Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An electrodeless technique for conductivity measurements Huntley, Christopher Ryland 1960

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1960_A7 H8 E4.pdf [ 2.84MB ]
Metadata
JSON: 831-1.0105212.json
JSON-LD: 831-1.0105212-ld.json
RDF/XML (Pretty): 831-1.0105212-rdf.xml
RDF/JSON: 831-1.0105212-rdf.json
Turtle: 831-1.0105212-turtle.txt
N-Triples: 831-1.0105212-rdf-ntriples.txt
Original Record: 831-1.0105212-source.json
Full Text
831-1.0105212-fulltext.txt
Citation
831-1.0105212.ris

Full Text

AN ELECTRODELESS TECHNIQUE FOR CONDUCTIVITY MEASUREMENTS by Christopher Ryland Huntley A Thesis Submitted i n P a r t i a l F u l f i l m e n t of the Requirements f o r the Degree of MASTER OF APPLIED SCIENCE i n the Department of PHYSICS We accept t h i s t h e s i s as conforming to the " re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September I960 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department of PHYSICS The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3 , Canada. Date September I960 Abstract The development of a two- t o r o i d system f o r e l e c t r o d e l e s s c o n d u c t i v i t y measurements i s o u t l i n e d . The sample i s formed i n t o a closed loop which couples two t o r o i d s , one of which i s d r i v e n by an audio-frequency source; the output voltage of the other t o r o i d i s then d i r e c t l y p r o p o r t i o n a l to the loop conductance. The system i s only u s e f u l f o r m a t e r i a l s with c o n d u c t i v i t i e s greater than 10 mho/cm. Apparatus was b u i l t and t e s t e d using both high (mercury) and low ( s a l t s o l u t i o n ) c o n u c t i v i t y samples; heating and c o o l i n g curves on the above agreed with r e s u l t s obtained u s i n g more conventional techniques. A unique o i l heater and pump was devised f o r uniformly heating low c o n d u c t i v i t y l i q u i d s to avoid the undesirable coupling e f f e c t s of heating w i r e s . i i i Table of Contents Page Chapter 1 I n t r o d u c t i o n 2 Chapter*2 Theory 4 2.1 Simple L i n e a r Theory 4 2.1.1 For w w e l l below resonant frequency of output cct 5 2.1.2 For w at resonance „r,5 2.2 Departures from L i n e a r i t y 6 2.2.1 Non l i n e a r i t y of the primary t o r o i d coupling 6 2.2.2 Non l i n e a r i t y of the secondary t o r o i d 7 2.2.3 S e l f inductance of the loop 6* 2.2.4 Skin e f f e c t 8 2.2.5 Summary 99> 2.3 Lower L i m i t on G 9 2.4 Choice of Frequency 10 Chapter 3 D i s c u s s i o n of Heating Techniques 11 3.1 I n t e r n a l Heating 11 3.1.1 D i r e c t current 11 3.1.2 A l t e r n a t i n g current 11 3.2 E x t e r n a l Heating 12 Chapter 4 Equipment 14 4.1 E l e c t r i c a l and Magnetic 14 4.1.1 The t o r o i d s 14 4.1.2 Source equipment 15 4.1.3 Detection apparatus 16 4.2 Heating Apparatus and Arrangement of Sample 17 4.2.1 Mercury 17 4.2.2 0.01N KC1 s o l u t i o n Id i v Page Chapter 5 Experimental Results 20 5.1 Primary T o r o i d 20 5.2 Secondary Toroid 21 5.3 Mercury Sample 22 5.4 0.01N KC1 s o l u t i o n Sample 25 5.4.1 C a l i b r a t i o n of sample tu b i n g 25 5.4.2 C a l i b r a t i o n of apparatus at resonance 25 5.4.3 E f f e c t of s t r a y coupling between t o r o i d s 26 5.4.4 Data on the s o l u t i o n 26 5.4.5 Results 27 Chapter 6 Conclusions 29 Appendix 1 E f f e c t of the coupling c a p a c i t y of uniformly spaced heating tape on the apparent conductance of the coupling loop. 30 B i b l i o g r a p h y 33 V L i s t of I l l u s t r a t i o n s Facing Page Fi g u r e 1 Experimental Apparatus 4 F i g u r e 2 Schematic Representation of Experimental Apparatus 4 F i g u r e 3 S e c t i o n through Primary T o r o i d 14 Figure 4 Automatic Mechanism f o r Heating and Co o l i n g Mercury Sample' 17 Figure 5 Photograph of Mercury Heating Apparatus 17 Figure 6 Cross S e c t i o n of Apparatus f o r Heating COIN KC1 S o l u t i o n 18 Figure 7 Photograph of Above Apparatus 18 Figure 8 S e n s i t i v i t y of Primary T o r o i d 20 Figure 9 E f f e c t of Magnetization on P e r m e a b i l i t y of Primary Toroid 20 Figu r e 10 S e n s i t i v i t y of Secondary To r o i d 21 Figure 11 E f f e c t of Magnetization on P e r m e a b i l i t y of Secondary Toroid 21 Figu r e 12 C a l i b r a t i o n of Apparatus' at 40c/s 23 Figu r e 13 Block diagram of Apparatus f o r Mercury Measurements 24 Figu r e 14 Photograph of Heating-Cooling run on Mercury 24 Figure 15 C a l i b r a t i o n of Apparatus at 1900c/s 26 Figure 16 Photograph of F i r s t Run on KC1 S o l u t i o n 28 F i g u r e 17 Photograph of Second Run on KC1 S o l u t i o n 28 Acknowledgements I would l i k e to tender my thanks t o the f o l l o w i n g without whose help success i n t h i s undertaking would not have been p o s s i b l e : The Defence Research Board of Canada which financed the p r o j e c t . The N a t i o n a l Research C o u n c i l f o r awarding the author a studentship and two summer supplementals from May 1959 to September I960. P r o f e s s o r R. E. Burgess f o r h i s c o n t i n u a l encouragement and wholehearted cooperation i n overcoming the many d i f f i c u l t i e s encountered during the work. Mr. John Lees without whose amazing glass-blowing s k i l l the i n t r i c a t e apparatus could not have been achieved.. 0 AN ELECTRODELESS TECHNIQUE FOR CONDUCTIVITY MEASUREMENTS Chapter 1. I n t r o d u c t i o n The o r i g i n a l two-electrode c o n d u c t i v i t y measurements gave way t o f o u r t e r m i n a l techniques when i t was r e a l i z e d t h a t the contacts o f t e n had appre c i a b l e r e s i s t a n c e ; however recent measurements on semiconductors i n d i c a t e undesirable doping of the specimen by the metal occurs at the areas of contact. The problem of s e l e c t i n g i n e r t e l e c t r o d e s becomes even more acute at higher temperatures; f o r example graphite e l e c t r o d e s are used i n molten selenium which d i s s o l v e s a l l metals, even gold and tungsten, q u i t e r e a d i l y . In t h i s e l e c t r o d e l e s s method the sample i s formed i n t o a closed loop which couples two t o r o i d s ; then w i t h a constant a l t e r n a t i n g current f l o w i n g through one t o r o i d , the voltage appearing across the other i s d i r e c t l y p r o p o r t i o n a l to the c o n d u c t i v i t y of the coup l i n g loop. The system described h e r e i n i s usable f o r loop conductances from 3 x 10~7mho t o 300 mho. The lower l i m i t i s determined by the s t r a y coupling between the t o r o i d s ; the upper l i m i t i s a r b i t r a r y and may be extended by us i n g very low frequency equipment. I t i s best s u i t e d f o r measurements on l i q u i d s since they may be enclosed i n a 3 c a l i b r a t e d loop of g l a s s t u b i n g ; however i t a l s o may be used on s o l i d s such as semiconductive organic f i b r e s as long as they may be formed i n t o a closed loop. Uniform heating of the specimen presents i t s own problems which are discussed l a t e r . Facing Page 4 PRIMARY •TOROID FIGURE 1. EXPERIMENTAL APPARATUS i ; Sin^wt M,. n,_ L i G G -AAAA-J FIGURE 2. SCHEMATIC REPRESENTATION OF EXPERIMENTAL APPARATUS Chapter 2. Theory 2.1 Simple L i n e a r Theory •Figure 1. shows the experimental set-up which may be represented s c h e m a t i c a l l y as i n Figure 2. The primary or d r i v e n t o r o i d c o n s i s t s of s e v e r a l t u r n s of copper wire wound around a t o r o i d a l core of high p e r m e a b i l i t y , the wire i s brought back t o avoid one net t u r n i n the plane of the t o r o i d ; the secondary or pick-up t o r o i d i s s i m i l a r but has a l a r g e number of turns to increase the output v o l t a g e . The core m a t e r i a l i s of high p e r m e a b i l i t y i n order t o e s t a b l i s h a w e l l defined magnetic f i e l d ; otherwise the t o r o i d a l c oupling would be a f u n c t i o n of the p o s i t i o n of the sample i n the t o r o i d . The symbols used i n Figure 2 are: 11 i s the current f l o w i n g through the primary t o r o i d of -:; frequency w/27f c y c l e s per second. n, i s the number of turns" of the primary winding. L| i s the s e l f inductance of the primary. /rL\ i s the r e l a t i v e p e r m e a b i l i t y of the primary t o r o i d core m a t e r i a l . M| i s the mutual inductance between the primary t o r o i d and the sample loop. ' e 2 i s the electr o m o t i v e f o r c e (e.m.f.) induced i n the loop. L 2 i s the s e l f inductance of the lo o p , with t o r o i d s i n place G i s the conductance of the loop. 1 2 i s the current i n the loop. 5 M£ i s the mutual inductance between the sample loop and the secondary t o r o i d . n 2 i s the number of turns on the secondary t o r o i d . L 3 i s the s e l f inductance of the t o r o i d . r i s the a.c. r e s i s t a n c e of the t o r o i d . i s the r e l a t i v e p e r m e a b i l i t y of the secondary t o r o i d core m a t e r i a l , eg i s the i n d u c e d e.m.f. i n t h e s e c o n d a r y i3 i s the current i n the output c i r c u i t C i s the shunt capacitance across the secondary R i s the input r e s i s t a n c e of the secondary l o a d , e^  i s the a c t u a l output v o l t a g e . 2.1.1 For w Well Below Resonant Frequency Of Output C i r c u i t e2=jwM, i , *2 (JwL 2 + l/G) e4= e3= j w M 2 i 2 Hense assuming G«l/wL 2 -w eM (M 2i, G . (1) 2.1.2 , For w At Resonance We have approximately, i f the induced r e s i s t a n c e s i n the loop and the second t o r o i d are r e l a t i v e l y s m a l l : e2= jwM, i t ( j w M 2 i 3 neglected) e 2=i 2/G (G«l/wL2. )• e3 = j w M 2 i 2 i 3 = e 3 / r at resonance where wa= 1/L 3C e^ = i 3/jwC 6 Combining the above: e = ,jwM, M zi,G 4 rC (2) I f the shunt load r e s i s t a n c e R i s a p p r e c i a b l e , t h i s becomes (3) e-a8.1wM,Mfti> G f . 1 rC 1 1+ 1 + U jwRC rRC Thus i n e i t h e r case the output voltage i s d i r e c t l y -p r o p o r t i o n a l to the loop conductance f o r given primary current and mutual inductances. 2.2. Departures From L i n e a r i t y In p r a c t i c e non l i n e a r i t y sets i n at high loop conductances due t o : (1) A change i n M, due t o an increase i n the loop current ±2 sinceyU, i s a f u n c t i o n of the magnetizing f o r c e which depends on i 2 as w e l l as i | . (2) A s i m i l a r change i n M2 due to i 2 . (3) Reduction of the loop current by i t s s e l f i n d u c t i o n . (4) Reduction of the loop conductance G from i t s d.c. value by the s k i n e f f e c t . 2.2.1. Non L i n e a r i t y 1 O f The Primary T o r o i d Coupling The i n i t i a l part of the e f f e c t i v e a.c. per m e a b i l i t y ^ c f ( H , ) c h a r a c t e r i s t i c may be approximated by the l i n e a r r e l a t i o n J ^ ^ f ^ o k l + a i H, ) where H, i s the r.m.s. value of the magnetizing f o r c e . The maximum value of loop conductance f o r a 5% change i n the mutual inductance M may then be evaluated. Since M, i s d i r e c t l y p r o p o r t i o n a l tOyu, i t may be w r i t t e n M,= M D 1 (i+a, H, ) and 'H, = *n, i^ t -J2~j where I, i s the e f f e c t i v e length of the f l u x path around the t o r o i d a l core. so e 2 = jwM,, [l+a, (n, i , - i 2 )j i , (4) as before i ^ G e ^ assuming G « l/wL 2 combining these and rearranging . . + f n« ± Z ^ 1 . - _J_ I%i'W'G ' . which d e v i a t e s by ^  5% from the l i n e a r r e l a t i o n s h i p 12 = jwM i, G • .. (5) when wG^ 0 . 3 2 lt a,M 0 li, . (6) where M I=M 0 I(1 + a, fi ^  ) the mutual inductance f o r low conductance loops at the d r i v i n g current i , . 2.2.2. Non L i n e a r i t y Of The Secondary Toroid The p e r m e a b i l i t y of the f e r r i t e i n the secondary t o r o i d i s al s o a f u n c t i o n of i t s magnetization and may be approximated by /** J>a 2 H z ) so the mutual inductance M2 may be w r i t t e n M2= M 0 4 (1 f a^Hji) and Up-iz -n 2 J 3 where 4 1 S t n e e f f e c t i v e f l u x path length around the t o r o i d a l core. 8 Only the non-resonant case w i l l be considered here so n 2 i 3 « i 2 ; f o r high loop conductances there i s no need f o r the e x t r a s e n s i t i v i t y obtained at resonance and the ass o c i a t e d magnetization of the secondary core may s h i f t the resonant frequency of the output c i r c u i t causing undue c a l i b r a t i o n problems, Thus: M2 = M 0 2 . [ l + a 2 i 2 . ] So M 2changes by ^5% when the loop current i s i 2 < .05 U a 2 s u b s t i t u t i o n f o r i 2 from (5): wG^ .05la agM^, (7) 2.2.3. S e l f Inductance Of The Loop The s e l f inductance L 2 of the coupling loop reduces i t s current at high conductances i . e . i 2 — ez  (jwL^ - r l/G) Thus f o r accuracy of the simple formula assuming, a purely r e s i s t i v e loop: wG *c Q.32 T~~~" (8) 2.2.4. S k i n . E f f e c t The general formula f o r the r e s i s t a n c e of a c i r c u l a r conductor of radius a as a f u n c t i o n of frequency due to 9 concentration of the current flow near the surface i s expressed i n terms of B e s s e l Functions; however, f o r small d e v i a t i o n s from i t s d.c. value R(0),,the. r e s i s t a n c e R(w) of a sample of length L may be approximated by: R(w) = R(0)-f UwV«rV].in M.K.S. u n i t s ' 1 192 •* . so f o r ^ 5% increase i n r e s i s t a n c e y«0wffaz*3.1 Rewriting t h i s i n terms of the conductance G of the sample loop of length I : wG ^  8 x^ 10* (9) 2.2.5. Summary (6), (7), (8), and (9) give f o u r upper bounds of wG i n order t h a t the output voltage b e - d i r e c t l y p r o p o r t i o n a l to the loop conductance w i t h i n 5%; i n general they are not the same so the l e a s t upper bound must be observed. There i s thus no r e a l l i m i t on G p r o v i d i n g s u f f i c i e n t l y low frequency i s used. The four l i m i t s are: wG^: 0.32 I,  > a. Me,i, 0.05 U f o r w w e l l below resonance a 2 M x i | 0.32 I * 8 x 10* I 2.3. Lower L i m i t On G The lower l i m i t on G i s determined mainly by the s t r a y 10 coupling between the two t o r o i d s , t h i s may be represented by a mutual inductance M 3. i . e . Mg= I e j l Wl.' where e^ i s the output voltage w i t h G= 0 The output e^ . due to the loop G i s given i n (,|<): |e4]= w^M.M^i, G Thus \e±l _ 1M3 le^l " M,M2wG For^5% e r r o r due t o st r a y coupling • le 4'k;05 le4r or wG > 20 M 3 M,M2 • - (10) M3 may be reduced considerably by c a r e f u l s h i e l d i n g and o r i e n t a t i o n ; i n c r e a s i n g the separation between the t o r o i d s i s unpromising as i t a l s o decreases G f o r a given sample cross section. A high gain low noise a m p l i f i e r feeding a narrow bandwidth d e t e c t o r i s a l s o e s s e n t i a l f o r very low c o n d u c t i v i t y measurements. 2.4. Choice Of Frequency The optimum operating frequency depends oh the loop c o n d u c t i v i t y and hence the sample composition. Since the output voltage i s p r o p o r t i o n a l to the square of the frequency the highest p o s s i b l e frequency i s d e s i r a b l e t o f a c i l i t a t e measurements. For high c o n d u c t i v i t y m a t e r i a l s the highest frequency compatible w i t h c o n d i t i o n s should be used. For other m a t e r i a l s the resonant frequency of the output c i r c u i t i s the l o g i c a l choice f o r maximum s e n s i t i v i t y . 11 Chapter 3 D i s c u s s i o n Of Heating Techniques There are two basic methods of heating the sample, i n t e r n a l and e x t e r n a l . 3.1i I n t e r n a l Heating In t h i s method he a t i n g occurs from a current f l o w i n g i n the sample; t h i s may be e i t h e r d i r e c t or a l t e r n a t i n g . 3.1.1. D i r e c t C u r r e n t In t h i s 1 case current i s s u p p l i e d t o the sample through two e l e c t r o d e s which are e q u i d i s t a n t to ensure uniform current i n the sample. A wire c a r r y i n g one-half of the t o t a l current through the sample must al s o pass through both t o r o i d s i n such a d i r e c t i o n t h a t there i s no net magnetizing f o r c e i n e i t h e r t o r o i d . Otherwise theeheating current i n the sample w i l l m a g n e t ically bias the f e r r i t e cores thereby changing the s e n s i t i v i t y of the system. T h i s method i s simple and makes r a p i d heating p o s s i b l e , however i t involves the use of e l e c t r o d e s which may contaminate the sample. In a d d i t i o n l o c a l heating may occur at the elect r o d e j u n c t i o n s , and e l e c t r o l y s i s may a f f e c t the composition of the sample as i n the case of i o n i c s o l u t i o n s . 3.1.2. A l t e r n a t i n g Current This may be accomplished i n a manner s i m i l a r to the d.c. method wi t h i t s a s s o c i a t e d disadvantages or p r e f e r a b l y by inducti©n. The l a t t e r avoids e l e c t r o d e s and a l s o ensures uniform heating of the sample assuming a uniform cross s e c t i o n , u n f o r t u n a t e l y i t a l s o has i t s disadvantages. The :presi?ence of a t h i r d t o r o i d around 12 the sample a f f e c t s the c a l i b r a t i o n as does the induced current i n the loop which magnetically biases the other t o r o i d s . T h i s current also induces a voltage i n the secondary t o r o i d which i n general i s . much l a r g e r than the s i g n a l voltage making d e t e c t i o n d i f f i c u l t , even though the heating and s i g n a l frequencies are d i f f e r e n t . Apart from these t r o u b l e s the high frequency high power necessary f o r appreciable heating makes t h i s * method r a t h e r i m p r a c t i c a l . The high frequency i s necessary f o r appreciable power t r a n s f e r through a f e r r i t e cored t o r o i d . 3.2. E x t e r n a l Heating The main problem with e x t e r n a l heating i s keeping the sample isothermal which excludes the use of heat lamps. The simplest method i s J w r a p p i n g the sample w i t h heating tape without forming a closed loop and making sure there i s no r e s u l t a n t net current through the t o r o i d s . I f no heating tape i s a v a i l a b l e then r e s i s t a n c e wire spaced uniformly over asbestos wrapped' around the sample makes a good s u b s t i t u t e . E i t h e r d i r e c t or a l t e r n a t i n g current may be used as n e i t h e r a f f e c t s the c a l i b r a t i o n or produces extraneous output s i g n a l s . The only disadvantages to t h i s system are the temperature v a r i a t i o n s due t o non uniform h e a t i n g and the c a p a c i t i v e coupling between the heating wire and the sample which becomes appreciable at very low c o n d u c t i v i t i e s . I f a sharp phase t r a n s i t i o n i s to be observed then isothermal heating i s e s s e n t i a l . The thermal c o n d u c t i v i t y of the sample c e r t a i n l y helps i n a t t a i n i n g isothermal c o n d i t i o n s but with the c o n f i g u r a t i o n used i n t h i s method i t i s not too e f f e c t i v e . An i d e a l but elaborate s o l u t i o n i s t o enclose the sample m a t e r i a l i n 13 the inner tube of a double walled g l a s s torus t h r e a d i n g both t o r o i d s ; a non conductive o i l i s then pumped through the outer part at the de s i r e d temperature. There does not appear t o be any o i l s s t a b l e much above 300°C. making t h i s inadequate f o r l i q u i d semiconductor measurements and the high o i l v i s c o s i t i e s make t h i s u n s u i t a b l e f o r low temperatures. However over t h i s l i m i t e d range and with a good o i l pump, which i s a problem i n i t s e l f , the temperature v a r i a t i o n i n the sample may be held within'one degree. The t r a n s l u c e n t g l a s s and o i l around the sample makes.this technique very u s e f u l f o r photo-conductive measurements. Facing Page 11+ SOLDERED JOIN IN COPPER SHIELD; WOODEN BOX COVERED WITH COPPER COIL WINDING -AROUND FERRITE FERRITE 'U-SECTION INNER PART OF CU- SHIELDING FERRITE INTERFACE SAMPLE LOOP-GLASS TUBING' JOIN IN CU. TO ALLOW REMOVAL OF SAMPLE RUBBER BANDS HOLD U-SECTIONS TOGETHER 4! FIGURE 3. SECTION THROUGH PRIMARY TOROID -§TULL SIZE 14 Chapter 4. Equipment 4.1 E l e c t r i c a l and Magnetic Apparatus 4.1.1. The Toroids Each t o r o i d a l core was made from two Ferroxcube #1F5 U-cores of manganese-zinc f e r r i t e ; they have a 1 inch square cross s e c t i o n and together form a rectangle 4"by 4 i " on the outside and 2" by 2|" i n s i d e which leaves ample room f o r the sample loop. The primary t o r o i d was wound with 63 turns of #18 enamelled copper wire on each h a l f ; a l l f o u r ends were returned to the same place not only to avoid one net t u r n i n the plane of the t o r o i d but a l s o t o enable the t o r o i d t o be clamped around the sample loop. The secondary t o r o i d was wound on a l a t h e with #28 enamelled copper wire; from i n d u c t i o n measurements each h a l f had about 1300 t u r n s . In order to reduce e l e c t r o s t a t i c coupling the primary t o r o i d was completely covered with copper f o i l , due precautions being taken to avoid a s h o r t - c i r c u i t t u r n around the core. The copper s h i e l d i n g through the centre p o r t i o n a l s o considerably reduced the e l e c t r o s t a t i c c oupling between the windings and the sample loop which was found to be otherwise quite appreciable at low c o n d u c t i v i t i e s . The s h i e l d i n g was arranged so that i t could be put i n place a f t e r the t o r o i d had been clamped around the specimen. The secondary t o r o i d was shi e l d e d i n a s i m i l a r manner wit h the a d d i t i o n of a l a y e r of aluminum f o i l wrapped around each U-section; t h i s f u r t h e r reduces i t s s e n s i t i v i t y to 60-cycle f i e l d s ever present i n the l a b o r a t o r y as w e l l as the s t r a y coupling between the t o r o i d s . A l l the s h i e l d s were grounded t o a convenient water tap at a common p o i n t . 4.1.2. Source Equipment The d r i v i n g apparatus must be capable of supplying s u f f i c i e n t constant current at the d e s i r e d frequency; f o r low c o n d u c t i v i t y measurements where a narrow bandwidth d e t e c t o r i s used the o s c i l l a t o r must al s o be very s t a b l e . The d r i v i n g current used does not exceed that f o r which the p e r m e a b i l i t y departs from i t s l i n e a r dependence yu,yx(l+a,H,); which roughly corresponds to the onset of s a t u r a t i o n of the f e r r i t e . In.the. case of the t o r o i d used , l i n e a r i t y extends t o around 50ma so a current of 30ma was chosen f o r a l l measurements. This current was su p p l i e d from a F i s h e r model 200A 55watt audio a m p l i f i e r driven by a Muirhead-Wigan model D-695-A/100 decade o s c i l l a t o r ; t h i s o s c i l l a t o r was chosen f o r i t s hourly frequency s t a b i l i t y ( b e t t e r than ,02%) as w e l l as i t s wide frequency range of lQc/s to 31kc/s. An E l e c t r o n i c s I n d u s t r i e s L t d . model 44 multimeter was used t o monitor the t o r o i d c u r r e n t . For low c o n d u c t i v i t y measurements the a m p l i f i e r may be roughly matched to the t o r o i d , however f o r high conductance loops the t o r o i d must be d r i v e n from a high impedance source since a change i n loop conductance a l t e r s the t o r o i d impedance which would change the d r i v i n g c u r r e n t ; t h i s was e f f e c t e d using a 220 ohm r e s i s t o r i n s e r i e s with the 100 chm output of the a m p l i f i e r . 4.1.3 Detection Apparatus The secondary t o r o i d output was a m p l i f i e d by a Challenger model.CHA-75 75watt audio a m p l i f i e r which has two high gain "Mic" inputs and one low gain "Phono" input as w e l l as bass and t r e b l e c o n t r o l s ; i t a l s o has 4,#,l6,and 100 ohm outputs. The a m p l i f i e r output was measured on a General Radio type 736-A Wave Analyzer i n order t o separate the de s i r e d s i g n a l from other s t r a y s i g n a l s picked up by the secondary t o r o i d and generated i n s i d e the a m p l i f i e r . For low conductance loops f u l l bass cut was used on the a m p l i f i e r to avoid overloading i t w i t h 60c/s picked up from a nearby motor used i n the heating system described l a t e r . When c o n d u c t i v i t y - temperature curves were d e s i r e d the d.c. output of the Wave Analyzer was fed i n t o one channel of a Mandrel model ER-90-1 X-Y graphic r e c o r d e r ; the other channel was fed from a Hewlett-Packard model 425-A d.c. micro-micro volt-ammeter connected t o a thermocouple monitoring the temperature of the specimen. I f the sample c o n d u c t i v i t y has a sm a l l temperature c o e f f i c i e n t , such as mercury, then the system s e n s i t i v i t y may be increased by bala n c i n g the d.c. output against a f i x e d voltage before feeding i t t o the recorder. In p r a c t i c e t h i s reference voltage was obtained by r e c t i f y i n g and f i l t e r i n g the v o l t a g e developed across a r e s i s t o r - i n s e r i e s with the Facing Page 17 CURRENT LIMITING RESISTOR lOonms 90ohm 2.2amp CENCO RHEOSTAT MOTOR DRIVEN [ C DRUM SWITCH FOR AUTOMATIC SHUTOFF HEATING WIRE. AROUND MERCURY 15onms FIGURE 4 AUTOMATIC MECHANISM FOR HEATING AND COOLING MERCURY SAMPLE FIGURE 5 PHOTOGRAPH OF MERCURY HEATING APPARATUS primary t o r o i d i n order t o reduce e r r o r s caused by primary current f l u c t u a t i o n s . 4.2. Heating Apparatus and Arrangement o£ Sample Several d i f f e r e n t h eating methods were i n v e s t i g a t e d , t h e i r v a rious merits and demerits have already been summarized i n chapter 3. Heating and c o o l i n g curves f o r a high c o n d u c t i v i t y m a t e r i a l (mercury) and a low c o n d u c t i v i t y s o l u t i o n (0.01N KC1 i n water) were obtained using two of these methods. .4.2.1 Mercury The mercury was contained i n 6mm I . D . pyrex gla s s t u b i n g bent to form a 28cm by 17cm r e c t a n g l e ; v e r t i c a l access tubes were attached on opposite sides f o r f i l l i n g and to a l l o w f o r expansion of the mercury. The loop was wrapped with a s i n g l e l a y e r of -L"asbestos tape over which was wound r e s i s t a n c e wire spaced uniformly at one t u r n per cm. with a t o t a l r e s i s t a n c e of 15 ohms; one of the lead i n wires was returned around the loop through both t o r o i d s t o avoid magnetizing them. This heating c o i l was fed from 110 v o l t s d.c. v i a a motor d r i v e n potentiometer arrangement (see Fig.4) f o r automatic heating and c o o l i n g at a very slow r a t e to keep the sample at a uniform temperature throughout the c y c l e . A run l a s t i n g s e v e r a l hours was found necessary to avoid thermal h y s t e r i s i s . A thermocouple was i n s e r t e d down each access tube i n t o the mercury to monitor the temperature; they were found to agree w i t h i n two degrees, the average being fed t o the chart recorder. Facing Page 18 RESERVOIR FOR OIL EXPANSION TOMG TUBING. 2-PHASE 4-POLE MOTOR WINDING FIGURE 6 .CROSS SECTION OF APPARATUS FOR HEATING .01N KC1 SOLUTION FIGURE 7. PHOTOGRAPH OF ABOVE APPARATUS 4.2.2 0.01N KC1 S o l u t i o n I t was hoped th a t t h i s technique could be t e s t e d on a l i q u i d semiconductor, however an exhaustive search d i d not r e v e a l any semiconductor or compound thereof that had a s u f f i c i e n t l y low m e l t i n g point arid high enough c o n d u c t i v i t y to be w i t h i n the range of the apparatus; i . e . a melting point below 250°C. and cr P-10~4 mho/cm. Thus i n order to t e s t the system f o r low c o n d u c t i v i t i e s a 0.01N KC1 S o l u t i o n i n water was used which has a c o n d u c t i v i t y around 103mho/cm and an appreciable temperature c o e f f i c i e n t . At the working frequency of 1900c/s >the resonant frequency of the output c i r c u i t , the heating c o i l setup as used f o r the mercury case has too much coupling c a p a c i t y to the sample (250 uuf t o t a l ) to be used f o r loop conductances l e s s than 10*mho ( f o r l e s s than 5% e r r o r ) see Appendix 1. Using the same loop dimensions as before t h i s means that the -4 specimen c o n d u c t i v i t y must be l a r g e r than 3x10 mho/cm, which i s smaller than that of the ,01N KC1 s o l u t i o n . However to e l i m i n a t e e r r o r s due t o uneven heating from the heating tape the apparatus shown i n Figures 6 and 7 was used; t h i s was not r e a l l y necessary but i t had been b u i l t i n the v a i n hope that the s t r a y t o r o i d c o u p l i n g could be reduced s u f f i c i e n t l y to allow measurements on l i q u i d s e l e n i u m - t e l l u r i u m mixtures whose c o n d u c t i v i t i e s (around 10 mho/cm) are too low f o r the use of h e a t i n g tape or i n d u c t i o n h e a t i n g . 1 9 In t h i s method the sample loop i s enclosed i n a l a r g e r g l a s s tube through which i s pumped o i l at the d e s i r e d temperature. The o i l pump was b u i l t t o operate at temperatures up t o 3 0 0°C which i s l i m i t e d by the o i l , A r o c l o r type 1 2 5 4 manufactured by the Monsanto Chemical Company; however the r a p i d drop i n i t s v i s c o s i t y at high temperatures caused the t u r b i n e to r a t t l e around i n i t s g l a s s bearings p u t t i n g an upper l i m i t of 2 0 0°C on the apparatus. A lower l i m i t of 5 0 ° C was caused by the o i l ' s very high v i s c o s i t y at lower temperatures, at t h i s temperature i t i s 1 0 0 , 0 0 0 times t h a t at 2 0 0 ° C . Over i t s temperature range the apparatus worked q u i t e s a t i s f a c t o r i l y maintaining the sample isothermal ; s i x thermo-couples around the system, immersed i n the o i l and i n the^. sample as w e l l as on the e x t e r i o r g l a s s surface, i n d i c a t e d a maximum temperature drop of only one degree i n the loop, due mainly to the high pumping r a t e of lOcc/sec. Facing Page 20 FIGURE 9. EFFECT OF MAGNETIZATION ON PERMEABILITY OF PRIMARY TOROID 20 Chapter 5 ' Experimental R e s u l t s In order to c a l i b r a t e the system and a s c e r t a i n i t s l i m i t a t i o n s caused by the n o n l i n e a r i t i e s already discussed each t o r o i d was t e s t e d i n d i v i d u a l l y . 5.1 Primary T o r o i d The c h a r a c t e r i s t i c s t o be determined were: n, , KOTI a, f t n, was counted during the winding and =126 t u r n s . In order to determine Mol and a, the emf e 2 induced i n a 7 , s i n g l e o p e n - c i r c u i t t u r n through the t o r o i d was measured as a f u n c t i o n of the primary current i , ' at an a r b i t r a r y frequency of lf00c/s; the r e s u l t s are shown i n Figure 8. From t h i s data the e f f e c t i v e a.c. p e r m e a b i l i t y y A . , was p l o t t e d as a f u n c t i o n of i , as shown i n Figure 9. S a t u r a t i o n of the core s e t s r i n around 60 ma while the l i n e a r f u n c t i o n y A . t y * { l * a, H,) drops o f f around #0 ma so a current of 30 ma was chosen f o r a l l measurements. At t h i s current we f i n d : NL -|e& I _ 47xl0" 3 = 0.62 mH lwi,l 2tr(/ +00)3xlO" z From the slope i n Figure 9: a.n. — 25 amp"1 hence a, - 0.20 amp"1 ~T . 7 , and s i n c e M^M^U + a, n, i , ) M0(= 0.36 mH 4-1 Facing Page 21 PRIMARY:-AMPERE TURNS FIGURE 10. SENSITIVITY OF SECONDARY TOROID FIGURE 11. EFFECT OF MAGNETIZATION ON PERMEABILITY OF SECONDARY TOROID Summarizing: V; n,= 126 turns M O I = 0 . 3 6 mH f i _ 0.20 amp"' M j*0.62mH I, 5.2 Secondary Toroid In t h i s case the c h a r a c t e r i s t i c s to be determined were: ' n 2 > M 0 2 > ^ »r > L 3 » a n d c The c o i l was wound on a lat h e without a c o u n t e r so n 2 was nOt known d i r e c t l y . I t s inductance was measured on a General Radio impedance bridge: L3 =17 Henries The e f f e c t i v e s e r i e s r e s i s t a n c e r was determined from impedanc measurements to be approximately 3,0k ohms at the resonant frequency of 1900c/s. The shunt capacity C was thus 400 u/if. In order t o determine M 0 2 , and ^  the output voltage e^ was measured as a f u n c t i o n of the loop current at a frequency of lOOc/s, w e l l below the resonant frequency 1900c/s. I t was not p o s s i b l e to d r i v e a s i n g l e loop with 15 amps from the d r i v i n g a m p l i f i e r so 20 turns of wire were passed through the t o r o i d and dr i v e n w i t h 0.75 amps thereby a c h i e v i n g the .same magnetizing f o r c e . The r e s u l t s are shown i n Figure 10 and the corresponding y U ^ ^ p l o t i n Figure 11. M 02 was defined f o r loop current ->-0 i.e.. M p p J e 3 l w i 2 - 3 . 4 = 5 . 4 mH iz-*0 200 TT From the slope i n Figure 11: = 0.20 amp-' Summarizing: r=30k ohm at 1900c/s * 2 = ° ' 2 0 a mP~' L3 = 17 Henries M 0 2 = 5.4 mH C = 400 jxjxf 5.3 Mercury Sample The apparatus has already been described i n section 4^.2.1 the mean sample length was measured to be 86cm - 1 while i t s inductance was estimated to be 8 uH from measurement of an equivalent shaped length of wire passing through both toroids and connected to an impedance bridge. t = 0.86 metre L 2 = 8 uH Substitution to determine the four upper l i m i t s on wG, summarized in section 2.2.5 gives: From (6): wG^ 0 .321/ 148,000 mho/sec a,Mfl,i, From (7): wG^ 0.05 lz = 13,400 mho/sec a 2Mxi, From (8): wG< 0 .32 = 40,000 mho/sec ' L 2 From (9): wG < 8x10* = 9,300,000 mho/sec T Thus wG should be less than 13,400 mho/sec for less than 5% deviation in the proportionality between output voltage and loop conductance. From the weight of mercury in the tubing the average cross-section area was found to be 0.300cm - 2%. Since the Facing Page 23 FIGURE 12  CALIBRATION OF APPARATUS AT 40c/s i n s i d e diameter was measured to be 0.61* .01cm g i v i n g an area of 0.293cm ±3%; the cross s e c t i o n was considered uniform and the "form f a c t o r " F, = length - 86cm - 287cm~'* 3% area ,300cm* From data published i n the Chemical Rubber Company's Handbook of Chemistry and Physics the conductance of the mercury loop was then estimated to be around 30mho. Hence w^450 sec"' and f«70c/s In view of t h i s an operating frequency of 40c/s was chosen; the o v e r a l l s e n s i t i v i t y may then be evaluated from equation (1): . |ej|= w2M| Magi, G = 0.0064 G v o l t s This s e n s i t i v i t y was checked using a S h a l l c r o s s #$17C r e s i s t a n c e box (O.Olohm to 10,000ohm) and a set of loops made from heavy stranded wire and c a l i b r a t e d by measuring t h e i r "IR" drop before s o l d e r i n g the ends together. The r e s u l t a n t data i s p l o t t e d i n Figure 12; from i t s slope: eg _ 0.0063 volts/mho ± 5% & which agrees with t h a t c a l c u l a t e d above. With the mercury loop i n place the output voltage was measured on the Wave Analyzer and found to be e 4= 230 mv at 25.2°C so the loop conductance G =• 37 mho if. and the c o n d u c t i v i t y cr — F,G * 1.06 x 10 mho/cm 4 compared to the handbook data: 1.04 x 10 mho/cm F a c i n g Page 24 MUIRHEAD FISHER OSCILLATOR AMP. —L500" 2Z0sTTrf AAAAAr— 1  THERMO-COUPLES IKHROlYHITE LOWPASS FILTER PHALLENGER AMPLIFIER ^LOO J-500 ;1^0° P.R. WAVE-ANALYZER HEWLETT-PACKARD uuvolt-ammeter 6 X 6 MANDREL X-Y CHART-RECORDER EIGURE 13 BLOCK-DIAGRAM OF APPARATUS FOR,.MERCURY MEASUREMENTS F a c i n g Page 24 DIFFERENTIAL OUTPUT /VOLTS) PHOTOGRAPH Figure 14 OF HEATING-COOLING RUN ON MERCURY In order to obtain the conductivity vs temperature c h a r a c t e r i s t i c the secondary t o r o i d output was passed through a zero-gain Krohn-Hite model 330A low pass f i l t e r and amplified by the Challenger amplifier to a t t a i n enough d i f f e r e n t i a l voltage to drive the Mandrel X-Y recorder. The low pass f i l t e r was found necessary as: even though the primary current waveform appeared very good on an oscilloscope the output was r i c h in harmonics since the system gain i s proportional to the square of the frequency. A block diagram of .the complete apparatus i s shown in Figure 13. A photographic r e p l i c a of an actual heating-cooling run i s in Figure 14; even with a cycle period of eight hours a small amount of thermal h y s t e r i s i s i s evident though i t may be noted that the i n i t i a l and f i n a l positions coincide. The s l i g h t positive curvature i s attributed to the nonlinearity of the Copper-Constantan thermocouple; i . e . 7mv to 8mv covers 19°C whereas lmv to 2mv covers 24°C. The mean slope of the trace i s 0.0060 volts/*C hence the temperature c o e f f i c i e n t of conductivity i s : -4 0.0060 = 9.0 x 10 /°C of the conductivity at 25°C. 6.7 This i s within 5% of the handbook data which y i e l d s : 8.9 x 10 /°C of the conductivity at 25°C 5.4 .01N KC1 Solution Sample 5.4.1 Ca l i b r a t i o n of Sample Tubing ' The solution was contained in the sample loop shown i n Figures 6 and 7 which was made from the same glass tubing and approximately the same shape as the loop used f o r the mercury. From the volume of solution in the loop the "average" cross-section area was computed to be 0.303cm ±2% which agrees with the previous estimates. However in t h i s case the mean loop perimeter was measured as 82.5cm ±1 giving a d i f f e r e n t "form f a c t o r " F 2 = 275 cm"' £ 3% • 5.4.2 Calibration of Apparatus at Resonance The s e n s i t i v i t y of the system at the resonant frequency of 1900c/s i s computed from (3): For no load, R = °° e 4 = j l l 9 G |e^ | = 119G For a VTVM load, R = 10 Megohm V -11190 |e4| = 104G * 1.14+ j.02 , For the phono input on Challenger amplifier, R =560kohm • 4 - .1490 . 3.46+- j.36 (This amplifier resistance was obtained experimentally by substituting a resistance box and adjusting i t to produce the same loading on the secondary toroid.) The Challenger amplifier was calibrated f o r i t s phono input under conditions of f u l l bass cut, f u l l t r e b l e boost, Facing Page 26 LOOP RESISTANCE OHMS FIGURE 15. CALIBRATION OF APPARATUS AT 19QOc/s 26 and f u l l g a i n . The output was measured across a 100 ohm load on i t s 100 ohm output: Phono voltage gain = 98 ±k% A S h a l l c r o s s #833 r e s i s t a n c e box was then used t o check the l i n e a r i t y of the system over a wide range, 100 ohm t o 10 Megohm, see Figure 15. The o v e r a l l s e n s i t i v i t y was thus found to be: 3500 volts/mho Hence the loop s e n s i t i v i t y i s 3500 ^ 35 volts/mho ~98~ This checks w i t h the computed value of 34- volts/mho 5.4 .3. E f f e c t of s t r a y c o u p l i n g between t o r o i d s With no loop through the t o r o i d s the r e s i d u a l output of the a m p l i f i e r was -z= 0.3mv. (The a m p l i f i e r noise i n the 4c/s bandwidth of the Wave-Analyzer amounted to only O.lmv.) Thus as defined i n 2 .3: = 3 x 10~6 — 8.2 x 10 - ,H 2ir{ 19001.03) so f o r ^5% e r r o r due to s t r a y coupling: wG ^ 20 MB = 0.05mho/sec . hence G ^ 4 x lo'mho M,M2-i . e . the loop r e s i s t a n c e should be l e s s than 250k ohms i n order that the apparent loop r e s i s t a n c e be w i t h i n 5% of the t r u e r e s i s t a n c e . 5.4.4 Data on the S o l u t i o n The .01N KC1 s o l u t i o n was prepared by d i s s o l v i n g 0.745 grams of potassium c h l o r i d e and d i l u t i n g t o make one l i t r e of aqueous s o l u t i o n . Accuracy &. 1%. The following conductivity table f o r t h i s solution was derived from the equivalent conductance data in A.A.Noyes' "The E l e c t r i c a l Conductance of Aqueous Solutions"; the author claims 0.3$ accuracy. Temp.°C 18 25 26 50 75 100 6- xlO" 5 122 141 145 213 287 362 So the conductivity increases l i n e a r l y with temperature: i . e . cT= 7 x 10"4+- 2.9 x l d V c mho/cm . (18°-100°) 5.4.5 Results Two heating-cooling curves f o r the solution were obtained, both using the sample loop of the oil-pump apparatus shown in Figures 6 and 7. The f i r s t run was obtained using two four-foot lengths of ha l f - i n c h heating tape (Electrothermal type HT-341) wrapped uniformly around the sample loop, the temperature being controlled by means of a Variac. This tape i s made of f i b r e g l a s s interwoven with resistance wire such that both terminals are 'at--the same end; there was thus no danger of coupling to the t o r o i d s . The t o t a l capacitance between the heating tape and the solution was measured as =50 uuf; hence loop resistances up to R = 3.2 = 6 Megohm . (see Appendix 1) wCT could be used before the heating tape would cause 5% error in the apparent loop resistance. A photographic copy of the run as traced out on the chart recorder i s ,in Figure 16. The high room temperature (30°C) was Facing Page 28 40. OUTPUT m v 20 i, = 30ma @ 1900o/s THERMOCOUPLE: COPPER-CONSTANTAN REF. JUNCTION 0°C OUTPUT: OF AMPLIFIER MEASURED ON WAVE-ANALYZER T25°C. 00oC* ' 21 ' TP-THERMOCOUPLE e . M . F . (mv.) Figure 16 PHOTOGRAPH OF FIRST RUN ON KCL SOLUTION kO -OUTPUT (mv) 30 _ V i, = 30ma @ 1900c/s THERMOCOUPLE: COPPER-CONSTANTAN REF. JUNCTION 0*C OUTPUT:AMPLIFIER INTO WAVE-ANALYZER t60°C 1 — — r ' 1 ' ' V THERMOCOUPLE E.M.F. (mv) Figure 17 PHOTOGRAPH OF SECOND RUN ON KCL SOLUTION a result of the hottest July on record in Vancouver. The slope of the trace is : Amplifier output V= 9.0+0.31T°C mv Since V=9#e4 and ei, = 35G therefore V = 3400G So: G= 2.6 x 10^-9.1 x l6"8T°C mho the loop resistance at 25°C was thus 200k ohm. Converting G to cr using = 275 cm"1 : CT = F2.G = 7.2xlO~V 2.5xl6"5T°C mho/cm which-is in reasonable agreement with Noyes' data. The second run on the .01N KC1 solution was taken heating the loop with the circulating hot o i l in the setup shown in Figures 6 and 7. The limitations of the o i l pump restricted the run to temperatures above 55°C while the boiling point imposed an upper limit of 95°C. A photographic copy of the trace is in Figure 17; the slope of the trace is : V= 9.2+0.33T°C mv and since 273 o c r ~ " t o o 0 -4- S o , ry = 7.4x10 4-2.7x10 T C mho/cm which is in reasonable agreement with Noyes' data. Chapter 6.  Conclusions This t w o - t o r o i d system f o r e l e c t r o d e l e s s c o n d u c t i v i t y measurements works very w e l l over a wide range of loop -6 conductances, 100 mho to 10 mho. The upper l i m i t may be extended using low-frequency apparatus (below 4-Oc/s) whereas the lower l i m i t depends on the s t r a y coupling between the t o x o i d s ; f o r a given separation f u r t h e r r e d u c t i o n of t h i s coupling would r e q u i r e extensive and elaborate s h i e l d i n g around the t o r o i d s . T h i s puts a p r a c t i c a l lower l i m i t on sample conductances of around 10 mho/cm. The complexity of the apparatus i n v o l v e d does not j u s t i f y i t s use over conventional f o u r - e l e c t r o d e techniques except i n cases where electrodes might contaminate the sample or where the sample does not lend i t s e l f to the a p p l i c a t i o n of e l e c t r o d e s as i n the case of semiconductive f i b r e s . The only p r a c t i c a l way of heating the sample i s w i t h heating tape wrapped unif o r m l y around the sample loop; i n t h i s method temperatures up to 200°C are e a s i l y achieved while much-higher temperatures could be reached using adequate i n s u l a t i o n . Care should be taken to avoid undue heating of the t o r o i d a l f e r r i t e cores as t h e i r p e r m e a b i l i t y i s temperature dependent. Appendix 1 E f f e c t of the Coupling Capacity of Uniformly Spaced .Heating Tape on the Apparent Conductance of the Coupling Loop For order of magnitude purposes consider the heating element to have zero impedance and the induced emf e^ t o be l o c a l i z e d i n both the sample and the heating tape. The c o n f i g u r a t i o n may then be represented s c h e m a t i c a l l y as shown: r i s the resistance- per u n i t length of the sample loop. C i s the capacitance per u n i t length of loop. L i s the t o t a l l ength of the loop. R i s the t o t a l r e s i s t a n c e of the loop. =• r J -c T i s the t o t a l capacitance coupling the heating tape to the sample loop, =? CL. w i s the s i g n a l frequency. X i s the d i s t a n c e from the- break i n the heating tape to the elemental s e c t i o n dx. In an elemental s e c t i o n x t o x + dx: i , i,+ d i , ig d i ^ V V + dV d i 2 = - d i j = -jwCVdx r e w r i t t e n V = .j d i 2 wC dx dV•= - i 2 r d x r e w r i t t e n dV — - i 2 r dx D i f f e r e n t i a t i n g (1) and equating to (2) gi v e s : d z iz = j w r C i 2 whose s o l u t i o n i s : i 2 = a )e* t X+ a 2e" 3' xand correspondingly: r «.X -ocx. -) = J£L 4 a . e - a 2 e f V _ wC where «L2=-jwrC and a, and az are a r b i t r a r y constants to be determined from the boundary c o n d i t i o n s . Boundary Conditions 1. i 2 (x= 0) = i 2 (x= L) hence: a, -*- a 2= a, e + a^e j / «t-l , x <x-L and: a? = a, (e - 1) = a. e t h e r e f o r e : i g a, |_e + e J 2. V (x= 0) = V (x = L) + ei hence: l * a , r { - - J * * ' t V " 1 - l l + e wC 1 J wC L thus: e; = 2 . 1 * a, ( V c ^ l wC L J 32 s u b s t i t u t i n g f o r a, i n (5) g i v e s : i 2 _ viCei le e J (6) 2j~ [ > e ~ L : i The current threading the secondary t o r o i d i s the sum of i i and i g which must, be constant around the loop. I = i i (x).+ i2. (x) = ig (x =: 0) since i i (x) — 0^ as x-*-0 S u b s t i t u t i n g from (6): 1 = w C e c 0 -|-e°cL) _ ,wCec . -2 J 0 0 C I - e~ L) ~ 2ioc °% When C = 0, the current i s I. = e: _ ec R • rL I _ jwrCL • , , which reduces t o : T oo, Coth ocl 1{> T I _ ocL i ' J — ~2~ Coth <Z± T h i s i s of the form x coth x whose expansion i s : 1 + x 2 _ x4" + 2*1 • • f o r x 2< Tr 2 3 45 945 since x 2= ocL 2 _ .jwRCT we have: 4 4 I - i + .iyR CT + (wRCrf I 0 ~" 1 i ! T 720 so f o r I I i . e . f o r a coupling capacity of 250 juuf at a frequency of 1900c/s the maximum loop r e s i s t a n c e f o r l e s s than 5$ e r r o r i n apparent r e s i s t a n c e i s : 3 2 6 R = 2?r( 1900) 250x10"'* = 1.2x10 ohm BIBLIOGRAPHY L i z e l l , B. The E l e c t r i c a l C o n d u c t i v i t y of L i q u i d Selenium and Selenium-Tellurium mixtures. 1 9 5 1 . Noyes, A. A. The E l e c t r i c a l Conductance of Aqueous S o l u t i o n s . 1 9 0 7 . Handbook of Chemistry and P h y s i c s . Chemical Rubber P u b l i s h i n g Company. 1954. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0105212/manifest

Comment

Related Items