A Gain Measurement in the Liquid Based on Friis Transmission Formula in the Near-Field Region

Transcription

1 A Gain Measuement in the Liquid Based on Fiis Tansmission Fomula in the Nea-Field Region # Takuhei Akagawa 1, Nozomu Ishii 2,3, Ken-ichi Sato 4, Lia Hamada 3 and Soichi Watanabe 3 1 Gaduate School of Science and Technology, Niigata Univesity, 2 Faculty of Engineeing, Niigata Univesity, Ikaashi 2-85, Niigata, , Japan 3 National Institute of Infomation and Communications Technology, 4-2-1, Nukui-Kitamachi, Koganei, Tokyo, , Japan 4 NTT Advanced Technology, , Midoi-cho, Musashino, Tokyo, , Japan nishii@bc.niigata-u.ac.jp Abstact In 3MHz-3GHz, a pobe used in measuing SAR (Specific Absoption Rate) of the mobile communication device is usually calibated by use of a ectangula waveguide filled with the tissue equivalent liquid. Above 3GHz, howeve, this conventional calibation has the possibility of inaccuate assessment because the diamete of the pobe is compaable with the coss-sectional dimension of the waveguide. Theefoe, an altenative calibation fo the SAR pobe based on anothe pinciple is equied and is developed by the authos. In ou poposed calibation, fist, the gain of the efeence antenna in the liquid is evaluated by use of two antenna method based on the Fiis tansmission fomula in the conducting medium, and then the electic field adiated by the efeence antenna is elated to the output voltage of the SAR pobe at a point in the liquid. Howeve, the fields ae significantly educed in the liquid and the gain is impossible to be calibated in the fafield egion. To ovecome this difficulty, the Fiis tansmission fomula in the conducting medium is extended in the nea-field egion. In the pape, some numeical and expeimental esults of estimated gain based on the Fiis tansmission fomula in the nea-field egion ae epoted and the validity of the new fomula is checked. 1. INTRODUCTION A new calibation method fo SAR (Specific Absoption Rate) pobes is equied to evaluate SAR values of vaious mobile communication devices above 3GHz. A conventional calibation method by use of with the waveguide filled with the tissue equivalent liquid [1], [3] is widely employed in 3MHz- 3GHz [1]. Howeve, the stonge effect of the pobe diamete cannot be ignoed at the highe fequency, especially the diamete is compaable to the coss-sectional dimension of the waveguide above 3GHz. Theefoe, an altenative calibation method have been developed by the authos [4] [6]. In ou poposed calibation, the gain of the efeence antenna in the liquid [7] is evaluated by use of two antenna method based on the Fiis tansmission fomula in the conducting medium [8]. Then, the SAR pobe can be calibated by elating the electic field adiated by the efeence antenna to the output voltage of the pobe at a point in the liquid [1], [2]. S paametes between two identical efeence antennas which ae faced and aligned with each othe ae measued to evaluate the gain of the efeence antenna. In othe wods, two identical antennas ae connected to pot 1 and 2 of the netwok analyze, and then the magnitude and phase of S 21 between two pots ae measued as a function of the distance between two antennas. Measued data ae fit into cuves deived fom the Fiis tansmission fomula in the fa-field egion and then the gain of the efeence antenna, the attenuation and phase constants of the medium, α, β, can be estimated. Howeve, the decay in the liquid is so enomous that the measuement of S 21 can be difficult in the fa-field egion [6]. Fo example, the attenuation constant in the liquid is α = 47dB/m at 2.45GHz. Above 3GHz, the decay is moe enomous, so that measuable ange fo S 21 is much shote than one expected by the scaling ule of the wavelength fo the fequency. At the time of witing, we have tied to study the following solutions. 1) To gain the input powe to the measuement system, P in, amplifies can be inseted in fome stages of pot 1 of the netwok analyze. 2) To ensue wide dynamic ange of the measuement system, the magnitude of S 21 can be measued as a function of the distance between the antennas by use of a spectum analyze instead of a netwok analyze. Since the phase of S 21 cannot be measued by use of the spectum analyze, the pemittivity of the liquid is equied to be measued by the contact pobe method. 3) To avoid the cuve fitting in the fa-field egion, measued S 21 should be fitted into the cuves based on the Fiis tansmission fomula which holds in the nea-field egion of the antenna in the liquid. This pape will epot some esults fo the item 3). The Fiis tansmission fomula in the Fesnel egion of the antennas in fee space had been studied to estimate the gain of elatively 1

2 lage-scaled apetue antennas; the gain is epesented by a function of the distance fom the antenna and detemined by its convegent point [13], [14]. Howeve, no efeences descibed the behavio of the fields o the Fiis tansmission fomula in the nea-field egion of the antenna in the conducting medium as fa as the authos know. In this pape, the Fiis tansmission fomula which holds in the nea-field egion of the antenna in the conducting medium is poposed by abstacting some points fom both the Fiis tansmission fomula in the nea-field egion of the antenna in fee space [14] and the Fesnel appoximations in fee space [15]. To discuss a physical aspect of new fomula, the distance dependence of the electic field in the nea-field egion is examined fo a dipole antenna and is extended fo an abitay antenna. And, taget functions fo cuve fitting ae ewitten on the basis on extended Fiis tansmission fomula in the nea-field egion. Finally, new cuve fitting is applied to calculated and measued data to confim the validity of the new fomula. 2. EXTENSION OF FRIIS TRANSMISSION FORMULA A. Fiis Tansmission Fomula in the Fa-Field Region Two polaization matched antennas ae aligned fo maximum diectional adiation in the liquid, as shown Fig. 1. If the distance between two antennas,, is sufficiently lage, the atio of tansmitted powe, P 1, to eceived powe, P 2, based on the Fiis tansmission fomula in the fa-field egion [8], can be expessed by S 21 2 = P 2 =(1 S 11 2 )(1 S 22 2 ) G 1 G 2 e 2α P 1 4β 2 2, (1) whee S ij ae S paametes between the pots of two antennas, G i ae the magnitude of the gain of antenna i, and α and β denote the attenuation and phase constants in the liquid, espectively. The subscipt i means the tansmitting o eceiving antennas fo i =1o 2. (1) can be ewitten in db fom as S 21 db = A 2 log α. (2) S 21 db can be measued as a function of the distance and a constant A and the attenuation constant α can be detemined by the cuve fitting fo S 21 db based on (2). Then, the sum of the gains in db fom can be expessed by (G 1 ) db +(G 2 ) db = A+2 log 1 (2β) (M 1 ) db (M 2 ) db, (3) whee (M i ) db = 1 log 1 (1 S ii 2 ) is a mismatch o eflection efficiency of antenna i. If two antennas ae identical, thei gains ae equal. Then, (3) can be eplaced by G db = 1 2 [A + 2 log 1(2β) (M 1 ) db (M 2 ) db ]. (4) In the conventional two-antenna method, the constant A o the gain G db can be detemined at a distance,, in accodance with (2) o (4). On the othe hand, ou method is moe accuate than the conventional one because A is detemined by use of the cuve fitting. Howeve, the phase constant β in the liquid is Connecto Tissue Equivalent Liquid Pot2 Pot1 Semi-ig Cable Refeence Antennas Fig. 1: A measuement system fo the gain of the antenna in the liquid. equied to evaluate the gain in accodance with (4). The phase constant can be also detemined by the contact pobe method. Howeve, the phase constant β is detemined by measuing S 21 as a function of the distance in ou method. When the wave tavels in the liquid, phase shift is expessed by a linea function of the distance as follows: S 21 = β + B, (5) whee B = ( G 1 + G 2 )/2 is a mean value of fictitious phases of the gains of two antennas. S 21 can be measued as a function of the distance, then the phase constant β can be detemined by the cuve fitting fo S 21 based on (5). The above cuve fitting can be pefomed without initial values in using the linea least-squae method. Now, the following equation can be obtained by combining (1) and (5): S21 2 =(1 S 11 2 )(1 S 22 2 ) G 1G 2 e 2γ 4β 2 2, (6) whee γ = α + jβ is the popagation constant in the liquid and G i = G i exp(j G i ) can be viewed as complex gain of antenna i. Thus, the distance dependence of S 21 between antennas can be expessed by f() = e γ. (7) The above expession is identical to the distance dependence of the electic field when the tansmitting antenna is a point souce located at the oigin. Howeve, it can not be igoously epesented as the behavio of S 21 in the nea-field egion of the antennas because the fields in that egion depend on the dimension of the antenna. B. Fesnel Appoximations in the Conducting Medium In geneal, an antenna can be viewed as the supeposition of point souces. Fo example, we conside two dipole antennas with a length of 2l, which ae faced and aligned with each othe. If the distance between two antennas is denoted as, the distance between the middle point of a dipole and the end of the othe can be expessed as R = 2 + l 2. If two dipole 2 Intenational Symposium on Antennas and Popagation ISAP 26

3 antennas ae much close, i.e. l/ 1, R can be appoximated as follows: ( ) { 2 l R = ( ) } 2 l = + l (8) Thus, the distance R between abitay points on two antennas may include the contibution of 1/ as well as. The tem of 1/ dominantly contibutes to R o the behavio of the fields as is close to zeo. To deive a geneal expession, we conside the case that the cuent distibution of the tansmitting antenna is expessed as J(). Then, the electic field in the cente of the eceiving antenna E() is given by [16] [ E() = jωµ Ī 1 ] γ 2 J( ) e γ v 4π dv, (9) whee and denote the position vectos on the tansmitting antenna and at the cente of the eceiving antenna, espectively. And Ī denotes unit dyad. Fo the Fesnel appoximations, the distance between two points R can be appoximated by R = ˆ + 2 (ˆ ) 2 2 = ˆ + a 1( ), (1) whee =, ˆ = /, =, and a 1 ( )={ 2 (ˆ ) 2 }/2. As is well known, fo the fa-field appoximations, R in the denominato of the integand in (9) can be appoximated by R and R in the exponential tem e γr in (9) can be appoximated by R = ˆ. Theefoe, the fa-field appoximation of (9) can be given by [ E fa = jωµ Ī 1 ] γ 2 e γ J( )e γˆ dv. (11) 4π v The above expession includes the popagation facto, e γ, and the fa-field tem of the spatial speading facto, 1/. Then, we conside the electic field E fo the Fesnel appoximations. If a constant b 1 which is satisfied with the following equation exists J( )e γˆ a 1( )/ dv = e b1/ J( )e γˆ dv, v v (12) (9) can be appoximated as follows: [ E nea = jωµ Ī 1 ] γ 2 e γ b1/ J( )e γˆ dv. 4π v (13) (13) includes the popagation facto, e γ, and the fa-field tem of the spatial speading facto, 1/, as is the case fo the fa-field appoximations. In addition, the electic field E includes the facto e b1/ which takes on a unique chaacteistic of the Fesnel appoximation. If Re(b 1 ) >, the magnitude of the electic field becomes smalle as is close to zeo, because of the existence of the facto e b1/. Moeove, the Fesnel tem of the spatial speading facto, 1/ 2, contibutes to the fields if is close to zeo. As mentioned above, fo the Fesnel appoximations, the distance dependence of S 21 between the antennas should be extacted as ). (14) f() = e γ e b1/ ( 1+ c 1 By adding a tem of the invese squae of the distance, c 2 / 2, the facto (1 + c 1 /) in (14) could be eplaced by (1 + c 1 /+c 2 / 2 ). Thus, this asymptotic expansion coesponds to the Fiis tansmission fomula in the nea-field egion of the antenna in the lossless medium, especially, fee space [14]. Although the added tem c 2 / 2 coesponds to the behavio of the fields in the eactive nea-field egion of the antenna, the measuement in this egion is pactically impossible because extemely smalle can not be assigned in ou measuement system. And the tem c 1 / coesponds to the behavio of the Fesnel fields so that its contibution cannot be ignoed in the measuable ange of ou system. On the othe hand, the facto e b1/ is exposed by the Fesnel appoximation of the exponential tem in (9). In the Fesnel appoximations in fee space, the phase nea the antenna is shifted by the contibution of the distance dependence of 1/ as well as [15]. In addition, as discussed above, not only phase shift but also exponentially decay ae included in the electic field in the nea-field egion of the antenna in the conducting medium. C. Fiis Tansmission Fomula in the Nea-Field Region As expected fom (14), S21 nea in the nea-field egion can be elated to S 21 in the fa-field egion as follows: ) 2 ( (S21 nea ) 2 = S21 2 e 2b1/ 1+ c 1 =(1 S 11 2 )(1 S 22 2 ) G 1G 2 e 2γ e 2b1/ ( 1+ c 1 4β 2 2 ) 2. (15) Thus, the distance dependence of S 21 db in the nea-field egion can be eplaced by S nea 21 db = A 2 log α + A log 1 1+A 2, (16) whee A 1 and A 2 ae constants. S 21 db can be measued as a function of the distance, and then A, α, A 1 and A 2 can be detemined by cuve fitting fo S 21 db based on (16). The sum of gains in db fom of two antennas is given by (3) as befoe. Similaly, the distance dependence of S 21 in the nea-field egion can be eplaced by S21 nea = β + B + B 1, (17) whee B 1 is constant. S 21 can be measued as a function of the distance, and then β, B and B 1 can be detemined by cuve fitting fo S 21 based on (17). The above cuve fittings in the nea-field egion must be employed by a nonlinea least squae method, which is often sensitive to its initialization. Intenational Symposium on Antennas and Popagation ISAP 26 3

4 ] B-5 S 21 [d -1 measued fa field fitting (4-6 [mm] Fig. 2: Measued S 21 db and its cuve fitting based on the Fiis tansmission fomula in the fa-field egion. -2 i] B-4 [d a in -6 G -8 fa field fitting =2mm) ( fa field fitting =5mm) ( -1 [mm] Fig. 3: Estimated gain by cuve fitting fo measued S 21 based on the Fiis tansmission fomula in the fa-field egion. 3. CURVE FITTING BASED ON FRIIS TRANSMISSION FORMULA IN THE NEAR-FIELD REGION A. Fa Field Fitting fo Measued Data The conducting medium assumed in this pape is the tissue equivalent liquid fo SAR estimation [1], [3] at 2.45GHz. The liquid has a pemittivity of 39.2 and conductivity of 1.84S/m at 22 C, which ae measued by the contact pobe method. Next, we discuss the measuable ange and level by measuing S 21 between two dipole antennas, which ae faced and aligned with each othe. The length of the dipole antenna is 13mm, which coesponds to a half of the wavelength in the liquid. Fig. 2 shows the measued S 21 as a function of the distance. The output powe level of a netwok analyze (Agilent N523A) is set up as 5dBm. The attenuation is mainly caused by conducting loss in the liquid as well as insetion loss in the RF cables. As shown in Fig. 2, the measued data hadly fluctuate at =7mm, howeve, they lagely do at = 1mm. This is because the noise level is smalle than the signal level at =7mm and is not at = 1mm. Fo efeence, the values of S 21 ae about 7dB and 9dB at =7mm and 1mm, espectively. Fig. 3 shows the estimated gain of the dipole antenna in the liquid detemined by the cuve fitting fo the measued S 21 based on the Fiis tansmission fomula in the fa-field egion, which is abbeviated to fa-field fitting in this pape. The hoizontal axis in this gaph c denotes the cente of the fitting ange [ c, c + ], whee denotes the width of the fitting ange. Fo simplicity, c is typed as in this figue. As expected, the estimated gain with no fluctuation can be obtained up to =7mm fo the cases of both =2mm and 5mm, and then it is noisie as is lage, and the gain can not be estimated when 1mm. Theefoe, the gain should be estimated by using measued data up to =7mm. Howeve, as shown in Fig. 3, the estimated gain at =7mm does not convege with the ideal gain in the fa-field egion. Theefoe, as descibed befoe, a stong solution is equied to ovecome the difficulty that the measuement is impossible in the fa-field egion due to the lage decay in the liquid. B. Fesnel Field Fitting fo Calculated Data Measuement of S paametes between two cente-fed halfwavelength dipole antennas which ae faced and aligned with each othe in the liquid can be simulated by the methods of moment fo the thin-wie stuctue in the conducting medium coded by Richmond [4], [17]. The code can output the pefomance of the antenna, the chaacteistics of wave popagation and the coupling effect between antennas in the liquid. In addition, it can calculate them in extemely shot time, in contast with the lage-scaled numeical computations, fo example, FDTD. The shot-time calculation is attactive in simulating the gain calibation which is equied to plot the field at many points. Of couse, this code simulates S 21 with no noise so that we can detemine the minimum distance c that the cuve fitting is valid in the fa-field egion. Now, we pesent an example of the cuve fitting fo the calculated data at 2.45GHz based on the Fiis tansmission fomula in the nea-field egion, which is abbeviated to Fesnel field fitting in this pape. A built-in function of a speadsheet softwae is used as a nonlinea least squae method. The cuve fitting is pefomed in the ange of [2mm, 6mm]. Then, the pemittivity and conductivity in the liquid is estimated to be and 1.8S/m, which ae almost equal to thei peset values. The estimated gain is.35dbi, which is equal to the gain obtained by the code developed by Richmond. Anothe estimated gain obtained by the fa field fitting is.56dbi, which is slightly lowe than the tue value of the gain. Fig. 4 shows the calculated S 21 and coesponding Fesnel field fitting cuve as a function of. Both cuves in Fig. 4 ae good ageement with each othe fo 1mm. Fig. 5 shows the estimated gain defined by (4) and (16), which is abbeviated to nea-field gain, of the dipole antenna in the liquid detemined by the Fesnel field fitting fo the calculated S 21. Also, Fig. 5 shows anothe estimated gain detemined by 4 Intenational Symposium on Antennas and Popagation ISAP 26

5 ] B-5 S 21 [d -1 calculated Fesnel field fitting (3 ] B-5 S 21 [d -1 measued Fesnel field fitting (3- [mm] [mm] Fig. 4: Calculated S 21 and its cuve fitting based on the Fiis tansmission fomula in the nea-field egion. Fig. 6: Measued S 21 and its cuve fitting based on the Fiis tansmission fomula in the nea-field egion. -1 i] -2 B [d -3 a in G-4-5 Fesnel field fitting (3 fa field fitting =2mm) ( -6 [mm] -2 i] B-4 [d a in -6 G -8 Fesnel field fitting (3-6mm) fa field fitting =2mm) ( -1 [mm] Fig. 5: Estimated gain of cente-fed dipole antenna by cuve fitting fo calculated S 21 based on the Fiis tansmission fomula in the nea-field egion. Fig. 7: Estimated gain of offset-fed dipole antenna by cuve fitting fo measued S 21 based on the Fiis tansmission fomula in the nea-field egion. the fa field fitting fo = 2mm, which is abbeviated to fa-field gain. The nea-field gain estimated by using the data nea the antenna conveges with the fa-field gain. This suggests that the gain can be estimated by fitting the measued data nea the tansmitting antenna to the Fiis tansmission fomula in the nea-field egion. The estimated gains ae good ageement with the gain calculated by the methods of moment. And it is found that the wide ange is equied fo the fafield gain to convege as shown in Fig. 5. Concetely, the fafield gain does not convege at =7mm, but it does at = 1mm. The level dops by 2dB fo S 21 measuement as the obsevation is moved fom =7mm to 1mm. In pactice, the level at = 1mm is as lage as the noise floo level in the measuement system. This is one of seious poblem to be solved when implementing ou poposed calibation. C. Fesnel Field Fitting fo Measued Data S paametes between two identical antennas which ae faced and aligned with each othe in the liquid ae measued and then ae fit to the cuves in the Fesnel field fitting. Fo matching, the half-wavelength dipole antenna is offset fed, that is, its ams ae 1mm and 12mm long. The Fesnel field fitting is pefomed in the ange of [3mm, 6mm]. Fig. 6 shows the measued S 21 and coesponding Fesnel field fitting cuve as a function of. The estimated fitting cuve is not identical to measued data fo 1mm. This is because the contibution of extemely nea-field field with distance dependency of 1/ 3 is lage than the contibution of Fesnel field with distance dependency of 1/ 2. The fluctuation is obseved in the data fo 1mm as befoe. Fig. 7 shows the nea-field gain and the fa-field gain fo =2mm. In the same manne as fitting fo the calculated data, the nea-field gain conveges with the fa-field gain even Intenational Symposium on Antennas and Popagation ISAP 26 5

6 if the cuve fitting is pefomed by the measued data in the nea-field egion. As shown in Fig. 7, the nea-field gain seems to convege at =5mm. Howeve, the fa-field gain does not convege at =5mm and the fluctuation is obseved in the data fo 7mm. The fa-field gain suffes fom the influence of the noise floo of the measuement system befoe it conveges so that the tue gain can not be obtained by the fafield gain estimation. Thus, the poblem that a measuement in the fa-field egion is impossible due to the lage attenuation in the liquid is boken down by the Fesnel field fitting in the nea-field egion, and then the gain can be accuately estimated. Fo efeence, the pemittivity and conductivity of the liquid ae estimated to be and 1.8S/m, espectively. And the estimated nea-field gain is 2.66dBi. 4. CONCLUSION A SAR pobe calibation method above 3GHz discussed in this pape can be pefomed by calibating the gain of efeence antenna in tissue equivalent liquid and elating the output voltage of the pobe to the value of the electomagnetic fields o SAR. To implement this calibation, it is equied to ovecome the poblem that the measuement in the fa-field egion is made impossible to estimate the gain because of lage attenuation in the liquid. As one of the solutions, we popose the method of estimating the gain of the efeence antenna in the liquid afte cuve fitting fo the measued data based on the extended Fiis tansmitting fomula in the nea-field egion of the antenna in the conducting medium. The poposed method is applied to both the calculated and measued data and then its validity is demonstated. [9] C. A. Balanis, Antenna Theoy Analysis and Design 3d ed., pp , John Wiley & Sons, Inc., New Yok, 25. [1] N. Ishii, Y. Yonemua, and M. Miyakawa, Simultaneous measuement of antenna gain and solution dielectic popeties, IEICE Tans. Commun., Vol. E88-B, No. 6, 25, pp [11] C. H. Wilcox, An expansion theoem fo electomagnetic fields, Commun. on Pue and Applied Math., Vol. 9, 1954, pp [12] M. K. Hu, Nea-zone powe tansmission fomulas, 1958 IRE Nat l Conv. Rec., Vol. 6, Pt. 8, 1958, pp [13] B. Y. Kinbe and V. B. Tseytlin, Measuement eo of the diective gain and of the adiation patten of antennas at shot ange, Radio Engg. and Electon. Phys., 196, pp [14] J. R. Pace, Asymptotic fomulas fo coupling between two antennas in the Fesnel egion, IEEE Tans. Antennas & Popagat., Vol. AP-17, No. 3, 1969, pp [15] R. E. Collin, Radiation fom simple souces, Chapte 2 in R. E. Collin and F. J. Zucke ed., Antenna Theoy pat 1, McGaw-Hill, New Yok, 1969, p.4. [16] J. A. Kong, Electomagnetic Wave Theoy, EMW Publishing, Cambidge, MA, 2, p.59. [17] J. H. Richmond, Compute pogam fo thin-wie stuctue in a homogeneous conducting medium, Nat. Tech. Infom. Sevice, Rep. NASA CR-2399, REFERENCES [1] IEC Intenational Standad , Human exposue to adio fequency fields fom hand-held and body-mounted wieless communication devices Human models, instumentation, and pocedues Pat 1: Pocedue to detemine the specific absoption ate (SAR) fo handheld devices used in close poximity to the ea (fequency ange of 3MHz to 3GHz), 25. [2] C. Peson, L. N. Ahlonsou, and C. Gangeat, New test bench fo the chaacteization of SAR measuement pobes used in tissue equivalent liquid, Abst. Bioelectomagnetics 2, Munich, Gemany, June 2, P-116, pp [3] K. Fukunaga, S. Watanabe, and Y. Yamanaka, Dielectic popeties of tissue-equivalent liquids and thei effects on specific absoption ate, IEEE Tans. Electomagnetic Compatibility, Vol. 46, No. 1, 23, pp [4] N. Ishii, K. Sato, L. Hamada, T. Iwasaki, and S. Watanabe, Simulation of SAR-pobe calibation using antennas in the liquid, Abst. Bioelectomagnetics 25 (CD-ROM), Dublin, Ieland, June 25, 1-6, pp [5] N. Ishii, K. Sato, L. Hamada, and S. Watanabe, Poposal of accuate SAR-pobe calibation using efeence antennas in the liquid at highe fequency, Poc. 28th Geneal Assembly of the Intenational Union of Radio Science (CD-ROM), New Delhi, India, Oct. 25, A11.7. [6] N. Ishii, K. Sato, L. Hamada, T. Iwasaki, and S. Watanabe, A gain measuement of antennas in the tissue equivalent liquid fo the SAR pobe calibation, Abst. Bioelectomagnetics 26 (CD-ROM), Cancun, Mexico, 26, PB-8, pp [7] R. K. Mooe, Effect of a suounding conducting medium on antenna analysis, IEEE Tans. Antennas & Popagat., Vol. AP-11, No. 3, 1963, pp [8] L. A. Ames, J. T. debettencout, J. W. Fazie, and A. S. Oange, Radio communications via ock stata, IEEE Tans. Commun. Sys., Vol. CS- 11, No. 2, 1963, pp Intenational Symposium on Antennas and Popagation ISAP 26