Tables of reactance integrals relating to the eddy current conductivity probe

(1)

(2)

(3)

'f..-NA TIO'f..-NAL RESEARCH LABORATORIES Ottawa, Canada

REPORT

Division of Mechanical Engineering Analysis Section

Pages - Preface - 5 Report: MK-ll

Text App. Tables Figures For: Subject: Submitted by: Approved by: SUMMARY

-

6 Date: January 1963

-

2 Lab. Order: 13852A

-

4 File: M2-26-7

-

7 Internal

TABLES OF REACTANCE INTEGRALS RELATING TO THE EDDY CURRENT CONDUCTIVITY PROBE

D. C. Baxter

Acting Section Head

D. C. MacPhail Director

Authors : F. T. Stock D. C. Baxter

Tables of the inductance and loss resistance change in a single-turn coil carrying an electrical current and surrounding a cylindrical tube in which is flowing an electrically conducting fluid are presented. These properties are a function of geometry, fluid velocity , electrical properties of the fluid and electrical frequency.

(4)

Page - (ii) MK-ll Summary List of nlustrations List of Symbols 1. Introduction TABLF OF CONTENTS

2. Physical Aspects of the Problem 3. Method of Calculation

4. Results 5. Conclusions 6. References

Appendix A: Evaluation of the Integrand at x =0

LIST OF TABLES

Effect of n on Accuracy Limits of Integration

Non-Dimensional Inductance Change (a) h = 1.5

(b) h = 2 (c) h = 3 (d) h=4 (e) h = 5

Non-Dimensional Loss Resistance (a) h = 1.5 (b) h = 2 (c) h = 3 (d) h=4 (e) h = 5 ti hEil

Page

(i) (Ui) (iv) 1 1 2 5 5 6 Table 1 2 3 4

(5)

LIST OF ILLUSTRATIONS

Integrand vs. x

Non-Dimensional Inductance Change vs. s (a) Q

=

0 (b) Q

=

0.2 (c) Q

=

0.4 (d) Q

=

0.6 (e) Q

=

0.8 (f) Q

=

1

Non-Dimensional Loss Resistance vs. s (a) _Q

=

0 (b) Q

=

0.2 (c) Q

=

0.4 (d) Q = 0.6 (e) Q

=

0.8 (f) Q = 1

Non-Dimensional Inductance Change vs. Q s

=

100, h

=

1. 5

Non-Dimensional Loss Resistance vs. Q

8

=

100, h

=

1. 5

Effect of Coil Diameter on Inductance Change (a) Q = 0

(b) Q

=

1

Effect of Coil Diameter on Resistance (a) Q

=

0 (b) Q

=

1 Page - (Hl) MK-ll Figure 1 2 3 4 5 6 7

(6)

Page - (iv) MK-ll SYMBOL a b c 1,c2,c3,c4

cr

E n F g h

1

n j K n k .6L M5 P Q R s un' v n v x LIST OF SYMBOLS DEFINITION Radius of coU

Radius of conducting core

See (5), (6), (7), (8)

.jk2

+

j(1~

(w - vk) Truncation error, (15)

Integrand of (1)

(a-b)/b

1 + 'g

=

a/b

=

radius ratio

Modified Bessel function of the first kind and order n (n

=

0, 1)

Imaginary unit

Modified Bessel function of the second kind and order n (n

=

0, 1)

x/b

Change in inductance due to flowing fluid

Maximum value of F v (x) in the interval (Xl' x 2) Modulus of

<ih,

(2)

v/bw

2g/Q

Real and imaginary parts of In' (3)

Velocity

kb, integration variabie

(7)

SYMBOL IJ.

(]

4> w

LIST OF SYMBOLS (Cont'd)

DEFINITION Magnetic permeability Conductivity Phase angle of db, (2) Circular frequency • . Ui ,f i j il "iiliiitWlI.IUU lI.h" • Page - (v) MK-ll 1",ii1l1llmlliiiiLl,II~

(8)

(9)

TABLES OF REACTANCE INTEGRALS RELATING TO THE EDDY CURRENT CONDUCTIVITY PROBE

1. INTRODUCTION

Page - 1

MK-ll

In Reference 1 an instrument is described which enables the measure-ment of the electrical conductivities of gases behind strong shock waves in shock tubes. The performance of the instrument depends upon a change in inductance of a coi! surrounding the tube, and is given by equations (13) and (14) of Reference 1 as 00

I

xl 10 lxi 11 (db) db 11

I

x

I

lo(db) ~L

f

K 2

-

h

I

hx

I

db Kl

I

x

I

dx J.ta _00 1 _10(db) ₊_{lxi KO}

I

xl 11(db) (1) 00

=

- h

f

F(xj Q, h, s) dx -00 with db

=

/x2

+

js (1 - Qx)/2

(Note that certain typographical errors in Reference 1 have been corrected. )

To enable use of this instrument a series of calculations of the inductance clrange as a function of flow parameters and coil geometry were made, and the results are presented here.

2. PHYSICAL ASPECTS OF THE PROBLEM

Equation (1) arises from the consideration of magneto-hydrodynamic flow of a conducting fluid in a circular cylinder. The cylinder is surrounded by a single-turn coil through which an alternating electrical current is passing. The non-dimensional parameters of the problem have the following physical significance: h is the ratio of coil diameter to the diameter of the conducting fluid cylinder, s is twice the product of applied current frequency, electrical fluid conductivity, magnetic pernieability and square of the cylinder radius, while Q (the magneto-hydrodynamic

(10)

Page - 2 MK-ll

term) is the ratio of fluid velocity divided by the product of frequency and cylinder radius. ~L is then the complex change in inductance experienced by the coil, due to the presence of the moving, conducting fluid. lts real part is the physical inductance change and its imaginary part the physical loss resistance divided by the frequency.

3. METHOD OF CALCULATION

Expressing the complex argument db of (1) as

dh

= P ej<p = P cos<p

+

j Psin <p and the Bessel functions of this argument as

I (db)

=

u

+

jv

n n n

equation (1) may he separated into real and imaginary parts 00 ~L /Ja

= -

h

J

Kl 2

I

hx

I

_00 where Cl

=

lxi

10

lxi

uI c₂

=

lxi 10 lxi vI c 3

=

Ix

I

KO Ix I uI

+

c₄

=

lxi Kolxl vI

+

P cos <p 1₁ 'x

I

U o

+

P sin <p 11 I x Ivo P cos

lP

1 1

I

x Ivo P sin <p 11 I x I Uo P cos <p Kl Ix

I

U

o

P cos

lP

Kl I x Ivo P sin <p Kl

I

x Ivo

+

P sin

lP

Kl

I

x

I

U o (2) (3) (4) (5) (6) (7) (8) 10

I

x

J,

1

I

x

I,

KO

1

x

I

and Kl

1

x

I,

modified Bessel functions with

real argument and of the first and second kind, were evaluated by a library routine

(11)

Page - 3 MK-ll

(Ref. 2). The following equations define the method used there to calculate I : n 00 10

I

x

I

=

6

{32r r=O (9) 00 11

Ix I

=

6

_{32r+1 (10) r=O {32r+1 = x/2 {32r (11) r+1 {32r+2 = x/2 {32r+1 (12) r+1

=

1 (13)

The method for calculating Kn is described in Reference 3. These calculations give aresult for Kn and In accurate to ten digits.

Bessel functions of the first kind with complex argument were evaluated

by Reference 4 using the series expansion (equations (10) and (11), page xxii of Reference 5).

=

u

+

jv

n

00

=

~

tI

(~+t)!

t=O ( p)n+2t

"2

cos (n+2t) + j

t~

t:

(n~): (~)

n+2t sin (n+2t) (14) MhiilliiUiliii iI

---=

•

(12)

Page - 4

MK-ll

The series are lerminated when the ratio of ti

(!-tt) i

(~)

n+2t to

lu

n

I

I. 1 •••

than 10-6, and this gives an accuracy of at least nine digits.

The infinite integral (1) was evaluated by considering finite subintervals (Tabie 2). In each subinterval the modified Gaussian two-point quadrature formula of Ralston (Ref. 6, 7) is used. That is, if the subinterval (xl' x

2) is divided into n equal parts of length ~, then

,. -Ó,X

=

3..[2

n/2-l +

~

(1+1/6

../2/7)

10 F(~

Gl

+1 +1/5

(.f2

-..(7)

+

2m

J)

n/2-1

+

ill<

(1-1/6

J2/7)

m~

F(ill<

[Xl

+1+1/5

(.fi

--/7)+

2m])

+

E (15)

n

where the truncation error is bounded by

(1

+

1/-/2) (x₂- xl) (6,x)5 M₅ 1125

and M5 is the maximum value of FV (x) in (xl' x 2).

A plot of a typical integrand of (1) (Fig. 1) shows that the higher derivatives and the function itself are larger in the vicinity of the origin; hence, for a given accuracy in the final answer, n should be higher for small values of x.

The n used in the calctilation of each subinterval was selected af ter comparing results obtained by the use of different n' s. The one was chosen which would give an accuracy of six places in the final answer. As an example, when

(13)

! I! '17 •• , I f l ! ! t I I

Page - 5

MK-ll

contribution to the total integral for various n is shown in Tabie 1. As aresult n

=

6 was used for the intervals shown. 'file values of n used in the various

subintervals are shown in Table 2.

'-The Iimits of integration were extended until six-place accuracy was achieved. The limits used depend on h, and are shown in Table 2. As an exampIe, for h

=

2, Q

=

0 and s

=

16 the results for the -9 to -6 range are 5.0 x 10-10

-9 -2

and 5.2 x 10 as compared with the final results of 3.1522 x 10 and

-2

2.1546 x 10 . Also, for h

=

4, Q

=

0, s

=

20 and range 3 to 6 we get

-12 -11 -1

8.1 x 10 and 2.0 x 10 as compared with 1.13816 x 10 and

1. 02343 x 10-1•

4. RESULTS

The computations were carried out on a Bendix G15 digital computer in a double-precision, floating-point mode. The range of variables used was

s

=

2(2)20, 40(20)100

h

=

1. 5, 2(1)5

Q

=

0(0.1)1

In addition a few, less accurate values were computed for s

=

100, h

=

1. 5 and Q

between 1 and 50. The results are given in Tables 3 and 4, and plotted in Figures 2

through 7.

5. CONCLUSIONS

Tables of the inductance and loss resistance change in a single-turn coU carrying an electrical current and surrounding a cylindrical tube in which is flowing an electrically conducting fluid have been presented. These properties are a function of geometry, fluid velocity , electrical properties of the fluid and electrical frequency.

mwu ...

(14)

'__.=a ' _ T ag . . . . , - -'2 _ _ _ - t ' 'cm Page - 6 MK-ll 6. REFERENCES 1. Savic, P. Boult, G. T. 2. Benson, G. C. Dempsey, E. Baxter, D.C. 3. Dempsey, E. Benson, G. C. 4. Stock, F. T. 5. 6. Czasak, V. Baxter, D. C. 7. Ralston, A. 8. McLachlan, N. W. /LES

A Frequency Modulation Circuit for the Measurement of Gas Conductivityand Boundary Layer Thickness in a Shock Tube.

N. R. C. Mech. Eng. Report MT-43, May 1961. Also available in Journal of Scientific Instruments, Vol. 39, June-1962, pp. 258-266.

Modified Bessel Functions of the Second Kind. Bendix Users I Project No. 349.

N. R. C. (unpublished).

Tables of the Modified Bessel Functions of the Second Kind for Particular Types of Argument.

Can. J. Phys., Vol. 38, No. 3, March 1960, pp. 399-424.

Bessel Function of the First KiIid with a Complex Argument.

Bendix Users I Project No. 558, N. R. C. (unpublished).

Tables of the Bessel Functions JO(z) and J1 (z) for Complex Arguments.

National Bureau of Standards, Columbia University Press 1943, p. xxii.

Evaluation of a Definite Integral. Bendix Users I Project No. 496, N. R. C. (unpublished).

A Family of Quadrature Formulas which Achieve High Accuracy in Composite Rules.

Journalof the Association for Computing Machinery, Vol. 6, No. 3, July 1959, pp. 384-394.

Bessel Functions for Engineers.

(15)

•• '_IW'''I~' . ! ! " " I ' I ' I'

APPENDIX A

EVALUATION OF THE INTEGRAND AT x

=

0

Since Kl (0) = 0() and 1 1(0) = 0 the integrand of (1) F

=

K 2

\hx\

_{\X\lo\X\11(db) - db 11 Ix}

\10

(db) 1 db KI\x1lo(ëfb) + IX\KOlxlll(db) Page - A-I MK-il (Al)

cannot he evaluated directly at the point x

= O. Exarnining the power series for

the Bessel functions about x

= 0 (Ref. 8):

2

=

1

+

x /4

+ ...•

1₁(x)

=

x/2 ••..•.

KO (x)

= -

In x/2 + ••....

Kl (x)

=

I/x + ..••..

and on substitution of these into (Al) it follows that

lirn F x-+O

As an exarnple, for h

=

2, Q

= 0, s

=

2 and using (A6)

F(O) =-0.002,529,099,4 - j 0.015,190,386,4 (A2) (A3) (A4) (A5) (A6)

(16)

Page - A-2 MK-ll

For comparison, direct substitution of 1 x 10-7 for x into (Al) gives the result

F(O)

=

-0.002,529,099,442,35 - j 0.015,190,386,398,5

The expression (A6) was not used by the computer, rather the value of 10-10 is substituted for x whenever x

=

O. 1111s wUI give a result accurate to ten places.

(17)

• • .•. / _ ' n t _ "IiJ"!' I ' 1 1 1 1 1 1 '1 , . P i l ! ! . I IjJm_a;;IIW!!ll''1IIIIlIlUw'UII'lII!.II_I.

( -0.

4,

-0. 1 )

(0.1, 0.4)

(-0.1,0.1)

TABLE 1

Table 1

MK-11

EFFECT OF

n

ON ACCURACY Q

=

0,

h

=

1.5,

s

=

6 n

X

₂

COMPUTED VALUE OF

J

F(

x)

dx

x

₁

REAL PART

4 -.007 568 345 3

6 -.007 568 327 9

8 -.007 568 325 6

4 -.007 568 305 6

6 -.007 568 322 0

8 -.007 568 324 1

4 -.006 427 168 5

6 -.006 427 156 8

8 -.006 427 146 0

lMAGINARY PART.

-.016 018 988

-.016

'

018 959

-.016 018 955

-.016 018 922

-.016 018 949

-.016 018 953

-.013 025 833

-.013 025 813

-.013 025 795

001

(18)

'"

Table 2

MK-11

I I I I I 'P

SUBINTERVAL

(x

_1,

x

₂₎

(-9,

-6)

(-6,

-3)

( -3, -2)

(-2, -1)

(-1, -0.4)

( -0.4, -0.1)

(-0.1, 0.1)

(0.1, 0.4)

(0.4, 1)

( 1 , 2)

(2, 3)

(3,

6)

(6, 9)

n

4

8

6

8

4

,u, TABLE 2

LIMITS OF INTEGRATION

SUBINTERVALS USED FOR THE FOLLOWING h's

1.5 1.5, 2, 3

1.5, 2, 3, 4

Allh

All h

Allh

1.5, 2, 3, 4

1.5,

2,

3

1.5

(19)

s Q = 0 0.1 2 0.007727 0.007755 4 0.028903 0.028990 6 0.058745 0.058871 8 0.092132 0.092247 10 0.125262 0.125311 12 0.155986 0.155934 14 0.183425 0.183255 16 0.207452 0.207164 18 0.228315 0.227917 20 0.246398 0.245909 40 0.344080 0.343286 60 0.386615 0.385818 80 0.412750 0.411942 100 0.431001 0.430188 TABLE 3( a)

NON-DlMENSIONAL INDUCTANCE CHANGE

(~~)

h

=

1.5 0.2 0.3 0.4 0.5 0.6 0.7 0.007839 0.007979 0.008174 0.008425 0.008730 0.009090 0.029250 0.029682 0.030284 0.031052 0.031984 0.033074 0.059251 0.059882 0.060762 0.061888 0.063256 0.064858 0.092594 0.093180 0.094016 0.095112 0.096479 0.098121 0.125471 0.125769 0.126249 0.126958 0.127944 0.129242 0.155801 0.155651 0.155578 0.155687 0.156080 0.156836 0.182783 0.182117 0.181413 0.180847 0.180582 0.180747 0.206354 0.205173 0.203845 0.202618 0.201723 0.201333 0.226797 0.225144 0.223244 0.221419 0.219964 0.219095 0.244518 0.242453 0.240054 0.237712 0.235782 0.234518 0.340938 0.337253 0.332804 0.328417 0.324809 0.322358 0.383386 0.379425 0.374605 0.369987 0.366379 0.364086 0.409444 0.405323 0.400370 0.395789 0.392363 0.390300 0.427646 0.423425 0.418439 0.413968 0.410735 0.408872 8 Q

=

1.5 2.0 5.0 10.0 20.0 ~O.O 100 0.4199 0.4324 0.4785 0.5091 0.5341 0.5559 0.8 0.9 0.009503 0.009969 0.034318 0.035709 0.066686 0.068731 0.100042 0.102234-0.130877 0.132855 0.158006 0.159611 0.181419 0.182628 0.201550 0.202405 0.218930 0.219495 0.234047 0.234388 0.321131 0.321011 0.363068 0.363121 0.389494 0.389661 0.408228 0.408442 - _ ... 1.0 0.010487 0.037241 0.070981 0.104689 0.135170 0.161644 0.184360 0.203872 0.220750 0.235484 0.321815 0.364044 0.390679 0.409551 ~-t ~Q _f7 1 -

,.

~

...

Q

....

- l 1"-' r' ::I: < z _rv.;; ;:;;,.. Cl:C c::1 -1 n"l 6~::::

c:go

ooG .... =:lé~ *~Q r-n _, _~ ~é:o zO o r -m C ' j r1"1 r-::4

(20)

s Q = 0 0.1 0.2 2 0.004456 0.004464 0.004490 4 0.016594 0.016617 0.016690 6 0.033508 0.033536 0.033626 8 0.052148 0.052161 0.052211 10 0.070323 0.070305 0.070266 12 0.086864 0.086807 0.086656 14 0.101358 0.101262 0.100997 16 0.113816 0.113687 0.113320 18 0.12!#+3 0.124287 0.123839 20 0.133505 0.133328 0.132819 40 0.179884 0.179700 0.179141 60 0.198924 0.198771 0.198295 80 0.210395 0.210254 0.209814 100 0.218310 0.218177 0.217761 TABLE 3( h)

(~;)

h

=

2 0.3 0.4 0.5 0.6 0.7 0.004533 0.004593 0.004670 0.004765 0.004876 0.016814 0.016987 0.017209 0.017480 0.017798 0.033780 0.033997 0.034278 0.034623 0.035032 0.052302 0.052438 0.052724 0.052868 0.053175 0.070215 0.070166 0.070136 0.070144 0.070208 0.086426 0.086143 0.085840 0.085552 0.085311 0.100585 0.100066 0.099487 0.098900 0.098355 .0.112747 0.112015 0.111187 0.110331 0.109513 0.123133 0.122225 0.121188 0.120105 0.119057 0.132011 0.130964 0.129760 0.128493 0.127257 0.178186 0.176845 0.175197 0.173389 0.171589 0.197445 0.196189 0.194591 0.192817 0.191051 0.209015 0.207811 0.206266 0.204552 0.202860 0.216996 0.215831 0.214331 0.212677 0.211058 0.8 0.9 0.005005 0.005151 0.018163 0.018574 0.035504 0.036039 0.053550 0.053998 0.070343 0.070562 0.085148 0.085085 0.097896 0.097556 0.108791 0.108206 0.118113 0.117325 0.126135 0.125184 0.169932 0.168508 0.189440 0.188067 0.201335 0.200037 0.209614 0.208375 3::-4

,,0

,~

_

..

CJI ~ 1.0 0.005313 0.019028 0.036636 0.054522 0.070875 0.085139 0.097358 0.107788 0.116725 0.12!#+0 0.167361 0.186977 0.199034 0.207461

(21)

s Q = 0 0.1 0.2 2 0.002007 0.002008 0.002014 4 0.007439 0.007444 0.007458 6 0.014927 0.014932 0.014946 8 0.023060 0.023060 0.023061 10 0.030862 0.030855 0.030834 12 0.037844 0.037829 0.037785 14 0.043864 0.043842 0.043777 16 0.048959 0.048932 0.048851 18 0.053245 0.053215 0.053123 20 0.056856 0.056823 0.056725 40 0.074697 0.074672 0.074598 60 0.081799 0.081780 0.081724 80 0.086046 0.086030 0.085980 100 0.088965 0.088949 0.088903 - - - - - ~-TABLE 3( c)

NON-DlMENSIONAL INDUCTANCE ClliU,GE

(!;)

h

=

3 0.3 0.4 0.5 0.6 0.7 0.002022 0.002034 0.002050 0.002069 0.002092 0.007481 0.007514 0.007555 0.007606 0.007665 0.014969 0.015002 0.015044 0.015097 0.015160 0.023064- 0.023068 0.023076 0.023088 0.023106 0.030801 0.030758 0.030707 0.030651 0.030593 0.037714 0.037618 0.037502 0.037371 0.037230 0.043671 0.043528 0.043354 0.043154 0.042937 0.048719 0.048539 0.048319 0.048063 0.047783 0.052973

I

0.0527h8 C.C52:;~3 0.052216 0.051888 0.056564 0.056342 0.056065 0.055739 0.055376 0.074468 0.074278 0.074020 0.073692 0.073300 0.081626 0.081478 0.081270 0.080994 0.080652 0.085894 0.085763 0.085578 0.085329 0.085016 0.088824 0.088704 0.088532 0.088301 0.088008 0.8 0.9 0.002118 0.002147 0.007734 0.007812 0.015233 0.015316 0.023131 0.023165 0.030538 0.030488 0.037085 0.036942 0.042710 0.042481 0.047487 0.047186 0.051538 0.051180 0.054987 0.054585 0.072856 0.072377 0.080253 0.079815 0.084648 0.084241 0.087663 0.087278 1.0 0.002180 0.007898 0.015411 0.023209 0.030447 0.036806 0.042257 0.046888 0.050823 0.054183 0.071881 0.079357 0.083816 0.086881 ~-i

,,0

I~ _/11 ~

n

(22)

s Q = 0 0.1 0.2 2 0.001141 0.001142 0.001143 4 0.004222 0.004223 0.004227 6 0.008444 0.008445 0.008449 8 0.012999 0.012998 0.012997 10 0.017335 0.017332 0.017323 12 0.021186 0.021181 0.021165 14 0.024483 0.024476 0.024454 16 0.027257 0.027248 0.027222 18 0.029577 0.029568 0.029539 20 0.031522 0.031512 0.031483 40 0.041022 0.041015 0.040996 60 0.044769 0.044764 0.044750 80 0.047007 0.047002 0.046990 100 0.048542 0.048538 0.048527 TABLE 3( d)

(~~)

h

=

4

0.3 0.4 0.5 0.6 0.7 0.001146 0.001150 0.001155 0.001161 0.001169 0.004234 0.004244 0.004257 0.004273 0.004291 0.008455 0.008464 0.008475 0.008489 0.008505 0.012995 0.012993 0.012991 0.012988 0.012986 0.017309 0.017291 0.017268 0.017241 0.O1721~ 0.021139 0.021104 0.021059 0.021007 0.020949 0.024418 0.024369 0.024307 0.024233 0.024150 0.027179 0.027120 0.027045 0.026956 0.026853 0.029492 0.029427 0.029343 0.029244 0.029129 0.031434 0.031365 0.031278 0.031172 0.031049 0.040962 0.040914 0.040848 0.040764 0.040659 0.044726 0.044690 0.044642 0.044578 0.044496 0.046969 0.046938 0.046896 0.046840 0.046769 0.048507 0.048479 0.048441 0.048390 0.048325 ' -0.8 0.9 0.001177 0.001187 0.004312 0.004336 0.008525 0.008547 0.012986 0.012986 0.017180 0.017147 0.020885 0.020818 0.024059 0.023961 0.026740 0.026619 0.029000 0.028861 0.030911 0.030760 0.040533 0.040386 0.044394 0.044273 0.046679 0.046570 0.048243 0.048142 1.0 ~~ ÄQ .2: _CD <".I

-

....

a. 0.001197 0.004363 0.008572 0.012988 0.017114 0.020748 0.023858 0.026490 0.028714 0.030599 0.040221 0.044132 0.046444 0.048025

(23)

s Q = 0 0.1 2 0.000735 0.000735 4 0.002716 0.002717 6 0.005423 0.005423 8 0.008331 0.008331 10 0.011088 0.011087 12 0.013528 0.013525 14 0.015608 0.015605 16 0.017353 0.017349 18 0.018808 0.018804 20 0.020025 0.020021 40 0.025937 0.025935 60 0.028261 0.028259 80 0.029647 0.029645 100 0.030597 0.030596 1 ____ _ ___ _ _ _ _ ~_._-- - - - ---

---

--

-

--

--.. ' 0.2 0.000736 0.002718 0.005425 0.008330 0.011083 0.013519 0.015596 0.017338 0.018793 0.020010 0.025928 0.028254 0.029641 0.030592 - - - -._ -TABLE 3( e)

NON-DIKENSIONAL INDUCTANCE CHANGE

(~~)

h

=

5 0.3 0.4 0.5 0.6 0.7 0.000737 0.000739 0.000741 0.000743 0.000747 0.002721 0.002725 0.002730 0.002737 0.002744 0.005427 0.005430 0.005434 0.005439 0.005445 0.008329 0.008327 0.008325 0.008322 0.008320 0.011077 0.011068 0.011057 0.011044 0.011029 0.013507 0.013492 0.013472 0.013449 0.013422 0.015581 0.015560 0.015533 0.015501 0.015465 0.017321 0.017296 0.017265 0.017227 0.017183 0.018774 0.018747 0.018713 0.018672 0.018624 0.019990 0.019963 0.019928 0.019885 0.019835 0.025916 0.025898 0.025875 0.025845 0.025808 0.028245 0.028232 0.028216 0.028194 0.028167 0.029633 0.029622 0.029607 0.029589 0.029566 0.030585 0.030575 0.030561 0.030545 0.030523 - - - -L-_____ _ _ _ _ _ _ _ _ L- ___ 0.8 0.9 0.000750 0.000754 0.002752 0.002762 0.005452 0.005460 0.008317 0.008315 0.011013 0.010995 0.013392 0.013359 0.015423 0.015378 0.017134 0.017079 0.018569 0.018508 0.019777 0.019712 0.025763 0.025711 0.028133 0.028093 0.029537 0.029502 0.030497 0.030465 1.0 0.000758 0.002773 0.005469 0.008313 0.010977 0.013324 0.015329 0.017020 0.018442 0.019642 O. 02561.~9 0.028044 0.029459 0.030427 ~-i Q ";:c:r 1;;--(Jol ct

(24)

-s Q

=

0 0.1 0.2 2 0.054486 0.054476 0.054446 4 0.102542 0.102474 0.102270 6 0.140318 0.140136 0.139594 8 0.167167 0.166844 0.165886 10 0.184563 0.184109 0.182761 12 0.194735 0.194181 0.192535 14 0.199810 0.199194 0.197360 16 0.201477 0.200831 0.198911 18 0.200961 0.200319 0.198386 20 0.199106 0.198484 0.196595 40 0.169502 0.169231 0.168242 60 0.149676 0.149533 0.148895 80 0.136179 0.136099 0.135631 100 0.125979 0.125953 0.125625 8 100 ~,. TABLE 4( a) NON-DlMENSIONAL 10SS RESISTANCE

(~)

wJ..ta h

=

1.5 0.3 0.4 0.5 0.6 0.7 0.054397 0.054328 0.054240 0.054132 0.054006 0.101934 0.101467 0.100875 0.100163 0.099338 0.138702 0.137476 0.135939 0.134116 0.132037 0.164317 0.162176 0.159518 0.156405 0.152906 0.180559 0.177571 0.173882 0.169598 0.164832 0.189846 0.186196 0.181698 0.176493 0.170738 0.194351 _{0.190250 0.185184 0.179319 0.172847} 0.195733 0.191367 0.185935 0.179622 0.172655 0.195155 0.190662 0.185016 0.178415 0.171113 0.193393 0.188869 0.183115 0.176334 0.168811 0.166080 0.162278 0.156720 0.149738 0.141883 0.147138 0.143567 0.138068 0.131149 0.123495 0.134040 0.130519 0.125038· 0.118242 0.110864 0.124122 0.120666 0.115240 0.108613 0.101526 - '- .. . . . . . _- .-Q = 1.5 2.0 5.0 10.0 20.0 I 0.0574 0.0427 0.0140 0.0068 0.0052 I I 0.8 0.9 0.053861 0.053699 0.098406 0.097377 0.129735 0.127243 0.149094 0.145041 0.159702 0.154321 0.164592 0.158209 0.165969 0.158876 0.165271 0.157693 0.163386 0.155487 0.160857 0.152750 0.133680 0.125519 0.115667 0.108029 0.103431 0.096297 0.094440 0.087769 .-3:~ " _,r:rCl CD ~

-

Cl 1.0 0.053518 0.096259 0.124594 0.140816 0.148795 . 0.151726 0.151732 0.150109 0.147621 0.144714 0.117650 0.100771 0.089544 0.081410

(25)

"" s Q = 0 0.1 0.2 2 0.029663 0.029658 0.029645 4 0.055495 0.055468 0.055394 6 0.075272 0.075208 0.075023 8 0.088730 0.088625 0.088318 10 0.096857 0.096721 0.096318 12 0.101039 0.100886 0.100428 14 0.102544 0.102387 0.101916 16 0.102343 0.102194 0.101742 18 0.101120 0.100986 0.100572 20 0.099324 0.099208 0.098844 40 0.080202 0.080211 0.080212 60 0.069134 0.069163 0.069233 80 0.061984 0.062019 0.062110 100 0.056755 0.056795 0.056903 TABLE 4(b)

NON-DlMENSIONAL LOSS RESISTANCE

(w~a)

h

=

2 0.3 0.4 0.5 0.6 0.7 0.029625 0.029599 0.029565 0.029525 0.029477 0.055273 0.055107 0.054895 0.054641 0.054345 0.074720 0.074302 0.069776 0.073146 0.072422 0.087814 0.087121 0.086251 0.085217 0.084036 0.095653 0.094740 0.093592 0.092231 0.090680 0.099671 0.098624 0.097303 0.095729 0.093930 0.101131 0.100034 0.098634 0.096952 0.095015 0.100978 0.099894 0.098489 0.096777 0.094782 0.099861 0.098832 0.097469 0.0957n 0.093780 0.098204 0.097252 0.095960 0.094319 0.092350 0.080131 0.079842 0.079207 0.078139 0.076626 0.069283 0.069178 0.068742 0.067863 0.066530 0.062205 0.062165 0.061802 0.061000 0.059757 0.057028 0.057029 0.056710 0.055958 0.054782 -0.8 0.9 0.029423 0.029362 0.054009 0.053637 0.071610 0.070721 0.082724 0.081302 0.088965 0.087114 0.091939 0.089793 0.092861 0.090532 0.092547 0.090118 0.091518 0.089045 0.090095 0.087614 0.074725 0.072527 0.064812 0.062807 0.058147 0.056279 0.053252 0.051501 1.0 0.029295 0.053229 0.069763 0.079786 0.085155 0.087526 0.088072 0.087548 0.086420 0.084969 0,070125 0.060615 0.054233 0.049564

s:

ë}

"cr

_I _~ -~

-~

(26)

s Q

=

0 0.1 0.2 2 0.012755 0.012754 0.012751 4 0.023733 0.023728 0.023712 6 0.031939 0.031926 0.031887 8 0.037304 0.037284 0.037224 10 0.040336 0.040311 0.040237 12 0.041691 0.041665 0.041588 14 0.041952 0.041928 0.041856 16 0.041548 0.041528 0.041466 18 0.040771 0.040755 0.040706 20 0.039806 0.039794 0.039757 40 0.031107 0.031115 0.031137 60 0.026468 0.026476 0.026500 80 0.023552 0.023559 0.023582 100 0.021454 0.021461 0.021483

....

TAaLE 4{ c)

NON-DIMENSIONAL LOSS RESISTANCE

(w!a)

h

=

3 0.3 0.4 0.5 0.6 0.7 0.012747 0.012742 0.012735 0.012726 0.012715 0.023686 0.023650 0.023604 0.023547 0.023481 0.031824 0.031736 0.031624 0.031489 0.031332 0.037125 0.036986 0.036811 0.036599 0.036352 0.040115 0.039945 0.039728 0.039465 0.039159 0.041460 0.041281 0.041050 0.040767 0.040435 0.041736 0.041564 0.041341 0.041064 0.040734 0.041361 0.041210 0.041008 0.040753 0.040442 0.040620 0.040494 0.040321 0.040095 0.039813 0.039692 0.039592 0.039450 0.039259 0.039009 0.031171 0.031210 0.031242 0.031250 0.031215 0.026539 0.026591 0.026647 0.026691 0.026703 0.023620 0.023672 0.023731 0.023783 0.023807 0.021521 0.021574 0.021635 0.0~1690 0.021721 0.8 0.9 0.012703 0.012690 0.023406 0.023322 0.031153 0.030954 0.036073 0.035764 0.038811 0.038424 0.040055 0.039629 0.040350 0.039915 0.040074 0.039650 0.039472 0.039071 0.038698 0.038325 0.031021 0.030955 0.026662 0.026556 0.023783 0.023698 0.021706 0.021634 3: -f " 0 I ~ 1\1 ~ 0' 1.0

_I

I 0.012675 0.023229 0.030736 0.035426 0.038002 0.039161 0.039433 0.039173 0.038612 0.037890 0.030715 0.026378 0.023546 0.021497

(27)

s Q = 0 0.1 0.2 2 0.007111 0.007111 0.007110 4 0.013198 0.013197 0.013191 6 0.017697 0.017693 0.017681 8 0.020586 0.020579 0.020560 10 0.022166 0.022159 0.022137 12 0.022823 0.022816 0.022794 14 0.022887 0.022880 0.022861 16 0.022598 0.022593 0.022578 18 0.022117 0.022114 0.022103 20 0.021546 0.021544 0.021536 40 0.016650 0.016653 0.016661 60 0.014108 0.014111 0.014118 80 0.012524 0.012526 0.012533 100 0.011389 0.011391 0.011398 TABLE 4( d)

(~a)

h =

4

0.3 0.4 0.5 0.6 0.7 0.007109 0.007107 0.007104 0.007101 0.007098 0.013183 0.013170 0.013155 0.013136 0.013114 0.017660 0.017631 0.017594 0.017550 0.017497 0.020529 0.020485 0.020430 0.020362 0.020282 0.022100 0.022049 0.021982 0.021902 0.021807 0.022757 0.022705 0.022638 0.022556 0.022458 0.022828 0.022782 0.022721 0.022645 0.022553 0.022551 0.022513 0.022462 0.022397 0.022317 0.022083 0.022054 0.022015 0.021963 0.021897 0.021523 0.021503 0.021475 0.021436 0.021384 0.016675 0.016693 0.016714 0.016737 0.016758 0.014131 0.014149 0.014172 0.014198 0.014227 0.012544 0.012560 0.012582 · 0.012607 0.012636 0.011409 0.011425 0.011445 0.011470 0.011498 0.8 0.9 0.007094 0.007089 0.013088 0.013059 0.017438 0.017371 0.020192 0.020090 0.021698 0.021575 0.022345 0.022216 0.022446 0.022321 0.022221 0.022108 0.021815 0.021717 0.021318 0.021235 0.016773

0.o167n

0.014253 0.014273 0.012663 0.012685 0.011526 0.011549 1.0 0.007084 0.013027 0.017297 0.019978 0.021440 0.022072 0.022181 0.021978 0.021601 0.021134 0.016767 0.014281 0.012698 0.011564 ~ -I ~ Q

,CT

/11 ~

--0..

(28)

s Q

=

0 0.1 0.2 2 0.004532 0.004532 0.004532 4 0.008400 0.008400 0.008397 6 0.011241 0.011240 0.011234 8 0.013047 0.013044 0.013037 10 0.014018 0.014015 0.014007 12 0.014405 0.0111402 0.014393 14 0.014419 0.014417 0.014410 16 0.014216 0.014214 0.014208 18 0.013896 0.013894 0.013891 20 0.013521 0.013521 0.013519 40 0.010394 0.010395 0.010398 60 0.008790 0.008791 0.008794 80 0.007794 0.007795 0.007797 100 0.007082 0.007083 0.007086 TABLE 4( e)

(w!a)

h

=

5 0.3 0.4 0.5 0.6 0.7 0.004531 0.004530 0.004529 0.004528 0.004527 0.008394 0.008388 0.008382 0.008374 0.008364 0.011226 0.011214 0.011198 0.011180 0.011158 0.013024 0.013006 0.012983 0.012955 0.012923 0.013992 0.013971 0.013945 0.013913 0.013875 0.014379 0.014359 0.014333 0.014302 0.014264 0.014397 0.014380 0.014358 0.014330 0.014296 0.014199 0.014186 0.014168 0.014145 0.014117 0.013884 0.013875 0.013862 0.013845 0.013824 0.013515 0.013509 0.013501 0.013490 0.013475 0.010404 0.010412 0.010422 0.ot0434 0.010446 0.008799 0.008806 0.008815 0.008827 0.008840 0.007802 0.007808 00007816 0.007827 0.007839 0.007090 0.007096 0.007104 0.007114 0.007126 0.8 0.9 0.004525 0.004523 0.008353 0.008341 0.011133 0,011104 0.012885 0.012843 0.013831 0.013781 0.014220 0.014169 0.014256 0.014210 0.014084 0.014044 0.013797 0.013765 0.013456 0.013431 0.010460 0.010472 0.008855 0.008871 0.007853 0.007869 0.007139 0.007154 J:-i ,.. Q I~ _ CD ~ ;--1.0 0.004521 0.008327 0.011073 0.012796 0.013726 0.014113 0.014157 0.013998 0.013727 0.013401 0,010482 0.008886 0.007884 0.007169 I _,

(29)

"'H"""'''' IIDW' " I 1111 . , 1'1 '111 !I IIIIIJIIIIIIIJI!IIIIIIIIIIIIIII!III! '11' "I "" '!IIuon"!!!'1 I 1'1 ,:.tl, ! I FIG.

MK- 11

r----r---~---~r---r---,---r_---~---~N ~--~---~~---~---+---~---~----_+~---__;O N V N 0 CD U) V N _{0 '} 0 0 0 0 0 0 0 0 0 0 0 0

...

) ( d

0

d 0 0 d u.. I )( fI)

>

C Z ~

a:

C) lIJ t-Z pil

(30)

(31)

ft! ' " ' "!!'II,, /1"'11 11, ! II I II UltuHWWII1JUIIII'YIIII11 lIylllIIIllIlUMM""_IIIW ''*'4,,1111' IU!lIIIIIlIIIIlIJ,111 I 0.5 0.3 ~L ~a 0:2 0.1 0.07 0.05 0.03 0.02 0.01 0.007 0.005 0.003 0002 0.001 0.0007 0.0005

o

h·1.5

~

/

V

h·2

,v/

V

~ I I I

I

J

I I

I

//

/1

1 1 11 11

11 I/

1

I _{h· 3} I

---

~

..,....-/

/

h=4

/

_V

r- _{h. 5}

/

~

l..---r-'

I

20 40 60 80 100

s

NON- DIMENSIONAL INDUCTANCE CHANGE VS. S

a :

0

FIG.

20

MK- IJ

120

(32)

(33)

0.5 0.3 ~L p.a . 0.2

O.

I 0.07 0.05 0.03 0.02 0.0 I 0.007 0.005 0.003 0.002 0.00 I 0.0007 0.0005

o

! ! " " " " . " ' . . . " M " Y . . . IIIIN'IIIIIIII" . . . ",·_nnn ... ,11I1I111111I111111111.111 h • 1.5

----

~

/

V

h • 2

V

~

/

I

/

h • 3 I I ~

r0-I

J I

I

/

/j

I 11 11

'I

/1

1

.,...-/

/

h·4

V

_V--

~ h·5

/

l.----

r-

V--V

20 40 GO 80 100 s

a

=

0.2

FIG. 2b MK-II

120

(34)

(35)

0.5 0.3 LlL /Ja 0.2 0.1 0.07 0.05 0.03 0.02 0.0 I 0.007 0.005 0.003 0.002 0.000 I 0.0007 0.0005

o

h c 1.5

...--

~

/

V

he 2

v/

V

~

j

J L h " 3 I I J 11

-

l -J ,."....

I I

L

,,-//

_/

h·4

I

/ V

...-

h· 5

Ij

_/

f...--"

I--'

~

/;

_V

1 I 11 11

11 rl

fl

20 40 60 80 100 s

NON -DIMENSIONAL INDUCTANCE CHANGE VS. S

o

=

0.4

FIG. 2c MK-II

(36)

(37)

0.5 0.3 ~L ~a 0.2

o.

I 0.07 0.05 0.03 0.02 0.0 I 0.007 0.005 0.003 0.002 0.001 0.0007 0.0005

o

h .. 1.5

---

~

/

V

_{h • 2}

,v/

l---

l

-V

I / I / h : 3 I V

-

~ I .."...-

...

I I

/ V I

I

/

h .. 4

'/

V

I---""' ha!5

ij

_/

~

1/

V

1 I 11 IJ

11 I1

U

1

20 40 60 80 100 s

NON - DIMENSIONAL INDUCTANCE CHANGE VS.

s

o

=

0.6

FIG. 2d I MK - I I

(38)

(39)

0.5 0.3 ~L fLo 0.2 0.1 0.07 0.05 0.03 0.02 0.01 0.007 0.005 0.003 0.002 0.001 0.0007 0.0005

o

h = 1.5

--

~

/

....

/

h • 2

V

----

~

/

/ /

h = 3 I 11

.

-I /

,

/

I

'j

/;

1 / 11

IJ

11

"

1 ...-

-/ .;'

/

h=4

I1

~

~ h-5

/

~

V

20 40 60 80 100 s

Q

=

0.8

FIG. 2e

MK- "

(40)

(41)

0.5 0.3

AL

Il a 0.2 0.1 0.07 0.05 0.03 0.02 0.0/ 0.007 0.005 0.003 0.002 0.001 0.0007 0.0005

o

h • 1.5

-

~

V

.-/

h = 2

I

_----

~

V--/

/

I / h = 3 I /

I

I1

-1

1 I

I /

//

_{1 J} 1 1 11

11 I1

1 ---

~

V

/

h=4

I1

~

~ _{h = 5}

/

~

t-" ~

V

20 40 60 80 100 s

NON- DIMENsrONAL INDUCTANCE CHANGE VS. S

a

=

I

FIG. 2f

MK -

Ir

(42)

(43)

-0.5 0.3 R WJJoO 0.2 0./ 0.07 0.05 0.03 0.02 0.01 0.007 0.005 0.003 0.002

!

I I / I /

I

'I

/

1/

~

I I

1J

1

I 0.00/

o

_n .. " ....

""'IIIII'''!!!

,..

--r---

-

r--

r---

r--

---r--

-r---

-- ----

...

-

r--

---

r--~

_r--

:---

-

~ r---20 40 60 80 5

NON-DIMENSIONAL LOSS RESISTANCE VS. S

a

=

0

h zo 1.5 h. 2 h = 3 h=4 h = 5 100

FIG. 30

MK-II

120

(44)

(45)

I I !!1I!lIJfW . _ f t '""M'D"""PHIIIMlIIMMI'II! 'I I I I !

1

1 '

I MI IIUIIlJll" . . . ,III.UWIIIUHtlMWY".' ·.WlIIIIIIIIWlhUlUIMI.LIII I

0.5 0.3 R WJl 0 0.2 0.1 0.07 0.05 0.03 0.02 0.01 0.007 0005 0003 0.002 0.001

I

I I / /

I

/

"

l(

I I I I

o

i'"

---I---

r---h • 1.5 i'"

-- --

_--

r--h : 2

-

_---

r----

r--.

-

h • 3 -...;;..

r---

î---.

--I---

_r---

--

_-

h .. 4

t--

r--h-5 20

40

60 80 100 s

Q

=

0.2

FIG. 3b

MK-II

120

(46)

(47)

I I I I I I U ' ' ' _ . , ,

..

0.5 0.3 R w,.,.o 0.2 0.1 0.07 0.05 0.03 0.02 0.01 0.007 0.005 0.003 0.002 0.001

!

1 I / I

I

/1

1/

/~

J I 1

I

,

o

IDOllllllllllllllllllllllllllllllll11 1IIIIIIIIIIIIIIIIIIIIII g l " . W .

'.*

....

V

---

r---....

-

---.

---

_----

~

-

_r--

r

--

--.

---

I---

~

r---

....

-

r

--20 40 60 80 s • • l!l!mlftlmlllll~u,lII, _ _ ---"LL--h. US

-

h·2 h .3

-

h • 4 h c 5 100

FIG. 3c

MK -11

120

(48)

(49)

0.5 0.3 R WfLO 0.2 0./ 0.07 0.05 0.03 0.02 0.0/ 0.007 0.005 0.003 0.002 0.001

/

I I / I

I

,

I

/ '

/i(

I I

JL

I

o

f"'"

--r--

---... t-.

-I----

---

_----

r---

-r----...

----

r---

...

---

...

_I"-...

-r----

r---

~

--

i

-FIG. 3d

MK-

II

h • Ui h • 2 h .3 h .4 h • 5 20 40 60 80 /00 /20 s

NON - DIMENSIONAL LOSS RESISTANCE VS. S

a

=

0.6

(50)

(51)

0.5 0.3 R "WJLO 0.2 .• "q 'I!" 11'1' "'ll't't" •

!~

r---

_r---r---

r---0.1 J I / I ~

I

0.07 0.05 I 0.03

I

0.02 1 0.0 _I

//

1 1 0.007

JJ

0.005 _I 0.003 0.002 0.00 I

o

---..

-

~

r---

... _~

--

_----

----

_--

_I'--

---

...

r--20 40 60 80 s

Q

=

0.8

h • I. 5 h a 2 ha3 - h. 4 h = !5 100 FIG. 3e MK-II 120

(52)

(53)

"!I" ! U !I!11y""ue'· 0.5 0.3 R wp. 0 0.2

o.

I 0.07 0.05 0.03 0.02 0.0 I

(

I I /

I I

I

1/

!

lit

11 0.00 7 11

I

0.005

,

0.003 ....

...

0.00 0.00 I

o

_ _ ,!lil 11 I _IIII_!I 111 lil! 1111 1 11 1I11"M.lllmlDlillDDI!IIIII11IIU1l1UnnqpemW",.rl l 11111111 11 lil!

-

_-

_r--.

r--

r---~

-

l---

---

i

--

---

-t--.

-

I-

----

---

f---

~

--r--...

-20 40 60 80 s

o

=

I

h • I.!5 h • 2 h • 3 h • 4 h • .!t 100 r _ _

FIG. 3f

MK-II 120

(54)

(55)

- - _ . _ . . --- -- -- - -0.60 0.58 0.56 6L

V

1'0 _{. /} 0.54

/ '

h = 1.5

/

s : /00 0.52

V

.,,-0.50 . /

/

, /

/

0.48

V

/

0.46

₇

/

0.44

/

~

--0.42

-...

_~

"""'-

-0.40 0.38 0.36 0./ 0.2 0.3 0.5 0.7 1.0 2 3 4 5 6 7 8 9 /0 20 30 40 50 Q ~:!!

NON - DIMENSIONAL INDUCTANCE CHANGE

VS.

Q

,.;:c)

I .

(56)

(57)

- - - -0.13

-'"

0./2

"'"

0./1

"-R

\

wp'o 0./0

\

0.09

\

0.08

\

h : 1.5

s :

/00 0.07

\

0.06

\

0.05

\

0.04 \

~

0.03

'\

'"

0.02

"

~

0.0/

r--r-

~

-r---

~" 0 r--.~-C') 0./ 0.2 0.3 0.5 0.7 /.0 _Q 2 3 4 5 6 7 8 9/0 20 30 40 50 I '

NON- DIMENSIONAL LOSS RESISTANCE VS.

0 =01

(58)

(59)

---_ ---_ ~ _ _ _ _ ~~ _ _ _ _ _ _ _ _ _ _ _ ~,J~"!I"'~II,,"~"!lU~_~I.~",,~' ~~ _ _ _ _ _ _ ~ _ _ _ _ _ _ _ _ ~~~ _ _ _ ~LI_ 0.5 0.4 0.3 0.2 0.1

o

... . / ' S : 100

--s : 40 ...-=::-s : 20 s

:-rc)

\

~\

\\

~

\\

f\\

_~

\

1\\

1\

~

_"

.~

~

"~

...

--:::::

_t3:::::

_~

r--

-

r--.:

2 3 h 4 5

EFFECT OF COIL DIAMETER ON INDUCTANCE CHANGE

Q

=

0

FIG.

60

(60)

(61)

0.5

0.4

S "

loe>'

1\ Ss

40 \

0.3

1\ \ \

....-::-

r\\

~

s

=

20 0.2 0./

~

\

i\~

_~

--

~

"

~

_~

s •

10

'"

~

~ ...

~

_r---

---

--::::::

_E:::::::::

_~

-

r---::

o

2 3 4 5 h

EFFECT OF COIL DIAMETER ON INDUCTANCE CHANGE

o

=

I

FIG. 6b MK-II

(62)

(63)

0.20 R wfLo 0.15 0.10 0.05

o

si:

2 0 / ~ Sa 10"""'" S .40 ...

~

1\

~

s

1100--

\

~

\

_~~

~

, 'II! l I I ! !'!''I!!!',',

\.

~

"

""

r--:

~

1 I I l ' I " ~

r---:::

~ .,,-FIG. 7 0

MK -11

t----:

~

t - :-==0

t----:

2 3 4

EFFECT OF COIL DIAMETER ON RESISTANCE Q=O

-5 h

(64)

(65)

•

0.20 R "'/La 0.15 0.10 0.05 "ti U " ! "II !! ! J l l S

.10--I

~

S.20 ...

I

~\

S·40---'

\

\~

S-100-r\

J ! J III!I ilIIIIIIWIllIIt"lIlUlIlIlIlIlllIIlIlllm!!IIIIMlIIIIM""""UI U' 1I11I1111I 1 1