Transmission Line Simulator: Analysis of Triangular Dipole Antenna with Impedance Matrix, Study notes of Information Technology

Download Transmission Line Simulator: Analysis of Triangular Dipole Antenna with Impedance Matrix and more Study notes Information Technology in PDF only on Docsity! I —. SENSOR AND SIMULATION NOTES Note 209 . ,.. 18 January 1974 :::.;:,::..”,’,:.; +y>-j? 5//?/97 NUMERICAL ANALYSIS OF A TRANSMISSION LINE EMP SIMULATOR Keith M. Soo Hoo The Aerospace Corporation . -- .. ABSTRACT A theoretical model is defined for an electromagnetic pulse (EMP) simulator for testing EMP effects on high altitude satellites. The simulator is composed of three flat-plate transmission line sections. The first and third sections are tapered to accommodate, respectively, a gen- erator and a terminating resistor. This problem Is analyzed in the frequency domain over those frequency components which are known to contribute most significantly to a typ- ical EMP waveform. The analysis uses a numerical technique to solve the basic problem of an unknown current distribution on a curved, tapered strip, excited by a known electric field. The unknown current is solved by the method of moments, using triangular basis functions. To check the computer program, input impedances are computed f“’or the triangular dipole. It is shown that these solutions compare quite favorably with experimental results. Variations in the solutions are also demonstrated for these same cases when edge singularities are not taken into account in the analysis. For the transmission line simulator, computed input impedance, VSWR, power dissipated in the terminating resistor, and the power lost to radiation are presented as a function of frequency. The computed current distributions are used . . to calculate the electric fields between and immediately a beyond the parallel plates. Although a considerable portion of power is converted to radiation at the higher frequencies, it is shown that at least within the working volume the ele- tric field maintains a reasonably constant level. 2 “ . I. INTRODUCTION Static field analysis has shown that the TEM mode of a parallel plate transmission line can simulate a free-space plane electromagnetic wave over a substantial portion of its interior region (Ref. 1). This method, however, does not account for the presence of higher order modes which may exist whenever the plate separation exceeds half a wavelength. For this case a dynamic field solution would be necessary. A high altitude EMP waveform consists of frequency components in the HF-VHF regime. Many wavelengths may therefore exist between the parallel plates of a transmission line designed to accommodate EMP testing of missiles or satellites. This paper considers the dyna]mic field analysis of a high altitude EMP simulator. The simulator consists of a parallel plate transmission line that is tapered at both ends to accommodate a generator and a terminat- ing load. The analysis is performed in the frequency domain, using the method of moments (Ref. 2). This numerical technique has the overriding advantage that the cur- rent everywhere on the transmission line can be accurately computed. lt is then reasonably straightforward to obtain all of the quantities of interest (e. g., the electromagnetic fields both interior and exterior to the transmis - sion line, the input impedance, the power absorbed by the terminal load, and the power lost to radiation). The principal disadvantage is that the method of moments cannot be applied to structures whose dimensions are consider- ably larger than a wavelength. This limitation is dependent on the storage capacity and running costs of the particular computer being used. For the purpose of the analysis, the basic element of the simulator if; a tapered conducting strip. The numerical analysis of flat strips and rectan- gular plates has been considered previously (Refs. 3, 4), and the tapered strip may be treated in generally the same manner. At the outset one shoulcl allow for two -dimensional variations for the unknown current density vector. * * However, this would lead to a system of two integral equations whose solution would be practical only at the extremely low end of the frequency band. In the discussion to follow, it is shown how the frequency range can be extended by making some a priori assumptions on the fo rrn of the unknown current. These important assumptions are checked by considering some special test cases for which there exists experimental data. 6 . ● II. THE GEOMETRY The simulator geometry relative to a cartesian coordinate system is shown in Fig. 1. In the “working volume” (i. e., the region in which the test object is usually placed) the two plates are of width WH and are separated by a distance H. Outside of the working volume the plates are tapered down to where the y can be made both physically and electrically compatible with the voltage suurce and the terminal load. For this study, both the source and the load are assumed to be uniformly distributed across the gaps at x = -(R + D/2) and x = +(R + D/2) respectively. $ -----’” ----- <- ‘H11—.------. —-,. ‘----- I 1 I TOP VIEW AZ A H ‘-—l___z’,..-.—.---—------ .—. T’___ “7 k“– ‘-‘Q ““““‘— –-D-----L--=3 SIDE VIEW Fig. 1. Transmission Line Simulator Geometry 7 9 ● + The first step is to define the basis function Jn. describe S parametrically in terms of a single variable It is convenient t* According to Fig. 1, the projection of S onto the x-z plane may be represented by the o to curve x = x(t) and z = z(t). If W is defined as the width of S, then W may also be expressed as a function of t. These three parametric expressions, which describe S uniquely, may be easily derived in terms of the dimensions of the simulator and are given in the Appendix. Both the starting and ending points (t = O and t = L, respectively) have been arbitrarily set at (X= -R-D/2, z= O). The total arc length L is also given in the Appendix. The basis function may then be expressed in terms of t and the cartesian coordinate y. The basis function should be relatively simple, so as to make the calculation of Z rnn as ‘ractable as POssible J but ‘t ‘hOuld also provide a reasonable approximation to the current. It is assumed to be of the following form: Tn(t) V(t, y) Fn(t, y) = w(t) ir(t, y) Tn is the triangular pulse function used by Barrington and Mautz (Ref. 5), and V(t,y).(f -#-)-’’2+(l+f”2”2 (8) w; + VW’; ;(t, y) = t ‘ Y J=RF where ;t and ~y are unit vectors along the t and y (9) directions respectively (&t is a function of t, and its parametric representation is given in the Appendix). 10 ● The triangular function provides variations of the current with t. at the edges of S (y = +W/2), and a piecewise linear approximation to The function V(t, y) is integrable singular the Sommerfeld edge condition is satisfied by this assumption (Ref. 6). The unit vector ;has been selected such that the basis function is directed along the edge of S at the edge and along the center line at y = O and is linearly distributed in between. To recapitulate, the basis function has been s elected such that variations of the current in the transverse clirection (y) are specified, varia- tions in the longitudinal direction (t) will be determined in a piecewise linear sense by solving Eq. (4) for [In] , and the direction of the current is also specified. Obviously we have not started with the most general basis func- tion. But the assunled function provides for greater than a factor of 2 reduction in N, the size of [Z -] , and therefore represents a significant savings in computer storage and computation time. Substitution of Eqs. (7) - (9) into (6) gives (io) r+ 1 ‘w‘(t) w‘(t‘ x Ut”:fl+yy w(t) W(t’) ‘1 / - A T’ (t) T~ (t?) $)dydy ’dtdt’ ~2 m n where [ .1 -1/2 K(t, y) = ~ V(t, y) I W2 “t (yw’)2 (ii) The integrations over y and y ‘ may be performed numerically. The integra- tions over t and t? may be handled in the same manner as shown by Barrington and Mautz (Ref. 5). Thus, further details will not be given here. 11 The generalized voltage matrix defined by Eq. (5) is quite easy to calculate. From the previous discussion it should be clear that there is a one-to-one correspondence between the subscripts m or n and the triangular pulse function. When m, n = i we shall center the triangular function at t = L/N; when m,n = 2 we shall center the triangular function at 2L/N, etc. ; then if it is assumed that a 1-volt source is applied across the feed gap The above matrix was infinitesimally small, siom of interest, this [1 o. [Vm] = ; 1 derived assuming that the gap height G is so that ~~an is a delta function. For the dimen- approximation is quite reasonable. Using the same previous arguments, the matrix to derive. Since the terminal Ioad is located at t = L/2, one gets [1zLmn 1 0.... “ “o .= ‘L . 0.. 1 .0 . . 0. .. ● “o (12) also easy o (13) Once [In] has been determined from Eq. (4) the electromagnetic fields may be computed. If the field point is far away (many wavelengths) from the simulator, the current may be approximated by N discrete sources whose amplitudes are In, and the field calculation is quite simple. If the field point is near or within the simulator the piecewise linear approximation must be EXPERIMENT (BROWN & WOOOWARD REPRESENTATION I REPRESENTATION II Ii!N20” ANTENNA LENOTH, dog (a) — EXPERIMENT (BROWN & WOOOWARD) ~ REPRESENTATION I O—Q——O REPRESENTATION II t .4X3--Jso ANTENNA LENGTH, dog (b) Fig. 3. Input Resistance (a) and Input of Triangular Dipole Antenna Reactance (b) 15 For the set of dimensions given in Table 1, the total arc length L is ii9.81 m. Calculations were performed for frequent ies ranging from 2 to 30 MHz. It was found that sufficient accuracy can be obtained for the pararn - eters of interest (most of which are near field parameters) when the subinter - val size is less than one-eighth of a wavelength. Therefore the number of subintervals N ‘was chosen to be 40 at the lower frequencies and was increased to 100 at the higher frequencies. Computed total currents as a function of t are shown in Figs. 4 and 5. The total current is defined by s?Vf2I(t) = ~t) ● :(t, y) dy-w/2 (14) These values result from a i -volt source applied across the feed gap. The current on the bottom plate is equal in value, but opposite in direction, and is therefore not shown. The load impedance ZL for all cases was 100 Q. It ,can be seen that the current decays with distance from the feed gap and that a the decay is faster at the higher frequency. This suggests that energy may be leaking away through radiation. The input-resistance and input reactance are shown as a function of frequency in Figs. 6 and 7 respectively. Since a perfectly matched line would have zero reactance, and the input resistance and the characteristic impedance of the line would be equal, it is possible that iOO G! is not the opti- mum load. However, as shown in Fig. 8, the impedance mismatch is minimal. It can be seen that the maximum VSWR (relative to 100 0) is i. 8, but that mostly it is below i. 5, which corresponds to a power reflection coefficient of less than 0.06. Also shown in Fig. 8 is the percent of incident power being radiated. This is easily determined from the input current and the current at the load. It can be seen that a large amount of power is radiated at the higher frequencies, which is consistent with the current decay shown previously. 16 180 — 150 — “lCHI — 50 z 4 #o –; ● x i a s -50 * -1(KI — -150 — -180 — td Fig. 4. Current Distribution at 6 MHz 12 [ — AMP LITLDE ----PHASE 10 Fig. 5. Current Distribution at 26 MHz 17 The electric fields within and just beyond the working volume are shown in Figs. 9 through i i. In Fig. 9 the ideal line is one in which the vertical electric field is 1 volt divided by the separation distance between plates. It can be seen that within the working volume the vertical electric field is very nearly ideal, and it is only beyond the working volume that the field drops off at the higher frequencies. In Fig. 10 the discontinuity in E= is due to the charge density on the plate. Actually Ez reverses direction when z > 6m, but this does not show up on a plot of absolute field strength. Fig. i 1 shows the vertical electric field plotted as a function of y. Radiation field patterns are shown in Figs. 12 and i 3. On the y = O plane the electric field vector is linearly polarized, since E~ = O from sym- metry. In both Figs. i2 and 13, tihe 0° and f80° directions correspond to viewing the simulator from the top and bottom plates respectively, while the 900 and 2700 directions correspond to viewing the simulator from the source end and the load end respectively. It can be seen that the simulator is tend- ing to be an end-fire antenna at the higher frequency. 20 I 24 20 E ~ 12 8 4 z --—- IDEAL LINE b \ \ \ \ \ -- I I I I I I 1 I I I -16 -12 -8 -4 0 4 8 12 16 20 X, meters Fig. 9. Electric Field Distribution at y = O, z = O 22 100 a—*4—A 20 MHz oo——e—e 5S MHz 80 VERTICAL COMPONENT {E)) ELECTRIC FIELD, mV/m 20 HORIZONTAL COMPONENT (E,) | 0 2 4 6 8 10 12 4 16 18 20 Z, meters Fig. 10, Electric Field Distribution at x=0, y=0 300° 29 280° 0° It270° 15-1$ i— A—-h’o--m=io + -igilss--io-l-s -0—5—10.15] 90°II I ~A -\ Y“0”%3325”0 190° 180” 170” Fig. 13. Radiation Pattern at 26 MHz (y= O plane) 25 V. CONCLUSIONS AND DISCUSSION The transmission line simulator is apparently an efficient radiato t at the higher frequencies. The radiation pattern resembles that of an end-fire, or traveling wave ~ antenna. The radiation, however, appears to be taking place beyond the working volume of the simulator; a large contribution may come from the discontinuity y between the end of the working volume and the beginning of the tapered section. Inside most of the working volume, the electric field was reasonably uniform and quite close to its ideal value. Furthermore, the power lost due to reflections was low. It can be con- cluded, therefore, that this geometry appears acceptab~e from the stand- point of EMP simulation. No attempt was made at optimizing the selected design, although it is obvious that the present computer program could be extremely useful for this purpose. Finally, some remarks should be made about the extension of the calculations to higher frequencies. For the calculation at 30 MHz the o arc length L was divided into 100 subsections (N = iOO), and the calculation required slightly less than 3 minutes on the CDC 7600 computer. The com- puter time increases faster than N2 because of the required integrations over y and yt . Therefore, even though there is sufficient storage capacity to handle factors of 4’ or 5 increase in frequency, which requires N to be increased by the same factor, such a calculation may be undesirable from the standpoint of cost. 26 REFERENCES i. C. E. Baum, “Impedarlces and Field Distributions for Parallel Plate Transmission Line Simulators, ” Sensor and Simulation Notes~, Vol. i, No. 21, June 1966; (also see: T. L. Brown and K. D. Granzow, “A Parameter Study of Two Parallel Plate Transmission Line! Simulators of EMP Sensor and Simulation Note 21 ,“ Sensors and Simulation Notes} Vol. 3, No. 52, April 1968). , 2.” R. F. Barrington, Fiel,d Computation by Moment Methods, The Macmillan Company, New York (1 968). 3. R. H. Ott, “The Scattering by a Two-Dimensional Periodic Array of Plate s,” Technical Report 2148-2, The Ohio State University Res earth Foundation, Antenna Laboratory, Columbus, Ohio (30 June 1966). 4. R. Mittra, W. A. Davis, and D. V. Jamnejad, “An Integral Equation for Plane Wave Scattering by Thin Plates, ” presented at 1972 Fall Meeting of the International Scientific Radio Union, Williamsburg, Virginia, 12-15 December 1972. 5. J. R. Mautz and R. F. Barrington, “Radiation and Scattering from Bodies of Revolution, ” Appl. Sci. Res. Q (6), 405 (June 19691). 6. D., S. Jones, The Theor ~ The Macmillan Company, New York (1 964), Chap. 9. 7. G. H. Brown and O. M. Woodward, Jr., “Experimentally Determined Radiation Characteristics of Conical and Triangular Antennas, ” RCA Rev. 13 (4), 425 (December 1952).—— 27’