# Memorandum on Impulse Winding Tester

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1 Memorandum on Impulse Winding Teser. Esimaion of Inducance by Impulse Response When he volage response is observed afer connecing an elecric charge sored up in he capaciy C o he coil L (including he inside resisance R ), he ransfer characerisic of he circui can be obained by he following formula. V = I 0 sc R + sl 0 V I From he above formula, he volage aenuaion will become as follows. V V + scr + s LC = + = scr + s LC scr + s LC The above formula is well known because i shows he frequency characerisics of oupu volage V agains he inpu volage V when i is subsiued for s. Here, leave s o be a complex number, and observe i on he complex plane. The value for s, which makes he numeraor of he formula 0 (zero), is generally called he Zero Poin, and he value for s, which makes he denominaor 0 (zero), is called he Pole on he complex plane. When he aenuaion is 0 (zero), infinie gain can be obained in he ransmission circui. However, under no inpu volage, he Zero Poin, which is placed on he complex plane, shows a peculiar waveform, and his is he impulse response. When he circui sars free oscillaion, how i behaves can be known by examining s a 0 (zero) of he numeraor. Also, he oscillaion waveform f() for he single frequency such as he resonance circui, ec will be indicaed by he following formula. f s ( σ + iω ) σ () = e = e = e (cosω + isinω) In his formula, becomes he aenuaion waveform in negaive, and indicaes he angular frequency. Now, le us esimae he inducance in he impulse response by he above wo formulas.

2 Firs of all, in he numeraor of he formula () below; + scr + s LC = 0 The roo of s squared will be found by he Formula of a Roo as follows. CR ± ( CR) LC 4LC Wheher he circui generaes oscillaion or no is deermined by he sign inside he roo, and i is called Criical Damping when he value inside he roo is 0 (zero). The value inside he roo should be negaive o generae oscillaion in he impulse response of he eser. And, he roo should have a muliple roo (complex numbers). These soluions of he roo can be divided ino Re (a real number) and Im (an imaginary number). Re : Im : ± CR LC R 4L = R L = LC LC R L i The above Re and Im are corresponding o and i respecively. Re indicaes aenuaion while Im indicaes oscillaion. By puing R=0, he aenuaion will become 0 (zero), and he imaginary number will be (/LC). The following formula will be derived. ω = LC = πf In order o find he inducance from he oscillaion waveform, he value of L should be found by applying he formula (4), afer seeking from he observed waveform by using he formula (3). However, because he value of R is unknown in he formula (4), he answer canno be found ou. Therefore, he value of R should be found by he aenuaion. The aenuaion curve is he inclusion lines of he cosine waveform. Accordingly, R can be obained by he following formula when he raio of he ime and size beween one end and he oher for he cosine waveform is known. R L A e =

3 A in he above formula indicaes he aenuaion value in he ime here. Transforming his formula, he following will be available. R L = ln A By subsiuing he above formula ino he formula (4), he value of L can be obained. This is explained by he observed waveform below. A in he formula (5) shows he raio of he ime and size beween hese wo poins (ime=0 & ime=) can be obained by he value of he ime axis The holding line of he aenuaion waveform Fig. (Impulse waveform when he R ingredien is fixed.) 3

4 -. Observaion of Real Waveforms The impulse waveform will appear as a coninual damped oscillaion consiss of wo kinds of frequency. The firs oscillaion of a half-lengh is mainly generaed by he inernal resonance capacior while he res of he waveform is by he parasiic capacior. As a resul, here is a suble difference in shape beween he acual waveform and he heoreical waveform. The Figure below shows he acual waveform. The inclusion line of heoreical waveform canno be exacly superimposed over he acual decay waveform even if he sag elemens are deduced. The reason why he above difference is made is ha he coil s resisive componen has frequency characerisics and he aenuaion becomes bigger in he high frequency band. Fig (Acual Impulse Waveform) The parasiic capacior is he sray capaciy of coaxial cables and/or connecors, and he sable oscillaion waveform will no be mainained because i drasically changes depending on he condiions. I may happen ha even he second half of oscillaion waveform canno be obained in he low inducance of he high core (permeabiliy of a ferrie core). However, he oscillaion waveform of he inernal capacior (he inernal capacior and he parasiic capacior) is sable. For his reason, i is desirable o use he firs half of oscillaion for esimaing inducance. 4

5 -. Esimaion Mehod of Inducance by he Timing Axis Informaion When finding he aenuaion raio from he formula (5), he value of he volage a he lef end of he wave canno be obained from he acual waveform for he following reason. The volage here indicaes he charged volage of he capacior, and he rise-ime a volage applicaion ino he coil will be delayed because of he ON ime required and he ON resisance of he SCR. Then, esimaing he inducance from he ime informaion insead of he volage informaion should be considered. The ime informaion here means he peak ime and he zero-cross ime. Now, if we apply he oscillaion waveform in he following formula, he waveform in he Figure 3 can be obained. f ( ) = e σ sinω Fig 3 By observing he above waveform carefully, i should be noed ha he peak value is deviaed o he lef. This deviaion does no occur wihou he aenuaion. The deviaion is closely relaed o he volume of aenuaion. Le us find his relaionship of he phenomenon. 5

6 Firsly, by differeniaing he formula (7), he followings will be available. f '( ) = σ e f '( ) = e σ σ sinω + ω e σ ( σ sinω + ω cosω) cosω The ime when he above soluion becomes 0 (zero) is he ime where he peak poin of he sine wave is available (shown as poins and ). And he following formula will be obained from he above. cosω σ = ω = ω an ω sinω In his formula (9), if is /, will become 0 (zero), and he waveform is observed wih neiher deformaion nor aenuaion. When he aenuaion is observed as in Figure 3, he peak poin of sine wave deviaes and is no longer /. The esimaion mehod by he iming axis uses he above formula, and he inducance will be obained by esimaing based on he quaniy of he deference from he poin of /. In he acual impulse response, he waveform sars from he poin. Therefore, if he ime o he nex zero-cross poin and he ime up o he nex peak poin are found, can be esimaed by he formula (9). Then he inducance of he coil can be obained by he formula (4). Needless o say, he ime informaion should be as accurae as possible in his case. However, he presen screen daa of 5 poins are insufficien, and he error range may be wider. In addiion o he above, he saring ime of he impulse injecion becomes very imporan for his purpose; herefore, he iniial calibraion should be performed sricly. Forunaely, i is possible o obain he accurae ime informaion because he sampling poins will become 64k a maximum by he new A/D converer employed for he new eser of DWX-XX. The ADC has only 8 bis informaion (7 bis a wors), bu accurae informaion can be obained by using some oher noise reducion algorihm joinly. -3. A Proposal of he Mehod for Obaining Time Informaion The ime informaion should be deeced on he presumpion ha he quanized noise exiss. Also, a leas 4k sampling will be required for seeking beer accuracy Deecion of Zero-cross Poin 6

7 Sar esimaing of he daa from he firs poin where + is deeced, and find he poin where he inegraed value becomes 0 (zero) while scanning he waveform. Appoin he average value of hese wo poins o be zero-cross poin. In his case, he noise reducion effec can be achieved by inegraing he daa Deecion of Peak Poin Scanning he waveform on and afer he zero-cross poin hrough digial filer, find he urning poin where he incline gradien becomes posiive. 7

8 . Evaluaion of Waveform In he presen evaluaion of he waveforms, when he es waveform is appoined o be f() and he maser waveform o be m(), he area comparison mehod deecs one of he following wo kinds of value in he observed waveform depending on which value is seleced. ( n here indicaes he number of sample frames) N n f ( ) (Average is seleced) max fn ( ) (Maximum value comparison is seleced) On he oher hand, he area deference mehod deecs he following value a all he observed flames. fn( ) m( ) max m( ) The area comparison mehod will no be a good mehod as an evaluaion funcion because i compleely disregards he waveform feaures. -. Correlaion Funcion Mehod A correlaion funcion is generally used for evaluaing he difference beween wo waveforms. When waveforms are appoined o be f() and m() respecively, he correlaion coefficien C( ) is indicaed as follows. C( τ ) = f ( ) m( + τ ) Here, appoining he maser waveform o be m() and he es waveform o be f(), he correlaion coefficien on he poin where =0 can be considered as he correlaion degrees. Of course, he correlaion coefficien o be compared is he maser waveform iself, i.e. he self-correlaion coefficien Cm(0). Cm( 0) = m( ) m( ) The value of he correlaion funcion always becomes posiive numbers because i is he same as he square when he waveforms are perfecly superimposed. If he phase difference, ec occurs, negaive ingredien will be generaed, and his affecs he correlaion coefficien. 8

9 -. Square Accidenal Error Mehod The square accidenal error, which emphasizes he difference of he funcions f() and g() as power, can be indicaed as follows. ( f ( ) g( )) This mehod is almos he same as he Area Difference Mehod excep ha he difference is squared. The smaller he value becomes, he more similar he waveforms are because his value is calculaed so as o minimize he value in he regression (he minimized square). The area deference mehod presenly employed is similar o he above mehod. -3. Deecion of One-urn Shor In he case of one-urn shor deecion, only he value for L changes while he value for Q drops very lile. Therefore, he one-urn shor will be able o be found by deecing he value of he zero-crossing poin and he peak poin??? 9

10 3. Elecric Discharge Deecion The presen discharge deecion evaluaion funcion is he following. ( f ( + ) f ( ) TH ) This is he sum afer subracing he cerain value of TH from he absolue value of he differeniaed waveform. The value becomes bigger when he second par of resonance has high frequency. Therefore, i is imporan o have enough sampling poins. 3-. Laplacian Procedure Mehod The laplacian procedure, which is called he curvaure, is he wo-sory differenial calculus of he funcion. L( ) = f ( ) f ( + ) + f ( + ) The value of he laplacian procedure is no big on he genle curve. Using his propery, suble discharge waves on he genle curve of he impulse waveforms can be deeced. The impulse waveforms have he simple increasing (or decreasing) quaniaive change of a mos do in he almos all areas. Therefore, if any value ha exceeds he above is deeced, such a value can be considered o he elecric discharge. 0

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