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UBC Theses and Dissertations

A study of the high-tension magneto ignition system with special reference to the source and elimination… Smith, Wilbert Brockhouse 1934

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COSTERS (coatiimea)w OSCIILOGI^»niSC yJ^ALYSIS (continued) 1. OAIJSBS c? I"TCH:?:'TIH:CTI: . . . . . . . . . . . . . . . . . . . . . . . 31 3. i i t i s E i ! r £ " i r m . I ^ ^ T I G A T I O E S <• 34 4. 353x0-1; OP -uss S P E C I A L SPARK PI'TG 36 CjEgIg3I021£ ...... ......•••..•«..•.....•..•.. ......... 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I t l a tho pur-pose o f tl.rl.6- t h e s i s to g*i%'3 BpBGinl a t t s m t i o j i t o thesis o - o e i l i s -ts ou3 f.nfi 'Bsve f c r a s r,ti4 in pnarticulnr to theme r e s p o n s i b l e f o r r r f i i o l o t e r f o r e a o G * I n f o l l o w sm Attempt hns l>eeit t o #-ivo a r a t i e & r s l thoujtfi «K>si0t3Gi6*i npprwdUaafcij, ©alhor«*:ticai 03cpl^nstiOTi o f t h e s e phosocsem* if? tonats o£ ©scGercMo c i r c u i t oonstaists. Tho production o f ene Ji aprrri; i n v o i v o a « c y c l e o f o p a r s t i o o e K h i c h , f o r the eonvoirlaaeo o f awelyuiB eaei r e f e r -ence, r.sy be r.ivitJefi i n t o e i x rath«r d f u t i n e t pexioaa, no •follows: ( l ) "Burin*? the £3ret p o r i o t l tho lot? v o l t s p o or primary ahortcircuifcod h;; tho ti£=Iii;? eontccfca, r.:»d no the ersBtare r o t a t e s o c a r t n i n curront Is peuarntec i n t h i s c o i l . 2 This current s tores its tho t&igmtlc c i r c u i t j o u l e s o f snorter which I s t a r i s uaod to supply tho currant; to tho sparfe plui?, together v?i th the I n c i d e n t a l loseos* (?) A few nacrofioeonds ho f o r e n spark i s desired s t the plug e ears opona tho tirelng contacts , thus fcresSdUig the? p i ' i a -s r y c i r c u i t . Th® pxim&vy c u r r e n t flocro^soa r a p i d l y , inducing high voltofpea So both p r i s e r y end tseeonczory wJnainga. *£Ms high secondary v o l t s go clisrgss up the a t r a y aoccnfisiry enpnci-t i e s to n s u f f i c i e n t l y hi{£h p o t e n t i a l to tsrask Sojns th© gss be tee cm the e l se t reses of thn spss& p l u g * # period 2 i s to fce fauna i s F « B « JUsbos**} pftpor: " r 3tlG , th« is? i t ieal ffheorsr of.,.Iftdue.Qa ¥ o I t S | M , . t l l . J ^ ^ (5) £ariod 5 c o n s i s t s of tho discharge o f tlw 3aec*idftrty r,feray csp&eifcles through tho icmcfe, «?bic!t ere incSacfcive, sue] the 3ine® u s u a l l y the va lo rs o f c a p a c i t y una laducfcene© 5 a velvet! nr© r e l a t i v e l y s a a l l & aiifl mileos •suppro&sox'a' art* used tlu) resiiitQDee i s a m l l , osei l le - t lcms of s vorj' high f r e ~ eacney w i l l take p lace • ( 4 ) Fol lowing tho '3i.-&ohnrrM c f pxiriod 3 tho reacdrrlng: energy i n the e l o c f c r i c and- asimofclc c i r c u i t s i s dlc-aipntotf i n thrr jip.nri:., between the e l e c t r o d e s of the spark pl'ug. (5) JI120 to high spoocls or improper stljastr^oafc o f tho ti m i n g c o n t a c t s , i t l a p o s s i b l e th:;t tho prijaory nay fcocoKc c h o r t c i r c u i t o a before «11 tho <cmarp;y i o ciisfilpntou i n the sp a r i i , I K r/hieh ense tho rosnirifuif? enor^* i s f.ividoo bets/eon tli© spark and .rotarninf: i n t o th© prlrssr;? c i r c u i t * (6) fiie c i r c u i t ramalm gtiVtftaz.tSall^ tiood Tor a con-s i d e r a i l a Sine, period 6, u n t i l tho srcmture hm r o t a t e s s i a f f i c i n n t l ^ fo r too cyc le to bogSn CYS* ag a i n f o r the produc-t i o n of the s s x i sparks Pr-aC-IilPTlOlT OF /.PH:^4?i),>, 'i'iio raagjieto usnc i s those i n v e s t i g a t i o n s vau s Id^ia model 40 ac- jifcosB en P i s to 1 8 w i t h the covers reiAOvao" cu8 mounted on the test ba^o. It sa.i d r i v e n at 900 rpm through' a gear t r a i n f r o a the sraehroaoua sjotor Kfcieh operate a fcbe t i m -i n g contacts fo r G CfsBorai E l e c t r i c <mtbo6e r/iy o s c i l l o g r a p h . Tliis eneuroa that tho va r i o u s phmzmmm. ss re nltcs^s in ayn-el'roniosa T;I th fehs c a c i l l o f - r a p h o J^ine sni.1uytr.cnt obtM nofl ~ DicrofijetQr erraiiger/mt t?fctnch*j<$ to tho rcjf-ialor t iming l e v e r tho msgnoto* Leads i r e : : the sagnato to tho apart: p3ttg Y&ZQ of o r c i i K s i j ; if?nitSon oa!>Io eseept whore oiherwiae £teto&« fhc nperJL plugs usofi t i c r e ; en AC nutoKobilo p l a ^ . as Cloo a v i a t i o n plug nre epecln.1 p l u ^ t o be ucscrlho£ la ter* Th5 J set-up, together vikh a per 11 en of the n n p l i f i o r ueod i s ciotorminiiift the current wave forio, i s shovrn on P i s t e 2<* 1/0%' froqueac;;- o s c i l l o g r a m v. ore token on a Meeting--house 6 element osc i l lograph* d i r e c t l y v;hon r o a r i h i e , c r rneans c f vacuus tube a m p l i f i e r s whea e i t h e r the voltage or e a r r e a t s ware herons tho £ept-e o l the tnotrurxmt, o r irhen t&e i n s e r t i o n o f tho vifcrntor oi-v.- ent V J O U I C a l t a r the c i r c u i t constants to too great an e x t e n t c L 0*meral h l o c t r i e cfithocio T^-i; o s c i l l o g r a p h tas used to cbfcnSn oucSlloirrans of the h i f r h frequoBcv components of "both the voltages am. c o r r a u t e . A PLATE 1. VXAH 2 r e s i s t a n c e c l a p - v o l t a p e n i r i t i o r isorvod to reduce the volte-pea to v a l u e s s u i t a b l e f o r a p p l y i n g t o tho o s c i l l o g r a p h d e f l e c t i o n p i s t e s . Current t r a c e s VQXQ t o k e n t>y osipli£^ii3£ the v o l t a g e drop s e r o a e a 3J?0 efea v & r i a o l o rooiatimeo srifch & ts-o s t a g e a m p l i f i o r * . f i i s above i t a n t i o B s f t a a p l i f i s r -woo coiiatructotl os f o l l o w s * vijo o s c i l l a t i o n c o i l s wore removed f r o n tho t i m i n g wave o s c i l l a t o r -on tho cathode r a y o s c i l l o g r a p h and -a p l u g i n orraBgerLsnt o u b s t i t u t o d , vrhlch i n s e r t e d a 15,000 ohia r a s i s t -nnce i n the p l a t o c i r c u i t o f the tuho and eonnoctod t h e R e f l e c t i o n p l a t e s serosa t h i s r e s i s t a n c e * ?he plat© cu r r e n t from on e x t e r n a l a m p l i f i e r c o n s i s t i n g o f a hl~tm tali©, tms . sent through the 10,000 0!® grid- losJs r e s i s t o r i n t h e o s c i l l a -t o r , coimoetion t w i i i g oMtdjood means o f t h i n l«go i n s e r t e d between the ends o f the m a l e t o r wi& ttia laotmtiag c l i p s , i l i i a c u rrent p r o v i d e d the g r i d b i o s f o r tho 50 watt turn froc. tho 13 drop i n the 10,000 oh a r e s i s t a n c e * A a e precaution s p o i n a t 6&asm t o the 50 w a t t s t a g e duo t o l a d * o f g r i d b i a s , tho p l a t e oarroat o f th e s x t o r n a l a m p l i f i e r •eras aloe sent t h r o u g h a r e l a j ? v*hich i n t e r r u p t e d tho AC power s u p p l y s h m i l t l t h i o c u r r e n t f o i l . , Other tip psrs. t u s i ncluded 3 B u r t e s t hod o ro.:* o s c i l l o -scope w i t h which o v i s u a l stttd> o f tho imva f o r n s wee fnaflo, s t h r e e tufeo r a d i o s e t w h i c h was? used f o r monitoring the r a d i o i n t e r f e r e n e o , nod miscellaneous apparatus usee" i n d o t G r o i n i n g the rnagnoto constants. 1. tausmc cimiiT,' The eons t r u e felon o f the segue t o xsae ouch t h a t tho e o i l a resslno-d s t a t i o n e r y one the r o t a t i o n o f the orr.oturo . a i r o c t o f l th© f l u x t h r o u g h t h e c o i l s f i r s t i n one? d i r e c t i o n una than i n tho o t h e r * ?hs t i d i n g c o n t a c t s were operated by e eara on t h e end o f the armature shntt ancl change o f t i i a i n g was securou by r o c k i n g tho e n t i r e c o i l aosocibljy about tho 0r13n.tu.ro a x i s * F i g u r e 1 shows a aehoreatic diagram o f the m i m e t i c c i r -c u i t shoeing: the. p r i n c i p a l dissensions.* fhe Icagth- .©£ th© a i r •.g&p was. O.OjJ. cm. •. . 2-. K&ECTBIC&T* CIS-GUI?. The e l e c t r i c a l c i r c u i t i s shcv?n c l i a g r a s a t i c a l l j ? i n f i g u r e 2« Tnlz c i r c u i t represents c o n d i t i o w o . p r e v a i l i n g to a f a i r oograa o r aeearaej tm& s i l l f o r e fcha b a s i c c i r c u i t f o r tho - s a t b o f t s t l o a l e n a X y a i a w h i c h f o l l o w s I n a l a t e r s e c t i o n * I t ma,;? oe v / e l l t o note- t h a t some o f tnea©- so c a l l e d constants &rs not c o n s t a n t hut are ftraofciosa o f c u r r a n t o r frequency* t h i s i n t r o d u c e s c o m p l i c a t i o n s I n t o t h e jsstfaesatlcal a n a l y s i s as mil &G tilting i t d i f f i c u l t t o o b t a i n s u i t a b l e numerical v a l u e s . 3. 3DS5;^IJIA?Io:i 0? SHS 3C0SI,AHY 3?HAY C/.MClrT. A stanflorft resonant e i r c o . i t c o n s i s t i n g o f a kiwmi inductance i n p a r a l l e l K i t h a v a r i a b l e coiidaosor o f known c o p a e i t r srao i n s e r t e d i n tho p l a t e c i r c u i t o f a ut 224 aereon-g r i d tucn connected as i n f i g u r e 3 to operate na a dj-notron o a e i l l o t o r , Tho r e s u l t i n g o s c i l l a t i o n r;as viov.oo on tho screen o f the cathode r a y o s c i l l o s c o p e o r l o c a t o r : w i t h the ro d i c s e t . Ths secondary o f the nagneto was t h e n connected i n p a r a l l e l r i t f c tho v a r i a b l e condonaor mm the l a t t e r adjusted f o r the esfss frequency s s p r e v i o u s l y * 9lm decrease i n c a p a c i t y cf th© s t a n d a r d condenser t?as t h e n nuKcr.i c a l l y equal t o tho s t r a y s e c o n d a r y c a p a c i t y a t t h a t freqaoncy. By s u i t a b l y c h o o s i n g veluea o f inductance and c a p a c i t y Dcasaraments wcyo nade over a ^ide. range o f f r e q u e n c i e s , and the valaos thus found p l o t t e d a g a i n s t frequency as in f i g u r e 4, The e f f e c t o f tho inductance o f the secondary B i n d i n g vm& n e g l e c t e d s i n c e i t weo vers? l a r g o as compared tvifch tho o s c i l l a t i n g inductances. bK-?;.i^i:r>?jo!? OF S E K %mra on m& sxiDxa&u A search c o i l c o n s i s t i n g o f 10 turns o f no. 30 MC t . l r e V-2S woantl on the l a m i n a t e s core a s n e a r s s p o s s i b l e t o the zsoir* c o i l s . ZBVO f o r o s o f tho r o t a t i o n a l v o l t a g e genera-toe I n t h i s c o i l and i n the p r i m a r y winding t?cre triewod s l c r a l -fcanoously on trie • leotlnfthouso o s c i l l o g r a p h sad the s e r i e s resistance© so adjusted t h a t the d e f l e c t i o n s vara t h e etna©. £s the s e n s i t i v i t y o f the v i b r a t o r s teas found t o be very n e o r l y i d e n t i c a l , tho t u r n s r a t i o was t a k e n as tho r a t i o o f the t o t a l a o r l o s r e s i s t a n c e s , tho value being p i v i n g s s the number o f p r i o r y t a r n s - 2 ? 3 * fho r a t i o o f t u r n s between tho p r i m a r y and secondary •»"fia found by applying fcncsu AC voltages to the respoet5 vo windings and ««asuring tho v o l t a g e g e n e r a t e d i n the other. 5?his r a t i o \-?sc found .to bo 50:1 which gave as tho number o f seconder;- turns, 15,7GG« i^mmsiBimm O F mmmicss* The s e l f inductance o f the s a g a s t o secondary was measured by aeana o f tm Anderson Inductance Bridge ,i; f o r a frequency range o f 300 t o 1000 c y c l e s e and. was found t o d i f f e r l i t t l e f rora j>0 henrys o v e r t h i s frequency range. t'he p r i m a r y inanetance was aieasured w i t h on AC a s s e t e r and v o l t m e t e r a t 60 c y c l e s f o r a current range o f 0.1 to 1.0 amperes and the value was found to be constant a t 0.02 henrya. Because of the very high r a t i o or* t u r n s and t h e f a c t t h a t the e o i l s were connected t o g e t h e r i n t e r n a l l y , conventional methods o f measuring the mutual inductance c o u l d not be used, so the c o e f f i c i e n t o f coupling K> was found f r o a the r e l a t i o n n ^ / n - x 1 - IC and the mutual inductance c a l o u l a t e d . A proof o f t h i s r e l a t i o n i s given i n iippendiK I . Proa t h i s r e l a t i o n K was found to be 0.9? ana K to be 0.75 henry. M^BCTKAfltffl 0? m SSUCSD/iEY !5FFE01?JfB HSSIS^AESIS* -In order to determine th© e f f e c t i v e r e s i s t a n c e o f tho secondary winding a t v a r y i n g f r e q u e n c i e s , tho scheme shown i n f i g u r e 5 «?as used, the v a r i a b l e condcaiser G was v a r i e d u n t i l the power f a c t o r of the c i r c u i t C, E^, r 8 and the o s c i l l a -t o r output, became u n i t y as i n d i c a t e d by the vacuum tube v o l t ~ r i o t e r , "he v a r i a b l e r e s i s t a n c e lip was then s u b s t i t u t e d f o r the nagneto secondary and v a r i a b l e condenser, I-U., Lj. and C, and adjusted so that the vacuum tube v o l t e e t e r read the ssiae as f o r m e r l y , when R p s 3-^ . I n t h i s nanusr the e f f e c t i v e r e s i s t a n c e of the magneto secondary was d e t e m i n e d f o r a frequency range o f 600 t o 5500 c y c l e s * Figure 6 i s a g r a p h showing the v a r i a t i o n , o f t h i s r e s i s t a n c e with f r e q u e n c y , f h e d i r e c t current r e s i s t a n c e uas o n l y 3000 oJaas but a s may be aeeu, the e f f e c t i v e r e s i s t a n c e i s encmously g r e a t e r , r e a c h i n g 700,000 ohras a t the resonant frequency of 3500 c y c l e s . r^easurcfasnt o f the d i r e c t current r e s i s t a n c e o f the windings was made w i t h an anneter and v o l t e s t e r , and the pri& ~ ary condenser V7as no a sure d by means o f a c a p a c i t y b r i d g e , as was the c a p a c i t y o f the spark p l a g e * So ure a l * p r e t e n c e a t p r e c i s i o n was made in. t h e determination o f any o f t h e magneto c o n s t a n t s s i n c e i t was f o u n d t h a t a l a r g e , unavoidable ex p e r i * -d e n t a l e r r o r was always p r e s e n t . Jtar i n s t a n c e , an iapereep-t i b l e movement of the armature caused, as nueh as 3% e r r o r i n the measurement o f inductance. Tlx?, fe-llos&sg t a b i c g i v e s a l i s t o f the c i r c u i t constants d s t e r a i n c d f o r the p o s i t i o n o f the a m a t u r e teae&iately f o l l o w i n g th© opening o f t h e t i m i n g c o n t a c t s . < Si z u 2*2 a 273 0.02 Henry 0.11 Oiooa 0.041 rafd. 0.57 0 . 75 Kenry 15.700 30 Henrys 5 9ooo Ohjss 700,000 Cans 100 3 10 10 -6 c m f d . t a n f d . miafd* Henry P r a c t i c a l l y constant 25.0. r e s i a t a n c e K l e s condenser Conputed from resonance P r a c t i c a l l y c o n s t a n t XuCo r e s i s t a n c e E f f e c t i v e r e s i s t a n c e a t 5500 c y c l e s Value a t 5500 c y c l e s AO spark p l u g 01eo spark p l u g Computed a p p r o x i m a t e l y i i s t h e magneto aroature r o t a t e s , the nagnetic f i n s : l i n k i n g tho w i r i n g s changes and i n d u c e s r o t a t i o n a l s a f e i n thera. I f the v a l u e s o f t h e maf o f the p e r o a n e a t taaguets, and the r e l u c t a n c e s o f the v a r i o u s p a t h s ho known» e q u a t i o n s can he w r i t t e n g i v i n g t h e f l u x d i s t r i b u t i o n and the v o l t a g e s i n -duced by i t s change w i t h r o t a t i o n . 'fhese equations w i l l p l o t i n t o s a v e forrao as i n f i g u r e ?. I n the a c t u a l magneto s how-e v e r , these wave f o r e s w i l l be c o n s i d e r a b l y m odified by f r i n g i n g and leakage , and srf.ll reseoble sore c l o s e l y the dotted curves. As may bo noted t h e v o l t a g e induced i n t h e c o i l s i s p r a c t i c a l l y scro f o r t h e g r e a t e r p a r t o f the c y c l e and r e a c h e s - a l a r g e value o v e r a.-very s h o r t p e r i o d , fh© & e s i ~ r a b i l i t y o f each a *?ave foa-ri i s a p p a r e n t f r o a t h e f c l l o w i n g c o n s i d e r a t i o n s , la© r o t a t i o n a l emf i s r e q u i r e d o n l y f o r t h e s h o r t t i a e n e c e s s a r y to e s t a b l i s h a c u r r e n t 1 i n t h e s h o r t c i r c u i t e d p r i m a r y , a f t e r which i t perform® no f a r t h e r u s e f u l f u n c t i o n . A g ain, when the t i m i n g c o n t a c t s c l o s e , i f t h e r e i s an a p p r e c i a b l e o a f i n the c i r c u i t , the r e s u l t i n g t r a n s i e n t may p o s s i b l y produce a spark a t the p l u g a t an improper t l c i s . The a c t u a l shape o f the r o t a t i o n a l v o l t a g e s a f e i s immaterial so l o n g a s i t i s a b l e t o e s t a b l i s h t h e r e q u i s i t e c u r r e n t I prece-ding the o p e n i n g o f the t i d i n g c o n t a c t s , but i t i s obvious t h a t a snooth wave shape i s t o be p r e f e r r e d . OPTSXIIG OF SIT, TlI'-nHG CQVZAG?:!. uhen a s p a r k i s d e s i r e d a t the p l u g , t h e t i m i n g c o n t a c t s are separated by the can, thus creaking the prietary G l r c a i t * ' Kef e r r i n g - t o f i g u r e 2, (mad o m i t t i n g a l l t o the Tight o f TI) i t i s seen t h a t opening the t i m i n g c o n t a c t s A i s the .same" as suddenly i n s e r t i n g the condenser i n t o the c i r -c u i t * S h i s nay be t r e a t e d -so. i f the c i r c u i t v/ere p r e v i o u s l y dead ana a reverse current 1 were suddenly a p p l i e d at the switch A, i n which case the cu r r e n t I w i l l d i v i d e between the condenser 0^ and the transformer i n the r a t i o o f t h e i r admit-tances. Let the current through the condenser 0^ and t h e r e -fore the in d u c t a n c e B at any t i n e a f t e r t = 0, be i • § Calculus i s need* . See A p g e M i s I I a ^ admittance, of- condenser -I adm'iTtanceof Trans-farmer an** Candcwsef" Using Heaviside's n o t a t i o n and l e t t i n g the operator d/dt be denoted by p„ the ad&itt&nee operator of the condenser C?2 i o i c i s e d i a t e l y seen to be C^P, end tha t o r the transformer . i s i . = / . impeda nte. 2T/ — Z-^/z.^ where 2 ^ i s the priiaary irapekl&nce s % -* L j P i s the mutual impedance s H 2P^ -S 2 i s the secondary inpedaaee s Eg *' LgF' + l/o^l -She transformer adnittanoe thus becomes on s u b s t i t u t i o n (_R, + L,P)(R± + UP -t '/cp) -MZPZ S u b s t i t u t i n g these admittances i n the r e l a t i o n t7o have, = ^ ^ c,p(R,i-L,p)(R^ft,p-hyrclp)~c,MlpJi-r^i-upi-^xP) • zP where yp = C,P(R, +LIP)(R^-fUP +tf*p) -C,MZP* a s d zP - C,P(R,+L,PX^ tup* '/Z^-aM^+Cfo-H-^p+'/w) "She ©filiation 2. s 0 i s o f t h e f o u r t h o r d e r i n p, hence t h e r e w i l l he two p a i r s o f r o o t s , which s a y he both r e a l , bo t i i coDplez, o r one r e a l and one complex. From t h e n a t u r e o f th© q u a n t i t i e s i n v o l v e d i t i s a p p a r e n t t h a t t h e r e a l p a r t s o f a l l tho r o o t s s r i i i be n e g a t i v e . I f the r o o t s o f ?^ s 0 a r e , t h e r e w i l l b e , i n g e n e r a l , t h r e e p o s s i b l e g e n e r a l s o l u t i o n s a c c o r d i n g a s t h e r o o t s a r c r e a l o r coiaplex. Case 1, wiiere ft and f^^- a r e r e a l : Case 2 S s h o r e and fi>- are i m a g i n a r y ; Case-3* "shore ^h, i s r e a l and i s i s s g i n & r y s 'i'he secondary i n d u c e d v o l t a g e , e=, c o r r e s p o n d i n g t o any o f the above e q u a t i o n s raay be o b t a i n e d from, the r e l a t i o n e^ s -K d i ^ / d t and w i l l have e s s e n t i a l l y t h e same wave f o r a as the p r i n a r y c u r r e n t . That p o r t i o n o f t h i s v oltage w h i c h appears a c r o s s t h e se c o n d a r y d i s t r i b u t e d c a p a c i t y i s t h o open c i r c u i t wave 'form o f t h e magneto and stay be evaluated i f the .magneto, c o n s t a n t s a r e known. F r o n e q u a t i o n can be o b t a i n e d two f r e q u e n c i e s o f o s o i l 1st ion., £j. s /#,/2 7r ami f ? s / I /a 7t, ©ad from 2 » a s i n g l e frequency % s falZKm '2'm h y p e r b o l i c f u n c t i o n s o b v i -ously g i v e u n i d i r e c t i o n a l • irapolses* If the c o e f f i c i e n t of c o u p l i n g is p r a c t i c a l l y unity, 2 then Ljt-g s II and the equation x 0 beeoces a cubic i n p 8 which will have on©, real root and one p a i r of roots v?iiioh say be either real or oonp£©3s:» 2f the f o l l o w i n g uaaerlcal values for the circuit constants are inserted i s equation (1) l% s 0»02 , % - 58 K s 0 -75 E 2 s 0.11 % s i s 1 0 5 • 0 1 r 0*041 at 'Cg s 100 X l o " 1 ^ when s i m p l i f i e d , the equation S~ s O becomes„ T h i s factors^® i n t o th© f o l l o w i n g terss • fp^Z.77W^(p+teOrlo+)fP+0-9*l^ which g i v e s the f o l l o w i n g r o o t s f o r p P( 1-7 /0-7 iio* P3 - .9 y.10*-j I-7 x IO* • ?L = -1-7 y-10^ -f. 10.7^/0^ P* = - 0 -9 * io*-tj /v* Sbsse roots isay be subs111uted i n the .appropriate s o l u t i o n , naoely ( 2 c). s in v/hleh case we have, •(, = 6-. l^toih 10-7 uo*t-tB, sink. I0-7+10^1-t-a I7*IC}t +B1S1* !-7i toHl fhe- constants J.^, 3 ^ , and B^,-are amplitude constants and nay be evaluated by Heavisides E&panalon TheoreE, i f r e q u i r e d . This s o l u t i o n gives only the one frequency o f o s c i l l a t i o n j f 2 = ftiiZ 7? s 1.7 x 10 4/2 K - 2700 c y c l e s per second. Shis frequency, though not i n ve r y c l o s e agreement - 1 5 -w i t h the observed value of 5500 c y c l e s per second 0 serves to i n d i c a t e the approximate n u n e r i e a l value to be expected* She discrepancy between the c a l c u l a t e d and observed v a l u e s i s p o s s i b l y due to the a&gneto o p e r a t i n g a t a d i f f e r e n t angle o f armature displaceiaent than t h a t f o r which the asgnetc con-s t a n t s >?ere laeasored. EF#3COTE EJKJISSAECE OF A 3PAK/C ETSGHAROE. An approximate e x p r e s s i o n f o r the e f f e c t i v e r e s i s t -ance o f an e l e c t r i c spark discharge can be obtained by making c e r t a i n aasoaptions r e g a r d i n g the maipsre o f the discharge and the mechanism by which the c u r r e n t i s c a r r i e d * I t i s assumed that the discharge i s c a r r i e d by n d r i f t o f e l e c t r o n s produced by ionise.I,ion c f the gases between the e l e c t r o d e s . This i o n i -z a t i o n aay be produoed i n i t i a l l y by the a p p l i c a t i o n o f a hig h p o t e n t i a l g r a d i e n t o r by soise o t h e r mesas f but i s maintained by c o l l i s i o n s o f the r a p i d l y moving e l e c t r o n s w i t h the gas molecules* She m i g r a t i o n c f the p o s i t i v e i o n s i s neglected* She d i f f e r e n c e between a spark and a r c discharge i s t h a t i n the former the discharge i s assumed to be c a r r i e d n s i n l y by e l e c t r o n s l i b e r a t e d from the gases between the e l e c t r o d e s , i v h i l e i n the l a t t e r the discharge i s c a r r i e d by e l e c t r o n s l i b e r a t e d f r o n the incandescent s u r f a c e s o f the e l e c t r o d e s . There i s 9 however, c o n s i d e r a b l e overlapping of the phenomena but f o r the sc>e o f s i m p l i c i t y t h i s i s neglected*# e s charge on o i e l e c t r o n fa r mass of an e l e c t r o n I* = number of e l e c t r o n s per u n i t volume d s s e e n f r e e path of an e l e c t r o n i n t h e d i r e c t i o n o f t h e . f i e l d v s ave r a g e v e l o c i t y o f t h e e l e c t r o n i n t h e d i r e c t i o n o f t h e f i e l d 12 s v o l t a g e between t h e e l e c t r o d e s s s d i s t a n c e he tire ea t h e e l e c t r o d e s R * f i e l d s t r e n g t h o r p o t e n t i a l g r a d i e n t & s c r o s s s e c t i o n a r e a o f t h e a pari: 2 s- c u r r e n t through t h e sparl: Each e l e c t r o n i n t h e f i e l d R i s a c c e l e r a t e d fey an amount a » Be * P r o s mechanics, t h e average v e l o c i t y v s</a£d ms S u b s t i t u t i n g f o r a, v s / 5 e d i' _ _ (1) The current through the sparfc i s g i v e n by I s I ' o A i ? _ _ _ _ _ _ _ _ _ _ _ _ _ _ , I f i t i s assumed t h a t each c o l l i s i o n o f an e l e c t r o n w i t h an u n i o n i s e d a o l e e u l e ' r e s u l t s i n the pr o d u c t i o n o f s o r e e l e c t r o n s , i t raay be shov;n t h a t t o a c l o s e approximation B ss 1 v S u b s t i t u t i o n , i n (2) g i v e s J s I A e v g - ft ft, e 2 d s g I'he r e s i s t a n c e o f t h e spark t h e r e f o r e becomes V St S » El 0 T — C3) The {juantitiee e ? tn, and K a r e c o n s t a n t s and e i s f i x e d b y the geometry o f the spark gap* She mean f r e e p a t h o f an e l e c t r o n , d, vri 11, honever, depend on the d e n s i t y and com-p o s i t i o n o f t h e g a s e s i n v o l v e d . I'or-j i t i s f o u n d by experiment that f o r a g i v e n s e t o f c o n d i t i o n s t h e volta.ee 13 a c r o s s the gap i s c o n s t a n t r e g a r d l e s s ' o f c u r r e n t changes so l o n g a s such changes are slow as cospared with the i o n i z a t i o n tic;e o f the gap* this then leads to the conclusion, that the current den-sity in an electric spark is constant f o r a g i v e n set o f . conditions t or expressed mathematically i s A or I s X» where K? s 2, IC E d e 2 . • - • - - (4) If the current through the spark changes faster than the i o n i z a t i o n time of the gap, the sectional area A does not have time to change and the q u a n t i t y m s becomes constants and the spark acta as s pure r e s i s t a n c e . If,, how-ever s the c u r r e n t changes are extremely r a p i d the nuraber of e l e c t r o n s per u n i t volume cannot change appreciably8 that i s , H remains f o r the moaent approximately constants and the e x p r e s s i o n f o r the current becoaes I M H e A a e &... # _ _ „ _ * • . . . « . ( 5) A closer study of (5) leads t o a very i n t e r e s t i n g r e s u l t , A sudden s n a i l increase in S will result in a large increase in 1* This abnoraal i n c r e a s e i n c u r r e n t w i l l i n t u r n produce an abnormal i n c r e a s e i n Is which will iahe place much Gore slo w l y depending on the i o n i s a t i o n ticse o f the spark. Mm I i s l a r g e and II i s l a r g e so i n order that (jj) be s a t i s f i e d there sust be a large decrease i n B w i t h i t s corresponding decrease i n current» Hence i t way r e a d i l y be appr e c i a t e d t h a t en o s c i l l a t o r y discharge w i l l take p l a c e through the spark„ the frequency and wave f o r s being cone co-apples f u n c t i o n o f the q u a n t i t i e s i n v o l v e d , including the i o n i z a t i o n t i n e o f the spark and the nature o f t h e gases b a t e o o n t h e e l e c t r o d e s * I t l a 1 . 7 e l l Jrnovrn that t h e v o l t a g e r e q u i r e d to break cown a Eparfc gap i s such higher thmi t h a t r e p a i r e d t o a a i n t a i i i a discharge through i t . I n view o f t h i s f a c t and t h o above doveiopstrnt i t f o l l o w s t h a t t h e r e i s considerable . l u s t l f i c a -t i o n f o r t a k i n g the v o l t a g e d r o p a c r o s s a epork t o be {n -f b i } 9 where' a and b a r e c o n s t a n t s depending on t h e c o n d i t i o n s under t h i o l s t he ®pwfk fc&fccs p l a c e wa& tho probable frequency o f t h o discbai»ge. The constant® a and b % i l x not n o o ^ s s s r i l y r a s a i n constant throughout tfec entir© d u r a t i c - s o f t h e spaM: e but may v a r y as the character o f t h o s p a r k changes. 2hafc i s * the valaes f o r a and © which h o l d f o r tho i n i t i a l broaftdocm o f the gap may bo d i f f e r e n t froja tho values which h o l d * f o r i n s t a n c e , f o r t-n o s c i l l a t o r y discharge© iiowevor, i t i s found oxperiasn*-t r . l l y t h a t a and b a r e p r a c t i c a l l y constant d u r i n g each s e p a r a t e p o r t e d o f the disc h a r g e . Hence„ f o r a n a l y t i c a l par*, pooos one taasfc bo c a r e f u l to choose values o f a and b w h i c h a r e a p p l i c a b l e * t o the problem 5a bench* H?SA£DG*?J: OP I S S us. Ebon t h o v o l t a g e a c r o s s t h e s p a r k i n g t i p s has b u i l t up to a s u f f i c i e n t value a, tho gap b r e a k s down and She s t r a y c a p a c i t i e s o f tho secondary c i r c u i t d i s c h a r g e throagh t h o gap* B e f e r r i n g to f i g u r e 2, brasltdoKn o f t h e gap i s tho see© as c l o s i n g s w i t c h B, whi c h i c t a r n i s t h e asms a s sudd only fcjjply-i n g a negative v o l t a g e a t tho sw i t c h E o f v?-iue equal t o that tsfcieh o s i s t c d a c r o s s tho s p i t eh p r i o r to i t s c l o s i n g . •-17-How, In taach greater than and £g Is jsuch grantor thou C, 'hones tvs are justified in c o n s i d e r i n g the com-ponent c i r c u i t s separately, Consider f i r s t the circuit con-s i s t i n g ef r , C, H t and that i s , ifasgln© YY t o be shorted,, Assuming the secondary voltage S to regain constant, and taking, the drop across the a park as a -t b l , we have, by equating v o l t a g e s aronnd the. c i r c u i t , E = a - - r t ? s € -+- I—7—' — : ry J A whence / = (£'-«) ( / '1 CR F -g /- C Pz) ^ CR-t-b)-f(bRC-tl-)P-i-bLCp^1 •P J&at t h e r o o t s of s - o be — <=<-^bLC y (Z£>LC)Z bLC I f /3 i s imaginary, tho general solution is •& darspod oscilla-tion o f frequency f ^ ~ , . I n order to g«tt sone 1ft es of- tte frequency e£ f,'Xat us s u b s t i t u t e n u ? 3 s r i e a l vsiu.es in the above eseprosBion f o r /3.. hot us tafco C : 5 x 10 , i s 10 , H - 10, and b - 5. x I t r . These v a l u e s , though o n l y approximate, are of tho order o f magnitude to bo found in m a c t u a l installation® Substituting these values in the equation /3 / (ZbLC)7- bi-c iso have —/z -18-whieh reduces to from which we o b t a i n f ^  - /Sfzic - 7 x 10* c y c l e s per second, which corresponds to a wave length o f about 4 factors » Consider nest t h e c i r c u i t c o n s i s t i n g o f r , K t L, safi C 2, th a t i s , n e g l e c t i n g c and a l l to the l e f t o l 3 » f l i e o p e r a t i o n a l solution in this case i s and tho general solutions, i f /3 i s Ss&gftHU^r, l e where * = &afl ^ ^ ^ ~ «c l - / ^ are aa bo fere the roots cf - -0 t h i s also i s a de taped o s c i l l a t i o n of frequency f 4 - /5f27\. Again l e t us s u b s t i t u t e s u i t a b l e numerical values i n the e x p r e s s i o n f o r . £ot I B tefc« Cp - 10"*^°, j, s 10**^ E -10, end b : 5 x 10^* »hen th esc values ere s u b s t i t u t e d i n the eonation. ••. have / (so/oy~ /6 =J ( z a w h i c h s i m p l i f i e s t o ^  = \T^hZfx /d'6 = j/-/x/a 7 ana we o b t a i a f 4 fro a the r e l e t i o n - ^  / 2 7T - 1*8 JC 10 c y c l e s per second, fhia frequoncy eoi«responaa t o a v?av© length o f about 16 ssetero. Prom tho above i t sppours that f ^ and f 4 are o f qu i t e h i g h r a d i o frequency, Since the frequency o f the so o s c i l l a -t i o n s depends to so {treat an extent on the s t r a y inductance r.nd c a p a c i t y o f the ;;ystes e and the soGiesfhat u n c e r t a i n q u a n t i -t y h 0 i t I s apparent that • these frequencies say he found almost e a r n e r s i n t h e s h o r t ??ave bancs, depending on t h e p a r t i c u l a r i n s t a l l a t i o n , , iJov?, when t h o gap brocks <IOT;n t h e r e i s i n a d d i t i o n t o the a b o v e , t r a n s i e n t s cue t o a v o l t r»ge B t a k e n m E c o s vtt„ a p p l i e d a t t h e s w i t c h B t o tho c i r c u i t c o n s i s t i n g o f r i n s e r i e s i-^ith Cg S D5 E i n p a r a l l e l , c l i e r e 2 i s t h e ccuivoXent .-impedoaco operator o f a l l to t h e l e f t o f JSC „ f i g a r s 2« E - (Ri+LipjK.+UP + '/ctP) ~ rf? f r o n t h e r e l a t i o n z - Z z Z' B o a s t i n g v o l t a g e s , we have froir. ?;Meh C = (Ct-PZ cosast -a) j dp i>Z •+- b -f-Z-Zl p I n o r d e r t o obfcnln the g e n e r a l s o l u t i o n f o r the abevg o p e r a t i o n a l . g e l a t i o n no G u&t f i a d t h e r o o t s o f £ - 0 where Z ~ GgbZl> b •+ 2 , 2 havS . B g t h e v a l u e a s s i g n e d above• It we t t ? k o b d i f f e r e n t frore sero i t i s a p p a r e n t t h a t " » o i s a f o u r t h degree ©quittioa i n p B aue- a c c o r d i n g l y T?XI1 have tea p e i r s o f r o o t s which assy be b o t h r e a l , b o t h e o s q s i O K , o r ono r o d a n d one c o r . p i o x . I n s e d i t i o n t h e r e w i l l be a p a i r o f i m a g i n a r y r o o t s f j t ? which apply t o t h e p o r t o f t h e s o l u t i o n i n v o l v i n g 5 c o s trfe but net t o t h a t i n v o l v i n g e.« S i n c e t h e s e r o o t s .t $w c o n t r i b u t e not his*? cf' I n t e r e s t t o our s o l u t i o n vo may die-eard the® and c o n s i d e r o n l y t h e -solution i n v o l v i n g & „ •-20-I f tho r o o t s o f zTi - o are —<*-< ± A mat -<**. * / ? 2 , . the g e n e r a l -solution t r i l l c o n t a i n tortus o f tho f o l l o t T i n c for© i f /$, and ^ 6/ are iEssgicary e This ^ i v e s t s o mors p o s s i b l e f r e q u e n c i e s o f o c c i l l n t i o n , f 5 s / ^ / / 2 7r nnfl f G - fap.n* • Proa m i i n s p e c t i o n o f the r e l a t i v e sognStufies c f tho c i r c u i t co l i s t cuts i n v o l v e s * i t my ho seen t h a t s n y p o s s i b l e o s c i l l a t i o n s w i l l probohly he o f audio frequency,, th a t io„ r«n^  current chfrnges w i l l he r e l a t i v e l y nlov.- n f t e r t h e i n i t i a l impulse a t t - 0* Sow, i t i o na e x p e r i e e n t s l f a c t t h s t the voltage a c r o s s e s p a r k i o j y o e t l c a l l ^ cosstmfc f o r r e l a t i v e l y clew c u r r e n t elvmgQS* Shla gay bo coiifirr>3u by r e f e r e n c e to the o a c i l l o ^ r a p b i e r e c o r d s tsftavn on p l a t e 8,„ Xomos t r a c e , sua p l a t e 13* letter t r o e e , t?bieh sre o f the spark c u r r e n t *n£ v o l t -sg© r e s p e c t i v e l y * ^hose t r e c c s a r e -explained nore f u l l y i n tho s e c t i o n ©s o s c i l l o g r a p h analysis*, itrom the above i t t o l l e r s a t once thot b must be garo s i n c e a b i l a c o n s t a n t witis r a s p c a t t o c u r r e n t * I f warn wo- put. h - 0 i n t h e agnation Sp - 0, the l a t t e r rofiacos t o "A - 0, w h i c h ishen m u l t i p l i e d out mil o i m p l i -f i e u y i e l f l s t h e ©quaties (c>LiLi. - Cr Mz) ?3~t (CIRT. L/ -f Ci R\< h z ) Pz -f (Lz -fRtR,C,)P -+ H z ~0 This i s a c u b i c .in p which w i l l have r o o t s —e< „ sna — - K j , shore fa m?y be e i t i i a r r e a l o r irssftin&ry* I f fa i s I s r ^ d o ^ r y os c i H o t Sons % * i l l occur o f frequency % - fal?. 7T, <ai£ i f r e n l , t h e r e u i l l be pro cert o n l y u n i S i r e c t ! onnl i a r u l s e s . i&% lis taho th© eano r/ho r a b - O and msbatituto numerical values for the fsogncto constants In tho above c u b i c ia pm have the follow lug m a s o n e d vaXuas s «*02 C x - G*C41 EX Z 0,11 h 2 - 50 if - 0*15 Eg * f x 1 0 5 r h i c h v?hen s u b s t i t u t e d i n tho above aquation and simplified This aquation f a c t o r ! s e a into tho following terms tthich gives tho f o l l o w i n g roots for p P, = - 3 -2S- x/a r Pz. = - z •Ss'x. SO *-h jz -6 X/d* Eonco the gassrsl solution cost Gins the following terrsn Tm frequency of the ahove oscillation is ~ f8j/2 7r . 2„6 x 10 4/2 7T s 4200 c y c l e s per second* This £ro-cuency arrived at by t s f c i n g for Jig the value f?fciefc i t has at 3500 cycles por second* By o x t r c p o l & t l n g frois the graph shotm i n figure 6, i t is x o o f i i l y soon that tho actual value of Sg should he soaowhst ahove this v a l u e , of tho order o f 10^ ohms 0 lit this case, i t in gaits p o s s i b l e that the Sacrament of the c i r c u i t w i l l bo above the c r i t i c a l value and o s c i l i a -t i e n s cannot take p l a c e * This conclusion is oonfirissS e x p e r i m e n t a l l y by tho oscillograms snotm on plates 16 sad 1? , 7,'hieb arc of tho voltage across tho sperh fitting- this period* These traces f a i l to she© any o s c i l l a t i o n s of tho above s o r t , but do shot? corresponding exponential t e r s e • shea the voltage aeroca She apart p l u g has decreased to tho c r i t i c a l valua a, ( n o t n e c e s s a r i l y the Q S K C value of a so at hreaMown) the aparh is osrsiugai ished ana a f u r t b o r a c t of t r a a s i e n t o say he s e t up. I f the c u r r e n t f l o w i n g lr» the aparh Juat before e x t i n c t i o n i a i 9 tho m t h e n s t i c a l develop-ment is i d e n t i c a l a i t h that given i n s e c t i o n .2 c n tho opening of the- t i m i n g c o n t a c t s , K i t h tho exception o f course of the interchange of the p r i a s r y and eocoudsry constants* -5?hus the operational solation for tho secendory c u r r e n t Is ' C.P(fo±L±P)(Rl-tLlP-fi/ClP)-C^MLf>3i_Ri-i.LlPi-l/£ip 1 fmm which, i f th© r o o t s of f»p s 0 are — oo t-/2, and -«*,. ±/5z , and ' /3, and are is^gtxiary^ the following: general e o l a t i o n is oMaiiiet t = A< s/'n(/*.t-0f) -r Az 6 1 s/n(/3_t-0^ The froauaneSaa of these o s c i l l a t i o n s are » /3,iZ~K and It wae found o o r Xtsentally that at tho e x t i n c t i o n cf the eparli the c u r r e n t d i e d away g r a d u a l l y , rather than suddenly as v:ae assessed ia this development* In this caee, s i n c e tho c u r r e n t at o:€tiiietloxi ia s«ro» it f o i l OK-a a t once t b r t tba o o c l l l o t i o t i s of irecu.oncisa £^ and f.^ s i l l be entirely ahacut* cay,, hesever^ esy-ect a voltage rS.an to-vrards e x t i n c t i o n of tho aparlc due to the removal of tho load fro® the nngnetc*. 0M)$mG O F ms muxm o o m c f a , It £ Q poaeihlo that ihm .timing' contacts cjsy elose while tiers f a .still as-.-sppyaslahle voltage stress- them* i n efatch case this auMmi change of c i r c u i t constants, amy result In farther trossionto* I f the total voltage osSeting* serosa the t l s i i B g contacts at the isatnn% of ©losing ho B„ the effect is tho sa«so m applying -3 to tho switch A astf emitting, tho conSossor ftom. the oiycait, figsro 2* Sifa^llfleatletj eaa he -mate by neglecting 7, sua C am! tho constant tors a of the spark' crop i n case the latt e r i s s t i l l i n pro^reaso From the relation can bo obtniaoQ tho operational solution for tha ceoouaary current i 3 ls = Q-h CzbP) MPES •__ , (Rl-t/-,P)(Hz-f-b -t ftzCz bP-tLzP i-1-2. c_ b Pz)-(n-ubp) MZP2 She vol tags appearing across fetes s-eeoisdsry is e , t'm opera-tions! solution being es = (R^-tUp) is _ (RL+LzP)(l+CzbP) MPE 1 (Rt+i-,p)(R^-tb -f fi2G bP-f LzP-i- LzC^bP2) - (1+CzbP) MZP2 If the roots of this cubic i n p, Z = c are -o< end — °c. i/^/ the gonef-sl solution t r i l l bo a£ tho 'fern. If ta isaglsary* /l<f s/n fp3,t~-•<?,) The froouaucy of this oscillation w i l l bo s (3,lz~K» In tha COB©, of the aagaeto usefi in theae invssfei--gstlsas e i t isss not p e s s i M s by S B J sorosl sfl.fustssnt or apoefi vsristio&« t o cmioo tho tising eesifcnefce to eloce vh$K th&re rsss s u f f i c i e n t wltsgo across fcfcoe to prefiaco O K otjssgmiolo fc2?w5o£cmt« Hoscnror, s?e !iave mt p l a t o 7 nn oscillographic f o& oeo i Z l a t j i o s ^ b i i f c c l o o s l y rooc&fcloo £ « 0ael.21«ti&i5 ic 81m t o «2Ci«£ at the timing eo&fecctc last prler to their opening, ane i s o f fyoqaoncy nbout 2.6 si 20^ c y c l e 0 per soe-OBfi* How, if??© o^rtrspola to f r a r , the grspfe is flftorc 6 f o r 3 l ? st 16 2t 20** c y c l e s vrc t r i l l obtain on enortioiia vnl"ao« A c c o r d i n g l y •«© ctsy ecusidor the socosdszy c i r c u i t es ©pea c i r c u i t e d * ( i n r e a l i t y there i o s t i l l ci p o r t i o n of Cp asaroas the 33oona*ypy Sua to too i n t e r n a l filstslbstafi c a p a c i t y ) r i h e o s c i l l a t o r circuit thus fceeOESfJ the pri&iij^ circuit ot3l$«. ees-d a t i n g of C j * Ej..s ©B4 SB. a&k&o%& t o r n tes to fins ere roBistS3CQ#. S£ an insp^eflcta o f tho. s a ^ B i t s f i o ©f the q&*mt£«. ti-es i n v s l ^ s a , . ©o far -jsassiolo^ eiii. all.0f?i«g. £©?• the i u t e r a s l saomito*? csp&ci% # i t i s ecsm thet feis eireait w i H . give sseillatioss c f the eesrsroet srSaa* o f aftgi&tttfie, BitaeXi^ / . 5 about 16 5 : 10 cycloG por &eoon£« fhs preceding aas&ye/ia hm nhovm. tbet there ©re 9 possible fraq&oacte© o f o e e i l l o t i o a comjoefcsS srlfeh the couplet© osgssto ignition system. Of theses,,, o s l g twe.B f ^ an3 ©re l i f e It? te ba o f *®8io £$QqmnB$* fhs- resninlBgr e«rven srfces fcfeoy «xl&t; ero to ba fstisafi in tbs asMs f r e c p t aey rang©.* Xn th© aeta&l m g s e t o only © vsr^r few of those frequencies are founfi, the others baing. e n t i r e l y absent OE reploeofi b^ h y p e r b o l i c o r &sg?anost£&l SiwGtiom* Of coarse,, u 3?mtrler im^Xriiie ray ]>c-c&o'e &f ssy of tt&BG ^m-Q fercs^ i s t?hich esse s l a r g e jamoor of harmtmte fregseaeiss m*? %@ o»§ei»s£# £sch em ssalj-sla i e , Ii0wa¥ore of l i t t l e TO&US cissec It is valid o n l ^ for the porioe co^oroa I'-*? fcbe f i r e t half G$Q1Q of tho fttjaaaca?atol* In c&Sition to tho eT»TO 9 pure oscSllofcloMQ, there is a possible oscillation of the apsrJ: itself^ trhich we ca;? •call fros the vex-f smtitre o f this osoillatioa i t i s sppsroist that the mm tax® s i l l hertll^" 1® sintteoiojaX* hosee i t is reaeo&stoXe to espsot a eosS es of BarsofiSc o s o S l l f c t i O B B e-f this frequency? also. It i s quite* gpparoat that th© forenoiug? analysis is of raeh «<s?a us© i n intorpretoDeorvoo" osoSilntlcsjic thare is predicting them* Is on*? csso tfc® nmlrsie holds rS g i e i ^ o s l ^ fcr trriiii5S«r»ts o f an osoilintor^ stature* tiinm tmat of tho circuit constants ser© &©$«rc&ne$ fos g*e?r£o£ic forces n»6 not iiEpalsce* Ale©, unless the r-etis&ttng trass isat-iJ oars Jaws, i t thecomas - estrone!;? filffiealt to pi 6? cttt saifesMo' nttfaericsl . values for etiijstitatios* ffcs. chief 5us£i£ie&tio& for «ai«g .this saothoc* of .analysie is the s c l o t i v o l ^ dimply cgaatioao which result* : An tmclycl© tretiM comit of a complete sixniertcsl eolutioa Bsr a l l possible tremicRts, involves the ueo ©f ?so6oi©s a a i heco-acs co eeiaplicatei se to he uoll^nigh Issclnole* So&eve?* sines this thesis l a coiicerDctf chle£l> t-?itb ©acillotcrjr transients, the acre sisjjip sss&lrsis is e,cito auQimf.te* en P 4 m:m n i a c i n ? E C J M I O B A L P i s t e 3 i s an .ooeXllogaroB fc alien as a o b o c h f a r tho c a r v e s chews, i n f i g u r e 7» "^ha t o p t r a c e i s that obtained from a 50 t a r n c o i l ©ound o i v o o t l y o n t h e pcaaaancn* a a g s e t g * 2ho s i e c l e t r a c e i a o f tho volSago g e n e r a t e a i n the 10 t a r n s c o r c h c o i l mntkti m t h o c o r e t i l t h the s a i n c o i l e * Tn& p y t o r y and secondary r o t a t i o n a l v o l t a g e Trove f O T K E s e r e found t o ho i a o n -t i c a i ; v i th t h i s t r a c e * She peak value on t h i s irravo form a o r r e s p s i i i a t o a aasissu© prim a r y volfeag© o f about v o l t s * The lot7er t r a c e i© a 60 c y c l e ticl»g isav©» OOm: I?ICO?J 01? OP>X QXBC9Z?. t l h e s the as&gnot© i s operating' •©» epos c i r c u i t 8 J © opcainp o f t h a timing contact© cuusa© a v e r y h i g h w i t age t o 00 i a f i u c e t ! i n tho secondary sdnniuge S h e - i s a r l a u a v a l u e o f t h i s v o l t a g e trae ssasrsrafl by a s p a r k ga-p sno* foufitl t o ho a p p r c T T i n-ately 9,000 v o l t s * P i s t e 4 shews fchs r e l a t i v e t i m i n g o f tha v a r i o u s phsnofasnu ©eourriag wx apesi c i r c u i t y a s w o l l m t h e aan-sral shape o f the lc?;ar frequency c o v e n a n t s o f tho wave form* S»c t o p t r a c e was obtained by a m p l i f y i n g a a s s a i l v o l t a g e f r o s s v o l t o g s S i v i a o r eDimsotefi a c r o s s the s e c t a r y B v?Ith t h e two stage a m p l i f i e r p r e v i o u s l y <?. e s c r i b e d * a no" atop-p i n g c i o ^ i i tho r o s a l b i n g v o l t a g e , w I t h a 2 0 s i p o t e n t i a l traxts* fcrraor* 'xho n i f i f i l e t r a c e l a o£ the a r i n a r y c u r r a n t , t h a value |aot a t hrooh being s t o a t 0»6 apposes. She l o s e r t r a c e so o f t h e p r i e n r y v o l t a i c . JFlate 5 i s an enlarges t r a c e o f the a o c o n c a r y opsn PLATE 6 . - 2 7 -oixeait volts-pe t okos % acspl i fy i r - t o u s s l l nreltnge f r o c i « w l t a g e fil-Jiaox c i t fc s VK 24$. m e m t n taoo s^ th the ^ S t e t o r elocsnfc ttiareetly i a the plofco c i r c u i t * t o t a l r e s i s t a n c e o f I t e T o l t e g s fiitrMer t?ns o f tho crfior of 12 s: 10^ cheo onS tho tcoe T?T50 B O biasofl on! operated" thot too s a t i r e x^svo fo r© • t?oo ^ o l l V3ltlil« the r ® g o o f l l m s r o p e r a t i o n * An i&epoe t io t i o f t h i s t r a c e . ©feows e t th e feogimjinr? o f • en o e c i l l a - . t S a n c f f requency 2160 c y c l e s per ascetic* Jte t h i s i s the recoi i i in t f reqaeacy o f tho o s c i l l o g r a m e lement, .St la d i f f i c u l t t o s a j ^ae ther o r .net t e i o f r e e « e n e y tseloags to fcftie cev© fona., Hos?evorp e « i n v e s t i g a t i o n tsy ,«seors o f the cothefie r a y o s c i l l o -g r a p h e s t o o l ioheS fcfce exlotoace o f oaeii an oBoi - H a t i o s i M t o f fr-etasaey B%m% J^OQ- c y c l e s par ooecaa.. ^ I t i t e 6 ..sne^s a csti3« oSe ray s & c i l l o c r s n o f the t j g g - i i i t i i s g o f the af&ova vmvo s p l a t e 7 5o tlAQ ea»e t?5tii the a a i s i ^ e c t i o s a l c o ^ p f j i i o s t gpeo t ly re£sco6 t<y c» extrer^ely a s w l X concenter i s ««rlo3 w i t h the v o l t e s ^ f!S¥icIai% /© -may fee s een , t h e r e i s i n i t i a l l y m o o c i l l r i t i o u o f la r g e nG-plitaSc?^ &*oojaonoy 5500 c y c l e a p e r eeeoofi w& &&umm&t. Q*#2 se X 0 * # «M>ch i s folletiaa* oy s asji-» ' f i i r o c t S o f i s l cGcponsnt nfeich S e e r e f i s o s mmh e o r e . s l o w l y * l a e t preced ing Use 358® oyelo o o c i l l s t i o n m p i n t o f tfeere e p F ^ a ? ® a n o t h e r o s c i l l a t i o n of £recjK.escy 16 s 10^ c y c l e s crafi Hecreoent 1*2 2-: 1G^« f M s o s c i l l a t i o n i s flue t o e r c i s i g e t the t i © l n g c o n t a c t s ana sppesre o;a s e v e r a l o f the o s c i l l o g r a m * £ h o r e i s no p r e v i s i o n aoac i n tee s t t t h c y s t i c a ! a ^ o l y i d e for fchie o s c i l l a t i o n "but i t i s apparent t h a t i t w i l l ©c v e r y n e a r l y idoafc iee l w i t h tee freq&eney ex io iag : on c l e a l j i g tho timing - 2 6 -CO?i'tSGtC a . • €02BJfI0"3 UK£3H LOAD. fhe p r i a o r y asd secoufioi-y currents uhen the eapnoto i s s u p p l y i n g energy fee t h e aporh plug- are s h G V u a en p l a t e to The top traeo i s o f tho p r i n a r y c u r r e n t the r.asinuia value 1>s i n g , a a before, oboat G.C atapcro* tho c d a o l e i s o f the v o l t -age noratea i n the 10 t u r n mtwch c o i X e and th© bottom t r a c e I s o f the c u r r e n t tlovizt® t h r o u ^ the apuzk p l u g , S h i s l e t t e r t r a c e uao obtained b y c e p l i f y i a g t h e voltage drop a c r o s s a 60 ohm r o s i s t a i c o w i t h the two stage a m p l i f i e r &»S a t o p p i a g <iows the* r o c u l t i o g v o l t a g e w i t h the 20s 1 p o t e n t i a l fcraasfornsra Tha wQ7rSa.ua v a l u e o f the car r e n t i a about 6*> reilliajspcrea* An extensive v i s u a l s t u d y o f the above oparh p l u g c u r r e n t t?avo v?aa ciaae v.*S th the cat ho tic r a y o s c i l l o s c o p e * S B S tho i b H oning- p o i n t s Botocl© She g e n e r a l shape o f t h e wave i^ae tha eat-B as t h a t sliowa on p l a t o 8 a f h o current rose audctciiiy t o about j?Q o i H i i i s p a r o s when an c a c i l l a t i o n o f frequency g r e a t e r t h a n 50 k i l o c y c l e a , ( t h e r e s o l v i n g powor o f the i n a t r u -cioat) t o o k p l a c e 0 ?he b e h a v i o u r o f t o l a o©dilation titm eo e r r a t i c t h a t i t urns i s p o a a i b l c t o f o r i a s u y conclu.noua ro{jart)»» i n g i t s - aaturo» J a i l e r l u g this a s e i l l a t i o n tho o u r r a a t e l o t s l y i a e r o a s a f i t o a siasiiaua o f about 6 5 a i H i rape-res a f t e r which i t doers&seti r a t h e r r a p i d l y to sere* £hero v;as no observable d i s c o n t i n u i t y o r o e c i l l e t i o e irhou th© c u r r e n t coo-sou* S e v e r a l t r a c o o t?ers takers o» tho cathoco r a y o a e i l -l o e r a p h o f the o s c i l l a t i o n s occurring; at the he$?:amivij o f tho above c u r r e n t wave* "hose a r e ahetm on p l a t e a 9 t o l i e P l a t o • 1 I I 1 I' I _ j _ | | [ o ( 2 3 * f 6 7 ' 8 9 Microseconds PLtfOE $+ tit/ntired M'lctvstreoneli — • 1 1 — 1 . ^_ i_ Micro ietanc/s " FLATS 10, F L A ? H 12. *8$u 9 ebot?a fee traces of tha v o l t a j p drop across- a 50*000 ohr-i r c a i c tar,eo i n a o i l o a K i t h the spark pluav i l a t e 10 shot: a the voltage drop across a oO ohm r e s i s t a n c e i n series sdtb tho spark p l u c f a m p l i f l a d vd th the two stage a m p l i f i e r * f he time axis for the lowsr trace is ore fourth o f that for the upper trace« *?his oscillation is apparently without decrement e*iu la certainly far froei a i i m c o i d a l , thus liici catiugr t h a t i t i s most probably duo to o e e i i l e t i o u a of tho aparh itself* Tho fuBdaeeataX frequency is about 160 k i l o c y c l e s but ss e:>ny ho S O S B , mii'im by so isuch as loo^U l*X&£e 11 shags s coapttiriaoa bettcaoa an dee aviation plug am as AG automobile p l i t f v Tlw latter plug mc observed feo give those i r r e g u l a r oseilletSejis eoatSmioaaiy srhile the forssr gmro tlism only oceaEioi3talljpf» el though both wore used unci or identical condiM 02 a» •Elate IS ehetis th© volt a g e across the aprirk plug corresponding ts the curren t o s c i l l a t i o n appear 5 r*£ on plate 10« uiifortauatsly the i u d i T i s u a l oscillations are isssstcofl by the width u£ the l i n e aria the dotted offset* P l a t e s 15 to 15 are IOT: frequency traces of tho v o l -tog© appear lag- aerea© tfcs aparls plug flaring o p e r a t i o n Those trace a vr-erc taken by d r i v i n g a Irsl 245 vacuoo tabe frorv; a voltage divider scrota tho plug eemliag the p l u t o car r e s t of tho tub© directly through the o s c i l l o g r a p h element* By s u l t a b l y r e g u l a t i n g the £L l a c oat cuvroat tho p l a t e essrresit WSB not alloived to excaea 150 r . i i i l i a a p 0 r o a B thus ft vim poaaiblo to boost up the lower part of tho curve without drainer to tho ' oscillograph element* She traces on plate 13 vrora tejcea irlth m m i> PJU?^ 14. PLATIi 15. - 3 0 -the e>:ci£etio» ao rsdueed t h a t the operation o f the toha v:?3 l i n e a r , An i n s p e c t i o n o f the l e x e r t r a c e s h c e a t h a t the v o l t -age -aerobe the opark i s p r a e t i e a l l y constant f o r the g r e a t e r p a r t o f t h e pe r i o d * "here i s 6 hcv?evnr e c s l i g h t i n c r e a s e i n voltoge ao tho current ait?pa t o very Itre values t l u o t preceding th e e x t i n c t i o n o f the apar£* Following- the e x t i n c t i o n o f the sperfe there i s a s a a l l modest Increase i n v o l t a g e a f t e r » M c h the carve f o l l o w s cXutoly tho corresponding p o r t i o n e f tho open c i r c u i t curve, ?!aftos l i ami 3.5 a r e the lower t r a c e on 13 Hotftod up cc that t h e w a l l e r v o l t a g e s f e l i m - i n g the esrfcinetS on o f the *ipnr£ eseewe v i e ! h i e , She upper t r u c e on p l a t s ' 13 i l l u s t r a t e s t'm e f f e c t , of a r c i n g at t h e t i c i n g con-t a c t s * Shew t h i s a r c i s extinguished" n e r v a l o peration i s r-;:£ur:;o«6 b a t owing- to the l i s t i t e o " eiiorgy ncsr a v a i l a b l e , tho ap j r h no est d i e s o u t * Tmiz n r d :*£ at the t i d i n g c o n t a c t s not en i y prod^cee harning cf th& ecu t a c t s hut rohs tho aparh. o f c o u s i o e r s o l s aueafcgr* H a t e s Ife ssM. 11 Skew the esrfeeefaely • - sharp %mvs form o c c u r r i n g the lre<&6oc?a o f the gape J?late Id i s of the voltage across tLo sp&^k plug eoa p l a c e 17 i s tns so^tj r l t h the a a i o i s e c t i o n o l cooponoate g r e a t l y reduces* 3bo voltage j n a t before T&o&M&m vaa shout 3*koo v o l t s , 'fhefcse o s c i l l o g r a m s do not shoe any r a d i o frequency o s c i l l a t i o n s f a l l e g i n g tho gap t r e a k a o s i i h a t fie show an exponential, tor© c f .spproj&asifcoly 10QO & *"1<5^ x a 0 J f l i o aosence o f o s c i l l a t i o n s s a y fcavo hoe a cue to the ikB.ap.im a c t i o n o f tho voltage d i v i d e r r e s i s t a n c e * heeaaao c s e i H s t i o n a were observed i n tho corresponds n# carron t PLAT:-; 18 MU3S3 OF I : J T X B F ^ U ? 5 C E . f h e r e a r c three way© i n tshieh- a cug&oto i f u i i t i o n s y a tesi ssi^ht cause r a d i o i n t e r f e r e n c e * F i r s t s o s c i l l a t i o n s and tiuvo f o r e s o f aaflio frequency s a y a f f e c t the r a d i o s o t fey i 2 i d a c t i o n or ceudactiou, Second,, o s c i l l a t i o n s o f r a d i o f r o -cmoccy c a y . . r a d i a t e tm£tialm.t.enevgp t o a f f e c t tho r a d i o f r o cadency c i r c u i t s o f v.a-n a c t * rhirfi„ oxtrer«ely aherp travo forms .T&y r a d i a t e p u i sG O e h i o h m y a l s o a f f e c t tho r a d i o f r e -quency c i r c u i t s * Qt tbeaOg the l a t t e r two oro the r;oot S r r o r taint s i B c c the r a d i o eat i s u s u a l l y - i c o n tod m f a r frosi the 5 c a t i o n uystOK that i n d u c t i o n i s n e g l i g i b l e and paths o f -conduct! on a r e u s u a l l y stbaisiit* Considering t f e r a d i o f r e q u e n c y o s c l l l f t l i o K S , i t ©as shorn i n the tsat h e c t i c a l m s l y e i e t h a t t h e r e wore two f r o -cueneiee* f x and f ^ t wM eh r.'ould p r o b a b l y ha o f t M e order o f ©altitude,, aud om i n d a t a r s i i n a t Q f r e q u e n c y #, ^%Q* &M t 0 t h e c s o i l i n t i o n o f tho spari:, a l s o probably i n t h i s range® The numerical values o f these* o a c i l l s t l o m s do,pp,D.d to such as ejr-tonfc upon e t r a y inductances capaoi feies o f tho c i r c u i t and condition© i n ths eparfe, t h a t any a t t e m p t t o evaluate thorn i s p r a c t i c a l l y hopelosa, riha h o s t wo c a n aaconplish i s t o f l « e H a l t i n g condition© f o r tbe. occurreuee o f th0.se o s 0 i l l n.tl.oua s?n thrt. i f I t *s Sees rod t o e3.iETl.PGte t h e n tr© can a d j u s t tbe c i r c u i t constants co t h a t t h e y cannot occur* Sharp vav« forns occur vbemovor there I s c oufidsn cheapo o f e i t h e r current or v o l t a g e , each as a t th© broaMovrn -o f the QpQfk gap e. troci ihe electromagnetic theory i t e s s ho iihoita that a change o'Z c u r r e n t ca-aaoe the r a d i i * t i e n o£ e isng-r ; c t i c p o i s e , or v e c t o r , the c&cnitu<!o o f t?hich i s p r o p o r t i o n s ! to the r a t e of change c i * current c upon r a d i a t i o n t h i s osgne-• t i e v e c t o r i c a s s d i & t s l y eeaceiefces i t o e l f ^ i f c h a n e l e c t r i c r e c t o r s t r i g h t a n g l e s t o i t t'rdi the tt;o becocs© a p o l o r i s e d electro-magnetic trr.vc« 3 i s i l a r l y a change o f voltege prcducoe on e l e e t r i e v e c t o r t?hieh i s p r o p o r t i o n a l to i t s r a t e o f ch&ngo, one z h i e h , when r& d i s t e d prodncse- e. perpend Sealer segno t i c v e c t o r and "becomes e l s e a p o l a r i s e d ©leotree&gnetie t?avo* •These v;sve& hove a var;? i n tie f i n i t e s f r e q a e n c j r ? hut provided thai* have enough energy ere quite, capable o f ar-tsotlsgt the r a d i o frequency c i r e a l e n c f a r a d i o ueb* -The i d e a l ronody f o r r a d i o i n t e r f e r e n c e i s o f coiircs to e l i m i n a t e the tjcuree* In t h e ease o f c s e i i l n f c i cue doe to a t r a v inteetsnoee. end c e p o e i t i e s t h i s , i s sfitita fomeihie,, hut o s c i l l a t i o n s o f tho aparh and e&arp %mm f e r n s nrecont c c n s i . d e r s h l y s e r e . . d i f f i c u l t y i n t h a t s e have no a o a & t i o n s f o r t h o s e phenomena froes r b i e h may compote p o s s i b l e l i m i t i n g c o n d i t i o n s * • I n s e c t i o n 4 o f t h ~ rasthssiafcieel a n a l y s i s ! -ore-developed eeuctionn f a r two p e s s l o l s froqaencies o f o s c i l l a -t i o n , Consider 'chs 'Xre-qr.o 120.7 in order th c't c. s c i 11a-t i m i s «sy t*:I:c p l a c e , the q u a n t i t y under t h e r o o t e a s t lie imeyinery, hone? f o r n c n - o s c i l i e t i c n t h i s q u a n t i t y suat bo e i t h e r r a o l o r earo, c r eEprscasa nn£hesiat;i c u l l y ( Z-bLC)2- ' bLC frc-s vrliSeh tre g o t the r e l a t i o n A p p l y i n g the oar.® racsoning t o f,, r.-o .f;ot the r o l o t i o n s r R) z _ / TUB above r e l a t i o n s i u a t i f y the use o f o o r l o o ro^Se-fcaneoi, eorauoreially 2OJOV;U aa *"i5Uppro«3ora?.B s i * a eoans o f reaucinj* r a d i o i n t o r f e r o n c s * 'rho one s e r i o u s orawfesefc t o t h i s p r a c t i c e i s t h a t f o r s a t i s f a c t o r y o p o r s t i o n tho vela© o f thouo cupyi'oa^oro i s <if n e c e s s i t y about 50,000 ohca, ami tho Inse r -ts en o f t f c i i i auch roeioteneo i n e o r i o a w i t h che £>pnr*: cansou a carious, ra duct Son i n tho cvxima currant florin.« tbrottffc tho gop ar?& hence o d e c r e a s e i n e f f i c i e n c y o f tho- e n t i r e i g n i t i o n .'u roipirda fcho sharp iv&vo £01« n* vory l i t t l e s con ho dons from a t h o o r a t i c u l j t u n c p o i i J t . Sharp curro a t nctvoc could bo smoothed" out by i m i a r f c l r v ; inductance i n t o tho c i r c u i t , t h n t I s i a e r a a a i a c 1, but th . :» i n turn v o u l d n.eccas:i tuto iat-roaaizip E which i s a l r e a d y t o o lnri?o f o r pood operation* 3 h a r p voltvm tsaves could cost "00 dool t - ; i t h by curofwl ohi n i c i n r - c f o i l o."Ulr-;:.o-it oulvGetcd to i;];son c-i;rf?6ae r.uch Ohio Id in** - 3 * - — . i n c i d e n t a l l y s c u l d a l s o provo a a e f u l i n reducing r a d i a t i o n -fro::: any other disturbances which c i d i b bo pronont* oi n e e t ho o s c i l l a t i o n o f the apart; i t s o l f i s duo o n l y to c o n d i t i o n s i n tho spark, I t i s e v i d e n t that r o can e 15rrali-n t o t h i s o s c i l l a t i o n o n l y b y a l t o r i n n thaaa confiitionso Un-f o r t u n a t e l y , tho o n l y c o n d i t i o n s over tsliich v.'o have enroot c o n t r o l are, tho c u r r e n t f l o w i n g and tho dirsonoSona o f tho gap, tho others boli)£t e i thor dependent on these o r f i n e d by tho con d i t i o n s o f operation* J u s t ivhat e f f e c t these q u a n t i t i e s have upon tha o s c i J l n t i o n o f tho anar2c s u n t bo determined by cjrpoi'lr>3ont sineo a t h e o r e t i c a l i uvea t i {rat i o n i a oat of the C j t l G S t i O S e ' TOT tho pnrpose o f monitoring the r a d i o i n t e r f e r e n c e a throe tube r a d i o eat raa i n a t a i l e d approximately 6 f o o t away fro*" tho I g n i t i o n oj/stco under t e s t e 5?his not eonaiafcod o f a regenerate vo detector v?5th pluft i n o o i l o covering a servo l e n g t h xrmm Stotn fipijsrosifcatciy 50 t o 500 wo t o r s , w i t h a tr,*o .-;tGt{ro O K p l i f i o r . IHfro c o r l & l c sroro provifiod, a 3 f o o t l o o p and r r?5 f o o t outoido vij:o« She act i t a e i f z:as c o m p l e t e l y un-a h l d d e d and ifas ©nrthod only thrcnyh tho l a b o r a t o r y b a t t e r y f r o o t?h J ch the p l a t o c u r r a n t vm drawn. «>hon tho I g n i t i o n aye to..< r w? f u n c t i o n i n g n o r n n i l y tha i n t e r f e r e n c e wsa uo bad that l o c a l broadcaatia,{.r a t u t i o n s I Toro completely drowned e u t s TUB i n t e n s i t y o f t h i s i n t e r f e r -ence r a s n e a r l y constant over t h y e n t i r e rau^o of the s o t , beir.p o n l y s l i g h t l y greater on tho shor t e r v,*aveOo xhoro sreus ~35~ no a p p r e c i a b l e i n t e r f e r e n c e r.hoii vim spork p l a ^ m e hi ceonnoc-t3d and tho wDg-nfito operated on opon c i r c u i t , , d n b n t i t u t i e n o f E .-inhere (*ap f o r the spnrk plug- r e s u l t e d i n t h o i n t e r f e r e n c e l e i u r ai>r.-ar;hfit l o e c l i r.ad i n t o d i s t i n c t x-mve l e n g t h hnnds, hut thsoc tvoro ot3crvod to c h i f t about i n a var;j e r r a t i c ^nuiiar, A v a r i a b l e r e s i s t a n c e WBG connected i n eorioo r r i t h tho sparic p l u g end i t s v a l u e v a r i e d f r e t s acre t o 50,000 ehsae* The - i nterforonee woe observed to de creese a t e n d i l j es the r e s i s t a n c e isns increased u n t i l i t n e a r l y vanloft*A ufc s v t l u e ol' 40,000 o h a j c Jfetcreaeinjr the reuiotnnee f u r t h e r hod nc n e U c e e b l c e f f e c t , u n t i l each a value was reached th n.t R oporfc as lender ocearrsd, i?hen tho Interference dropped euddenl:.- to & n e g l i g i b l e v s l n o * M t h no r e s i s t a n c e i n s e r i e s tho sp&rh •rsts b r i l l i e n t l y white hat 2 3 r e s i s t a n c e ?res Inoertee i t become blue "aid l e e s b r i l l i a n t u n t i l at 50,000 ©tow i t v-«os n foehlo blue l i n e * ' A eoneontrS e aMolded c a b l e t?a- t r i e d e.a e l o r d f r e e the s-.<?£noto to tho aparh p l u g 2i:d t h i s a f f e c t a a «, eensidoroblo r o c u e t i o n i n the i n t e r f e r e n c e , but duo to the uusbioiced p o r t i o n s ofc the ondu aorse energy «ee s t i l l r a d i a t e d , * i t ©as found t h a t s o l i t t l e as an i n c h e f anshielded l o a d van e u f f i -cic?".t to cause ob'octicn-uclo i n t e r f e r e n c e o Koivovar, *ehoii t h i o i c e d uoo need t?lth a 50,000 elm suppressor nfc the p l t t f , th© interfere*.ice r n.: i i c i r c o l ; ; i;oticnnblo oven thourrh & poa t i o n of the c a b l e regained unuhiolded, F l s t o 1G jhou-n the e f f e c t of she 50,000 ohis sapereesor on tho /»0 p l u g current* -2he l o w e r trace i s o f the a w n l i f l e f i voltage drop neross e 60 ohn r e s i s t a n c e without tho . s u p p r e s s o r I n c i r c u i t * i s s a y he noted t h e r e i t - an o s o i no-t i o n o l s i l c r t o those e p r o s r i on p i c t c a i O and 11 and i s rrest p r o b a b l y duo to o s c i l l a t i o n o f tho sporh:* The s p p e r tre.ee i s o f tho sane v o l t s g e c r o p hut K i t h the 50„ 000 ohm . s u p p r o ~ u c r i n s e r i e s * *s GCJ he aeon the - s u p p r e s s o r ime etx j -p i o t o i ^ e l i m i n a t e d the o e e i l l e t i c n a p p e a r i n g on the l o t ; o r t r u c e , h o t t h e r e i s s t i l l p r e s e n t a v s r v m a i l o s c i l l a t i o n o f f r e q u e n c y about 45 k i l o c y c l e s , A v i s u a l i n s p e c t i o n £&va p r e c i s e l y tho saae r e s u l t s f o r the O l c o p i n g * 13ES1SH OF 2l£h S F B C l A h i S P J m P 3 £ S , -I t appears free* the p r e c e d i n g o s r p e r i & e n t o l i n v o o U -•gsfcions t h a t the r a d i o i n t e r f e r e n c e a r i s i n g f r o a t h e s p a r k p l u g s p a r k isev hs r e d u c e d t o a s a t i s f a c t o r y l e v e l i f n o t i i l i f a i n a t o d e o c a p l c t e l r , by tho mm o f e 50,000. oha s e r i e s r e s -i & t a u o o o r »sappre»eor f, eoraVineS v s i t h r-. complete o y s t e r o f s h i e l d i n g " . However t h e r e i s n s e r i o u s o b j e c t i o n t o tho us© o f v lerpro s e r i e s r e s i s t a n c e i n t h a t tho e f f i c i e n c y o f tho opnrh i s .thorefcj- r e d u c e d * I t wag p o i n t e d o u t i n . t h e l a s t h e c a f c i e e l o n s l y e i c t h a t t h e c r o s s s e c t i o n e r e n o f the spar!: i s p r o p o r -t i o n a l -to the ©arrant f l o w i n g t h r o u g h I t , hence a r e d u e t i o n i n c u r r e n t causes a p r o p o r t i o n a l r e d u c t i o n i n the siste o f the ^perfc* :.Viih t h e s u p p r e s s o r i n tho c i r c u i t tho nsxiratis c u r r e n t i s n e v e r each gsfo&ter t h e n t h e m e n c u r r e n t and t h e r e f o r e tho .. S X E S o f the sp ^ r f c i s isore o r l e s s d.etextiSned' by t h i s me&n c u r r e n t * Zi t h o u t tho s u p p r e s s o r t h e r e i s on i n S t i u l r a s h o f c u r r e n t due t o the d i s c h a r g e o f the s e c o n d a r y s t r u y c a p a c i t i e s , w hich i o many eiscos the v a l u e o f the fcoan c u r r e n t , T h i s r.sy be - 3 7 -SC-CJS by r c f a r a n e o t o p l a t e s 9 iniu 22. STJii-s l a r g o i n i t i a l c u r r e n t w i l l r e s u l t i n cna c r both o f two t h i n g s ? tho r r o c o f tho s p a r k t ? i l l be i s i t i a l l y l a r cto o r tho t o n p o r a t o r e w i l l ho hiphe e i t h e r e f f e c t asking, f o r a g r e a t e r e f f i c i e n c y o f t h e s p a r k . I f now a condenser o f v a l u e e q u a l t o t h e s t r a y seeen&ery e u p n e i t i a a be p l n c e a a o r o a a tho aparh: pin;; hctu'aesx the ur.ppreaaor ant! tho gap, St 3a ap p a r e n t t h a t tho i n i t i a l coT.'tlit! o-:s -111 be . n s s r l y t h o aaro a s w i t h o u t t h a auppsmasor* That I s f t h i c eondouaor i v i l l a&rgo up ihhrouyn tho auppreiinor roo5 *taacp u n t i j tho voltn/te *roco^oii M y h e?iou£h t o brawl: c o r n tho pups r l n n i t w i l l d i a c h a r y a through, the ca-ap r i t h n c u r r e n t c-l 7 o a s t a,a f r a a t aa t h a t d"o to t h a diacharr-e o f tho secondary G«racItie3o * h i a i n i t i a l c!5schr?rao r i l l bo f c l l o r a d by tho u s y - l ana 11 c u r r e n t i f n l o h t h a auppreaaor p e m i t s t o f t o r ; , r i t h tho above earslAoratf ons i s s t u d the s p a n : p l u r abotru J u section §r» f i f y r r o f r a n c o n s t r a o t m i , The i a a t i r p o r -t i o n i s tho - r e g u l a t i o n nparlc p l i a r baeo * ? i t h the e x c e p t i o n o f the uph^ro r a p o l o c t r o d o a , vhe nppor p o r t i o n contain-: the b u i l t * * i n eoauenss£ 0 2 h l a c o n s i s t s o f a s t a c k o f a l t e r n a t e l y l a r g e cud s c u l l a l i M i n n a r r s s l i a r s s s p s r & t o a by Ki-ea ca.sk eves* ^hc* .M'.nll crashers connect to tha c o n t r o l r o d VMC t h a I r i r o cues eonnact t o t h o o b e l i * 2ao c o r e 1 B o f isles, and. l a h e l d t o g e t h e r hy 'the c o l l a r threaded onto she c e n t r a l r o d * 3h& capaalt;.. o f tJio condenser aa r-osaurea by a c o a d o r a o r b r i d g e r a n fount, t o bo S5 r-; c r o r , i e r o f a r a d s , u l i f r h t i y l o s a t h a n tho s t r a y s e c o n d a r y c-ip -'C11;.o i t v.'ii-u a o o i r l c c t"; \iz,c- job-: r l ^ c l ^ p n r J r i ny a l o c t r o d o a P t n e e i t r a a found t h a t o s c i l l a t i o n s o f the s p a r k aoldom o c c u r r e d tri'sh s l i i a type o f e l e c t r o d e «?hoi3 the current v-as rol'.-:15 v e l y o a a l l * P l a t o .19 shows t r a c e s of the current through t h i s s r a c i a l p l u g , tho Upper t r a c e h^in^, o f tho a o p l i f i e e voltage drop across a ofesj r e s i s t e u c e w i t h o u t a suppressor, e«d the l o s e r ooo ths SOD© hat v-ith o 50,000 ohr< suppressor in s e r i e s * £s asp be ceen, there i s a t no time .on o s c i l l a t i o n o f the spar!: u ; K i the t r a n s i e n t uhich coos appear o n the upper t r a c e i s g r e a t l y reduced on the l o r c r t r a c e v. hen tho suppressor i s i n t ^ s c i r c u i t * rue c u r r a s t trove forsi *?as nuver o&servod. to •deviate to any exteat from t h a t shewn on t h i s p l a t s * fhe v o l t a g e s across the s p e c i s i l p l u g and the /.0 p l u g with tho ^0 ,000 ohm suppressor i n a e r i e s , at the hroahdovca o f the ffap„ are show on p i s t e u 20 and <?1 r e s p e c t i v e l y * t r o t ; -'those t r a c e s i t appears t h a t breakdown o f the gap i s r e l e t i v o ~ l y l a t e r i n the esse o f tho s p e c i a l plug by about ,$G ujicro-eecoods, otherwise the t r a c e s are very s i m i l a r * Txat* v i s u a l ohsorvatiens i t was found t h a t o s c i l l a t i o n s s i m i l a r to those sppoari^g on p l a t e IS o c c a s i o n a l l y appeared I n the volfearo 'oeroos the AC plug even P i t h the suppressor i n s e r i e s but ouch o s c i l l a t i o n s 'g-oro c o v e r oboarved In t h e case o f the s p e c i a l plug* Plate 22 shows a ceesporXsoji o f the currents f l o u i n y i n the th.roo plugs used 'in chose i n v e s t i g a t i o n s * -...'hose traces are a l l o f the voltage drop across a j?0,000 ohia u a r i e s r e s i s t a n c e * the top t r a c e being- from the .AO plu£, tho n i a d l e *ron tho Oleo plur- S'jc the bot ton f r o o the er.oeioi .plug* irhoao. t r a c e r do not represent say p a r t i c u l a r c o u c i t l o c u oecorria-* i n operation sines tho placing: of th© o s c i l l o g r a p h d o f l o e t i o n p l a t e s across the j>0 0000 ohm r e s i s t a n c e introduced considerable extraneous c a p a c i t y i n t o the circuit« fhey ec» however, i l l u s t r a t e the l a r g e i n i t i a l c u r r e n t a t the breakdown o f t h e gap a s w e l l a s the tisao tahen to charge the condenser of the s p e c i u l plug to n s u f f i c i e n t v o l t a g e • Ine o s c i l l a t i o n s shown on t h e s e t r a c e s d i d not take place nhen the o s c i l l o g r a p h p l a t e s were d i s c o n n e c -t e d frcrri the c i r c u i t , a s wag a s c e r t a i n e d hy aeans o f the r a d i o E C t o B i s spar]: p r o v i d e d hy the s p e c i a l p l u g was t y p i c a l o f the diocharging o f a c o n d e n s e r , being b r i l l i a n t l y white and accompanied, hy the u s u a l snap*,' f h a r e was o n l y a s l i g h t de-crease i n h r i l l l a n c y v/hen the suppressor was i n s e r t e d i n t o the c i r c u i t a She r a d i o i n t e r f e r e n c e frons t h i s plug rras a l m o s t a s bad a s from the other p l u g s , without the s u p p r e s s o r , hut w i t h i t the i n t e r f e r e n c e dropped t o a n e g l i g i b l e v a l u e , and. when the s h i e l d e d c a h l e was used* a l s o , v a n i s h e d e n t i r e l y , . In t h e E n t h c a a t i c a l a n a l y s i s i t v;as shown t h a t there were i n a l l 10 p o s s i b l e frequencies o f o s c i l l a t i o n f o r the complete oagneto i g n i t i o n system. Sane o f these frequencies were observed, cone v/ere p r e s e n t but i n p o s c i b l e to l o c a t e , and cone \7ere e n t i r e l y 'absent* The f o l l o w i n g t a b l e (jiven a summary o f these o s c i l l a t i o n s * iYbsent ao an o s c i l l a t i o n bat appears oa p l a t e 6 as an exponential# Appears oa p l a t e s 6 and 7 , treoueney about 3500 c y c l e s . Cno o r both were present "out c o u l d not be l o c a t e d , d e f i n -I t a l y t the f r e q u e n c i e s were p r o b a b l y between loV and 10© c y c l e s * Could n ot be observed as o s c i l l a t i o n c but equivalent exponential terras appear on p l a t e a 16 and 17* E n t i r e l y a b s e n t . • E n t i r e l y absent but a s i m i l a r f r e q u e n c y a p p e a r s on p l a t e Appears on p l a t e s 10 , 11 , ana 1*2, about 1 8 0 k i l o c y c l e s * Radio i n t e r f e r e n c e f r o a t h e i g n i t i o n system without s h i e l d i n g o r suppressor r c a i s t a n c e c rms found to be extremely b a d 0 Such, s h i e l d i n g no was t r i e d p r o v e d t o be q u i t e inade-quate , but t h e i n t e r f e r e n c e c o u l d be very n e a r l y e l i m i n a t e d by the use of a 5 0 , 0 0 0 ohm r e s i s t a n c e i n s e r i e s w i t h the spark plug. T h i s r e s i s t a n c e had* H o o v e r , a v e r y d e l e t e r i o u s e f f e c t upon the sporl;, which c o u l d i n a measure be eonpensatefc f o r by the use o f a s p e c i a l spark p l u g i n >.?hioh the e l e c t r o s t a t i c • c a p a c i t y had been-increased, by a b u i l t - i n condenser. S h i s p l u g t t o g e t h e r w i t h a J50,000 oho s u p p r e s s o r and r e a s o n a b l e shielding„ reduced the i n t e r f e r e n c e t o such a lent l e v e l as t o r e a d e r i t inaudible« without an a p p r e c i a b l e r e d u c t i o n i n t h e b r i l l i a n c y o f t h e s p i r i t * In c o n c l u s i o n the w r i t e r w i s b e s t o thank B r * SI* T i c k e r s s Eead o f th© B e p a r t s e n t o f E l e c t r i c a l Jlngiiaeerins a t the U n i v e r s i t y o f B r i t i s h Columbia, f o r h i s many h e l p f u l sagcestions and c o n s t r u c t i v e criticise©; P r o f e s s o r V.'.B« Cool-t h a r d , a l s o o f the Separfcnenv o f E l e c t r i c a l Tiigineeriur;, f o r h i s h i n d a s s i s t a n c e i n t h e l a b o r a t o r y ; and I3r« &«!£• Shram, o f t h e aepartraent o f Physio© t f o r .the l o a n o f the Cathode Ray OacS 11 escape w i t h wfeiea a v i s u a l s t u d y ot t h e v a r i o u s wave farras v?as aade* BXBL10GMHIT. FsBc 3 i l a b e e Tciylor-Jones S.J* Berg J * J * Thomson -*» i!ath6i?mtieal Sheory o f Induced Voltage i n the High-tension Ilagneto* UeGo Bureau o f S t a n d a r d s S c i e n t i f i c l a p o r a 0 IIo. 424. ~ E l e c t r i c a l Yoaaurononts-o K e G r s w I l i l i , 191?* - Theory o f i l i e I n d u c t i o n C o i l * P i t n a n , 193%* - i l e a v i a i d e f s Operational C a l c u l u s . l i C G r a ^ - i l i l l " , 1929" Conduction of E l e c t r i c i t y t h r o u g h Gases. Cambridge U n i v e r s i t y P r e s s , 2$0p<, 42» The follov/lng I s a p r o o f o f t h e r e l a t i o n used i n de-t e r m i n i n g the c o e f f i c i e n t o f co u p l i n g between the p r i n a r y and secondary o f t h e nagneto, n ^ 2 / n ^ 2 « 1 » K 2 , sho r e 21-. i s t h e s e c o n d a r y r e s o n a n t frecu.en.cy w i t h t h e p r i m a r y open, n^ i s t h e secondary r e s o n a n t f r e q u e n c y v?ith the primary shorted and K i s the c o e f f i c i e n t of co u p l i n g * i s t he secondary s e l f i n d u c t a n c e i s the• ©eeoMary c a p a c i t y E P ^B the s e c o n d a r y r e s i s t a n c e I>2 i s t h e p r i m a r y s e l f inductance Hi i s t h e primary r e s i s t a n c e V/hsa the primary i s open c i r c u i t e d t h e r e s o n a n t f r e -cueney i s £iven by 2 oft^ where s 1 s i n c e a t /ZTcT rescnanco the power f a c t o r o f the c i r c u i t i s u n i t y 9 that i s CO< L z — , ! ^ CO 1 Oz l e t u s aaow w r i t e t h e g e n e r a l d i f f e r e n t i a ! e q u a t i o n s f o r the t r a n s f o r m e r a l o n e with t h e pri i s a r y s h o r t e d , l e t t i n g T,g represent the v o l t a g e across the secondary. Peso »-v a r r s i v j i n s equation 2, we have MJ = -(±L<%- + f 3 i i i N ) ^ A Substitute t h i s v a l u e o f i n equation 1 . Git Ez = R,C2 t U±± - J f i # - R, L-+ at Li d-t Li -43-' R e a r r f i s g i s g s a n ation j5, s?e'-.'bav& \ *~< J ait ?ro» equation 4 i t i s seen t h a t t h e secondary 'inductance i s u e e r e a s e d hy the i n t r o d u c t i o n o f t h e s h o r t e d p r i m a r y hy an unount H c / l ^ . He a l s o have the r e l a t i o n s K l l ^ t * -"^aing t h i s , -m o b t a i n f o r the new secondary inductance '•••Q are now a b l e t o g e t an expression f o r the resonant frequency o f the secondary v?ith the p r i a a r y shorted* We have Z 7 T where I f sow'we acinars cO, mmi u>z and t a k e 'the r a t i o of. co> Z/(vzz we o b t a i n a t once th© d e s i r e d r e l a t i o n 8 m follows. - 4 4 Ke&viside's o p e r a t i o n s ! c a l c u l u s i s baaed on t h e use o f the .operator d/dt, which i s w r i t t e n *p*, a s an. a l g e b r a i c Quantity o p e r a t i n g on u n i t f u n c t i o n , w r i t t e n "be u n i t f u n c t i o n i s a f u n c t i o n a h i c h baa t h e v a l u e sero f o r a i l t i n e s b e f o r e t s 0, and the v a l u e u n i t y f o r a l l t i m e s a f t e r t » 0*. At t s 0 the f u n c t i o n r i s e s suddenly from s e r o t o u n i t y . I f we c o n s i d e r the sudden a p p l i c a t i o n o f an essf IS t o a s e r i e s c i r c u i t c o n t a i n i n g -resistance It and inductance 1, ®e o b t a i n t h e w e l l knovm d i f f e r e n t i a l equation I n K e n v i s i d e v s n o t a t i o n t h i s i s w r i t t e n It we now t r e a t the v a r i o u s Q u a n t i t i e s a s a l g e b r a i c symbols ere raay s o l v e f o r 1 T h i s i s c a l l e d the o p e r a t i o n a l s o l u t i o n . I f t h e r e i s a condenser G i n s e r i e s srith oar hypo-t h e t i c a l c i r c u i t , the d i f f e r e n t i a l equation bocoiaes E1 - RI ~t l~ | f rt jf'Ueft limi Keayieide d e f i n e s Jdt as i d e n t i c a l eritfa the o p e r a t o r l / p when they b o t h o p e r a t e on u n i t f u n c t i o n * U s i n g t h i s f a c t , the above d i f f e r e n t i a l a g n a t i o n becoses - L.r-'c'p)1-f r o a which ;ve o b t a i n the o p e r a t i o n a l s o l u t i o n 1 7 / -f-Rcp-f-LCpZ-An abbreviated, sray of w r i t i n g t h e above o p e r a t i o n a l s o l u t i o n ^hers i n c a l l e d t he admittance o p e r a t o r and. ZP i s c a l l e d t h e impedance operator,, I n t h i n case these a r e r e s p e c t i v e l y YP = CP and Z P = / -P RC P -f- LC P^ S h i e aatiiod, s a y be extended, t o any c i r c u i t s no matter h o e o m p l e s g p r o v i d e d we always reccnber that the opera t o r p' must e v e n t u a l l y operate on u n i t f u n c t i o n * 2n o r d e r t o o b t a i n the g e n e r a l s o l u t i o n f ror-i the o p e r a t i o n a l e o l a t i o n sre m y as s K e a v i s i d e ' s E&paneien 2heoret& She a c t u a l p r o c e d u r e i s a s f o l l o w s t ( ! ) t ? r i t e mwx t h e v a l u e o f Zn from the o p e r a t i o n a l s o l u t i o n * (<c) 3 c t Sp s 0 and solve f o r p, which g i v e s a s r o o t s , r ^ , e t c * (3) l l f f e r e n t i a t c w i t h respect to p; l*e« f i n d d2p/dp» C O : - r i t e out P-lf* and — j / l y -(5) S u b s t i t u t e i n (4) t h e values o f p found I n ( 2 ) , (6) V r i t e and s u b s t i t u t e p s G„ f h i o £ives tho c o n s t a n t t e r n o f tne expansion., i i (7) V:"rite o u t the remainder o f the s o l u t i o n by p u t t i n g f o r each c f the stunts t ion t e r e s the v a l u e s found I n (>)» t h u s , E X I So 2ZP\ -+ ~ 6 -t-— ^_ / -46-I f v,'s are not i n t e r e s t e d i n the amplitude o f the t r a n s i e n t s hat v/isU only to hue?/ t h e i r character sasy d i s -pense with steps p% 4 s > s and 7, since those steps serve only to f i n d the constant c o e f f i c i e n t s of € Pn1~ « I f we knov? the r o o t s of the equation 2. s 0, we may w r i t e i n n e c l i n t e l y the general s o l u t i o n a s , p.t p*_t- ?jt X = Aa -h A,6 -t- Az€ -i- A? £ -(-••-• p,t HQVJ e say he w r i t t e n , i f i s p o s i t i v e 9 e ' = cosh Pit -t Sihh\ P,t o r i f i s n e g a t i v e , -fit • , £ =. coshr,t -s/.nhPxt o r i f i s imaginary 8 J a r . £ - /^ t -t s/n Pit Also i f and are conjugates, nay c( i J / 3 r;e have, A,ep,t-t A , 6 P*t= € ~-r(A,£ m-tA,e'J/3t) e ^ArtKicaspt ^A^As/n/3t) where $ i s a phase angle determined" f r o a the • constants Aj_ t .and -Af..---I n a c t u a l problems i t i s u s u a l l y p o s s i b l e to group terms so that the general s o l u t i o n aay he expresses i n one o r nore o f the f o l l o w i n g forms, I = A* + A, €• P,t-f AL£ Pztr-t-1= Ac -/- e^'^A'Cash/S-t-p B,sthh/3,t^ -p 6 *' (AiCosh/3Ltf BzSmh^ij-t--j = A a -p £K,t(A>cos/Z,T-+B, s/n{5,t) -f £ ^\Atcc>s/3zt i-S/k/5zt) P • I f we arc i n t e r e s t e d only i n the character o f the t r a n s i e n t s present, uinee the q u a n t i t i e s A« 3, and. 0 arc m e r e l y aaplit&de constants, we sey writ© one of the above g e n e r a l g e l a t i o n s a s soon as we Kiio?? the va2a.es o f the r o o t s o f Zv s 0» A l s o we may change f r o m one s o l u t i o n t o a n o t h e r by use o f e i s p l e t r i g o n o a e t r i c o r h y p e r b o l i c r e l a t i o n s and t h e o p e r a t o r * y, l a c a s e the a p p l i e d v o l t a g e i e n o t o f u n i t f u n c t i o n shape, i t o a s t be converted i n t o i t s e q uivalent operator be-f o r e the general s o l u t i o n c a n be obtained* . f o l l o w i n g i s a l i s t o f some eomaon e q u i v a l e n t o p e r a t o r s s,»(cut±0)7~ Ptoa*fc£'h<t co*C«**<t)fs ^ % \ % l ' h ± 1 k c o r e complete l i s t o f eq u i v a l e n t o p e r a t o r s ao w e l l an cany o t h e r o p e r a t i o n a l r e l a t i o n s and s o l u t i o n s , and a p r o o f f o r the E x p a n s i o n Sfaeoren, a r e t o be fo u n d i n ' R c a v i c l f i e ' s C / p e r a t i o i i a l ( l u l c u i u s * by E . t ?« Berc, ! r c f r r a ! a - ; i i l l 0 -48-ivpP3I2Jl£ I I I . I n a roathcaatieal s i a l y e i s o f coupled c i r c u i t s i t i s o f t e n n e c e s s a r y to have expressions f o r the e q u i v a l e n t inpe-donce and the t r a n s f e r i n pe dance o f the c i r c u i t u nder cousider-a t i c n . I f a l l prirsary q u a n t i t i e s be denoted by the s u b s c r i p t U r , a l l secondary q u a n t i t i e s b y t h e s u b s c r i p t ? 2 ? B and a l l ctatual q u a n t i t i e s by.no s u b s c r i p t s t h e d e s i r e d r e l a t i o n s a r e a s f o l l o w s , Equivalent Impedance s Ze = -p- = Z / Z z ~ z ^ ? r a a o f e r Impedance s Z r = = — z ^ where the " - t e r n s a r e i ope canoe o p e r a t o r s ^ that i s * t h e y a r e f u n c t i o n s o f th© c i r c u i t constants and the operator ? p f »(&/dt}* She p r o o f o f these r e l a t i o n s i s as f o l l o c / S o Suppose both primary and secondary t o c o n t a i n B i n a e r i e s 8 r e s i s t a n c e , induetance and c a p a c i t y , ( t h e p r o o f i s e q u a l l y v a l i d f o r any c i r c u i t c o n s t a n t a p r o v i d e d t h e c o r r e c t impedance o p e r a t o r s a r e tahen)• By equating eefe a r o u n d b o t h p r i e e r y and secondary c i r c u i t s t?e o b t a i n the f o l l o w i n g d i f f e r e n t i a l equations, E.-X.Z+t.g +J.yW -Mg* * , Ifow, e q u a t i o n s (1) end (2) can a l s o be w r i t t e n a s O = (Rz.-f-L-.P Xt-MPl, ^Zt-Xi-Zml, -< A. -From (2a) we get the r e l a t i o n - r Z /Jt _ -49-Sufestitate t h i s i r a l n e o f X 2 i s ( l a ) f l i e e q u i v a l e n t iopsdemeo f o l l o w s a t once I- z z A g a i n , f r o m (2a) *re have t i i a t z Zfr, S u b s t i t u t e t h i s v a l u e o f X* i n ( l a ) - Z / Z i T - 7 - r - _ / ' 2 " / Z z — -2"/ r^ - -7 *—2- -7- ~ ~ T~ LZi I ~ — i z —Z-fn-t-i. — , fixe t r a n s f e r imps fiance t h u s become* -I'M 

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