutive parameters of the AMM layer are defined appropri-ately for each scatterer. In other words, the same mesh in Figure1(b) can be employed for arbitrary scatterers (such as rectangularor elliptical scatterer) by simply changing the parameters of theAMM layer which are calculated with respect to the dotted bound-ary of the scatterer. Hence, the proposed technique provides a greatflexibility to handle various low-frequency scattering problems ina more efficient and favorable manner by decreasing the memoryand processing power and by employing a simpler mesh genera-tion algorithm.

It is known that AMM layers can tune the spatial variations ofelectromagnetic waves in a desired manner by manipulating theirstructural features. The constitutive parameters of the AMM layerin this article are obtained by using the coordinate transformationtechnique, which is often utilized in the design of PMLs [16-19].The coordinate transformation approach is based on the fact thatMaxwell’s equations are form-invariant under coordinate transfor-mations [20]. In other words, Maxwell’s equations are still valid,but with appropriately defined constitutive parameters that conveythe effect of the coordinate transformation to the electromagneticfields. The permittivity and permeability parameters turn out to beanisotropic as well as spatially varying. Therefore, the designmethodology introduced in this article is theoretical, and intends todevise a simulation tool for the purpose of efficient solution oflow-frequency electromagnetic scattering problems with fewerunknowns. Recently, AMM layers based on the concept of coor-dinate transformation have been proposed to produce materialspecifications for the purpose of obtaining electromagnetic invis-ibility [21], reshaping objects in electromagnetic scattering [22],rotating electromagnetic fields in a closed domain [23], miniatur-izing waveguides [24], and spatial domain compression in finitemethods [25].

This article is structured as follows. In Section 2, we introducethe design procedure of the AMM layer for PEC scatterers. InSection 3, we present finite element simulations of some 2D TMz

electromagnetic scattering problems at low frequencies. Finally,we present our conclusions in Section 4.

2. AMM LAYER DESIGN

The AMM layer is designed via the coordinate transformationapproach, as illustrated in Figure 2. Initially, the spatial regionoccupied by the AMM layer is constructed at an arbitrary butsufficiently large distance from the small scatterer whose boundaryis ��S (it is assumed that the spatial domain occupied by thescatterer is a convex subset of �3). In this figure, �A denotes thespatial domain occupied by the AMM layer with the inner andouter boundaries ��A,in and ��A,out, and �e denotes the emptyspatial domain between ��S and ��A,in.

While designing the AMM layer, each point P inside the AMMlayer (�A) is mapped to P inside the transformed region �� A � �e enclosed within the boundaries ��A,out and ��S. Thismapping, which can be interpreted as a space expansion, is defined

as a coordinate transformation T:�A3 � as follows:

r� � T�r�� r�o � r�s��r�o � r�i�

�r� � r�i� � r�s (1)

Figure 1 AMM layer design in low-frequency scattering: (a) Originalproblem; (b) equivalent problem with AMM layer

640 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 50, No. 3, March 2008 DOI 10.1002/mop

where r� and r� are the position vectors of the points P and P in theoriginal and transformed coordinate systems, respectively, and� � � represents the Euclidean norm. In addition, r�s, r�i, and r�o arethe position vectors of Ps, Pi, and Po which are determined on ��S,��A,in, and ��A,out, respectively, along the direction of the unitvector a. The unit vector a is calculated emanating from a pointinside the innermost domain occupied by the scatterer (such as thecenter-of-mass point, which can be designated as the origin) in thedirection of the point P inside the AMM layer. It is worth men-tioning that the coordinate transformation in Eq. (1) is continuoussuch that two closely located points r� and r�* in �A are mapped toalso closely located points r� and r�* in �, due to the definition ofthe transformation.

As a result of the coordinate transformation in Eq. (1), theoriginal medium turns into a spatially varying anisotropic mediumwhere the original forms of Maxwell’s equations are still preservedin the transformed space. That is, Maxwell’s equations are form-invariant under space transformations, and a general coordinatetransformation leads to the following expressions for the permit-tivity and permeability tensors [20].

�� (2a)

�� (2b)

where � and � are the constitutive parameters of the originalisotropic medium (usually free-space in scattering problems), and

�� (det J�)(J�T�J�)�1 (3)

where J� is the Jacobian tensor defined as (in Cartesian coordinates)

J� �� x,y,z�

��x,y,z�� x/�x � x/�y � x/�z

� y/�x � y/�y � y/�z� z/�x � z/�y � z/�z

�. (4)

It can be concluded that the constitutive parameters of the AMMlayer in Eq. (2) can be directly calculated from the Jacobian of thetransformation in Eq. (1).

In conjunction with the coordinate transformation inside theAMM layer, the fields are transformed as [26]

E� �r��3 E�˜

�r�� J�T � E� �r�� (5a)

H� �r��3 H� �r�� J�T � H� �r�� (5b)

and the electric field satisfies the following vector wave equation

� � �� 1 � � � E�˜

�r�� k2�� E�˜

�r�� 0 (6)

where k is the wave number of the isotropic medium. This equationderived from Maxwell’s equations in original coordinates is basi-cally equivalent to the following vector wave equation in trans-formed coordinates due to the form-invariance property of theMaxwell’s equations under the mapping r� � T�r��

� � � � E� �r�� k2E� �r�� 0 (7)

where � is the nabla operator in the transformed space. In a FEMcode, the equation in Eq. (6) is solved in original coordinates usingthe mesh in Figure 1(b). Then, the original fields in transformedcoordinates can be directly calculated from the expressions in Eq.(5), which are rewritten for convenience as follows

E� �r�� J�T��1 � E�˜

�r�� (8a)

H� �r�� J�T��1 � H� �r��. (8b)

Since the transformed space includes both the domain inside the

AMM layer and the empty domain around the scatterer (i.e., �� e), the expressions in Eq. (8) provide exactly theoriginal near field values for the scatterer. For instance, in the 2DTMz case where E� �r�� azEz�x,y�, the Jacobian tensor in Eq. (4)reduces to

J� � �� x/�x � x/�y 0� y/�x � y/�y 0

0 0 1� (9)

implying that Ez� x,y� � Ez�x,y� because the 3,3-entry of �J�T��1 isequal to 1. This fact is illustrated in Figure 3 for the sake of beingprecise in how to calculate the near fields of the scatterer.

It is worth mentioning that while solving the wave equation in

Eq. (6), the field E�˜

�r�� can be split into two parts: one is the knownincident field �E� inc� produced in the absence of the AMM layer, and

the other is the scattered field �E�˜

s�. In other words,

E�˜

� E� inc � E�˜

s. (10)

Then, the vector wave equation for the scattered field is expressedas follows

Figure 2 Design of the AMM layer with coordinate transformation


� � �� 1 � � � E�˜

s�r�� k2�� E�˜

s�r�� 1 � �

� E�˜

inc�r�� k2�� E�˜

inc�r��. (11)

As in a usual FEM procedure, the weak variational form of thevector wave equation in Eq. (11) is derived using the weightedresidual method and solved by discretizing the computationaldomain using some number of elements. After solving Eq. (11) forthe scattered field, the total field is calculated by Eq. (10). Finally,as mentioned previously, the original field inside the computa-tional domain is calculated by Eq. (8). It is evident that, inside thefree-space region, these equations transform to the expressions ofa free-space scattering problem, since �� becomes the identitytensor.

In the 2D TMz case, the vector wave equation in Eq. (6)reduces to the scalar partial differential equation

� � �� Ez� x,y�� k2�33Ez� x,y� � 0 (12)

where

�� 11 �12 0�21 �21 00 0 �33

� and �� 11 �12

�21 �21�. (13)

Then, a similar FEM procedure is followed as in the 3D case tocalculate the field values in both original and transformed coordi-nates.

All above-mentioned expressions can be employed in a generaldesign of the AMM layer for any geometry in a straightforwardmanner. As a special case in the cylindrical coordinate system,assuming that the scatterer is a circular cylinder and the AMMlayer is a circular shell, the coordinate transformation in Eq. (1)reduces to the following simple form

� ��o � �s

�o � �i�� i� � �s (14a)

� (14b)

z � z (14c)

where �s, �i, and �o are the radii of the boundaries ��S, ��A,in, and��A,out, respectively. Then, the tensor in Eq. (4) reduces to thefollowing diagonal form (in cylindrical coordinates)

�� 11 0 00 �22 00 0 �33

� � ��

�0 0

0�

� � 0

0 0 ��o � �s

�o � �i�2� �

�

�(15)

where

� �i � �o

�o � �i

�o � �s. (16)

Then, the wave equation in Eq. (12) is expressed as

1

�

�

��11

�Ez

�� 22

�2

�2Ez

�2 � k2�33Ez � 0. (17)

It is useful to emphasize that the calculation of the constitutiveparameters of the AMM layer using Eq. (2) can be carried out inthe preprocessing phase (i.e., before the matrix construction phase)in a computer code. Thus, the computational effort to implementthe coordinate transformation creates almost negligible burden onthe processing power of the computer, compared to some otherphases of the code (such as usual matrix construction and solutionphases).

3. NUMERICAL EXPERIMENTS

In this section, we report the results of some numerical experi-ments to validate the design procedure of the AMM layer in some2D TMz electromagnetic scattering problems involving infinitelylong cylindrical PEC scatterers. All simulations are performedusing our FEM software employing triangular elements. In allexamples, the wavelength in free-space (�) is set to 1 m (i.e., k is2�). The element size is approximately set to �/40 in the equiva-lent problem, but it is decreased gradually toward to the boundaryof the scatterer starting from �/40 in the original problem. Inaddition, the computational domain is terminated with a PMLabsorber which is implemented by the locally conformal PMLmethod [19] whose parameters are � 7k and m � 3. Thethickness of the PML region is chosen as �/4. In all examples, thesame mesh structure is employed in the solution of the equivalentproblem, irrespective of the shape of the scatterer. That is, in allexamples, the AMM layer in the equivalent problem is a circularshell as shown in Figure 1(b), where the radii of the inner and outerboundaries are �/2 and � respectively, and the thickness of theAMM layer (dAMM) is �/2. Moreover, the incident plane wave isassumed to be in the form of E� inc � azexp jk�xcos inc

� ysin inc�� where inc is the angle of incidence with respect tothe x-axis.

Figure 3 Field transformation in the equivalent problem


In the first example, we consider a scattering problem where aplane wave ( inc � 180 ) is incident to a circular cylinder whoseradius (rs) is �/20. We first simulate the original scattering problem[see Fig. 1(a)], and plot the contour of the electric field inside thecomputational domain in Figure 4(a). This plot represents the realpart of the electric field to better visualize the field behavior insidethe computational domain. Then, we simulate the equivalent prob-lem with AMM layer [see Fig. 1(b)] by solving the wave equationin Eq. (12), and we plot the contour of the electric field in originalcoordinates inside the computational domain in Figure 4(b). Weexpect that the two simulations in Figure 4(a,b) must yield iden-tical field values inside their common free-space region (free-spaceoutside the transformed space). Therefore, a way to measure theperformance of the proposed method is to introduce a mean-squareerror criterion as follows

E1 �

��FS

E� zeq�r�� Ez

org�r��2

��FS

E� zorg�r��2

(18)

where Ezorg and Ez

eq are the electric fields calculated in the originaland equivalent problems [i.e., the fields in Fig. 4(a,b)], respec-tively, inside the common free-space region. Then, we calculate E1

as 0.1256%. Although the error term in Eq. (18) must ideally beequal to zero, the small error value observed in the simulation isbecause of FEM modeling errors which may be decreased furtherby refining the FEM mesh. We also note that we have removed thetilda from the term Ez

eq�r�� in Eq. (18) because the equivalent

problem provides original fields inside the free-space region (i.e.,Ez

eq�r�� Ezeq�r�� in �FS).

Then, we transform the field values calculated inside the AMMlayer in the equivalent problem using the fact that Ez

eq� x,y�� Ez

eq�x,y� (see Fig. 3). That is, we calculate the field values intransformed coordinates in the equivalent problem, and we plot thecontour of the electric field inside the computational domain inFigure 4(c). We conclude that the field distribution in Figure 4(c)is almost identical to the one in Figure 4(a), as expected, implyingthat both problems are actually equivalent. Finally, we calculatethe analytical field values using the Mie series expansion [27]inside the computational domain to validate the results of theequivalent problem. For this purpose, we plot the contour of theelectric field calculated by the Mie series in Figure 4(d) [which isalmost identical to Fig. 4(a,c)], and introduce a mean-square errorcriterion as follows

E2 �

��FS

E� zeq�r�� Ez

mie�r��2

��FS

E� zmie�r��2

(19)

where Ezmie and Ez

eq are the electric fields calculated by the Mieseries and equivalent problem, respectively, inside the whole free-space region. This error value basically compares the field valuesin Figure 4(c,d). Then, we calculate E2 as 0.3154%. In addition, inFigure 5, we plot the field values of Figure 4(c,d) along their dottedcut-lines (x � 0 and y � 0 cuts).

Figure 4 Contours of real part of electric field for circular cylinder: (a) Original problem; (b) equivalent problem in original coordinates; (c) equivalentproblem after transforming the field values (in transformed coordinates); (d) Mie series. [Color figure can be viewed in the online issue, which is availableat www.interscience.wiley.com]


To measure the amount of the reduction in unknowns in thisexample, we use the following expression

Nreduce �Norg � Neq

Norg (20)

where Norg and Neq are the number of unknowns (or the number ofnodes) employed in the original and equivalent problems, respec-tively. Then, we calculate the reduction in unknowns as 40% inthis example. Thus, we conclude that the proposed techniqueprovides a considerable reduction in the memory requirement.

In the next example, we consider a circular cylinder whoseradius (rs) is �/10. We follow the same procedure as in theprevious example, and we plot the bistatic radar crosssection(RCS) profile in Figure 6, which compares the results of theoriginal and equivalent problems and the Mie series, to measurethe far-field performance of the proposed technique. In addition,for the same scatterer, we vary the thickness of the AMM layer(dAMM) and we tabulate the E2 values for different thicknessvalues of the AMM layer in Table 1. We conclude that as the

thickness of the layer decreases, the results start to deteriorate, butthey are reliable even for electrically thin layers.

In the next simulation, the radius (rs) of the circular scatterer isvaried between �/40 and 0.45� to visualize the performance of theAMM layer with respect to the electrical-size of the scatterer. Weplot the E2 values and the monostatic RCS values as a function ofthe radius of the scatterer in Figure 7(a) and Figure 7(b), respec-tively. We conclude that as the electrical-size of the scatterer isdecreased, the error increases because the spatial variations in theentries of the permittivity and permeability tensors of contiguouselements increase due to the definition of the coordinate transfor-mation. However, we can assert that the errors are at an acceptablelevel and they can be further improved by refining the FEM meshor by increasing the thickness of the AMM layer to handle rapidfield variations. The monostatic RCS profile in Figure 7(b) alsoreveals that the proposed method provides reliable results even forelectrically very small scatterers at the far-field.

Finally, to demonstrate the applicability of the method in scat-terers with arbitrary crosssections, we consider the scattering prob-lem where a plane wave ( inc � 45 ) is incident to a squarecylinder whose edge length is �/10. We first simulate the originalscattering problem, and plot the contour of the electric field insidethe computational domain in Figure 8(a). Then, we simulate theequivalent problem with AMM layer, and the plot the contour ofthe electric field in original coordinates inside the computationaldomain in Figure 8(b). We calculate E1 as 0.2260% comparing theresults in Figure 8(a,b) inside the common free-space region. Then,we transform the field values calculated in the equivalent problemin Figure 8(b), and we plot the field values in transformed coor-dinates in the equivalent problem in Figure 8(c). We also plot thefield values of Figure 8(a,c) along the dotted line (x � y cut) to

Figure 5 Real part of electric field along cuts in Figure 4(c,d)

Figure 6 Bistatic RCS profile for circular cylinder

TABLE 1 E2 Values for Different Thickness Values of theAMM Layer (rs � �/10)

dAMM E2 (%)

0.5� 0.15390.4� 0.23510.3� 0.42210.2� 1.2842

Figure 7 Performance of the AMM layer as a function of the radius ofthe scatterer: (a) E2 profile; (b) monostatic RCS profile


show the equivalence of the field values in Figure 8(a,c). Finally,we calculate the reduction in unknowns as 37% in this example.

The numerical simulations in this section demonstrate that theperformance of the AMM layer in low-frequency scattering prob-lems is in conformity with the theory.

4. CONCLUSIONS

In this article, we have introduced a new design technique em-ploying an AMM layer for the efficient solution of low-frequencyscattering problems. The AMM layer is implemented by a suitablydefined coordinate transformation. We have concluded that theproposed method makes the solution of the low-frequency scatter-ing problems feasible by reducing the memory requirement and theprocessing power. We have numerically investigated the applica-bility of the method in various configurations by means of finiteelement simulations and we have validated the theoretical predic-tions.

REFERENCES

1. L. Rayleigh, On the electromagnetic theory of light, Philos Mag 12(1881), 81-101.

2. R.E. Kleinman, The Rayleigh region, Proc IEEE 53 (1965), 848-856.3. L. Rayleigh, On the incidence of aerial and electric waves upon small

obstacles in the form of ellipsoids or elliptic cylinders and on thepassage of electric waves through a circular aperture in a conductingscreen, Philos Mag XLIV (1897), 28-52.

4. T.B.A. Senior, Scalar diffraction by a prolate spheroid at low frequen-cies, Can J Phys 38 (1960), 1632-1641.

5. A.F. Stevenson, Solution of electromagnetic scattering problems as

power series in the ratio (dimension of scatterer)/wavelength, J ApplPhys 24 (1963), 1134-1142.

6. D.P. Thomas, Diffraction by a spherical cap, Proc Cambridge PhilosSoc 59 (1963), 197-209.

7. J. van Bladel, Low frequency scattering by cylindrical obstacles, ApplSci Res 10 (1963), 195-202.

8. D.R. Wilton and A.W. Glisson, On improving the electric field integralequation at low frequencies, URSI Radio Science Meeting Digest, LosAngles, CA (1981) 24.

9. J.R. Mautz and R.F. Harrington, An E-field solution for a conductingsurface small or comparable to the wavelength, IEEE Trans AntennasPropag 32 (1984), 330-339.

10. Y. Zhang, T.J. Cui, W.C. Chew, and J.S. Zhao, Magnetic field integralequation at very low frequencies, IEEE Trans Antennas Propag 51(2003), 1864-1871.

11. A. Bayliss, M. Gunzburger, and E. Turkel, Boundary conditions for thenumerical solution of elliptic equations in exterior domains, SIAMJ Appl Math 42 (1982), 430-451.

12. J.P. Berenger, A perfectly matched layer for the absorption of elec-tromagnetic waves, J Comput Phys 114 (1994), 185-200.

13. M. Kuzuoglu and R. Mittra, Mesh truncation by perfectly matchedanisotropic absorbers in the finite element method, Microwave OptTechnol Lett 12 (1996), 136-140.

14. J.D. Moerloose and M.A. Stuchly, Reflection analysis of PML ABCsfor low-frequency applications, IEEE Microwave Guided Wave Lett 6(1996), 177-179.

15. O. Ozgun and M. Kuzuoglu, Locally-conformal perfectly matchedlayer implementation for finite element mesh truncation, MicrowaveOpt Technol Lett 48 (2006), 1836-1839.

16. W.C. Chew, and W. Weedon, A 3D perfectly matched medium from

Figure 8 Real part of electric field for square cylinder: (a) Original problem; (b) equivalent problem in original coordinates; (c) equivalent problem aftertransforming the field values (in transformed coordinates); (d) along cuts in (a) and (c). [Color figure can be viewed in the online issue, which is availableat www.interscience.wiley.com]


modified Maxwell’s equations with stretched coordinates, MicrowaveOpt Technol Lett 7 (1994), 599-604.

17. O. Ozgun and M. Kuzuoglu, Multi-center perfectly matched layerimplementation for finite element mesh truncation, Microwave OptTechnol Lett 49 (2007), 827-832.

18. M. Kuzuoglu and R. Mittra, Investigation of nonplanar perfectlymatched absorbers for finite element mesh truncation, IEEE TransAntennas Propag 45 (1997), 474-486.

19. O. Ozgun and M. Kuzuoglu, Non-Maxwellian locally-conformal PMLabsorbers for finite element mesh truncation, IEEE Trans AntennasPropag, 55 (2007), 931-937.

20. G.W. Milton, M. Briane, and J.R. Willis, On cloaking for elasticity andphysical equations with a transformation invariant form, New J Phys 8(2006), 248 1-20.

21. J.B. Pendry, D. Schurig, and D.R. Smith, Controlling electromagneticfields, Science 312 (2006), 1780-1782.

22. O. Ozgun and M. Kuzuoglu, Electromagnetic metamorphosis: Reshap-ing scatterers via conformal anisotropic metamaterial coatings, Micro-wave Opt Technol Lett 49 (2007), 2386-2392.

23. H. Chen and C.T. Chan, Transformation media that rotate electromag-netic fields, Available at http://arxiv.org/abs/physics/0702050v2(2007).

24. O. Ozgun and M. Kuzuoglu, Utilization of anisotropic metamateriallayers in waveguide miniaturization and transitions, IEEE MicrowaveWireless Compon Lett, in press.

25. O. Ozgun and M. Kuzuoglu, Domain compression in finite methodsusing anisotropic metamaterial layers, IEEE Trans Antennas Propag,submitted for publication.

26. I.V. Lindell, Methods for electromagnetic field analysis, Oxford Uni-versity Press, Oxford, 1992.

27. R.F. Harrington, Time-harmonic electromagnetic fields, McGraw-Hill,New York, 1961.

© 2008 Wiley Periodicals, Inc.

OPTIMIZATION OF CASCODECONFIGURATION IN CMOS LOW-NOISEAMPLIFIER

Ickhyun Song, Minsuk Koo, Hakchul Jung, Hee-Sauk Jhon,and Hyungcheol ShinSchool of Electrical Engineering and Computer Science, 301-1015Seoul National University, San56-1, Sillim-dong, Gwanak-gu, Seoul151-744, South Korea; Corresponding author: [email protected]

Received 7 August 2007

ABSTRACT: In this paper, design consideration of the cascode config-uration in low-noise amplifiers (LNA) using 0.13-�m CMOS technologyis presented. Performance factors of LNAs such as signal power gain,noise factor, and power consumption are analytically expressed in de-vice parameters from its small-signal equivalent circuit. The effect of thecommon-gate transistor in each performance factor is evaluated at thetarget frequency of 17-GHz ISM band. At this frequency, power gain andnoise factor are degraded, which result from the common-gate transistor.Figure of merit of LNAs is also optimized. © 2008 Wiley Periodicals, Inc.Microwave Opt Technol Lett 50: 646–649, 2008; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23163

Key words: low-noise amplifier (LNA); cascode; figure of merit (FoM);CMOS; noise figure

1. INTRODUCTION

As the scaling down of CMOS technology continues, the cutofffrequency, fT, and the maximum oscillation frequency, fMAX, of aMOS transistor have exceeded far more than 100 GHz, which

extends the operation limit of CMOS circuits for wireless appli-cations. With better RF performance advantages obtained fromscaling, CMOS radio-frequency circuits are integrated with base-band analog, mixed, and digital circuits of system. In the perspec-tive of system integration, CMOS is more cost-effective thancomparative bipolar or compound technology.

A low-noise amplifier (LNA) is one of the most critical circuitblocks in a wireless transceiver. As the first stage in the receiverarchitecture, noise figure of LNA dominates overall noise perfor-mance of the system [1]. Hence, an LNA should add little noise tothe next stages while providing enough signal gain for signalprocessing in the following stages. Low power consumption andinput and output impedance matching should also be considered[2].

In this paper, the effect of cascode configuration in LNAs withthe target frequency of 17 GHz is presented. A common-gatetransistor in cascode configuration has been usually ignored incircuit analysis, since it is regarded as a simple current buffer. Atvery high frequency of above 10 GHz, however, the cascodetransistor has significant degradation effect which should be con-sidered for evaluating accurate performance [3].

2. RF LOW-NOISE AMPLIFIER

The conventional circuit schematic of CMOS LNAs is shown inFigure 1. Core of the LNA consists of cascode transistors M1 andM2. The cascode configuration has some advantages against thesingle transistor case. First of all, cascode amplifiers suppress theMiller effect caused by gate-to-drain capacitance of the M1 tran-sistor. Reduced Miller effect improves signal gain and noise per-formance. Also, cascode configuration shows better circuit stabil-ity and isolation between input and output ports. Ci is the couplingcapacitor and Rb is a resistor for transistor gate biasing. Sourcedegeneration inductor, Ls, with small inductance relative to thegate inductor provides real part of input impedance, which is usedfor simultaneous input and noise matching. Gate inductor, Lg,makes imaginary part of input impedance zero. Lo and Co areoutput matching elements, and they are tuned to the operationfrequency.

Impedance matching is important for maximum power transfer.To achieve impedance matching, the conjugate-matching condi-

Figure 1 A conventional LNA schematic


efficient finite element solution of low-frequency scattering problems via anisotropic metamaterial...

Documents