2009 SBMO/IEEE MTT-S International Microwave & Optoelectronics Conference (IMOC 2009) 73

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1 Current Distribution of a Printed Dipole with Arbitrary Length Embedded in Layered Uniaxial Anisotropic Dielectrics Benjamin D. Braaten Electrical and Computer Engineering Department North Dakota State University Fargo, North Dakota, USA benbraaten@ieee.org David A. Rogers Electrical and Computer Engineering Department North Dakota State University Fargo, North Dakota, USA david.rogers@ndsu.edu Robert M. Nelson Engineering and Technology Department University of Wisconsin - Stout Menomonie, Wisconsin, USA r.m.nelson@ieee.org Abstract The current distribution of a printed dipole with arbitrary length embedded in several layers of anisotropic dielectrics is evaluated using the spectral domain immittance functions. In particular, the currents on a printed dipole on a single anisotropic layer and in three anisotropic layers are determined. A good comparison between the isotropic cases, commercial software and published literature is observed. Finally, general comments are made on how each component of the anisotropic permittivity affects the current distribution of the dipole. dn+ dn+2 dn+ dn y Air µ, ε Anisotropic layer N+2 Anisotropic layer N+ Anisotropic layer N JK µ, [εn+2]ε µ, [εn+]ε µ, [εn]ε Printed antenna I. INTRODUCTION d3 Anisotropic layer 3 µ, [ε3]ε The study of printed antennas in layered anisotropic dielectrics has appeared in many recent areas of research []- [3]. This is because it is well known that many microwave substrates actually possess anisotropic properties [4]. For example, the far-field radiation of a Hertzian dipole with an arbitrary orientation has been studied in the presence of layered anisotropic dielectrics []. Subsequently the numerical aspects relating to the convergence of numerical solutions for electromagnetic waves in stratified bianisotropic media was investigated [2]. But, in much of the research associated with anisotropic material, the dipole is Hertzian or the current distribution is assumed. Many times these current assumptions are not sufficient [5]. This work investigates several problems for a printed dipole of arbitrary length embedded in layered anisotropic dielectrics. In particular, the current distribution is determined for a thinwire printed dipole excited with a delta source. It is shown how each component of the permittivity in the anisotropic layers directly affects the current on the embedded printed dipole. Knowing this component information is essential when designing resonant-type printed antennas on manufactured microwave substrates [4], [4]. Precise knowledge of the current distribution on a printed dipole can be very important. For example, because of the narrow bandwidth of a printed dipole, it is very important to be able to model the dipole and the surrounding environment with great accuracy to ensure the dipole performs well in Fig.. d3 d2 d2 d Anisotropic layer 2 Anisotropic layer µ, [ε2]ε µ, [ε]ε A printed antenna embedded in N+2 anisotropic layers. the intended application. This is especially important for the following applications: avoiding scan blindness in arrays [6], radio frequency identification [7] and intentional radiators in electromagnetic compatibility [8]. II. THE SPECTRAL DOMAIN IMMITTANCE FUNCTIONS Consider the general problem shown in Fig.. The structure consists of N+2 layers of anisotropic dielectrics with a printed antenna on the third layer. It is assumed that each layer has an optical axis in the y-direction and the k th layer has a thickness of d k, a permeability of µ and is uniaxially anisotropic with permittivity [ε k ] = ε k2 ε k ε k2 ε. () The Hertz vector potentials, Πe and Π h [4]-[5], are introduced to solve for the fields in each anisotropic layer and the region above the top layer. Π e is denoted as the electric Hertz potential, and Π h is denoted as the magnetic Hertz potential. Since the optical axis is chosen to be in the y-direction, the /9/$26. 29IEEE 72

2 y-components of Π e and Π h are chosen to yield a TM or TE mode with respect to the optical axis. This then gives and Π e = Π e â y (2) Π h = Π h â y. (3) The electric Hertz potential will result in a TM-to-y solution and the magnetic Hertz potential will result in a TE-to-y solution. Then, the total solution in each region will be the sum of the TM and TE solutions [9]. Next, the fields in the k th region are written in terms of the Hertz vector potentials in the following manner: Ē k = jωµ Π kh (4) H k = jωε Π ke. (5) Moreover, Πe and Π h are solutions, respectively, to the following wave equations [3]-[5]: and 2 Πke + ω 2 µ ε ε k Πke + (ε k ε k2 ) ε k2 2 Πke y 2 = (6) 2 Πkh + ω 2 µ ε ε k2 Πkh =. (7) Furthermore, the Fourier transform is applied to (6) and (7) to greatly simplify the numerical integration. This then leads to the spectral (or transform) domain wave equations [3]-[5] for Π ek and Π hk, the Fourier transforms of Π ek and Π hk, respectively. Then, by using the assumed solutions to the spectral domain wave equations, enforcing the boundary conditions and factoring, final expressions for Π ek and Π hk are obtained in terms of the structural parameters of Fig.. Then Π ek and Π hk are substituted into the spectral domain expressions of (4) and (5) to represent the field in the k th layer. This results in the following spectral domain immittance functions in the k th layer [3]-[5]: Ẽ xk (α, y, β) = J xk Zxxk (8) where J xk represents an x-directed dipole current on layer k and Z xxk are referred to as the spectral domain immittance functions. Z xxk represents the Green s function in the spectral domain. Once the expressions for Zxxk are derived, the spectral domain moment method [2] is used to solve for the unknown current Jxk The steps leading to (8) are quite extensive and are beyond the scope of this paper. A thorough derivation of Z xxk can be found in [3] and [5]. To simplify the problem, spectral domain immittance functions are used to evaluate a three-layer version of Fig.. In particular, the current on a printed dipole embedded in three anisotropic layers is determined. The three-layer printed dipole problem is shown in Fig. 2. The dipole on layer 2 has a length L, width W, is driven with a delta source at the origin and is assumed to be a thin-wire. This then requires only the tangential component of the electric field in the x-direction to be enforced by (8) to satisfy the tangential boundary conditions Fig. 2. y x z Printed Dipole Anisotropic Substrate W L d2 d d3 Anisotropic Superstrate Anisotropic Substrate Ground Plane A printed dipole embedded in three anisotropic layers. (B.C.) on the printed dipole. For this work, Jx will be used to represent a dipole current on layer (k = ) and J x2 will be used to represent a dipole current on layer 2 (k = 2). III. THE SPECTRAL DOMAIN MOMENT METHOD A. Deriving the Matrix Elements As mentioned in the previous section, the spectral domain immittance function Z xx (for this discussion the subscript k is dropped from Z xxk in (8)) represents the Green s function in the spectral domain. Therefore, the Green s function in the spatial domain, Z xx, is the inverse Fourier transform of Z xx. The x-component of the electric field in each region can be written in the spatial domain as [2] [ ] E x (x, y, z) = Z xx (x x, z z )J x (x, z ) dx dz. z x (9) Since Z xx in (9) is in the spatial domain, the two-dimensional inverse Fourier transform Φ(x, y, z) = + + Φ(α, y, β)e j(αx+βz) dαdβ. () will need to be applied to Z xx. This will then result in representations of the Green s functions in the spatial domain. Proceeding in this manner, the x-component of the electric field can be written as [ E x (x, y, z) = Z xx (α, β) z x ] e j[α(x x )+β(z z )] dαdβj x (x, z ) dx dz. () Next, the current in terms of basis functions is defined as J x (x, z ) = N I xn r xn (x, z ) (2) n= where I xn is the unknown magnitude of the n th x-component of the surface current. The known expansion functions are 29 SBMO/IEEE MTT-S International Microwave & Optoelectronics Conference (IMOC 29) 73

3 r xn (x, z ). Since Galerkin s method is being used, the weighting functions w xm (x, z) will be defined to be the same as the expansion functions r xn (x, z ). The weighting functions will be indexed by the variable m and are defined at x. Substituting (2) into (), taking the inner product w xm (x, z), E x (x, y, z) and factoring gives where w xm (x, z), E x (x, y, z) = ] [K Zxx (α, β)k 2 dαdβ (3) K = z x w xm (x, z)e jαx e jβz dxdz (4) and K 2 = r xn (x, z )e jαx e jβz dx dz. (5) z x Notice that the following expression is contained in (3): r xn (x, z )e jαx e jβz dx dz. (6) z x This expression is the two-dimensional Fourier transform of the basis function r xn, denoted as r xn. Therefore, if the basis functions are chosen appropriately, an analytical expression for (6) could be derived and substituted directly back into (3). Since the Galerkin s method is being used, this substitution could be done for both the weighting and expansion functions. Continuing in this manner, (3) simplifies to the following: w xm (x, z), E x (x, y, z) = w xm ( α, β) Z xx (α, β) r xn (α, β)dαdβ.(7) The steps leading to (7) are key to implementing the spectral domain immittance functions in this work. Notice that the integration of (7) is entirely in the spectral domain and the electric field values are in the spatial domain. These steps simplified the numerical implementation of the spectral domain immittance functions greatly. Notice that the expressions in (7) do not contain a numerical derivative. This is another great advantage of this technique. B. The Basis Functions ) Spatial domain: The basis functions defined are for the printed dipole in Fig. 2. Since the surfaces in this work are planar, a two-dimensional piecewise sinusoidal (PWS) basis function is chosen. It is assumed that the current on the dipole in Fig. 2 does not vary with respect to the width of the conductor and only varies with respect to the length. Therefore thin-wire assumptions are enforced. This then allows the PWS to be constant with respect to the z-axis and vary sinusoidally with respect to the x-axis. The PWS basis functions are shown in Fig. 3 and are defined as r i (x, z ) = sin[k( x x i+x )] sin(k x) U(z ), x i d x x i ; sin[k( x x +x i)] sin(k x) U(z ), x i < x x i + d;, otherwise (8) xi- xx xix xi+ x x x Fig. 3. Piecewise sinusoidal basis functions used in the numerical computations. +k Region of polar integration poles (from pole (at the origin) basis functions) pole (from immittance functions) -k Region of rectangular integration (grid) Region of rectangular integration (grid) k α k β x Integration is not performed on the dotted lines to avoid the singularity associated with the pole Fig. 4. The numerical integration mesh definitions (polar and rectangular) and the location of the poles on the α β plane. where U(z ) is a unit pulse defined over the width of the conductor (i.e., in the z-direction) and zero otherwise. Equation (8) will be used to represent the unknown surface current Js in the dipole problems. 2) Spectral domain: In this section the two-dimensional Fourier transform Φ(α, y, β) = + + Φ(x, y, z)e j(αx+βz) dxdz (9) will be applied to (8) to determine the basis function in the transform domain. This gives ( 2 r i (α, β) = sin(k x) k + α + ) e jαx i k α ( ) ( ) x(k + α) x(k α) sin sin 2 2 z sin(β z/2) jβzi e e jβ z/2 (2) zβ/2 where z (i.e., the width of the dipole) has been substituted for W to generalize the expression. A closed form solution for the basis functions in the transform domain has now been derived. This can now be substituted directly into (7) and then be used to represent the unknown current in the spectral or transform domain. Notice that the last expression in (2) is in the form of a sinc function and needs to be defined at the origin. Also, notice that a singularity exists at α = ±k. The two-dimensional numerical integration (Fig. 4) will have to be modified to avoid these poles. 29 SBMO/IEEE MTT-S International Microwave & Optoelectronics Conference (IMOC 29) 74

4 Re(I) (ma) =3.25, ε x =3.25 [5], ε x =3.4, ε x, ε x =3.4 Re(I) (ma) , ε x2, ε y3, ε x3 =9.4, ε x2, ε y3, ε x3 =9.4, ε x2 =3.4, ε y3, ε x3 ε, ε, ε =9.4, ε ε, ε, ε =9.4, ε = Fig. 5. Re(I) of the.5 dipole on a single anisotropic substrate. Fig. 7. Re(I) of the.25 dipole embedded in three anisotropic layers with ε = =3.25, ε x =3.25 [5], ε x =3.4, ε x, ε x = ε, ε, ε, ε =9.4, ε x2, ε y3, ε x3 ε =9.4, ε =3.4, ε, ε ε, ε, ε =9.4, ε Im(I) (ma) Im(I) (ma) 2 5, ε x2, ε y3 =9.4, ε x3 =3.4.5 Fig Im(I) of the.5 dipole on a single anisotropic substrate Fig. 8. Im(I) of the.25 dipole embedded in three anisotropic layers with ε = IV. NUMERICAL RESULTS A. Single-Layer Results In the first problem the complex current distribution, I as a function of x, was computed for the printed dipole in Fig. 2. In this problem the dipole was placed on a single anisotropic substrate (i.e., d 2 = d 3 = ). The dipole was defined to have a length of L =.5 and a width of W =.4. The thickness of the substrate was set at.6 and the dipole was driven at the center with a -V delta source. The calculated current distribution is shown in Figs. 5 and 6 for various values of [ε ]. The isotropic results for ε x = = 3.25 were compared to commercial software and the results published by Rana and Alexopoulos [5] and reasonably good agreement was observed with the exception of the currents near the port. The real part of the current reported in [5] and the values computed in this paper for the isotropic case in Fig. 5 do not agree quite as well as the rest of the results. The difference between the values are approximately 9% or about 8 µa. This discrepancy has also been observed by others who have reported the difficulty of modeling the current at or near the delta source location [2]-[22]. Several comments can be made about the results in Figs. 5 and 6. ) When the isotropic substrate was increased from 3.25 to 5.2, the imaginary part of the current at the feedpoint approached zero. This corresponds to the dipole approaching resonance for larger values of permittivity. 2) Both the real and imaginary parts of the current are reduced by increasing the permittivity in the direction of the optical axis (i.e., the component). This is because the dominant field in the substrate is the T M mode, which has a y- component in the anisotropic substrate. 3) When the permittivity in the direction of the optical axis was reduced such that < ε x, the real part of the current was almost unchanged (less than.5 ma) while the imaginary part of the current was increased by approximately.5 ma. This shows that if it is desirable to individually modify the 29 SBMO/IEEE MTT-S International Microwave & Optoelectronics Conference (IMOC 29) 75

5 imaginary part of the current on the dipole and not the real part, a designer can do this by choosing different values of. The same cannot be said about changing the component of the permittivity orthogonal to the optical axis (i.e., ε x ). B. Triple-Layer Results The next study considered the current distribution of the printed dipole embedded in the three-layer structure shown in Fig. 2. The dipole on layer 2 was defined to have a length of L =.25 and a width of W =.83. The thickness of each layer was set at.26 (a 6 mil thickness at 5 MHz), and the permittivity of layer was defined at ε = 3.25 (isotropic). The dipole was driven at the center with a -V delta source, and the values of [ε 2 ] and [ε 3 ] were varied. The current distribution from these calculations is shown in Figs. 7 and 8. Several comments can be made about the results in Figs. 7 and 8. ) Both the real and imaginary parts of the current are reduced by increasing the permittivity in the direction of the optical axis in the layer immediately below the dipole (i.e., the component). The reason for this was discussed in the singlelayer results section above. 2) Both the real and imaginary parts of the current were reduced when the permittivity in the direction of the optical axis in the anisotropic superstrate (i.e., ε y3 ) was increased. 3) When the permittivity orthogonal to the optical axis in the anisotropic superstrate (i.e., ε x3 ) was reduced, both the real and imaginary parts of the current were significantly increased. This is because along the anisotropic boundary containing the printed dipole, the components of the electric field are orthogonal to the optical axis [5]. V. GENERAL COMMENTS In general, it is difficult to make broad comments that apply to printed antennas in various structures and applications. But, several key comments can be made about the results in Figs. 5 through 8. ) The currents are intimately related to the individual components of the anisotropic permittivity of the dielectric layers both above and below the dipole. This is because the T M mode is dominant in the anisotropic layers around the printed dipole. The T M mode has an electric field component in the direction of the optical axis (i.e., the y-direction) in the layers below the printed dipole and a component orthogonal to the optical axis in the layers above dipole. 2) When the values of permittivity are increased in the direction of the optical axis in the layers below the printed dipole, the imaginary part of the current at the feedpoint approached zero. This corresponded to the dipole approaching resonance. 3) When the value of permittivity in the direction orthogonal to the optical axis in the layer above the printed dipole is reduced, the imaginary part of the current is increased. These characteristics agree with the results observed in [5] and should be considered during the design process. VI. CONCLUSION The current distribution of a printed dipole with arbitrary length embedded in several layers of anisotropic dielectrics was evaluated using the spectral domain immittance functions. In particular, the currents on a printed dipole on a single anisotropic layer and in three anisotropic layers were determined. Both cases were discussed and good agreement between the isotropic cases, commercial software and published literature was observed. Finally, the effects of each component of the anisotropic permittivity on the current distribution of the dipole were noted. These results demonstrate that the anisotropic properties of manufactured microwave substrates should be considered in the design process, especially for resonant type antennas. REFERENCES [] A. Eroglu and J. K. Lee, Far field radiation from an arbitrarily orientated hertzian dipole in the presence of layered anisotropic medium, IEEE Trans. Antennas Propag., vol. 53, no. 2, pp , December 25. [2] J. Ning and E. L. Tan, Hybrid matrix method for stable analysis of electromagnetic waves in stratified bianisotropic media, IEEE Microw. Wireless Comp. Lett., vol. 8, no., pp , October 28. [3] B. D. Braaten, Modeling Multiple Printed Antennas Embedded in Stratified Uniaxial Anisotropic Dielectrics, Ph.D. dissertation, North Dakota State University, Fargo, ND, 29. [4] B. D. Braaten, R. M. Nelson and D. A. 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6 [7] B. D. Braaten, G. J. Owen, D. Vaselaar, R. M. Nelson, C. Bauer-Reich, J. Glower, B. Morlock, M. Reich and A. Reinholz, A Printed Rampartline antenna with a dielectric superstrate for UHF RFID applications, IEEE Int. Conf. on RFID, The Venetian, Las Vegas, NV, April 6-7, 28. [8] C. R. Paul, Introduction to electromagnetic compatibility, 2nd. ed., John Wiley and Sons, Inc., Hoboken, NJ, 26. [9] R. E. Collin, Field Theory of Guided Waves, 2nd ed., IEEE Press-John Wiley and Sons, Inc., New York, 99. [2] D. B. Davidson and J. T. Aberle, An introduction to the spectral domain method-of-moments formulations, IEEE Antennas Propag. Mag., vol. 46, no. 3, pp. -9, June 24. [2] T.K. Sarkar, A study of the various methods for computing electromagnetic field utilizing thin wire integral equations, Radio Sci., vol. 8, pp , 983. [22] G.P. Junker, A.A. Kishk, and A.W. Glisson, A novel delta gap source model for center fed cylindrical dipoles, IEEE Trans. Antennas Propag., vol. 43, no. 5, pp , May SBMO/IEEE MTT-S International Microwave & Optoelectronics Conference (IMOC 29) 77