EFFICIENT FINITE ELEMENT SOLUTION OF LOW-FREQUENCY SCATTERING PROBLEMS VIA ANISOTROPIC METAMATERIAL LAYERS Ozlem Ozgun and Mustafa Kuzuoglu Department of Electrical and Electronics Engineering, Middle East Technical University, 06531 Ankara, Turkey; Corresponding author:
[email protected] Received 4 August 2007 ABSTRACT: We introduce a new technique which remedies the drawbacks in the Finite Element solution of low-frequency electromagnetic scattering problems, through the usage of an anisotropic metamaterial layer which is designed by employing the coordinate transformation approach. The usual finite element method should utilize a “challenging” mesh generation scheme to accurately simulate the “small” objects in scattering problems; on the contrary, the proposed technique provides a considerable reduction in the number of unknowns, and requires a more convenient and simpler mesh structure inside the computational domain. The most interesting feature of the proposed method is its capability to handle arbitrarily shaped “small” scatterers by using a “single” mesh and by modifying only the constitutive parameters inside the matamaterial layer. We report some numerical results for two-dimensional electromagnetic scattering problems. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 639 – 646, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop. 23167 Key words: low-frequency electromagnetic scattering; anisotropic metamaterials; coordinate transformation; finite element method (FEM) 1. INTRODUCTION
The scattering of electromagnetic waves from electrically small objects (viz., objects whose dimensions are small compared to the wavelength) has been investigated in various fields of science and engineering (such as physics, electrical engineering, geophysics, astrophysics, and biology) for more than a century. The lowfrequency 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 contribution. In the Rayleigh scattering problem, the unknown field quantities are expressed as a convergent series in positive integral powers of the propagation constant k, and then the unknown expansion coefficients are determined from Maxwell’s equations and boundary conditions [2]. These expansion coefficients are functions of the direction of incidence and observation, as well as the geometry of the scatterer. Because of the dependence on the geometry of the scatterer, the calculation of the unknown coefficients in the Rayleigh series is usually a difficult task especially for objects of arbitrary shape. The Rayleigh technique has been utilized in the literature to solve low-frequency scattering problems for scatterers of some specific shapes [3-7]. Apart from the above-mentioned analytical approaches, some approximate computational methods have been devised to solve low-frequency scattering problems on account of the advances in the computer technology. However, accurate solution of lowfrequency scattering problems is still a challenging task in the context of numerical approaches [such as the method of moments (MoM) and the finite element method (FEM)]. In the MoM approach employing the electric field integral equation (EFIE), the matrix is known to be ill-conditioned at low frequencies, and some methods have been proposed to overcome this problem [8, 9]. In addition, the MoM approximation of the magnetic field integral
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equation (MFIE) has also some drawbacks at low frequencies in terms of the stability of the solution, as reported in [10], which states that small errors in the real part of the current distribution causes significant errors in the far-field profile, and which proposes a method to resolve this issue. In the usual (FEM), accurate simulation of the low-frequency scattering problems may not be manageable especially due to the high memory requirement to take into account the fine sections of the “small” objects with high numerical precision. That is, although the FEM is a powerful method to handle any type of geometry and material inhomogeneity, the FEM mesh may require a large number of unknowns to define properly the geometry of the object whose size is only a fraction of wavelength inside the computational domain. Furthermore, it is known that to employ the FEM to the solution of scattering problems involving spatially unbounded domains, the physical domain must be truncated by an artificial boundary or layer to achieve a bounded computational domain. The most common approaches in mesh truncation are absorbing boundary conditions (ABCs) [11] and perfectly matched layers (PMLs) [12]. However, to employ the ABC, the truncation boundary must be located sufficiently far away from the scatterer (at least in the order of wavelength) to reduce spurious reflections of the propagating waves at the outer boundary. The consequence of this requirement results in an increase in the computational domain, especially at low frequencies, due to a large number of elements inside the white-space (i.e., usually free-space in scattering problems) which is not occupied by the objects. On the contrary to the ABC, the PML region may be designed very close to the scatterer, thus is capable of minimizing the white-space especially in high frequencies [13]. However, at low frequencies, the distance between the scatterer and the PML boundary should also be sufficiently large so that the evanescent waves become negligible at the outer boundary [14]. In addition, the PML may yield ill-conditioning in the FEM matrix equation at low frequencies [15] due to the large propagation constant k. Therefore, the FEM mesh size increases enormously at low frequencies for the mesh truncation techniques to work well, and to achieve a fine discretization of small objects inside a relatively large computational domain. In this article, we propose a new technique to solve efficiently low-frequency electromagnetic scattering problems via FEM using an anisotropic metamaterial (AMM) layer, as illustrated in Figure 1. The original scattering problem for a two-dimensional (2D) electrically small, perfect electric conductor (PEC) cylinder of circular crosssection is shown in Figure 1(a) together with its possible mesh structure employing triangular elements and PML region inside a square computational domain. As mentioned previously, 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 computational domain requires a large number of unknowns even if a nonuniform mesh generation scheme is employed (i.e., the element size is gradually increased from the boundary of the scatterer toward the outermost boundary). Proper development of such a nonuniform mesh algorithm may be a challenging task if the scatterer is of arbitrary shape, especially in three-dimensional (3D) problems. However, in Figure 1(b), we design an arbitrarily shaped AMM layer (⍀A) which is located at an arbitrary distance from the scatterer, and we solve this problem which turns out to be “equivalent” to the original problem in Figure 1(a). It should be noted that the “empty” region in Figure 1(b) is no longer included in the computational domain. The two configurations in Figure 1(a,b) are equivalent in the sense that they yield identical field values in their common free-space regions, and that the field values calculated in
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low-frequency problem into a relatively high-frequency problem, and makes the solution of the problem more feasible by decreasing the number of unknowns considerably and by employing a more convenient, even uniform, mesh generation scheme inside the computational domain. Another special feature of the proposed technique is that low-frequency scattering of waves from arbitrarily shaped scatterers can be solved using a single mesh if the constitutive parameters of the AMM layer are defined appropriately for each scatterer. In other words, the same mesh in Figure 1(b) can be employed for arbitrary scatterers (such as rectangular or elliptical scatterer) by simply changing the parameters of the AMM layer which are calculated with respect to the dotted boundary of the scatterer. Hence, the proposed technique provides a great flexibility to handle various low-frequency scattering problems in a more efficient and favorable manner by decreasing the memory and processing power and by employing a simpler mesh generation algorithm. It is known that AMM layers can tune the spatial variations of electromagnetic waves in a desired manner by manipulating their structural features. The constitutive parameters of the AMM layer in this article are obtained by using the coordinate transformation technique, which is often utilized in the design of PMLs [16-19]. The coordinate transformation approach is based on the fact that Maxwell’s equations are form-invariant under coordinate transformations [20]. In other words, Maxwell’s equations are still valid, but with appropriately defined constitutive parameters that convey the effect of the coordinate transformation to the electromagnetic fields. The permittivity and permeability parameters turn out to be anisotropic as well as spatially varying. Therefore, the design methodology introduced in this article is theoretical, and intends to devise a simulation tool for the purpose of efficient solution of low-frequency electromagnetic scattering problems with fewer unknowns. Recently, AMM layers based on the concept of coordinate transformation have been proposed to produce material specifications for the purpose of obtaining electromagnetic invisibility [21], reshaping objects in electromagnetic scattering [22], rotating electromagnetic fields in a closed domain [23], miniaturizing waveguides [24], and spatial domain compression in finite methods [25]. This article is structured as follows. In Section 2, we introduce the design procedure of the AMM layer for PEC scatterers. In Section 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
Figure 1 AMM layer design in low-frequency scattering: (a) Original problem; (b) equivalent problem with AMM layer
Figure 1(b) are inherently related to those in the original problem in Figure 1(a). In other words, the equivalent problem in Figure 1(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 contain elements at the close vicinity of the scatterer which is shown by a dotted-curve in Figure 1(b). The problem in Figure 1(b) may be interpreted as a scattering problem involving a “larger” PEC scatterer (shown by ⭸⍀A,in) coated by an AMM layer. Therefore, the equivalent problem in Figure 1(b) transforms the original
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The AMM layer is designed via the coordinate transformation approach, as illustrated in Figure 2. Initially, the spatial region occupied by the AMM layer is constructed at an arbitrary but sufficiently large distance from the small scatterer whose boundary is ⭸⍀S (it is assumed that the spatial domain occupied by the scatterer is a convex subset of ᑬ3 ). In this figure, ⍀A denotes the spatial domain occupied by the AMM layer with the inner and outer boundaries ⭸⍀A,in and ⭸⍀A,out, and ⍀e denotes the empty spatial domain between ⭸⍀S and ⭸⍀A,in. While designing the AMM layer, each point P inside the AMM ˜ layer (⍀A) is mapped to P˜ inside the transformed region ⍀ ⫽ ⍀A 艛 ⍀e enclosed within the boundaries ⭸⍀A,out and ⭸⍀S. This mapping, which can be interpreted as a space expansion, is defined ˜ as follows: as a coordinate transformation T:⍀A 3 ⍀
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ជr˜ ⫽ T共rជ兲 ⫽
储rជo ⫺ rជs储 共rជ ⫺ rជi兲 ⫹ rជs 储rជo ⫺ rជi储
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(1)
It can be concluded that the constitutive parameters of the AMM layer in Eq. (2) can be directly calculated from the Jacobian of the transformation in Eq. (1). In conjunction with the coordinate transformation inside the AMM layer, the fields are transformed as [26] ˜ Eជ 共rជ 兲 3 Eជ 共rជ 兲 ⫽ JញT 䡠 Eជ 共rជ˜兲
(5a)
˜ ជ 共rជ˜兲 ជ 共rជ 兲 3 H ជ 共rជ 兲 ⫽ JញT 䡠 H H
(5b)
and the electric field satisfies the following vector wave equation ˜ ˜ ញ ⫺1 䡠 ⵜ ⫻ Eជ 共rជ 兲其 ⫺ k 2⌳ ញ 䡠 Eជ 共rជ 兲 ⫽ 0 ⵜ ⫻ 兵⌳
(6)
where k is the wave number of the isotropic medium. This equation derived from Maxwell’s equations in original coordinates is basically equivalent to the following vector wave equation in transformed coordinates due to the form-invariance property of the Maxwell’s equations under the mapping rជ˜ ⫽ T共rជ兲 ⵜ˜ ⫻ ⵜ˜ ⫻ Eជ 共rជ˜ 兲 ⫺ k 2Eជ 共rជ˜ 兲 ⫽ 0
(7)
Figure 2 Design of the AMM layer with coordinate transformation
where rជ and rជ˜ are the position vectors of the points P and P˜ in the original and transformed coordinate systems, respectively, and 储 䡠 储 represents the Euclidean norm. In addition, rជs, rជi, and rជo are the position vectors of Ps, Pi, and Po which are determined on ⭸⍀S, ⭸⍀A,in, and ⭸⍀A,out, respectively, along the direction of the unit vector aˆ. The unit vector aˆ is calculated emanating from a point inside the innermost domain occupied by the scatterer (such as the center-of-mass point, which can be designated as the origin) in the direction of the point P inside the AMM layer. It is worth mentioning that the coordinate transformation in Eq. (1) is continuous such that two closely located points rជ and rជ* in ⍀A are mapped to ˜ , due to the definition of also closely located points rជ˜ and rជ˜* in ⍀ the transformation. As a result of the coordinate transformation in Eq. (1), the original medium turns into a spatially varying anisotropic medium where the original forms of Maxwell’s equations are still preserved in the transformed space. That is, Maxwell’s equations are forminvariant under space transformations, and a general coordinate transformation leads to the following expressions for the permittivity and permeability tensors [20]. ញ ញ ⫽ ⌳
(2a)
ញ ញ ⫽ ⌳
(2b)
where and are the constitutive parameters of the original isotropic medium (usually free-space in scattering problems), and ញ ⫽ (det Jញ)(JញT䡠Jញ)⫺1 ⌳
(3)
where Jញ is the Jacobian tensor defined as (in Cartesian coordinates)
Jញ ⫽
冋
册
⭸x˜/⭸x ⭸x˜/⭸y ⭸x˜/⭸z ⭸共x˜,y˜,z˜兲 ⫽ ⭸y˜/⭸x ⭸y˜/⭸y ⭸y˜/⭸z . ⭸共x,y,z兲 ⭸z˜/⭸x ⭸z˜/⭸y ⭸z˜/⭸z
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(4)
where ⵜ˜ is the nabla operator in the transformed space. In a FEM code, the equation in Eq. (6) is solved in original coordinates using the mesh in Figure 1(b). Then, the original fields in transformed coordinates can be directly calculated from the expressions in Eq. (5), which are rewritten for convenience as follows ˜ Eជ 共rជ˜ 兲 ⫽ 共JញT兲⫺1 䡠 Eជ 共rជ兲
(8a)
˜ ជ 共rជ˜ 兲 ⫽ 共JញT兲⫺1 䡠 H ជ 共rជ兲. H
(8b)
Since the transformed space includes both the domain inside the ˜ AMM layer and the empty domain around the scatterer (i.e., ⍀ ⫽ ⍀⌳ 艛 ⍀e), the expressions in Eq. (8) provide exactly the original near field values for the scatterer. For instance, in the 2D TMz case where Eជ 共rជ兲 ⫽ aˆz Ez 共x,y兲, the Jacobian tensor in Eq. (4) reduces to
冋
册
⭸x˜/⭸x ⭸x˜/⭸y 0 Jញ ⫽ ⭸y˜/⭸x ⭸y˜/⭸y 0 0 0 1
(9)
implying that Ez 共x˜,y˜兲 ⫽ E˜z 共x,y兲 because the 3,3-entry of 共Jញ T兲⫺1 is equal to 1. This fact is illustrated in Figure 3 for the sake of being precise 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 Eជ 共rជ兲 can be split into two parts: one is the known inc incident field 共Eជ 兲 produced in the absence of the AMM layer, and ˜ the other is the scattered field 共Eជ s 兲. In other words, ˜ ˜ Eជ ⫽ Eជ inc ⫹ Eជ s.
(10)
Then, the vector wave equation for the scattered field is expressed as follows
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˜ ⫽
o ⫺ s 共 ⫺ i兲 ⫹ s o ⫺ i
(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 the following diagonal form (in cylindrical coordinates)
冋
册
⌳ 11 0 0 ញ ⫽ 0 ⌳ 22 0 ⫽ ⌳ 0 0 ⌳ 33
冤
⫺␣
0
0
0
⫺␣
0
0
0
冉
冊
冥
o ⫺ s 2 ⫺ ␣ o ⫺ i
(15) where
o ⫺ i ␣ ⫽ i ⫺ o . o ⫺ s
(16)
Figure 3 Field transformation in the equivalent problem
Then, the wave equation in Eq. (12) is expressed as
冉
˜ ˜ ញ 䡠 Eជ inc共rជ 兲兴. ⫻ Eជ inc共rជ 兲其 ⫺ k 2⌳
(11)
As in a usual FEM procedure, the weak variational form of the vector wave equation in Eq. (11) is derived using the weighted residual method and solved by discretizing the computational domain using some number of elements. After solving Eq. (11) for the scattered field, the total field is calculated by Eq. (10). Finally, as mentioned previously, the original field inside the computational domain is calculated by Eq. (8). It is evident that, inside the free-space region, these equations transform to the expressions of ញ becomes the identity a free-space scattering problem, since ⌳ tensor. In the 2D TMz case, the vector wave equation in Eq. (6) reduces to the scalar partial differential equation ញ ⬘ⵜE˜ z共 x,y兲兲 ⫹ k 2⌳ 33E˜ z共 x,y兲 ⫽ 0 ⵜ 䡠 共⌳
(12)
where
冋
册
⌳ 11 ⌳ 12 0 ញ ⫽ ⌳ 21 ⌳ 21 0 ⌳ 0 0 ⌳ 33
冋
⌳ ញ ⬘ ⫽ 11 and ⌳ ⌳21
册
⌳12 ⌳21 .
(13)
Then, a similar FEM procedure is followed as in the 3D case to calculate the field values in both original and transformed coordinates. All above-mentioned expressions can be employed in a general design of the AMM layer for any geometry in a straightforward manner. As a special case in the cylindrical coordinate system, assuming that the scatterer is a circular cylinder and the AMM layer is a circular shell, the coordinate transformation in Eq. (1) reduces to the following simple form
642
冊
⭸E˜ z ⌳ 22 ⭸ 2E˜ z 1 ⭸ ⌳ 11 ⫹ 2 ⫹ k 2⌳ 33E˜ z ⫽ 0. ⭸ ⭸ ⭸2
˜ ˜ ញ ⫺1 䡠 ⵜ ⫻ Eជ s共rជ 兲其 ⫺ k 2⌳ ញ 䡠 Eជ s共rជ 兲 ⫽ ⫺ 关ⵜ ⫻ 兵⌳ ញ ⫺1 䡠 ⵜ ⵜ ⫻ 兵⌳
(17)
It is useful to emphasize that the calculation of the constitutive parameters of the AMM layer using Eq. (2) can be carried out in the preprocessing phase (i.e., before the matrix construction phase) in a computer code. Thus, the computational effort to implement the coordinate transformation creates almost negligible burden on the processing power of the computer, compared to some other phases of the code (such as usual matrix construction and solution phases). 3. NUMERICAL EXPERIMENTS
In this section, we report the results of some numerical experiments to validate the design procedure of the AMM layer in some 2D TMz electromagnetic scattering problems involving infinitely long cylindrical PEC scatterers. All simulations are performed using our FEM software employing triangular elements. In all examples, the wavelength in free-space () is set to 1 m (i.e., k is 2). The element size is approximately set to /40 in the equivalent problem, but it is decreased gradually toward to the boundary of the scatterer starting from /40 in the original problem. In addition, the computational domain is terminated with a PML absorber which is implemented by the locally conformal PML method [19] whose parameters are ␣ ⫽ 7k and m ⫽ 3. The thickness of the PML region is chosen as /4. In all examples, the same mesh structure is employed in the solution of the equivalent problem, irrespective of the shape of the scatterer. That is, in all examples, the AMM layer in the equivalent problem is a circular shell as shown in Figure 1(b), where the radii of the inner and outer boundaries are /2 and respectively, and the thickness of the AMM layer (dAMM) is /2. Moreover, the incident plane wave is assumed to be in the form of Eជ inc ⫽ aˆz exp关 jk共xcosinc ⫹ ysininc 兲兴 where inc is the angle of incidence with respect to the x-axis.
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Figure 4 Contours of real part of electric field for circular cylinder: (a) Original problem; (b) equivalent problem in original coordinates; (c) equivalent problem after transforming the field values (in transformed coordinates); (d) Mie series. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com]
In the first example, we consider a scattering problem where a plane wave (inc ⫽ 180⬚) is incident to a circular cylinder whose radius (rs) is /20. We first simulate the original scattering problem [see Fig. 1(a)], and plot the contour of the electric field inside the computational domain in Figure 4(a). This plot represents the real part of the electric field to better visualize the field behavior inside the computational domain. Then, we simulate the equivalent problem with AMM layer [see Fig. 1(b)] by solving the wave equation in Eq. (12), and we plot the contour of the electric field in original coordinates inside the computational domain in Figure 4(b). We expect that the two simulations in Figure 4(a,b) must yield identical field values inside their common free-space region (free-space outside the transformed space). Therefore, a way to measure the performance of the proposed method is to introduce a mean-square error criterion as follows
冘 E1 ⫽
⍀FS
problem provides original fields inside the free-space region (i.e., E˜zeq共rជ兲 ⫽ Ezeq共rជ兲 in ⍀FS). Then, we transform the field values calculated inside the AMM layer in the equivalent problem using the fact that Ezeq共x˜,y˜兲 ⫽ E˜zeq共x,y兲 (see Fig. 3). That is, we calculate the field values in transformed coordinates in the equivalent problem, and we plot the contour of the electric field inside the computational domain in Figure 4(c). We conclude that the field distribution in Figure 4(c) is almost identical to the one in Figure 4(a), as expected, implying that both problems are actually equivalent. Finally, we calculate the analytical field values using the Mie series expansion [27] inside the computational domain to validate the results of the equivalent problem. For this purpose, we plot the contour of the electric field calculated by the Mie series in Figure 4(d) [which is almost identical to Fig. 4(a,c)], and introduce a mean-square error criterion as follows
兩Eជ zeq共rជ兲 ⫺ Ezorg共rជ兲兩2
冘
兩Eជ zorg共rជ兲兩2
E2 ⫽
⍀ FS
where Ezorg and Ezeq are the electric fields calculated in the original and equivalent problems [i.e., the fields in Fig. 4(a,b)], respectively, inside the common free-space region. Then, we calculate E1 as 0.1256%. Although the error term in Eq. (18) must ideally be equal to zero, the small error value observed in the simulation is because of FEM modeling errors which may be decreased further by refining the FEM mesh. We also note that we have removed the tilda from the term Ezeq共rជ兲 in Eq. (18) because the equivalent
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冘
(18)
⍀FS
兩Eជ zeq共rជ˜兲 ⫺ Ezmie共rជ˜兲兩2
冘
兩Eជ zmie共rជ˜兲兩2
(19)
⍀ FS
where Ezmie and Ezeq are the electric fields calculated by the Mie series and equivalent problem, respectively, inside the whole freespace region. This error value basically compares the field values in Figure 4(c,d). Then, we calculate E2 as 0.3154%. In addition, in Figure 5, we plot the field values of Figure 4(c,d) along their dotted cut-lines (x ⫽ 0 and y ⫽ 0 cuts).
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TABLE 1 E2 Values for Different Thickness Values of the AMM Layer (rs ⴝ /10) dAMM
E2 (%)
0.5 0.4 0.3 0.2
0.1539 0.2351 0.4221 1.2842
where Norg and Neq are the number of unknowns (or the number of nodes) employed in the original and equivalent problems, respectively. Then, we calculate the reduction in unknowns as 40% in this example. Thus, we conclude that the proposed technique provides a considerable reduction in the memory requirement. In the next example, we consider a circular cylinder whose radius (rs) is /10. We follow the same procedure as in the previous example, and we plot the bistatic radar crosssection (RCS) profile in Figure 6, which compares the results of the original and equivalent problems and the Mie series, to measure the 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 thickness values of the AMM layer in Table 1. We conclude that as the
thickness of the layer decreases, the results start to deteriorate, but they are reliable even for electrically thin layers. In the next simulation, the radius (rs) of the circular scatterer is varied between /40 and 0.45 to visualize the performance of the AMM layer with respect to the electrical-size of the scatterer. We plot the E2 values and the monostatic RCS values as a function of the radius of the scatterer in Figure 7(a) and Figure 7(b), respectively. We conclude that as the electrical-size of the scatterer is decreased, the error increases because the spatial variations in the entries of the permittivity and permeability tensors of contiguous elements increase due to the definition of the coordinate transformation. However, we can assert that the errors are at an acceptable level and they can be further improved by refining the FEM mesh or by increasing the thickness of the AMM layer to handle rapid field variations. The monostatic RCS profile in Figure 7(b) also reveals that the proposed method provides reliable results even for electrically very small scatterers at the far-field. Finally, to demonstrate the applicability of the method in scatterers with arbitrary crosssections, we consider the scattering problem where a plane wave (inc ⫽ 45⬚) is incident to a square cylinder whose edge length is /10. We first simulate the original scattering problem, and plot the contour of the electric field inside the computational domain in Figure 8(a). Then, we simulate the equivalent problem with AMM layer, and the plot the contour of the electric field in original coordinates inside the computational domain in Figure 8(b). We calculate E1 as 0.2260% comparing the results in Figure 8(a,b) inside the common free-space region. Then, we transform the field values calculated in the equivalent problem in Figure 8(b), and we plot the field values in transformed coordinates in the equivalent problem in Figure 8(c). We also plot the field values of Figure 8(a,c) along the dotted line (x ⫽ y cut) to
Figure 6 Bistatic RCS profile for circular cylinder
Figure 7 Performance of the AMM layer as a function of the radius of the scatterer: (a) E2 profile; (b) monostatic RCS profile
Figure 5
Real part of electric field along cuts in Figure 4(c,d)
To measure the amount of the reduction in unknowns in this example, we use the following expression N reduce ⫽
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Norg ⫺ Neq Norg
(20)
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Figure 8 Real part of electric field for square cylinder: (a) Original problem; (b) equivalent problem in original coordinates; (c) equivalent problem after transforming the field values (in transformed coordinates); (d) along cuts in (a) and (c). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com]
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 the performance of the AMM layer in low-frequency scattering problems is in conformity with the theory.
6.
4. CONCLUSIONS
8.
In this article, we have introduced a new design technique employing an AMM layer for the efficient solution of low-frequency scattering problems. The AMM layer is implemented by a suitably defined coordinate transformation. We have concluded that the proposed method makes the solution of the low-frequency scattering problems feasible by reducing the memory requirement and the processing power. We have numerically investigated the applicability of the method in various configurations by means of finite element simulations and we have validated the theoretical predictions.
7.
9.
10.
11.
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OPTIMIZATION OF CASCODE CONFIGURATION IN CMOS LOW-NOISE AMPLIFIER Ickhyun Song, Minsuk Koo, Hakchul Jung, Hee-Sauk Jhon, and Hyungcheol Shin School of Electrical Engineering and Computer Science, 301-1015 Seoul National University, San56-1, Sillim-dong, Gwanak-gu, Seoul 151-744, South Korea; Corresponding author:
[email protected]
extends the operation limit of CMOS circuits for wireless applications. With better RF performance advantages obtained from scaling, CMOS radio-frequency circuits are integrated with baseband analog, mixed, and digital circuits of system. In the perspective of system integration, CMOS is more cost-effective than comparative bipolar or compound technology. A low-noise amplifier (LNA) is one of the most critical circuit blocks in a wireless transceiver. As the first stage in the receiver architecture, noise figure of LNA dominates overall noise performance of the system [1]. Hence, an LNA should add little noise to the next stages while providing enough signal gain for signal processing in the following stages. Low power consumption and input and output impedance matching should also be considered [2]. In this paper, the effect of cascode configuration in LNAs with the target frequency of 17 GHz is presented. A common-gate transistor in cascode configuration has been usually ignored in circuit analysis, since it is regarded as a simple current buffer. At very high frequency of above 10 GHz, however, the cascode transistor has significant degradation effect which should be considered for evaluating accurate performance [3]. 2. RF LOW-NOISE AMPLIFIER
The conventional circuit schematic of CMOS LNAs is shown in Figure 1. Core of the LNA consists of cascode transistors M1 and M2. The cascode configuration has some advantages against the single transistor case. First of all, cascode amplifiers suppress the Miller effect caused by gate-to-drain capacitance of the M1 transistor. Reduced Miller effect improves signal gain and noise performance. Also, cascode configuration shows better circuit stability and isolation between input and output ports. Ci is the coupling capacitor and Rb is a resistor for transistor gate biasing. Source degeneration inductor, Ls, with small inductance relative to the gate inductor provides real part of input impedance, which is used for simultaneous input and noise matching. Gate inductor, Lg, makes imaginary part of input impedance zero. Lo and Co are output matching elements, and they are tuned to the operation frequency. Impedance matching is important for maximum power transfer. To achieve impedance matching, the conjugate-matching condi-
Received 7 August 2007 ABSTRACT: In this paper, design consideration of the cascode configuration in low-noise amplifiers (LNA) using 0.13-m CMOS technology is presented. Performance factors of LNAs such as signal power gain, noise factor, and power consumption are analytically expressed in device parameters from its small-signal equivalent circuit. The effect of the common-gate transistor in each performance factor is evaluated at the target frequency of 17-GHz ISM band. At this frequency, power gain and noise 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 in Wiley 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 cutoff frequency, fT, and the maximum oscillation frequency, fMAX, of a MOS transistor have exceeded far more than 100 GHz, which
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Figure 1
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A conventional LNA schematic
DOI 10.1002/mop
