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High frequency method of locating power cable faults 1947
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Title | High frequency method of locating power cable faults |
Creator |
Nalos, Ervin Joseph |
Publisher | University of British Columbia |
Date Created | 2012-03-26 |
Date Issued | 2012-03-26 |
Date | 1947 |
Description | The Location of a fault in a power cable is by no means a simple task and the techniques used at present warrant further improvement. Many methods of fault location have been devised but each has its particular limitations. Some of these methods include D.C. and A.C. bridge methods, echo-ranging methods, and high-frequency methods. The limitations and impracticability of these various techniques has been the main reason for this research. In this thesis, a review of the high-frequency method has been made, resulting in the development of a method of locating high-resistance faults. Expressions, permiting the use of this improved method in instances where cable potheads are relatively inaccessible have also been developed and checked experimentally. Briefly, the method consists of determining the input impedance of the cable with its remote end terminated in its surge impedance. This is done by simultaneously measuring the voltage drops across a standard resistor and across the cable. The ratio of these drops is an indication of the cable impedance at that frequency. Observations are taken on a band of frequencies on a faulted cable and on a good cable, both terminated in the characteristic impedance. The difference of these two effects is attributed to the reflections from the fault. From the plot of the impedance as a function of the frequency, by a short graphical computation, the distance to the fault may be obtained. Faults as high as twenty times the surge impedance have been successfully located on relatively short lengths of cable. The distance to the fault has been estimated well within [page missing] |
Subject |
Cables Electric Lines |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2012-03-26 |
DOI | 10.14288/1.0105076 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/41763 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0105076/source |
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I £3 #7 h«-j fly High Frequency Method of Locating Power Gable Faults Ervin Joseph Nalos A Thesis submitted i n Par t i a l Fulfilment of the Require- ments for the Degree of Master of Applied Scince i n the Department of E l e c t r i c a l Engineering The University of B r i t i s h Columbia August ,1947 1 Table of Contents Page I. Introduction 3 II. Mathematical Analysis A. Symbols used 6 B. Analysis of Transmission Lines . . . . 7 C. Analysis of Faulted Line, end O.C. . . . 8 D. Analysis of Faulted Line, end S.C. . . . 9 E. Analysis of Faulted Line, end char.imp. . 10 F. Analysis of Double Faults 11 G-. Effect of Long Instrument Leads . . . . 12 H. Alternate Approach,far end char. imp. . . 13 III. Discussion of Experiments A. Circuit Diagram and Apparatus . . . . 15 B. Calculations 1. Characteristic Termination . . .21 2. Far End Open-circuited . . . .22 3. Far End Short-circuited . . . . 23 4. Double Faults 24 5. Effect of Long Lead-in Cable . . 25 6. Alternate Method,Charact.Termin. . 26 C. Dicussion of Results 1. Effect of Harmonics . . . . . .30 2. Additional Errors 31 3. Accuracy,Limitations,and Conclusion 33 4. Bibliography 35 5. Acknowledgement . . . . . . . 36 2 L i s t of Illustrations / Page Pig.l Transmission Line 7 Pig.2 Faulted Line,far end O.C. . ' . . . . . 9 Fig.3 Equivalent Circuit of Faulted Line . . . 9 Fig. 4 Line with Double Fault correctly terminated 11 Fig.5 Equivalent Circuit of Line with Double Fault 11 Fig.6a Line with Lead-in Cable.. . . . . . .12 Fig.6b Circles of constant Attenuation,Frequency . 14 Fig.7 Video-frequency Amplifier 16 Fig.7a Layout of Apparatus . ... . . . . . . . 17 Fig.7b Schematic Diagram of Layout . . . . .17 Fig.8 S-function Method of Measuring Attenuation 18 Fig.9a,9b Reflected voltage and' current wave . .19 Fig.10 Connection to Amplifier . . . . . .20 Fig.10a A r t i f i c i a l Fault . . . . . . . .20 Fig.10b Characteristic Termination . . . . . 20 Fig. 11a,lib,lie Effect of Harmonics . . . . 30 Fig. l i d High-pass F i l t e r Circuit . . . . . 3 2 Fig.12 Tuned Circuit to,eliminate Harmonics . 33 Li s t of Tests facing pages shown Test 1,2,3 s Input Impedance,line correctly terminated . . . 21 Test 4,5 : Input Impedance,Line O.C. and S.C. 22 Test 6,7 : 'Effect of Harmonics . . . . . 30 Test 8 : Double Fault . . . . . . . 24 Test 9 : Effect of Mismatch at Far End . . 24 Test 10 : Effect of Long Input Cable . . 26 Test 11 : Alternate Method with Char.Term. . 28 3 HIGH FREQUENCY METHOD OF LOCATING POWER CABLE FAULTS I. Introduction The location of a fault i n power cable i s by no means a simple task and the techniques used at the present warrant further improvement.In practise,power cables are burried in'dry or wet-ground,placed i n conduit,or l a i d i n troughs,separately or i n groups; their path may cross an open f i e l d or a busy intersection. Frequently,they are ad- jacent to non current-carrying e l e c t r i c a l conductors.Cables may be terminated i n potheads, i n manholes,or near the top of a line pole.Occasionally,one end of a cable may be inaccessi- ble as for example i n certain types of ring-buss switch bo- xes.Due to the various conditions under which cables are used,methods of fault location successful under one set of conditions w i l l not necessarily work on others. Many methods of fault location have been devised but each of them has i t s particular limitations.The simple D.C. bridge techniques-' have been superseded to some extent by 1 7 e l e c t r i c a l pick-up methods. ' These,however, are of l i t t l e use i n locating faults i n armored cable,in lead-sheath cable located amongst a number of other cables,or i n cables adja- cent to non-electrical current-carrying structures because the audio signal employed does not always leave the cable at the point of the fault but rather passes along the cable armor or other conductors i n a devious path. The echo-ranging technique ^ i s at present limited to telephone work because the lengths of power cable used are rel a t i v e l y short.For po- wer cables,where the velocity of electro-magnetic waves i s approximately 500 ft/microsecond,the pulse returns too rapid- l y to be detected unless the cable i s longer than 100 f t or so. Moreover,the echo-ranging method requires elaborate trans- mitting and receiving equipment which at present makes i t impractical. A l l numbered references are given i n bibliography. 4 The limitations and impracticability of these various techniques led to the development of a supplementary method 5 employing high frequencies. The method,utilizing frequen- cies from 0.1 Mc to 32 Mc,is based on the principle that standing waves may be established on e l e c t r i c a l l y long lines and that reflections are produced at discontinuities such as open and short c i r c u i t s where the fault resistance does not equal the characteristic impedance of the cable.Since the cable input impedance i s a function of the frequency.it i s possible to ascertain the frequencies where maximum impe- dances occur by recording the frequencies corresponding to maximum input voltages. By proper interpretation of these frequency differences between adjacent impedance peaks,the distance to the fault may easily be calculated. This method i s limited to faults whose resistance i s less than a third of the characteristic impedance.The most accurate determinations are obtained when the fault r e s i s - tance i s zero;however,such cases are rarely found i n practi- se and i t i s thus necessary to burn faults down u n t i l they can be detected.This requires time as well as special equip- ment. Frequently, the resistance of a burnt down fault w i l l rise during the test to a value which makes the method i n - applicable without further carbonization of the fault. The object of this research has been twofold: F i r s t l y , t o review the high frequency method i n an endeavor to develop a method of locating high-resistance faults; secondly, to develop expressions permiting the use of this method i n instances where cable potheads are r e l a t i v e l y i n - accessible, such as on top of poles.In effect,this means ta- king into account long instrument leads* The f i r s t of these objectives has been achieved by the development of a simple method of measuring the cable input impedance.Briefly,the method consists of terminating the cable at the remote end i n i t s characteristic impedance and measuring simultaneously the voltage drops across a 5 standard resistor and across the cable. The ratio of these drops i s an indication of the cable input impedance at that frequency.Observations are taken on a band of frequencies on a faulted cable and on a good cable,both terminated i n the characteristic impedance.The difference of these two impor- tant effects i s attributed to reflections from the fault. Prom a plot of impedance as a function of frequency,by a short graphical computation,the distance to the fault may easily be obtained.To f u l f i l l the secondary objective neces- sary expressions permiting the use of long instrument leads have been developed and are presented i n the main body of the thesis.Faults as high as twenty times the characteristic impedance have been successfully located on re l a t i v e l y short lengths of cable.The distance to the fault has been estimated well within 5$ and higher accuracy i s to be expected on longer cables. 6 I I . MATHEMATICAL ANALYSIS A. Symbols used In Mathematical Analysis. x - distance from sending end E - rms voltage at distance x from sending end I - rms current at distance x from sending end z - series impedance of cable per unit length y - shunt impedance of cable per unit length c - propagation constant* a - attenuation constant- c « (a* jb) 88 /zy 1 b - phase constant k - constants satisfying boundary conditions E - sending end voltage E 3- receiving end voltage 1^- sending end current I - receiving end current Z r- input impedance Z 3- equivalent receiving end impedance 1 - total length of given cable K = VR R - Fault resistance 3 = ( a l/cos2 bl) d = (R/R*Z Q ) = (l/lok) d 2 = ^/Rg+Zj ( R l V R l V R 2 Z o V R l Z 0 ( R 2 t Z 0 ) (R,R~*R~z * Z 2 ) , _ _ _ x v 1 2 2 o o V R_Z (Rrt+Z ) ' 1 o 2 o u = V = Z Q" Characteristic Impedance of Lead-in Cable Z = Sending end Impedance appearing at Sending End of 3 Lead-in Cable Z 0 Rc+JXc, 3 " 3 3 I.(r • j x ) r l = r a = Z Z 3 3 o o 1/2 itanh {al+fal^+coth Dal-»(al}y = coth 2 Qal+Cal)^ 1/2 ^coth [al*(al)l-tanh [al+(al)^= csch 2 ^.l+Cal)^ x~ • 1/2 Jtan [bl+(bl)g-cot tbl+(hl)lf= -cot 2 fcl+(bl)0-] r b = 1/2 jtan [blt(bl)g«cot [bl+(bl) 0ij= esc 2 [bl*(bl)0-} V r = Voltage drop across fixed standard resistor r which is a measure of the r f current through the cable V c - Voltage drop across cable which, when divided by r f current gives the cable impedance r = Standard resistor used i n measuring r f current through cable * Notation used here i s not standard. 7. B. Analysis of Transmission Lines, dE - T — - Iz dx dl = Ey dx d 2E d x 2 d2l_ dx2 z d l = I z y dx - v dE Izy X —o fist Solving these two differential equations, we obtain: E = k-^cosh cx + kgSinh cx I = k_cosh cx * k^sinh cx (1) (2) The constants k are obtained by boundary conditions, as follows: At Sending end, where x = 0, E = E r = kx I * I p = k 3 Also, - ck^sinh cx «*• ckgcosh cx 5 Iz iSH • cE^sinh cx + ckocosh cx dx r d = I pz cosh cx + k^z sinh cx Hence, ck cE r « kij.z 2 s I rz from which kg = I r Z Q k V B E r / z , E s = E pcosh c l I rZ 0sinh c l I s = I rcosh c l + E r / Z c sin* 1 c l z - Eg a E rcosh c l I rZ 0sinh c l 3 " Is I^cosh c l + E * / z 0 sinh c l (3) (4) But Z r « E r/1. thus 8. z . z Z rcosh c l * Z 0sinh c l (5) Z Qcosh c l Z rsinh c l For Open-circuited end, Z r i s in f i n i t e , for Short-circuited end, i t is zero. Hence, Z o c - Z 0coth c l Z n„ = Z^tanh c l sc o (6) C. Analysis of Faulted Line with far end open-circuited. The c i r c u i t i s shown i n Fig. 2 and the equivalent ci r c u i t i s shown i n Fig. 3; from this, we obtain, Z r s RZ Qcoth clg R + Z 0coth clg Rcoth cl2Cosh c l i R+Z0coth clg sinh cl- cosh c l i + Rcoth clgsinh c l ! R + Z Qcoth clg 1 + tanh c l ^ (k + tanh clg) K + tanh cl.,+ tanh c l ~ (7) 1. If attenuation is negligible, c and i n the above expression, "~ tanh c l = tanh j b l • jtan b l Hence, Z 3 a j(l-tan hl]^tan big) + jktan h i ! ' K + j (tan b l ^ t a n big) a + jb i s appr. jb (8) If b l ! i s a n l n t e S r a l multiple of big, the above ratio equals l / 2 for b l g a 0,^2113... , this being a pure resistance, 2. If attenuation is small but not negligible, a l i s small sinhg2al = 2al cosh a l - 1 tanh c l sinh 2al+jsin 2bl _ a l 2 (cosh 2al-sin 2bl) cos 2hl + jtan b l 9. X, o— R 3 Fig. 2 Fig. 3 a l . a l . 1 + <k + cos2 b i 2 * jtan b l 2 ) ( c o s 2 ^ * Jtan b l ^ (k+ a l - al. cos 2 bl- cos^ bl- -) * j (tan bl}+ tan b l 2 ) Jl-vs1(k««-s2)-tan b l ^ tan b l 2 l + j £(k+s2) tanbl^+s^tanbLgl (K + s^ + s 2) + j (tan b l i +tan b l 2 ) . . . . ( 9 ) D. Analysis of Faulted Line with far end short-circuited. The only difference between this and the previous case i s In the value of the equivalent impedance which now becomes RZ tanh c l 0 Z a -O d. R + Z Qtanh c l 2 ' Rtanh clocosh cl - i v * - + sinh c l ! Ri-Z0tanh c l 2 Rtanh closinh cl - i i cosh c l - i * £ i J R*Z0 tanh c l 2 Z s (tanh c l 2 + tanh cl]_(l+ktanh c l 2 ) 1 «• ktanh cl 2+ tanh cl^tanh c l 2 1. If attenuation i s negligible, again substituting tanh c l s jtan b l , then Z s (-ktan bl^tan b l 2 ) + j(tan bl^+tan b l 2 ) Z 0 (l-tan bl^tan b l 2 ) 4 jktan b l 2 (10) (11) 2. If attenuation is small but not negligible 10. (s-j + s 2 (1+ks ]_) -k tanbl-L tanblgl+jfcanblg (1+ksj) + tanbl^ (l+ksg)] \.l*ks 2*sis 2- tanblitanbl 2]+ j[(k+ s])tanbl 2* s 2 tan bl-jl (12) E. Analysis of Faulted Line with far end terminated In 3urge impedance. The receiving end impedance of the equivalent c i r c u i t becomes, Z r = R+Z0 = dZ, Z r + Z 0tanh c l ^ d *• tanh c l ^ • • - i - n i P . i» — • — m 8 3 I I I • i • • — i .^^.mmmm.m • Z Q Z rtanh c l ^ 1 + d tanh c l ^ ..(13) 1. If the attenuation i s negligible, then tanh c l s tanhjbl a jtan b l Z s d+jtan bin _ _ 8 S C Z D 1+jdtan b l i d(l+tan 2bl 1) + j( l - d 2 ) t a n b ^ 1 + d^ tan^ bin • • e e • (X^f) When bl-j_ = 0,T,2.n;.. , Z s / Z 0 B d (resistive) When b l a > Zs/Zo = Vd( " ) 2. If the attenuation is small but not negligible, [(d*3j) (l^s^) + dtan2bl£l + j (l-d2) tan b ^ (15) ( l * d s i ) 2 + d 2 tan 2 bl- When b l j » 0,%*!} When bin =f *T -1- *•) 2.) — ( d < 3 i > ( i * a . , ) , Z s / a l/d which can easily be veri- fied by substituting and taking the limit as b l ^ approaches "72. 11. F. Analysis of Double Faults Fig. 4 shows the ci r c u i t diagram for the case of charac- t e r i s t i c termination which i s the only case considered. The equivalent c i r c u i t is shown i n Fig.. 5 . In most cases at- tenuation i s small and for simplification It Is neglected i n this consideration. If not small, attenuation may be ac- counted for i n a manner described previously. Again, Z s i S Z r+Z 0tanh c l ^ J r Z 0*Z rtanh c l i Z o R 2 / Z . R + J Z o t a n b l 2 zo + 0 z o R 2 / Z o 4 R 2 t a n b l 2 d 2+jtan b l 2 l+jdgtan b l 2 Fig. 4 Hence, Z Fig. 5 - Z s R l I T — Z s R l R-L (d 2+jtan b l 2 ) ^ d 2 + R l / z ^ + ^ 1 + d 2 R l / z ) t a n b l 2 dg+jtan b l 2 u+jvtan b l 2 (16) ^s _ Zo (d 2-vZ Qtan bl^tan b l 2 ) + j (tan bl 2*-uZ 0tan hl^) (uZ D-tan bljtan b l 2 ) - j(vZ 0tan bl2+-d2tan D I 2 ) . . . ( 1 7 ) As an example, i f R-|_« 100 ; R2= 50 ; ZQ= 50 l x= 200 f t . ; 12= 75 f t - the above expression becomes ( . 5 -tan 2 f tan .75f) + j (tan.75f+1.25tan2f) (l . 2 5 - t a n 2 f tan .75f) + j (tan.75f+ -5tan2f) which i s plotted in Test 8. This analysis is merely of academic value and the occurence of this case In a power cable is extremely unlikely. It is included merely to complete the study and might be used to advantage i n telephone work. 12. G. Effect of Long Lead-In Cable on Impedance of Cable under Test Often faulted cables are overhead or not directly accessible and an auxiliary lead i s brought out to the apparatus from the end of the•faulted cable. This cable is not usually short e l e c t r i c a l l y at frequencies of 2-3 Mc and hence i t s effect has to be calculated. From Fig. 6awe obtain by a process analogous to previous deri- vations, Js _ Z cosh cl+zlsinh c l s o z'coshcl * Z sinh c l o s Zs/Zp ¥ tanh c l 1 + Z s/Z 0 tanh c l JL A Fig. 6a (18a) With negligible attenuation, this reduces to: - • Z s / Z 0 4 J t a n b l 1 + O'Zs/z1 t a n D l (18b) Alternately, from ( l8a) and ( l8b) , z ; ( Zs/Z Q -Iten WL) - « Z 0 (l-jZa/z« ^ bl)„ (19) The plot of expression (l8b) for 25 f t . of Cable RG8/U is given i n Test 10 together with the curves obtained by a test on a cable with a 100 Ohm fault a fixed distance from the sending end. 13 H. An Alternate Approach of Analysis of Cable Faults with far End Terminated i n Characteristic Impedance, This method may be used to advantage i f apparatus is available for measuring reactive component of cable im- pedance with reasonable accuracy. The tedious calculations encountered i n Eq. 14 and 15 are avoided by the use of this method. Since the far end of the cable i s correctly termi- nated, no reflections return to the sending end. The input Impedance of the cable behaves as though i t were terminated at the fault with an impedance Z R = R Z 0 / R + Z which is fixed. If Z . — ~ ,= cosh ( c l ) 0 and Z r s sinh (cl) 4 Z i - Z l V Zo- zr' o then Z g Z rcosh c l + Z csinh c l Z Z cosh c l + Z sinh c l ^o o r coshcl sinh ( c l ) Q 4 sinh c l cosh ( c l ) Q ) cosh c l cosh ( c l ) Q + sinh c l sinh ( c l ) 0 i z s r ~i R s + J ' X s - tanh|cl4(cl)J = — ; * r 4jx s ( 2 o) Z Q u „ J ^o As before, i f c = a*jb, tanh c l = ] ( s l n n &1 c o s n a l ) * 3* (sin b l cos bl) cosh 2al cos 2bl+sixth 2 a l s i n 2 b l If i n the above equation a value of c l 4- (cl) Q= [al+(al)^J + j£bl+(bl)J], i s substituted for c l , keeping [al-»-(al)0"] constant (rg-r,) 2 t x 2 = r | which is a circ l e i n the (r sxj Tlane. 14. Similarly, i f [bl * (bl) 0 lconstant, r s + ( x s " x l ) 2 • * f (21, 22) In Eq. (21) and (22) *1 - eoth ^ [ a l + f a l j j s coth 2 ( a l ) Q i f attenuation is negligible r a = csch 2 [al+(al)0~J= csch 2 ( a l ) Q " " x x = cot 2 p x L 4(bl) 0"] r b = esc 2[bl4(blj 0"] Fig. 6b shown below gives a clearer idea of the symbols enumerated above. Further plots of similar figures with neg- l i g i b l e attenuation are shown i n connection with Test 11 on a Lead covered Cable* with a 100 Ohm Fault and terminated In i t s Surge Impedance. tV - const I Es»2l - Coogfcmt Atentdhbri CM const 4f«K circles pass+VircooV) 0,o). ^ ^ ^ Note: If attenuation Is small, a l « 0, and Eq. (21), i.e. circ l e of constant "attenuation" depends only on ( a l ) Q which is a constant for a given line and fault. Hence the radius vector from origin to a point on the circumference of the ci r c l e represented by Eq. (21) is the sending end Impedance * 2 - No. 8 conductor, paper-insulated, lead-covered, HKv Cable 15. with i t s appropriate angle. To determine what impedance corresponds to a given frequency i t is noted that ( h l ) c i s a constant and that (hi) is a linear function of frequency, the constant depending on the type of cable under test. Know- ing this constant, any circle of constant (bl) also represents a c i r c l e of constant frequency, cutting the other c i r c l e at 2 points (usually), and giving the expected impedance at that frequency. I l l DISCUSSION OF EXPERIMENTS A. Circuit Diagram and Apparatus. 1. The general method of locating faults was des- cribed in the latter part of the introduction and the ci r c u i t diagram i s given below i n Fig. 7a s-nd 7b. The ap- paratus consists of a Signal Generator with variable Frequency up to 5 Mc, an amplifier covering the video- frequency ranges, a power-pack giving 300 volts D.C., a filament transformer with, 6.3AC volts output, a standard dropping resistor, and 2 vacuum-tube voltmeters. A Sig- nalyst or Triplet Signal Generator are satisfactory for this method. The c i r c u i t diagram of the Video-frequency Ampli- f i e r is shown i n Fig. T and the arrangement of measuring the drop across the standard resistor and the cable are shown enlarged in Fig. 10. A Cathode-ray Oscilloscope i s also required i f the waveshape is to be examined. The Cossor double beam oscilloscope enables the examination of waves up to 3 Mc. The output of the generator was amplified to about 3 volts at an intermediate frequency thus making i t possible 16. to measure accurately the drop across the cable. AO • B o o -SDK .OI 6 0 K .01 5K \<ao • o i 8" Pig. 7 Video-frequency Amplifier 2. Tests were made on 2 types of cables. Tests 1 to 10 were made on cable FT&R AN Type RG8/U and test 11 was made on a 2-conductor, oil-impregnated, lead-covered cable, loaned especially for this test by the British Columbia Electric Railway Co. Ltd. Por the purposes of comparing calculated and experimental results, attenuation was neg- lected in the mathematical expressions derived before. This is well justified as the calculations below show. The at- p tenuation was calculated by the S-function method and checked by measurements of a shorted quarterwave section. Cable: Type RG8/U, surge impedance=52 ohms, velocity=60$ app. Apparatus: Bontoon Q Meter • • Frequency of test: O.935 Mc determined by cable length of 100 f t . Cable length = l44vel/freq.in Mc 17. Fig.Ta Layout of Apparatus Sample oable Under Test See "Fig. 10 Fig.7b Schematic Diagram of Layout 18. With c o i l only, Q .- 200 C 1 = 281.5 uuf L]_ = 100 uh With c o i l and cable i n series, Q 2 = 78 C 2 = 255 uuf Co= Capacity to resonate cable inductance of 11 uh at O.935 Mc i s 1 10 12 Lw since C [ ( 2 ) ( 3 . 1 4 ) ( . 9 3 5 ) ] 2 ( 1 1 ) 1 0 6 2635 uuf 1 1*9 - = —=- = 1.11 and hence 1^ = 111 uh and cable in- C 2 L l ductance i s (111-100)'= 11 uh Coil resistance = X C 1/Q = l/C^Q^ = 3.02 ohms Coil and cable resistance in series = l/CgWQg e 8.55 ohms Cable resistance = (8.55-3.02) r 5.53 ohms Q of cable = X c 2 ~ X c l m 6 3 _ n ^ R 2-R I 5.53 ^ From Graph shown i n Fig. :8-., at f Z Q C 0 : (-935)(52)(2635)10 6 = .128, S = 275 and attenuation a = S/Ql = 275 /H40 = .24 db/ 1 0 0 f t # Ito. Fig. Q When a sample of quarter-wave section was tested with far end short-circuited, the impedance was found to be 2000 ohms 19. From Fig. (;S.sa) and (b) below Fig. :9̂ a Fig. :9b I? = 2000 and -X = . Hence k = 1/20 k i l . o Thus the reflected current wave is 1/20 less than the input current wave due to attenuation. db = 201og ±i/±2 s .44 db per 200 f t = .22 db/100 f t . Since the attenuation varies approximately as the square root of the frequency, even at 10 Mc this value is small compared to the phase-shift constant and hence the omission of attenuation is justifi e d i n our tests provided the cable is not unreasonably long. The attenuation of the lead-covered cable of the BCER Co. Ltd. is of the same order, i n fact somewhat smaller, and can therefore be neg- lected. 3. The a r t i f i c i a l fault and the characteristic Termination are shown in Fig. 10. The size of the resistors must be kept small physically and a l l connecting leads must be as short as is practically possible i n order to minimize stay capacities to ground. To prevent leakage effects, a shield is put over the entire resistor as is indicated i n Fig. 10. B. Calculations 1. Far end terminated i n characteristic termination In test l,a 1000 ohm fault was injected at a 20. F i g . \o Connection of Cable to A m p l i f i e r For accurate measurements t h i s connection may be i n - s t a l l e d i n case where i t i s s h i e l d e d . The cable i s then connected to the a m p l i f i e r by a standard connector. The leads go to the Vacuum Tube Voltmeter. F i g . \oa F i g . lob P i g • ' o a This f i g u r e shows the a r t i f i c i a l f a u l t between conductor and ground. A l s o shown Is a s h i e l d which i s normal- l y placed over the cable to a s s i m i l a t e a f a u l t i n a power cable. F i g . iob C h a r a c t e r i s t i c Termination on Cable FT & R AN Type RG8/U. The sheath i s terminated by 4 p i g t a i l s whose ends are soldered to the ground end of the r e s i s t o r . A s h i e l d f i t s over the e n t i r e t e r mination. R e s i s t o r s should not be wire-wound i f p o s s i b l e , to minimize the c a p a c i t i v e reactance a t h i g h frequencies. Small carbon r e s i s t o r s g i v e best r e s u l t s . TESTS ON CABLE FT4 R AN TYPE- RG8/U 5OJQ. Otcud diagram o— MOTE. : Oi\cu\a\ei Qvtves do not account ôf a^eooahon wliicli i s negligible at \o\tt<{tt<\oenc\e$. Characteristic- lamination 1= Z o o «• -Vefmination : 7 f u fi--i H ' r : t f f f tmt Te»r2 Tomce%2\. (2) (?e{th impend'VBIIJ. © Calculated Wt feftttiofl ® Fao^ £ Imprf-fcct-termination © Calculated Cef tetibn ' © Faoft^ Imf rtfftH +ef initiation IllilllillllllH 1 1 • 111111111 • 1111111 llllllllllllllllllllllllift Wm @ Reflation -\fom xm ' ' © Calculated faoHrPe^Ktpn mtttttmttt tttt P R I N T E D Ifsf U.S.A. 3 fPCO. IN 3 i E N E D I E T Z 6 E N C O ^ N O . 3 ^ 6 pee*?. IM MC. 2 3m « distance of 100 f t . and i t was observed that the difference between any two minimum values of Impedance i s 3.2 Mc. Since the cable velocity is approximately 60$ that of light, i.e. 5.9«10^ ft/sec, the phase-shift constant, b = w/v « (.01 f) appr., where f is the frequency i n Mc. Hence, corresponding to half wavelength, bl = ( .01)(3.2) 1 and distance to fault 1 = 3«l4 _ g Q f t ( c f 1 0 Q f t ) ' (.01) (3.2) " _ _ « _ To estimate the size of the fault, which can also be done by other methods,' i t is found that peak of curve occurs when Zs/Z - 1.045 = R at maximum j hence R = 52/ .045 • = 1158 ohms (cf 1000 ohms) In test 2, the peak to peak separation s t i l l cor- responds to half wavelength and equals 3.15 Mc and hence the distance to the fault, 1 = 3.14 _ 99.8 f t . (cf 100 ft) (.01) (3.15) Also maximum observed value of Zs/Z 0 is 1.17 and hence si'ze of fault is 52 / .17 = 306 ohms (cf 340 ohms.) In test 3, peak to peak i s 3.20 Mc, giving distance to fault as 3*14 = 98 f t (cf 100 f t ) (.01)(3.20) Also size of fault is 52/ .49 = 106 ohms (cf 100 ohms) since maximum value of Z3/z0 is 1.49 In tests where faulted line is terminated i n surge impedance i t is often d i f f i c u l t to estimate the correct value of Z Q. In tests 1, 2 and 3,- a value of resistance O RINTCDI(J.U 2 3 4 O | 2 J E U G E N E D I E T Z G E N C * N O . 3 4 6 B X Fff&p. IN MC FREM <N MC- 22. close to Z q i s put on the far end of a good conductor which is often available i n the same sheath and a curve of frequency against Z s/ Z i s run. These are shown as black dotted curves on the graphs. Then a similar curve is run with the same resistor on the far end of the faulted cable. These two curves are then subtracted and the result Is due to the fault reflection only, provided the mismatch at the far end is not too great. It is readily seen that faults twenty times the surge impedance can be detected by this method. The reason why this i s possible is due to the fact that the observed reflection i s from the fault alone and not, as i n the following tests, due to a combination of reflections from the fault as well as the end of the cable. In the above cal- culations, use was made of Eq. 14 whose maximum value i s R/R+Zj. 2. Far end open-circuited In test 4 i t is to be noted that as frequency ap- proaches zero, Z s /2 approaches l/k and when the second minimum i s reached at 1.57 Mc, Z g^ z i n Eq. 8 becomes 2-fjk tan bl-j^ ) -zs k+j (tan bl-j-1/tan b l 1 ) ( Z ° From graph in test 4, 4+k 2tan 2bl- L /.36 = k2»(tan b l 1 - l / t a n b l 1 ) 2 • Thus tan b l n r (12.86 + V164' „ 5 5 1 i\ .61 Thus bl'2 1.42 radians, and since second minimum occurs at 3/4 wavelength i.e. 3 T / 2 , b l 2 = ( 3 T / 2 - M^) = 3.29 radians. k - .5 Z s / Z 0 = .60 f = 1.57 Mc.) 23 Thus 1± - 1.42/(.01)(1*57) « 90.5 f t ( c f 100 f t ) 1 2 = 3.29/(.0l)(l.57) - 209 f t ( c f 200 f t ) since h = .0l(f)/Mc/ f o r t h i s c a b l e . Size of f a u l t i s determined from k * Z Q/ R; thus R r (52) (2) = 104 ohmsCpf 100 ohms) The f i r s t minimum of the graph could have been used as well had the second harmonic signal not been so large. In case of the open-circuited end, i t i s preferable to use the f i r s t maximum or the second minimum point of the curve but not the f i r s t minimum, since the harmonic distortion at that point is most pronounced. Attenuation of higher harmonics at the increased frequency of the second minimum increases the ac- curacy of results. 3» Far end short-circuited k = .5 from 0C Test Z s / Z c " 2.55 f = .53 Mc" In test 5, when the f i r s t maximum is rea,ched at .53 Mc, Eq. (12) becomes: f s _ s ; Zo 1 / 2 x (1 + cot b l 2 ) *• j cot b l 2 k From experimental data above and from above equation i n ( c o t 2 b l 2 ) , we obtain, cot^bLg + 2.25 c o t 2 b l 2 - .630 = 0 from which we get, cot b l 2 = ,502 and big .« 1.103 radians Wow at the f i r s t maximum (bl +bl^ ) = ~^/2 since the entire length of the cable i s a quarter-wave length at this fre- quency. Thus, FREC? IN M C 24- b l 2 = 1.103 radians = .47 " Hence, since.b = .01 f /Mc/, 1-L - .47/ 001) (.53) = 89 f t ( c f 100 ft) ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ * and 1 2 = 1.103/(.01)(.53)= 208 f t ( c f 200 ft) Size of Fault Is estimated i n the same manner as for the open ci r c u i t test. In this case "k" may also be obtained from a reading of the fault resistance and characteristic impedance. For this test i t is best to use the f i r s t maxi- mum or f i r s t minimum / not counting zero frequency/. The undesirable second harmonic effect does not seem to be as pronounced since i t does not occur at low-impedance peaks. The explanation for this is given i n a later paragraph. 4. Double Faults In test 8 calculated and obtained curves are drawn for a 100 ohm fault 200 f t from sending end and a 50 ohm fault 275 f t from sending end. The true reflection from these two faults Is again obtained by putting the surge impedance / resistance i n this case/ at the far end of the cable and subtracting any small discrepancies due to mis- match. For this, a good conductor is required i n the same sheath. Curve 1 i s a graph of a special case of Eq. 17: £s _ Zo ~ (.5- tan 2f tan .75 f) + 3 (tan.75f+1.25 tan 2f) (1.25-tan 2f tan.75f) «* j (tan.75f*.5 tan 2f) An expression of this type i s not easy to plot but fortunately such faults are not met In power cables. 26, The experimental curve is rather d i f f i c u l t to analyze without having a f a i r l y definite idea where the faults are beforehand. The curve is plotted merely as an i l l u s t r a t i o n of the workability of the method for any number of faults. It should be noted that above 3 Mc., the peaks of the observed curves are diminished due to attenuation which has not been taken into account i n the calculated curves. If the variation of attenuation with frequency and the decreased gain of the amplifier at these frequencies had been ac- counted for, the peaks of the calculated curve would have diminished as well. It i s well to keep i n mind that at this stage the accuracy of the method is impaired due to stray capacities. 5» Effect of long lead-in cable In practise, a cable pothead may be located on a pole; i n this case the leads to the cable under test cannot be made short unless the test apparatus Is brought up on top of the pole. This is clearly impractical and i t Is neces- sary to develop an expression such as Eqs. (18) and (19) which take into account a fixed length of input cable. A low-loss cable with negligible attenuation is obviously the most advantageous. If a cable whose characteristic imped- ance Is approximately the same as that of the cable under test is used, one need only subtract the length of the Input cable from the total length to the fault to get the actual length to the fault. This is the situation which un- intentionally arose i n Test 10, i n which the values of the e&r\o Zvrzcr Oris F t o F lapo-rC/vetE Rse/u ON I w g p f l h f c g OF^OLTEP C a b l e - > CmcotT' \ \QOJl FAUrr : 2^56^2. 2 * ' 184.5' \ IK. m © Calcolated Exult* Iwput Cable (3) ^efkctvon^om fault, W^-f^t'l^mma-br^ -̂\«̂ uTCflbW Impcf-fectTermmtrtt'on ^ In^ut Cable (No^oH) P R I N T E D f t t U . S . A . E U G E N E D I E T Z S E N C O . N O . 3 4 6 B X 26, characteristic impedances are close together. If this vere not the case, i t would he necessary to investigate further the expressions In Eq. 18 and 19. It is noted from the calculated curve in test 10 that as long as the characteris- ti c impedances are within 10$ of one another, we may assume the cable to be merely lengthened. From curve (2) i n test 10, the observed difference between adjacent peaks Is 1.32 Mc. The approximate distance to the fault i s hence ( 1 ' ? 7 ) ( 2 ) . 216.5 f t . (.oil)(1.32) since for the lead-covered 2-conductor cable, b « (.011) f /Mc/ Thus actual distance to fault i s (216.5 - 25) = 191.5 f t (cf 184.5 ft.) Again, i f one wishes to investigate the size of the fault, the f i r s t maximum of the curve i s desirable, since at- tenuation errors are least there; thus, 1.53 =(R+Z)^R and R (56/.53) = 106 ohms (cf 100 Ohms) 6. Alternate method with characteristic termination at far end. A close study of expressions developed i n Eq. (21) and (22) open up a different approach to this problem by incor- porating the effect of a sending end impedance Into the propagation constant as a fixed additional value. Circles of constant frequency 'can then be drawn and checked against the frequencies obtained for various input impedances. Two alternate tests are proposed here. In both, one f i r s t draws the circle of zero attenuation / true for power cables / which gives the input Impedance with i t s angle at a l l frequencies. This c i r c l e can be drawn by knowing the characteristic impedance of cable and the fault resistance, since the minimum and maximum input impedances are Ry{R+Z0\ andfR+Z^/R respectively. If apparatus measuring the re- actance component of the impedance Is available, one method would be to determine the point of maximum reactance and recording the corresponding frequencies. This i s shown i n Test (11) by circles (2-2) and (2 ,-2 t). Prom this the dis- tance to the fault may be determined. Another pos s i b i l i t y would be to use a dropping resistor " r " i n Fig. 7b whose value Is ZQ for the cable. In this case one records the frequencies when the voltmeters V,, and V. read the same values i.e. when Z s = Z 0. Circle (.1-1.) corresponds to these values of the input impedance and by knowing at what fre- quency the meters read the same values, the distance to the fault may be determined. This method is only of value when accurate readings of reactance are to be taken. Preliminary tests indicated the p l a u s i b i l i t y of this method but not very accurate results are obtainable without additional refine- ments. This method is unfortunately rather complex when applied to shorted and open lines. In these cases one ob- tains spirals instead of circles and their analysis is too complicated to be of practical value. For the lead-covered cable, with c i r c u i t as shown i n test 11, ^ + = t a ^ | " j b l / ( a l ) o ] Z tanh ( j . O l l ' f l «- .76) with negligible attenuation and 100 ohm fault at a distance o f " l " f t from sending end. These values automatically check with the circl e of constant attenuation. The observed point of maximum reactance was about 1.12 mc. The cir c l e (2-2) , passing through this point and the point (1,0) having i t s centre on the x„ axis Is thus uniquely determined. Hence, from Eq. (21) and (22) - -cot2[(bl+(bl) 0)]= -.48 When dealing with resistances only, ( c l ) 0 is real and hence b l 0 is zero. Thus cot 2b l = .48 bl - .56 radians or (T /2+.56) rad. - 2.13 rad. The latter value is chosen since c i r c l e (2-2) crosses the point of maximum reactance for the second time /capacitive/ i.e. half wavelength from f i r s t point of maximum reactance. For this cable, b " (.011) f Hence distance to the fault is 1 = — 2 . 1 g _ 1 7 3 f t ( c f l 8 2 t f t ) (.011) (1.12) Alternately, when a dropping resistor was used whose value was the characteristic impedance of the cable / 5 6 Ohms/ and frequencies were recorded consecutively at which the two drops across the resistor and across the cable were equal /Z s = ZQ/, the values should correspond to ci r c l e ( l - l ) From experiment, these frequencies are: ?, 1.11, I .87, 2.65,....Mc. For a l l these values cot 2b l = 0 since the centre of the cir c l e ( l - l ) l i e s on the origin. Thus bl = A , 5*/*, 7" A , .... corresponding to fre- quencies f = ? , 1.11, 1.87, 2.65,efe Hence distance to the fault i s : 22C = 195 f t (cf 184.5 f t ) 4(.Oil)(1.11) ' SI s 191 f t ( C f 184.5 f t ) 4(.011)(1.&7) — — — Z_ _ 188 f t (cf 184.5 f t ) 4(.Oil) (2.65) • • — - — . The size of the fault, which i s relatively unimportant, can be found by similar means as i n previous tests. The accuracy of these results is well within 10$ and can be relied on for similar tests. If desired, any particular point on the zero attenuation cir c l e can be checked by drawing circles such as (3-3) for a given frequency and the actual input impedance at this frequency should equal the radius vector from the origin to the point where the circ l e (3-3) cuts the zero attenuation c i r c l e . This was also checked In test (11). ft N T E D i p L . 2 g 4 0 I JL J U 6 E N E D I E T Z 6 E N C4 N ( Fee~> In MC . Ffc6S>. IN We 3o C. D i s c u s s i o n of R e s u l t s 1. E f f e c t of Harmonics The output v o l t a g e of the s i g n a l generator i s "by no means s i n u s o i d a l and contains second harmonic v o l t a g e to a consi d e r a b l e degree.With t h i s i n mind, i t i s d e s i r a b l e to i n v e s t i g a t e which of the discussed t e s t s i s l e a s t c r i t i c a l to t h i s predominant second harmonic. F i g . 11 a , l i b , a n d 11c show the e f f e c t of t h i s second harmonic on the input imped- ance as the frequency i s v a r i e d . Par e n d :open-circuited S h o r t - c i r c u i t e d Terminated i n Z Q I n Fig.11a,maximum impedance to second harmonic occurs at the p o i n t where the impedance to the fundamental i s a m i n i - mum; hence at t h i s frequency a bump i s observed.This bump i s c l e a r l y seen i n t e s t (6) and i n t e s t (4j . I t i s e s p e c i a l l y prominent at the f i r s t quarter-wavelenght because the a t t e - n u a t i o n i s n e g l i g i b l e at t h i s low frequency.The bump at the next minimum i s almost n e g l i g i b l e compared to the f i r s t bump. In F i g . l i b the s i t u a t i o n i s more f o r t u n a t e because,for the t e s t w i t h the f a r end s h o r t - c i r c u i t e d ; t h e minimum impedan- ce to the fundamental does not occur at a p o i n t of maximum im- pedance to the second harmonic.By adding the impedances due to the f i r s t and second harmonic i t i s noted t h a t the bumps occur h a l f way between f r e q u e n c i e s of maximum and minimum input impedance.Thus i n t h i s t e s t , t h e percentage of second harmonic i s not so c r i t i c a l . This bump can be e a s i l y obser- ved i n test ( 5 ) a n d [ l \ i n d i c a t i n g a l a r g e second harmonic content. 31. Pig. l i e shows the effect of slight mismatch at termination of cable. If the terminating resistor i s too high, the deviation from perfect termination /curve 1/ has the properties of a small "open ci r c u i t " i.e. maximum impedance occurs at same points as for open c i r c u i t . Alternately, i f the terminating resistor is too small, the curve of impedance tends towards that of short-circuited far end. This follows from the fact that a small resistance approaches zero /short-circuit/ in the l i m i t . In this case, the effect of the second harmonic is almost negligible at a l l times since the variations of impedance are small at a l l frequencies. This leaves the curve of input impedance versus frequency almost sinusoidal. It is concluded that i f the signal contains considerable distortion, this test is by far the most advantageous from a l l points of view. The elimination of harmonics Is discussed i n a later section. 2. Additional errors Aside from errors due to distorted waveshape, the simplicity of the method does not lend i t s e l f to many errors except at high frequencies when stray capacities between various parts of the c i r c u i t and ground reach sizable values. To prevent the large capacities, short leads are used throughout the c i r c u i t i n addition to carbon, not wire-wound resistors. It is thus important to keep the value of the dropping resistor low, in order to minimize the effect of the capacitive reactance across i t s terminals. Another source of error is 60-cycle hum .coming from 32. the amplifier. The small amount which came through was easily removed by a high-pass f i l t e r c i r c u i t shown i n Pig. lld. Vr ( -dj—- ) Amplifier Calk uooerTisi Pig. 11 d. -pass High f i l t e r c i r c u i t to eliminate 60-cycles This f i l t e r c i r c u i t is not required for tests with far end short-circuited or correctly terminated since i n those cases the cable i t s e l f provides an effective short for the 60- cycle. An effort was also made to eliminate the higher harmonics from the generator. By beating the signal gener- ator against the carrier of a radio station i t was found • that harmonics as high as the 5th were detected, the second being by far the worse. When a tuned c i r c u i t i s interposed between the signal generator and the amplifier, higher harmonics are eliminated because the impedance of the tuned ci r c u i t is high when not at resonance. Thus the impedance to the fundamental is much lower than the Impedance to a l l harmonics. When the tuned c i r c u i t shown i n Fig. 12 was in- cluded i n the circuit, a considerable difference was obtained between the impedances. To be of practical value this c i r c u i t would have to be designed to remain at resonance as the frequency setting on the signal generator. 33. is varied. •o o o- Pig. 12 Tuned ci r c u i t to eliminate harmonics Some improvement In waveshapewas also obtained by ad- justing the bias i n the oscillator so that the output would be decreased but more sinusoidal. With short leads and carbon resistors, the only con- siderable error remaining is due to capacity of the volt- meter to ground. This probably accounts for the larger discrepancies i n the calculations. It i s also important to connect the voltmeters in such a fashion that they both measure the same part of the wave. Por a sine wave, both positive swings and negative swings are equal but for a distorted wave a good measure of impedance is s t i l l obtained i f both meters measure the same part of the wave. 3. Accuracy, limitations and conclusions In a l l the tests carried out, an accuracy of better than 10$ was obtained i n estimating the distance to the fault. An accuracy of better than 5$ was obtained for tests with far end terminated In surge impedance, indicating that this is the best method to be used. Faults as high as twenty times the characteristic impedance are easily detected by this method. With the present amplifier and size of resistors, the upper li m i t - of frequency i s about 5 Mc hence tests for lines shorter than 25 f t . do not yield ac- curate results. It is to be noted here that improved accuracy is obtained when faults farther away are being tested. The tests with long "instrument leads" gave results well within experimental errors and their appli- cation to measurements where cable potheads are located on poles i s very useful. The procedure for tests with far end correctly terminated is as follows: a. Set up ci r c u i t shown i n Pig. 7b b. Connect Z Q across far end of cable as in Fig. 10b. c. Record readings of V r and V c as frequency i s varied for faulted cable. d. Record readings of V r and V c as frequency is varied for a good cable. e. Plot curves of cable input impedance against frequency. f. Subtract curve for good cable from curve for faulted cable, getting net result due to fault alone. g. Difference i n frequency between adjacent peaks of resultant curve represents a wavelenght. 1. From propagation constant of cable, find distance to fault. Repeated tests on same cable yield the same result in- dicating the r e l i a b i l i t y of the method. The range of frequency used with the present type of equipment is 100 Kc to 4 Mc approximately. 35 Bibliography 1. Blankmayer,W.H. Power lin e Fault Locator, p.166 Electronics.Vol.17 No.l January 1944. 2. Chandler, Steward,Jr. The"S"-function Method of Measuring Attenuation of Coaxial R.f. Cable, pp.616-19*Electrical Engineering.Yol.64 Ho.9 September 1945. 3. Golding, E.W. E l e c t r i c a l Measurements and Measuring Instruments, Pitman & Sons,ltd., 1944. 4. Honnel, M.A. Location of Line Faults. Electronics. Vol.17 No.11 November 1944. 5. Noakes.F. High Frequency Method of Location of Faults on Power Cables, pp.46-7 Elect r i c a l News.Vol.53 No.13 July 1 1944. 1 " 6. Slater, J. C. Microwave Transmission. McG-raw-Hill Book Co.,Inc. 1942. 7. Vahey,J. A. Eecent Developments i n Cable Fault Locating. p.96. Edison E l e c t r i c Institute Bulletin. Vol.7 March 1939. 8. Woodruff, L. F. Principles of E l e c t r i c Power Transmis- sion. John Wiley & Sons,Inc. p.97. 1938. 36 ACKN OWLEDG-EMENT The author wishes to express his appreciation to Dr. P. Noakes i n suggesting the topic and for his helpful assistance during the course of the research. An Abstract of the High Frequency Method of Locating Power Cable Faults By Ervin Joseph Nalos A Thesis submitted i n Partial Fulfilment of the Requirements ments for the Degree of Master of Applied Science i n the Depart0ment of E l e c t r i c a l Engineering The University of Bri t i s h Culumbia August 1947 The Location of a fault in a power cable is by no means a simple task and the techniques used at present war- rant further improvement.Many methods of fault location have been devised but each has i t s particular limitations.Some of these methods include D.C. and A.C. bridge methods,echo-ran- ging methods,and high-frequency methods.The limitations and impracticability of these various techniques has been the main reason for this research. In this thesis,a review of the high-frequency method has been made,resulting in the development of a method of lo- cating high-resistance faults.Expressions,permiting the use of this improved method i n instances where cable potheads are re- latively inaccessible have also been developed and checked ex- perimentally .Brief ly, the method consists of determining the input impedance of the cable with i t s remote end terminated in i t s surge impedance.This is done by simultaneously measuring the voltage drops across a standard resistor and across the cable.The ratio of these drops Is an indication of the cable impedance at that frequency.Obao'vations are taken on a band of frequencies on a faulted cable and on a good cable,both ter- minated in the characteristic impedance.The difference of these two effects is attributed to the reflections from the fault. From the plot of the impedance as a function of the frequency, by a short graphical computation,the distance to the fault may be obtained.Faults as high as twenty times the surge impedance have been successfully located on relatively short lenghts of cable.The distance to the fault has been estimated well within
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