THE FAULT LOCATION ALGORITHM BASED ON TWO CIRCUIT FUNCTIONS

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1 U THE FAULT LOCATION ALGORITHM BASED ON TWO CIRCUIT FUNCTIONS Z. Czaa Char of Electronc Measurement, Faculty of Electroncs, Telecommuncatons an Informatcs, Techncal Unversty of Gañsk, Polan The paper presents an algorthm for etecton an locaton sngle parametrc faults n analogue electronc crcuts base on makng use of nput-output measurements. It utlses two crcut functons (for nstance: voltage transmttance K U an nput opene amttance Y n measure on the same frequency for the fault locaton. Each of crcut functons s regare as blnear transformaton. When we put together these transformatons nto a three-mensonal space (for nstance: Re(K U, Im(K U, Y n t s wreste a famly of curves representng the changes of respectve elements' values. Ths composton causes ncreasng selectvty of the fault locaton, because the curves o not cut themselves an furthermore they are more separate, than t has place n classcal applcatons of blnear transformaton. The algorthm conssts of two parts. Frst part s ame at etermnaton of the optmum measurng frequency for a smultaneous measurement of two crcut functons. Secon part acheves the fault locaton. It was chosen 3-orer low-pass Butterwoth Flter to verfy the algorthm. Keywors: Blnear transformaton, Detecton sngle parametrc faults INTRODUCTION For many electronc crcuts as well as electrc moels representng techncal, physcochemcal an bologcal obects, there s no possblty of access to the nteror of crcut. Therefore agnostc base on nput-output measurements has growng mportance. One of the methos for ths agnostc s blnear transformaton [][]. It s use for locaton of the parametrc faults n not very complex two -, an four termnal networks RC. Geometrc nterpretaton of blnear transformaton F(p, where p- element's value, s complex plane, on whch we raw up famly of arcs representng changes of nvual elements' values (Fg.. The locaton of the sngle fault conssts n measurng of functon F(p an plottng the measurement pont on plane []. If the measurement pont les of one of these arcs, than the fault element s the one, for whch curve has been mae [][3]. From Fg. showng the famly of arcs rawn up for 3-orer low-pass Butterwoth Flter (Fg. we see, that the curves are near each other, an n some places they are crss-crosse, makng mpossble n ths way unambguous R C R R3 C C3 + _ U locaton of the sngle fault. Atonally, locaton of the fault element wll be more ffcult, because measurements are burene wth errors, whch make, that the measurement result oes not necessary nee to le on the curve representng the fault element (Fg.. Fgure. 3-Orer Low-Pass Butterwoth Flter uner nvestgaton (DUT, where: R=R=R3=0kΩ, C=4.3nF, C=3nF, C3=0nF. In ths artcle we propose an algorthm that solves scusse problem. In our conseratons we assume followng presumptons: - Dagnose crcut s lnear. - There can be only one parametrc fault n crcut. - We know crcut typology an nomnal values of all elements.

2 - We make test at only one measurement frequency. - There s a measurement error of crcut functons parameters. The works that have been mae so far, analyse problem on complex plane, usng for t measurement of one crcut functon [][][3][4]. In ths artcle we propose smultaneous measurement of two crcut functons: F(p - voltage transmttance an G(p - nput amttance. Usng both crcut functons, we take moellng of changes of elements' values from plane to three-mensonal space. Pcture 3 shows n 3D space the famly of threemensonal curves representng changes of elements' values of flters presente on pcture. As we can see on pcture, we result wth lack of crss-crossng of curves representng changes of elements' values an ncrease of stance between them. It has mae the fault locaton more unambguous (selectve what wll be presente n chapter 3. Analyse was mae wth program Matlab. Fgure. The graphc representaton of the bllnear transformaton F(p - the arc famles. Fgure 3. The graphc representaton of the composton of two crcut functons F(p an G(p - the curve famles.

3 . AN ALGORITHM FOR DETECTION OF THE SINGLE FAULTS Locaton of the sngle parametrc fault s base on the followng thess: The fault element s the one for whch three-mensonal curve s the nearest to the measurement pont. The algorthm s compose wth two parts. The task for frst pre-testng component s to etermne optmal frequency wth whch there are the best crcumstances for smultaneous measurement of two crcut functons an etecton of the sngle faults. After choosng proper frequency, there s atabase generate contanng value of optmal measurement frequency an co-ornates of the nomnal pont. Secon component of the algorthm localses the fault element by choosng curve n space, whch s the nearest (accorng to etermne crtera to the measurement pont. In three-mensonal space each curve can be escrbe by parametrc equaton. For analyse case form of mensonal curve corresponng wth changes of value of -th element s gven analytcally n the followng way: x y z = Re( F ( p = G ( p ( where: =,...N, N - number of elements n crcut (For DUT N=6, F- the transmttance voltage, G - the nput amttance of crcut Both algorthms pre-testng an testng use propose escrpton of curve.. Determnng of optmal measurement frequency The man pre-teste problem s etermnng optmal measurement frequency wth whch take place suffcent senstvty of both crcut functons n epenence on all elements of teste crcut. For etermnng of optmal measurement frequency we assume crtera escrbe n [4]. Because t has concerne only case on plane, we have ntrouce changes connecte wth transton to threemensonal space. The algorthm s followng. On the begnnng we set startng frequency, frequency step f step an number of steps S. Next, for each frequency f=f step s, where s=,...,s we make the followng actons: - Calculaton of co-ornates of the nomnal pont (x nom,y nom,z nom, where all curves (escrbe by equaton ( representng changes of each elements' values crss-cross. - Calculaton of co-ornates of the fnals of these curves (x,y,z (x,y,z, where =,..,N, - Determnng of a stance between the nomnal pont an each of the fnals of curves: - stance from the frst, +N - stance of the secon fnal of three-mensonal curve from the nomnal pont = ( ( x xnom + ( y ynom + ( z z nom = N ( x xnom ( y ynom ( z z nom (3 - Determnng of evaton coeffcent length between curves á(s [4]. Next, we etermne á mn as mnmal value among calculate set of á: mn{ ( s} α = α. On ths bases we can efne optmal measurement frequency for whch wll be mae measurements of crcut functons, whle testng of the crcut. For teste crcut t amounts to f opt =700Hz. Locaton of the sngle fault n the crcut The am of algorthm s locaton of the sngle faults n the crcut. An algorthm s lookng for curve, whch s the nearest to the measurement pont plotte n three-mensonal space. An algorthm s compose of the followng steps:. From the pre-testng algorthm we take the optmal measurement frequency f opt (for the teste crcut f opt =700Hz an co-ornates of the nomnal pont (x nom,y nom,z nom. We etermne set of ponts, represente by three vectors of co-ornates x, y, z for -th curve. mn s

4 3. We calculate mnmal stance of the measurement pont P m wth co-ornates (x m,y m,z m from -th curve calle a coeffcent of nearness â(. 4. Steps an 4 are mae for N curves. 5. From the set of calculate coeffcents of nearness {â(}, =,..,N we etermne mnmal value â mn On the bases of t we etermne whch curve s the nearest to the measurement pont, an n ths way, whch element s fault. The key element n ths algorthm s step 3. From the earler calculatons each curve s represente by three vectors of co-ornates x, y, z. We can also escrbe each of them by equaton (. It s theoretcally possble to etermne each equaton escrbng epenence on change of each coornate gven curve, by blnear transformaton. But the forms of result functons are very complex (the level of complexty rse wth amount of elements, an beses ths, each functon we shoul etermne analytcally for each element separately. There are 8 equatons for teste crcut. Secon, very mportant savantage of ths approach s lack of unversalty. The etermne set of functon fts only to one crcut typology, therefore propose algorthm woul be not unversal. We ece to use algorthm searchng for mnmal stance of the measurement pont from gven curve - escrbe below. The ponts representng the curve are place wth uneven ensty. Therefore for ncreasng precson of etermnng coeffcent of nearness â, we ece to use parabolc nterpolaton, mae for three of curve's ponts whch are the nearest to the measurement pont, an next we etermne mnmal stance between curve (4 an the measurement pont:. We search for a pont P (x,y,z from the set of co-ornates x, y, z representng gven curve, the pont, whch s the nearest to the measurement pont P p an two ponts P k (x k,y k,z k P l (x l,y l,z l surrounng pont P, where pont P k s nearer to the measurement pont P p than pont P l, where k,l=±.. We make parabolc nterpolaton for range (P, P k on the bases of ponts P, P k, P l an we result wth the followng escrpton of curve [5]: x = x y = Ax + Bx + C z = Dx + Ex + F (4 Ths approxmaton s suffcent, because from the property of testng algorthm follows, that for locaton of faults t s enough, that the measurement pont s nearer to gven curve than other curves, an moreover, the shape of curves s smlar to parabolc (Fg Next, we etermne absolute error Ä that woul be mae by mpeance meter HP 49A at measurement of the value corresponng wth pont P (n verfcaton of ths metho use of the meter HP 49A s planne. 4. On the bases of Ä we etermne amount of ponts approxmate curve (4, lyng between P a P k, accorng to formula: m = ( x x k + ( y y k + ( z z k M (5 where: M=0 coeffcent of ecreasng approxmaton error n relaton to the measurement error. 5. At the last stage on the bases of (4 we etermne co-ornates of ponts lyng between P a P k an stances between these ponts an the measurement pont. Next, from the set of calculate stances we fn mnmal value, whch s coeffcent of nearness â for gven curve. 3 AN INFLUENCE OF ADDITIONAL DIMENSION ON QUALITY OF THE SINGLE FAULTS LOCATION By connecton of two blnear transformaton F(p an G(p n one, represente by threemensonal space (Fg. 3 we get ncrease stance between curves representng changes of nvual elements' values (curves are more separate each from other. In ths way we acheve fact, that the locaton of sngle faults n crcut s more unambguous. Atonally, we elmnate possble crss-crossng of curves (Fg., what make unambguous etermnng of the fault element mpossble.

5 Here we ntrouce the proof that by escrbe metho we acheve fact, that locaton of the sngle faults s more unambguous than n methos base on blnear transformaton [][3] analysng only on plane.. Fact, that locaton of the sngle faults s more unambguous s connecte wth ncrease stance between curves (between any ponts on any two curves.. Let be gven two blnear functons: F A p + B ( p = (6a C p + D G E p + F ( p = (6b H p + K where: A, B, C, D, E, F, H, K complex coeffcents for functons mae for -th element, where =,..,N, N number of elements n crcut. In the same way we can escrbe functon for -th element, where, =,..,N. 3. Because each from complex coeffcents for functons rawn up for -th an -th element has the followng form: that A A, B B etc. From that, we get functon n form: an A = A ( p,.., p A = A ( p,.., p F (p, p +, p,.., p + F (p N,.., p N where: (7 (8 G (p G (p (9 4. We can wrte parametrc equatons for curves on plane, representng changes of value of -th element an -th element n the followng way: [4] x y = Re( F ( p (0a x y = Re( F ( p (0b From ths, we can efne square of stance between any pont lyng on curve rawn up for -th element, an any pont lyng on curve rawn up for -th element: l = ( x x + ( y y ( 5. In 3D space we get followng parametrc equaton of curves representng -th an -th element: x y z = Re( F ( p = G ( p (a x y z = Re( F ( p = Im( F ( p = G ( p (b From ths we can wrte square of stance between two any ponts lyng on nvual curves representng -th an -th element n ths way: = ( x x + ( y y + ( z z (3 6. Usng formula: ( an (3 we can wrte n form: = l + ( z z (4

6 Because from the formula (9 an ( we can wrte, that An n ths way from (4 an (5 follows, that: (z z 0 z z so from ths: (5 l (6 So stance between the same ponts n three-mensonal space s always larger than ther stance n two-mensonal space. These stances are equal only for the nomnal pont, what means lack of faults n crcut, an n ths case amounts to 0. 4 CONCLUSSION Presente algorthm for locaton of the sngle faults n teste crcut DUT, base on smultaneous measurement of two crcut functons solves the problem of locaton of faults n case, when the measurement pont oes not le on any of curves. It s characterse by smplcty of work an n ths way spee of work an s easly mplemente n programmng envronment. Moreover, ntroucton of 3D menson makes locaton of sngle faults n crcut more unambguous (even a few ozen tmes (Fg. 4. It s the next step [4] for mplementng presente metho n practce, because all measurements are buren wth errors an the elements n teste crcuts have tolerance. Of course escrbe algorthm wll be evelope further, because t oes not stll concern all possble cases that can occur whle testng crcuts, for example: when all elements are n lmts of tolerance. Introuce algorthm wll be base for methos an algorthms for etecton of the sngle faults wth measurements at one frequency. The am of these methos s to be mplemente n algorthms for multfrequency measurements makng possble etecton any amount of faults n teste crcuts. Fgure 4. Dstance between ponts lyng on curves R an R as well as R an C3, an equally stant from the nomnal pont, n functon of stance from the nomnal pont R. Where: - stance n three-mensonal space, I - stance n plane REFERENCES [] MARTENS. G. Fault entyfcaton n electronc crcut wth the a of blnear transformaton, IEEE Trans. On Relablty, No, May 97, pp [] CATELANI M., FEDI G.,A fully automate measurent system for fault agnoss of analog electronc crcuts, XIV IMEKO Worl Congress, Tampere, Fnlan, June , Topc 0,CD- ROM [3] CZAJA Z., ROBOTYCKI A.,Dagnoss of lnear electronc systems usng neural network an blnear transformaton, IMEKO TC-4, Naples, Italy, September , Vol., pp [4] CZAJA Z. The parametrc fault etecton algorthm n electronc crcuts base on blnear transformaton, IMEKO TC-0, Wroc³aw, -4 September 999, pp [5] STARK M. Geometra analtyczna z wstepem o geometr welowymarowe, PWN Warszawa 974 AUTHOR: Char of Electronc Measurement, Faculty of Electroncs Telecommuncatons an Informatcs, Techncal Unversty of Gansk, ul. G. Narutowcza /, Gansk, Polan, Phone: Int , Fax Int , E-mal: zbczaa@pg.ga.pl

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