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SENSOR AND SIMULATION NOTES Note 209 18 January 197H 04751 57597 NUMERICAL ANALYSIS OF A TRANSMISSION LINE EMP SIMULATOR Keith M 800 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 39This 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 for 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 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 CONTENTS ABSTRACT I INTRODUCTION II GEOMETRY III THEORY IV RESULTS V CONCLUSIONS AND DISCUSSION REFERENCES APPENDIX U 00 qu 114 26 27 28 FIGURES Transmission Line Simulator Geometry Symmetry Axes in Generalized Impedance Matrix Input Resistance and Input Reactance of Triangular Dipole Antenna Current Distribution at 6 MHz Current Distribution at 26 MHz Input Resistance vs Frequency Input Reactance vs Frequency Reflected and Radiated POWer vs Frequency Electric Field Distribution at y 0 z 0 Electric Field Distribution at x 0 y 0 Vertical Electric Field at x 0 z 0 Radiation Pattern at 6 MHz Radiation Pattern at 26 MHz TABLE Simulator Dimensions 15 17 17 18 18 19 21 22 23 21 25 114 I INTRODUCTION Static field analysis has shown that the TEM mode of a parallel plate transmission line can simulate a freespace 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 HFVHF 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 dynamic 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 It is then reasonably straightforward to obtain all of the quantities of interest eg 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 is 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 should allow for twodimensional 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 form of the unknowu current These important assumptions are checked by considering some special test cases for which there exists experimental data II THE GEOMETRY The simulator geometry relative to a cartesian coordinate system is shown in Fig 1 In the quotworking volumequot 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 they can be made both physically and electrically compatible with the voltage source 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 DZ and X 2 R DZ respectively TOP VIEW SIDE VIEW Fig 1 Iransmission Line Simulator Geometry Ill THEORY Let S be the surface defined by the conducting plates and the non conducting source and terminal gaps of the transmission line Then one can obtain the following integral equation for the unknown current density J jcupo ffJ r39 dSquot jw ffvg 39 J r39 dS39 S S gti amp 9 Etan Zsr J on S 1 where e jkIF F4 1 w m m 0 Eq 1 is known as theelectric field integral equation and stems from the boundary conditions on the total electric field Its derivation is given elsewhere Ref 5 In Eqs 1 and 2 F and E are vectors from the origin to the source and field points respectively Also is the tangential component of the incident or excitation electric field which is zero eVeryWhere on S except across the source gap where it is assumed known The load impedance Zs is for a unit length and unit width and is nonzero only across the terminal gap Finally Vt is the gradient operator 2 is the freespace wave in the direction tangent to S and k mule 601 number The induced current actually resides on both sides of S However because of the assumption that S is infinitesimally thin Eq 1 correctly involves only 3quot which is the sum of the currents on both sides and the integration is performed over only one side of S The current density is next expanded in a set of basis functions Jn to be defined later as follows N 3 Z In 31 3 n1 Where N represents the number of subintervals to be defined later If the inner product of Eq 1 is taken with Tm Eq 1 may be transformed into a set of algebraic equations given by zmn zL In va 4 mn where 0 Vm ffm Egan dS 5 zmn jwoffff3 mgt rum iiiVt 3 mvt 311 dS dS 6 S S Zmn In and Vm are the generalized impedance current and voltage matrices respectively and 39ZLmn is a matrix determined by the terminal load Again the detailed steps leading to Eq 4 are well documented Ref 5 This technique is known as Galerkin39s method and is a Special case of the method of moments Eq 4 may be solved on a computer using a standard matrix inversion algorithm Once has been determined the problem is essentially solved However we need to investigate more fully the deter mination of Zmn since this is the most difficult portion of the calculation The first step is to define the basis function 3n It is convenient to describe S parametrically in terms of a single variable t According to Fig 1 the projection of S onto the xz plane may be represented by the curve x xt and z zt 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 0 and t L respectively have been arbitrarily set at x R DZ z 0 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 Zn 1 as tractable as possible but it should also provide a reasonable approximation to the current It is assumed to be of the following form Tnt Vty 310 y W u t Y 7 Tn is the triangular pulse function used by Harrington and Mautz Ref 5 and 12 12 Vt y 1 1 8 w t yW39Tf uty X VWZ yw 2 Where 1 1 and uy are unit vectors along the t and y directions respectively 13t is a function of t and its parametric representation is given in the Appendix 9 10 t Where Kw y wt y w2 yW 212 11 The integrations over y and y may be performed numerically The integra tions over t and t may be handled in the sande39manner as shown by Harrington 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 mn 1 we shall center the triangular function at t LN when mn 2 we shall center the triangular function at ZLN etc then if it is assumed that a 1volt source is applied across the feed gap va 12 o The above matrix was derived assuming that the gap height G is infinitesimally small so that Etan is a delta function For the dimen sions of interest this approximation is quite reasonable Using the same previous arguments the matrix ZL Jis also easy mn to derive Since the terminal load is located at t LZ one gets 0 7 o 39o z z L 39 13 Lm 0 o 0 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 12 1D 1D retained and an integration over S must be performed The integration may be approximated in the same manner that Eq 10 was approximated and the details will not be presented here We include one final Word on Zmn Since the simulator is sym metric about both the z 0 and x 0 planes it should not be necessary to compute all of the elements of Zmn This is illustrated in Fig 2 Only the matrix elements in the shaded area need to be calculated It can be proven that A B C and D are axes of symmetry D is an axis of symmetry because of reciprocity Therefore all matrix elements in the unshaded region may be generated by making four successive mathematical reflections about A B C and D This results in a significant reduction in computer time 1 B C 2an A c A39 B D Fig 2 Symmetry Axes in Generalized Impedance Matrix 13 IV RESULTS Calculations were first made on a triangular dipole antenna i e a center fed symmetric dipole whose radiating arms are triangular plates since these results can be compared with existing experimental data Ref 7 The calculated and measured input impedances are shown in Fig 3 Representation I is the calculated result using the theory presented in Section III It can be seen that agreement with experiment is excellent Representation II is the result of using a simplified basis function For this case the basis function was assumed to be uniform in the transverslt direction and in each subinterval the width of S was assumed to be constant Thus instead of Eqs 8 and 9 We now have Vt y 1 and at y Representation II although inferior to Representation I gives reasonably good agreement with the experimental results Thus Representation 11 may be used whenever it is considered desirable to trade off accuracy with mor favorable computer costs Most of the calculations on the simulator used Representation II For this study a single set of dimensions was selected Table I The feed terminal gap was designed such that the characteristic impedance at the input output is 100 2 according to the formula Zo lo6012 GWG Table I Simulator Dimensions H 12m G 02m WH 18m WG 0754m R 20m D 18m 11L CI INPUT RESISTANCE ohms INPUT REACTAKE ohms m 600 I O 500 r 400 P 300 2m EXPERIMENT BROWN 3 wooowmo 0 0 0 REPRESENTATIONI woo o o o REPRESENTATION u 39 so 100 150 200 ANTENNA LENGTH dog a 39 EXPERIMENT 200 39 BROWN 8 WOODWARD 000 REPRESENTATION I O O O REPRESENTATION II m m g L39s39oL 41017 150 200 ANTENNA LENGTH dog 0 Fig 3 Input Resistance a and Input Reactance b of Triangular Dipole Antenna 15 For the set of dimensions given in Table I the total arc length L is 119 81 m Calculations were performed for frequencies ranging from 2 to 30 MHz It was found that sufficient accuracy can be obtained for the param eters of interest most of which are near field parameters when the subinter val size is less than oneeighth of a wavelength Therefore the number of subintervals N was chosenito be 40 at the loxver 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 WZ 10 fwZ m my dy 14 These values result from a 1 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 S2 It can be seen that the current decays with distance from the feed gap and that the decay is faster at the higher frequency This suggests that energy may be leaking away through radiation The input39resistance 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 100 SE 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 2 is 1 8 but that mostly it is below 15 which corresponds to a power reflection coefficient of less than 006 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 16 180 150 400 ISO 480 PHASE dog 40 420 460 AMPLITUDE mA AMPLITUDE mA B 71 5 5 v 4 T a t v 39 I 2 quotquot AMPLITUDE was 1 n l l l I 3 01L an 03L 04L 05L t D Fig 4 Current Distribution at 6 MHz 12 AMPLITUDE PHASE 005L 01 L 015L 02L 025L 03L 035L 04L 045L 05L i gt Fig 5 Current Distribution at 26 MHz 17 180 o 160 o o 2 40 o g 8 0 z a o lt 039 gm o 62 0 I o 39 O o O 0 o 39 I 0 o loo 1 39 o f C C C o o 80 o o I E I L 1 A l I O 4 8 12 16 20 24 23 32 FREQUENCY MHZ Fig 6 Input Resistance vs Frequency 40 20 o m o E 0 o a o m g o o g o o 0 gZO o o 0 C o C O o o a 4o O O O o I 60 k l i I l L l l 0 4 B 12 20 24 23 32 16 FREBBENCY MHZ Fig 7 Input Reactance vs Frequency 18 6T VOLTAGE STANDING WAVE RATIO FREQUENCY MHz Fig 8 Reflected and Radiated Power vs Frequency w 100 RADIATED POWER 80 60 4o VSWR 20 I I I I I I I o 4 8 12 16 20 24 28 32 RADIATED PIIJWER PERCENT The electric fields within and just beyond the working volume are shown in Figs 9 through 11 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 Ez is due to the charge density on the plate Actually Ez reverses direction when 2 gt 6m but this does not show up on a plot of absolute field strength Fig 11 shows the vertical electric field plotted as a function of y Radiation field patterns are shown in Figs 12 and 13 On the y 0 plane the electric field vector is linearly polarized since Ecp 0 from sym metry In both Figs 12 and 13 the 0 and 180 directions corre3pond to viewing the simulator from the top and bottozm plates respectively while the 90 and 270 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 39EZ VERTICAL ELECTRIC FIELD mVm Q l 8 T Fig 9 X meters Electric Field Distribution at y 0 z 0 20 MHz 10 MHz 5 MHz IDEAL LINE 1 l 22 ELECTRIC FIELD mVm 100 80 20 Fig 10 HORIZONTAL COMPONENT Ex H 20 MHz o o o 5 MHz VERTICAL COMPONENT Ez I 6 8 10 12 14 16 2 meters Electric Field Distribution at x 0 y O VERTICAL ELECTRIC FIELD mVm PLATE EDGE h HZOMHZ 5MH2 W 24 25 0 V CONCLUSIONS AND DISCUSSION The transmission line simulator is apparently an efficient radiator at the higher frequencies The radiation pattern resembles that of an endfire 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 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 acceptable 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 arc length L was divided into 100 subsections N 100 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 y39 Therefore even though there is sufficient storage capacity to handle factors of 439or 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 C E Baum quotImpedances and Field Distributions for Parallel Plate Transmission Line Simulators Sensor and Simulation Notes Vol 1 No 21 June 1966 also see T L Brown and K D Granzow quotA Parameter Study of Two Parallel Plate Transmission Line Simulators of EMP Sensor and Simulation Note 21 Sensors 2 Simulation Notes Vol 3 No 52 April 1968 I R F Harrington Field Computation by Moment Methods The Macmillan Company New York 1968 R H Ott quotThe Scattering by a TwoDirnensional Periodic Array of Plates Technical Report 21482 The Ohio State University Research Foundation Antenna Laboratory Columbus Ohio 30 June 1966 R Mittra W A Davis and D V Jamnejad quotAn Integral Equation for Plane Wave Scattering by Thin Platesquot presented at 1972 Fall Meeting of the International Scientific Radio Union Williamsburg Virginia 1215 December 1972 J R Mautz and R F Harrington Radiation and Scattering from Bodies of Revolution Appl Sci Res 20 6 405 June 1969 D S Jones The Theory of Electromagnetism The Macmillan Company New York 1964 Chap 9 G H Brown and O M Woodward Jr quotExperimentally Determined Radiation Characteristics of Conical and Triangular Antennasquot 32 my 4 425 December 1952 3927 APPENDIX The parametric equations which describe 5 can be derived from Fig 1 First the total arc length L can be expressed in terms of the dirhensions of the simulator as follows L2GD 4R2HGZ Al Next let x L 245 D mm 02 lm As 4WH G AA awL2GD Then xt zt Wt and 626 are given as follows xltR Zt 0 tlt A39s W39WG xR a may tug z 2 z 2 b j t lt122 A6 WWGQWt 6 x xazaz J 28 xt H z gt Q tltI12 A7 4 Z 4 Z W WH gtgt utux D 2 x 2axt 4 2 H lt L D z a t 2 z 4 2 gt 122 tlt A8 g D W39WH wt394 2 Ea 1 a u t x x z z xR3912 SigtgLJ A9 2 4 2 zLt 2 WWG utuzj where 3x and 1 are unit vectors in the x and 2 directions respectively Finally the expressions for LZ S t S L may be generated from Eqs A5 through A9 since from symmetry xltt xt AiO 24 gt zt AL11 CD 729 3O AtZ A13 A 14 SENSOR AND SIMULATION NOTES Note 209 18 January 197H 04751 57597 NUMERICAL ANALYSIS OF A TRANSMISSION LINE EMP SIMULATOR Keith M 800 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 39This 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 for 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 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 CONTENTS ABSTRACT I INTRODUCTION II GEOMETRY III THEORY IV RESULTS V CONCLUSIONS AND DISCUSSION REFERENCES APPENDIX U 00 qu 114 26 27 28 FIGURES Transmission Line Simulator Geometry Symmetry Axes in Generalized Impedance Matrix Input Resistance and Input Reactance of Triangular Dipole Antenna Current Distribution at 6 MHz Current Distribution at 26 MHz Input Resistance vs Frequency Input Reactance vs Frequency Reflected and Radiated POWer vs Frequency Electric Field Distribution at y 0 z 0 Electric Field Distribution at x 0 y 0 Vertical Electric Field at x 0 z 0 Radiation Pattern at 6 MHz Radiation Pattern at 26 MHz TABLE Simulator Dimensions 15 17 17 18 18 19 21 22 23 21 25 114 I INTRODUCTION Static field analysis has shown that the TEM mode of a parallel plate transmission line can simulate a freespace 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 HFVHF 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 dynamic 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 It is then reasonably straightforward to obtain all of the quantities of interest eg 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 is 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 should allow for twodimensional 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 form of the unknowu current These important assumptions are checked by considering some special test cases for which there exists experimental data II THE GEOMETRY The simulator geometry relative to a cartesian coordinate system is shown in Fig 1 In the quotworking volumequot 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 they can be made both physically and electrically compatible with the voltage source 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 DZ and X 2 R DZ respectively TOP VIEW SIDE VIEW Fig 1 Iransmission Line Simulator Geometry Ill THEORY Let S be the surface defined by the conducting plates and the non conducting source and terminal gaps of the transmission line Then one can obtain the following integral equation for the unknown current density J jcupo ffJ r39 dSquot jw ffvg 39 J r39 dS39 S S gti amp 9 Etan Zsr J on S 1 where e jkIF F4 1 w m m 0 Eq 1 is known as theelectric field integral equation and stems from the boundary conditions on the total electric field Its derivation is given elsewhere Ref 5 In Eqs 1 and 2 F and E are vectors from the origin to the source and field points respectively Also is the tangential component of the incident or excitation electric field which is zero eVeryWhere on S except across the source gap where it is assumed known The load impedance Zs is for a unit length and unit width and is nonzero only across the terminal gap Finally Vt is the gradient operator 2 is the freespace wave in the direction tangent to S and k mule 601 number The induced current actually resides on both sides of S However because of the assumption that S is infinitesimally thin Eq 1 correctly involves only 3quot which is the sum of the currents on both sides and the integration is performed over only one side of S The current density is next expanded in a set of basis functions Jn to be defined later as follows N 3 Z In 31 3 n1 Where N represents the number of subintervals to be defined later If the inner product of Eq 1 is taken with Tm Eq 1 may be transformed into a set of algebraic equations given by zmn zL In va 4 mn where 0 Vm ffm Egan dS 5 zmn jwoffff3 mgt rum iiiVt 3 mvt 311 dS dS 6 S S Zmn In and Vm are the generalized impedance current and voltage matrices respectively and 39ZLmn is a matrix determined by the terminal load Again the detailed steps leading to Eq 4 are well documented Ref 5 This technique is known as Galerkin39s method and is a Special case of the method of moments Eq 4 may be solved on a computer using a standard matrix inversion algorithm Once has been determined the problem is essentially solved However we need to investigate more fully the deter mination of Zmn since this is the most difficult portion of the calculation The first step is to define the basis function 3n It is convenient to describe S parametrically in terms of a single variable t According to Fig 1 the projection of S onto the xz plane may be represented by the curve x xt and z zt 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 0 and t L respectively have been arbitrarily set at x R DZ z 0 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 Zn 1 as tractable as possible but it should also provide a reasonable approximation to the current It is assumed to be of the following form Tnt Vty 310 y W u t Y 7 Tn is the triangular pulse function used by Harrington and Mautz Ref 5 and 12 12 Vt y 1 1 8 w t yW39Tf uty X VWZ yw 2 Where 1 1 and uy are unit vectors along the t and y directions respectively 13t is a function of t and its parametric representation is given in the Appendix 9 10 t Where Kw y wt y w2 yW 212 11 The integrations over y and y may be performed numerically The integra tions over t and t may be handled in the sande39manner as shown by Harrington 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 mn 1 we shall center the triangular function at t LN when mn 2 we shall center the triangular function at ZLN etc then if it is assumed that a 1volt source is applied across the feed gap va 12 o The above matrix was derived assuming that the gap height G is infinitesimally small so that Etan is a delta function For the dimen sions of interest this approximation is quite reasonable Using the same previous arguments the matrix ZL Jis also easy mn to derive Since the terminal load is located at t LZ one gets 0 7 o 39o z z L 39 13 Lm 0 o 0 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 12 1D 1D retained and an integration over S must be performed The integration may be approximated in the same manner that Eq 10 was approximated and the details will not be presented here We include one final Word on Zmn Since the simulator is sym metric about both the z 0 and x 0 planes it should not be necessary to compute all of the elements of Zmn This is illustrated in Fig 2 Only the matrix elements in the shaded area need to be calculated It can be proven that A B C and D are axes of symmetry D is an axis of symmetry because of reciprocity Therefore all matrix elements in the unshaded region may be generated by making four successive mathematical reflections about A B C and D This results in a significant reduction in computer time 1 B C 2an A c A39 B D Fig 2 Symmetry Axes in Generalized Impedance Matrix 13 IV RESULTS Calculations were first made on a triangular dipole antenna i e a center fed symmetric dipole whose radiating arms are triangular plates since these results can be compared with existing experimental data Ref 7 The calculated and measured input impedances are shown in Fig 3 Representation I is the calculated result using the theory presented in Section III It can be seen that agreement with experiment is excellent Representation II is the result of using a simplified basis function For this case the basis function was assumed to be uniform in the transverslt direction and in each subinterval the width of S was assumed to be constant Thus instead of Eqs 8 and 9 We now have Vt y 1 and at y Representation II although inferior to Representation I gives reasonably good agreement with the experimental results Thus Representation 11 may be used whenever it is considered desirable to trade off accuracy with mor favorable computer costs Most of the calculations on the simulator used Representation II For this study a single set of dimensions was selected Table I The feed terminal gap was designed such that the characteristic impedance at the input output is 100 2 according to the formula Zo lo6012 GWG Table I Simulator Dimensions H 12m G 02m WH 18m WG 0754m R 20m D 18m 11L CI INPUT RESISTANCE ohms INPUT REACTAKE ohms m 600 I O 500 r 400 P 300 2m EXPERIMENT BROWN 3 wooowmo 0 0 0 REPRESENTATIONI woo o o o REPRESENTATION u 39 so 100 150 200 ANTENNA LENGTH dog a 39 EXPERIMENT 200 39 BROWN 8 WOODWARD 000 REPRESENTATION I O O O REPRESENTATION II m m g L39s39oL 41017 150 200 ANTENNA LENGTH dog 0 Fig 3 Input Resistance a and Input Reactance b of Triangular Dipole Antenna 15 For the set of dimensions given in Table I the total arc length L is 119 81 m Calculations were performed for frequencies ranging from 2 to 30 MHz It was found that sufficient accuracy can be obtained for the param eters of interest most of which are near field parameters when the subinter val size is less than oneeighth of a wavelength Therefore the number of subintervals N was chosenito be 40 at the loxver 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 WZ 10 fwZ m my dy 14 These values result from a 1 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 S2 It can be seen that the current decays with distance from the feed gap and that the decay is faster at the higher frequency This suggests that energy may be leaking away through radiation The input39resistance 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 100 SE 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 2 is 1 8 but that mostly it is below 15 which corresponds to a power reflection coefficient of less than 006 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 16 180 150 400 ISO 480 PHASE dog 40 420 460 AMPLITUDE mA AMPLITUDE mA B 71 5 5 v 4 T a t v 39 I 2 quotquot AMPLITUDE was 1 n l l l I 3 01L an 03L 04L 05L t D Fig 4 Current Distribution at 6 MHz 12 AMPLITUDE PHASE 005L 01 L 015L 02L 025L 03L 035L 04L 045L 05L i gt Fig 5 Current Distribution at 26 MHz 17 180 o 160 o o 2 40 o g 8 0 z a o lt 039 gm o 62 0 I o 39 O o O 0 o 39 I 0 o loo 1 39 o f C C C o o 80 o o I E I L 1 A l I O 4 8 12 16 20 24 23 32 FREQUENCY MHZ Fig 6 Input Resistance vs Frequency 40 20 o m o E 0 o a o m g o o g o o 0 gZO o o 0 C o C O o o a 4o O O O o I 60 k l i I l L l l 0 4 B 12 20 24 23 32 16 FREBBENCY MHZ Fig 7 Input Reactance vs Frequency 18 6T VOLTAGE STANDING WAVE RATIO FREQUENCY MHz Fig 8 Reflected and Radiated Power vs Frequency w 100 RADIATED POWER 80 60 4o VSWR 20 I I I I I I I o 4 8 12 16 20 24 28 32 RADIATED PIIJWER PERCENT The electric fields within and just beyond the working volume are shown in Figs 9 through 11 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 Ez is due to the charge density on the plate Actually Ez reverses direction when 2 gt 6m but this does not show up on a plot of absolute field strength Fig 11 shows the vertical electric field plotted as a function of y Radiation field patterns are shown in Figs 12 and 13 On the y 0 plane the electric field vector is linearly polarized since Ecp 0 from sym metry In both Figs 12 and 13 the 0 and 180 directions corre3pond to viewing the simulator from the top and bottozm plates respectively while the 90 and 270 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 39EZ VERTICAL ELECTRIC FIELD mVm Q l 8 T Fig 9 X meters Electric Field Distribution at y 0 z 0 20 MHz 10 MHz 5 MHz IDEAL LINE 1 l 22 ELECTRIC FIELD mVm 100 80 20 Fig 10 HORIZONTAL COMPONENT Ex H 20 MHz o o o 5 MHz VERTICAL COMPONENT Ez I 6 8 10 12 14 16 2 meters Electric Field Distribution at x 0 y O VERTICAL ELECTRIC FIELD mVm PLATE EDGE h HZOMHZ 5MH2 W 24 25 0 V CONCLUSIONS AND DISCUSSION The transmission line simulator is apparently an efficient radiator at the higher frequencies The radiation pattern resembles that of an endfire 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 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 acceptable 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 arc length L was divided into 100 subsections N 100 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 y39 Therefore even though there is sufficient storage capacity to handle factors of 439or 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 C E Baum quotImpedances and Field Distributions for Parallel Plate Transmission Line Simulators Sensor and Simulation Notes Vol 1 No 21 June 1966 also see T L Brown and K D Granzow quotA Parameter Study of Two Parallel Plate Transmission Line Simulators of EMP Sensor and Simulation Note 21 Sensors 2 Simulation Notes Vol 3 No 52 April 1968 I R F Harrington Field Computation by Moment Methods The Macmillan Company New York 1968 R H Ott quotThe Scattering by a TwoDirnensional Periodic Array of Plates Technical Report 21482 The Ohio State University Research Foundation Antenna Laboratory Columbus Ohio 30 June 1966 R Mittra W A Davis and D V Jamnejad quotAn Integral Equation for Plane Wave Scattering by Thin Platesquot presented at 1972 Fall Meeting of the International Scientific Radio Union Williamsburg Virginia 1215 December 1972 J R Mautz and R F Harrington Radiation and Scattering from Bodies of Revolution Appl Sci Res 20 6 405 June 1969 D S Jones The Theory of Electromagnetism The Macmillan Company New York 1964 Chap 9 G H Brown and O M Woodward Jr quotExperimentally Determined Radiation Characteristics of Conical and Triangular Antennasquot 32 my 4 425 December 1952 3927 APPENDIX The parametric equations which describe 5 can be derived from Fig 1 First the total arc length L can be expressed in terms of the dirhensions of the simulator as follows L2GD 4R2HGZ Al Next let x L 245 D mm 02 lm As 4WH G AA awL2GD Then xt zt Wt and 626 are given as follows xltR Zt 0 tlt A39s W39WG xR a may tug z 2 z 2 b j t lt122 A6 WWGQWt 6 x xazaz J 28 xt H z gt Q tltI12 A7 4 Z 4 Z W WH gtgt utux D 2 x 2axt 4 2 H lt L D z a t 2 z 4 2 gt 122 tlt A8 g D W39WH wt394 2 Ea 1 a u t x x z z xR3912 SigtgLJ A9 2 4 2 zLt 2 WWG utuzj where 3x and 1 are unit vectors in the x and 2 directions respectively Finally the expressions for LZ S t S L may be generated from Eqs A5 through A9 since from symmetry xltt xt AiO 24 gt zt AL11 CD 729 3O AtZ A13 A 14

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