Non-resonant Permittivity Measurement Methods

Transcription

1 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March Non-resonan Permiiviy Measuremen Mehods Sergio L. S. Severo, Álvaro A. A. de Salles, Federal Insiue of Science and Technology IFSUL, Peloas RS, Brazil, Elecrical Engineering, Federal Universiy of Rio Grande do Sul UFRGS, Poro Alegre, RS, Brazil, Bruno Nervis, Braian K. Zanini Elecrical Engineering, Federal Universiy of Rio Grande do Sul UFRGS, Poro Alegre, RS, Absrac The measuremen of he dielecric properies of maerials has been applied in non-desrucive ess, humidiy measuremen, soil analysis and even cancer deecion. The mehods have been developed for over 70 years based on he ineracion of he elecromagneic waves wih he maerial under es. This work presens a general model of scaering parameers for non-resonan mehods of ransmission/reflecion and single-por reflecion. Equaions for deermining permiiviy are obained. New equaions for shor-circuied load and coupled load in he double reflecion mehod are presened. Index Terms Microwave measuremens, permiiviy, shor-circui ransmission line mehod, ransmission / reflecion mehod. I. INTRODUCTION The dielecric properies characerizaion is fundamenal in engineering. This is employed in nondesrucive es and evaluaion [], moisure measuremens [2], soil analysis and umor issue deecion. The physical conceps and echnological aspecs arelaed o deermining he dielecric properies from he ineracion of he elecromagneic fields wih he maerial. These fields mus be generaed, guided or radiaed over he sample (MUT maerial under es) and deeced afer he ineracion. Tradiionally, hese asks were performed using microwave insrumenaion echniques in laboraory [3]. Simulaneously, measuremens mehods [4] and mahemaical mehods for propagaion, radiaion and scaering of microwaves were developed [5]. These mehods can be divided in resonan and non-resonan. The non-resonan mehods are suiable for broadband measuremen. Among hese mehods he mos imporan ones are he SCTL (shor-circui ransmission line) [3] and he NRW (Nicholson-Ross-Weir) [6][7]. The purpose of his work is o generae explici equaions for he permiiviy using a sraighforward scaering parameers model for load-erminaed samples. II. PERMITTIVITY MEASUREMENT METHODS A. Hisorical Developmen In 946, Robers and Von Hippel [8] presened a mehod for he measuremen of permiiviy using an air-filled recangular waveguide wih a sample of MUT in he end of he waveguide. By

2 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March comparing he sanding wave paern of he parially sample-filled waveguide and ha of a shorerminaed air-filled waveguide i is possible o deermine he permiiviy. Such mehod is known as SCTL reflecion mehod. I obains he line impedance from he peaks and valleys of he volage sanding wave paern. This mehod was sill widely used in 96, when [9] repors uncerainies of 2% for he permiiviy and 5% for he loss angen. The use of chars for hyperbolic funcions was avoided by having sample lenghs of ¼ e ½ of he wavelengh inside he maerial. In 974, a compuer program was developed aiming o increase he precision of he Robers-von Hippel mehod [0]. In 970, Nicolson and Ross [6], using a sampling oscilloscope, a sub nanosecond pulse generaor and he Fourier ransform, obained he scaering parameers of a sample. Wih S and S 2, expressed as funcions of heflecion coefficien in he maerial-air inerface and he ransmission coefficien beween wo faces of he sample, and measured by he aforemenioned seup, hey obained he permiiviy and permeabiliy of he maerial. In 974, Weir [7] obained he scaering parameers direcly from he frequency domain by using an auomaic nework analyzer, solving he phase ambiguiy generaed by larger han half wavelengh sample lengh and measuring he group delays in differen frequencies, assuming ha he permiiviy does no change significanly for small variaions in frequency. In [], he problems of he mehod in dispersive maerials are discussed. Regardless of hese problems, he mehod is widely acceped and known as Nicholson-Ross-Weir (NRW) algorihm. An explici equaion for he permiiviy as a funcion of he ransmission and reflecion parameers is presened in [2]. The auhors show ha i is possible o obain he uncerainy of he permiiviy as a funcion of he sample lengh, wih he lowes uncerainy being obained when he sample lengh is a quarer of he wavelengh inside he maerial. The mehod becomes unsable when he sample lengh is a muliple of half wavelengh. Thesonan mehods are inadequae for characerizaion in he frequency domain. The reflecion mehods, also known as single-por mehods, which measure heflecion coefficien of a guided wave or a wave in free-space [3], can be used for specroscopy. In [4] such mehods are reviewed and possible configuraions for he measuremens are presened. Among hese, he mehod wih wo arbirary erminaions can be highlighed. In heflecion mehods, he explici equaions for he permiiviy are obained hrough wo or more measuremens in wo differen configuraions, as i is shown in [5]. B. Transmission-Reflecion mehods sae-of-he-ar The work in simulaneous measuremen of ransmission and reflecion coefficiens of a sample o obain permiiviy is consolidaed in [6], in which explici equaions independen of reference plane or sample lengh are shown. The half wavelengh uncerainy is discussed and he measuremen uncerainies are deermined. Works aiming o solve he half wavelengh problem are also referenced. In [7] a new mehod o solve problems relaed o dispersive maerials is presened. A complee

3 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March review regarding he ransmission-reflecion mehods is also done in [7]. C. Single-por reflecion mehods Reflecion mehods which employ he measuremen of wo reflecion coefficiens were already presened in [3]. These use which uses a shor-erminaed ransmission line (as he SCTL mehod) and an open-circui erminaed ransmission line. Alhough he equaion for he permiiviy is simple [4], he mehod only works a specific frequencies since, o creae an open circui, i is necessary o creae a shor-circui a a quarer-wavelengh disance. In [5] an explici equaion wih he S parameer (measured wih a coupled load or free-space and a shor-circui) is shown. Oher approach is described in [8], using only he ampliude of heflecion coefficien. Two disan frequencies (in non-dispersive media) or hree near frequencies (in dispersive media) can be used. The simpliciy of hequired insrumenaion makes he mehod very aracive. III. DIELECTRIC SLAB SCATTERING PARAMETERS MODEL A. Reflecion coefficien Γ and propagaion facor T Consider an uniform dielecric slab, wih complex permiiviy 2 and hickness d immersed in a dielecric wih permiiviy o he lef and 3 o high, spliing he space ino regions and 3, as shown in Fig.. E E2 T γ %Γ, ε, η γ 2, ε 2, η 2 γ 3, ε 3, η 3 E ( E 2 ( E20 E3 E 20 ( E3 ( d z=,d$ z=0$ Fig.. Sample elecromagneic wave ineracion Le us assume an elecromagneic wave, which is perpendicularly inciden a he inerface (z=-d). The inciden elecric and magneic waves a he inerface are E and H, respecively. Boh are parallel o he inerface and are parially refleced o he medium and parially ransmied o he inerior of he slab. E - and H - are hefleced waves, which ravel in he medium in he negaive z direcion. From z=-d, he ransmied waves E 2 and H 2 ravel in he posiive z direcion. On he inerface beween he slab and he medium 3 (z=0) here are he fields E 20 and H 20. These fields are parially ransmied o medium 3, indicaed as waves E 3 and H 3 and parially refleced back o medium 2, he waves E - 20 and H The propagaion consans of he maerials are, 2 e 3. The elecric and magneic fields arelaed in each medium by he inrinsic impedance of hese media: η,

4 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March η 2 e η 3. If he medium 3 is infinie, here will be no propagaion in he negaive z direcion in his medium (no reflecion) and E 3 - =0. If medium is vacuum and medium 3 has he same permiiviy of medium 2, he value of Γ, heflecion coefficien a he inerface, is given by: m r G = E - - E = m r where μ r and ε r are helaive permeabiliy and permiiviy of medium 2, respecively. If he media have he magneic permeabiliy of vacuum (μ r = ), he coefficien is simplified o: G = - (2) The propagaion consan γ in a dielecric wih negligible conduciviy and wih magneic permeabiliy equal o he vacuum can be approximaed o: j w c e (3) r where c is he velociy of ligh in he vacuum. Therefore, he propagaion of a TEM (ransversal elecromagneic) wave hrough he disance d in a maerial wih he propagaion consan γ can be expressed by he propagaion facor T [7]. Using (3) is possible o define: Some auhors call T he ransmission coefficien [4]. To avoid confusion wih he ransmission coefficien hrough a slab, he original erm propagaion facor will be kep. B. MUT (Maerial Under Tes) scaering parameers. The inrinsic impedance variaion beween he wo differen media will resul in ha par of he inciden wave o be ransmied and par of i o befleced. In a dielecric slab, as shown in Fig. 2, here are wo inerfaces and hen mulipleflecions will happen inside he slab. Using harmonic analysis, his is simplified in he case of a high loss sample, because he mulipleflecions add up o an aenuaed sanding wave paern. () (4) S2 S S η=$η0$η2$$$$$$$$$η3=$η0$ Fig. 2. MUT scaering parameers The oal fields can be obained hrough he complee soluion of he wave equaion inside he slab

5 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March when he dielecric characerisics of he maerial and he sample dimensions are known. The oal field, for 0 > z > -d, is: E 2 ( z) = E 20 e -g 2 z E - 20 e g 2 z (5) Theflecion coefficien a he inerface beween he wo media, when he sample is infinie, allows for he subsiuion of E 20- hrough Γ since: E 2 G = E - E = - E - 20 E 20 (6) ( z) = E 20 e -g 2 z - Ge g 2 ( z ) (7) The same procedure is applicable o he magneic field. The oal magneic field can be wrien as: H 2 ( z) = H 20 e -g 2 z H - 20 e g 2 z (8) Since he magneic fields arelaed o he elecric fields hrough he inrinsic impedance of he medium, (8) can bewrien as: Applying (6) in (9): H 2 ( z) = E 20 e -g 2z - E 20 h 2 - h 2 e g 2z (9) H 2 ( z) = E 20 h 2 ( e -g 2z Ge ) g 2z (0) Since he elecrical and magneic fields are angenial o he inerface, i is possible o wrie, for z=-d: H 2 E 2 = E E - () = H H - = E - E - (2) h h Considering he propagaion facor T along he slab, he oal fields a z = -d, obained from (7) and (0), are: E 2 ( z = -d) = E 20 ( T - - GT) (3) H 2 ( z = -d) = E 20 h 2 ( T - GT ) (4) From () (3) and (2) (4), he boundary condiions allow o wrie: E E - = E 20 ( T - - GT) (5) ( ) (6) E - E - = h E h 20 T - GT 2 Assume ha he inciden elecric field in he slab a z=-d is E. Thefleced elecric field is E -.

6 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March Since he media and 3 have he same inrinsic impedance, a scaering parameers model can be applied. Therefore, heflecion coefficien of he slab, as seen by he inciden wave (inpu), will be he S parameer iself, or: By expressing E - as S E and dividing (6) by (5): S = E - E (7) ( - S ) ( ) = h h 2 S ( T - GT ) (T - - GT) If η = η 3 = η 0 and he medium 2 is non-magneic, haio η / η 2 is equal o he squaroo of he relaive dielecric permiiviy of he medium 2. From (2), is possible o isolae his squaroo as a funcion of Γ and hen obain heflecion coefficien a he inpu of he slab: ( ) ( - T ) 2 S = G - T 2 Thelaion beween he inciden elecromagneic wave a he inerface a z=-d and he emerging wave a he inerface a z=0, when he medium 3 is equal o he medium is he parameer S 2 iself: S 2 = E 3 E (20) A he inerface z = 0, he oal angenial fields mus be equal in boh sides. For he elecric fields, assuming ha here are no fields raveling in he negaive z direcion in he medium 3: (8) (9) E 3 = E - 20 E 20 (2) Γ relaes he fields E 20 e E 20 -, herefore: E 3 = E 20 ( - G) (22) Subsiuing E 20, from (22), and E 3 by S 2 E in (20), ino (5) and (6) and replacing haio beween he characerisic impedances by heflecion coefficien Γ: E E - = S E 2 - G E - E - = - G ( ) T - - GT) ( ) S 2 E ( G) ( - G) T - GT ( (23) ( ) (24) Adding (23) and (24), eliminaing E - e E i is possible o isolae he ransmission coefficien hrough he slab: S 2 = T - G2 ( ) - T 2 (25)

7 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March Given ha he dielecric slab is symmerical and he maerial is isoropic and homogeneous, he scaering parameers marix is compleely specified by making S 22 =S and S 2 = S 2. IV. SIMPLE MODEL FOR NON-RESONANT METHODS A. NRW algorihm Consider a sample, as shown in Fig. 3, inside a coaxial cable wih a erminaion impedance Z 0$ Z L$ conneced immediaely afer he sample (d L =0) or he medium 3 d$ wih an inrinsic impedance differen dl$ of ha of he medium in free-space. In boh cases, i is Γin$ possible ΓL$ o model he sysem as a slab represened by is scaering parameers and loaded by impedance Z L or an infinie medium of inrinsic impedance η L. η 0$ η$ ηl$ Z 0$ Z L$ Γin$ d$ ΓL$ dl$ S ZL Γin$ ΓL$ η 0$ η$ ηl$ Fig. 3. MUT in ransmission line and free space wih load. S ZL Theflecion coefficien a he inpu can be obained from he scaering parameers and he reflecion coefficien a he load from [9] [20]: ΓL$ G in = S - G s L (26) - S 22 G L Where Δ S = S S 22 S 2 S 2. Considering he sampleciprociy: G in = S - S S ( 2 )G L (27) - S G L If he load impedance is made equal o he characerisic impedance of he line loaded wih he sample, heflecion coefficien a he inpu will be ha of an infinie sample. This is due o he fac ha, wihou reflecion a he second inerface, he wave will only exis in he posiive direcion from he inpu of he sample. Any load or infinie slab wih he same impedance as he medium being measured will presen he sameflecion coefficien Γ when considered in relaion o he inpu medium. From hese consideraions, i follows ha if he subsiuion Γ in = Γ L = Γ is done in (27), i is possible o obain heflecion coefficien a he inerface as a funcion of he slab scaering parameers: ( ) = S - S G - S G Γin$ ( ) 2 2 ( - S 2 )G \ - S 2 - S 2 2 G = 0 S

8 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March ( ) 2-4S G = S 2 - S 2 2 ± S 2 - S (28) 2S The sign in (28) mus be chosen in a way ha Γ. Defining: (29) Equaion (28) can hen be wrien as: G = K ± K 2 - (30) Equaions (29) and (30) are presened in [6] and [7] as a fundamenal par of he NRW algorihm. The scaering marix relaes he elecric fields in he sample, as shown in Fig., in he form: E - = S E S 2 E 3 - E 3 = S 2 E S 22 E 3 - The inciden field in he load is E 3 and hefleced is E 3 -. Theflecion coefficien a he load, Γ L is given by: G L = E - 3 (3) E 3 Since he sysem is symmerical S 22 = S and he maerial is isoropic and homogeneous, hen S 2 = S 2. From (3), E 3 - =Γ L E 3, he above equaion sysem can be wrien as: E - = S E S 2 G L E 3 E 3 = S 2 E S G L E 3 We can add hese wo equaions and obain an equivalen sysem wih he same soluion. The sum resul is ha: E 3 E - = ( S S 2 )E ( S 2 S )G L E 3 (32) Since in he proposed siuaion, here is no a refleced wave inside he sample and Γ in = Γ L = Γ, i follows: Similarly, he propagaion facor is given by: G = E - E (33) T = E 3 E (34) From hese equaions we can evaluae expressions for E 3 e E -, which are subsiued in (32), wih Γ L = Γ: TE GE = ( S S 2 )E ( S 2 S )GTE

9 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March hen: T = S S G 2 - ( S S 2 )G (35) Equaion (35) shows he propagaion facor as a funcion of he scaering parameers and he reflecion coefficien a he inerface presened in [5] and [6]. The NRW algorihm o deermine he permiiviy from he MUT scaering parameers is fulfilled when (2) is considered and hen: And from (4): æ = - G ö è ç G ø 2 (36) æ c = j wd ln ( T ö ) è ç ø (37) The equaions (36) and (37), isolaed or combined, can be used for permiiviy deerminaion [2]. The use of (36), as described in [2] will resul in a permiiviy explici expression, independen of he sample size. However his leads o indeerminaions when he sample lengh is a muliple of half wavelengh in low loss maerials. The auhors conduced an uncerainy analysis as a funcion of he permiiviy of he measured maerial and of he sample size. Equaion (37) does no show hese problems, bu i depends on he sample lengh, which leads o phase ambiguiy problems since T is complex and is logarihm may have infinie soluions [22]. B. Reflecion only mehods If in fig. 3, since d L = 0 and he load is a shor-circui, we have he mehod known as SCTL (shorcircui ransmission line). This mehod is also applied o he free-space [4] where he shor-circui is made hrough a meal back (meal-back mehod). Oher load ypes are possible. The model in fig. 3 can be used wih any load. The sample scaering parameers, as funcions of he propagaion facor T and of heflecion coefficien a he inerface Γ, are given in (9) and (25). In a disinc approach from he deducion of he NRW algorihm, which is inended o wrie Γ as a funcion of scaering parameers only, we now wan an expression for he inpu reflecion coefficien Γ in, given by (27), as a funcion of he facor T and of he coefficien a he inerface Γ. When subsiuing (9) and (25) in (27) (obained from (26)), hen: G in = G - T 2 ( ) - G L ( - T ) 2 ( ) - T 2 - G L G - T 2 2 (38) C. Doubleflecion mehods same size samples and differen loads. I is possible o obain an explici equaion for he permiiviy from (38) hrough he double reflecion mehod [][5] (also known as double impedance mehod [4]) or when considering he

10 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March same sample wih wo differen loads. These measuremens resul in he inpu reflecion coefficiens Γ e Γ 2 from hespecive loads Γ L e Γ L2. For each one of he loads he propagaion facor T can be isolaed in (38) wih Γ given by (2): ég L - T 2 = ég L ( ) ( ) - ( ) - ( ) ù é ûú G in ù é ûú G in - Thus, if Γ L =Γ L =- (shor circui) in he firs measuremen and Γ L =Γ L2 = (open circui) in oher measuremen, are applied o equaion (39) and compared, he permiiviy as a funcion of wo reflecion coefficiens Γ and Γ 2 is: ( )( -) ( )( ) = G - G The normalized inpu admiance of a ransmission line is given by [9]: y = Y Y 0 = - G - G Therefore, he permiiviy of a sample, when obained from wo measuremens, one erminaed in a shor-circui and he oher erminaed in an open-circui, is given by: ù ûú ù ûú (39) (40) (4) = y a y c (42) The permiiviy as a produc of a shor-erminaed line admiance (y c ) by an open-circui erminaed line admiance (y a ) has already been shown in [4]. D. New doubleflecion explici equaions When measuring heflecion coefficien of a shor erminaed sample and, hen, of an impedance mached erminaed sample (Γ L =- e Γ L2 =0), i is possible o obain anoher explici equaion from (39): = G - 3 G G G Applying he same procedure, bu wih an open-circuied load in place of he shor-circuied one and, hen, of an impedance mached erminaed sample (Γ L = e Γ L2 =0), he permiiviy is now given by: (43) = G - - G G 3 - G (44) This equaion has been derived earlier [5], bu is derivaion uses a differen procedure. The equaions (43) and (44) are paricular cases of a general equaion. Given any wo loads Γ L and Γ L2 (wih wo measuremens Γ and Γ 2 being done wih hese wo loads), he general explici equaion for permiiviy is:

11 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March = G L G L2 G - G L G L2 - G L G G L2 G 2G L - 2G L2 G - G L G L2 - G G L G L2 G - G L G L2 - G L G G L2 G - 2G L 2G L2 G - G L G L2 - G (45) E. Doubleflecion mehods same loads and differen size samples Using wo samples wih differen lenghs, arranged on a shor-circuied line, as shown in Figure 4, i is possible o obain he permiiviy and he permeabiliy. This procedure appears in [] and [23]. d Γ d 2 Γ L =- Γ 2 Fig. 4. Shor-circuied lines wih differen sizes samples. Γ L =- An explici equaion for he permiiviy can be obained considering ΓL=- in (39): ég T 2 = - ég - ( ) - ( ) By measuring wih sample widhs d2 = α d he squared propagaion facor is given by: ù ûú ù ûú (46) æ é T 2 = ç - ç é - è ( ) - ( ) where α is a scaling faor. Knowing helaionship beween he widhs and measuring heflecion coefficiens, i is possible, for cerain α values, o obain explici expressions for he permiiviy considering he equaion: ( ) - ( ) æ ég ç - ç ég - è ùö ûú ù ûú ø a é = - é - ùö ûú ù ûú ø a ( ) - ( ) In [], he widhs are se as d2 = 2 d (or α = 2). Equaion (48) can hen be solved explicily, obaining he permiiviy: ( )( G - 3G 3 -) ( G ) 2 ( ) = G - Employing he same procedure, saring from equaion 39 bu forcing he samples o end wih a mached load or an absorbing maerial in free space, anoher explici equaion for he permiiviy can be obained: ù ûú ù ûú (47) (48) (49)

12 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March 207 ( )( G - 2G ) ( )( G 2G - ) = G - G 308 (50) V. ACCURACY To esimae he uncerainy of he new equaions, he Mone Carlo mehod is applied. The error sources considered are he finie accuracies of he measured reflecion coefficien (wihin 3% of he nominal value for ampliude and phase) and of he load impedance (aken o be wihin % of nominal value). The combined effec of hese error sources is compued for a populaion of 5000 samples in a recangular disribuion. A low-loss maerial wih ε= 4 0,2j and 25 mm widh was used. The sandard deviaion in permiiviy generaed by hese error sources when applied o equaions (36) (obained from (28), NRW mehod), (40) and (43) are shown in Fig. 5 and Fig. 6, for heal and imaginary pars of he permiiviy, respecively. When he same errors sources are applied o (49) and (50), hesuls are shown in Fig. 7 and Fig. 8, for heal and he imaginary pars of he permiiviy, respecively. Fig. 5. Same size differen loads. Sandard deviaion of heal par of he permiiviy. Fig. 6. Same size differen loads. Sandard deviaion of he imaginary par of he permiiviy. In he Fig. 5 and Fig. 6 i can be observed ha he new explici equaion, (43), has smaller uncerainy in he frequencies which are muliple of half-wavelenghs (3, 6 and 9 GHz) in comparison

13 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March o he radiional mehod of (40). Minimal uncerainy in frequencies.5, 4.5 and 7.5 GHz is found for (43), whereas (40) has an insabiliy. I also can be noed ha (43) has, in he enire band, a lower uncerainy for he imaginary par (when compared o he NRW mehod). The uncerainy for heal par of permiiviy in quarer-wavelengh frequencies (.5, 4.5 and 7.5 GHz) is slighly higher for (43) han for he NRW mehod, bu for half-wavelengh frequencies he precision of (43) is higher han he NRW. Fig. 7. Same load differen sizes. Sandard deviaion of heal par of he permiiviy. Fig. 8. Same load differen sizes. Sandard deviaion of he imaginary par of he permiiviy. In Fig. 7 and Fig. 8, i can be observed, ha he differen-size-samples mehod, which uses mached loads, shows a lower error along mos par of he band. For boh equaions he uncerainies ge smaller (and closer o one anoher) as he frequency increases. This can be due o he larger number of wavelenghs inside he maerial sample widh (a virual hickening), resuling in larger aenuaion and less signal being refleced a he erminaion. VI. CONCLUSION A new general model for he non-resonan permiiviy measuremen mehod was presened. From

14 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March his model he equaions for classical NRW algorihm and SCTL mehod can be derived. Explici equaions o deermine he permiiviy were obained from he new model. In addiion o his, wo new equaions for he doubleflecion mehod were evaluaed. One of hem uses a shor circui load and a mached load and he oher uses an open-circui load and a mached load. A new equaion is also obained for he mehod wih differen sizes erminaed in he same load, in his case, a mached one. The uncerainy of he new equaions is calculaed using he Mone Carlo mehod and, in boh cases, i is lower han he classical mehods. These mehods were used for TEM waves in he freespace and in ransmission lines. However, hey could be easily exended for waves and samples in recangular waveguides. REFERENCES [] S. Kharkovsky and R, Microwave and Millimeer Wave Nondesrucive Tesing and Evaluaion, IEEE Insrumenaion and Measuremen Magazine, vol. 0 (2), pp , April [2] K. Kupfer, Elecromagneic Aquamery Elecromagneic Wave Ineraion wih Waer and Mois Subsances, Berlin: Springer-Verlag, 2005, 529p. [3] A. Von Hippel, edior. Dielerics Maerials and Applicaions. Cambridge,MA: Technology Press of MIT., 954, 438p. [4] L. F. Chen, e al., Microwave Elecronics Measuremen and Maerials Characerizaion. Chicheser: John Wiley & Sons, p. [5] D. M. Pozar, Microwave Engineering Third Ediion. New York, NY: John Wiley & Sons p. [6] A. M. Nicolson, and G. F. Ross. Measuremen of he Inrinsic Properies of Maerials by Time-Domain Techniques. IEEE Trans. Insrum. Meas., vol. IM-9, No. 4, pp , Nov [7] W. B. Weir. Auomaic Measuremen of Complex Dielecric Consan and Permeabiliy a Microwave Frequencies, Proceedings of he IEEE, vol. 62, No., pp , Jan [8] S. Robers and A. von Hippel. A New Mehod for Measuring Dielecric Consan and Loss in he Range of Cenimeer Waves. J. Appl. Phys., vol. 7, pp , April 946. [9] M. G. Corfield, J. Horzelski and A. H. Price. Rapid mehod for deermining v.h.f. dielecric parameers for liquids and soluions using sanding wave procedures Briish Journal of Applied Physics, vol. 2, pp Dec. 96. [0] S. O. Nelson, L. E. Seson, and C. E. Schlaphoff. A General Compuer Program for Precise Calculaion of Dielecric Properies From Shor-Circuied-Waveguide Measuremens. IEEE Trans. Insrum. Meas., vol. IM-23, No. 4, pp , Dec [] U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya. Resolving Phase Ambiguiy in he Inverse Problem of Refleciononly Measuremen Mehods. Progress In Elecromagneics Research, vol. 29, pp , June 202. [2] S. S. Suchly and M. Mauszewski. A Combined Toal Reflecion-Transmission Mehod in Applicaion o Dielecric Specroscopy. IEEE Trans. Insrum. Meas., vol. IM-27, No.3, pp , Sep [3] D. K. Ghodgaonkar, V. V. Varadan, and V. K. Varadan. Free-Space Measuremen of Complex Permiiviy and Complex Permeabiliy of Magneic Maerials a Microwave Frequencies. IEEE Trans. Insrum. Meas., vol. 39, No. 2, pp , April 990. [4] M. A. Suchly, and S. S. Suchly. Coaxial Line Reflecion Mehods for Measuring Dielecric Properies of Biological Subsances a Radio and Microwave Frequencies a Review, IEEE Trans. Insrum. Meas., vol. IM-29, No. 3, pp.76-83, Sep [5] S. L. S. Severo, Aquameria por microondas: desenvolvimeno de ransduor em microfia, Programa de pós-graduação em engenharia Elérica, Disseração de Mesrado, Universidade Federal do Rio Grande do Sul. Poro Alegre [6] J. Baker-Jarvis, E. J. Vanzura, and W. A. Kissick. Improved Techique for Deermining Complex Permiiviy wih he Transmission/Reflecion Mehod, IEEE Trans. on Microwave Theory and Techniques, vol. 38, no. 8, pp , Aug [7] U. C. Hasar, J. J. Barroso, M. Bue, Y. Kaya, M. E. Kocadagisan, and M. Erugrul. Aracive mehod for hicknessindependen permiiviy measuremens of solid dielecric maerials. Sensors and Acuaors A: Physical, vol. 206, pp , Feb [8] U. C. Hasar and M. T. Yurcan, "A microwave mehod based on ampliude-only reflecion measuremens for permiiviy deerminaion of low-loss maerials," Measuremen, vol. 43, no. 9, pp , Nov [9] R. J. Weber, Inroducion o microwave circuis Radio frequency and design applicaions, New York, NY: IEEE Press, 200,432p. [20] D. M. Pozar, Microwave Engineering Third Ediion, New York, NY: John Wiley & Sons p. [2] A. Boughrie, C. Legrand and A. Chaponon, Nonieraive Sable Transmission/Reflecion Mehod for Low-Loss Maerial Complex Permiiviy Deerminaion, IEEE Transacions on Microwave Theory and Techniques, vol. 45, no., pp , January 997 [22] J. J. Barroso, and H. C. Hasar, Resolving Phase Ambiguiy in he Inverse Problem of Transmission /Reflecion Measuremen Mehods, Journal Infrared Milli. Terahz Waves. Vol. 32. pp , 20

15 Journal of Microwaves, Opoelecronics and Elecromagneic Applicaions, Vol. 6, No., March [23] J Baker-Jarvis,M. D. Janezic, J. H. Grosvernor and R. G. Geyer. Transmission / Reflecion and Shor-Circui Line Mehods for Measuring Permiiviy and Permeabiliy. NIST Technical Noe 355-R - NIST Naional Insiue of Sandards and Technology. Boulder, CO, 236 p. December 993.