a b c Fig. S. The anenna consucion: (a) ain geoeical paaees, (b) he wie suppo pilla and (c) he console link beween wie and coaial pobe.
Fig. S. The anenna coss-secion in he y-z plane.
Accoding o [], he adiaion esisance as follows of he EFD above he gound can be epesened Q sin sin sin dd, (A) sin Q cos( cos ) cos sin( cos )sin cos.5cos( ( cos ))( cos ).5cos( ( cos ))( cos ), (A) whee kl, kh. To obain he closed-fo epession of (A) {his epession was no deived in []}, he suppleenay sybols have been inoduced o denoe he consans: s sin, s sin( ), c cos, c cos( ). Loweing he degee of sine by he sandad foula, we ewie (A) as 6 Q ( cos( sin sin )) dd (A3) sin Using he well-known foula fo Bessel funcion [], we obain J n ( z) cos( zsin n ) d, (A4) and he epession (A3) ay be ewien as follows: 6 Qd Q J ( sin ) d d sin sin (A5) n [9] he epession fo adiaion esisance whee sin ( ) sin. Q.5 d sin of he EFD locaed in fee-space is given 6 6.5 ln.5ci(4 ) Ci( ) s (sin ( ) ), (A6) Afe siplifying (A5), i leads o Q J ( sin ) 6 d sin, (A7)
whee.57757 (Eule consan), and he cosine inegal is defined as cosu Ci( ) ln( ) du. (A8) u Now, we obain in (A7) one inegal only. s inegand is syeical. Le us denoe i hen / Q J ( sin ) d Q J ( sin ) d sin (A9) sin Duing he inegaion seveal changes of vaiables will be ade. The fis vaiable change is cos, cos,, d d sin (A) Afe he inegaion liis ecalculaion, we ge uppe lii a cos( ), lowe lii b cos (he eaple of inegaion wih he changes (A) is pesened in [9]). Then, accoding o (A), we obain: J ( ).5cos( ( - ))( + ) The conen of he backes is squaed, and we ewie (A) as follows: J cos( ( - )) ( + ).5cos( ( + ))( - ) d (A) cos( ( + )) ( ) - cos( ( + ))( - ) cos( ( - ))cos( ( + ))( - ) cos( ( ))( )d (A) Le us conside he inegal as follows: J cos( ( - ))cos( ( + ))d (A3)
Then epession (A) ay be ewien in he fo J cos( ( - )) ( + ) cos( ( )) ( ) cos( ( + ))( - ) cos( ( ))( )d (A4) and epession (A7) is epesened as 6( + ) (A5) When using foula [] a J b a cos( c) d sin a b c (A6) a epession (A3) akes he fo J d.5sin 4.5c sin ( ) (A7) The following aheaical anipulaions and he use of (A6) lead o epessions J ( s sin( ) 4ssin( )) 3 J Applying he well-known epession fo he Bessel funcion [3] c cos( ) 4ccos( )d, (A8) d s sin 4 sin ( ) (A9) and he following equaion v ( ) (.5) J v () Г( )Г( ) + v+ +
!! ( ) d ( )! we can now ewie (A8) and (A9) as ( ) (.5 ) ( ) c cos( ) Г( +)Г( +) ( s sin( ) 4ssin( )) 4ccos( )d (A) ( ) 3 ( ) d 3 ( )!, (A) s sin 4 sin ( ) whee Г( ) ( )! is he gaa funcion fo inege. A saighfowad aheaical anipulaion using he posiion change of suaion and inegaion leads o s (.5 ) sin( ) ( ) d Г( +)Г( +) 4s (.5 ) sin( ) ( ) d Г( +)Г( +) c (.5 ) cos( )( ) d Г( +)Г( +) 4c (.5 ) cos( )( ) d (A) Г( +)Г( +) The following changes of he vaiables ae used inside he las epession (A): - fo he fis and second inegals cos, cos, d sind (A3) Afe he inegaion liis ecalculaion, we ge uppe lii a cos, lowe lii b cos ;
- fo he hid and fouh ones sin, sin, d cosd (A4) afe he inegaion liis ecalculaion, we ge uppe lii a sin, lowe lii b sin. As a esul, we can ewie (A) in he fo s / sin( cos ) cos (sin ) d ( ) 4 Г( +)Г( +) 4s / (.5 ) sin( cos ) cos (sin ) d Г( +)Г( +) c / (.5 ) cos(sin )(cos ) d Г( +)Г( +) 4c / (.5 ) cos( sin )(cos ) d (A5) Г( +)Г( +) n ode o avoid he division by zeo, he coesponding eleens of seies in (A5) ae shifed in inegal and, afe such anipulaions, he coesponding Poison inegals ae used [3] / cos( zsin )(cos ) d Г( )J ( z).5.5 (.5 z) / sin( z cos )cos (sin ) Г( )J ( z).5.5 (.5 z) Boh las epessions in conflaion wih he classical equaion ( ) z ( )! Ci( z) ln( z) lead o he following esuls fo (A) and (A5): / sin( cos )cos s d sin / / sin( cos ) cos cos(sin ) 4s d c d cos sin / cos( sin ) 4c d 3 ( ) d 3(Ci( ) cos
, (A6) ln( )) s sin 4 sin ( ) Г( +)J.5( ) s 4 Г( +)Г( +).5 4s Г( +)J.5( ).5 4 (.5 ) Г( +)Г( +) Г( )J.5( ) c.5 4 Г( +)Г( +) Г( )J.5( ).5 4 (.5 ) Г( +)Г( +) (A7) 4c Ne, accoding o (A), hee is a change of he vaiables in he fis and second inegals. The analogous change wih sin will be used in he hid and fouh ones. Afe such changes by applying paial facion epansion (A8) ( ).5 ( ) ( ) o he denoinaos of all five inegals, we can ewie (A6) in he fo of 3(Ci( ) ln( )) s sin 4 sin ( ).5 3 ( s sin( ) 4ssin( )) cos( ) 4 cos( ) d c c (A9) Using he spheical Bessel funcions [3] j( ) J.5( ), j ( ) J.5( ) we can ewie (A7) as follows: j ( ) j( ) s s 4!! j ( ) c 4! j ( )!. (A3) c
The epession fo he geneaing funcion of Bessel funcions is known [3]. Using he appoach descibed in [4], his geneaing funcion ceaes new equaions o be used in ansiions fo he seies o he inegals in epession (A3) j ( ) cos( ) cos z z z z d! z, ( ) sin ( ) sin ( ) j z z z z d! Using he las epessions we can ewie (A3) in he fo of /4 sin ( 4 4 ) sin ( ) s d s / sin ( ) sin ( ) d /4 cos( 4 4 ) c c d / cos( ) c c d (A3) Ne, hee is a change of he vaiable in (A9) accoding o u( ), ( u ), d du, a, b (A3) As a esul, equaion (A9) is ewien as 3(Ci( ) ln( )) s sin 4 sin ( ) 3 s sin(( u )) 4ssin( u ) du c cos(( u )) 4ccos( u ) (A33) u The saighfowad aheaical anipulaions lead o 3(Ci( ) ln( )) s sin 4 cos( u) 4 4cosu sin ( ) du (A34) u
Afe he change of vaiables and conflaion of he inegals, we can ewie (A3) as /4.5 s sin ( 4 4 ) c cos( 4 4 ) s sin ( 4 ) Accoding o (A5), o obain he epession fo cos( 4 ) d c (A35) i is necessay o use (A6), (A34) and (A35). Tansfoing (A34) accoding o (A8) and using he changes of vaiables we can ewie (A5) in he fo of 4.5 6 sin cos.5 ln.5ci(4 ) Ci( ) (sin ( ) ) s s c d 4 /4 s sin c cos d 4 d.5 (3Ci( ) Ci( ).5Ci(4 ).5 s.5 3 sin 4 sin ( ) ln (A36) To deduce he final epession fo, i is necessay o pefo he inegaions on he basis of (A8), (A), (A8), (A3) (on he analogy descibed in [9]) sin d a.5[sin ( a)(ci( a) Ci( a)) cos ( a)(si( a ) Si( a ))], cos d.5[cos a(ci( a) Ci( a)) a sin ( a)(si( a ) Si( a ))], whee cos ( ) cos, sinu Si( ) du ; afe he use of hese inegals and beviies, he las u epession (A36) akes he fo of 63.5ln(4 ).5Ci(4 ) Ci( ) 3Ci( ) Ci
Ci.5 Ci 4 s Ci 4 sin ( ) sin ( ) sin 4 (A37) whee li Ci 4.5ln( ).5Ci 4 4.5 ln 4 Thus, in he final sage we can deduce he following epession fo he adiaion esisance of he EFD above he gound.5 3Ci( ) Ci 6.5 ln( ).5Ci(4 ) Ci( ) Ci.5 Ci 4 s Ci 4 sin ( ) sin ( ) sin 4 (A38) The inpu ipedance of he EFD above he gound he in efeed o cuen in on he inpu einals can be obained by a ansfe elaion sin, o as follows: in he in sin, (A39) whee is he cuen aiu of cuen disibuion along he ends-fed dipole adiao [9].