Focal Waveform of a Prolate-Spheroidal IRA

Transcription

1 Sensor and Simulation Notes Note 59 February 6 Focal Waveform of a Prolate-Sheroidal IRA Carl E. Baum University of New Mexico Deartment of Electrical and Comuter Engineering Albuquerque New Mexico 873 Abstract This aer develos some analytic aroximations for the focal waveform roduced at the second focus of a rolate-sheroidal reflector due to a TEM wave launched from the first focus. This is extended to consider the sot sie of the eak field near the second focus. ` This work was sonsored in art by the Air Force Office of Scientific Research. R

2 . Introduction Two revious aers [9, ] have discussed the basic concet of using one or more rolate sheroidal reflectors to focus a fast-transient electromagnetic ulse on some targets of interest. Another aer [] has shown that an inhomogeneous sherical TEM wave launched on guiding conical conductors from one focus is converted by a double stereograhic transformation to a second (reflected) inhomogeneous sherical TEM wave roagating toward the second focus. Both waves have the same temoral waveforms before other scattering (from feed arms, etc.) can reach the observer. The resent aer is concerned with analytic calculations of the waveform at the second focus. For this urose let us consider the geometry in Fig... Let there be two thin erfectly conducting cone wave launchers with electrical centers lying in the x lane. With resect to the negative axis they are oriented at θ θc (.) in the wave-launching sherical system ( r, θ, φ ). These are related to cylindrical ( Ψ, φ, ) and Cartesian (x, y, ) coordinates as ( θ ) ( θ ) ( φ ) y ( φ ) Ψ rsin, rcos x Ψ cos, Ψ sin φ φ, Ψ Ψ + (.) The rolate sheroid is described by Ψ + b a / a b (.3) The thin-cone electrical centers are described by the angle θ c with Ψ c r sin ( θ ) c rcos c ( θ ) c (.4) At the reflector we have (subscrit )

3 x Ψ b S S a C L -a θ c θ c + Vut _ () y x a aerture lane stereograhic rojection lane Ψ b Fig.. Wave Launcher in Prolate-Sheroidal Reflector 3

4 ( θ ) Ψcot c tan ( θc ) + b a tan b ( θc ) + a (.5) Given θ c one can solve for and Ψ. One can also secify and comute Ψ and θ c. As a secial case one may choose the symmetry lane for which (.6) This simlifies the equations to Ψ b, r a sin ( θ ) sin ( π θ ) c c π < θc < π b a (.7) For the resent let us truncate the reflector at the lane. The ortion used is S, to the left. This is consistent with the traditional truncation of araboloidal reflector imulse radiating antennas (IRAs). More sohisticated truncation contours can be considered, but are beyond the scoe of this aer. For later use the truncation lane will be taken as an aerture lane. The ortion of this lane inside the rolate shere is designated S a. It is this surface which will be used for integrating over the reflected TEM wave to find the fields at the second focus, r. 4

5 . Preulse The stereograhic rojections in [] can be used to calculate the fields. Let wave have the form r E r θ ( ), φ V θ, φ ut c θ, φ θ + φ θ sin ( θ) φ V( θ, φ) θ, φ sin θ ( θ ) +, φ sin ( θ ) θ θ sin ( θ ) φ (.) c [ με ] / wave seed in medium Let the wave have V ± V on the two cones. The stereograhic rojection of this wave is θ Ψ [ a ] tan, φ φ φ (.) The electrical center of the thin wire on this rojection lane is θ [ ] tan c Ψ c a, φc, π r w radius of thin wire on rojection lane (.3) For the two thin wires we have the well-known solution [, 5] y x + + c c V ( x, y) V arccosh Ψc Ψ Ψ n r w y x + Ψc Ψc Ψ arccosh c Ψ n c rw rw (.4) From this we have 5

6 Zc Z fg characteristic imedance Z fg / μ ε wave imedance of medium (.5) Ψ arccosh c Ψ n c π rw π rw It is often convenient to secify arameters. In this case we have Z c (around 4 Ω ), and from this determine the various transmission-line fg 4 Ψ.6, c 4 (.6) 377 rw for tyical numbers. Converting (.4) to ( θ, φ ) coordinates we have Ψc Ψ + cos( φ ) + V Ψ (, ) arccosh c rw Ψc V θ φ n r w Ψ Ψ cos( φ ) + Ψc Ψc θ Ψ [ a ] tan (.7) for the otential in wave. The gradient in (.) then gives the electric field. A secial case has φ V ± π / (.8) which is a symmetry lane (the x lane). Another symmetry lane (the y lane) has φ (for Ψ + x) Ψ + Ψ V ( θ, φ) Varccosh c Ψ n c rw Ψ Ψc 6

7 Ψ c + V arccosh Ψc Ψ n Varccosh Ψc Ψ c Ψ O c + rw Ψ c rw Ψ Ψ Ψ V arccosh Ψc Ψc θ cot + rw a (.9) For θ near π this is Ψ ( ), arccosh c Ψc π θ θ, [ ] tan c V θ φ V + Ψ c a rw a (.) Ψ V (, ) Varccosh c θ θ φ tan c Ψ [ π θ], π f arccosh c Ψ n c g r + w rw rw From (.) we have at the second focus r E Eut x c V arccosh c c V E Ψ Ψ θ tan c 4 rw a π f g (.) This is the reulse orientation and magnitude (for ste excitation). It lasts for a time [] / [ a ] / a b Δ t a a b c c c a (.) after which the reflector signal arrives at the second focus. For the secial case as in (.6) and (.7) for which the launcher intersects the reflector at the symmetry lane we have / θ cos( ) tan c θc a b + sin( θc ) b a + b V a + E π fg b [ a ] (.3) V b E Δ t π fg c (time integral or area of reulse) 7

8 3. Fields on Aerture Plane Now we consider the wave heading from the reflector toward the second focus. There is no set of conical conductors guiding the wave there. So we consider this second sherical TEM wave [] on the aerture lane, which we can use in turn to find the fields at r (and other ositions as well). For resent uroses we take the aerture lane as, as in Fig... The reflected wave illuminates S a, a disk of radius Ψ. Note that the reflector is truncated at the aerture lane. This is because the field from the wavelauncher reverses sign for the wave on the other side of the launching conductors. This is the same truncation used for tyical reflector IRAs, based on a self-recirocal geometry []. There are more sohisticated (nonlanar) truncation geometries which may be considered in the future. In [] the reflected wave was related to the first wave by a double stereograhic transformation. They are equal (excet for a minus sign) on the stereograhic rojection lane ( a) for which ( Ψ, φ ) V ( Ψ, φ ) V ( Ψ, φ ) cylindrical coordinates on rojection lane θ θ Ψ [ a ] tan [ a+ ] tan φ φ φ (3.) The second TEM wave takes the form r ( ) a E r θ, φ V θ, φ μt + c c (3.) Note the factor giving an incoming (on r ) ste-function wave. For the aerture integral we need the tangential electric field on S a. First we have Ψ Ψ + cos( φ ) + V c c V( θ, φ) arccosh Ψc Ψ Ψ n rw Ψ Ψ cos( φ ) + Ψc Ψc θ θ [ a ] tan, [ ] tan c Ψ + Ψ c a (3.3) 8

9 For θ near ero we have V V ( θ, φ) arccosh Ψc Ψ 4 cos( φ) + rw Ψc arccosh c a V Ψ + θ cot c θ cos( ) rw a φ + arccosh Ψc π fg rw V a+ θ cot c r a E ut + x rπ fg a c c (3.4) Comaring this to the reulse amlitude in (.) we have ( θ near ) a+ θ cot c r a E E x u t r a + (3.5) c c For the secial case (as in (.3)) that the launcher intersects the reflector at the symmetry lane we have (for θ near ero) V b r a b r a E ut + x E ut + x π fg r a c c r [ a ][ a+ ] c c r a E ut + r c c (3.6) a fairly simle result. At the center of the aerture lane we have r a, E Ea ut + x c c Ea V a+ θ cot c π fg a a+ θ E cot c a (3.7) For this reduces to Ea E (3.8) 9

10 Wave number is focused on r. Without guiding conductors (3.5) cannot hold all the way as r. So we are considering the fields on S a for later integration. On the center of S a we have the electric field E olaried in the x direction. In IRA-related calculations [5, 6] it has been seen that, for circular aertures, the field at the center is an imortant arameter. The boresight radiated field can be found by integrating the TEM field over the aerture, or by integrating a uniform field of the center value (including olariation) over the same circular aerture. Seen another way, one can exand the field in cylindrical coordinates and note that terms with cos( mφ ) and sin( mφ ) for m integrate to ero (for observation field oints on the axis). (There is no m term.) This is basically a symmetry result. Similarly here, let us consider a uniform field on the ( Ψ, φ) rojection lane. This corresonds to a otential function ( ) V Ex EΨ cos φ (3.9) for a uniform field E olaried in the x direction. Matching this field to the second wave at r a + (center of rojection lane) a E E ut x c E V θ cot c π fg a E cot θc a (3.) For this becomes E E V (3.) a + π f g b Now convert V on the rojection lane to ( θ, φ ) coordinates as θ V E a + tan cos [ ] ( φ ) (3.) giving an electric field

11 ( φ ) ( θ ) ( φ ) ( θ ) ( φ ) ( θ ) a + θ θ ( ) sin r a E E sec cos θ θ tan φut + r sin c c a + cos sin r a E θ φut + r + cos + cos c c (3.3) On the aerture lane we have r / Ψ + cos( θ ) r r+ cos( θ) r + φ θ i Ψ cos φ, φ φ ( θ ) (3.4) The tangential electric field is then a + Et E cos( φ) Ψ sin( φ) φ r + r r a ut + c c (3.5) Converting to Cartesian coordinates we need only the x comonent (due to symmetry) as Ψ i x cos sin ( φ) φ i y ( φ) E x a + E cos ( φ) + sin ( φ) r + r Ex r a E x ut + c c (3.6)

12 4. Fields at Second Focus We are now in a osition to evaluate the fields at r by integrals over the fields on S a. A revious aer [] has develoed the formulae. From [ (3.3)] we have (in time domain for ste excitation) a a E f Efx x, Efx Eδ δ t + Es ut c c Eδ E x ds, Es E 3 x ds πc e r π Sa r Sa / r + Ψ (4.) have Using cylindrical coordinates on S a we can evaluate the integrals. Considering first the φ variable we Eδ Ψ π E r a+ cos ( φ) sin ( φ) dφd πc + Ψ Ψ r r + r Ψ E a + d c + Ψ Ψ r r + r Ψ E Ψ [ a ] d c + Ψ 3 r Es Ψ π E a + cos ( φ) sin ( φ) dφd π 3 + Ψ Ψ r r + r Ψ a + E 3 + ΨdΨ r r + r E Ψ Ψ [ a+ ] dψ 4 r Next we have / r, υ +Ψ Ψ +, dυ ΨdΨ Ψ + E 3/ Eδ [ a ] υ dυ c + (4.)

13 3 [ ] [ ] [ ] [ ] [ ] / / r s E a c E a c E E a d E a υ υ Ψ + + Ψ + Ψ Ψ + E a E a Ψ Ψ (4.3) Note that s E has units V/m, while E δ has units Vs/m corresonding to the time integral (or area) of the δ function.

14 5. Some Design Considerations Summariing, we have Eδ / / E Ψ V a + θ [ ] cot c Ψ a c π fgc a Es E a + V a + θ cot c + Ψ π fg a + Ψ E V θ tan c V, a E θ tan c t π fg Δ π fgc E V θ cot c π fg a (5.) One can comute θ c from Ψ b, giving (or Ψ ) in (.5). For the secial symmetric case of, merely relace Eδ E b V b E a+ b V, a+ b Es c a π f gc a a π fg a E V a + V, b E Δt π fg b π fgc E V π fgb (5.) A first observation concerns the common factor V /( π f g ). For large fields one needs large voltage and low wave-launcher imedance. Note that an extra factor of increase in the fields is obtained by going to a 4- arm feed in the usual sense of a IRA with arms at 45 [4], and a little more can be achieved by arms at about 6 from the horiontal (the x lane) as rojected on a constant- lane [7, 8]. This is just another common factor with which we can deal searately. The δ -function art of the field does not have infinite amlitude when the incident ste-like wave has a nonero rise time. For simlicity let us imagine that the δ -function is relaced by δ () t u() t u( t tδ ) t (5.3) δ 4

15 i.e., a gate function of width t δ and height t δ. This makes the imulsive art of the field have amlitude E t δ δ, showing the imortance of a small ulse width. For convenience define T [ ] Δt a (5.4) tδ ctδ We will want this to be large, which argues for large a. Considering the simler case we have Es E b < (5.5) a If we want the ste from the reflector (the ost ulse if you like) to cancel the reulse ste then one would like b to aroach a, but this has other consequences. Next comare the main (δ ) ulse amlitude to the reulse amlitude giving E δ b b T [ a+ ] T (5.6) tδe actδ a ct a+ a The factor / a is less than one, but it needs to be large to give a large main ulse. This is further aided by a large T imlying large a and small t δ. So, intermediate values of are called for, i.e., at the maximum of [ a ] which is a 3, b a θc 3 π 35 4 b 3 a 4 (5.7) 5

16 6. Sot Sie Given that the imulse has some small width t δ, the maximum fields will exist in some small region around r. So let us estimate a ulse width of this imulsive art for ositions near r. For the δ -function ulse we have the wave from every osition on S a arriving at exactly the same time at r. We can then estimate a ulse width near r by the disersion in the arrival times from all arts of S a at the observation oint. For this urose, consider the lengths of the ray aths in Fig. 6. indicating the maximum time differences from the edges and center of S a to the observer. For an observer at + Δ on the axis we have tc arrival time from center Then we have ctc a +Δ te arrival time from edge cte / / + +Ψ + +Δ +Ψ / / + + Ψ + + Δ + Ψ / / Δ + + Ψ + + Ψ + + Ψ Δ a + / + Ψ (6.) t ulse width with resect to tc te ct Δ (6.) / [ r] +Ψ For the secial case of we have ct Δ a (6.3) Large / a (small b/a) minimies this. For an observer radially dislaced ΔΨ from r we have t arrival time from near edge 6

17 Ψ S a ΔΨ x r + Δ Ψ Fig. 6. Disersion of Imulse Near r. 7

18 Then we have ct / / + +Ψ + + [ Ψr ΔΨ] / / + +Ψ + +Ψ ΨΔΨ / / Ψ rδψ + +Ψ + +Ψ +Ψ Ψ ΔΨ a / +Ψ t arrival time from far edge ΨΔΨ ct a + (6.4) / +Ψ t Ψ ulse width with resect to Ψ t t Ψ ΔΨ ctψ / + Ψ (6.5) For the secial case of we have b ctψ ΔΨ ( ΔΨ ositive) (6.6) a Small b/a also minimies this. Comaring these ulse widths to t δ, the width at r, we can note that the ulse widths in the direction are tδ + t, and in the Ψ direction tδ + tψ. The hysical sot sie is then given by (counting width in both directions from r ) t Ψ t t δ (6.7) Given by (6.) and (6.4). For the secial case of we have a Δ ctδ, ΔΨ ctδ a (6.8) b Of course, these are just rough estimates. Detailed waveform calculations will give more accurate results, including actual waveforms instead of bounds on ulse widths. 8

19 7. Concluding Remarks This modest study has found some analytic aroximations useful for estimating the focal waveforms and focal sot sie for the rolate-sheroidal IRA. One may think of this as analogous to the first of the aers dealing with the IRA using a araboloidal reflector [3]. Considering the sohisticated design aers which followed the introduction of the IRA concet, there is much yet to be done for the rolate-sheroidal version. Here we have found some esecially simle results for the case of reflector truncation at. More detailed numerical treatment of the results for may lead to yet more insight into an otimal design. 9

20 References. C. E. Baum, Imedances and Field Distributions for Symmetrical Two Wire and Four Wire Transmission Line Simulators, Sensor and Simulation Note 7, October C. E. Baum, Focused Aerture Antennas, Sensor and Simulation Note 36, May C. E. Baum, Radiation of Imulse-Like Transient Fields, Sensor and Simulation Note 3, November C. E. Baum, Configurations of TEM Feed for an IRA, Sensor and Simulation Note 37, Aril C. E. Baum, Aerture Efficiencies for IRAs, Sensor and Simulation Note 38, June C. E. Baum, Circular Aerture Antennas in Time Domain, Sensor and Simulation Note 35, November C. E. Baum, Selection of Angles Between Planes of TEM Feed Arms of an IRA, Sensor and Simulation Note 45, August J. S. Tyo, Otimiation of the Feed Imedance for an Arbitrary Crossed-Feed-Arm Imulse Radiating Antenna, Sensor and Simulation Note 438, November C. E. Baum, Prodicing Large Transient Electromagnetic Fields in a Small Region: An Electromagnetic Imlosion, Sensor and Simulation Note 5, August 5.. C. E. Baum, Combining Multile Prolate Sheroidal Reflectors as a Timed Array with a Common Near-Field Focus, Sensor and Simulation Note 54, November 5.. C. E. Baum, Prolate Sheroidal Scatterer for Sherical TEM Waves, Sensor and Simulation Note 58, January 6.. E. G. Farr and C. E. Baum, Radiation from Self-Recirocal Aertures, ch. 6,. 8-38, in C. E. Baum and H. N. Kritikos (eds.), Electromagnetic Symmetry, Taylor & Francis, 995.