National uureau <j Library, K-V. FEB 2 3 1966

^ecUnlcaL ^^^ote

Of «TANDABDfi

MAR 6 ]973

3

^c.3ff

COMPUTATION OF THE PERMEABILITY AND

PERMITTIVITY OF A RELATIVELY SMALL RING

SAMPLE IN A TOROIDAL COIL

ERIC G. JOHNSON, JR.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

wecAaicat ^^ote 311

ISSUED JANUARY 25, 1966

COMPUTATION OF THE PERMEABILITY AND PERMITTIVITY

OF A RELATIVELY SMALL RING SAMPLE IN A TOROIDAL

COIL

Eric G. Johnson, Jr. Radio Standards Laboratory

institute for Basic Standards

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington D.C. 2Q402

Price: 30((:

CONTENTS

Page

Abstract . o

1 . Introduction ^

2. Procedure • • ^ ( ;

3. Analysis °

4. Conclusion ^^

Appendix I.... ^^

Appendix II

Ill

Computation of the Permeability and

Permittivity of a Relatively

Small Ring Sample in a Toroidal Coil

Eric G. Johnson, Jr.

The derivation and the FORTRAN machine program are presented for a formula which gives the inapedance of a partially

filled toroid coil . This formula shows the relationship of the

complex permeability and permittivity of the ring material,

the dimensions of the coil and sample, and the frequency of

the applied electromagnetic field. Limitations of the formula

due to proximity of the ring and coil, range of permeability

and permittivity of the ring, and uniformity of the coil winding

are considered. The cost of the program in terms of computer

time is also considered. The principal value of the formula

is to accurately determiine the permeability and permittivity

of ferrites.

Key Words: Complex permeability and permittivity, FORTRAN,

impedance, toroid coil.

1 . Introduction

In this technical note we develop the theory and the associated computer

program for the impedance, Z, of a toroid sample with rectangular

cross section imbedded in a toroid shaped frame of similar cross

section around which a wire is wound. In order that the problem be

amenable to theoretical solution it is necessary to require that the

wire coil and the ferrite sample both have a rectangular cross section.

The expected range of validity of this theory can be estimated by the

Df

requirement that 2 tt — n e < 1. Here D is the largest length in

c meters of the toroid cross section, the f is the frequency in hertz, and

the £, |JL are the relative pernaittivity and permeability, respectively,

of the sample. The c is the velocity of light which is in M. K.S. units

Q

3x10 m/sec.
2. Procedure

Assume a ferrite sample with the following boundary conditions:

1. The experimental circumstances are such that the surface

of a rectangular cross -sectioned toroid can be assumed to be a spacially

uniform current sheet.

2. A second rectangular cross -sectioned toroid consisting of the

ferrite material inside the current sheet toroid is to be constructed so

that its size is electrically small compared with the current sheet

toroid. This means that the ferrite sample is small enough that nonuniformity of a uniforna current sheet will produce no essential difference

in the determination of the pernaeability and permittivity of the ferrite

sample.

3. Two complex frequency dependent parameters --permeability

|UL and permittivity e are used to characterize the ferrite.

4. The excitation mechanism of the toroid system is such that

no angular field dependence exists. This means that the basic electromagnetic field equations will be reduced to a single equation. The basic

equations that are used are

V X E = -i U) M H, ^^^ VX H = i (U e E .

(natural units are used, namely c = 1)

5. The coordinates of interest are given in Figure 1

(1)
r =

1

I

I

1 R

Figure 1

The r , r , r , and r are the radial coordinates. The z , z , z , and z are the "z" axis coordinates. The crosshatched area characterizes the

ferrite sample.

By defining the function

P(r,z) = (r - r^) - e(r - r^)] [0(z - z^) - e{z - z^)

with 6 (x) = 1 and e (x) =

if X ^

if x<

the permeability and permittivity may be written as |jl = 1 + {[\^ - l)P(r, z),

and s = 1 + (£ , - l)P{r, z). The u. , e which are the vacuum permeability M o o and permittivity, respectively, are set equal to one. The |jl , £ are

the ferrite 's permeability and permittivity, respectively. The \i and £

give the permeability and permittivity for the entire cross section of

Figure 1. Upon reducing (1) with assumptions 1 to 5 and letting

f = rH , the following differential equation results:

-8 — 8f+-a - d f + ^^£ = ,

fjL r lr£ r |jl z £ z

(2)
Here fo =_^ ^ Pr 'a

\ r / z/

In order to get rapidly convergent results in the machine calculation, we

extract the quasi -static part from this equation, and to make the function continuous to its first derivative, we write

f = I (l + eg{r.z)) . (3)

This transformation assumes that the discontinuity of the material

surface tends under the condition of the experimental system to form

a surface current.

V = iO)

I -tl^ dr dz = i UJI -( 1 + e g) dr dz (4)

and that the total current is given ^ by I , = ZnH r; where r = r, and ^ total cp 1 H is the magnetic field intensity along the azimuthal direction. Since

I = 2TTf(r , z) = Nj, where j is the current in each wire loop, we

xoLai 1 Nj deduce that I = —^ . Therefore, the measured impedance may be

written as

NV !^ OJN

Z =-r- = irr--

J 2rT In z . - z

r /( 4 1 + \i - 1 Inl — llz - z

' "^M 1 1 r / 1 3 21 (5)

+ / dr dz -t—

g (r, z)

Because natural units are used, Z is dimensionless. The first and

second terms represent the quasi-static approximation. The last term

gives the corrections. To evaluate Z we need g. The differential

equation for g is obtained by substituting (3) into (2). The

following equation is obtained:

r r r z

(6)
Assume the following form for the solution:

g = SR(r)Z(z) (7) n n

in which g = on the surface of the current toroid.
3. Analysis

Once the above steps have been taken, the rest of the analysis

uses in the usual manner normal modes and perturbation theory.

Because of the cylindrical symmetry, the best orthogonal set to use is

Bessel functions for the radial dependence. This gives integrable

results.

The Bessel functions used are inaplied by the equation

^r 8 - 8 + k

^ r r r n

R = , n

(Sa)

which can be obtained from the homogeneous version of (6), with

(7) substituted for g and with the boundary conditions R (r ) = R (r ) = 0. The dual function to R is given by R = — R , where R satisfies

n n r n n

R

n

*- 1 *- 2 r 8 - 8 + k

r r r n

= , (8b)

where the 8 denotes differentiation to the left. The orthogonality

condition is

JJ. n n

dr = 6

nn

(8c)

where 6 =1 n = n'

nn

= n 7^ n' .

Z (z) obeys the following differential equation:

8

,^ 2 + 0) 2,2- k

z n

Z + S A , P(z) Z , ^ -B - D P(z). (9) n nn n' n n

P. M. Morse and H. Feshback, Methods of Theoretical Physics

(McGraw Hill, New York, 1953), Vol. II, Chapter 9, pp. 1001-1025.
This is derived by taking (6), multiplying by R , and integrating from

^1 '° ^4- In (9),

^nn'" J^

drR^(r)R^,(r) ^ (f^^^^j^-D^

B = iu n J n

(r)dr ,

and

^2

p(z) = e (z - z^) - e (z - z^).

2 2 2 2 2 Define the variables X = - uu + k , and A. = X - A /, or

n n n n nn

2 2—2 2

X (z) = X + (^ - ^ )P(z), and use the Green's function defined by

n n n n

'

n

*- 2 2

8 - \ (z) z n

= -6(z - z)

6(z - z) is the Dirac delta function

to solve the differential (9), the resulting integral equation is

3 4

Z (z ) = S I A ,G (z,z) Z , (z)dz +/ G (I,z )B dz "^ n?^n-^z.'''' ^

^ ^z "^

(10)

+ D G (z,z)dz.

n J n

"2
By considering only first order mode correlation, Z (z) may be

approximately written as

Z^(z)2= I G^ (z,z)dzB^ + D^J G^(z,z)dz|

z z^ / 1 2

(11)

/3

+ S A„„, G (z,z)dz

, J nn n

n'T^n z_

r ^ =

^ G ,(z,z)B ,dz + i D ,G ,(z,z)dz Inz n inn

z_

•- 1 2

This approximation implies that the sample is electrically small. G is

n

G (z,z) = A C (z ) S (z ) , where z is the larger of z, z pair and

z^ is the smaller of z, "z pair. A is the normalization. It is such that

n

(12) is true:

A C ,S - C S , = -1 , n n n n n

(12)

where z = z. The functions S , C are defined as follows:

n n

\ = exp[x^(z-zj)] - exp[-X^(z-Zj)]

C = C exp X (z - z_) +C exp -X (z - z_)

n n L n 2 J n L n 2^

,\ z^^zs Zj

S^ = si exp[x^(z - z^)]^ S \ exp [-X^(z - z^)]]

C = C exp X" (z - Z-) + C exp A (z-z_)

n nl-n 3 J n^Ln 3 J

z 2: z ^ z

3 2

S = S

n

C^ = exp[x^ (z - Z4)] - exp[-X^(z - z^)] .
Here

,

and

\ = + n

X = + n

, 1 2 2

- ^ ^ k

n J

X 2-A >

n nnl

(13)

1 2—1—2 1 2—1—2

The coefficients C , C , C , C and S , S , S , S are

n n n n n n n n

defined with the aid of the function f ;

Thus

f (X . X 3, 4) = i inn < c

-1

\J^ +l|a^.+

J. n

34

=ri? - 1 ia'^

and

C^^=f,(X X 3.4)

n inn

^\-^^ t^"\'3,4) n Inn

s^„ = f, (X .r 2,1)

n Inn

S^ =f (X ,-X .2, 1) . n Inn

(14)

By defining a second fvmction

i^iX

"^" -.->.;t1 ^2,_1 I— 1,, . n.n Tr2., n.n ^,X^,2.3,C ^.C J

= - c ^(l+5^)a 33+C _^(l-i^)a

«• n n

32

the C , etc. , are:

n

n ^ n n n n

C^ =fJ-X ,X ,2,3,C^ .C^ ) n 2 n n n n

S^ =fJX ,X ,3, 2,S^ .S ^

) n 2 n n n n

(15)

s\ =f (-X X 3, 2,S ^^,S \). n ^ n n n n
n The a ^ ., etc. are defined as

34

n

a 34=exp[x^(z3 - z^)] = (a^^3)-^

n

a"23 = exp[X^(z2.Z3)J=(a^3/^

and

^^2 = ^"p[^<^ -^2\

Using the above definitions, the nornnalization A^ is

n'2n n 12 n i<i

and the impedance reduces to

2 r

Z 3 i (z^-z^)ln(r/r^) + (Mj^-l)(-3-^2)ln(r3/r^) !^ iWN"

ZTT

+ E

n=l{

+ [W-V^l^] ^\^^n)

(h... - 1) T (rJ - T (rJ M

W

/ n

Here we have further defined

. n 3 n 2 J n

4

V = Z dz , n J n

"1

W = Z dz , n J n

(16)

and have noted that the T is a solution to the equation

n

2 1 2 d+-d +k

r r r n

dT

n T = 0. The R is related to T by R = r—— N . The

n n n n dr n

dT

k is fixed by the condition that at r = r, and r ,, n 1 4 dr

n

= 0. The N is

n

10
the normalization factor needed to have (8c) true. We can write T in

n

terms of Bessel functions of the zero and first order as

T =[J (k r)Y.,(k r ) - J (k r )Y (k r)l . (17a) nLo^ Inl Ini onj

The coefficients A ,, B , D , and N are given in terms of the Bessel

nn n n n

functions of zero and first order as

B = N iw n n [V4)- V^-i'] -

D = N U) (|i - I) T (r ) - T (r ) n n n Lnj n^

and

A , =^

nn'

i

n n

N N , [-rk ^T d T + rk /t d T nnL nnrn n nrn

where n r n'\ when

n = n', we have

A =0)^ (jj, - 1) N

^ nn '^M n

(17b)

r /^nrn T 3 T + knl.2nrn ^[9 ^(T )a T -Tnr^Zn 3 9 _(T )

The N is given by

n

N = 1 / n

k r,T (rJ8 8 ,T (rJ-r.T (r.)9 9 _T (r.)

n 1 1 nM' r , 2 nM ' 4 n' 4' r , 2 n' 4'

k k

If we use the recurrence relationships

d J (k r) = -k J,(k r)

r o n n 1 n

d Y (k r) = -k Y,(k r)

r o n n 1 n

drln J (k r) = knon J (k r) - -rlnJ fk r) ,

11
and

drln Y (k r) = k Y (k r) - i Y,(k r), non' r In

we find that N and A become

n nn

N ^= 1/

n

k T (r) r n I n

rT +r lY (rJT (r) - Y (r ) J (r ) n loll 1 ol

(18)

41

1

•

A =^^(»i^^ - 1) N

^ nn ^ '^ M n

d T r r(T

n

d

r T

n +-V^2

rd T k r 1 Y(k r )J (k r) rnnl»o nl o n

-Y (k r)J (k r ) on o n 1

- Y.(r)J (r ) 1 o ^

n , Z

2 n

3 ^

2 -»

rT +r lY (r )J (r

)

n 1 » o 1 1

(19)

In order to evaluate V and W , we note the definitions

n n

and

B' = T (r .) - T (r J , n n 4' n* 1

n XI 5 x\. c

(20)

and explicitly insert into the impedance, the G for V and W . The

n n n

impedance finally becomes

Z = N \^

2rT (z^- Zj)ln(ryrj) + (H^-1 ){'z. ^--l ^ \r, {r ^/

r ^ \ (21)

oo 4

+ S ANnnnlnn B'<B(R^+R ) + D(R +R )) + n nn n'

n=l L °° 2 2 5 7 5 7 + SAN B' S N , A ,A J B , (R ,+R ,+R , +R , ) . n n n ^.^ n nn' n' \ n' nn' nn n'n n'n

n=l n -Ti \

+ D , (R^ ,+R^ ,+R , +R^ , \ n nn nn n n n n(

12
oo 2 4 3 + (fJL,, - 1) S A D' N (D R + B R ) M __, nnnnn nn

CX) + (fi^^ - 1) S D' A N^ S A ,A ,N /<J B ,{r'^ ,+r'^ , ) ^M n n n , nn n n \ n nn n n

n=l n >n

8 8 + D ^Ir"" ,+R , ) n nn n n

Here we have explicitly extracted the normalization factors N , N , from 12 "" "" B , D , and A ,. The R , R , etc. , are defined as follows:

n n nn n n

z z. R

n

(z)dz + / S (z)dz /_ C dz

Zj ^1 "

+ / C dz I S dz + / S dz / C dz +!

I S dz i

i n J n

z ^ z

3 3

n /— n

Z3

/ C dz + / C dz 1 + C dz ; I S dz + / S dz

,

J. n J^ n ' 'j^ n "J n j^ n '

\^' z R =! S {z)dz

n \ / n I n

Jz Z-.

1 2

C dz +1 C dz S dz

n ' / n

z^ ^ z

3 2

R"

n

3 3 3 I 3

\ C dz [ S dz + / S dz f C dz + f C d7,

n / n j n I— n M n 'M z -^z -'^Z -'Z z -^z

2 2 2 2 1

S dz

n

+11 S dz

n

'"2

C dz

n

13
R njnjn = C dz S dz + S dz C dz J^nJ-n

"2 "2 "2

R , = nn

z r z r z z z

^z. L z_ z_ -'z •'z.

+ S , C ,dz

^3

+ C dz

14^ n .

f S^(z)dz C^,J S^,dz

""2 L "2

+ S , C ,dz + C , n J— n n

z S ,dz + S , C ,dz

n

6

R , nn

r r = S dz C dz C , S ,dz + S , C ,dz

n \ n J n n J— n

LT Z-. * z^ z ' 2 2

r z. + C dz I S dz C S dz + C dz

n n J n J — n

Z^ * Z^ Z ; 2 2

R , = C (z)dz S (z)dz

nn J n J n

"2 "2

C S dz +S C d2 + n J n n J n

^2

- \ + C , S ,dz +S J C ,dz

n J n' nM J n' ^1 ^3

+ JS^(z)dz /_C^(z)dz

"2

14
/ rZ

C , S ,dz + S , C ,dz + C , n' in' n' J n' n'

^Z

S ,dz + S , C ,dz

Z Z p i R^ , =

I C (7)d7 I S (z)dz c , I C S ,dz +S , C ,dz

n' in' n' J n'

"2

^3 ^3

r 3

/ S (I)d7 C (z)dz C , S ,dz + S ,

I C ,dz

I n J — n ' n' Jz_ n' n' J

n'

By making use of common factors in R , etc. , the above may be reduced to

n

R

n n nl d nJ4 n

(22)

n nl2 n23 n23 n34

(23)

R"

n n n

(24)

nn

nn

L \ 2. 3 ^ nn' "^^nn'^nl2^n'12'^^ nn' ^n34^n'34 "^ ^nl 2^n'34^ nn'

(25)

R

nn

^^n34^n'l2" nn'

8 4 5

nn' n'12 nn' n'34 nn'

L ,F ._ + L ,G . . nn nl2 nn n34

(26)

(27)

The remaining definitions are

n

C dz / S dz + / S

n J n

dz C dz + C dz S dz

J— n J n J n

z, z z^ z^

1 3 3

r 4 _ r 4

+ S dz |_ C dz

^3 "

15
H , nn

nn

z.

C C ,dz

n n

S S ,dz

n n

nn

C S ,dz

n n

nn

S C ,dz

n n'

nn

C dz

n

^2

nn

S dz

n

^2

nn

C dz S C ,dz + S dz C C ,dz

I n J n n J n J— n n

"Z ^2 ^2

nn

C S S ,dz + S C S ,dz

n J n n n J— n n

^2

dz

nl2 S dz

n

n23

r^3

S dz

n

16
^n23

and

C dz , n

n34

4

C dz . n

The results of integration of the above definitions are listed as follows

The single integral expressions become

nl2

n23

n23

a ,, + (a .S^ - 2

nl 2 nl2

n

n ^

L »"n23

=_-i- ?C' (-l)'-"' fl -(a „)'''

X i n L n23

i + 1.

n

n34

_ 1 -\

n

K we 6(sfine

2 - a - (a ) n34 ^ n34^

•1"!

(28)

(29)

(30)

(31)

^^,(i,j)^ [i-l)'''\^i'l9''\.] (3 2a)

PjCi.J) =

and

P^dJ) -

i + 1 l.(a )<-^)''\a ^<"^^

1+1

/^^i(i.j) ,

<Vz3''-'^-<^a3''"'

1 + 1 ml (3 2b)

17
then the remaining single integral becomes

H^ , = E C ' C^^, P,(i,j)

H , = 2 S

' S^ , P,(i-1. j-1)

nn' j^ j n n' 1

nn i j n n 2 n'n

H^„^ = ?. S' CJ P (i-1, j-l) =H^ , . nn 1 j n n <s n n

(33)

The double integrations give

r' = S

c s

-* n n

n

j \i

(-I)'

,j+i

n

-^^(^23)'' +<-n23, ^^

+ '^2 - ^3> (^nZs''""" [(-1)^ -(-!)'

n ^ i 1 n ^ J L

i + 1

(34)

^3^ C>1,^- ,.X,i-[-2Ma„,,,<-' M^„.2>'"'>„

+ <^l-^2'(\l2'

(-1)^

i-if -(-1)'

n

i + 1 <-)^'"[— 'w'"' ^(w*"n/^„

+ (Z3 - .4)<^„34)*'"' [(-i^<-n'] (35)

Upon further definition

i + 1 P^d) <i'0-i.^J''[ ]/\i'it'' (36)

18
and

P^a.j) P^d) (a^,23^

(-1)'

(37)

the two-fold integrations give

_ 2

= S

n n' n

nn i j i

3

nn'

^n'

S

' C^ ,S

J

, n n^ n

In'

_ / (-lyj - (-1)4 P^(i,j) +

(-1)J Pj(je,i)

+ {-lf-^^'P^{U3) (38)

-|(-i)^ .(-i)yp^(i. i,j) + (-i)^

P2(i-l,i + l)+(-l)'"^^Pj(je-l.j-l)

(39)

If we define

m2 n J +'^„/(-i)

n

^m3<^'i'JJ ^ V <-^) +

(-iyJ+l+(-l)i + ll ^T

^ + ^ (-1)J+^J +1 (-1)

Jt +1 ,

1 . - , , J + 1 n

(40)

(41

and

R^{i»hi) kJ-'^-kJ-'''*'k'J-''''' /X^.d.J.i).

(42)

the remaining two-fold integrations are

nn i j J? n h n'

(p3(i,j,i) -p^(i,i+i)) /^^lihn

+ (-P3(iJ.i)+P4(J-l)) / ^^ii^*n (43)

19
L^ , = S C ' S' s\ nn i j i n n n' P3{j-l,i + l,j? -1) -P^(i)) / ^^jO,^)

+ (-P3(j-l, i + 1, i -1) + P^(j-1, i)) / ^^i{i.^)

(44)

The triple integration gives

_8

nn' i j i m

n n n n

\^,

(-1)

m

(p^(i,j,i,na) - P^(i,j+1)) /^^3(^J»ni)

-P3(i,i,j) + p^(i,j + i)) /^^i(jej)

(-ir

(-1)

m

-P^(i,j,i,m) +P^(i,j+1))/X^3(i,j,m)

(P3(i-l,i+l,m-l) -P^(i))/\^j(i,m)

(-P5(iJ,je,m) +P^U -l,m))/?.^3(i,j,m)

P3(i,i,j) -P^(i -l))/^^i(iJ)

(-ir P^(i,j,je,na) - P^(i -l,m))/\^3(i,j,m)

(-P3(i -l,i + l)+P^(i -l,m))/X^j(i.n:i)l

Here the definition

(45)

P^(i,j,i,m)

is used.

(-1/ (-1)" (-!)''' (-1>''^'

(\23' (V23' - (^n23) <V23'

n L -> L n

(46)

20
4. Conclusion

The derivation of the formula for the impedance of a toroid

sample in a toroid has been completed. In (21), one can note

that the Z is a dimensionless quantity. To make it have the dimensions

of ohms, we must multiply Z by Z , where Z =„ / — = 37^ 7Q ig the

o o y £

free space impedance. In addition, if one examines (9)

and (10), one can see that under the circumstance that the ferrite

sample has r,12= r^ and r^ = r ^, that A , = when n' 7^ n. This means

3 4 nn

that the formula (21) is now mathematically exactly correct. However,

the physical effects of the closeness of the winding on the outside toroid

becomes a major source of error. This means that the condition under

which the large disk toroid may be used must be such that the effects of

the winding can be neglected. Such condition can be accurately determined only if one used a theory for which the non -uniformity of the

windings has been explicitly accounted for. Rather than to have to

develop such a theory, we avoid such needs by using the present formula

in a way that will make them sufficiently accurate. This may be done

by mathematically varying the sample size relative to the outside

current sheet toroid. When the double sum correction in the impedance

formula (21) is less than some figure of merit, say 2%, of the total

correct impedance, one can be reasonably sure that the resulting

impedance values are known to 0. 5%.

Note - The criteria used to establish the 0. 5% figure is developed in

the FORTRAN subroutine IMPED. Here the double sum series is

truncated by the requirement that the last correction to the double sum

correction be less than 10%. The single sum series is truncated by the

requirement that the last correction to the single sum part be less than

0. 5%. Because the two series have alternating signs and apparently

converge like l/n^ for the double suna and like l/n-' for the single sum,

the error made in the evaluation of the impedance is deternained by the

value of the largest finally accepted correction - namely 0. 5%.

21
In the theory and experiment the effects of the reactive coupling

of the wire coil to the sample can be minimized by placing the sample

as far as possible from the wire coil- -namely the exact center of the

coil. To make explicit comparison of the theory with the experiment,

one must have the following operational procedure: first, the experiment is done with and without the sample. The two resulting impedances,

are then subtracted to produce an effective impedance Z . The

reason for this subtraction is to attempt the removal of the effects of

external fringing in the impedance results. These effects are due to

the coil arrangement which allows magnetic flux to exist outside the

toroidal cross section. For this operation to be meaningful the

assumption is made that the fringing field is the same for the system , with or without the sample. This condition is the same as the theoretical

assumption that one can use a uniform current sheet to characterize the

internal impedance of the toroid. However, to have a more precise

relationship to the experimental procedure, one must produce a Z

(for the theory) in exactly the same manner as was the Z produced

for the experiment. This nneans that we compute the case where the

ferrite core has u. = £ = 1 to obtain Z and then calculate for the case

of [JL, e equal the desired values to get Z . The effective impedance

is given by Z ^- = Z - Z . This result may now be compared with

eff t vac ' ^ the experimental effective impedance. To get a reliable comparison

Note - The experiment produces two numbers, while we need four

numbers to fix the theoretical impedance. It should be obvious that to

fix the four numbers it is necessary to do a second measurement under

known conditions, so that we have sufficient experimental data to get

the |j. and e. The second measurement is to use a parallel plate

capacitor.

22
one should have several sample sizes for the same material. Byassuming that the material is sufficiently homogenous, this comparison

will give a check on the accuracy of the above correspondence.

The appendix I gives the FORTRAN and FLOW chart needed to

calculate the correct impedance. The main program, TOROID, gives

the definitions and indicates the needed input. The number appearing

on the far right of the listing corresponds roughly to the number of the

same equation in the text. This program takes about 0. 1 minute per

point. K one is to do very many points, say 6000 or more, it would be

desirable to make the program more efficient. The dimensions of the

1Z - Z \ r 4 1 ^ ^ , Largest number allowed

3n s. In I r - r I 4 1 /

in the conaputing machine. The n is the largest number of modes

needed to satisfy the figure of merit. (Usually n < 7. ) Because of the transcendental nature of the Equation (21), we

cannot solve for the \i, e, explicitly. This means that the best one can

do is to prepare a table of impedance in which the values of |jl and £

are varied over ranges of interest. Such a table will be forthcoming

when the needs of the various personnel have been determined.

To aid in the use of this program, sample output is included in

appendix II. The reader should note that the 1. 5 x 10 Hz case

strongly violates the region of validity of the theory.

23
APPENDIX I

•EQUIP»5=60

"•EQUIP»6=61 " •FTN,X.*

C MAIN PROGRAM

CTOROID

FOR TOROID CALCULATION TOROID

TYPE COMPLEX CZI I »CF IX tCZ .CZ I ,CZI 1 1 tZCCR, CV ,DUM,DELV »W COMMON R.Z,Vj^FIX, FR »Z I »Z 1 1 » I T » IC » ITT »FR2,ICYLE >DUM»ZC

DIMENSION IC( 6) ,FR2( 5004)

DIMENSION ZII(2)»FIX(2)»Z(8),CZ('^),ZI (2)»ZIII(2) »ZC(58) ,CR(5) f

TOROID

EQUIVALENCE

R(10),CV(2),V(4) ,DUM(10),DELV{2) tWCZ) UM(20)

(CZII»ZII ) >(CFI X »FIX) »(CZ»Z)>(CZI»ZI)»(CZIII,ZIII )*

980

1 2 "999

101

200

"i02^

cc

cc

cc

L (CR,R) (CV,V) »(DUM»UM)

DO 980 I=l,5796

R( I )=0.

READ(5.2) (R( I ) , 1^1 ,8.2J_.iZJ IJ » I =1 » 8 »2 ) ,TURN * IC ( 5 ) » IC( 6)

FORMAT ( AE18.5/4E18.5/E18,5»4I4)

I F (EOF* 5)100,999

"CONTINUE

IT=-1 _ READ INPUT TAPE 5 102, ER, IC ( 4 ) , IS» I TES

IF(EOF, 5)100,200 _ CONTINUE

FR=ER*(2.0944E-10)

TOROID

F0RMAT(E18,5,3I4)

ITHE INPUT IS THE RADIUS VARIABLES , THE Z AXIS VARI ABLES>THE MAGNET I TOROID

2C PERMIBILITY,AND DIELECTRIC PERM I T IV I TY, TURN COUNT, AND FREQUENCY TOROID

BALL DIMENSIONS ARE ASSUMED TO BE IN CM, TH^ FREQUENCY IS HERTZ,

FREE SPACE MU AND EPSILON IS ONE. THE INPUT FORMAT IS FLOATINGTOROI

D

V(1)+IV(3) IS THE DIELECTRIC PERMITIVITY AND V(2)+IV(4) IS THE

COMPLEX PERMIBILITY

DO 4 1=2,8,2

Z( I) = R{ I ) = 0.

FIX(1)=0. _, _

) WRITE(6,7) IS,ER»TURN,(R{ I ) 1=1,8,2) , (Z( I ),I = l,8r2)

FORMAT ( 44 H1 IMP E DANC E RESULTS FOR TOROID GEOMETRY PAGE 14/

i8H THE FREQUENCY IS ~E18.4,25HHZ . THE TURNS COUNT

TOROID

ITOROID

27H THE

37H THE

41H THE

5E REST

6

7

8

IS F4.0,/

RADIAL DIMENSIONS ARE 4E18,4 / Z AXIS DIMENSIONS ARE 4E18.4 //

LENGTH MEASUREMENTS ARE IN CM. IMPEDANCES ARE IN OHMS.

DIMENSIONLESS.

QUASI

2T0R0ID

2T0R0ID

9T0R0ID

THE

ARE

EPS I LON^ _

DBL SUM

Zl

"//115H

STATIC IMP

MU

CORRECT IMP

IMP /125H

+ 1 Z2

Ul + 1 U2 El + 1 E2

ZCl + 1 ZC2 DZl

71

72

211

9 +1 DZ2 CNf " ) ITT = 1 IS=IS+1

I F ( I T| 71 ,7 1,13

IV=IC(4)

FIX(2)= FR»(59.95 )»(TURN**2)

DO 13 11 = 1, IV

FORMAT (4E18. 5)

READ (5,211 )V(1),V(3),V(2),V(4)

CALL IMPED

TOROID

TOROID

TOROID

TOROID

CZII=CFIX*CZII

24
^Z^I^=CZII+CF2^X*CZI

CZIir=ZC(r)*CFIX

^ WRITE(6f8)V(3),V{^),V(l),V(2)»ZIII»ZI,ZII»ICYLE

ITT=ITT+1

FORMAK E13.4,9E12.4,I4)

TOROID

13

IF( ITT-46)13.6,6

CONTINUE

IF(ITES)1»1,103

103 IT=0_

GO TO

100 END

101

TOROID

TOROID

TOROID

DIMENSION

DIMENSION

2,R,Z,

XI, XJ,

^G34,A

TDdOO

,R(5),

10

1

SUBROUTINE IMPED

TYPE COMPLEX CFR

i AN

_2 _ F12

IC(6),^

FR2(2)

1 TTdT

2_ UM ( 2

2 CB(

EQUIVALENCE (CFR

"1(DUM,UM)

COMMON R»Z,V,FIX

1ALAM»CB»A23,A34,

2.TW»SV»VE

ICYLE=0

_ IT _OyE R FLOW FAULT 10 »10

~

I j=^( IT) 1,101,2

Z34=Z(3)-Z(4)

IMPED

VE,Z12»Z23,Z34,ZAl,ZA»SV,FR3,ZC,AD,AE,DUMj

A4,Bl,bl,A5,A12,A34,A23,CB,ZB,V,FIX,ZI ,ZII,

LAM

,2) ,WK(100) ,T(100,2),ADI (100) ,TW(8)

Z(4) ,ZC(58) .DUM(10),AN(100»5) ,A4RL(2) ,B1(

IMPED

100

(100),A5RL(2) A12(100,2),A34(100,2) ,A23( 100,2)

,

0)^

100,8)"

2,FR2)

,V(2) ,F12(100) ,ALAM( 100,2) ,G34( 100)

,(A4,A4RL) , (XUXIRL) , (XJ.XJRL ), (A5,A5RL) ,

, FR,ZI,ZII,IT,IC,ITT,FR2,FR3,B1,D1,AN, F12,G34.

A12 ,ICYLE,DUM,ZC,TD,WK,T,ADI ,Z12,Z23,Z34

IMPED

IMPED

IMPED

IMPED

IMPED

Z12=Z(1)-Z(2)

Z23= Z(2^-Z(3)

Z A 1 = Z 2 3 *C"LOG ( R "(3 ZA= (Z(4) -Z( l))

FR2(2)=0,0

IA=IC(5)

)/R(2)

)

*CL0G(R(4)/R(1)

)

IMPED

IM^ED

IMPED

IMPED

IMPED

IMPED

T^JT F R2( 1) = (FR)**2

CALL TXX(1,1)

It = l SV = V ( 2 ) - ( 1 . , 0. OJ

_

VE = V ( 1 ) *V ( 2) - ( i ,0 , ,"0 ) FR3=CFR2*VE

IMPED

IMPED

IMPED

IMPED

IMPED^

N2 = l CALL SCX(N 2,1)

TF(N2T42,42,201

DO 3 1=2,58

TU2""

201

"3 ZC(I ) = (0. 0,0.0)"

ZC(1 )=ZA-ZA1*SV

IMPED

IMPED

IMPED

IMPED

IMPED

IMPED

301

"303

150 40 N = 1,IX"

N1=N+1

IB=IC(6) +N

~TF ( IB .GT." 100"

DO 303 L=25,28

) I 8=1 00

IMPED

IJ»1PED IMPED'

IMPED

IMPED

"ZC(L)=TD".O,0.0) IMPED

25
DO 30 M=N1»IB IMPED

ICYLE=ICYLE+1

M2=M

IMPED

IMPED

CALL TXX(N2.M2)

CALL SCX(N2,M2)

IMPED

TMPFD

103 IF{N2)^2»42»304

304 AD= AN(M,1)«DUM(8)

IMPED

IMPED

AE= AN(M»2)*DUM(8)

ZC(35)= DUM(4)»AD»AN(Ntl)

IMPED

IMPED

ZC(36)= DUM(5)»AE*AN(N»1)

ZC ( 37 ) »DUM ( 6 ) *AE«AN ( N , 2

)

IMPED

IMPED

ZC(38)= DUM{7)»AE*AN(N»2)

ZC ( 26 ) =ZC { 35 ) +ZC ( 36 ) *SV+ ( ZC (37 ) +ZC ( 38 ) «SV ) *VE

IMPED

IMPED

ZC(25)=ZC(25)+ZC(26)

1000 PRINT 1001. M AO»AE,AN{M»l)»AN(M.2)»SV»ZC(25),ZC(26).(ZC(I).I=35

IMPED

»38

X8)» (DUM(I)»I=4,8)

1001 F0RMAT(6H 1001 12 .SC { E12«4tE12.4) /( 8X,5C(E12.4 ,E12.4) ) ) XI=CABS(ZC{26))

XJ=CABS(ZC(25))

XK=XJRL»(0.1)-XIRL

IF(XK)30,31,31

30 CONTINUE

ITT=ITT+2

IMPED

PTEST

A4=FR3*CFR2»F I X*ZC ( 26 ) WRITE OUTPUT TAPE 6,11,N2»A4RL

11 FORMAT (54H WARNING CONVERGENCE IS SUCH THAT ON DBLE SUM CYCLE

1 I4/25H THE LAST IMPEC COR. WAS E14.4, 4H +1 E14.4 ) PTEST

PTEST

31 ZC(5)=ZC(5)+ZC(25)

ZC(12)= AN(N,1)*B1(N)*DUM(1) IMPED

ZC(13)= AN(N.1)*D1(N)*DUM(2)

ZC(14)= AN(N.2)*D1{N)*DUM(3)

IMPED

IMPED

ZC ( 4 ) =ZC ( 12 >+ZC ( 13 ) *SV+( ZC ( 1 3 ) +ZC ( 14 ) SV ) VE

ZC(2)=ZC(2)+ZC(4)

XI=CABS(ZC(4) ) XJ=CABS(ZC(2))

XK=XJRL*(0,005)-XIRL

IF(XK)40,41,41

40 CONTINUE

ITT«ITT+2

IMPED

A5=CFR2»FIX*{ZC(4)+FR3*ZC(25)

)

WRITE OUTPUT TAPE 6»22.IA»A5RL

22 FORMAT(57H WARNING TOTAL CONVERGENCE IMPLIES THE

1 I4/25H THE LAST IMPED COR. WAS E14.4. 4H +1

SINGLE SUM

E14.4)

COUNT PTEST

PTEST

41 ZI=ZC(1)+CFR2»ZC(2)

"'II = FR3»CFR2*ZC(5)

43

iF t. .ERF LOW FAULT 43»42

WRITE OUTPUT TAPE 6,44

44

42

FORMAT (47H ACCUMULATOR OVERFLOW NEXT

RETURN

LINE HAS NO MEANING ) END

SUBROUTINE TXX(N,M)

IMPED

TXX

TYPE COMPLEX TWO»ONE.AVE,ZR0

1 vALAM

CX22»CR,.B1»D].,AN.ATEM,A12»A34. G34,F12

DIMENSION X22(2)»CR(5)»R(10) BKIOO), B1RL{200)»D1(100).01RL(200)»

26
AN(100,5),A12(100»2)»A34(100,2).F12(100),G3^(100)»

2 ALAM{100»2),ANRL(1)

DIMENSION IC(6)»TD(10Q»2 ) ^WKC 100 ) ,T ( 100,2 ) ,ADI ( 100) IMPED

1,TB(2) ,T11(4),T12(2)

EQUIVALENCE (CX22 »X22 ) ,

(

CR»R ) » ( Bl ,B1RL) »

(

D1«D1RL) »(AN,ANRL)

COMMON R.Z.V»FIX»FR,ZI»ZII»IT,IC»ITT,FR2,FR3»B1»D1,AN. F12,G3^, IMPED

1ALAM»CB,A23,A3A,A12 t ICYLE ^DUMtZC .TD,WK.T. ADI »Z12,Z23,Z3A IMPED

2*0NE,TW0*ZR0>AVE

COMPLEX Z»V»FIX»ZI,ZII,FR2»FR3.CB»A23,DUM»ZC,Z12»Z23»Z3A

DIMENSION Z(4)»V(2),CB(100»8),A23(100«2).DUM(10)»ZC(58)

IF(IT )1.101,2

1 Nl = l TWO=( 2.0,0.0)

TXX

TXX

0NE=(1. 0,0.0)

AVE=(0. 5,0.0)

TXX sex ZRO=(0. 0,0.0)

X22(2)=0.0

sex TXX

R(9)=1.5/(R(7)-R(1) ) R(10)=0.1 *ni9)

101 N2=l

2 IF{M-N1)41,3,3

3 N1=N1+1

B1RL(2»M)=D1RL(2»M) =ANRL< 2*M+400 ) « 0.

TXX

WK(M)= BESSEL(5,1)

WK1=WK(M)

TXX

TXX

X2=BESSEL(1,A)

X3=BESSEL(1,2)

TXX

TXX

X4= BESSEL(1,1)

Xl= BESSEL(1,3)

TXX

TXX

DO 33 1=2,3

K=I-(-l)**I

TXX

TXX

J=I-1

TD(M,J) = (BESSEL( I ,4)*X1-BESSEL( I ,3 )»X2 )*WIC1

TXX

TXX

TB(J)=BESSEL(K:,1)*X2-BESSEL(IC,2)*X1

T{M,J)=8ESSEL(I,l)*X2-BESSEL(I,2)*Xl

T12(J)=BESSEL(I,1)»X3-BESSEL(I»2)»X4

TIK I)=BESSEL( I,3)*X3-BESSEL(I,A)*X4

33 Tll{K)=BESStL(K.,3)*X3-BESSEL(K,4)*X4

AN2=0.0 TXX

B1(M)=TB{2)-TB(1)

D1(M)=T(M.2)-T(M,1)

TXX

TXX

WIC2 = WK(M)»*2 TXX"

ADI(M)=0,0 TXX

DCnT 1=2,3 TXX

IFd.EQ. 2)340, 341

~T40" TC:^KK =1 $^ rr=3 $ go TO 342

341 K:=4 $ K.K = 7 $ 11=5

342 J=l-1

AN2 = AN2- (-1 .Q)»*I*(TB( J ) *R(KK ) ) * ( TB( J )»R ( KK ) -t-R ( 1 )»T11 ( K )) TXX

35 ADITM ) = ACn 1 M ) - ( - 1 . U ) *I*RTTT)1K^ (T(M,J)»WK.2*(RlII)*T(M,J)+R(l)»Tll ( TXX

II ))+TD(M,J)»(2.0*T(M,J)+R(II)*TD(M,J)+ W(C(M)»R(1)*T12 ( J m TXX

ADTfM )=0.b*ADlTMl TXX

AN(M,3)= 1.0/( WK2»AN2) TXX 18

AD1(M)=AD1(M)*AN(M,3) TXx"

27
38

39

X22(l )= WK 2::F R2^

GO TO 39

IF(M-N2)4.38»38

X22(1)=WK(M)*»2-FR2

ALAM(M»1)= CSQRT(CX22)

TXX

N2=N2+1

A12(M » 1)= CEXP(ALAM(M,1)*Z12)

A3A(M»1)= CEXP(ALAM(M,1)»Z34)

A34(M.2)=0NE/A34(M,1)

A 12 ( M . 2 ) = ON E / A 1 2 ( M , 1 ) G34 ( M ) = ( TW0-A34 ( M , 1

) -A34 ( M*2 ) ) /ALAM ( M 1

)

F12(M) = (A12(M,1)+A12{M,2) -TWO ) /ALAM ( M , 1

)

ATEM =AN(M»3)/ALAM(M,1)

AN ( M 1

) =B 1 ( M ) *ATEM ^(_M 2 ]_= D 1 ( M) *ATEM

AN(M»4)=0NE

RETURN

TXX

TXX

TXX

TXX

TXX

END

FUNCTION BESSEL(

I

*K)

DIMENSION R(10) ,6 (12,4)

DIMENSION S(19)_

COMMON R, S.I

T

I F(IT) 1QQ » 1 10,1010

TXX

BESSEL

BESSEL

100 A5= R(9)

R{10)=5,0*R( 10)

1010

114

141

IT = 1 ISV = 2 IM = 1 A4= l.OE+04 BESSEL

GO TO {10,10,10,10,1),I

DEL=R(10)

BESSEL

BESSEL

R(9)=R(9)+DEL

X=R(9)*R(1

)

Y= X/3.0

IF(Y-l.O) 101,110,110

BESSEL

BESSEL

101 11 = 5 GO TO 3 BESSEL

BESSEL

15 X=R(9)*R{7)

Y= X/3.0

102

IF(Y-1,0) 102,103,103

11=6

BESSEL

BESSEL

BESSEL

103

GO TO 3 11 = 5

2T

GO TO (41,400) , ISV

PH2 = PHr

F2=F1

11 = 7

BESSEL

BESSEL

16

GO TO 4

ANULL=B( 5,3)*B(6,4)-B(6,3)*B(5»4)

BESSEL

BESSEL

20

110

GO TO 71

ANULL=(B(5,3)»C0SF(PH1)+B(5,4)*SINF(PH1))*(-F1)

BESSEL

BESSEL

GO TO 71

11 = 6

BESSEL

BESSEL

Y=1.0/Y BESSEL

BESSEL

BESSEL

28
X=R(9)»R(7)

Y* 3.0/X BESSEL

GO TO (^1»^00) , ISV

22 ANULL= PH1-PH2 BESSEL

ANULL=F2*F1«SI NF ( ANULL

)

2 GO TO 71Y= X/3.0 BESSEL

3

IF(Y-1.0)3,3,4 BESSEL

Y= Y**2 BESSEL

910

IF ( X ) 910*910,912

X4=0. $ GO TO 91A

912

914

X4=0,63661977»LOGF(X/2,0) BESSEL

B(I1»3)= X*(0.5-Y*(0.56249985-Y*(0,21O93573-Y*(O.03954289-Y*(O.00BESSEL

1443319-Y*(0. 00031761-Y*(0. 00001109) ) ) ) ) ) ) BESSEL

B(I1»4)= X4»B(I1,3) -((0.6366198-Y*(0.2212091+YBESSEL

1*( 2. 1682709-Y*(1.3164827-Y*(0.3123951-Y*(0.0400976-Y*(0. 0027873 )))BESSEL

2)))))/ X BESSEL

"4U0

GO TO (6,6,6.6,15,16),I1 BESSEL

Y = 1 . / Y BESSEL

Xl= 17a/SQRTF(X) BESSEL

Fl= XI *(0.79788456+Y»(0.00000156+Y*(0.01659667+Y*(0.00017105BESSEL

BESSEL

BESSEL

1-Y*(0.00249511-Y*(0.00113653-Y»(0. 00020033) ))))))

41 PH1= X-(0. 785398 16-Y*(0.12499612+Y*(0. 0000 5650-Y*(0.00637879-Y*(

10.00074348+Y*(0.00079824-Y*{0,00029166) ))))))

GO T0(42,42,42,42»20»21»22)»ll

^42" B(I1,3)= F1»SINF(PH1)

B(I1,4)=-F1»C0SF(PH1)

BESSEL

BESSEL

BESSEL

BESSEL

GO TO 7 BESSEL

B(I1,1)=1.0-Y*(2.2499997-Y»(1.2656208-Y*(0.3163866-Y*(0.0444479-Y*BESSEL

1(0,0039444-0. 0002100*Y) ) >

)

^ BESSEL

B(I1,2)= X4*B(I1,1)+0,36746691+Y*(0.60559366-Y»(0,74350384-Y*( BESSEL

10, 25300117-Y*(0.04261214-Y*(0.00427916-Y»(0. 00024846) )) ))

)

GO TO 8

PHO=X-(0.78539816+Y*(0,04166397+Y*(0,00003954-Y»(0.00262573-Y*(

10«00054125+Y*(Q.00 029333-Y*(0, 000135 58) ))))))

"~B (TI , 1 ) =FO*XOSF( PHO ) B(I1,2)=F0*SINF(PH0)

~"G0

71

TO 8

I F ( S IGNF ( ANULL A4 ) -ANULL ) 741 , 74 » 741

TM^l

A4=ANULL

BESSEL

BESSEL

~T0= XI* (0.79788456-Y*( 0.00000077+Y*( 0,00552740+Y»{ 0.000095 12-Y*( BESSEL

10, 00137237-Y»(0.00072805-Y*(0. 00014476) )))))) BESSEL

BESSEL

BESSEL

BESSEL

BESSEL

BESSEL

740

GO TO 14

741 GO^ TO (742*740), I

~7i*T RT9 j =R (9 ) +1 D^L*ANULL ) / ( A4-ANULL ) DEL=(-0,5)*DEL

BESSEL

BESSEL

A4=ANULL

ANULL=ABSF( ANULL)

BESSEL

BESSEL

BESSEL

751

"76-

IF { ANULL-T, 0E-05)75,75,141

B(5,1)=R(9)

GO TO (77,751 J7TSV

IF(R (9

)

*R(l)/3, 0-1,0) 77,76 »76

"TSV^T

BESSEL

BESSEL

29
Fl=1.0

77 DP 8

I1=J

J = l,^ BESSEL

BESSEL

JJ=J+J-1

X=R( JJ)*R(9)

GO TO 2 CONTINUE

BESSEL

BESSEL

10

IM=2

R<^) = BESSEL

RETURN

R(9)+ A5

BESSEL

=B(I,K)

BESSEL

END

SUBROUTINE SCX(N»M) sex TYPE COMPLEX A12 t A23, A3^,ADD. AH.AHl ,AH2

t

L AK,6,AK7,AIC8,AL2»AL3,AL4,AL5

AH3»AH4,AK1»AK2.AK4,AK5,

,ALAM,ALM,ALM2,ALM3.AN,AVE

»B1» CB» D1»D

.R1»R2,RD1,RD

IV»DUM» F12,F23

2»RS1»RS2, TWO*

,FIX,FR2.FR3, G23»G34,0NEC

V» X1*X2»XA»XAL»XP1»XP2»

XP3»XP4,XP5,XP7.XPP,XT»XY, Y2» Z,Z23 »ZB»ZC»Z I tZ 1 1 »ZRO

.Z12,Z34

DIMENSION DUMRL(20) ,R(10)

A12(100,2),A23(

ALM2T2»2,2),ALM

Dl( 100) ,DUM( 10)

DIVRL(2)

100*2) »A 34(1 00»2) ,ALAM(100>2) *A LM(2»2),

3(2*2*2) .AN( 100

.F12(100) G34(l

»5),B1(100)*CB(100*8)

Q0)*XP1(2*2)»XP2(2*2)

XP3(2*2.2) XP4(

IC(6)*TD(100,2)

2) XP5(4*4) *XP7

DIMENSION WK(IOO) »T(100,

EQU'I VALENCE (DUM*DUMRL) ,(DIV»DIVRL)

1 *(ON EC*ONE)

COMMON R Z"* V * F I X FR * Z I * Z

I

1ALAM.CB»A23*A34*A12 jICYL

12>2) Z{4) »ZC(58)»V(2)

2)*ADI(100) IMPED

I*IT,IC*ITT*FR2

E*DUM*ZC.TD,WK*

*FR3,B1.D1*AN* F12»G34» IMPED

T,ADI *Z12»Z23*Z34 IMPED

7001

"1

;.ONEC.TWO*ZRO,AVE

IF( KYLE ) 1*1 *2

"N3=i

IT=IT+1

IF(M-N3)4»3*3

IF DIVIDE CHECK 31*31

"31"

IT

"L2

SCX

N3=N3+1

IF( IT-2)12*11_,J.2

AN(M*51=(5NE

GO TO 311

"AN ( M 5 ) =ONE/ AN ( M * 4 ) IF DIVIDE CHECK 310,311

310

311

AN(M*5)=(1. 0*0.0)

ADp=ADIjM)

ALAM(M.2)= CSQRT(ALAM(M*1

A^23(M*1)= C£XP( ALAM(M*2)

A23 ( M * 2 ) = ONE/ A23 ( M * 1

)

IF DIVIDE CHECK 5,32

)**2-ADO*FR3)

*Z23)

31 RT5 ALAM(M*1)/ALAM(M,2)

SCX

SCX

RSl=

RD1 = RS2 = RD2 =

ONE/Rl^

Rl+ONE

Rl-ONE

R2+0NE

R2-0NE

SCX

SCX

SCX

SCX

SCX

SCX

CB(M*1)= AVE*( RS1*A34(M*1)+RD1*A34(M,2) ) SCX 14

30
CB(M»2)°(-AVE)» (RD1*A 34(M»1)+RS 1*A3A ( M*2) ) SCX

CB(M,3)=(AVE)*( RS1*A12(M,2)+RD1*A12{M,1)

)

SCX

CB(M,4) = (-AVE)*<RD1*A12(M,2)+RS1*A12(M,1) ) SCX

CB(M,5)=(AVE)*(CB(M,1)*RS2*A23(M,1)-CB(M,2)*RD2*A23(M,2)

)

SCX 15

CB(M»6 ) = (AVE)*(CB(M»2)*RS2*A23(M>2)-CB(M»1)*RD2*A23(M»1 ) ) SCX

CB(M»7)=(AVE)*(CB(M»3)*RS2*A23(M.2)-CB(M,4)*RD2*A23(M,1)

)

SCX

CB(M»8)=(AVE)»(CB(M,4)»RS2*A23(M,1)-CB(M,3)*RD2*A23(M»2) SCX

XA= AVE/ ( CB(M,5)*A12(M,1)+CB(M.6)»A12(M,2))

IF DIVIDE CHECK 5,3A SCX

3^ AN(M.4)=CSQRT(XA)

AN(M.5)=AN(M.6)*AN(M,4)

DO 35 1=1,8

35 CB(M »I ) = C B {MjJ. ) * A Nj M »4^____

F 12 ( M ) =F 12 ( M ) *AN ( M 5

)

G34( M) =G3»(M)*AN(M»5)

4 IF(M"-N-1)500, 41*430 SCX

41 G23={CB(N»2)*( A23 (N»2 )-ONE ) -CB (N » 1 ) » ( A23( N, 1

) -ONE ) ) /ALAM(N, 2 ) SCX

F23=(CB(N»3)»{ A23(N»2)-ONE)-CB(N»4)*(A23(N,l)-ONE) )/ALAM(N,2) SCX

AK1= ZRO SCX

AK4=ZR0 SCX

AK2=F12 (N)»G23+F23*G34(N) SCX 23

DO 43 1=1.2 SCX

11=1+4 SCX

DO 43 J=l»2 SCX

J2=J+6 SCX

Jl=J+2 SCX

K= 3-J SCX

XY=( (-ONE)**J-(-ONE)*»I

)

SCX

AKl = AKl-( ( (-0NE)**J)*CB(N,I1)*(A12(N,K)*Z12*XY+ SCX 35

1((-0NE)*»( I+J) )*(A12(N,I)+A12{N»K)-TW0)/ALAM(N,1) )SCX

2+( (-0NE)**I)*CB(N»J2)*(A34(N,K)*Z34*XY+ SCX

3( (-ONE)** (I+J) )*(A34(N,I ) +A34( N,K.)-TWO) /ALAM( N » 1

)

) )SCX

43 AK4=AK4+CB(N,I )*CB(N,J1)*(A23(N»K)*Z23*XY+ SCX 34

l((-ONE)»*(I+J))*(A23(N,I)+A23(N»IC)-TWO)/ALAM(N,2))

DUM(3)=AK4/ALAM(N,2)

SCX

SCX

DUM(2)=DUM(3)+AK2

AK1=AK1*AN(N.4)

SCX 24

430

DUM ( 1

) =DUM ( 2 ) +AK2+TW0*F 12 ( N ) *G34 ( N ) + AK 1 /ALAM ( N . 1

)

WK2 = WK.(N)**2

SCX

SCX

22

WK.2P=WK(M)**2

DUMRL(15)=((R(5)*T(N.2)*TD(M»2)-R(3)*T(N,1)*TD(M»1) )*WK2

SCX

4308

1 -(R(5)*T(M,2)*TD(N,2)-R(3)*T(M»1)*TD(N,1) ) *WK2P ) / ( WK2-WK2P)

DUMRL(16)=0.0

SCX

431

DO 431 1=4,7

DUM(I)=ZRO

SCX

SCX

DO 499 11=1,2

GO T0(432,433),I1

SCX

SCX

6 432 N1=N

N2=M

SCX

SCX

5 DO 4321 12=1,2

DO 4321 13=1,2

SCX

SCX

4 ALM( I2,13)=({-ONt)**(I2+l)J*ALAM(Nl,2)+((-ONE)**(I3+l))*ALAM(N2

XP1( I2,I3) = (0NE - A23(N1, I2)*A23(N2, 13) )/ALM( 12,13)

»2)SCX 32A 1 SCX32A 1

3 1P=3-I3 SCX

31
4321 XP2( I2tI3)»(A23(N2,IP)-A23(Nl»I2) )/ALM(I2»I3)

15=3-13

17=13+2

sex

AHA^=ZRO

AH1=ZR0

sex sex AH2=ZR0

AH3=ZR0

sex

sex

DO 4322 12=1.2

16=12+2

sex sex 14=3-12

DO 4322 13=1,2

sex sex sex

sex AH2=AH2 + (eB(Nl,I6 )*eB(N2.I7 )«XP1

(

I4t 15 ) ) AH 3 = AH 3 +

(

eB(Nl» I2r*CB(N2»I7 ) *XP2( I2»I3))

AH4=AH"4+(CB(Ni,I6 ) *eB( N2» 13 )*XP2 ( 14.15) ) 4322 AH1=AH 1 + (

e

B

(

Nl , I 2 ) *eB ( N2 . 1 3 ) »XP1( 12,13) ) IF ( ALAM(N2.2)-2.*ALAM(N1,2) ) 4341 IF((0»»1. )*( ALAM(N2,2)-2.*ALAM(N1,2) ) 438,4341,438

) 438,4342,438

sex

sex

33 sex

sex

4342 ISET=1

GO TO 4381

438

4381

ISET=2

DO 45 1=1,2

IP=3-I

DO 45 J=l,2

sex sex

sex

sex

sex

sex JP=3-J

DO 45 L=l,2

sex sex LP=3-L

L2=2*(L-1)+I

XP2(2,

XA=XP2

2)= XA

(1,2)

XP2(1,

XP2(2,

AH=AH3

2)« XP2(2,1)

1)= XA

sex sex

Y2= ( -ONE ) «* I +

( -ONE ) *J

ALM2 ( I , J ,L ) =- ( ALAM ( Nl , 2 ) *Y2+ALAM ( N2 , 2

)

( -ONE

)

**L ) sex

sex 40

ALM3 ( L , I , J ) =- ( ALAM ( N2 , 2 ) *Y2+ALAM ( Nl , 2

) * ( -ONE ) »L

)

DO 45 K«l,2

sex

sex

41

<2=2»(K-1)+J

KP=3-K

sex

sex

DIV= ( ALM(I,J)+ALM(L,K))

IF( eABS(DlV) .GT. l.E-38 ) GO TO 449 $ DIV=0.

sex

448 XP5(L2,K2)=A23(N1,LP)»A23(N2,KP)*(-Z23)

GO TO 45 sex

sex

46

449

45

XP5(L2,K2) = (A23{N1,LP)»A23(N2,K.P)-A23(N1,I)*A23{N2,J) )/DlV

eONTINUE

sex sex

433

GO TO 65

N2=N

sex

sex

ALM( 1,2)=-ALM(1,2)

ALM(2,1)=-ALM(2,1)

sex sex N1=M

XA=XP1(1,2)

sex sex XP1(1,2)=XP1(2,1)

XP1(2,1)=XA

sex

sex XA=XP2(1,1)

XP2(1,1)= XP2(2,2)

sex

sex

sex sex sex

sex

32
AH3=AH4 sex AH4=AH

DO 62 I»lf3 sex

sex

IA=4-I

DO 62 L=1,IA

sex

sex

J = I + L XPP=XP5(I,J)

sex sex XP5(I,J)=XP5(J,I)

XP5(J»I)=XPP

sex

sex

62 CONTINUE

DO 64 J=l,2

sex

sex

DO 64 1=1.2

DO 64 IC=1,2 sex

sex

XAL=ALM2(I.J,IC)

ALM2(I»J,K)=ALM3(K,I,J)

sex

sex

64 ALM3(K»I,J)=XAL

ISET=2

sex sex

65 DO 70 1=1,2

IP=3-I sex sex

XP4( I)=(0NE-A23(N1,I ) ) / ( ALAM( Nl .2 ) « (-ONE )**( I+l)

)

DO 70 J=l,2

sex

sex JP=3-J

XP7( I.J)=A23(N2,JP)*XP4( I

)

sex sex 37

DO 70 K=l,2

GO TO (69.692),ISET

sex sex

69

691

IF({-1.0)**I+(-l,0)»«J+2.0*(-1.0)»*K) 692*691,692

XP3( I »J,K)=-A23(N1,JP)*Z23

sex sex 42

692

GO TO 70

XP3( I»J»K)=(A23(N1,JP)-A23(N1,I)*A23(N2,K) )/ALM2(I ,J,K) sex

sex

70 CONTINUE sex AL2=ZR0 sex AL3=ZR0 sex AL4=ZR0 sex AL5=ZR0 sex AK8=ZR0

DO 4b 1=1,2 sex"

iP=3-i sex

^150 48":j^i , 2

" sex"

Ji=J+2 sex

XT =((-ONE)**I-(-ONE)»*J) sex

DO 48 L=l,2 SCT

LP=3-L sex Li=L+2 sex"

L2»2* (L-1)-H sex AL2=AL2+(C6TN2,r> ) *eB ( Nl , L) * ( XT*XP7 ( L » J ) + ( ( -ONE ) »»J*XP1 ( L .SCX

II)) - (-0NE)**I*XP2(L,J )) *eB(N2,Jl) SCx_

AL3=AL3+(CB(N2,I ) ) »CB( Nl ,L1 )» (XT*XP7 ( LP , J )+ (

( -ONE ) »J»XP2 (SCX

ILP, I P )) -( (-0NE)**I*XP1(LP,JP) ) ) *eB(N2,Jl ) SCX_

AL4=:AL4+eB(Nl,I)*eB{Nl,Jl)*eB(N2,L)*( (XP3( I,J,L)-XP7( I ,LP)) SCX

1/ALM(J,L)+( XP4(JP)-XP3(I ,J ,L) )/ALM(I,L) ) AL5^AL51^BTN 1,1) *CB ( N 1 , Jl) *C8 ( N 2 , L 1 ) * ( ( XP3 ( JP , I P , LP ) -XP4 (I ) ) /ALM ( SCX

1J,L)+(XP7(JP,L) -XP3(JP,IP,LP) )/ALM(I,L) ) SCX

DO 48 K=l,2 SOT

33
_ J<F_? ^< sex Kl=K+2 sex K2=2* ( K-1)+J sex AK8=AK8-(eB(Nl,I )*CB(N2,J) ) *eB ( N2,K1 )(( -ONE ) «<*((( XP5 ( L2Sex ^5

1»K2)-XP7(I ,JP) )/ALM3(L«J»K) ) + (-XP3( I»L»J)+XP7( I,JP) ) /ALM ( L , J

) -( ( XPsex

25(L2,K2)-XP7(LP.K ) )/ALM3( I»J,K) )+ ( XP3 ( I .L . J )-XP4 ( LP ) )/ALM(I,J)) SCX

3+( (-ONE)**J*(-( (XP5(L2»K2)-XP7( I»JP) ) /ALM3 ( L» J »K ) ) + ( XP3 ( LP » I P»KP ) SCX

4-XP4(I ) )/ALM(L,K)+( ( XP5 ( L2 »K2 ) -XP7 ( LP,K ) )/ALM3( If J,K))+( (-XP3( sex 5LP,IP,KP)+XP7(LP»K ) )/ALM( I.K) ) ) ) ) »CB(N1,L1) SCX

^8 CONTINUE sex

AK8 = AK8/ALAM(N2»2) SCX

AL2=AL2/ALAM(N2.2) SCX

AL3 = AL3/ALAMJjsi2jr2J SCX

AK6= F12(Ni)*AL2+G34(Nl)*AL3 SCX 27

AK5= AK6+F12(Nl)*(AHl*F12(N2)-fAH3*G34(N2) )-t-G3A ( Nl )* ( AH2*G34( N2 )+ SCX 25

1 AH4*F12(N2) ) SCX

AK7 = AK8-t-F12(N2)*AL4->-G34(N2)*AL5 SCX 26

DUM(4)=DUM(4)+AK5 SCX

DUM( 5)=DUM(5)4-AK6 SCX

DUM(6)=DUM(6)+AK7 SCX

499 D UM ( 71= DUM ( 7 2+AK8 SCX

DUM(5)=DUMr5>+DUM(7) SCX

DUM ( 4 ) = DUM ( 4 ) +DUM ( 6 ) SCX

500 RETURN SCX

5 N=-N

WRITE OUTPUT TAPE 6,51 SCX

51 FORMAT (54H OMEGA IS SUCH THAT STRUCTURE IS RESONANT-NOT ALLOWED )SCX

RETURN sex

END

SCOPE

34
INPUT DATA DEFINED TOROID

R 5 RADIUS VALUES

Z ="Z"AXIS VALUES TURN = NUMBER

OF WIRE TURNS

ER = FREOUENCr

IN HERTZ

ISslNITIAL

PAGE COUNT &

V(l) = €l/e5

V(2) =

H-'/f^o

V(3) = -e'/'eo

V(4) = -/x"//i.o

&

R,Z, TURN,

IC(5),

IC(6)

ER, IC(4),IS

ITES

CALCULATION

CALCULATION

SINGLE SUM < 1C(5) DOUBLE SUM < IC(6)

<D

<D

USUALLY

IC(5) = 40 IC{6)-40

,^X-,o^

•' ITES > I

35
o o o

tU lU o o

•r » m u> • • o o O

N0>

XPH

Ph

IPH > o I 01 UJ o o m cI ir\

r ai oI

UJ UJ o oo o o oo3

^<^ >-

lU Ml

a. ac < < e z 2 Z

LU LJ z z 3 OI : <. a "SI

36
o o

m in

-» o

o c

I tU UJ

IT* 6

hi ri N c^

• • o o

oo

UJ UJ o o

a: o

o o ^ in

Ul 3 O O

Z UJ < X o u.

UJ

o o

I

UJ UJ < <

o o

z i: UJ UJ s s

_1 lO

•« •- -^ X

c <

rNj CD U

M O

u e

rsl O

fM

37


U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERC

OFFICIAL BUSINESS
TECHNICAL NOTE NO. 312 - UNASSIGNED