A Model for Predicting the Reflection Coefficient for Hollow Pyramidal Absorbers

Transcription

1 A Model for Predicting the Reflection Coefficient for Hollow Pyramidal Absorbers Christopher L. Holloway nstitute for Telecommunication Sciences/NTA 325 Broadway, Boulder, CO 80303, USA (303) , Martin Johansson Core Unit Antenna Technologies Ericsson Microwave Systems SE MGlndal, Sweden Edward F. Kuester Dept. of Electrical and Computer Engineering University of Colorado Boulder, CO Robert T. Johnk and David R. Novotny RF Fields Division National nstitute of Standards and Technology Boulder, CO Abstract - n this paper, we present an expression for the effective material properties of electromagnetic hollow pyramidal absorbers. This expression can be used to efficiently calculate the reflection coefficient of this absorbing structure. The reflection coefficients based on this model have been compared to and closely agree with calculated finite element results. We show how the reflection coefficient varies as a function of the absorber wall thickness. Measurement data for a hollow absorber obtained using time-domain measurement techniques are also presented. NTRODUCTON n recent years there has been an increase in the demand for low frequency ( MHz) electromagnetic absorbers [l]. This has spawned an emergence of new absorber designs and structures; three examples are alternating wedges, multiple layer ferrite tiles, and hollow pyramids [l, 21. The hollow pyramidal absorber, in its simplest form, consists of four planar slabs of absorbing material joined together to make a pyramid with constant wall thickness [2] (see Figure 1). Compared to the solid pyramid, the hollow pyramid offers an additional degree of freedom. This can be advantageous when optimizing the (broadband) performance of an absorber geometry. The reflection coefficient of these structures can be obtained by full numerical means; however, while this approach is accurate, it can be very time consuming. n previous work [l, 3, 41, it was shown that when an incident wave encounters an absorber with electrically small period (an absorber whose period is small compared to the wavelength in the material), it does not see the fine structure of the hollow absorber. nstead, the incident wave behaves as though it has encountered a solid medium whose effective conductivity and permittivity vary only with the depth z in the medium. These effective material properties~ are different than the conductivity and permittivity of the actual material used to construct the absorber. References [l, 3, 41 demonstrate that for this case of an electrically small period, the zeroth order average fields are related by and v x (E ),, = -3-w [P ] * (H ),, v x (T ) a g =.iw [Eh] - (E%g - These equations state that the average fields satisfy Maxwell s equations in an anisotropic inhomogeneous medium characterized by the tensors [ E 1 and [ph]- These effective material properties are referred to as the homogenized permittivity [eh] and permeability Ph- Since the average fields treat the periodic absorb: ing medium as an effective anisotropic inhomogeneous region with tensor permittivity and permeability [eh] and [p ], the reflection coefficient of the composite (or homogenized) structures can be efficiently obtained with either classical layered media approaches or by classical transmission line methods [4]. The inhomogeneity in z results from the variation in the geometry of the absorbing structures in Z. There has been a great deal of attention in the past toward determining the effective properties of com- 861

2 posite regions and a survey of this work can be found for the static fields given in [3] for a periodic hoi: in [5]. Expressions for the effective material proper- low absorber to obtain the numerical exact effecties of some of the more common types of absorbing tive material properties. Using these exact results, structures can be found in [l, 31. n this paper we we investigated different approximate expressions to present expressions for the effective material proper- determine the best equation to represent the effecties of a periodic structure (or cell) that resembles the tive material properties of the hollow absorbing struccross-section of hollow pyramid absorbers. tures. t is shown in [9] that the effective properties for EFFECTVE MATERAL PROPERTES this structure can be approximated by an empirically adjusted version of the Lichtenecker upper bound Eq. A cross-sectional view of the hollow absorbing struc- G% ture is shown in Figure lb. n [3], it is shown that Et N WP l--2(t+d)lp [ &+2d &--il /p E^l the effective material properties of a periodic cell (or Wp -1 (3) structure) are governed by the solution of a static +- c1+2(t+d) E^z-Zl ) p 1 ' fields problem. For any arbitrary-period cell, such a 21 and 22 are given by static solution cannot be obtained analytically, and must therefore be solved numerically. The effective properties can also be approximated E^r = EL [Q-J(2-5ds + 5di - dg) from one of the following four methods (which furnish upper or lower bounds when the materials are loss- -EL (do - d: + d;) less [5]). For transversely isotropic lattices, the unit cell of the doubly-periodic structure contains symme- -do,/a2 (el2 + eu2) - 2 B el cu tries so that E,, = eyy - Et and ezy = eyz = 0. The 1 Hashin-Shtrikman (HS) upper and lower bounds [6] are the best obtainable bounds using only the mate- do)[w(l- do)- ~~dol} rial parameters EO and E, and the fill factor g (volume A fraction of space occupied by the bulk material 6,). e2 = q, [QJ (1-2du + 2d; - d;) They bound the value of the (real) transverse permittivity according to cgs 5 et 5 cgs, where - cl (1 - ad, + 2d; + dz) +(l - do),/a2(cl2 + cu2) - 2 B cl cu 1 / Wok4 - do)- EL~O]}, where do = 2dlp, The Lichtenecker (Li) bounds [7] depend on the B = 1-2do + d; + 2df, - d;, specific geometry of the unit cell. They have the form and E& 5.st 5 ekii, where A=l-do+d;. ki n this expression Q-J and EL are Hashin-Shtrikman upper and lower bounds (Eq. (1)) evaluated at the maximum and minimum fill factor of the hollow ab- (2) sorber for a given value of d; that is, where p is the period of the absorber array. Using a numerical waveguide simulator model, based on the finite element (FE) method [8], we solved EL E ~d=o (5) EU F t=p/2--d 862

3 where the fill factor is given by g = 4 (2td f d2) /p. RESULTS Figures 2 and 3 show the magnitude and phase of Et (Eq. (3)) f or a hollow pyramidal absorber with E, = (4 - j4)~c for three values of d. This expression for Et agrees very well with the FE data and the maximum relative error is less than 3 %. This expression is much more accurate than the alternative approximation given in [lo]. Using et as the effective material parameter, we calculated the reflection coefficient of a hollow pyramidal absorber using the effective layer model presented in [l, 3, 41. Figure 4 compares these results with FE results for the actual three-dimensional hollow pyramidal geometry. The accuracy is good, with a relative error of less than 3 % for p < X/2. For higher frequencies (x/2 < p < X) the agreement is quite good, but the lack of higher-order modes in the effective layer model eventually makes the error large. t is interesting to observe how the effective properties vary as the wave propagates into the hollow absorber. Figures 5 and 6 show the real and imaginary parts of ct obtained from Eq. (3) as a function of the depth in the absorber. The results in this figure are for an absorber with a taper length of 1 m, p = 30 cm, E, = 2 - j2, and various values of d. The effective properties are very close to those of free space for the first portion of the absorber. Similar results are shown for other types of pyramidal shaped absorbers. This indicates that the very tips of the absorber have little effect on the overall low frequency performance. This is the justification for cutting off the tips of absorber, as is in some of the commercially available absorbers. The effect of varying the absorber thickness d for the hybrid absorber (hollow absorber combined with ferrite tiles, see Figure 7) is illustrated in Figures 8 and 9. These figures show results for the reflection coefficient of a hybrid with various layers (whose dimensions are given in Figure 7) and various values of d for two different carbon loadings in the bulk cone absorber material. These figures also show results for the ferrite tile alone. These figures show that the solid absorber greatly improves the high frequency performance over that of the ferrite tile alone; however, it does so at the expense of the low frequency performance. As the thickness of the hollow absorber decreases, the low frequency performance improves at the expense of that at high frequencies. Also notice that as the thickness becomes smaller and smaller the performance of the hybrid approaches the performance of the ferrite tile. These results show that it is possible to improve the low frequency response of the absorber with certain values of d, however, at the price of the high frequency performance. This illustrates that compared to the solid pyramid, the hollow pyramid offers an additional degree of freedom. This can be advantageous when optimizing the (broadband) performance of an absorber geometry. t should be emphasized that the results presented here are based on commercially available materials [l]. f the material properties of both the ferrite and urethane absorber are varied, along with the geometries (i.e., dimensions) of the different absorber configurations, then optimally performing absorbers can be achieved. Finally, using time-domain measurement techniques developed at NST [ll]-[14], measurements were performed on a thin wall hollow absorber. Figure 10 shows the measured backscatter coefficient of the hollow absorber backed with a ferrite tile. The figure also shows the results of a ferrite tile alone. Unfortunately it was not possible to compare the measured data to the model presented here because the material properties of the hollow absorber and ferrite tile were not known. However, a qualitative comparison can be made. Notice that these results behave in the same manner as those in Figures 8 and 9. That is, an increased reflection in a range of frequencies is depicted in exchange for a lower maximum reflection over the entire band. CONCLUSON n this paper, we have presented expressions for the effective material properties of hollow pyramidal absorbers. The expressions given here have been compared to and closely agree with calculated finite element results. These expressions can be used as material properties in effective layer models to calculate efficiently the reflection coefficient of hollow pyramidal absorbers. We show how the reflection coefficient varies as a function of the absorber wall thickness d. 863

4 t needs to be stressed that the model presented here is valid as long as the wavelength in the material is small compared to the spatial period of the cones. Once either the frequency of operation or the material properties of the absorber become too high the model presented here must be modified. Hence, this model may not be valid for some of the newer hollow absorbers constructed of thin resistive sheets. We are presently investigating methods to improve the model for these situations. We also are developing time-domain operators that will allow the effective material properties of the hollow absorber Eq. (3) to be implemented into the finite-difference time-domain model presented in [15]. a) period cell: g = (4ad-4d *)/p * REFERENCES P P 131 [41 [ P PO1 [111 W P31 P41 P51 C.L. Holloway, R.R. DeLyser, R.F. German, P. McKenna, and M. Kanda, EEE Trans. Electromag. Compat., P(l, pp , Marly, B. Baekelandt, D. De Zutter, and H.F. Pues, EEE Trans. Antenna Propagat., 43(H), , E.F. Kuester and C.L. Holloway, EEE Trans. Electromag. Compat., 36(4), , C.L. Holloway and E.F. Kuester, EEE Trans. Electromag. Compat., 36(4), , E.F. Kuester and C.L. Holloway, EEE Trans. Microwave Theory Tech., 38(11), , Z. Hashin and S. Shtrikman, J. Appl. Phys., 33, , K. Lichtenecker, Phys. Zeits., 25, , N.M. Johansson and J.R. Sanford, in Proc EEE Antennas Propagat. Symp., Seattle, WA, M. Johansson, C.L. Holloway, and E.F. Kuester, Effective electromagnetic properties of honeycomb composites and hollow pyramidal and alternating wedge absorbers, submitted to EEE Trans. Antennas and Propagat., C.L. Holloway and M. Johansson, in Proc EEE Antennas Propagat. Symp., Montreal, Canada, , R.T. Johnk, A.R. Ondrejka, H.W. Medley, Natl. nst. Stand. Technol., Tech. Note, to be published. R.T. Johnk and A. Ondrejka, in Proc. of the EEE 1997 ntern, Symp. on Electromagnetic Compatibility, Austin, TX, pp R.T. Johnk, A.R. Ondrejka, H.W. Medley, in Proc. of the EEE 1998 ntern, Symp. on Electromagnetic Compatibility, Denver, CO, pp R.T. Johnk, A.R. Ondrejka, C.L. Holloway, in Proc. of the EEE 1998 ntern, Symp. on Electromagnetic Compatibility, Denver, CO, pp C.L. Holloway, P. McKenna, and D.A. Steffen, in Proc. of the EEE 1997 ntern, Symp. on Electromagnetic Compatibility, Austin, TX, pp b) Figure 1: a) Hollow pyramid sectional view. absorber and b) cross HS upper Figure 2: Magnitude of the effective transverse permittivity Q] for a hollow pyramidal structure for three wall thicknesses: 2d/p = 0.1, 0.2, and 0.4; E, = (4 - j4)eo. 864

5 ---- d=gcn, ---- d=,a,, +!r d=5cm g Figure 3: Phase of the effective transverse permittivity Let for a hollow pyramidal structure for three wall thicknesses: 2d/p = 0.1, 0.2, and 0.4; ea = (4 - j4)eu. 0.W oo z b-0 Figure 5: Real part of ct for an absorber with a taper length of 1 m, p = 30 cm, ea = 2 - j2, and various values of d E g.a, tii s cz.g C a ui- E d = p/ d=gcm ---- d=i m -+r d=5cm + d=3cm @- d=lcm loo.m) Frequency Figure 4: Magnitude of reflection coefficient for a hollow pyramidal absorber; plane wave incidence along normal to lattice plane: p = 30cm, d = 2 cm, height = 95 cm, base thickness = 5 cm, and E, = (2 - j2)qj. (MHz) -2.0 t, z (m) Figure 6: maginary part of ct for an absorber with a taper length of 1 m, p = 30 cm, E, = 2 - j2, and various values of d. 865

6 95cm 2.54cm 2.54cm 6mm -1.27cm -E a, -20 D E s g d=,m d=,cx,, ferrite tile only loo0 Frequency (MHz) Figure 7: Geometry of a hybrid hollow absorber. Figure 9: Reflection coefficient for a hybrid with dimensions given in Figure 7 for a 10 % carbon loading, see [l]. - d = ~ d=,c,,, + d=5cm + d=zcm ff d=ocm ---Lf ferrite tile only hollow absorber backed by ferrite tile Frequency (MHz) Figure 8: Reflection coefficient for a hybrid with dimensions given in Figure 7 for a 7 % carbon loading, see [l]. 100 Frequency (MHz) Figure 10: Measured backscatter coefficient for a hollow absorber. 866