035675-2-T USER'S MANUAL TEST DESCRIPTION AND REPORT FOR MOMJET/AIMJET CODES H. Anastassiu, M. Smelyanskiy, D. Filipovic, J. Volakis Wright Laboratory WL/AACT Bldg. 23 WPAFB, OH 45433 January 1998 35675-2-T = RL-2496

USER'S MANUAL TEST DESCRIPTION TEST REPORT FOR THE MOMJET/AIMJET CODES CONTRACT NO. F33615-97-C-1000 DEMACO Subcontract Agreement No.1300 Prepared for: Wright Laboratory WL/AACT Bldg.23 WPA FB, OH 45433-700 Prepared by: H. T. Anastassiu, M. Smelyanski, D. S. Filipovic, J. L. Volakis Department of Electrical Engineering and Computer Science University of Michigan 1301 Beal Avenue Ann Arbor MI 48109-2122 volakis engin.umich.edu January 15, 1998 1

Contents 1 Scope 5 1.1 Identification.................................. 5 1.2 System overview..................................... 5 1.3 Document overview................................... 5 2 References 6 3 Execution procedures 8 3.1 Code structure.............................. 8 3.2 User inputs................................. 12 3.3 Initalization.................................15 3.4 Compilation and running............................16 4 List of subroutines 17 4.1 Initialization codes:............................... 17 4.2 Subroutines in makefile:............................17 4.3 A uxiliary files:................................... 19 4.4 Output files:....................................... 19 4.5 Notes - theory...................................... 22 5 Test example and initialization 27 5.1 Test exam ple.................................. 27 5.2 Initalization files.................................. 27 6 Software input 29 6.1 Interactive input.................................. 29 6.2 Input file.......................................... 29 7 Software output 31 7.1 Screen output.................................. 31 2

7.2 Output files.................................................32 3

List of Figures 1 2 3 4 5 6 7 8 9 10 11 Cylindrical inlet terminated by a cylindrical hub........ A quarter of a hub geometry................... A quarter of a blade geometry...................... Decomposition of the computational domain............ Mesh with node, edge and triangle numbering.......... Sampling points on the hand-off plane.............. Simplified model of a jet engine.................. RCS for both polarizations.................... Modulation pattern for rotating blades-phi polarization..... Modulation pattern for rotating blades-theta polarization... E and H1 fields on the hand off plane................ 9..:10........ 11....... 12....... t4...........14....... 21...... 23........ 34....... 35........ 36....... I41 4

1 Scope 1.1 Identification Momjet is based on a free space Moment Method code written by Pamela Haddad at the Radiation Laboratory in 1991. Momjet code version 1.0, was written at the end of 1996 begining 1997 by H.T. Anasatssiu, modified in summer 1997 by D.S. Filipovic (added some output calculations and I/O features). List of abbreviatons: MoM- Method of Moments; RCS- Radar Cross Section; CPU- Central Processing Unit; DBOR- Dicrete Body of Revolution; FEM- Finite Element Method; PEC- Perfect Electric Conductor; EL- Elevation; AZ- Azimuth; VV- Vertical/Vertical polarization; HH- Horizontal/Horizontal polarization. 1.2 System overview Momjet is a FORTRAN77 code which uses the Moment Method (MoM) to model electromagnetic scattering from perfectly conducting jet engine models, at both, far-field and near-field of the engine. 1.3 Document overview Section 3.1 describes the execution procedure and code operation. Section 4.5 of this manual gives a brief description of MoM applied to the engine problem, and section 5 presents a few examples of Radar Cross Section and modulation calculations. 5

2 References References [1] H.T. Anastassiu, Electromagnetic Scattering from Jet Engine Inlets Using Analytical and Fast Integral Equation Methods, Ph. D. Thesis, University of Michigan, 1997. [2] D. C. Ross, J. L. 'Volakis and H. T. Anastassiu, "Hybrid Finite Element-Modal analysis of jet engine inlet scattering" IEEE Trans. Antennas and Propagation, vol. 43, no. 3, pp. 277-285, March 1995. [3] D. C. Ross, J. L. Volakis and H. T. Anastassiu, "Overlapping modal and geometric symmetries for computing jet engine inlet scattering" IEEE Trans. Antennas and Propagation, vol. 43, no. 10, pp. 1159-1163, Oct. 1995. [4] H. T. Anastassiu, J. L. Volakis and D. C. Ross, "The Mode Matching Technique for electromagnetic scattering by cylindrical waveguides with canonical terminations" Journal of Electromagnetic Waves and Applications, vol. 9, no. 11/12, pp. 1363-1391, Nov./Dec. 1995. [5] C. Ross, J. L. Volakis and H. T. Anastassiu, "Efficient Computation of Radar Scattering Modulation from Jet Engines" Radio Science, vol. 31, no. 4, pp. 991-997, July-Aug., 1996. 6

USERS MANUAL

3 Execution procedures 3.1 Code structure Momjet models the scattering from metallic cylindrical jet engine inlets terminated by an arbitrary, PEC, cylindrically periodic structure. A simplified geometry is illustrated in Fig. 1, where the termination is a cylindrical hub. A plane wave is assumed to illuminate the open end and couple to waveguide modes in Region 1, propagating towards the termination (Region 2). Each of these modes is used as an excitation for the MoM applied to one periodic sector of the termination (see Figs. 2, 3). Solving the MoM linear system yields the equivalent currents on the termination, and in turn the scattered field within the inlet. To extract the coefficients of the scattered modes propagating towards the open end an integration [1, 2] is performe d at a hand-off plane, which is perpendicular to the inlet axis and lies at a fraction of a wavelength in front of the termination (see Fig. 4). Each of the scattered modes propagates to the open end and radiates in free space. The overall modal contribution yields the Radar Cross Section (RCS) of the engine. To exploit the inherent periodicity of the engine, the waveguide modes must have exponential, and not trigonometric dependence on (P. Immediate consequence of this is that the modal orders may be negative, positive or zero, and there is no distinction between "odd" and "even" behavior, as opposed to the Mode Matching code that has been developed in the past [4]. Additionally, this representation enables calculation of RCS modulation (caused by blades rotation) in a simple manner as a postprocessing job [5]. 8

y z=0. Figure 1: Cylindrical inlet terminated by a cylindrical hub 9

i Figure 2: A quarter of a hub geometry. 10

"/ i Figure 3: A quarter of a blade geometry. 11

Modal or Ray region Fields to Ray/Modes Integration Surface (Hand-off Surface) mmw Z Z:Z:: Z Z:i:1:: 1:i: ii. 1 Z iii i;;11......................iiii?..........iiiiiiiiiii,-Engine mi Numerical or Rigorous Modeling Figure 4: Decomposition of the computational domain. 3.2 User inputs The input file comprises of a discretized sector of the termination, including the waveguide wall up to the hand-off plane. Typical input geometries are illustrated in Figs. 2, 3. The current version of Momjet requires an edge-based format of the geometry file (interior nodes and corresponding edges are listed first). To illustrate the file format we consider a 0.2A x 0.2A flat plate (see Fig. 5), lying on the xy plane and centered at the origin. This geometry is not typical for the code, but its simplicity is exploited to clarify the meshing format of the input file. The geometry file has the form 12

98.1.1.0.1 -.1.0 -.1.1.0 -.1 -.1.0.1.0.0.0.1.0 -.1.0.0.0 -.1.0.0.0.0 16 8 7 8 56 59 69 39 79 8 9 29 4 8 47 16 15 37 36 2 8 25 1 9 10 2 1112 234 5 6 13 4 5 14 1 6 7 8 15 7 16 8 3 (Nnodes = 9, Ntriangles = 8) (the next Nnodes entries are the Cartesian coordinates of the nodesinner nodes first followed by outer (total number of edges Nedges) (number of exterior edges Next) (next Nedges- Next entries give the pair of nodes defining the edges) (the remaining entries give the triangle connectivity. Each line entry contains 3 numbers corresponding to the edge numbers forming the triangle in a clockwise order) 13

N3 N6 N1 E14 Ell T5 T2 E5 E2 E13 E4 E12 T4 T3 -. N7 N9 ' N5 T6 'T8 E1 E8 E10... E7. E16 T1 i T7 N4 N8 N2 -.,, Figure.............. 5: Mesh with...........node edge and triangle numbering. Figure 5: Mesh with node, edge and triangle numbering. 14

To obtain this format: * Create the mesh geometry using a meshing package such as IDEAS and export it to a universal file called, for instance, geo-file.unv. * Convert geo-file.unv to a converter file, called cnv, by running program u2c-new2.f. * Convert cnv to the desired format geo-file by running program c2p-fas t.f. Both u2cnew2.f and c2p-fast.f are available in the same directory with the main code. Momjet also prompts user for outer radius (in terms of A), inlet length up to the hand-off plane (in terms of A), measurement frequency ( choose 0.3d0 if want RCS in dB/(A2)). Inclusion of Ufimtsef currents and evanescent modes is optional. All input parameters may be specified in the input file. 3.3 Initalization Prior to running the code, several parameters must be set appropriately: * In module dinm.inc set maxnod greater or equal to the number of nodes in geo-file. * In module param.inc set *in=number of nonnegative modal orders in hollow section (Region 1). * ii=number of modes per order. Chosen to include at least all propagating modes. * ngas=number of points per subinterval for hand-off plane integration. * nsym=number of periodic sectors (slices) of engine termination. * nangle=number of different incidence angles (0ic). * bgas=Gaussian integration tolerance (at the hand-off plane). * exrho=2intrho=nunmber of integration subintervals along the p directi on. * exphi=2intPhi=number of integration subintervals along the p directi on. * zhandoff=z location of the hand-off plane (the tip of the termination is assumed at z = 0. *irotmax=max rotational angle ( (sugested integer value). 15

*irotst=-rotation step (suggested integer value). *irott=number of rotation steps. *kstepp=0 step for RCS calculation. *kfica=number of sampling points on the hand-off plane for unique engine ID repr esentation. *dejo=l: near-field on the hand-off plane; 2: near-field on the inlet mouth. If one wants field somewhere between, put 1 < dejo < 2, and make it real. In order to define accurate number of propagating modes (in and ii), two runs of the code have to be done, according to following steps: 1) initialize in and ii sufficiently high. 2) uncomment line in the Makefile for compilation without optimization, and comm ent the line with optimization included. 3) compile and run the code. 4) check max in and ii when modes are propagating, stop execution, and change parameters in the param.inc file. 5) uncomment line with included optimization and comment line without optimization. 6) compile and run the code. 3.4 Compilation and running * Compile the code by typing make (there is only one Makefile in the directory). * Run the code by typing momj. The interaction is self-explanatory. See section 5 for further details. If you want to use input parameter redirection, then type momj < input.dat. 16

4 List of subroutines 4.1 Initialization codes: u2c-new2.f: converts geo-file.unv to a converter file, called cnv. c2p-fast.f: converts cnv to the desired format geo-file. meshpr.f: generates some data for auxiliary files (dim.inc, param.inc...). 4.2 Subroutines in makefile: Momjet.f: main code. ac-intcyl2.f: performs a double surface integration over two triangles of a function in local area coordinates. an-intl.f: performs an analytical double surface integration over the singular part of the integrand. back-subst.f: solves the MoM linear system using LU decomposition. edge-len.f: updates the edge length table given the edge numbers of the source and observation triangles. excitcyl.f: computes elements of excitation vector (right hand side of linear system). fillzcyl2.f: calculates elements of impedance matrix Z corresponding to incident modal order ninc get-R.f: reads in triangle resistivity values from file specified by user. Accor ding to Pamela Haddad, author of the basic MoM code, the algorithm has not been validate d for non-metallic scatterers. get-meshl.f: reads in mesh data from file specified by user. get-verts.f: returns the vertices of a triangle. normal.f: constructs normal vector to a triangle. ops.f: various small functions and subroutines performing auxiliary mathematical calculations. 17

pot-ints-ss.f: computes the potential integrals for uniform and linearly varying surface sources distributed on a planar polygon S. Dependent subroutine in the same module: i-params. pwcyl5.f: returns the value of (r - rm) E., where E is the incident cylindrical mode and rm is the position vector to the mTth vertex of the observation triangle. Dependent subroutines in the same m odule: integration, discreteap, fieldap, vectintcyl, ffargcyl, dyadfree, cGsldx, GAULEGX, cGsldy, GAIULEGY. See code listing for more information on the function of each subroutine. resfnc.f: returns the value of the integrand of the resistive term (r - rm) (r- rn), where rm, rn are the position vectors to the mth/nth vertices of the given observatio n triangle. xsintcyl.f: performs a single surface integration of a function over a triangle in local area coordinates. solve2.f: linear system solvers. update-signs.f: updates the triangle sign and edge use tables for an input observation interior edge and an input source interior edge. usr-datacyl.f: prompts user for, and reads in, data pertaining to file names, incident field, rcs observation cut via standard I/O. vectint.f: performs a single surface integration of vector function over a trian gle. cylinsubr2.f: includes several subroutines and functions related to the modal analysis of the hollow section (duct). Subroutines included: excitvec, radcrosec, vemult, modest. See code listing for more information on th e function of each subroutine. blockdata.f: initializes common block which contains the local area coordinate integration points and weights used in all integration routines. 18

4.3 Auxiliary files: const.inc: includes various constants. dim.inc: includes uniform array dimensioning. param.inc: includes modal and integration parameters. 4.4 Output files: Program generates different forms of output data, regarding both near field and far field calculation. Some output files are memory consuming (depending on the users specification and configuration under test), so that obtaining all output files upon only one run might be impossible. rcsphi.txt: RCS = RCS(Oinc) for 0 incident polarization. rcsth.txt: RCS == RCS(,inc) for 0 incident polarization. modul.txt: RCS = RCS(Orot)lonc=const modulation file. unwaight.txt: total unweighted scattered electric field on the hand-off plane (I D for every engine termination). efield.txt: total electric field on the hand-off or inlet mouth plane. hfield.txt: total magnetic field on the hand-off or inlet mouth plane. imped.txt: wave impedance on the hand-off or inlet mouth plane. engine.field: output file in the XPATCH format. incrot.txt: scattered parameters on the hand-off or inlet mouth plane (for both polarizations and all incident angles). incrotcomp.txt: complex values of incrot.txt. firot.txt: scattered parameters with modulation on the hand-off or inlet mouth p lane (for both polarizations and all incident angles). firotcomp.txt: complex values of firot.txt. Number and position of sampling points on the hand-off plane is determined by the 19

parameters ngas, exrho and exphi (see param.inc file and Fig. 6). 20

ngas=6,intrho=intphi=2 ngas=3,intrho=intphi=3 v :v ./ / Sc'', - - - - -,17 \`ZI`\ \ \\, \\ // /7// 3' // // \\ \ / / / / / / // \ \ \\\\\ /// I / \ \ \\\\ \\\ i r \ Il I lII/ I I I //// \ \\\ \ I I I I I I / / \ \ \ \I\ 111 \ \\\ \ I I I I I I \ \ \ /// Y \\ I I I,, I I I I \ \ i, \ \ \ \ \ / / I I /II I \\I \\\ \\\ \\ I \ / / // / \ // / I \ \\\ \ s/ I / \\\\ ' / \\\ / \ r \ U \\ // \,I / / \ r/ /I // r/ \\\ \\\ i// /I Figure 6: Sampling points on the hand-off plane. 21

4.5 Notes - theory A major reason for an emphasis on the EM characterization of jet engines is their critical importance in determining the airplane's reflectivity and for target identification. Even though the engine is a small component of the overall structure, its complexity and visibility in the front sector of the aircraft make its contribution a large component of the overall airframe scattering. Moreover, the rotating engine blades cause unavoidable doppler shifts which are important for target identification purposes. The latter can be an important discriminatory for both civilian and military applications and more reliable than other methods. As applied to electromagnetic scattering problems, the Moment Method (MoM) has proven to be a very accurate integral equation technique. However, its brute force application to the jet engine problem is not feasible due to excessive electrical size of the structure. To reduce the CPU time and storage requirements down to manageable levels, the inherent blade periodicity of the jet engine was exploited to show that the computational domain can be reduced down to a single blade or engine "slice". This discrete body of revolution (DBOR) approach reduces the number of unknowns and storage requirements by a factor equal to the number of blades. In [I] we applied the DBOR concept in the context of integral equation methods for modeling the complex jet engine configurations. By considering mode by mode excitation [1, 2, 3] it is shown that the analysis over the entire engine can be reduced down to a surface integral equation over a single blade. It is further demonstrated that the resulting modal scattering matrix is sparse, leading to additional storage and CPU time reductions. To demonstrate the reduction of the computational domain we consider a cylindrically periodic scatterer (Fig. 7). We define the current column vector {J}(m5) of the mrh slice by 22

Figure 7: Simplified model of a jet engine. 23

j}(ms) [- ) r(m) Ms.)] T (l,J [.( 1 > ^ (1) 2 where Ims) are the elementary currents on the msh slice. Brute force MoM yields the linear system [A](O~){J(O)+ [A](1){J}() +... + [A](ONs-){J}(N-1) = {b}(~) [A](10){J}J() + [A](1){J}(1) +... + [A](INs-1) {J(Ns-l) = {b}(1) [A](N"-I'O){J}( +... + [A](N -1IN-){J}(Ns-1) = {b}(N-l) (2) where A(pq jkZ ~ fp(r) R sG(R r R r) R fq (r')Sd2 ' (3) So oSo are the entries of the submatrices [A](nsms) representing the interactions between the fp and fq elementary currents located on the mrh and the nh slices, G is the cylindrical waveguide dyadic Green's function and R is the rotation dyadic [1]. Also, {b}j(") is the excitation column vector of the n~h slice defined by {b)}T) [v ~),..., v] (4) where V(n= Eti (R r)-Rn-. fp (r) dS (5) In the latter equation, E' is the incident field, So is the outer surface of the slice within the volume Vo. Clearly, the index m, indicates the slice where the source point is located, 24

whereas ns is the index representing the slice containing the observation point. Both indices run from 0 to Ns - 1. Following the analysis in [1] it can be shown that it is sufficient to determine only the currents on the basic slice by solving [[A](~~) + [A](Ol)ejS +... + [A]O(N-l)eJ(Vi-)nis] {J}(o) = {b}(O) (6) If the cylindrical waveguide dyadic Green's function is to be used, ( 6) can be further simplified, but that procedure is mainly of theoretical importance; the cylindrical waveguide dyadic Green's function is very difficult to handle computationally, and therefore the free space Green's function is used throughout the code. Additional elementary currents are placed on the waveguide walls to account for the appropriate boundary conditions. An additional result of [1] is that for a given order of the incident mode, only a limited set of scattered modes is excited. Namely, only the scattered modes with orders n that satisfy n - ni + vNs, v Z are reflected back, and this is in agreement with the result given by the FEM analysis of a similar problem [3]. The importance of the periodicity analysis cannot be exaggerated. Since the problem essentially reduces to modeling only one slice of the scatterer, the number of equations or unknowns is reduced by a factor of Ns. For a typical NA = 40, this implies a CPU time and memory reduction each by a factor of 1600. For large scatterers with large periodicity numbers Ns, such as jet engines, the problem can thus be scaled to a tractable size. Moreover, the limited coupling among the scattered modes results in sparse modal scattering matrices which are much easier to store and handle. 25

SOFTWARE TEST DESCRIPTION AND REPORT 26

5 Test example and initialization 5.1 Test example This example is of the 4 blade termination described in Fig. 3. The outer radius is 2A, the hub radius is A, the termination is 0.5A deep, the length of the hollow region is 3A and the hand-off plane is located 0.1A in front of the termination. The number of nodes for this geometry is 656. 5.2 Initalization files the dim.inc file should read Integer nmax,maxnod,maxedg,maxtri,maxz,maxrcs Parameter (nmax=4) Parameter (maxnod=657) Parameter (maxtri=2*maxnod) Parameter (maxedg=3*maxnod) c Parameter (maxz=3*maxnod) Parameter (maxz=-L777) Parameter (maxrcs=200) and the param.inc file should read integer ii,in,iis,ii2,ii4 integer ngas,nsym,nangle,kfica,irott,irotmax,itotst integer intrho,int phi,exrho,exphi,kstepp real*8 bgas,zhandoff parameter(in= 11,ii=4) 27

parameter(iis=inL*ii) parameter(ii2=2 *iis) parameter(ii4=2 'Ii2) parameter(ngas=:3) parameter (nsym= 4) parameter (nangle=121) parameter(bgas=:1.Od-2) parameter(intrho=2) parameter(intphi =2) parameter(exrho 2 **intrho) parameter(exphi =2**intphi) parameter(zhandoffO=. dO) parameter(irotmax=91) parameter(irotst:=1) (irotst=10 for firot.txt and incrot.txt) parameter(irott=:irotmax/irotst + 1) parameter(kstepp=1) (kstepp=2 for firot.txt and incrot.txt) parameter(kfica=:exrho*exphi*ngas*ngas ) parameter(dejo=1) We emphasize that in and ii are chosen so that all propagating modes are included. The code output yields the propagation characteristics of each mode, hence running the code itself for a few test cases eventually provide s the correct values of in and ii. 28

6 Software input 6.1 Interactive input For this geometry the input reads: Enter inlet radius in wavelengths 2.0dO Enter mesh file name: blade-2w_4 Enter length of hollow section in wavelengths 3.0dO Enter 1 for inclusion of rim contribution and 2 for non-inclusion 2!(not including Ufimtsef currents on the rim) Enter 1 for inclusion of evanescent modes and 2 for exclusion 2 Enter measurement frequency in GHz 0.3d0!(to obtain RCS in dB/A2) 6.2 Input file For running the code by using redirection, the input file is: 2.0dO blade_2w-4 3.OdO 29

2 2 0.3d0 30

7 Software output 7.1 Screen output The output reads: TM modes (n,m.krho,prop./evan.) 0 1 1.20241288350811 1 0 2 2.760039056538 1 0 3 4.32686395109136 1 0 4 5.89576721702858 1 1 1 1.91585298512173 1 1 2 3.50779348764358 1 1 3 5.08673402355608 1 1 4 6.661845747896'92 -1 2 1 2.56781132204712 1 mode indices, eigenvalues in A-1, 1 for propagation and -1 for evanescence TE modes (n,m,krho,prop./evan.) 0 1 1.91585298512173 1 0 2 3.50779348764358 1 Incident mode order= -10 Order of system -= 1777 No. of triangular elements = 1216!(working with mode order=-10) 31

Slice 0!(working with slice 0) 7.2 Output files Output file rcsphi.txt is: Phi-phi Polarization Incident angle,phi-phi,phi-theta 1.000000000000000E-02 29.63130309987938 -33.7006615992966.51 29.61263172062007 -33.6934223356198 1.01 29.55830862102566 -33.671398471112 1.51 29.4690832676185 -33.6322516048832 2.01 29.34619755314773 -33.5727299658859 2.51 29.19138128-112484 -33.4894831967613 3.01 29.00683892922878 -33.3799718084723 3.51 28.79522126110019 -33.2432625945877 Output file rcsth.txt is: Theta-theta Polarization Incident angle,theta-theta,theta-phi 1.000000000000000E-02 29.63128937395119 -33.7006771055563.51 29.57655924653133 -33.7333963422247 1.01 29.41605365137127 -33.8246024976252 1.51 29.14821273587283 -33.9607797097266 2.01 28.77040147133607 -34.1182654337324 2.51 28.27883999812705 -34.2625784532682 32

3.01 27.66851579327062 -34.3501517110011 3.51 26.93309717096325 -34.3348619021295 Output file modul.txt (rotation angle,RCS) is: Phi-phi Polarization theta= l.OOOOOOOOOOOOOOOE-02 1 29.63130309269709 2 29.6313030730663 3 29.63130304108263 4 29.63130299690191 5 29.63130294073939 Theta-theta Polarization theta=.OOOOOO00OOOOOOOOE-02 1 29.63128938113347 2 29.63128940076.427 3 29.63128943274794 4 29.63128947692864 5 29.63128953309115 Output file unweight.txt is: X,Y,Z,Unwaited E-Field at the Handoff Plane 5.641662856629827E-02 1.015140670462751E-02.1 81.08614154147517 4.049268616987463E,-02 4.057375264402420E-02.1 80.9713012643719 1.003855320833504E-02 5.643681853501174E-02.1 80.26172131340697 -1.015140671619875E-02 5.641662856421617E-02.1 79.68034874500992 33

Radar Cross Section 35 30 25 20.................... I........................ '\ I^^~ ^ ^"""......... \............/................................................... \. I...........\ ^: / v \: / -X /;\ s~. 0) cr 15S o10 5S - phi-phi polarization - - theta-theta polarization I, 0 0 10 20 30 Incident angle theta 40 50 60 Figure 8: RCS for both polarizations. 34

29 28 Phi-phi modulation. I I...... -=......................................................................................... -..................................................................................................: 27 - 26 25 - C) 0 24 Cc I............. A 23 - - 22 - 21 -\ * * * -:- - * * j-: / \. /.. \. / /..::.. theta=5 -- theta=15 /................ /.:.. ': J 20 theta=25 l II 19' 0 I - I,,, 10 20 30 40 50 Rotation anglE 60 70 80 90 100 Figure 9: Modulation pattern for rotating blades-phi polarization. 35

Theta-theta modulation 30 I I..................................... 28 - 26 24 C) OC CC.........'.: /....\...:. ~/.:/ \' 22 - 20-......... /: 18 - /. 1................... I I I I '* / ' ' \ / theta=5 -- theta=15 theta=25 \.:... \........... 1/ I *I... 1E -, I II I I I. I I I I I 0 10 20 30 40 50 60 Rotation angle 70 80 90 100 Figure 10: Modulation pattern for rotating blades-theta polarization. 36

-4.057375265232938E-02 4.049268616155283E-02.1 78.92981225914005 Output file efield.txt (X, Y, E[dB]) is: Phi polarization Incident angle theta is.0 5.641662856629827E-02 1.015140670462750E-02 -25.6136436860384 4.049268616987463E-02 4.057375264402420E-02 -25.6263851407603 1.003855320833505E-02 5.643681853501173E-02 -25.5845661159781 -1.015140671619874E-02 5.641662856421617E-02 -25.6136436882361 -4.057375265232938E-02 4.049268616155282E-02 -25.6263851430904 Theta polarization Incident angle theta is.0 5.641662856629827E-02 1.015140670462750E-02 12.26351927746214 4.049268616987463E-02 4.057375264402420E-02 12.25082615166598 1.003855320833505E-02 5.643681853501173E-02 12.26903042801212 -1.015140671619874E-02 5.641662856421617E-02 12.26351927652879 -4.057375265232938E-02 4.049268616155282E-02 12.25082615094696 Output file hfield.txt (X, Y, H[dB]) is: Phi polarization Incident angle theta is.0 5.641662856629827E-02 1.015140670462750E-02 -135.934187068036 4.049268616987463E-02 4.057375264402420E-02 -135.987570212459 1.003855320833505E-02 5.643681853501173E-02 -135.99616979836 -1.015140671619874E-02 5.641662856421617E-02 -135.934187071225 37

-4.057375265232938E-02 4.049268616155282E-02 -135.98757021614 Theta polarization Incident angle theta is.0 5.641662856629827E-02 1.015140670462750E-02 -95.2241346546987 4.049268616987463E-02 4.057375264402420E-02 -95.2297448855929 1.003855320833505E-02 5.643681853501173E-02 -95.219767234333 -1.015140671619874E-02 5.641662856421617E-02 -95.2241346556522 -4.057375265232938E-02 4.049268616155282E-02 -95.2297448862359 Output file imped.txt (X, Y, modZ) is: Phi polarization Incident angle theta is.0 5.641662856629827E-02 1.015140670462750E-02 218.6452468368924 4.049268616987463E-02 4.057375264402420E-02 249.1510287069808 1.003855320833505E-02 5.643681853501173E-02 249.7799134972396 -1.015140671619874E-02 5.641662856421617E-02 248.6452468492195 -4.057375265232938E-02 4.049268616155282E-02 249.1510287238088 Theta polarization Incident angle theta is.0 5.641662856629827E-02 1.015140670462750E-02 215.806613029913 4.049268616987463E-02 4.057375264402420E-02 21]5.730199783424 1.003855320833505E-02 5.643681853501173E-02 215.8189546096599 -1.015140671619874E-02 5.641662856421617E-02 215.8066130301293 -4.057375265232938E-02 4.049268616155282E-02 215.7301997826038 38

Output file engine.field (incident:EL, AZ; observation:EL, AZ; frequency [GHz]; VV; HV; VH; HH) is: 30.0.0 30.0.0.3 (4.99521787318461,.9731762510585861) (-2.285742277847609E-03,2.063994260126811E-04) (-1.575715769867148E-03,8.954386064833249E-03) (-1.30286653617953,3.40646251893064) 30.0 1.0 30.0 1.0.3 (4.97867761320453,1.07756716164062) (-.234753947300065,.2457638251759209' (-.599056785821603,.2768031011598088) (-1.33348562145874,-3.4231551755766) 30.0 2.0 30.0 2.0.3 (4.92405101980507,1.37583167577948) (-.459165704963816,.468591104543572) (-1.16817454290087,.5111160198199839) (-1.41684678891507,-3.47068275177345) Output file incrot.txt (X, Y, E, H, modZ) is: Phi polarization Incident angle theta is.0 29.63130309987938.05642.01015 -25.614 -135.934 248.645.04049.04057 -25.626 -135.988 249.151.01004.05644 -25.585 -135.996 249.780 -.01015.05642 -25.614 -135.934 248.645 -.04057.04049 -25.626 -135.988 249.151 Theta polarization Incident angle theta is.0 29.63128937395119.05642.01015 12.264 -95.224 215.807.04049.04057 12.251 -95.230 215.730.01004.05644 12.269 -95.220 215.819 -.01015.05642 12.264 -95.224 215.807 39

-.04057.04049 12.251 -95.230 215.730 Output file firot.txt (X, Y, E, H, modZ) is: Phi polarization Incident angle theta is.0 Rotation angle is 1 FField RCS is 29.63130309269709.05642.01015 -25.614 -135.931 248.606.04049.04057 -25.627 -135.991 249.191.01004.05644 -25.584 -135.993 249.747 -.01015.05642 -25.614 -135.931 248.606 -.04057.04049 -25.627 -135.991 249.191 Theta polarization Incident angle theta is.0 Rotation angle is 1 FField RCS is 29.63128938113347.05642.01015 12.263 -95.224 215.804.04049.04057 12.251 -95.230 215.731.01004.05644 12.269 -95.220 215.821 -.01015.05642 12.263 -95.224 215.804 -.04057.04049 12.251 -95.230 215.731 40

* ~~i: i. *. i ~*: W vsl.0....... I.................... iI: ii i ii ii~i i: i:...........i- _i;i~;...;:I, " ~-.''....?. **S 5.. Figure 11: E and H fields on the hand off plane. 41