11311-1-F = RL-2216 Sensor and Simulation Notes Note February 1973 ON THE TRANSIENT WAVEFORMS RADIATED BY A RESISTIVELY LOADED LINEAR ANTENNA Yu-Ping Liu, Dipak L. Sengupta and Chen-To Tai The University of Michigan Radiation Laboratory Department of Electrical and Computer Engineering Ann Arbor, Michigan 48105 Abstract The transient waveforms radiated by step voltage excited linear antennas, loaded non-uniformly and continuously with resistance, are investigated by numerical means. The input time varying excitations considered are step voltage, Gaussian pulse and gamma pulse types. Current distributions on the harmonically excited antenna are obtained as functions of frequency for different values of the loading. The transfer functions of the antenna and the spectral densities of the radiated waveforms are obtained as functions of frequency and the loading for two values of the antenna thickness and for different directions in space. Finally the time dependent radiated waveforms produced by the antenna are obtained by using fast Fourier inversion technique. The effects of the various antenna parameters on the radiated waveforms are also investigated. Some of the results are compared with available approximate analytical results.

TABLE OF CONTENTS Page Abstract 1 List of Figure Captions 3 I Introduction 11 II Background 12 m Basic Relations 14 IV A Note on the Loading 18 V Brief Outline of the Numerical Method Used 20 VI Current Distribution I(z, w) 24 VII Transfer Function of the Antenna f0 (0, ) 31 Vm Spectral Density e0(0G w) 40 IX Time Dependent Radiated Waveform e0 (00 t) 50 X Conclusions 63 Appendix A 64 Appendix B 138 References 165 2

LIST OF FIGURE CAPTIONS Figure 1: Amplitude of the current distribution along a non-reflectively L wL loaded antenna. a= 156, - = 5r, r = 318. O Figure 2: The phase variation of the current along a non-reflectively loaded L wLL antenna. = 156, w- 5r, r = 318. C 0 Figure 3: The amplitude of the current distribution along the antenna as a L wL function of the loading parameter. -=156, = 37r. a c Figure 4: The phase of the current distribution along the antenna as a function L wL of the loading parameter. a- 156, - 37r. a 'C Figure 5: The amplitude and phase of the current distribution along a nonreflectively loaded antenna for two different frequencies. = 156, a r =600. O Figure 6(a): Amplitude and phase of fe (2- w) as functions of normalized freL quency with r as the parameter. - = 156. o a Figure 6(b): Amplitude and phase of f (3, w) as functions of normalized freL quency with r as the parameter. - = 156. o a Figure 6(c): Amplitude and phase of f (~ 4w ) as functions of normalized fre4' L quency with r as the parameter. - = 156. o a Figure 6(d): Amplitude and phase of?(r, ) as functions of normalized freL quency with r as the parameter. -= 156. O a Figure 7 (a): Amplitude and phase of f (-. 2 ) as functions of normalized freL quency with r as the parameter. = 100. o a Figure 7(b). Amplitude and phase of (r, o) as functions of normalized fre0 3 L quency with r as the parameter. L= 100. o a 3

Figure 7 (c): Figure 7 (d): Figure 8(a): Figure 8(b): Figure 8(c): Figure 8(d): Figure 9 (a): Figure 9 (b): Figure 9 (c): Figure 9(d): Amplitude and phase of fi (4 W) as functions of normalized freL quency with r as the parameter. - = 100. 0 a Amplitude and phase of f ('6, t) as functions of normalized fre96' L L quency with r as the parameter. - 100. 0 a Envelope of IeO (2, w) as a function of normalized frequency L with r as the parameter. = 156. o a Envelope of | (, W) as a function of normalized frequency 0 I3 L with r as the parameter. -= 156. o a Envelope of e ( t, a) as a function of normalized frequency L with r as the parameter. -= 156. o a Envelope of j E(, ) as a function of normalized frequency L with r as the parameter. = 156. o a Envelope of |e (, to) as a function of normalized frequency L with r as the parameter. a 100. o a Envelope of e (3, Wo) as a function of normalized frequency with r as the parameter. = 100. o a Envelope of ee (4, W) as a function of normalized frequency L with r as the parameter. -= 100. o a Envelope of j (, W)| as a function of normalized frequency L with r as the parameter. a= 100. o a 7r Figure 10(a): e ( 2 t) as a function of t/T with r as the parameter. T= 3. 33 ns, T = 20.94 ns and - 156. a 7r Figure 10(b): e (i, t) as a function of ns, T =20.94 ns and L = a t/T with r as the parameter. T 3.33 156. 4

Figure 10(c): Figure 10(d): Figure 11 (a): Figure 11(b): Figure 11(c): Figure 11(d): e (7, t) as a function of t/r with r as the parameter. ns, T = 20.94 ns and = 156. a e (0 t) as a function of t/T with r as the parameter. 66' LL ns, T = 20.94 ns and a = 156. e (2 t) as a function of t/T with r as the parameter. u 2 a 0 ns, T = 20.94 ns anda = 100. ee (7, t) as a function of t/T with r as the parameter. ns, T = 20.94 ns and - 100. 7T ee (-, t) as a function of t/T with r as the parameter. L 4 o ns, T = 20.94 ns and -L = 100. a e (-, t) as a function of t/Tr with r as the parameter. ns, T = 20.94 ns and -= 100. a T = 3.33 T 3.33 T 3.33 T=3.33 3.33 T 3. 33 Figure 12: Figure 13: Figure A-1: Figure A-2: Figure A-3: Figure A-4: t /T as a function of r with 6 as the parameter. T 3.33 ns, L L T = 20.94 ns; L.= 156, ----— L= 100. a a Early time behavior of e (0, t) as a function of time. T = 3.33 ns, T = 20.94 ns, r = 480, -a 156. numerical;. --- —asymptotic results of Latham and Lee. (List of Figure Captions from Appendix A) Division of the antenna into N-sections. Real (I ) and imaginary (I ) parts of the current distribution on r C uniformly loaded dipole. L = 7r/2, / = 9.92, = 2X/r/. Current distribution on non-reflectively loaded monopole. Q = 11.5, tL/c = 27r, Oa = 5. 3. Current distribution on exponentially loaded monopole. f a 11.5, Oa w 5.3, wL/O w 2w. 5

Figure A-5: Phase variation of the current along the antenna. Figure A-6: Current distribution on non-reflectively loaded antenna. / = 11.5, kL = 5r, *ar 5.3, R (z) = 60 *a/L -I z. Figure A-7: Phase variation of the current along the non-reflectively loaded antenna. 1= 11.5, kL =5r, ca = 5.3, R (z) 60 *ta/L -Iz. Figure A-8: Transfer function f (0 w) of the non-reflectively loaded antenna. = 11.5, kL=5r, 5,a =5.3, R(z) 60 Ia/L -|z| Figure A-9: Amplitude of the current distribution on resistively loaded dipole antenna as a function of loading. s = 11.5, kL = 57r, hOa variable. Figure A-lOa: Amplitude of the current distribution II(z, A)l vs z/L on the loaded antenna with kL ( uL/c), as the parameter, C = 60 x 4. Figure A-lOb: Amplitude of the current distribution II(zow) vs z/L on the loaded antenna with kL (= wL/c), as the parameter, C X 60 x 5. Figure A-lOc: Amplitude of the current distribution I (z, u) vs z/L on the loaded antenna with kL (= wL/c), as the parameter, C = 60 x 8. Figure A-lOd: Amplitude of the current distribution II(z, w) vs z/L on the loaded antenna with kL (= wL/c), as the parameter, C z 60 x 10. Figure A-lla: Magnitude and phase of the transfer function f0 (r/2w), of the loaded antenna in the direction ra r/2 as a function of kL = (tcL/c) for different loading. C = 60 x 4. Figure A-llb: Magnitude and phase of the transfer function f (7r/2,w) of the loaded antenna in the direction = 7r/2 as a function of kL = (wL/c) for different loading. C = 60 x 5. Figure A-11o: Magnitude and phase of the transfer function f (7r/2,w) of the loaded antenna in the direction 9 = r/2 as a function of kL i (wL/c) for different loading. C = 60 x 8. 6

Figure A- ld: Figure A-12a: Figure A-12b: Figure A-12c: Figure A-12d: Figure A-13a: MKlliltldtlo 1ll phlto of the transfer function f0 (/2, w) of the loaded antenna in the direction 0 = 7r/2 as functions of kL = (wL/c) for different loading. C X 60 x 10. Magnitude and phase of the transfer function f (r/3, w) of the loaded antenna in the direction 0 = r/3 as functions of kL = (wL/c) for different loading. C = 60 x 4. Magnitude and phase of the transfer function f (Tr/3, w) of the loaded antenna in the direction 0 = r/3 as functions of kL = (wL/c) for different loading. C = 60 x 5. Magnitude and phase of the transfer function fle (r/3, u) of the loaded antenna in the direction 7r = 7r/3 as functions of kL a (wL/c) for different loading. C x 60 x 8. Magnitude and phase of the transfer function f (7r/3, t) of the loaded antenna in the direction 0 7r/3 as functions of kL (wL/c) for different loading. C = 60 x 10. Magnitude and phase of the transfer function f0 (r/4, O) of the loaded antenna in the direction of 90 r/4 as functions of kL = (wL/c) for different loading. C = 60 x 4. Figure A-13b: Figure A-13c: Magnitude and phase of the transfer function antenna in the direction 0 a r/4 as functions different loading. C = 60 x 4. Magnitude and phase of the transfer function antenna in the direction 0 = r/4 as functions different loading. C = 60 x 8. f (r/4, to) of the loaded of kL (wL/c) for f (x/4, w) of the loaded of kL (o wL/c) for Figure A-13d: Magnitude and phase of the transfer function f/(w/4, a) of the loaded antenna in the direction 0 i r/4 as functions of kL (a wL/c) for different loading. C " 60 x 10. 7

Figure A-14a: MIllgnltudo and phase of the transfer function f0 (r/6, w) of the loaded antenna in the direction 0 = 7r/6 as functions of kL (wL/c) for different loading. C a 60 x 4. Figure A-14b: Magnitude and phase of the transfer function antenna in the direction e = 7r/6 as functions different loading. C = 60 x 5. f (rr/6, w) of the loaded of kL (wL/c) for Figure A-14c: Figure A-14d: Figure A-15a: Figure A-15b: Figure A-16a: Figure A-16b: Figure A-17: Figure A-18a: Figure A-18b: Magnitude and phase of the transfer function f0 (/6, w) of the loaded antenna in the direction 0 = 7r/6 as functions of kL = (wL/c) for different loading. C = 60 x 8. Magnitude and phase of the transfer function f (7r/6, o) of the loaded antenna in the direction 0 rwr/6 as functions of kL ( (wL/c) for different loading. C " 60 x 10. I ee (7r/2, w) vs wL/c for the loaded antenna excited by a Gaussian pulse with = 0.471 ns. ee (r/2, t) vs t for the loaded antenna excited by a Gaussian pulse with a w 0.471 ns. e (7r/2, co) vs wL/c for the loaded antenna excited by a Guassian pulse with a X 1 ns. e (7r/2, t) vs t for the loaded antenna excited by a Gaussian pulse with a 1 ns. e (7r/2, t) vs t and e (r/2o, t) vs wL/c for the loaded antenna excited by a Gaussian pulse with a = 3.33 ns. e (r/3, j) vs wL/c for the loaded antenna excited by a Gaussian pulse with a= 0.471 ns. e (r/3, t) vs t for the loaded antenna excited by a Gaussian pulse with a= 0.471 ns. 8

Figure A-19a: Figure A-19b: Figure A-20a: Figure A-20b: Figure A-21: Figure A-22a: Figure A-22b: Figure A*-23a: Figure A-23b: Figure A-24a: Figure A-24b: Figure A-25a: Figure A-25b: |e (X/4, )| vs WL/c for the loaded antenna excited by a Gaussian pulse with a = 0.471 ns. ee (r/4, t) vs t for the loaded antenna excited by a Gaussian pulse with a = 0.471 ns. e (7r/6, to) vs wL/c for the loaded antenna excited by a Gaussian pulse with a = 0.471 ns. ee (r/6, t) vs t for the loaded antenna excited by a Gaussian pulse with a X 0.471 ns. Gamma pulse with d = 1.7 x 10 /sec obtained from Fast Fourier Transform program. eo (vr/2, c) vs wL/c for the loaded antenna excited by a gamma 9 pulse with d " 1.7 x 10 /sec. e0 (r/2, t) vs t for the loaded antenna excited by a gamma pulse with d X 1.7 x 10 /sec. ' e(7r/3, w) j vs wL/c for the loaded antenna excited by a gamma pulse with d = 1.7 x 10 /sec. e6 (7r/3, t) vs t for the loaded antenna excited by a gamma pulse with d = 1.7x 10 /sec. | e (/4, W)I vs wL/c for the loaded antenna excited by a gamma pulse with d = 1.7 x 10 /sec. e (7r/4, t) vs t for the loaded antenna excited by a gamma pulse with d 1.7 x 109/sec. e (7r/6, W) vs wL/c for the loaded antenna excited by a gamma pulse. C = 60 x 4, d " 1.7 x 10 /sec. e (r/6, t) vs t for the loaded antenna excited by a gamma pulse C 60x4, d=1.7x 109/eeo. 9

Figure B-la: Figure B-lb: (List of Figure Captions from Appendix B) Far field waveform in the broadside direction, a = 1 ns; 7 = 3, 33 ns; i = 5. Far field waveform in the broadside direction. a 1 ns; r = 3. 33 ns; O = 8. Figure B-2a: Figure B-2b: Figure B-3a: Figure B-3b: Figure B-3c: Figure B-4a: Figure B-4b: Figure B-4c: Far field waveform in the broadside direction? 3.33 ns; 0= 5. Far field waveform in the broadside direction, =3.33ns; 0x8. Far field waveform in the direction 6 i r/6. ns; =8, Far field waveform in the direction 0 a r/4. ns; 0 = 8. Far field waveform in the direction 60 r/3. ns; b u 8. Far field waveform in the direction 69 r/6. ns; O =8, Far field waveform in the direction 0 m r/4. ns; * 8. Far field waveform in the direction 0 = r/3. ns; 0 m 8.. a 0.471ns; a= 0.471 ns; al ns; T-3.33 a 1 ns; T 3.33 a 1 ns; T 3.33 a D. 471 ns; = 3.33 aw 0.471 ns; T = 3.33 a a 0.471 ns; r7 3.33 10

I INTRODUCTION The transient far field waveforms radiated by linear antennas, loaded non-uniformly and continuously with resistance, are investigated in the present technical note. The antennas considered are finite length thin cylinders excited symmetrically by unit slice generators having unit step function type of time dependence. Emphasis is given here to a specific type of resistive loading in which the amount of loading increases continuously towards the antenna end points where it becomes infinitely large. Resistively loaded long and thin dipoles may be used as simulators (RES) to radiate intense electromagnetic pulses of desirable shapes. Basic design considerations and limitations of pulse radiating long dipoles have been discussed 1 2 by Baum '. In any pulse radiating finite antenna, the reflection effects at the antenna end points produce undesirable features and distortions in the radiated waveforms. One way to reduce such distortions is to attenuate the outward traveling current waves on the antenna to an insignificant amount by the time they reach the end points. A possible method of obtaining this is to load the antenna with resistance; reactive loading being frequency sensitive is ruled out from the wave distortion point of view. It is also clear from physical considerations that uniform resistive loading is incapable of eliminating the wave reflections at the antenna end points although it may reduce the effects of such reflections to a certain extent. It has been shown by Wu and King and by Baum that with a special type of non-uniform resistive loading it is possible to obtain a reflection free antenna, i. e., the antenna sustains only single outward traveling current 4 wave. As discussed by Baum, such an antenna may be used to radiate intense electromagnetic pulses of desirable shape. The motivation of the present research has been to investigate and obtain quantitative results for the various effects of the resistive loading on the transient radiation from such an antenna. 11

II BACKGROUND Rigorous analytical solutions of similar time dependent boundary value problems are extremely difficult, if not impossible, and are available only for some highly restricted cases. Wu and Morgan have derived theoretical expressions for the transient current distribution on unloaded infinitely long linear antennas excited by a step voltage across an infinitesimal center gap. Brundell treated analytically the more general problem of determining the fields anywhere in space for a step excited unloaded dipole. Transient radiated fields for unloaded and uniformly loaded infinite linear antennas with similar excitations have been discussed theoretically by Latham and Lee '. The waveforms radiated by a Gaussian pulse excited infinitely long dipole have been obtained 10 numerically by Harrison and King. The time dependent far fields radiated by unloaded and uniformly loaded finite linear antennas with Gaussian pulse excitation have been obtained by Bennett and Auckenthaler by applying direct time domain numerical analysis. Transients in step voltage excited unloaded cylindrical antennas have been investigated both theoretically and experimentally 12 by Schmitt. Schmitt's theoretical time domain radiated waveforms are of limited value because of the fact that he arbitrarily truncated the frequency domain results at kL ^- 5, where 2L is the length of the antennas and k = c is the free space propagation constant. By applying the moment methods in the 13 time domain Sayre obtained the waveforms radiated by unloaded and loaded 14,15 dipoles for step excitation. Taylor and his group obtained by numerical means some results of limited application for the case of discretely loaded linear antennas excited by step voltage. Transient waveforms radiated by discretely loaded linear antennas with step excitation have also been studied numer16, 17 ically by Merewether Transient waveforms radiated by a non-uniformly loaded finite linear 12

antenna with step voltage excitation have first been investigated analytically 4 by Baum who used the transmission line model for the antenna. The transmission line model simplifies considerably the analytical investigation of the antenna and led to the development of the special non-reflective loading which 4 significantly improves the radiated waveform. However, because of the basic approximations involved the results obtained from the transmission line model are rather approximate. In the following sections we investigate numerically the waveforms radiated by a step voltage excited linear antenna which is continuously and non-uniformly loaded with resistance along its length. At first the radiation fields produced by the harmonically excited antenna are obtained. The far field waveforms produced by the step voltage excited antenna are then obtained numerically with the help of Fast Fourier Inversion technique. The effects of the loading and other physical parameters of the antenna on the radiated waveforms are also studied in detail. Numerical investigations of the waveforms radiated by similar antennas with short pulse excitations and analytical results for the Gaussian pulse excitations were discussed in our two previous reports. In view of their limited accessibility,and also for completeness, these two reports are included here as Appendices A and B without any significant editing. 13

HI BASIC RELATIONS Let us assume that a linear antenna of length 2L be aligned along the z-axis of a rectangular coordinate system with the origin located at the center of the antenna and that it is excited by a unit slice generator located at the origin. The far electric field produced by such an antenna is directed entirely in the 0-direction. In the time harmonic case the far electric field is given by the following: F =0(r, 9, w)ewt (1) where, F (r,, )= 4c e I (zW) e dz, (2) = a is the intrinsic impedance of free space, 1 c = is the velocity of light in free space, o is the radian frequency of excitation, r, 0, 0 are the spherical polar coordinates of the far field point with the origin located at the center of the antenna, I(z', w) is the current distribution on the antenna due to the harmonically time dependent unit slice generator. In Eq. (2), F (r, 0, w) is the time independent far electric field produced by the antenna. For convenience we define the antenna transfer function in the following manner: 14

I r f(, a)= r F(r, O, W) e juro sin 0 j z' Cos 4=rc I(z',w)e dz'. (3) Notice that in the transfer function given by Eq. (3) the dependence of the field amplitude on r as well as the phase shift suffered by the field in traveling from the antenna to the far field point are both suppressed. As a result of this all the field quantitites in our subsequent discussion will be independent of r. Let the slice generator have arbitrary time dependence such that the input signal voltage in time is represented V (t). It is now assumed that V (t) is Fourier transformable. This means that the following relations hold: V() V(t) e-jt dt, (4) V(t) = (w)e dw. (5) 2 O After making use of the linearity of the system along with the superposition theorem and the concepts of Fourier transform technique, it can be shown that the time dependent far field produced by the antenna excited by a unit slice generator signal V (t) is given by the following: e(0, t) = i(' w)V(w )e dw = dw 15

In Eq. (6) the quantity 'e(0, w) = f (0, W)V(w) may be looked upon as the spectral density of the far field waveform. Notice that, by definition, e (0, t) and e (6, t) are related to each other by the following transform relationship: e (0e, W)T(0, )) ) = e(O, t) e -jt dt. (7) In the present problem the transient voltage at the 6-gap located at the center of the antenna is represented by V(t)= f(t), (8) where, r/T s O<t<T f(t) = (9) 0, otherwise For sufficiently large T (usually T = few times r, where T= L = the transit time c on the antenna) the radiated waveforms may be identified with those radiated by the same antenna when excited by a unit step function voltage in time, i. e., V (t) = U (t). For convenience of numerical computation, the rectangular pulse type of excitation has been used here to obtain the step voltage response from the antenna. With the input signal of the form given by Eqs. (8) and (9) we have,, ( sin wT/2 -jwT2 V (S) = e (10) as the spectrum of the input voltage signal. As can be seen from the above, the first step in the first step in the andetermination of the current distribution T(z', t) on the antenna when it is excited by a harmonically time dependent unit slice generator. Let us assume that the an 16

tenna is loaded with distributed resistance such that its internal impedance may be expressed as r (z') ohms/meter for 0 < z'J < L. Under this condition it - s can be shown that the current distribution I(z', u) on the antenna satisfies the following modified Hallen's integral equation: _ Fzz +a2] 4[ (.z - zz)+a2 1/2 dz' = Bcoskz -- sin kl|z 20o + — iT(,w)r () sink(z —)d?, (11) 0 where, k - is the propagation constant, C B is a constant to be determined from the end condition I(+L,= 0, a is the radius of the antenna element. 17

IV A NOTE ON THE LOADING The antenna is assumed to be resistively loaded such that its internal impedance may be expressed as: r r (z') = i ohms/meter, -L < z' < L, (12) where r = r (0) ohms/meter and is referred to as the loading parameter. Eq. 0 5 (12) predicts that the internal resistance of the antenna increases continuously from the value r at the input end to infinity at the antenna end points. The rationale behind this particular type of resistive loading is as follows. It is 3 known that for a linear antenna loaded according to Eq. (12) there exists a 2L critical value of r, depending on the thickness parameter Q = 2tn of the o a antenna, for which the loaded antenna excited by a harmonic slice generator sustains a single wave of current traveling in the direction of increasing z', i. e., from the generator towards both ends of the antenna. In other words, for this type of loading there is no reflected wave on the antenna traveling in the opposite direction. For this reason the antenna loaded according to (12) and with this critical value of r is sometimes referred to as the reflectionless 4 0 antenna. Baum arrived at the same conclusion from his transmission line analysis of the same antenna. For r less than the above critical value the antenna stops being reflectionless and sustains a standing wave type of current 18R distribution. From theoretical considerations Shen and Wu have found that for r larger than the critical r the antenna sustains a progressive wave of current whose distribution may be expressed as a hypergeometric function. Our previous numerical investigation of the problem (Appendix A) essentially confirmed the above observations. In the present study we investigate in detail 18

the effects of the variation of the loading parameter r0 on the transient far field waveforms radiated by the antenna. 19

V BRIEF OUTLINE OF THE NUMERICAL METHOD USED As mentioned in Section II we have used numerical techniques to obtain the solution of the problem outlined in Section Im. Detailed discussion of the numerical methods is given in Appendix A. In this section we give only a brief outline of the numerical techniques used to obtain the various quantities. 19,20 Standard numerical techniques 20 are used to solve Eq. (11) for I(z', c). For this purpose the integral equation (11) is reduced by moment methods to the following set of N simultaneous algebraic equations: N I(z' ) G(z., z) d z n=1 z n B cos kzj- 2 sinkz 3 II klz l +-i- (z'o)r (z')sink (z-z')dz' (13) n oZ n j=1, 2,.....,N, where, 2 2 1/2 e-j k zJ- Z)+ a G(z rZ 2 1/2 (14) )2 2 4r (z-z) +a N/2 - o for j 2 2' n= f (15) n N, for J 2 ' N n-+ 1 2 20

and it is assumed that N is an even number and Z' EA. Eq. (13) implies n that the antenna of length 2L is divided into N sections, the numbering of the sections increasing from 1 to N along the antenna length from -L to L. It is now necessary to make an appropriate approximation to the current distribution I(z') (for simplicity we use the notation I(z') for T(z', )) in each of the sections A z. We make the following quadratic approximation to the unn known current in each section: T(z)=A +B (z- )+C (z'-z ), for z'EAz n n n n n n (16) 0, otherwise, where An, Bn, C are the three unknown constants. These constants are n n n determined by requiring that the continuations of I(z') expression given by Eq. (16) into the centers of the adjacent sections give the appropriate current values there. After evaluating the constants it can be shown (Appendix A) that the current in each section is given by the following recurrence relation: I(z')=I X (z)+I Y (z) + I Z (z') z' Az (17) nin nn nii n n where,, -z Z )2 X zW + -- n zIAZn (18) n 2Az 2Az2 n 2Az n 2 ' n z -z (z'-z )2 Z(z ) = n + 2 z eAz (20) n 2Az 2 n 2Az '(z )= I etc., n n Az is the length of each section, 21

n 1, N. For the two end sections A z and A zN the currents are given by the following (Appendix A):i (z 1 Y1 (z')+ I Z (z'), for z'EAz1 (21) ()= INY (z')+ +IN X'(z') for z'eAzN (22) N N N-N N N where, -2 (z' -1 (z'-z ) Y (z') a 1+ 2 for z'eAz1, (23) 1 Az 2 1 (z'-z1) 2(z'-z )2 (z'-zl) 2(z'-z )2 '? (Z') = - J. + 2. for z' EA z1 (24) Y(z') 1- N - 2, for ZEAZN (25) N Az 2 N LAz (z -zN) 2(z' -N)2 X' (z')= - + - for z'EN. (26) N 3Az 3A2 3Az algebraic equations involving the N unknown current coefficients I1, I,..., I. The extra unknown constant B is now determined by applying the end condition I (L, ) = 0. By using Taylor series expansion for the currents at the centers of the last four sections and retaining four terms in the series, we obtain the following extra equation: -5N-3 + 21N2 - 35 IN_-1+ 35IN ~0 (27) We now have N+ 1 equations for the N+1 unknowns and the system of equations are solved by usual matrix methods for I, I2,..., IN and B. 22

The transfer function f0(0, c) given by Eq. (3) can now be evaluated numerically with the help of the following relation: wt sin9 N j- z cos0 f(O, w) =j 4 I Az (28) n=l where we have used the notation T(z, w) = I and z' = z is the coordinate of n n n the center of the section A z n The spectral density eO (0, w) of the radiated waveforms is obtained by using Eqs. (7), (10) and (28). Finally the time dependent radiated waveform e (0, t) is obtained by carrying out the integral given by Eq. (6) with the help of Fast Fourier Inversion technique. During the numerical computation the antenna of length 2L has been divided into N equal length sections. Time domain results have been obtained for L = 1 meter so that the antenna is 2 meters long. The frequency domain calculations have been truncated at the highest frequency f such that Az = 6 ~ where x is the free space wavelength corresponding to f. In the next two sections we discuss the various results obtained. 23

VI CURRENT DISTRIBUTION (z, w) It is instructive to study how the loading effects the current distribution on the harmonically excited antenna. Figures 1 and 2 show the amplitude and the phase distribution, respectively, of the time independent current on the linear antenna of length 2L = L L X 156 and r C 318. Under these condia o tions the antenna approximately satisfies the non-reflecting criterion3'4'21 The approximate theories of Wu and King3 and Baum4 predict the existence of a pure traveling wave of current with linearly decaying amplitude distribution on the antenna. The results shown in Fig. 1 indicate that the amplitude distribution of current may be considered to be approximately linear except near the feed region where it is significantly different. The linear variation of phase in Fig. 2 indicates the existence of a pure traveling wave of current in the antenna. Figures 3 and 4 show the amplitude and the phase distribution of the current on the harmonically excited antenna of length 2L = 3 X, L = 156 and for a different values of the loading parameter r. For r < 318, the reflection effects on the current distribution become appreciable; the results for r = 120 clearly indicate that the antenna sustains a standing wave type of current distribution. For r > 318, the amplitude distribution of the current resembles an ex0 ponentially decaying function and the phase distribution, although not linear, is progressive along the length of the antenna. In this sense it is proper to say that for r larger than the critical value of the loading parameter (i. e., larger than 318 in this case) the antenna maintains its non-reflecting properties, but it supports a progressive wave of current. It should be noted that the critical value of the loading parameter r is different for an antenna with different value of L. 0 L a For a given non-reflecting antenna with -L = 156, r0 = 600, the amplitude and phase of I(z, W) are shown in Fig. 5 for two values of frequency w such 24

3 7 3 0 I I I I I I I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 1: Amplitude of the current distribution along a non-reflectively M L wL loaded antenna - a ' r j 25... I

180 100 - \ \ 0.1 0.2 0.3 0. 0.6 0.7 0.9 1.0 0 -100 \ \ \ -180 Figure 2: The phase variation of the current along a nonreflectively loaded antenna. a= 156, 5r, a c r =318. 26

7 6 l 5 W /o 3 2 0. 0 0.25 0.50 0.75 1.0 L Figure 3: The amplitude of the current distribution along the antenna L wL as a function of the loading parameter. a 156, c- 37r. -.... 27

180 160 - 140 120 100 - 80 o 40 20 0 0. 25 0.50 0. 75 1. O -20 -40 - -60 -80 -100 r 120 - 120 -140 -160 r = 240, r = 360 360 0 - 180 480 r = 48V 0 Figure 4: The phase of the current distribution along the antenna L wL as a function of the loading parameter. — = 156, = 37r. a c 28/ 0!0 = 240 o

wL -..- -.-..r I(z, )) for -= 10 ----.- Arg I(z, w) for 10 I-... |I(z, M)| forL = 30 -L - -- - Arg I(z, ) for - = 30 C 4.0 I I I I \ -] 120 \\ \ \ I I CI 1 1 1 - 1 I I - \ \ \ 60 g i~o \ \ 2.0- \ 1 \ \\ \1tlo 0- I \ I 1f~ 1 1. I I \ \ I \ O\ \ l 180 0 0.25 0.50 0.75 Iz/L -_ L Figure 5: The amplitude and phase of the current distribution along a nonreflectively loaded antenna for two different frequencies. L 156, r = 600. a o

that _L = 10 and.L= 30. It is found that the antenna retains its non-reflecting c c characteristics within the range of frequencies considered in Fig. 5. 30

VII TRANSFER FUNCTION OF THE ANTENNA f (6, E) In this section the magnitude and phase of the transfer function of the antenna are given as functions of L/c and for different values of the loading parameter r. Figure 6 a gives the variations of the magnitude and phase of 0 f (0, ) in the broadside direction (6 = 7r/2) of the antenna of length 2L and - = 156 and for three values of the loading parameter. Similar results are a shown in Figs. 6b-d for the same antenna and for three different directions. The corresponding results for an antenna of length 2L and - 100 are shown in Figs. 7 a-d. In all the curves, f0(0, W)j approaches zero as w approaches zero which is consistent with the fact that there is no radiation at zero frequency. For higher frequencies the mean value of jf (0, W)I tends to increase with an increase of frequency. In general, the mean value of the phase of f(0, w) 6' appears to decrease rapidly for small values of L/c and then assumes a constant value. In each case for r = 240 both the amplitude and phase of f0 (0, ) oscilo 6 late with OL/c. With increase of r these oscillations are smoothed out. It is o interesting to observe that the oscillations in the amplitude and phase of f (6, w) appear to be smoothed out for r > the critical loading parameter. The increase of the parameter L tends to increase the amount of oscillation in the curves. 31 i

300 200 100 150~ r = 240 o *-% D f: kC r= 480. O 100~ AMPLITUDE r0= 480 Figure 6 (a): Amplitude and phase of '(' w) as functions of 2 L normalized frequency with r as the parameter. -= 156. r = 240 L/ a 0 wl/c 510 15 I20 5 10 15 20

300 200 100 3 PHASE 150~ 10 100 r = 240 0 r = 600 o AMPLITUDE Figure 6(b): Amplitude and phase of f0 (-, w) as functions of normalized L frequency with r as the parameter. a= 156. Ot a I 10- 5 2 5 10 15 20

300 r = 240 o 1500 AMPLITUDE PHASE 200 100 3 4i, to.29) ~ 100~ 50~ r = 600 0 Figure 6(c): Amplitude and phase of f(T, W() as functions of 04 4 L normalized frequency with r as the parameter. = 156. %... A' O ~ ' a' 5 10 15 20

PHASE r = 240 0 3 t:.-I %O 100~ 50~ r = 600 o r =480 o r =240 0 Figure 6 (d): Amplitude and phase of f0 (6, so) as functions of normalized qu as 56 frequency with r as the parameter. = 156. o a 5 10 15 20

300 150~ AMPLITUDE /PHASE /dz 200 r = 240 r 480 ~3 o r = 720 r = 480 0 100 r = 720 r 240 0 Figure 7 (a): Amplitude and phase of f ([, W) as functions of nornzaized L frequency with r as the paranmete- -= 100. a if - ' A.0L/ 5o 100~ 3 k 50~ /c 5 10 15 20

150C 300 200 100 AMPLITUDE - 240 PHASE,,d 3 r = 720 0 tz I co) CZ) 14-4 100c 50~ N 7T Figure 7(b): Amplitude and phase of f (-, w) as functions of normalized L frequency with r as the parameter. - 100. 0 a 0 Z 0 5 10 15 20

300 AMPLITUDE r = 240 PHASE 200 CO 100l r =- 720 o r = 480 0 N 7r Figure 7 (c): Amplitude and phase of f0 (, t) as functions of normalized frequency with r as the parameter frequency with r as the parameter. -= 100. 0a uL/w /c 10 15 20 0 5

,' 7r Figure 7 (d): Amplitude and phase of f (6, a) as functions of 500 normalized frequency with r as the parameter. L L 100. a PHASE 400 r =720 0 r =480 O 300 - r =240 zl*t r — 480 200 r =240 o AMPLITUDE 100 Br = 720 WL/ Q 4 -bD~ Sk - 100~ 50~ 0 5 10 15 20

vmI SPECTRAL DENSITY e (0, ( ) The spectral densities e (0, w) of the transient waveforms radiated by the step voltage excited antenna under various conditions are discussed in the present section. For three given values of r and - = 156, Figs. 8 a-d show the envelope of le(0, w)j vs. wL/c for four values of the observation angle 9. It is found from Fig. 8 that in general the envelope of e'(0e, w)| vs. wL/c has a dominant peak at the low frequency end for all values of 0 and r. After the initial peak | e (0, w)| decays at first rapidly and then slowly with increase of WL/c. Depending on the values of r and 0 there also appear some minor peaks in| e (0, )\ for large values of wYc*. For the purpose of discussing the behavior of l e(09, W)j let us define the frequency regions larger and smaller than the frequency where the initial peak appears as the high and low frequency regions, respectively. Critical study of the results shown in Fig. 8 reveals the following observations: (a) As 0 decreases away from the broadside direction 0 = 90, the low frequency content of len(0(, W) decreases and the high frequency content decreases. Also for a given loading, as 0 decreases the amplitude of the initial peak decreases. For a given r, the position of the initial peak appears to be independent of 0. As 0 decreases from 90~ the minor peaks in 'e (, a) become appreciable. (b) As r increases, the low frequency content of |e (0, w) and the initial peak in e (0, W) decreases significantly. The rate of decrease of le'(0, W)I in the high frequency end appears to be almost independent of r. (c) In the broadside direction the position of the initial peak in le (0, /2) increases with an increase in the loading parameter r. For example, from Fig. 8 a it is found that the initial peaks are located at 'L/c ' 1.56, 2.4 and 401

160 140 120 r = 240 0 100 r =480 - 80 10 --- r = 600 60 Figure 8 (a): Envelope of |e (, w) as a function of normalize( 40 L frequency with r as the parameter. a= 156. -— a_____. 20 0 0 10 20 30 40 I UJ/c

120O r = 240 o r = 480 o r = 600 0 Figure 8 (b): Envelope of le ( 3, c) as a function of normalized frequency with ro as the parameter. - = 156. o a 20 t 10 20 30 40 wL /c

100 80 60 40 20 0o = 240 r = 480 0 Figure 8(c): Envelope of fe0(,,,o) as a function of normalized L frequency with r as the parameter. a — 156. 0 r = 600 0 I I I - I I. 10 20 30 40 CL/

r - -240 0 -- 4( 1-3. P- Ir-I o wr — %-.# C> I r = 600 0 ji Z e aL ol laolIlA,! W) -aL S a III 'actiola L - 15-0. i ( 61,,ete v * a. E.,velope 0 0.,,, the Pat. h Fjure 9 (d) - t r Irequency NVI 0 1 V 30 20 0

2. 6 for r = 240, 480 and 600, respectively. o The above observations will have implications in the corresponding time dependent waveforms. Figures 9 a-d show the corresponding results for the antenna with L= 100. The general behavior of these results is similar to that of the results shown in Fig. 8. On comparing the results of Figs. L 8 and 9 it is found that the decrease of -a tends to make the minor peaks in l'e (0, to) less pronounced. 45

160 140 120 = 240 -r -480 o -~ O-f* %-I Q> ZGp r =720 0 40 Figure 9(a): Envelope of ('e ( w) as a function of normalized frequency with r as the parameter. 100 frequency with r as the parameter. ' 100. oa 20 0 10 20 30 40

140 120 100 80 3 t-, D 60 40 20 0 r = 240 = 480 r = 720 o Figure 9(b): Envelope of e (?, w) as a function of normalized frequency with r as the parameter. -a= 100. O a I I 0 10 20. 30 40

100 80 oo 3 60 hts| I-z 40 20,r = 240 0 r =480 0 Figure 9(c): Envelope of l e(4 )J as a function of normalized L frequency with r as the parameter. - = 100. o a r = 600 o 0 10 20 30 40

70 6C 51 3 tzI a 40 30 2C 11 I r =240 0 wa r=720.7 ucino Figure 9 (d): Envelope of\( 6) as asfunthef ofamte L normalized frequeflcy Wvit asteprmee.0 0 1-1-M..-w I kL

Ix TIME DEPENDENT RADIATED WAVEFORM e (0, t) Figures 10 a-d and 11 a-d show the time dependent waveforms in different directions radiated by loaded linear antennas for L- 156 and 100, respectively, a and for various values of the loading parameter r. In each case L = 1 meter 0 so that T t 3. 33 ns and the width of the input rectangular voltage pulse T = 20. 94 ns. It has been found that for T > 3 T, the time dependent radiated waveforms due to the discontinuities at the two ends of the input pulse do not interfere with each other. Hence, the results shown here for T = 20. 94 ns may be identified with the transient waveforms radiated by the antenna for step voltage excitation. The undulations in the waveform for r = 240 (for example, Fig. 10 a) are attributed to the existence of reflected current waves on the antenna. In general, the existence of reflection effects causes the waveform to cross the zero axis more than once, as can be seen in the r = 240 waveform in Fig. 10 a. For r > the critical value, the radiated waveform has only one zero crossing. It has been found that for a given angle e, the increase of r has the following effects on the radiated waveforms: (i) decreases the initial amplitude e (0, t). This conclusion is based on the numerical sense. In the asymptotic results obtained from analytical 9 1 studies it is found that e (0, t) j as t —0. (ii) increases the rate of decay of e (0, t) after the initial rise, (iii) decreases the first zero crossing time t and consequently decreases the pulse width, (iv) decreases the amplitude e0(9, t) for t > /T. For a given value of the loading parameter r, the decrease of the observation angle 0 from the broadside direction (0 = 7/2) has the following effects on the radiated waveforms: (i) increases the rate of decrease of e (0, t) from its initial amplitude and decreases the zero crossing time t. Consequently, the pulse width decreases, 50

150 i 120 90 60 I- " 0 0 I ro= 240 r = 480 - t X r I * * * BAUM'S RESULTS FOR r = 690 o 30 t 't - 4 2! 3 I 5 I 0 r r. XI.C I LI K r ='600 0 0 or_ - -30 Figure 10(a): e (-, t) as a function of t/r with r as the parameter. T = 3. 33 ns, 0 2 0 T L -T 6 and a=156. T a

150 120 90 's -60 Flpr =240 r = 480 0 Figure 10(b): e (, t) as a function of t/T with r as the 63' 0 \parameter. T = 3.33 ns, Tu —u 6 and |r = 600 =156. 0 a 1 2 3 0 t/T

150 120 90 tn C13 s 60 El r = 240 0 O = 480 Figure 10(c): ee(-, t) as a function of t/r with r S parameter. T= 3. 33 ns, -T 6 and 30 T L = 156. a 1 2 3 0 A1 --- — t/T as the r = 600 0

r = 240 o 01 IR'- r =480 80 o 8l o= Figure 10 (d): e( -, t) as a function of t/r with r as the T ) \a parameter. T= 3.33ns, -V6 and L -- 156. 40 a r = 600 0 1 2 3 0 C - ' -4(

180 150 120 cai CAI -3 4c 90 Figure 11(a): 60 30 0 = 240 0 e (, t) as a function of t/Twith r on the 2 T o parameter. T 3.33 ns, - 6 and L 100. 3 r 720 0 1 2 t/T

180 <* 150 120 01 90 60 30 0 240 'r =480 0 Figure 11 (b): e6 (3 t) as a function of t/r with r on the parameter. T 3.33 n, T 6 and L I L 100. 3 r =720 o 1 2 t/T

210 180 150 120 90 60 Figure 11(c): e (, t) as a function of t/T with r on the r 240 T = o240 parameter. T 3. 33 ns, T" 6 and 0 100. r =480 30 0 r 720 0 2 3 0

260 240 200 160 e0 120 = 240 r =480 0 I, Figure 11 (d): e ( t) as a function of t/T with r as the T parameter. r = 3.33 ns, 6 6 and LT = 100. a 2 3 -r = 720 o 1 o

(ii) the initial amplitude of e (0, t) increases. The shape of the waveform radiated by a non-reflecting antenna can be adjusted by adjusting the zero-crossing time t. For this purpose it is of interest to study the variation of t as a function of the various physical parameters of the antenna. Figure 12 shows the normalized zero crossing time as a function of loading with the observation angle 9 as the parameter. Notice that results are shown for values of - = 156 and 100. As far as the zero crossing time is concerned, it is found from Fig. 12 that the decrease of 0 has similar effects of t as the increase of r, i.e., t decreases with a decrease of 0. In the broadside direction, t is found to decrease with the L 0 decrease of -. In this direction the spread between the t values, for the L 0 two values of - shown, increases with an increase of r. a o It is interesting to compare the time dependent waveform obtained by numerical means with that obtained by analytical means from the transmission line model of the same antenna as done by Baum. For comparison the waveform produced by the loaded antenna with r = 690 are superimposed in Fig. 10 a. It appears from Fig. 10 a that the transmission line model predicts slightly slower decay of ee (7/2, t) for t < t than our values. For t > t, the approximate model predicts a higher value of e (/72, t). These discrepancies may be attributed to the fact that the transmission line model assumes a linearly decaying current amplitude distribution on the antenna. Our numerical investigation indicates that the amplitude of the current distribution of the antenna is, in general, approximately exponentially decaying. The early time behavior of ee (0, t) is compared with Latham and Leers asymptotic results for uniformly loaded infinite dipole antennas. It is assumed that the infinite dipole antenna is loaded uniformly in such a manner that the internal impedance of the antenna is r ohms per meter. We compare the early time behavior of the waveform radiated by this infinite antenna with that 59

1.0 lb "*-f 0.6 - 414 0.2 6 0 10 200 300 400 00 60 700 T IN OHMS/METER - 0 Figure 12: t /T as a function of r with 0 as the parameter. Tr 3. 33 ns, T= 20.94 ns; L __ _ _ _ a a

of the waveform produced by the non-reflectively loaded finite antenna for t << T. It is assumed that the paramater 'a' is the same for both antennas. The results are shown in Fig. 13 which indicates that the Latham and Lee theory predicts faster decay of e (8, t) for early times. With the assumption that the antenna is loaded uniformly with r 0, the decay rate of the field for early times should have been slower. This discrepancy indicates that the nonuniformly loaded finite antenna may not be approximated by a uniformly loaded infinite antenna for t << T. 61

150 150 120 OI 90 9 90 180 \kl 90 60\ I I 60 120 -0 30 60 0.5 \ 0 30 60 0.5 0.5 0- 0 t/T- - 0. 5 t/T — 0 L t/TFigure 13: Early time behavior of e (0, t) as a function of time. T 3. 33 ns, T = 20. 94 ns, r =480, = 156. --- numerical; --- asymptotic results of Lathaml and Lee.

x CONCLUSIONS Transient waveforms radiated by a step voltage excited resistively loaded linear antenna have been investigated by numerical means. Results have been obtained in different observation angles from the antenna and for different values of the loading parameter. It has been found that for an antenna of given L ratio there exists a critical value of the loading parameter. If the loading parameter is equal to or larger than the critical value, the antenna becomes non-reflecting. The waveform produced by the non-reflectively loaded antenna shows only one zero crossing. The zero crossing time as well as the radiated pulse width depends upon the loading parameter and the observation angle. It is hoped that these observations may be found useful in designing a pulse radiating dipole antenna. It would be interesting to compare our time dependent results with those obtained by the singularity expansion method (SEM). In particular, the relationship of the critical loading parameter with the natural resonant frequency (or frequencies) of the antenna should be investigated. 63

APPENDIX A NUMERICAL INVESTIGATION OF WAVEFORMS RADIATED BY A PULSE EXCITED RESISTIVELY LOADED LINEAR ANTENNA ABSTRACT The waveforms of the radiation field produced by non-uniformly resistanceloaded finite linear antenna excited by pulse signals are investigated by numerical means. The antenna model considered is a thin cylinder loaded symmetrically and continuously with resistance and is assumed to be excited symmetrically by a slice generator supplying a time dependent signal of arbitrary shape. Current distributions and the transfer functions of the antenna are obtained as functions of frequency for different values of the loading. Spectral density of the radiated waveform produced by the antenna is obtained as a function of frequency for two different types of input pulse and for different values of loading and widths of input pulse. Finally the radiated waveforms produced by the antenna for the particular input pulse are obtained by using Fast Fourier inversion technique. Far field waveforms are obtained in 0 = Xr/2, r /3, r /4, r /6 directions and for different values of the loading. Three selected values of the ratio of the input pulse width to the\ transit time on the antenna have been used for a Gaussian pulse, while one specific ratio value has been used for a Gamma pulse. pulse. 64

APPENDIX A 1. INTRODUCTION The waveforms of the radiation field produced by non-uniformly resistanceloaded finite linear antennas excited by pulse signals are investigated in the present Appendix. The antenna model considered is a thin cylinder loaded symmetrically and continuously with resistance. Emphasis is given to a specific type of loading in which the amount of loading increases continuously towards the antenna end-points. The antenna is excited symmetrically by a slice generator supplying short pulse type time dependent signals. Analytical solution of such a boundary value problem is extremely difficult. 4 A similar problem with step input excitation has been analyzed by Baum from the transmission line point of view. Analytic results for both unloaded and uniform 9a 9b resistively loaded antennas with step excitations are given by Latham and Lee a' only for cases when the antenna lengths are infinite. Bennett and Auckenthaler reported some results for uniformly loaded finite linear antennas obtained by applying numerical technique directly in the time domain. By applying the moment 13 methods in the time domain Sayre3 obtained results for unloaded and uniformly 14, 15 loaded linear antennas of finite length. Taylor and his group 1 obtained some results of limited application by numerical means for the case of discretely loaded linear antennas. In the present Appendix our approach to the oproblem has been numerical. At first the radiation field produced by the antenna excited by harmonically time dependent slice generators are obtained numerically as a function of frequency. The far field waveforms produced by the pulse excited antenna are then obtained numerically with the help of Fast Fourier Inversion technique. 65

2. METHODS OF ANALYSIS In this section we give a brief discussion of the method of analysis and also define a few terms that will be used throughout. Let us assume that the linear antenna of length 2L be aligned along the z-axis of a rectangular coordinate system with origin located at the center of the antenna and that it is excited by a slice generator located at the origin. In the harmonically time dependent case the voltage signal supplied to the antenna by the unit strength slice generator may be represented by ejwt volts, where w is the radian frequency. Let the current distribution on the antenna due to this source be I (z, w) e. The far electric field produced by the antenna under such conditions, will have only a O -component and may be written formally as follows: F ( r, w ej, (A. 1) where, j n wsin e j k r kz' o F I(r)' 4 e 4i(z, w) ek, c dz,.(A.2) -L where, I n ~ -e intrinsic impedance of free space, 0 1 c - = velocity of light in free space, k W - propagation constant in free space, c (r, 0, 0) = the spherical coordinates. Let the slice generators have arbitrary time dependence such that the input signal voltage envelope in time is represented by V (t). It is now assumed that V (t) is Fourier transformable i.e. V(t)e* V(S) (A. 3) 66

which means that the following relations are true, OD ()= I V(t) e tdt, (A. 4) -00 OD V(t) 1 e V ) e. (A.5) 2r J -00) Assuming the linearity of the entire system and using the superposition theorem, it can be shown that the far field produced by the antenna when excited by a slice generator having arbitrary time dependence is given by the following expression: oD (r, c 2t- F(rw)V(w)e d (A. 6) -oD Notice that E, as given by Eq. (6) is dependent on the parameter r, and also that it is delayed in time with respect to the input by c which is the retarded c time taken by the signal to reach the far field point from the antenna. It is found convenient sometimes to remove both of these effects from the final result. This can be done as follows. 'v Let us define a quantity f9 which is related to F0 in the following manner: f7(0o, u) ( r0(r, x) e Jn w sin 0 L c 47r c4r |F I(zh ',)eejkz csdz' (A. 7) -L Similarly we can remove the dependence of r and r from Eq. (A. 6) and define C the following modified field quantity produced by the antenna excited by arbitrary 67

time dependent input signals: p jr e(t) = j rFe(r, ) e c V(w) e dw -ao f(' fo ( )(w) d. (A.8) -00 Thus, formally if the field! (0, O ) produced by harmonically time dependent excitation of unit strength is known, then the field produced by any other time dependent signal can be obtained with the help of Eq. (A. 8). Of course, it is assumed that the time dependent signal must have a Fourier transform. From the analogy with signal transmission through linear system, we shall call f (0, W) given by Eq. (A. 7) as the frequency response or the transfer function of the antenna and e ( 0, w) = ft (0, ) V (tw) as the spectral density function of the radiated signal for arbitrary time dependent input signal. Note that according to this definition e0 (, t) and e' (, w) constitute a Fourier transform pair, i.e. e (6, t) e'(, () (A.9) 3. BRIEF OUTLINE OF THE REPORT It can be seen from Eq. (A. 8) that the knowledge of the transfer function of the antenna under consideration is necessary for obtaining the waveform radiated by the antenna for arbitrary signal input. Equation (A. 7) indicates that the current distribution I (z, O,) on the antenna for the harmonic time dependent excitation must be known so that f0 (0, w) may be evaluated. In the following sections we at first determine the current distribution I (z, W). This is done by numerically solving a modified form of Hallen's integral equation, appropriate for the antenna 68

under consideration. The transfer function f0 (, w) is then obtained by numerically evaluating Eq. (A. 7). The spectral density function e" (, w) is obtained by multiplying the transfer function by the Fourier spectrum of the input signal. Finally the waveform of the radiated signal is obtained by numerically evaluating Eq. (A. 8) with the help of Fast Fourier Inversion technique. Two input waveforms have been assumed in this work. They are the Gaussian and gamma pulse represented by 2 2 -t2/a V (t) = e /2 (A. Oa) -dt V2(t) te U(t) (A. 10b) where the constants parameter a and /d are proportional to the width of the input pulse and U(t) is the unit step function. The Fourier spectra corresponding to the signals given above are, respectively, 22 V1() 4 aJe- w2e c /2 (A.lla) V2(~) (+ d) (A. llb) 4. INTEGRAL EQUATION FOR THE CURRENT DISTRIBUTION I(z, W). In this section we discuss briefly the integral equation for the current distributions on thin cylindrical antennas continuously loaded with resistance. As before we consider a linear antenna of length 2 L oriented along the z-axis of the Cartesian coordinate system such that z = 0 is at the center of the antenna. Assuming azimuthally independent excitation, Hallen's integral equation for the current distribution on a linear antenna excited by harmonically time dependent slice generator of unit strength is given by: L z I(z', )G(z,z')dz' "Bcoskt-j/n E () ink(z-Jo)d _L (A. 12) 69

where, E is the electric field on the surface of the antenna due to the induced sz currents. G( z, z' ) is the free space Green's function and B is a constant to be determined from the end condition I (+ L) = 0, a is the radius of the antenna element and E is the free space permittivity. Since the dipole is electrically thin, i.e., a < 0. 01, we can assume the current to be located at the axis of the antenna. This implies that the Green's function is approximately given by: 1/2 -Jk [(z-z) + a2] G (z, z') ' e.-.1/ (A.13) I 2 2 1/2 4r [(z-zV) +a J If the antenna is loaded with distributed resistance R (z) ohms/meter, then the total tangential electrical field on the surface of the antenna is given by: E (z) = I(z,w)R (z) = E (z)+ E (z) (A.14) z S oz sz where E (z) is the field due to the externally impressed source. In the oz present case the external source is assumed to be a unit slice generator with harmonic time dependence, i.e., E (z) = 6(z), where 6 (z) is the Dirac delta function. oz Under these conditions, the integral equation for the current distribution on a symmetrically loaded linear antenna is given by: ) ezk (z-z') +a fI(z' t)eik-_F -2-kJ1 dz' -L 4w [(zz 2+a /2 4,r (z - z') + a2, B coskz - -J- sink zl + 70

+ J | I(,w)R (e)sin k (z - )d (A. 15) Eq. (A. 15) is the desired integral equation for the current distribution on the loaded antenna when excited by slice generator of unit strength having harmonic time dependence e. In the next section Eq.(A. 15) will be solved numerically for some assumed values of R (z). The loading function of special importance to us is of the form: C R (z) L Iz (A.16) Note that according to the previous notation of Eq. (12), r = C/L. For some special value of C, the above loading gives rise to a pure outward traveling wave 9 at a specific frequency on the antenna as discussed by Wu and King. For this reason the antenna with this special loading is sometimes referred to as the 1 reflectionless antenna. Baum arrived at the same conclusion from his transmission line model analysis of this antenna. We shall consider in detail the effects of the loading of the type given by Eq. (A. 16) for various values of C including the value corresponding to the reflectionless case. 5. EVALUATION OF THE CURRENT DISTRIBUTION I(z. w). 19 Standard numerical technique is used to solve Eq. (A. 15) for I(z, o). For this purpose the integral equation is reduced by moments method to the following set of N simultaneous algebraic equations: N f J I(z', t)G(zj z)dz' n l Az n Bcoskz -2inkiz 71

t + J/roX' n f I(z', w)R (z') sink(z -z')dz' 5 J Az n j 1, 2,...,N (A. 17) where the summation on the r. h. s. of Eq. (A. 17) is interpreted as:.n N/ n=j for < N/ (A. 18) N/+ 1 nu N/2 + 1 for > N/2 and it is assumed that N that the antenna of length sections increasing from shown in Fig. A-1. is an even number and z' E Az. Eq. (A. 17) implies n 2 L is divided into N sections, the numbering of the 1 to N along the antenna length from - L to L as z a -L z=0 zO 1 n2 n N/I -- - nn=l n=2 n= N/2 n=N L FIG. A-i: Division of the antenna into N-sections. It remains now to make an appropriate approximation to the current distribution I (z') in each of the sections A z. When the antenna length is small n 72

electrically the usual pulse approximation0 to the current in each section provides sufficient accuracy. Since our preliminary results have been obtained by this method we give here a brief discussion of the appropriate expressions used in this method. Pulse Approximation Expressions In this method the unknown current in each section is assumed to be a rectangular pulse, i.e., I(z) = I Z'Az n= 1, 2,...,N n n (A. 19) = 0 elsewhere Using Eq. (A. 19) it can be shown that the general integral equation given by Eq. (A. 15) can be transformed into the following N simultaneous equations: N In G| G(zi,z')dz' B coskz - l-sink(z)+ Az + /lo n I R (g)sink(z -~)d9 n s n (A. 20) jul, 2,...,N where the meaning the summation is as explained before. In general the n unknown current I (z) and the unknown constant B are complex quantities. Let us assume: I(z) = IR(z) + JIc(Z) (A. 21) B BR+ JBc 73

Separating Eq. (A. 20) into real and imaginary parts we obtain: N n=l f cos k (z - z')2 + a] n 47r (zj - )2 + a dz'- BR coskz R j 1/ri D nl Azn R 8(9) sin k (z i -9) d N n=l In J n [() 2 '/2 sin k z-z) +a 4w [(z - z)2 +a2 dz' a 0 (A. 22) N n=l I cos k (z - z )2 + a2] n 4r [(z- z)2 +a2] dz'- Bc coskzj C 1 / nro Rn n Az Q) sin k (z -9) d9 8. i p a n N n1l IR n sin k [(z -z)2+a Ir 2z dz' Azn 4w (zj -z)2+a2 /2 [,2 ] 1 *- sin 2 r0 sink Izl 2 n (A. 23) 9 74

where the upper (lower) sign is used for j < N/2 (j > N/2 ) respectively. The above sets of equations along with the end conditions have been solved numerically for the unknown currents. The results will be discussed later. Pulse approximation method provides farily accurate results for small antenna lengths. However, if the antenna length is long electrically, to obtain sufficiently accurate results, N must be chosen very large. Hence to obtain accurate results without taxing the computer capability a different type of approximation should be used. In the next section we discuss such a method. Quadratic Approximation Expressions As mentioned before, when the antenna length is large, the computer capability makes it inappropriate to use the pulse approximation method. For the present problem of resistively loaded linear antenna, it is known from theo1o 9 retical considerations ' that the current amplitude decreases linearly towards the ends of the antenna. For this reason, we make the following quadratic approximation to the unknown current in each section: I(z') A +B (z'-z )+C (z'-z ) for z' Az n n n n n n (A. 24) 0, otherwise, where An, B, C are three unknown constants. These constants are determined n n by requiring that the continuation of I( z') expressions given by Eq. (A. 24) into the centers of the adjacent sections give the appropriate current values there. Thus we obtain the following: I(z ) I A n n n n-l 1-l n n - n - n 2 I(znl ) InlAn+Bn(Zn Zn)Cn(Zn Zn) (A.25) nI(z+l )I+lA +B (Zn -z )+c (zn, -z ) n+1n n +1 n n n+1n 75

After eliminating An, Bn, C from Eq. (A. 5) with the help of the relations given n by Eq. (A. 6) we obtain the following recurrence relation for the current in each section: I(z')-I X (z')+I Y (z')+In+ Z (z'), n-1 n n n Ini n for z' Az n (A. 26) where z'-z (z'-z ) n+ n x (z = 2z 2 Az2 ~ Z'EAZ n (A. 27) (z'-z )2 Y (z') 1 -.2 n n Az2 z ' A z n (A. 28) z'-z (z'-z )2 n n z (z') — a + A n 2A z -. 2 2Az 9 Z' EAz n (A. 29) where it has been assumed that each subsection is of equal length A z. In view of the fact that in Eq. (A. 26) the value of the current in section A z is related to n the currents In+1 I 1 in the centers of the adjacent sections AZ +1 and A 1 n+1' n 1 n-1 respectively, the current values at the center and hence the entire two end sections A z1 and A ZN should be treated separately. For this purpose we make use of the two sets of current coefficients I-L, I1, 12 and IN_ 1 IN' IL for obtaining the currents in the sections Azl and AzN respectively. Using these two sets of current coefficients we obtain the following from Eq. (A. 24) to determine the currents in the two end sections: I(z')I= Yl '(z')+I2 (z') ( I(z')=INY(z- + X ) I(z') = INYN(z'N for z 'EAz for z'EAzN N (A.30) (A. 31) where, 76

(z'- z) (z'- z )2 Yl(z ) 1 + 2Az 1 2 V (')- zZN2(zt zN)2 N 3Az 2 3Az N 3Az 2 N 3Az 3AZ2 Z'A1 (A. 32) (A. 33) (A* 34) (A. 35) Z IE AZ N After substituting Eqs. (A. 26H-A. 35) into Eq. (A. 17) the taneous equations are obtained: following set of N simul I [J Y' (z')G(z',, z ) dzt + f X(z'I)G(z', Zj )dz I 1Az2I + 12 1 IZ(z')G(z',, )dz' + f Y(z')G(z', Zj )dz' + f X(z ')G(z ', z ) dzj1 At1Az 2 Az3 n=3 I f' Z n1 (z9)G(z', zi)dzi + f Yn(z')G(zu zi)dz' + f Xn~1(z')G(z',zi)dz ~n-1 An An+i N-2 N-2 j fYN- (z')G(z', z )dz' + AZN-i + f i x I)Gz',,z'.zdz'] + Az N 77

+IL IY' (z9)G(z',, z )dz' + N J Az N I Z l(z ')G(zI 0,z1 )dz]I A N -1l + H 2 j12[ iv( Az I sin kIzI +i~ 0 Z'(z9)F(zl~dz' + YV (z')F(zl~dz' + 1 Az2 (z')F(z')dzj I 2 12 Az 2 Xz 3(z')F(zt)dzj N/2 n=3 H 1I~ Zn1 (z9)F(zl~dz' + Az Yn (z')F(z')dz' Aznm1 n Az n1 I for j ( N/2 + j/% H I f n=N +1Azn1 2 r n1 (zt)F(zt)dzt + fY n z')F~z'dz' + f: AI +HN-11, IN-i z L AZ N-2 iN2(z')F(zl~dz' + IYN-1(z')F(z')dz' Az N Az N + HNJL'N Az N- Z~(z)F(zl)dz + ISr' (z9)F(zl~dzll N ji Az N (A, 36) 78

for j >N/2 j (as the subscript) 1, 2,3,..., N where 1, for j m H ( mj 0, for j <m C F(z')= L sink(z -z'). (A.37) L-Iz'j J The above is a set of N algebraic equations involving the N unknown current coefficients I, I2,...., IN and the extra unknown constant B which is to be determined by applying the end condition I(- L) 0. By using Taylor's expansion for the currents at the centers of the first three sections we obtain the following: (z +L)2 I(zl) I1 (z+L)I'(-L)+ 2 I"(-L), (A.38) (z2+L)2 I(2) I (z2 +L)I'(-L)+ 2 I"(-L) (A.39) (z +L)2 I(z3) = I3 (z3+L)I'(-L)+ 2 I"(-L), (A.40) where we have already used the fact I(-L) = 0. The derivative terms in (A. 38)-(A. 40) can be eliminated and we obtain the following extra equation: 313 - 102 + 151, 0. (A. 41) Thus Eqs. (A.36) and (A. 41) constitute a set of N+ 1 equations for the N+ 1 unknowns (i. e., I1, I2.. IN, B). The system can now be solved by standard means. If the end condition is applied at the other end of the antenna, i. e. I(+L) * 0, then the following equations should be used instead of Eq. (A. 41): 79

15 IN-10 lN- + 3IN-2 0. (A.42) Eqs. (A. 41) and (A. 42) will be referred to as the 3-point end condition formula. In order to estimate the accuracy of the results, computation has also been done by using 4-point and 2-point end condition expressions. These have been obtained by applying the end condition I(+L) = 0 and retaining 4 and 2 terms in the Taylor's expansion respectively. The relevant expressions for these two cases are: 52N13+- 1N-351 +35I1 0 (A.43) N N-1 * (A. 44) The above completes the theoretical discussion on the numerical procedure to be followed in the determination of the current distribution for the loaded linear antenna. 6. EVALUATION OF THE TRANSFER FUNCTION f (0, w) In the previous section we discussed the numerical method of obtaining the current distribution I (z, w) on the antenna for the harmonic time dependence case. After introducing the sampled values of I(z, c) in Eq. (A. 7), f (0, w) v is obtained numerically with the help of the following equation: Wy YJ Z w "zCos e evosinet cn f (e ) = - 2I e n Az (A.45) 0 c 4ir n n n where we have used the notation I( z ) I and z' = z is the coordinate at n n n the center of the section Az.n The pulse approximation is good for transfer n function calculation because the current distribution is linear as mentioned before. 7. PRELIMINARY NUMERICAL RESULTS In this section we discuss briefly some preliminary numerical results 80

obtained for a few simple cases. The motivation behind this section has been mainly to ascertain the correctness and accuracy of the numerical technique used for later investigation. 7. 1 Uniformly Loaded Antenna L = A/2. For the case of uniformly loaded dipole of total length 2L s X, there is no variation of loading along the antenna, i. e., R (z) = r ohms/meter. Fig. A-2 S shows the real and imaginary parts of the current distribution on a uniformly i loaded dipole for two values of r. These results have been obtained by using Eqs. (21)-(23) with N = 30. In Fig. A-2 we have used the notation 0= 2Xr/n where X is the wavelength and Yo is the intrinsic impedance of free space. 12 The results shown here compare very well with those of King, et. al.. Thus it proves the correctness of the computation followed here. 7.2 Non-Reflecting Loaded Case L X. Here consider a one wavelength long linear antenna non-uniformly loaded according to Eq. (A. 16). The radius a of the antenna element is chosen such that = 2 In — = 11. 5. This value of Q is chosen so that our results may a 3 be compared with some available published results. Wu and King predicted from theoretical considerations that a purely outgoing traveling wave of current is sustained on the antenna loaded according to Eq. (A. 16) provided the constant C is chosen to be equal to 60 ta with P=aa 5.3. In other words the antenna considered in this section is loaded as follows: R (z) X 60. (A.46) s L-I z with ca = 5. 3 (Note: in this notation, r = 60 Oa/L). The cpa notation is used 0 here to correspond to that used in Wu and King. Figure A-3 gives the amplitude of the current distribution on the antenna as obtained by numerical computation using pulse approximation with N = 30. The current distributions obtained for the same antenna by Taylor and Shen18 are also shown in Figure A-3 for comparisame antenna by Taylor and Shen are also shown in Figure A-3 for compari 81

z/L A I C I%,0 0i= 1 =i1 I r -2 -1 2 mA/V FIG. A-2: Real (I ) and imaginary (I ) on unifrormly loaded dipole. parts of the current distribution L=r/2. (=9.92, pi 2X ri/ y0. 82

III 6 mA 5 4 3 2 1 O O ~ Shen 0 0 0 Shen --- Taylor X — X Result 0 0 0 0 0 IZI/L 0.25 0.50 0.75 1.0 FIG. A-3: Current distribution on non-reflectively loaded monopole. f = 11.5, wL/c = 2r, V$ a = 5.3.

son. The results of Shen have been obtained experimentally and those of Taylor have been obtained by numerical solution of Pocklington integral equation for the current distribution. Figure A-3 indicates that our computed results agree fairly well with Shen's experimental values. 7.3 Exponentially Loaded Case Fig. A-4 shows the current distributions on a one wavelength long monopole loaded exponentially. The loading function used is R (z) = ce ZI, (A.47) where c' is a constant and a is another constant which determines the rate of loading. In order to compare the results of Fig. A-4 with those of Fig. A-3 the constant c' is chosen to be c' = 604a 60 x 5.3. Fig. A-4 shows the current distributions obtained numerically with the help of Eqs. (A. 21)-(A. 23) with N = 30 for two cases with a = 2. 2 and a = -2. 2. The case with a = 2.2 corresponds approx18 imately to Shen's and it clearly shows the existence of a traveling wave type of current distribution. For a = -2. 2, the loading decreases towards the end and the current distribution obtained is of standing wave nature. 7.4 Phase Distribution of the Current Fig. A-5 shows the phase variations of the current along the length of the antenna for the different non-uniformly loaded cases considered above. The progressive linear variations of phase for the non-reflecting case indicates the existence of a pure traveling wave of current in the antenna. It is interesting to observe from Fig. A-4 that for the exponentially loaded case with c' = 60 x 5.3, a = 2. 2 the antenna may be considered to be approximately non-reflecting. This observation may have significant implications for theoretical analysis of such antennas. No such conclusion can be made from the phase variation for the exponentially loaded antenna with a = - 2. 2. 7.5 Results for a Long Antenna 2 L = 5 In the previous sections numerical results have been given for an antenna 84

III mA 5 4 3 co 2 1 — O60,ae-2-2 IZI/L -X e2 2 21zl/L X:1 -X 60 a e 0.25 0.5 0.75 1.0 FIG. A-4: Current distribution on exponentially loaded monopole. Q = 11,5, bca = 5.3, wL/c = 2 r.

180 160 140 120 100 80 60 40 20 0000 e22 IzI/L 1 XXXx 1- z/L A A A A-2.2 z|/L e 0 0 0.8 0.9 1.0 (I) IQ) (a) C1) "0 p4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 z /L 0 -20 -40 -60 -100 -120 -140 -160 -180 "'A N)N FIG. A,,5: Phase variation of the current along the antenna. 86

having total length 2 L = 2). During the process of obtaining the transfer function of the antenna as a function of frequency, the effective length of the antenna becomes very large at the high frequency end. The computer capacity as well as the accuracy of the numerical results restrict the highest frequency for which the transfer function can be evaluated. For long antennas, the computer program should be modified for retaining satisfactory accuracy of the results without taxing the computer. For small antenna length the usual pulse approximation for the current in each section A zn provides sufficient accuracy as has been found in the previous sections. As discussed in Section 4, for long antennas we use the quadratic approximation to the current during the numerical evaluation of the current. We study the accuracy of this procedure in this section. In addition to this we also investigate the use of 4-, 3- and 2-point end conditions for the long antenna case. The results of a sample computation done for the case with 2 L = 5X with different values of the loading parameter in Eq. (A. 16) are discussed in the present section. Figs. A-6 -- A-8 show the current amplitude distribution, the current phase distribution and the transfer function respectively of the non-reflecting loaded linear antenna of length 2 L = 5 x with the loading parameter C = 60 x 5. 3. All these curves have been obtained by using Eqs.(A. 36)with N = 30. In each case the results obtained by using 4-, 3- and 2-point end condition expressions are also shown in Figs. A-5 -A-7. It can be seen from Figs. A-5 - A-7 that the results are not appreciably different among the three cases. However, near the end of the antenna the results are found to differ with each other slightly. From a study of Figs. A-5 and A-6, in particular, the phase variation near the end of the antenna as shown in Fig. A-6 it is concluded that the 4-point end condition expression given by Eq. (A. 43) is more accurate and hence should be used during the numerical computation of the transfer function of the antenna for high frequencies. Figure A-9 shows the amplitude of the current distribution on the antenna as a function of the loading parameter C = 60 obtained by Eq. (A. 36) with N = 30 and the 4-point end condition. 87

(milliampvolt) III ------ - 4 point end condition * * 3 point end condition -& - - -A- 2 point end condition 7 6 5 6 points/k quadratic interpolatj 3 - 2 0 A I I I I, 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 FIG. A-6: Current distribution on non-reflectively loaded antenna, Q1 = 11.5 kL = 5a, R a= 5.3, R (z) " 60 @L/L - zI. 88 ion

X -X 4 point phase 180~ 100 0.1 0.2 0.3 0* 3 point 2 point Izl 0.9 1.0 L 0.5 0.6 -180~ FIG. A-7: Phase variation of the current along the non-reflectively loaded antenna, 0 = 11.5, kL 5S, vr = 5.3, R(z) = 60r/L-lzl. S 89

*-Kr 4 point * 0 3 point ~ 2 point kf (ODpW)I 600 500 400 300 200 100 9 100 200 300 400 500 60 0 700 800 900 FIG. A-8: Transfer function f0 (O,w) of the non-reflectively loaded antenna. ~2=z11.5, kL=m5wi, Oba= 5.3, R (z = 60Ote/L-IzI. g0

X) X 6041 0' —0 60%3 -— ' 60'4 m ---4 6OX5 *- - 60, I I mA 7 6 -I au 4 3 2 - 1 - L I I I I I I I I a A~ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FIG. A-9: Amplitude of the current distribution on reslstively loaded dipole antenna as a function of loading. fn * 11.5, kL = 5 r, ta variable. 91

It s linteteatlng to obsorvo hero that on the basis of King's work, the expansion factor / is a fixed value for one particular size and frequency. That means only at this specific value of loading (60k1a) and specific frequency, we can have a reflectionless current wave on the antenna. For C < 60 x 5, the reflection effects on the current distribution become quite appreciable as expected. While C > 60 x 7, Fig. A-9 tells us the nonreflective nature. The higher value of loading on the antenna supresses the small amount of reflection that occurs due to the use of a value of kb slightly different from the specific value referred to above. On the basis of the results given in this section, we have decided to use the 4-point end-condition expression during the numerical computation. Six subsections per wavelength to divide the antenna is required to guarantee the accuracy. 8. CURRENT DISTRIBUTION I( z, w) ON THE LOADED ANTENNA In this section we give the numerical results obtained for the amplitude of the current distribution I( z, w) for the harmonically excited loaded antenna. The loading used is of the form given by Eq. (A. 16) with C variable. Figures A-lOaA-lOd show the amplitude of the current distribution on the antenna with antenna length as parameter for different values of the loading. The amplitude of the current in general increases as the frequency is increased. The value of the magnitude at the same frequency is surpressed by the higher loading as expected. The current distribution is not strictly linear as those of the transmission line found by Baum. It is due to the factor & which is a function of the thickness ratio of the antenna and the frequency used. So it is not possible to excite a traveling current wave for all frequencies on an antenna of fixed size. 9. TRANSFER FUNCTION OF THE LOADED ANTENNA f ( 0, ). In this section the magnitude and phase of the transfer function of the antenna are given as functions of wL/c for different values of the loading parameter C. 92

III X X wtL =30 C mA 3. 3. 3. 3. 3. 2. 2. 2. 2. 2. 1. 1. 1., 1.: 1.' 0. 0.1 0.4 0.: wL C - = 20 C = 0.25 0 Z/L FIG. A-lOa: Amplitude of the current distribution I I (z, ) vs z/L on the loaded antenna with kL (s wL/o ), as the parameter, C = 60 x 4. 93

I1I mA 3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 X —X —x-x = 30 C wL = 20 of0 c wL 0.25 c Z/L 0 0.25 0.50 0.75 FIG. A-10b: Amplitude of the current distribution I I (z,) | vs. z/L on the loaded antenna with kL (= wL/c), as the parameter C = 60 x 5. 94 1

III mA 4.2 4.0 3.8 X XX L - 30 c 00 = 20 0 0 c -L- 0.25 C 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 Z/L FIG. A-lOc: Amplitude of the current distribution II (z,w)|vs z/L on the loaded antenna with kL (= oL/c ), as the parameter. C = 60 x 8. 95

Ill mA 4.0 3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4 -2.2 2.0 1.8 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 L XX 30 X —X —X c 0-0-o 0 w L - = 20 c w L L- =0.25 C Z/L 0.25 0.50 0.75 1 FIG. A-lOd: Amplitude of the current distribution II (z, w)jvs z/L on the loaded antenna with k L = wL/c), as the parameter. C = 60 x 10. 96 1

These have been obtained numerically with the help of Eq. (A. 45) and the sampled values of the current distributions discussed in the previous section. Figures A-lla-A-lld give the variations of the magnitude and phase of the transfer function in the broadside direction ( = T /2) of the antenna and for four values of the loading parameter c. Similar results are given for 0 = -r/3, 0 = r/4, 0 x r/6 in Figs. A-12 — A-14 respectively. Each figure contains four different values of the loading factor C. In all the curves shown,l1 f( 0, w)| approaches zero as w approaches zero, which corresponds to the fact that there is no radiation at zero frequency. For higher frequencies, |f( 0, w)| appears to be an oscillating function and tends to decrease with increase of the frequency. The 3 phase of f (0, w) falls steadily to a positive constant. Wu and King proved that the asympotatic value of the transfer function is a constant. However, within the range of the computations covered here the transfer function in Figs. A-ll —A-14 does not reach its asymptotic value especially for 0 = 7r/6. For an unloaded thin linear antenna, we know that the transfer function would be zero for L = nk at the broadside direction (where n is an integer) when the current is sinusoidal. Fig. A-lla shows that the values at kL = 2 -r, 47r, 67r, 87r are minimal. For higher values of C, the loading reduces the ringing which is due to the reflection from the end. It is appropriate to mention here that the impulse response of the antenna may be obtained by numerically carrying out the inverse Fourier transform. However, the data for fo (, U) obtained so far is not sufficient to get reasonably 14 good results. Taylor has shown that the transferlfunction reaches its asymptotic value at a certain frequency which is beyond the value wL/c = 25 considered here. 10. NUMERICAL RESULTS In this section we discuss the numerical results for two different shapes of voltage input. Spectral density is obtained by multiplying the transfer function with the input pulse spectrum. The waveform of the radiated signal is then 97

( 7T " w ) 2 '"ON C = (iOK4 0 100 - mag. ~0 -3 100 phase 0 1 2 3 5 6 9 1 0 1 1 12 13 Iq 1Is ) 8 1 9 2')0 2 1 wlL C FIG. A-11a: ~Magrnitudo {-ind phaso of tre, traiirsfer funiction f0 (2r/2 w), of the loaded antenna in the direction 0 = 2 as a function of IdL (ol /c )Ifov- differenc"t loadling.c: — 60.x 4,

) Oj V * 0 0 0 0 * 0 t 0 * 0 *D0 C'. _ * 0 c3 I0 I C 0 C l * 0 CIl * 0 - i i 0,0 * 0 3 0 0 0 0 - * o 0 0o 0 -4 I 0 -1 0* C - c )0 - O da 0 ) 0* -o - o C. I) ~ o - 3 * 0 - o o C * o - -4 0 0 '.-4 * 0 L 0 0- ' - * * 0 0 '. 0 0 C* 0 4 0 0 -i 02 0 c 0 0 0 0Z 0 C-] 0 0 * 0 0 0 0 0- 0 0o * 0 0 0 r-' ' 0 0 0 ' 0 00 0 * CY0) d 0' ' 0 * 0 0 0 - 0 0 0 o o o o 0 - o - - - I I I ' —\ — 9 9 99

0 300 130 0 ^I-li 7T f (-.0 w ) 0 2 0 0 0 200 1 - 0 1000 - - 10 0 0 nmag. 0 0 0 0 0 *0 0 0 0 0 0 * ~0 **, g 1 00 r phase )00000000000000000000000 ooooo 00O000000 00O 000 00 000100 0 a o0 0 0 000 f I I I I III I I I I I I a I I I I I I I I I I I-, - - I I 1 — - 1 I I I I - - - -I I 1 1 I - I I I I -II I __j - —., i 0 FIG. 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 13- 16 17 1 8 19 20 21 22 2 3 A-iic:Mnnitdeandphse f te ranfe fucton (i /2, w)of the loaded antenna in the direction 0 7. /2 as a function of kIL = (w.L/c) for dIffocrent 1oaditn-. C= 60 x 8. 24 C

It'o(,w) 300 C = 60A 10 15ID"0 3 -ICN mag. phase w L /C 0 r2 3 6 7 8 9 1 0 1 1 12 13 14 15 1 6 17 1 8 19 20(~ 2 1 22 2~ FIG. A-1 id: Maniud ad p-ha'Se, Of the- tran11Sfer fUnction f0 (7 2, w. of thie loaded antenna in the direei ion 0 = 7r /2 as functions of kI,-T (= / ) for- uiftcrunt 10loaingiCl6)x10 kL

0 0 130 0 - 0 0 0 - 200 mnag. 7..0 100 0_ 0 0 00 0 0a 0 0 0 0 0 -100 phase 0000 000000 0 00 0 0 0 50 0 0 0 0 0 0 0 0 0 00 000 0000 0 0 00 00 0000 0 0 0 n 0000 0 0 0 0 0000 0 0 I I I I I I I I I I )o I A. I. - I --- I I- - I - I I - I J - I 0 1 2 3 4 5 6 7 6 9 1011'12 13 FIG. A-12a: Mvagniltude and phase of tiie transfler I L I io J.C1,11 /,w of f Unction-S o f IkL1 ( = CL /c ) 101 u~i J,;r er Io2(t J 14.1 " I ir C: 60 x4. I I I I I I --- I - I I I I I I t a 14 15 16 17 18 19 20 21 22 23 the lolade,-Id antenna in the direction 0 = 74 /3 as 294

0 f0~ 3 ) 0 0 0 a0 0 * 0 - 200 mag. 1... *100 0-, a 0 - 0 0 0 0 0 0 00; 0 4, 0 0 0 0 0 0 0 0 0 - 100 Phase 0000 0 0 0 0 Oo 0 30 - 0 0 0 0 00 000,0000 000 00000 0 00 00000 S 0 wL /C f 10 I I I I --- - I I - - I I I ---- I t.. -A, I I -A - 1. I I I - I1 - - - -. t - I I I 1 %-R0 1 FIG. 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 47 16 1 7 1 8 1 9 20 2 1 22 A-12b: -Mag-nituide, and phlase of the transfer functionl f0(1,, /3,w) of the loaded antenna in the direction 0 = 7r3as functions of LL (= ujL/c ) for dlifferenit loaIdinig. C 60 x 5 I 23 2 -17- -- k L

Z) 3 IL I 0~ Q) to ' —4.~4 0 0 -0 C0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -]C 'o 0 b~ 0 0 0 0 0 0 0 0 0 0 0 0 ( 8 co 0 Cd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 * 0 0* 0 0b 00 00 0 3 -1 -1 -1 CO / C\] Cli] Cl l 0 Cy) Cl\2 '4 d 3 Z)q4 0 0 0 0 a 0 0 0.0 0 0 0) V-4 0 a --- U') 104

o ~ 0 0 0 0 0 mag.S. 0 0 0 00 0 0 00 00000 Oc -100 * 0 phse 000000000000000000 000000000000000 0 00 0000 wL1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2021 22 23 FIG. A-12d: Mlagnitude and phase of the transfer function %o(7r/A w) of the loaded antennia in the direction 0 = 7T,/3 as 1500 100 __ 3 a — 24EJ4 functions of kL(= cwL/c ) f~or different lending. C =:60 x 10.

4~,/I, C = 60x4 0 1 50( - mag. 11 00/ (J0 0 50 - phase wL/ % 1 2 3 FIG. A-13a: 4 5 IVla(ofitud1(1 0 = 71/4 as 6 7 8 9 10 11 12 13 14 115 1 6 1 7 1 8 10 9 21 22 23 aIndl peae o t trnfe unto T(ir /4, w-) of the loadled antenna in the irection of funt-ctionI- Of kl, (= UL/c ) for dliff-rent ioadlinc. C GO6 x 4.

e 7 o 300 0 150 - 0.*0.g * 9 *00 mag.. - *. 004, 0 -200 e.* 0 0 * * 0. 100 - 0. * * 0 3 t~l.I b t 4 -(b ''4 S4 ~ O 0 0 0 *0 0 0 6 * 0 o0 * 000000 ~.. 50~50o - - 100 O0 <0 v.0 * 0 0 0 o ~ phase O000 000 00 00 0 00 0000000 0 ~~o0000 0~o Oo0 00 00 0 0 0 0 OO t 00 0 0. wL1 ~ l. I I- I I I I I I I I I - I I I I I 1 1 1 1 1 I s 1 2 FIG. A-13b: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 MIag'nitude and phase of the transfer function fO("r/4, w) of the loaded antenna as functions of kL (= aL/c ) for different loadLing. C = 60 x 4. ~ I.............. 18 19 20 21 in the direction 0 22 = t/4 23 24

e%..# 7r f e ( p w ) 0 -300 0 150 - 0 0 mag. 0 0 0 0 ~-200 0 a 0 100 - 0 0 * 0 ", a 0 a 4, * 0 0 0 0 @ 00 00000 3 t,- I CD Z4-4 U 4 0 ~-100 0 50 - 00I0-phase @0000000000000000Oa00 00 000coo 00000 % OL1000 ->00 —.-..-I 1 I L 1~I I I I I I_ I -- I I I- I I I I- - I I t I I 0 1 2 FIG. A-13c: 3 4 D 6 7 8 9 10 1 1 12 13 14 MagnitUdle and phiase of the transfer function f0(~-i /4, w) of the functions of kL, (= wL/ c) for dlifferent loadingv. C = 60 x 8. - - - -. - - -. 21-, - I I 11. 11 1-1.-I. 15D 1 6 1 7 1~ 8 1 20 21 22 loaded antenna in the direction 0 = 7r/4 as 23 24

T l 4 I,} o 150 - C = 60 10 I 50 5 0 /c i 1 2 3 4 5 6 7 8 9 10 11, 12 13 14 15 16 17 18 19 20 21 22 23 FIG. A-13d: Magnitude and phasc of the transfer function f0(T/4, w) of the loaded antenna in the direction 0 = 7-/4 as function of kL(=- L/c) for different loading. C = 60 x 10.

06 0 0 150 - -500 100 0 0 o 00 0 0 0 -400 0 0 0 000 0 0 00 oOOO 0 00 0 0 0 Oc 3 bfl phase 0000000j 00 I00 00 0 @00 0 9 9 -300 * 0.. -200 niag. I... 0 0 *.0 & 00 000 0 0 50 0 a 0 0 0000 000 00. -100 0 0 0 0 00000000 0 004 0 0 -I — -1 IL - 1 - -- 1 - - I I I I I I I I I I I I I I I I I I I,. -. I -. I I a a -. L 1 FIG. 2 3 A -14a: 4 5 6 7 8 9 10 1 1 12 13 -14 1 Magn9L ~ IIi tuL de and phase of the transfer function?( r, /6, L) o f thlI as function of k (= 6uL/c ) for differca;t loading-, C= 60 x 4. c5 16 17 1 8 19 20 21 Le loaded antenn~a in thle (lirection 22 1 0 = 7/ 230" 24

0 0 0 - 300 150 0 0 - 250 0 00000 0 0 0 200 000 0 00000o0000 0 00 o c mag. phs 0 -00 00 000/ 00 100 0 -0I 3 t~kz I ~-150 0 0 ~- 100 000 00 0 0 0 00 0 00 - 50 0 0 0 0 0 0 0000 Oo w L /C I I I I I I I 1. I I. I I - I I I I I I I I I I - 1 FIG. 2 3 A - 14b: 4 5 6 7 8 9 10 11' 12 13 14 15 16 MlagnJitulde anad phase of the transfer func'tionl" f(7;-/6,(O) of the loaded as function of u (= coL/c ) for (iivJ~l loiis;.3 C = U x 5. 1 7 1 8 19 antenna in thfe 29 21 direction 0 22 = 77, 6 23 24

01 -300 0 0 0 150 0 1A 0 '- 200 0 100 - 0 0 mag. 0000000phase 00000000 00 0 00 00.00 0 00* 0 0 00 0 0 00000 -6 0 000 0 00 0 a 000 0.0.0 's * 0 0 00 0 - 100 0 0 0 0,0 00 0 50 - 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0000 wL/ k 0 0c 000 0 1 FIG. 2 3 4 5 6 7 8 A-14c: MNagnitude;,and lph1a se(, of the as fUnction of k'L ( = w/c ) u 10 ii1' 12 1 3 14 1 5 16 17 1 8 19 20 21.22 transfer futiction f0" (r / 6, cu) of the loaded antenna in the direction 0 = 7T /6 for different loadling. C = 60 x8. 26)3 24

0 0 0 200 0 00 phase mg 00 0000000 000 000000 000 -000. 000 00 1000 000 Oo 0 00 0000 000 00000 00 0 1 2~~~~~ 0. 3 4 6 78 91 11l,~,131451611819 20212 2 FIG A14d Mgntud ad has o te tanfe fuctonf0(/6~) f heloaedanenn i te dretin 0= 0/ as L~ctia o hE wE/) fr difcrnt ladin. C= 60x 1 1500 0 / 100 500 1000 3 24

obtained by using Fast Fourier inversion technique. 10. 1 Spectral Density e' (0, w) and the Time Domain Solution e (0, t) of the Radiated Field Excited by Gaussian Type Pulse In this section, results are given for the spectral density 'e ( 0, w ) and the time domain solution e (0, t) excited by a Gaussian pulse input. The form of the input pulse and its Fourier transform are given by Eqs. (A. lOa) and (A. lla). For the time domain results shown here, the antenna length is taken to be L = 1 meter. This implies that the transit time on the antenna for centerfed case is 7r L/c = 3. 33 nanosecond. The radius a of the antenna element is chosen such that = 2 In a = 11. 5 in all cases. The spectral density e(0, wO) of the radiated waveform shown here has been obtained by multiplying the transfer function of the antenna by the Fourier spectrum density function of the input signal. The waveform e (0, t ) of the radiation field produced by the antenna excited by the Gaussian signal has been obtained by numerically carrying out the integral in Eq. (A. 8) with the aid of Fast Fourier inversion technique. The results shown here have been calculated in the directions 0 = r /2, 7r/3, 7r/4 and 7r/6 from the antenna, where 0 = ir/2 corresponds to the broadside direction. Three different values of the width of the Gaussian pulse have been considered for 0 = ir/2, while only a narrow pulse has been obtained for the other directions. These pulses are chosen such that the spectral density will converge to zero at high frequencies. Four different values of the loading constant are used in each case. Figs. A-15 —A-17 show I'e(v/2, t)I versus kL and e (7r/2, t) verrsus t for these cases with a = 0.471 nsec., 1 nsec., 3.33 nsec. respectively. From the frequency domain results, we observe that for a wider pulse only the low frequency portion of the transfer function is responsible for the overall response. For example, when a = 3. 33 nsec., we only have to consider the frequency spectrum up to kL = 3. 5 as shown in Fig. A-17. For a narrow pulse, a wider frequency domain of the antenna transfer function has to be considered. However, if the width of the pulse is too narrow, then the computing time re 114

140 120 100 80 60 40 20 C = 60x 4 C = 60x 5 - C = 60 8 - C = 60x10 o =0. 471 nsec e-n c~n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 FIG. A-15a: Je (,/2,) vs L/c,, for the loaded antenna excited by a Gaussian pulse with cr = 0.471 ns. 23 w L C

e(E, t) 0 2' C = 60x4 a= 0.471 nsec C = G0X10 '-A a)* t 2. 6 nsec FIG. A-15b: e (2t) vs t for the loaded antenna excited by a Gaussian pulse witlh 0.471 ns.

w)J(f 120 ax= nsec 100 C G0'x5 80 60 C = 6OxlO 40 20 I t I I I- wL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 kL C FIG.Ae16: j(7r/2, to jvs wL for the loa ded antemnn excited by a Gaussian pulso'.-. a = 1 ns. I I

;2t) ee(~, t 121 =l1 nsec 100 C=60 4 C=60 10 t oco* nsec -20 -40 -60 -80 -100 -120 -140 FIG. A-16b: e(7r /2, t) vs t for the loaded antenna excited by a Gaussian pulse. a = l ns'.

e (E, t) 0 2' C = 60x4 C = 60x10 t nsec -4 -6 250 ) 50 -8 -10 -12 FIG. A-17: e (x/2,t) vs t and ee(r /2, w) vs w L/ for the loaded antenna excited by a Gaussian pulse with a=3.33ns. C = 60x 4 - C = 60O10 -14 -16 -18 wL c 0.5 1 1.5 2. 2 kL -20

quired becomes excessive. For practical consideration of the computing time, we we have chosen a = 0. 471 nsec. such that the upper frequency spectrum limit corresponds to kL = 25. Fig. A-15ashows e 0(7r/2,w)| for a =0.471 nsec. Forvaluesof C =60x4 and C = 60 x 5, the ringing phenomenon enters the picture. For C exceeds 60 x 8 the ringing disappears. From the time domain results shown in Fig. A-15b, it appears that the initial part of the waveform represents predominantly the time derivative of the input Gaussian pulse. The general shape of the waveform depends on the radio /r as expected. It is also seen that after the second zero crossing, e( r/2, t) remains positive for all positive values of t and the magnitude decreases with the increasing C of the resistive loading. Corresponding results are shown in Figs. A-18 — A-20 for three other directions with a = 0.471 n sec. and for four different resistive loading. The general characteristics are similar to those discussed above. Due to the difference of the path length, the reflection occurs at different frequencies I e ( 0, w )|. The strength 0 of the far field is maximum at 9 = -r /2 and decreases as 0 deviates from the broadside direction. In every case, the effect of loading in general reduces or eliminates the ringing. 10.2 Spectral Density eo(0, w) and the Time Domain Solution e (0, t) of the Radiated Field Excited by a Gamma Pulse In this section, we consider an input voltage function represented by a Gamma pulse. The equations which define this pulse and its transform are given by Eqs. (A. 10b) and(A. lib). Unlike the Gaussian pulse, the Gamma pulse is defined here for positive I values of t only. The spectral density 'e (0, w) shown here, again, has been obtained by multiplying the transfer function of the antenna with the Fourier Spectrum of the input signal. Then Fast Fourier inversion technique is used as before. The results shown in Figs. A-22 -A-25, correspond to one particular pulse width 120

I I 140e ( /3, w) 120 100 80 60 40 20 O 1 2 3 4 5 6 7 FIG. A-18a: (r/3,u))vs uL/ C = 60x4 C = 60 x 5 C = 60x 8 C = 60 x 10 wL C 21 22 23 for the loaded antenna excited by a Gaussian pulse with (r = 0. 471 ns.

e 0(-r / 3, t) 400 300 tN, 1.3 2.6 3.9 5.2 t n sec -24 FIG. A-18b: e 0(7T/3, t) vs t for the loaded antenna excited by a Gaussian pulse wvith a =0. 471 ns.

100 80 a =0. 471 nsec o = 6OA4 o — = 6 0"5 C = 60x8 I.~. 60 C -60K-10 40 I - - I I-I I I I I. I I. - I I I I I I 1 D 7 8 9 1 1 2 1 4 1 16 17 18 19 20 21 22 wL kL FIG. A-19a: [ 0(7r /4,w)J Vs wL for the loaded antellnna excited by a Gaussian pulse with ca = 0. 471 ns.

e 4(,t) e,(4 (= 0.471 nsec 100 C =60x4 C = 60 10 t 2.6 3.9 nsec -100 FIG. A-19b: e (/4, t) vs t for the loaded antenna excited by a Gaussian pulse with a = 0.471 ns. -200

\^16,\ C = 60 x 4 C = 60 x5 80 - 60 C = 60 x Ul C = 60 x 10 02 3 - 4 5 6 7 FIG. A-20ao o(l/IW1 vs wL/o 8 9 10 11 12 13 14 t = 0. 4 1 n for the loaded antenna excited by a Gaussian pulse with u 0.471ns

e 6 t300. 300 C = 60x 4 C- 601 0 t nsec FIG. A-20b: e0(7r/6,t) vs t for the loaded antenna excited by a Gaussian pulse with a = 0.471 ns. -21

1.2 1.0 0.8 0.6 0.4 0.2 gamma function with d = 1.7 x 10 /sec tO 41 n sec -0.2 -0.4 -0.6 FIG. A-21: Gamma pulse with d = 1.7 x 10 /sec obtained from Fast Fourier Transform program. -0.8

,-i('r1 2,LO IC -. 60 X 4 - C - 6OX5 /C - 60 x 10 Lj L c 23;5 24 3 4with d 17 to-I o2 I~3 —4 13l pulse L ee712,fo ~leloclec, tntennae:ctd 4 for the 10 Isee. VIG. A-.22a.

e(r /2, t) 100 80 60 40 20 C = 60 x4 C = 60 x 10 t 3.26 n sec -20 -60 FIG. A-22b: ea(r/2, t) vs t for the loaded antenna excited by a gamrma pulse with d 1.7 x 10 /sec.

40 30 0 =60 x 5 20 0 2 4 6 8 10 12 14 1 6 1 8 20 22 24 wL C riU. A-i-a:, -e0 '/7T/63W) vs w L/C tor the ioaatea antenna exciteca by a gramma pulse with ai =~. i x iu / sec,

e(r /3, t) 120 100 80 C = 60 x 4 40 C = 60 x 10 I-A CA) I-A 20 3.26 t n sec -20 -40 -60 FIG. A-23b: e (7/3, t) vs t for the loaded antenna excited by a gamma pulse with d = 1.7 x 109/sec.

I (IT 1 4, U)) 0 c = 6( %. 4 (; -.- 6 0 - x 5 CZ 6oxg 30 c = 60 x 10 V-A 0 0 U) L.c 2i i.0'? 2 lb A. - 9 ---- — I-. - " 6 II %, 10 /see. 14. th Cl = 1. q 2 13 se NO 11 I, no, Pill 0 garn.,"cjtecl by ' 8 C, lot the lo,,Icled 'c"I 6 -V s L,) 1, i/ c 0 pi kg A,2,4e,': eo -FIG -

a120 100 80 60 40 20 ee( /4, t) C = 60x4 C = 60 x 10 3.26 t n sec -20 -40 -60 FIG. A-24b: e (r/4, t) vs t for the loaded antenna excited by a gamma pulse with d = 1.7 x 109/sec.

~~e20 30 12 13 14 10 ii) I C 1 7 18 19 20 21 22 23 24 wI C the loaded antenna excited by a am us 0x,d 17x1 sc 2 FIG. A-~25a: &-l(,g1 6 L) vsW /cfor

ee( /6, t) 100 80 60 40 20 C = 60 x 4 C = 60x 10 CA) 1.63 3.26 t n sec -20 -40 FIG. A-25b: -60 e(7r /6, t) vs t for the loaded antenna excited by a gamma pulse C = 60 x 4, d = 1.7x 10 /sec.

at four different directions. For each direction in space, four values of loading have been considered. The width d of the gamma pulse is chosen to be 1. 7 x 10 /sec. which approximates the same width as that of a Gaussian pulse with a = 0.47. Figure A-21 shows the input gamma pulse reproduced by using the Fast Fourier inversion program. This computation is to show the round-off error involved in the F. F. T. program for sharply rising pulse. The truncation of the high frequency portion of the spectrum also induces error. The error results in the oscillating portion for negative value of t and the shift of the starting point to the left instead of at zero. Similar error would be introduced in computing the time domain solutions based on gamma pulse input. Since gamma pulse does not converge to zero as fast as Gaussian pulse, the spectral density shows small oscillation at high frequencies as shown in Figs. A-22 —A-25. The truncation of a gamma pulse also introduces a larger error than that of a Gaussian at the same truncation frequency. Except for these minor variations, the general behavior of the time domain solution presented previously for the Gaussian pulse also applies to the gamma pulse. 11. CONCLUSION The waveform of the far-zone field radiated by a non-uniform resistively loaded linear antenna excited by a voltage pulse has been investigated by numerical means. Results have been obtained for a Gaussian pulse with three different pulse width and for one particular gamma pulse at four directions of observation. Various values of the loading parameter have been considered. In general, the resultant waveform corresponds to the convolution of the harmonic response and the input signal. The initial portion of the resultant wave however, appears to be proportional to time derivative of the input signal. The result shows that the amplitude of the current distribution on the har- \ monically excited antenna is almost linear, being independent of the loading resis-1 1.... 136

tance when the latter exceeds certain value. The phase of the current represents the characteristics of a traveling wave. Thus, the analysis of Baum based on the transmission line model appears to be a valid description of the current for a resistively loaded antenna too. Our study also verifies the work of Wu and King who examined the problem only for one particular loading at a single frequency. In the broadside direction, the waveform exhibits a compression in pulse width as the loading resistance is increased. No general trend is observed in other directions. For a Gaussian pulse, our numerical results check very well with the analytical solution which indicates that the computing program has been properly executed. To simulate practical problems, we have considered the excitation to be presented by gamma pulse. In general, much of the result obtained using the Gaussian pulse also applies to the gamma i pulse. One significant difference concerns the computing error involved in the two cases. For a gamma pulse, the truncation error appears to be prominent. As a whole, it appears that a "critical" damping can be achieved when the loading resistance attains a certain sufficiently high value, corresponding to C > 60 x 10. 137

APPENDIX B ANALYTICAL INVESTIGATION OF WAVEFORMS RADIATED BY A RESISTIVELY LOADED LINEAR ANTENNA EXCITED BY A GAUSSIAN PULSE ABSTRACT The far field waveforms radiated by linear antennas loaded non-uniformly and continuously with resistance are investigated analytically. The antennas considered are symmetrical and excited at the centers by slice generators having Gaussian pulse type of time dependence. Expressions have been developed for the far field waveforms produced by such antennas in any direction. The analytic results have been compared with those obtained by direct numerical means. The general agreement between the two results has been found to be satisfactory. It is thus concluded that the various expressions given may be used to study the behavior of waveforms radiated by a resistively loaded linear antenna excited by Gaussian pulse type signals. 138!

1. INTRODUCTION The present report discusses analytically the far field waveforms produced by symmetrical linear antennas loaded non-uniformly and continuously with resistance. The antennas considered are thin cylinders and excited at the centers by slice generators having Gaussian pulse type of time dependence. Emphasis is given here to a specific type of loading in which the amount of loading increases continuously towards the antenna end-points. Direct numerical investigation of such a problem has been discussed in Appendix A. Here we derive theoretical expressions for the waveforms produced by the loaded linear antenna excited by Gaussian pulse type time dependent signals. Fourier transform technique is utilized in obtaining the final time dependent results. Numerical results are obtained by computing the various expressions derived. These results are then compared with those obtained by direct numerical means. 2. BASIC RELATIONS Let us assume that the linear antenna of length 2 L be aligned along the z-axis of a rectangular coordinate system with the origin located at the center of the antenna and that it is excited by a slice generator located at the origin. The far electric field produced by the antenna when excited by a harmonically time dependent slice generator of unit strength consists of only a 0-component and is given by the following expression: F = F (r, 0, ) e, (B.1) where, j jwr sin0 jc L jiz'os0 F(r, 0, w) 4 I I(z,))e dz', (B.2) -L 139

r - is the intrinsic impedance of free space, c I ~ is the velocity of light in free space, o o r, 0, 0 are the spherical polar coordinates of the far field point with origin located at the center of the antenna, I (z', w) is the current distribution on the antenna due to the harmonically time dependent slice generator. F ( r, 0, w) is the time independent far electric field produced by the antenna and may be looked upon as the transfer function of the antenna. For convenience we define the following modified form for the antenna transfer function: i r fe (0, ) = rF (r, 0, w) e jwrT sin0 L Zi zcos 47 I(z', w)e dz', (B.3) -L Notice that in the modified transfer function given by Eq.((B. 3), the dependence on the factor r as well as the phase shift suffered by the signal in traveling from the antenna to the field point are both suppressed. Let the time waveform of the input voltage signal be given by: -t / 2 V(t) = e, (B.4) where a is proportional to the width of the input Gaussian pulse. The Fourier transform V ( ) of V ( t) is obtained as follows: Go 22 j'-' 0'/2 V(U) = V(t) e- dt /2 a e 2 (B.5) -a14 140

By making use of the linearity of the system along with superposition theorem and the concepts of Fourier transform technique, it can be shown that the modified time dependent far field produced by the antenna excited by a Gaussian voltage pulse is given by the following: w-oo -= 2 f0 (0,6w) ). (B.6) 2 7f, et I-00 In Eq. (B. 6) the quantity e(,0 f (9, O ) V () Vw) may be looked upon as the spectral density of the far field wave. Notice that by definition e8 ( 0, t) and e (0, 0o) are related to each other by the following transform relationship e( 0, w) = eO(, t) e dt -00 coo = fo(0, w) V(W). (B.7) 3. TRANSFER FUNCTION OF THE ANTENNA fr(0, (O). In this section we discuss the evaluation of the modified transfer function of the antenna as given by Eq. (B. 3). To obtain f (e, c) it is necessary to know the current distribution I (z, U ) on the antenna excited by the harmonically time dependent unit slice generator. Let the antenna be resistively loaded in the following fashion: C r (z) L C ohms/meter, (B.8) S L-z I I 141

where C is a constant expressed in ohms. For some special value of C, the loaded linear antenna sustains a pure outward traveling wave of current for the time harmonic 3 4, 24 case 4. In particular, for the non-reflecting case the necessary loading is, o 0 r(z)~.." 1. (B.9) which means that the constant C = 2,r, where k is obtained from the expansion parameter as defined by King and Wu 2.(Note: in this notation r = C/L). As discussed in the references cited, the current distribution on the antenna loaded according to Eq. (B. 8) and excited by a harmonically time dependent unit slice generator is given by: 27r jW -[ Lz'I I (xz, a~ - I [1 w L e * (B.10) We shall not go into more discussion of the current distribution on the antenna. The 3 parameter ~ in Eq. (B. 10) is very slowly varying function of U and will be assumed 2L from now on that f is a constant for an antenna of given length L and 2 = 21 n - a where a is the radius of the antenna element. For the non-reflecting case,5.3. Thus Eq. (B. 10) gives the complete description of the current distribution on the non-reflectively loaded linear antenna excited by a harmonically time dependent slice generator of unit strength. After introducing Eq. (B. 10) into Eq. (B. 3) the following expression is obtained for the antenna transfer function: f(0,) = sin 0 (jwr) (ijwr) f0Z 2Y2 1+jwT x 1 142

where, L T = -C is the transit time of the signal on the antenna. The integral involved in Eq. (B. 11) can be carried out in closed form and is given by the following: 1 (1 -| vi ) e iVeWV o 0 dv l-1 1 j T(1 -Cos 0) + 1 1 e-J w (1+cose) 1 + - + cos 1 ) jUT (1+COSO) jar (11 + cos ) ) i (B. 12) After introducing Eq. (B. 12) into Eq. (B. 11) we obtain the complete expression for the antenna transfer function fe (, w). 4. SPECTRAL DENSITY OF THE FAR FIELD WAVEFORM: e' (., w). In this section we obtain the spectral density function e' (, U) for the antenna when excited by a Gaussian pulse. This is done by making use of Eqs. (B. 5), (B. 7) (B. 11) and (B. 12) and the final result is given by the following expression: e (0,c ) = Or 9 0 v 1 x jW 22 sin (jWT) (jW) O /2 1+ juT -jwT(1-cose) 1-os -(jUT) (1-COSO) +1 1 - cos j T ( (-cos ) + 1 (e-jw 14+cos0)-T +, l+cos0 0 (jWT) (1+C080) 1j (B.13). 143

In particular, in the broadside direction of the antenna 0 = 7r/2 and we obtain the following simplified expression for the spectral density of the far field waveform: ~ *r W) = ae e 't w - e 0 2 q/ 2 2 a W/2 (iwrT) (iw) x 1+ jwT sin w /2 w r/2 (B. 14) x jwT JUT All the important frequency domain results pertinent to the antenna may be obtained from Eq. (B. 13). It is useful to investigate the high and low frequency behavior of e (0, ) analytically. After studying the behavior of ee (6, W) as given by Eq. (13) in the low and high frequency limits it is found that 2 2 9 '' a/2 e (,w) - " (jWT) sin O0 e 22, 1 -a /2 e.0 ' " f sin e for r << 1 for T>>1. (B. 15) (B.16) The above two relations indicate that for all 0 0, %e (, o)- 0 as w-P0 and oo. After study Eq. (B. 13) for 0 0 it is found that ee(0(, W) 0. Thus it is concluded that for all values 0, the spectral density function tends to zero in both the low and high frequency limits. 5. EVALUATION OF THE FAR FIELD WAVEFORM e (0, t). After introducing Eq. (B. 13) into Eq. (B. 6) and some algebraic manipulations the far field waveform can be written in the following form: e ( 0,t) = I 1 sin e ( v,t)+I(vf',t)o (B. 17) 9 144

where 1 22 ejW 1 I (v t)- =i e J + 2 edw, (B 18) I '' (J2 v2 1 -c0 V = 1-cos 0 'I 1 + Cos 0 (B. 19) The integral in Eq. (B. 18) may be carried out by applying the convolution theorem. For this purpose it has been found convenient to write Eq. (B. 18) in the following form: I (VI O = I1 (vv O +I2 (v, (B. 20) where, I (v t) = 2 1 ' 2 7r O 22 -ao W/2 (jTw) (ijW) 1 ejTd e - je dw 1+ jWT jtT V 22 -a /2 (jir)T( ) (JT) e -1 JTrrd e 1+jT r)2 2 e d (B. 21) (B.22) Goo I (v o t =1 Let us consider the integral given by Eq. (B. 21) first. It can be shown that I1 (v, t) can be written as follows: I 1( V$ t) = I1 (vp t) + I1"(Vs, O) (B. 23) where, 00.1 n1 2 Il' (v, t)* = r 2 O-a) 22 1 -7O/2 jTd ( jw)- e /2 eJW dw + 2 T (B. 24) aI( (=v, t) In' (vt)w at We now make use of the following two Fourier transform relations: (B. 25) 145

2 -2 2 2 e, ) 2 r a ee (B. 26) - t/ 7/ Te 1 1 2 2 + T (B.27) After using the relations ( B. 2 7) an d malking use of the convolution theorem Eq. (B. 24) can be transformed into the following form: I1' (v,t) = 2 av at I ( t-'a) 22 2 -ea//T, e e da -oD (B. 28) From Eq. (B.25) we obtain I" (v, t) as follows: 1 I,, /.\ IaT a2 i(v, tO = - - 1 2 av at2 ID eit-a) 2 2 e11al/d -COD (B. 29) Before evaluating the integrals in Eqs. (B. 28) and (B. 29) we discuss the integral I2 (v, t ) given by Eq. (B. 22). It has been found convenient to write I2 (v, t) in the following form: I2 (v, t) I21 (v, t)- I21 (v, t-v) where, (B.30) (B. 31) 22 oo -ea o/2 I (vth -( e dw 21 2(vv t) 2 1 + -COD It can be shown that I21 (v, t ) can be written as follows: I21 (v, t) = I2 (vt)+I (v t) (B.32) 146

where oO It (Vj t) = i 1 21(V t)' 22 v - 00 22 e ej r do 22 e d 1+ Tr (B. 33) (B.34) I ( t) (v, t) 21 at 21Y~t Following the similar procedure as before, Eq. (B. 33) can be transformed into the following form: 00 21 21 2 a I -V -00 - t-ea/22 -H/ e e da (B. 35) I 1 ( v, t) can be evaluated by making use of Eqs. (B. 33) and (B. 34). It can be shown that I (v, t) as given by Eq.(B. 28) maybe expressed in the following form: 2 r 1 a /27 (v, t) =.. I1 2 vT x + r ~'7t/, [- l+erf 1-+ ()+ e 1-s" 2 a T 1/ (B. 36) where the error function is defined as follows erf x - 2 -vra-j _t2 e dt (B37 (B. 37). 147

After using Eqs. (B. 36) and (B. 29), I" (v, t) is obtained as in the following 1 1 '2 V7 2 7 -- e 2 - /2 T - e ((l+erf ( a - t/T - ))e 7 + ( -erf1 ( (B. 38) Introducing Eqs. (B. 38) and (B. 36) into Eq. (B. 23), we obtain the following expression for I (v, t): 1(v, t) V-= r 1 va 2 2 a /27 - T C1' + erf t/ e 1 - e,r2 - -t/2 -/2 a I]. (B. 39) I' (v, t) as given by Eq. (B. 35) can be evaluated in closed form and is given 21 by: 7 1 I21 ) 2 2 V T 2 2 a /2 T e [1 erf ( tl a)) erf ~ (+ 7) e %) a t -t / e (B. 40) After using Eqs. (B. 34) and (B. 40) I"l(v, t) is obtained as follows: 2i \ I2 (v, t) = 21 r 1 ~- - 2 2 V T 2 1 e 2 - + e rf1 + (l-erf -(t+ N 2 a T t -a -t / (r- - )J e e/T e 0 (B.41) 148

By making use of Eqs. (B.40), (B. 41) and (B. 32) we obtain:j 2 1 /272 I21(v, t) = - -i 2e V 2 V T 1 t a -t -V 1 + erf 1 (-_ f)) e r' JT (B. 42) Introducing Eq. (B. 42) into Eq. (B. 30) we obtain: /2 2 e 7 a 12 (v, t) -=.. 2 OrV VT [-t - e a/T + + erf-( ( 'r _ t-T,...,. - a _.. +(l+erf -(t v - ) e T. (B.43) After introducing Eqs. (B. 42) and (B. 43) into Eq. (B. 20) we finally obtain the following: I(v, t) - VVT L 2 1 t1 2 a 1 2e — (l+ ) l+erf(- (- -t) e q T r a T"/7; + 2 U+ o r22 e -erf t-TV a VT r Y o T ~~er~( t-Tv T e (B.44) Changing v to v ' in Eq. (B. 44) we obtain: I(v', t) = via 1 -t2 2 ~ 727 T V1 1 + erf ( ) ~ - o T / 2 -t/ a /2 2 e e t-Tv]T T e 4 (B.45) 2 a/2 T e + 1c v' T 1 + erf ( t -v' ) V^/T —^ Thus we obtain the following expression for I (v, t) + I (v', t): I(v, t) + I (v', t) = ' i r2 1 (1 t,22 Lr2 (v+vI) e L~IV 149

2 e e cr - - T 1 + erf t a-l.I'v+1~f e + a T 1 + erf-(,r2 t-vT t-v T a)) e a Tr 2 v t-Vt T7 T [ + (1 +erf - t-V'T a ( T a' T (B. 46) Introducing Eq. (B. 40) into Eq. (B. 17) we obtain the following expression for the far field waveform produced by the antenna: e(e, t) = 9, 2r 0 W sin0 O7 (-+-v ) -t2 t/2 2 e/T /2 a 1 t ca e e + erf (- - -) a 2 2 + T T t-vT e" T 2 t-v' T T e V2 v' 2 2 e/22 e (1+ 1 t-v'? 7a1 erf (- ^-2) a TJ,r2)) I (B. 47) where, v= l+cos, vu = 1 - cos, In the direction 0 = /2, v = v' = 1 and we obtain from Eq. (B.47): 150

e (- t) = v2 -0 2' 2 2 T e 1+ er t CFa-t/?e aF T" a2 2 a/2 2 e a c 7" (t- T) ar ar I " e (B, 48) At time t = 0 the Eq. (B. 47) reduces to the following: e (0,q 0) = 0 2 sin 0 1 +1 oe. (-+-,)v +a 7" a /2 ~2 e a/22T e a22 a/2 T2 e 2- c 21 K 3k "V/.(B, 49) The broadside field produced by the antenna at time t = 0 is obtained by setting 9=ir/2 and v=v'=1 in Eq. (B. 48)anls given'by: e (-7 0) = 0 2'0 V a 1 2 2 a 2r -e 'A7" /27" (1 - erf ( cr ) %[2- T ' -7 + cr ) T + erf-I ( (B. 50) 151

For both large positive and negative time, i. e., t >> T, and T > a, it can be shown that Eq. (B. 48) can be approximated by the following: -t2/ 2 a/ 2 t -t/7 t + e 2T {+erf t (e-2) e T. T T (e-2) e ltl>> T T>> 7,-) It can be seen from Eq. (B. 50) that e0 (7r/2, t)-0 as t-)oo which is as it should be. It can be seen from Eqs. (B. 47) and (B. 48) that the radiated waveform is different from Gaussian and also that it is a function of the parameter a/T. It is thus anticipated that the radiated pulse shape will strongly depend on the ratio of the input pulse width to the transit time on the antenna. 6. TIME DEPENDENT CURRENT DISTRIBUTION i(z, t) In some cases it may be of interest to know explicitly the time dependent current distribution on the Gaussian pulse excited loaded antenna. The current distribution on the antenna excited by a harmonically time dependent unit slice generator is given by Eq. (B. 10). For a Gaussian input signal of the form given by Eqs. (B. 4) and (B. 5), the time dependent current distribution on the antenna is obtained by using the following relations: -o i(z,t)= (2 7r -0O where I( z, w) and V(w) are given by Eqs. (B. 10) and (B. 5) respectively. The integral in Eq. (B. 52) can be evaluated in closed form in a manner discussed in Section 5. The final expression for the time dependent current distribution is given by the following: 152

i( t ll)/c) /2 a2 X~ L \/ - T e /2 l+erf (t- )} e T. (B.53 The input current i ( O t) on the antenna when excited by an input signal of the form -t 2 V(t) e /2 is given by:.7. r a2 a /2 a2 Cr /22 1 t a t1"r i(o, t) am - b e r a t erf -th a B. 54) 7. NUMERICAL RESULTS In this section we discuss some of the selected results obtained by numerical computation of the waveform expressions derived in section 5. Figs. B-l and B-2 show e ( 72, t) vs t for a 1. 0 ns and a 5 0. 471 ns respectively and for two different values of the parameter. In both cases the transit time on the antenna is T= 3. 33 ns. The corresponding values of e ( 7, t) obtained by direct numerical investigation 0 2 are also shown in the same figures for comparison. The direct numerical investigation of the problem has been discussed in a Appendix A. However, it is appropriate here to give a very brief outline of the numerical method. In this method, at first, the current distribution I( z, w) on the harmonically excited loaded antenna is obtained by numerically solving an appropriate form of Hallen's integral equation. The transfer function f( 6, w ) of the antenna and 153

the spectral density e( 0, w ) of the radiated waveform are then obtained by numerically evaluating Eq. (6) with the help of fast Fourier inversion technique. The results shown in Figs. B-1 and B-2 indicate that the agreement between the analytical and numerical solutions may be considered to be satisfactory. In particular, in the main pulse region the two results agree very well. Figs. B-3 and B-4 show the radiated waveforms obtained from Eq. (B. 47) for three different values of 9 and for a = 1 ns, and a = O. 471 ns. The transit time on the antenna is T= 3. 33 ns.I 8: CONCLUSION In the above we have developed analytic expressions for the waveforms radiated by a resistively loaded linear antenna excited by Gaussian pulse. The antenna is assumed to be loaded such that it sustains a pure traveling wave of current. The analytic results have been compared with those obtained by direct numerical means. The general agreement between the two results has been found to be satisfactory. It is thus concluded that the various expressions given above may be used to study the behavior of waveforms radiated by a resistively loaded linear antenna excited by a Gaussian pulse. 154

ee(, t) X X analytical solution numerical solution 0n -3.26 -1. 63 1.63 4.89 6. 52 8.15 9.78 11.41 13.04 14.67 t in nanoseconds -- -8( I? FIG. B-la: Far field waveform in the broadside direction. a =1 ns; T= 3. 33 ns; = 5. -12( - 14(

e (, t) V - analytical solution " x — *- numerical solution 150 100 Un Om -3.63 -1.63 -50 -100 ( 4.89 6.52 8.15 9.78 11.41 13.04 t in nanoseconds -- FIG. B-lb: Far field waveform in the broadside direction. a = 1 n 333s; =3.33ns; 8. - 150

e (, t) I x n x analytical solution numerical solution 300 U I- 2.6 -2. 6 -1.3 1.9 5.2 6.5 7.8 9. 1 10.4 11. 7 13. 0 - 100 t in nanoseconds --- A -200 I\ / FIG. B-2a: Far field waveform in the broadside direction. =0.471 ns; T= 3.33 ns; t/ =5.

e(2, t) 92' I analytical solution,,, numerical solution 300 200 c-" en -1.3 3.9 5.2 6.5 7.8 9. 1 - 100 / t in nanoseconds - > -200 FIG. B-2b: Far field waveform in the broadside direction. a= 0.471 ns; T= 3.33 ns; = 8.

ee(, t) o 6 401 300 200 100 -3.9 -2.6 -1.3 1.3 -100 -200 - 300 Z 9.T 78 9.1 10.9 11.7 5.2 6.5. t in nanoseconds FIG. B-3a: Far field waveform in the direction 0 = 7/6 a 1 ns; T=3.33 ns; p=8.

e (, t) 400 300 200 100 3.9 -3.9 -1.3 1.3 5.2 6.5 7.8 9,1 10.4 11.7 t in nanoseconds -- -300 -400 FIG. B-3b: Far field waveform in the direction 0 = r/4. a= 1 3.33ns; 3.33ns; =8.

e (3, t \ 500 400 300 200 100 -3.9 -2. 6 -1. 3 1.3 -100 -200 -300,-400, 5.2 6.5 7.8 9.1 10.4 11.7 t in nanoseconds -- FIG. B-3c: Far field waveform in the direction 9 =,Tr/3. a- Ins; T= 3.33 ns; 0 = 8.

eo(, t) 800 600 400 200 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 -3.9 -2.6 -1.3 -200 - 400 - 600 -800 -1000 t in nanoseconds -- FIG. B-4a: Far field waveform in the direction 0 = 7/6. a = 0.471; 333ns;= 3.33 = 8.

e(-, t) T 1000 800 600 400 200 -3.9 -2.6 -1.3 2. 6 3.9 - 200 co -400 5.2 6.5 7.8 9.1 10.4 11.7 t in nanoseconds > — Far field waveform in the direction 0 = 7r/4. r = 0. 471 ns; T= 3. 33 ns; t = 8. - 600 - 800 FIG. B-4b: - 1000

ee(Z, t) 039 / 1 800 600 400 200 I.. 4^ -2.6 -1.3 -200 -400 -600 -800 -1000 t in nanoseconds - FIG. B-4c: Far field waveform in the direction = Tr/3. a=0.471 ns; T= 3.33 ns; = 8.

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