Contact impedance of grounded and capacitive electrodes

Size: px

Start display at page:

Download "Contact impedance of grounded and capacitive electrodes"

Chad Allen
5 years ago
Views:

1 Abstact Contact impedance of gounded and capacitive electodes Andeas Hödt Institut fü Geophysik und extateestische Physik, TU Baunschweig The contact impedance of electodes detemines how much cuent can be inected into the gound fo a given voltage. If the gound is vey esistive, capacitive coupling may be supeio to galvanic coupling. The standad equations fo the impedance of capacitive electodes assume that the halfspace is an ideal conducto. Ove esistive gound at high fequencies, howeve, the contact impedance will depend on the electical popeties, i.e. electical conductivity and pemittivity, of the subsuface. Hee, I eview existing equations fo the esistance of a galvanically coupled, spheical electode in a fullspace. I extend the theoy to the geneal case of a sphee in a spheically layeed fullspace which may display both galvanic and capacitive coupling. Fo a capacitively coupled electode, the common assumption of an ideally conducting fullspace (o halfspace) beaks down if the displacement cuents in the fullspace become as lage as the conduction cuents. Fo a modeately esistive medium with m this is the case fo fequencies lage than khz. Fo vey high esistivities aound M, the tansition fequency educes to Hz. Thus, in pinciple, one may detemine electical esistivity and pemittivity by measuing magnitude and phase of the electode contact impedance. Intoduction DC esistivity measuements ae usually caied out with fou electodes. This way, the atio between measued voltage and inected cuent is independent of the gounding esistance of the electodes. Howeve, calculation o estimation of the electode esistance may be impotant in some situations. If the gound is vey esistive, technical issues may limit the cuent that can be inected into the gound. When tying to decease contact esistance, fo example by wateing electodes, the exact dependence on gound esistivity o geomety is impotant to find an optimum stategy. Finally, the contact esistance itself might be used to obtain infomation on the gound esistivity (Dashevsky et al., 5). Fo galvanicall coupled electodes, equations descibing the inected cuent as function of voltage have been deived fo diffeent electode geometies by Kaew (957). Capacitive electodes nomally consist of sheets close to the gound with no diect contact. They ae used with an altenating cuent of sufficiently high fequency such that the impedance is sufficiently low. They may be paticulaly useful if the gound is vey esistive and galvanic coupling is not feasible, o if fast measuents with a moving system ae to be caied out. Kuas et al. (6) descibe the theoy behind 4-point esistivity measuements with capacitive electodes and discuss the conditions unde which inductive cuents may be ignoed. To estimate the contact esistance of capacitive electodes, the halfspace is nomally assumed to be an ideal conducto. Ove vey esistive gound, howeve, the assumption of an ideal conducto is no longe valid, and the contact esistance of capacitive electodes will depend on electical conductivity and dielectic pemittivity of the halfspace.. Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 64

2 Hee, I eview the equations fo galvanically coupled electodes and extend the theoy to capacitively coupled sphees in a fullspace. I investigate unde which conditions the assumption of an ideally conducting halfspace beaks down and -point measuements might be feasible to detemine conductivity and dielectic pemittivity of the gound. The basic setup is sketched in figue. A DC voltage is applied to galvanically coupled electodes (top panel) o an AC voltage to capacitively coupled electodes (bottom panel). The aim is to deive equations fo the esistance R, equied to calculate the cuent I fom the applied voltage U via: R U / I () whee R depends on esistivity fo galvanic coupling, and on esistivity and electic pemittivity fo capacitive coupling. U I U ~ I, Figue : Sketch of the basic setup. Top panel: DC voltage applied to galvanically coupled electodes. Bottom: AC voltage applied to capacitively coupled electodes. Galvanically coupled spheical electode in fullspace The calculation of the esistance of abitay electodes ove a halfspace depends on the shape of the electodes and equies numeical solution. Theefoe, I simplify the poblem by consideing spheical electodes in a fullspace. This stongly deviates fom the situation sketched in figue, but in ode to obtain physical insight, simple analytic equations ae desied. The equation fo the contact esistance of a single galvanically coupled spheical electode in a fullspace was given by Kaew (957): R () 4 whee is the esistivity of the fullspace and is the adius of the sphee. One impotant implication is that the esistance is invesely popotional to the adius, and not to the suface of the electode. This will apply to othe types of electodes as well, in a sense that the spatial dimension of the electode entes linealy into the esistance. The linea dependence might be counteintuitive, because one could expect the esistance to decease with the suface aea of the sphee. The impotant point is that the electic field at the suface of the sphee deceases. Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 65

3 with /, which compensates one spatial dimension, as can be seen fom the deivation in appendix. Anothe useful assumption is that the distance between the two electodes is lage compaed to the size of the electodes. In that case, each electode may be teated independently. The distance between the electodes dops out of the equations and the total esistance will simply be the sum of the two single electode esistances (Kaew, 957). Equation () can easily be extended to the situation whee the electode is suounded by spheical shells. The paametes fo the case of two sphees, which will be sufficient to descibe most of the pactical situations, ae defined in figue : Figue : Geomety of a spheical electode, adius with potential, suounded by a spheicall shell with adius and esistivity, in the fullspace with esistivity. The esistance of the spheical electode is given by: R 4 4 (3) A deivation slightly deviating fom that of Kaew (957) is given in Appendix. Equation (3) may be used in diffeent foms to study the dependence of esistance on the esistivity distibution of the volume suounding the electodes. It is common pactice to decease contact esistance by pouing wate into the gound nea the electode, and we may estimate the amounts of wate and the esistivity contast which is equied to achieve a cetain eduction in esistance. We assume that the wate fills a spheical shell of adius and educes the esistivity to compaed to of the undistubed fomation. The decease of contact esistance is then expessed as R R (4). Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 66

4 whee R (5) 4 denotes the esistance in a fullspace with esistivity, which would exist if no wateing was applied. Figue 3 illustates the eduction of electode esistance by a conductive spheical shell suounding the electode. The esistance quickly deceases with the size of the conductive shell, but fo adii lage than times the electode size, a futhe incease is not efficient any moe. The behavio with espect to esistivity contast is simila: Once a easonable esistivity contast of : is eached, a futhe incease does not lead to a significant decease of esistance. R R.5... Figue 3: Reduction of electode esistance as function of adius of the conductive shell fo diffeent esistivity atios between oute fullspace and conductive shell. Note the logaithmic adius axis. Capacitively coupled sphee Fo the capacitively coupled sphee, it is useful to use electical conductivity instead of esistivity. We may use the same equations deived fo the static case if we eplace the electical conductivity by a complex conductivity defined by: * i (6) whee is the dielectic pemittivity. This substitution is ustified in detail in Appendix. One assumption which is not expanded on hee is that induction effects may be ignoed. This aspect was discussed in some detail by Kuas et al. (6). The complex electode impedance is obtained by ewiting eq(3) with the substitution defined in eq. (6): * * * * Z * * 4 * (7). Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 67

5 A capacitively coupled electode may be studied by setting conductivity and elative dielectic pemittivity in the inne shell to the values of ai ( =, =). If the fullspace suounding the electode is sufficiently conductive, the common ideal conducto assumption will hold, and the esistance will not depend on the electical paametes of the fullspace. This can be seen by witing eq. (7) in the limit : o Z (8) i 4 o which may be compaed with the impedance of a plate ove an ideally conductive halfspace: Z d (9) i o A whee d is the distance between the halfspace and the plate, and A is the aea of the plate. Obviously, the thickness of the inne shell ( - ) coesponds to d, and 4 coesponds to the aea A. Howeve, fo a esistive fullspace, this appoximation will not be valid any moe. The tansition is illustated in figue 4, which shows the esistance fo a spheical capacitive electode with mm sepaation between electode and fullspace, calculated fom eq. (7). The cuve fo = S/m epesents the ideally conducting fullspace. The esistance follows a / fequency dependence ove the entie fequency ange, and does not depend on conductivity o pemittivity of the fullspace. The uppe limit is set by the cuve fo vey low conductivities ( = - S/m) which epesents a spheical electode in the ai. Z Fequency (Hz) Figue 4: Amplitude of the complex impedance as function of fequency fo diffeent electical conductivities (in S/m) of the fullspace. The adius of the spheical electode is =.m, the shell between the fullspace and the electode is mm thick ( - =.m), and the elative pemittivity of the fullspace is =3.. Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 68

6 If the fullspace is modeately esistive (i.e. = -3 S/m), the electode esistance stats to deviate fom the ideal conducto limit at appox. khz. If the fullspace is vey esistive (i.e (i.e. = -6 S/m), the tansition stats at elatively low fequencies aound Hz. Of couse, the tansition fequency coesponds to the point whee displacement cuents stat to become as lage as conduction cuents. Thus, if a capacitive electode system is used ove pemafost aeas, ove vey dy ock, o on space missions landing on asteoids o comets, the ideal conducto equations will beak down. Figue 5 illustates the behavio of the phase of the impedance. In the limit of an infinitely conductive o esistive fullspace ( = o - S/m), the impedance behaves like that of an ideal capacito, and the phase is -9 degees. Fo finite fullspace conductivities, the phase will be sensitive to vaiations in conductivity (and pemittivity, not illustated), which may in pinciple be used to detemine those paamtes. Measuing amplitude and phase of the inected cuent elated to the souce voltage gives two equations which ae equied to solve fo the two unknowns and. In pactice, howeve, the additional dependence on the distance between electode and fullspace o halfspace, and capacitive coupling between cables and the measuing device may ceate difficulties. Dashevsky et al., (5) suggested to measue the diffeence of the impedance fo two diffeent heights in ode to emove coupling effects, and used this appoach to evaluate pavement quality , Figue 5: Phase in (degees) of the contact impedance of a capacitive electodes fo diffeent conductivities in S/m) of the spheical fullspace. Paametes ae the same as in figue 3. Conclusions Fequency Fo a single, galvanically coupled sphee, the esistance deceases with the adius of the sphee, and not, as one might expect, with the aea of the sphee. Thus, if in pactice the contact aea is inceased by using many metal sticks in paallel, the decease of esistance will be popotional only to the squae oot of the numbe of sticks. When educing contact. Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 69

7 esistance by wateing, thee is a satuation effect with espect to both esistivity contast and volume. Below a esistivity contasts of. between wate and undistubed gound, the esistance does not futhe decease. Thus, thee is no point using excessive amounts of salt to ceate extemely conductive wate. Fo capacitively coupled electodes, the common assumption of an ideal conducto beaks down fo esistive gound and high fequencies. Depending on electode size and geomety, the electode impedance may be undeestimated by two odes of magnitude if the finite conductivity is neglected. In pinciple, two-point measuements to detemine electical paametes of the subsuface with capacitive electodes ae feasible. Howeve, the penetation depth of such measuements is only in the ode of the size of the electodes. Moeove, distotion effects by capacitive coupling between cables and the measuing device have to be caefully contolled. Refeences Dashevsky, Y.A., Dashevsky, O.Y., Filkovsky, M.I., Synakh,.S., 5, Capacitance sounding: a new geophysical method fo asphalt pavement quality evaluation, J. Appl. Geoph. 57, Kuas, O., Beamish, D., Meldum, P.I., and Ogilvy, R.D., 6, Fundamentals of the capacitive esistivity technique. Geophysics, 7, G35-G5. Kaew, A.P., 957, Gundlagen de Geoelektik, BM elag Technik, Belin. Appendix : Deivation of the electode esistance fo the spheical shell model. A. Spheical electode in a homogeneous fullspace We use the geomety sketched in figue. We assume a constant potential on the spheical electode with adius. At any distance fom the cente of the electode, the potential fo > must follow: (A) because fom potential theoy it will decay with /, and ( )= has to be fulfilled. Theefoe, the electic field at is: E (A) and in paticula: E (A3) This allows us to calculate the cuent density at the suface of the electode and the total cuent by integating ove the aea of the sphee:. Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 7

8 E (A4) and I 4 4 (A5) Finally, we obtain the electode esistance fom the atio between potential and cuent: R I 4 (A6) which is equal to eq. (). A. Spheical electode within a spheical shell in a fullspace In ode to fulfil Laplace s equation fo the potential, in the oute fullspace (> ) it must follow: b ( ) (A7) whee b is a yet unknown constant to be detemined fom the bounday conditions. Within the inne shell ( << ) we use the fom: ( ) a (A8) Obviously, this fom of () fulfils Laplace s equation, and the constant a must be detemined fom the bounday condictions. At the edge oute of the inne shell (= ), the two potentials must be equal: ( ) b a (A9) and thus: a (A) b The second condition esults fom the continuity of cuent density at the bounday. Inside the bounday (< ), the electic field is: E a (A) and outside (> ) it is :. Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 7

9 b E Continuity of cuent density at = equies that E E (A3) and thus b a (A4) We now have two equations (A and A4) fo the two unknowns a and b, and we obtain a (A5) The solution allows us to calculate the cuent density, which may be expessed as: (A6) whee (A7) is the cuent density of the sphee in a fullspace without a spheical shell. We finally obtain the esistance in the fom given in equation (3) though I R 4 (A8) Appendix : Deivation of the potential equation in the complex case Ampee s law states that: t D oth (A9) whee H is the magnetic field, is cuent density and D is the electic displacement. By taking the divegence, we obtain: t D div (A). Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 7

10 With E (A) and tansfomation to the fequency domain, such that the time deivative becomes a multiplication with i, we get: E i E div (A) If we intoduce the complex conductivity * i (A3) (A) wites: * E div (A4) Finally, Faaday s law states that B ote (A5) t If induction effects can be ignoed, then ot E (A6) and the electic field may be obtained fom a scala potential : E gad (A7) We thus obtain the basic equation fo * gad div (A8) which is the basis fo the deivation of the electode esistances. It is identical to the equation used in the static case, the only diffeence being that it is complex and the DC conductivity was eplaced as defined in eq. (A3). Thus, all aguments apply fo the complex case as well, and eq. (3) may diectly be tansfeed into eq. (7).. Kolloquium Elektomagnetische Tiefenfoschung, Hotel Maxičky, Děčín, Czech Republic, Octobe -5, 7 73

Review: Electrostatics and Magnetostatics

Review: Electrostatics and Magnetostatics Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion