An Improved Modeling of TDR Signal Propagation for Measuring Complex Dielectric Permittivity

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1 Jounal of Eath Siene, ol. 26, No. 6, p , Deembe 215 ISSN X Pinted in China DOI: 1.17/s An Impoved Modeling of TDR Signal Popagation fo Measuing Complex Dieleti Pemittivity Chih-Ping Lin 1, Sh-Hong Tang 2, Chun-Hung Lin 3, Chih-Chung Chung 4 1. Civil Engineeing, National Chiao Tung Univesity, Hsinhu 374, Taiwan, China 2. Infomation Management, St. May s Mediine, Nusing and Management College, Yilan 2653, Taiwan, China 3. Disaste Pevention and Wate Envionment Reseah Cente, National Chiao Tung Univesity, Hsinhu 374, Taiwan, China 4. Civil Engineeing, National Cental Univesity, hongli 321, Taiwan, China ABSTRACT: Time domain efletomety (TDR) is a measuement tehnique based upon tansmission line theoy. The solutions of tansmission line equations ae efomulated in tems of independent physial popeties, instead of oupled pe-unit-length iuit paametes. The omplete TDR esponse is effetively modeled by a non-unifom tansmission line using the non-eusive ABCD matix appoah. Appoahes to alibate line paametes and pefom TDR measuements based upon suh model ae intodued with an example on dieleti spetosopy. TDR modeling in tems of deoupled physial paametes and non-eusive algoithm allows moe onvenient alibation of line paametes and failitates intepetation of TDR measuements. KEY WORDS: time domain efletomety, tansmission line theoy, tansmission line measuements, dieleti measuements, spetosopy, soil measuements. INTRODUCTION Time domain efletomety (TDR) is a measuement tehnique based upon tansmission line theoy. The measuement utilizes able ada (a pulse-eeive devie fomally known as time domain efletomete, also abbeviated as TDR) and sensing waveguides. It has wide appliations anging fom defomation monitoing (Lin C-P et al., 29; Lin M W et al., 25; Dowding et al., 22, 1988), level measuement (Cataldo et al., 28a, b, 27; Di Sante, 25; Nemaih, 21), to measuement of dieleti pemittivity (Heimovaaa, 1994; Topp et al., 198; Clakson et al., 1977). The pulse geneato sends an eletomagneti pulse along a lead able and the sensing waveguide, whih then diets the eletomagneti wave into the mateial unde test o envionment to be monitoed. The sensing waveguide an be a oaxial able (e.g., fo monitoing of loalized shea defomation) o a speiallydesigned multi-onduto waveguide (e.g., fo monitoing of dieleti pemittivity, eletial ondutivity, and mateial level). Impedane hange ous when the measuement waveguide is subjeted to defomation o eletial popety of the suounding mateial hanges. Refletions fom the impedane hange ae eoded and used to intepet engineeing paametes. Despite damati developments in eletoni sensos, wieless ommuniation, and automati data aquisition Coesponding autho: plin@mail.ntu.edu.tw China Univesity of Geosienes and Spinge-elag Belin Heidelbeg 215 Manusipt eeived May 21, 215. Manusipt aepted Septembe 7, 215. systems in the aeas of instumentation tehnologies ove the past twenty yeas, efletomety method ompetes well due to its elevant pefomane haateistis, in tems of high flexibility, high sensitivity, lage appliation onditions as well as a distibuted detetion appoah. The deployment of TDR measuements anges fom shot able and sophistiated waveguides in the laboatoy to long able and ugged, unmathed waveguides in the field. The TDR measuement system is geneally a non-unifom tansmission line, onsisting of a lead able, onnetos and a sensing waveguide. The TDR esponse (measued efleted wavefom) is detemined by the length, oss-setional geomety, eletomagneti popeties of the insulating mateial, and esistivity of ondutos in eah unifom tansmission line setion. TDR typially measues one of the above physial popeties in the non-unifom line while the emaining ae alibated and kept onstant. Fo example, hange in oss-setional geomety of the sensing able is detemined fo slope monitoing using oaxial ables, length of etain unifom setion is detemined fo level measuement, and dieleti popety of the mateial in the sensing setion is detemined fo mateial haateization. A TDR esponse an be ompliated due to vaious phenomena, inluding dieleti dispesion (fequeny-dependent pemittivity), ondutive loss, esistive loss, as well as multiple efletions. Thee is a gowing demand fo modeling of TDR wavefom to bette undestand the measuement system and seve as the fowad model fo model-based invesion (Lin et al., 29; Cataldo et al., 28a; Lin, 23; Dowding et al., 22; Feng et al., 1999; Fiel and O, 1999; Heimovaaa, 1994; Yanuka et al., 1988). The TDR esponse an be modeled by finite element o Lin, C.-P., Tang, S.-H., Lin, C.-H., et al., 215. An Impoved Modeling of TDR Signal Popagation fo Measuing Complex Dieleti Pemittivity. Jounal of Eath Siene, 26(6): doi:1.17/s

2 828 Chih-Ping Lin, Sh-Hong Tang, Chun-Hung Lin and Chih-Chung Chung finite diffeene appoah (Bolvin and Chambael, 27; utilizes a moe effiient appoah based upon the tansmission line theoy (Paul, 1994; Ramo et al., 1994). Some onsideed multiple efletions but negleting dieleti dispesion (Yanuka et al., 1988), wheeas othes foused on dieleti dispesion (inluding ondutive loss) by minimizing impedane mismathes (Fiel and O, 1999; Heimovaaa, 1994). Moe eently, both multiple efletions and dieleti dispesion have been onsideed (Lin, 23; Feng et al., 1999). The effet of onduto esistane has mostly been negleted due to ompliation in fomulation and alibation of esistane fato. But its impotane and a moe ompehensive modeling of TDR wavefom have been eently pesented (Lin and Tang, 27; Lin et al., 27). In fomulating the solution of tansmission line equation, on the othe hand, most studies have unanimously utilized the lumped iuit model in tems of pe-unit-length esistane, ondutane, indutane, and apaitane. These lumped-iuit paametes ae all funtions of the physial popeties of the tansmission line (i.e., the oss-setional geomety and popety of ondutos and eletomagneti popety of the medium suounding the ondutos) and hene inte-oelated, obsuing the poess of system alibation and model-based invesion onsideing all wave phenomena. A deoupled paameteization of the tansmission line paametes has been epoted (Lin and Tang, 27). But it should be futhe genealized. The final solution of TDR measuement in a non-unifom tansmission line was deived as a eusive equation omputing in a bottom-up fashion in tems of satte funtion S11 (Feng et al., 1999) o input impedane (Lin, 23). Tansmission line paametes of diffeent setions ae tangled up in the eusive omputation, making it diffiult to sepaate leading setions fom the sensing waveguide and esulting in a omplex alibation poedue (Cataldo et al., 29). This pape is aimed to fomulate a moe geneal and non-eusive solution of tansmission line equation in tems of independent physial popeties and ompletely model TDR esponse aounting fo all wave phenomena inluding multiple efletion, dieleti dispesion, and onduto esistane. The alibation of system paametes and TDR measuement based upon suh model ae intodued with an example on dieleti spetosopy. 1 DECOUPLED FORMULATION OF TRANSMISSION LINE EQUATIONS The govening equations fo the tansmission line (i.e., the telegaph equations) in the phaso fom (i.e., in the fequeny domain) an be deived as d ( z) ( j2π fl) I( z) (1a) dz d() I z ( g j2π f) ( z) (1b) dz whee is the voltage between ondutos, I is the uent in the tansmission line, z is the position along the line, and f is the fequeny. The pe-unit-length paametes,, l, g, and, ae funtions of the oss-setional geomety of the tansmission Dowding et al., 22). But the majoity of the liteatue line and eletomagneti popeties of the media between ondutos. The eletomagneti popeties of a mateial is haateized by its dieleti pemittivity (), eletial ondutivity (), and magneti pemeability (). In geneal, these paametes ae funtions of fequeny. The dieleti pemittivity and magneti pemeability ae typially expessed elative to that of fee spae, =/ and =/, in whih is the elative pemittivity, is the elative pemeability, = F/m is the vauum pemittivity, and =4 1-7 H/m is the vauum pemeability. The pe-unit-length paametes Ramo et al. (1994) an be witten in genei foms fom as s ( f) (2a) l (2b) 2πf g (2) ( f) (2d) whee s () is the sufae esistivity of onduto, (m) is the geometi fato fo esistane, and (dimensionless) is the geometi fato fo indutane, ondutane, and apaitane. The sufae esistivity is a funtion of fequeny and an be appoximated by s = s 1-7 f 1/2 () fo good ondutos, whee s (s.5 ) is the haateisti of the onduto fo the skin effet. alues of s fo vaious typial ondutos an be found in Ramo et al. (1994). The genei foms of pe-unit-length paametes in Eq. (2) ae impotant fo deiving deoupled fomulation. The geneal solution of Eq. (1) an be witten as (Ramo et al., 1994) z e e z z (3a) z (3b) z e e I z whee + and ae the two unknown onstants in the geneal solution, is the popagation onstant, and is the haateisti impedane. The tems and an be witten as j2π flg j2π f (4a) j2πfl g j2πf (4b) Equation (4) an be found in most textbooks on eletomagneti waves. But the pe-unit-length paametes,, l, g, and an be analytially detemined fom oss-setional geomety only fo speial tansmission lines (i.e., oaxial lines). Futhemoe, these paametes ae not independent, as depited in (2). To deive the solution in tems of un-oupled paametes, substitute (2) into (4) j2πf 1 1 j s j2 f j2πf j2πf (5a)

3 An Impoved Modeling of TDR Signal Popagation fo Measuing Complex Dieleti Pemittivity 829 j2πf 1 1 js j2πf j2πf j2πf (5b) Binging out ommon fatos and letting ε jσ/2πfε =ε, Eq. (5) an be eaanged as j2πf 1 j s 2πf 1 1 s 1 j 2πf (6a) (6b) Bette-paameteized foms fo and an be futhe expessed as j2πf A (7a) v p A (7b) whee v is the speed of light, j /(2π f) is the omplex dieleti pemittivity, p is the geometi impedane defined as the haateisti impedane in fee spae, and A is the esistane oetion fato aounting fo the effet of able esistane. p and A an be witten out as 1 p (8a) s R A (1 j) (1 ) 7 j 2π 1 f f (8b) Notably, p is a funtion only of the geometi fato (), and A is a funtion of magneti pemeability (equal to 1. fo most mateials), the geometi fatos, and sufae esistivity. The esistane loss fato R (s -.5 ) is defined to epesent the ombined effet of geometi fatos and sufae esistivity. Equation (7) is the geneal fom fo popagation onstant and haateisti impedane in TDR modeling. If able esistane is ignoed (i.e., s =), A beomes 1. and and p have expessions idential to that deived in pevious studies (Lin, 23; Feng et al., 1999; Heimovaaa, 1994; Clakson et al., 1977). Futhemoe, the solution of the Telegaph equation is expessed in Eq. (7) in tems of independent physial paametes (geometi fato p, mateial popeties ε and, and esistane loss fato α R ), instead of oupled line paametes (, l, g, and ). Equation (7) is deived to expliitly sepaate effets of geometi haateisti (i.e., p ), mateial popety (i.e., ), and able esistane (i.e., ) on the popagation onstant and haateisti impedane. Sine A is fequeny dependent, R is defined as the ontolling paamete fo able esistane. Both p and R depend on waveguide dimensions ( and ). Although the geometi fatos ( and ) may be alulated theoetially fom pobe dimensions fo simple onfiguations (i.e., oaxial line), it is best and moe onvenient to dietly alibate p and R fom TDR measuements. Thus, the tansmission line is uniquely haateized by p,, and R. While length L, geometi fato p, mateial popeties ε and, and esistane loss fato α R epesent the physial paametes of a tansmission line, the popagation onstant () and haateisti impedane ( ) ae the two esultant paametes in the solution of the tansmission line equation. The popagation onstant ontols the speed and deay of a wave taveling along the line. Fo a line with setions of diffeent haateisti impedanes, efletion and tansmission of wave will ou at the setion intefaes. The phase shift of wave popagation is a ombined effet of L and, while the efletion magnitude is affeted by p and. Theefoe, all the physial paametes annot be uniquely detemined by invesion of the measued efleted wavefom alone. One of the thee paametes (L, p, ) needs to be known so that the othe two paametes and R an be detemined fom the measued TDR wavefom. This holds fo both system alibation and TDR measuements. Fo leading setions pio to the pobe, the exat values of (L, p, ) ae not neessay as long as they have the same effet on measued wavefom. Typially, L an be dietly estimated and p and an be alibated by mathing the measued wavefom. To peisely detemine the pobe geometi paametes L and p, mateials of known dieleti popety an be used to alibate L and p. Duing measuements, eithe one o two of the vaiables (L, p, ) ae vaiant and taget(s) of TDR measuements. Fo example, in level measuements, L is hanging while p and ae invaiant. It is also possible to simultaneously measue hanging L and while p emains onstant (Cataldo et al., 28a, b, 27). In mateial haateization, is to be measued while L and p of the pobe emains onstant. 2 NON-RECURSIE MODELING OF TDR MEASU- REMENTS Conside a TDR measuement system shown in Fig. 1 fo dieleti haateization of mateials. The mold is fist filled with mateials to be measued and the ental od with a onneto ap is then inseted to fom a oaxial ell. A long lead able is used to epesent possible field deployment, in whih the oaxial pobe is eplaed by a two-od o thee-od waveguide. Suh a TDR measuement system onsists of a lossy non-unifom tansmission line, whih an be modeled as a asade of unifom setions as shown in Fig. 2. The onneto ap shown in Fig. 1 is extemely thin; hene a minimum of two setions an be used to model the TDR measuement system, one fo the lead able and the othe fo the pobe setion. Additional setions an be added if the onneto ap is thik and ontains mismathes. Assuming =1 fo all setions, eah unifom tansmission line setion is haateized by thee independent paametes, p, ε, and α R, espetively epesenting the effet of oss-setional geomety, mateial popety, and line esistane loss. Theefoe, eah unifom setion is haateized by L i, p,i,,i, and R,i. Equations (3) and (7) epesent the geneal solution fo a unifom tansmission line. The unknown onstants in Eq. (3) an

4 83 Chih-Ping Lin, Sh-Hong Tang, Chun-Hung Lin and Chih-Chung Chung Figue 1. Configuations of the TDR measuement system and alibated paametes. Figue 2. Repesenting a non-unifom line as a asade of unifom setions, eah uniquely haateized by its setion length (L i ), geometi impedane ( p,i ), dieleti popety (,i ), and esistane loss fato ( R,i ). The effetive teminal input impedane in (z i ) is defined as the equivalent impedane when looking into the iuit at position z i. be detemined by applying bounday onditions. But fo non-unifom tansmission line as shown in Fig. 2, waves an be efleted and tansmitted at the intefaes of the setions, adding omplexity to the solution. Feng et al. (1999) and Lin (23) intodued a multi-setion wave popagation model based on the onept of input impedane, in whih the oveall input impedane of the measuement system is deived eusively in a bottom-up fashion stating fom the teminal impedane L (equal to in an open loop). ( z ) in n L ( z ) in n1, n ( z ) in n2, n1 () L, ntanh nln n, Ltanh nn l ( z ) tanh l ( z )tanh l in n1, n1 n1 n1 n, 1 in n1 n1 n1 ( z ) tanh l in 1,1 1 1 in,1,1 in( z 1)tanh 1 l 1 (9) whee the input impedane in (z i ) is defined as the equivalent impedane when looking into the iuit at position z i ;,i, i, and l i, ae the haateisti impedane, popagation onstant, and length of eah setion, espetively; and L is the teminal impedane. The whole eusive omputation has to be epeated fo eah wavefom simulation. A moe staightfowad appoah is utilized in this study to avoid epeating omputations in a model-based invesion and sum up effets fom all leading setions fo possible oveall alibation of line paametes. Seveal popagation maties have been intodued fo the satteing poblem (i.e., wave popagation in a non-unifom media) in geophysis and eletomagnetis, suh as the satteing matix, the tansfe matix, and the ABCD matix (Klein and Swift, 1977; Claebout, 1976; Potonotaios and Wing, 1967). In the ontext of two-pot measuement, Goiti and Slob (25) adopted the tansfe matix to expliitly sepaate the sample holde fom the tansition setions. Howeve, fo TDR modeling, the ABCD matix is found to be the simplest. It an be eadily explained in the ase of a unifom line. The voltage and uent at the two ends (z=, z=l) an be obtained fom Eq. (3) as

5 An Impoved Modeling of TDR Signal Popagation fo Measuing Complex Dieleti Pemittivity I l l e e l l l e e I l (1a) (1b) The elationship between the line voltage/uent at z= and the line voltage/uent at z=l an be obtained by equating + and in Eq. (1) as whee l sinh l osh l 1 I l sinh l osh l I os h l (11) l l e e (12a) 2 l l e e sin h l (12b) 2 Theefoe, + and ae eliminated by putting the solution Eq. (3) in the fom of Eq. (11). The 22 matix elating voltage and uent at two ends is alled the ABCD matix (). l sinh l l l osh A B 1 C D sinh osh (13) This epesentation elates the line voltage and uent at one end of line, z=, to the line voltage and uent at the othe end of the line, z=l. Fo a unifom tansmission line, the voltage and uent at the two ends (fou vaiables) an be solved by Eq. (11) and two bounday onditions. In geneal, the ABCD matix an elate the line voltages and uents of the two ends of any ith unifom setion of a non-unifom tansmission line by eplaing the l with l i in Eq. (13) z i z i 1 i I zi I zi 1 (14) The oveall ABCD matix () of the entie line is the matix that elates the voltages and uents at the two ends of the entie line. l I l I (15) whee an be obtained as the podut (in the appopiate ode) of the ABCD maties of the individual unifom setions as A B n n 1 1 C D (16) Now that the ABCD matix that elating the voltage and uent at the two ends has been obtained, teminal onstaints an be inopoated to solve fo teminal voltages and uents. At the soue end, z=, thee is an independent voltage soue S and a soue impedane S. The line is teminated at the load end, z=l, with a load impedane L. Thus, the bounday onditions (teminal onstaints) an be witten as () I() (17a) S S l () Il () (17b) L The fou unknowns (teminal voltages and uents) an be solved by fou equations, (15) to (17). The solution of patiula inteest is () in a TDR measuement. DB L L s L DBC A S (18) whee L is the load impedane, S is the soue impedane, S is the soue voltage, and A, B, C, and D ae fom Eq. (16). One the oveall ABCD (hain) matix of the entie line is obtained by use of Eq. (16), the sampling voltage an be omputed by Eq. (18). The solution aounts fo multiple efletions (by Eq. (16)) and able esistane (by Eq. (7)) and an aommodate dieleti dispesion and eletial ondutivity by fomulations in the fequeny domain. Sine all the above fomulations ae in the fequeny domain, the wavefom simulation would stat with a Fouie tansfom of the soue voltage signal. Equation (18) is then used to ompute the TDR esponse (i.e., sampling voltage) in the fequeny domain. And finally a Fouie invese tansfom of () gives the simulated TDR esponse in the time domain. 3 CALIBRATION OF TDR MEASUREMENT SYSTEM The sensing setion of a TDR measuement system is often times the last setion (n) shown in Fig. 2. Othe leading setion onsists of onnetos, leading able, and loal impedane mismathes within the TDR devie. To alibate the measuement system is equivalent to detemining L i, p,i,,i, and R,i fo i=1~n 1. As disussed ealie, it should be noted that diffeent ombinations of (L, p, ) an esult in the same TDR wavefom. One of the thee paametes (L, p, ) needs to be known o assumed so that the othe two paametes and R an be alibated fom the measued TDR wavefom. Calibation of these line paametes utilizes model-based invesion by wavefom mathing. The numbe of unknowns ineases with the numbe of unifom setions. When the numbe of setions is lage, the tansmission line maybe divided into seveal detahable subsetions and wavefoms ae measued by suessively adding up the subsetions. Line paametes of eah subsetion an then suessively alibated to avoid lage numbe of unknowns in the invesion. When n is lage, the alibation of the measuement system beomes a fomidable and tedious task. Futhemoe, it is often diffiult to detemine the appopiate numbe of setions (n) when paasiti efletions aused by loal impedane mismathes and impefet onnetos ae onsideed. An altenative effiient alibation poedue is devised heein based on the deoupled and non-eusive fomulation. The effets of all leading setions (i=1~n 1) may be summed up by a system ABCD matix, as shown in Fig. 3. And the sensing setion an be sepaated fom the system and teated as a new load

6 832 Chih-Ping Lin, Sh-Hong Tang, Chun-Hung Lin and Chih-Chung Chung Figue 3. The sensing setion is sepaated fom the system and teated as a load impedane in (z n-1 ); and the effets of all leading setions (i=1~n 1) ae summed up by a system ABCD matix. impedane L = in (z n-1 ). Equation (18) of the TDR esponse () is ewitten as in( zn 1) D B S in( zn 1) DBs in ( zn 1) C A in( zn 1) ( B/ D) in( zn 1) ( B/ D) s in( zn 1)( C / D) ( A/ D) (19) It should be noted that A, B, C, and D annot be uniquely detemined using only the efletion measuement without tansmission data. But all that ae needed fo TDR wavefom omputation ae the atios A/D, B/D, and C/D as indiated in Eq. (19). These atios an be alibated by shot-open-load (SOL) measuements (i.e., a measuement by shoting at the beginning of the sensing setion, a measuement by emoving the sensing setion, and a measuement of known dieleti pemittivity in the last sensing setion). Regadless the total numbe of setions, the atios A/D, B/D, and C/D ontain all popagation effets within the leading setions. 4 EXAMPLE-CALIBRATION AND MEASUREMENT BASED UPON TDR MODELING As an example shown in Fig. 1, the line paametes fo the leading able and the oaxial pobe need to be detemined befoe the system an be used fo measuing dieleti popety. The measuement ell was manufatued at the time when thee was no quik solution fo alibation of multiple impedane mismathes. It was puposely onstuted with an ulta thin ap to minimize impedane mismathes. Theefoe, the measuement system an be appoximated by a minimum of two setions, a lead able and the pobe setion. This measuement ell is by no means the best design fo soil measuement and thee wee no othe diet measuements (e.g., veto netwok analyze) available fo diet validation of the measued dieleti values. The following exeise is to demonstate dieleti measuement based upon TDR modeling using the new fomulation pesented. Line paametes fo the two unifom setions ae deoupled and epesented by distint physial paametes L, p,, and R, espetively epesenting length, geometi impedane, mateial popety, and line esistane, as illustated in Fig. 2. Sine only two setions ae involved in this exeise, the line paametes ae dietly alibated. The length of the lead able is dietly measued (i.e., L=1 m) and a mateial of S known dieleti popety (i.e., wate) in the pobe setion is used to obtain measued wavefom fo alibating the emaining paametes (i.e., eah setion has thee unknowns). Fo the example shown in Fig. 1, six unknown paametes ( p,, and R in the fist setion and L, p, and R in the seond setion) ae dietly alibated using one measued wavefom in tap wate. The dieleti pemittivity of wate is well known fom the liteatue (Klein and Swift, 1977) and the eletial ondutivity is dietly measued by a ondutivity mete. The devie used fo TDR measuement is a Tektonix 152C. TDR wavefom simulations and alibation of the system paametes by optimization wee implemented using Matlab. The alibated paametes ae listed in Fig. 1. The pedited wavefom using the alibated paametes mathes vey well with the measued wavefom, as shown in Fig. 4. Due to the lak of simple method to aount fo able esistane, the effet of able esistane was mostly negleted in dieleti haateization based on TDR. When ompaed with the pedited wavefom negleting able esistane ( R =), also shown in Fig. 4, signifiant effet of able esistane on the TDR wavefom is evealed. The esistane loss fato R affets the wavefom though the fequeny-dependent tem A. The ise time of the efleted pulse and the plateau of the step pulses inease as R ineases. The deoupled paameteization makes it easy to onside and alibate able esistane. This is extemely useful sine TDR is ineasingly used in the field Figue 4. Simulated and measued wavefom of tap wate with alibated paametes, whee measuing pobe is onneted to 1 m lead able.

7 An Impoved Modeling of TDR Signal Popagation fo Measuing Complex Dieleti Pemittivity 833 onditions whee long ables ae ommon. One the line paametes ae alibated, the TDR system in Fig. 1 an be used fo dieleti haateization of mateials. Thee diffeent soil samples wee pepaed at about the same volumeti wate ontent to demonstate the dieleti spetosopy of soils. Table 1 listed the soil type, bulk dy density (weight of soil patiles divided by the total volume), and volumeti wate ontent fo the thee samples. The appaent dieleti onstant fom time-domain tavel time analysis using the tangent line method is also listed in Table 1 to povide efeene data fo the following dieleti spetosopy. Afte the system alibation, the only unknown is. The omplex dieleti pemittivity an be dietly solved fo eah fequeny using Eq. (19). The esults ae shown in Fig. 5. The dieleti values at high fequenies ae lose to the appaent dieleti onstants listed in Table 1 as expeted. Figue 6 shows the pe dited wavefoms ompaed with the measued wavefoms. The exellent math between the simulated and measued wavefoms in Fig. 6 indiates that the model is eliable and auate fo the TDR measuement. Figue 5 shows simila dieleti pemittivity at fequenies geate than 1 MHz fo all soil types at about the same volumeti wate ontent, but distint dieleti elaxation phenomena an be obseved at lowe fequenies. Theefoe, the dieleti pemittivity at high fequenies an be used to estimate soil wate ontent, wheeas the dieleti dispesion at lowe fequeny is elated to patile size and wate-patile inteation. These measuements ae useful fo haateization of poous mateials. It is noted that measuements at fequeny geate than 2 MHz ae eoneous fo the high plastiity lay (bentonite). This is beause the lay is moe ondutive (lossy) and the signal to noise atio is downgaded at highe fequenies. The dieleti measuement may be altenatively onduted by Table 1 The bulk dy density (ρ d ), volumeti wate ontent (θ), and appaent dieleti onstant (K a ) of soil samples Soil type ρ d (g/m 3 ) θ K a Ottawa sand Silty sand Bentonite Figue 5. Real pat and imaginay pats of omplex dieleti pemittivity fo the thee soil samples. Im. Imaginay pat; Re. eal pat. Figue 6. Deoupling wave popagation model fo TDR wavefom simulation. onstaining the omplex dieleti pemittivity (f) via some mathematial model and pefoming invesion of the model paametes in the time domain by wavefom mathing. It is undestood that epesentation of the measuement system in Fig. 1 by two unifom segments is only an appoximation sine possible mismathes in the leading setion and finging effet due to geomety hange at the beginning of the sensing setion is negleted. Futhe impovements patiulaly at high fequenies may be ahieved by avoiding abupt geomety hange at the intefae between the leading setion and the sensing waveguide. Mismathes and finging effets within the leading setion an then be taken into aount by oveall alibation using Eq. (19) and SOL measuements. 5 CONCLUSION Time domain efletomety (TDR) is a measuement tehnique based upon tansmission line theoy. It is finding moe and moe appliations in engineeing measuements and monitoing in both laboatoy and field woks, diving the demand fo an auate TDR wave popagation model. In this study, the solutions of tansmission line equation wee efomulated in tems of independent physial popeties paameteized by geometi impedane ( p ), mateial popety ( and ), and esistane loss fato ( R ), instead of inte-elated iuit paametes (, l, g, and ). The deoupled paameteization of tansmission line equations allows moe onvenient alibation of line paametes and failitates intepetation of TDR measuements. When ombined with the ABCD matix appoah, the non-unifom tansmission line in a TDR measuement system an be effiiently modeled aounting fo multiple efletions, dieleti dispesion, ondutive loss, and able esistane all togethe. In patiula, the effet of able esistane whih was often ovelooked an have signifiant effet on the measued TDR wavefom. Effetive appoahes to alibate line paametes in a TDR system and pefom TDR measuements wee intodued. An example was given to demonstate dieleti spetosopy based upon TDR modeling. The exellent math between the simulated and measued wavefoms indiates that the model is eliable and auate. The intodued fomulation and modeling appoah povides a simple, yet theoetially sound foundation fo vaious engineeing measuements based on TDR.

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