Tables of Chebyshev Impedance-Transforming Networks of Low-Pass Filter Form

GEORGE L. MATTHAEI, MEMBER, LEEE

Summary-Tables of element values are presented for lumped- element Chebyshev impedance-transforming networks, as are tables giving the Chebyshev passband ripple of each design. These circuits consist of a ladder network formed using series inductances and shunt capacitances; they give a Chebyshev impedance match between resistor terminations of arbitrary ratio (designs with resistor tenni- nation ratios from 1.5 to 50 are tabulated). The responses of these networks have moderately high attenuation at dc (the amount of attenuation depends on the termination ratio) ; their attenuation then falls to a very low level in the Chebyshev operating band, and then rises steeply above the operating band in a manner typical of low- pass filters. Designs having operating-band fractional bandwidths ranging from 0.2 to 1.0 are given. These impedance-transforming networks can be realized in lumped-element form for low-frequency applications, and in semilumped-element form (such as corrugated waveguide) at microwave frequencies.

N GENERAL

UMEROUS papers have appeared on the design of quarter-wave step-transformer structures. Among these papers is one by Young, which con-

tains tables of designs for quarter-wave transformers having Chebyshev transmission characteristics.' Since the accurate calculation of such transformer designs can be very tedious, the existence of extensive tables of step-transformer designs has proved very valuable. Such transformers are extremely useful a t microwave frequencies, but for applications involving frequencies much below 1000 \IC, the size of step transformers can become impractically large.

Herein tables of element values are presented for a form of lumped-element low-pass filter structure that has impedance-transforming properties similar to those of a step transformer but which can be constructed in very compact form, even a t low frequencies. Though it is anticipated that this type of impedance-transforming structure will find greatest application a t frequencies below the microwave range, i t will be seen that the tabulated designs presented here will also be useful as prototypes for the design of semilumped-element im- pedance-transforming networks a t microwave frequen- cies.

Fig. 1 shows the general form of the impedance-trans- forming structures under consideration. It should be noted that the structure is of the form of a conventional

Manuscript received February 12, 1964. This research was s u p ported by the U. S. Army Electronics Research and Development Laboratory, Ft. Monmouth, X . J., under Contract DA 36-039-

Calif. The author is with The Stanford Research Institute, Menlo Park,

Leo You;g, "Tables of cascaded homogeneous quarter-wave transformers, IRE TRANS. ON hlICROWAvE THEORY AND TECH- NIQUE, vol. MTT-7, pp. 233-237; April, 1959.

AMC-00084(E).

low-pass filter structure. The main difference between these structures and those of conventional low-pass filters is that conventional low-pass filters have termi- nating resistors of equal (or nearly equal) sizes a t each end. In the case of the filters discussed here, the termi- nating resistors may be of radically different size, which means there will be a sizable reflection loss a t zero fre- quency. As a result of this sizable attenuation LA^^ a t zero frequency, the transmission characteristics of Chebyshev filters of this type have the form shown in Fig. 2. Sote that there is a band of Chebyshev ripple extending from 0,' to w b ' , and that above cob' the attenu- ation rises steeply in a manner typical of low-pass filter structures. It should be noted that the attenuation L.4

indicated in Fig. 2 is transducer attenuation expressed in decibels; L e . , it is the ratio of the available power of the generator to the power delivered to the load, expressed in db.

Xs mentioned above, the designs tabulated herein should prove useful for semilumped-element microwave structures, as well as for lumped-element low-frequency structures. An example of a semilumped-element wave- guide structure is shown in Fig. 3. The structure shown is of the corrugated waveguide filter form's3 in which steps or corrugations in the guide height are used to simulate the shunt capacitors and series inductors of the structures in Fig. 1. In Fig. 3, where the guide top and bottom walls come close together, the effect is largely like that of a shunt capacitor (as is suggested by the dashed-lined capacitors in the figure). Where the guide top and bottom walls go far apart to form a groove, the effect is predominantly like that of a series inductance (as is suggested by the dashed-line induct- ance in the figure). In this manner, semilumped-element impedance-transforming structures can be designed from the prototype designs given herein. .A structure such as that in Fig. 3 might prove desirable for wave- guide applications, say at L-band or below, where a con- ventional step transformer might be larger than can be accommodated for some given application. Such struc- tures can also find application a t higher frequencies for use as structures for coupling to magnetically tunable yttrium-iron-garnet ferrimagnetic resonators. Struc- tures of the form in Fig. 3 have advantages for such ap-

IRE, vol. 37, pp. 651-656; June, 1949. S. B. Cohn, "Analysis of a wide-band waveguide filter," PROC.

3 G. L. Matthaei, Leo Young, and E. hl. T. Jones, "Design of

Structures,' Stanford Research Inst., Menlo Park, Calif., prepared ?*licronave Filters, Impedance-Matching Networks, and Coupling

on SRI Project 3527, Contract DA 36-039 SC-87398, ch. 7; 1963.

939

940 PROCEEDINGS OF THE IEEE August

, R;+l G;, , RESISTANCE AT END n + l RESISTANCE AT END 0

(a>

G:. l~ ' In t l --- , , ,....+, Rh , CONDUCTANCE AT END n + I

CONDUCTANCE AT END 0

(b) Fig. 1-Definition for normalized prototype element values for im-

pedance-transforming networks of low-pass filter form. (The tabulated element values are normalized SO that go=l and h' = 1. [ S e e Fig. 2 .I)

I I

Fig. 2-Definition of response parameters for low-pass impedance- transforming filters. (The frequency scale for the tabulated proto- type design is normalized so that urn'= 1, as indicated above.)

IUPEDANCE-TRAUSFORUINC CTRVCTYRE

Fig, 3-A corrugated-waveguide structure for matching between guides of different heights.

plications because the inductive section at the low- impedance end of such a structure (forming L4 at the right end in Fig. 3) provides needed room for locating a ferrimagnetic resonator while at the same time giving the impedance transformation required for obtaining tight coupling to the ferrimagnetic r e ~ o n a t o r . ~

PARAMETERS OF THE ATTENUATION CHARACTERISTICS

The frequency scale of the networks tabulated here has been normalized as indicated in Fig. 2 so that the arithmetic mean-operating-band radian frequency,

Wa' + Wb' Wm' = 1

2 (1)

is scaled so that

Wm' = 1.

Herein the frequency variables and element values of the normalized prototype circuits will be primed to indi- cate that they are normalized, and corresponding un- primed quantities will be reserved for the same param- eters scaled to suit specific applications. With the nor- malization in ( 2 ) ,

and w

Wb' = 1 +- '

2 ( 3

In most impedance-transformer applications, the pa- rameters of the normalized prototype circuit that will be of importance are the impedance or admittance transformation ratio r (see Fig. l ) , the fractional band- width w , and the db passband attenuation ripple Lar (see Fig. 2 ) . After a designer has determined the value of r , the minimum value of w, and the maximum allow- able value of Lar for his application, the next step is to determine the number n of reactive elements required in the circuit in order to meet these specifications. The required value of n can easily be determined with the aid of Tables 1 t o 5 (pp. 944-948). For example, suppose a designer desires an impedance transformer to give an r = 3 impedance ratio over the band from 500 to 1000 Mc with 0.10-db or smaller attenuation ripple ratio in the operating band. The required fractional bandwidth is given by

f b - fa - 2 ( f b -fa) -w=-- 1 (6) f m . f b + f a

4 G. L. Matthaei, et al., "Microwave Filters and Coupling Struc- tures,'' Stanford Research Inst., Menlo Park, Calif., Final Rept., SRI Project 3527, Contract DA 36-039 SC-87398, Sec. 111; February, 1%3.

1964 Xatthaei: Tables of Chebysheo Impedance- Transforming Networks

which for this example gives and

2(1000 - 500)

1000 + 500 - - = 0.667.

This value of fractional bandwidth lies between the w=O.6 and w =0.8 values in Tables 1 to 5 , so the w =0.8 value will be used. This will give an operating bandwidth somewhat larger than is actually required, which is often desirable. However, if this is objection- able, the desired bandwidth can be achieved by inter- polating between the values in Tables 1 to 5 in order to determine the required value of n for w =0.667 and LA? sO.10 db, and then interpolating between numbers in the element-value tables (to be discussed later), to obtain a prototype with w=O.667, r =3, and the given value of n. Reducing the bandwidth from 0.8 to 0.667 would mean that a smaller value of LA? could be achieved for given values of r and n.

Assuming that use of w = 0.8 is satisfactory, we deter- mine the required value of n as follows. From Table 1 (which is for n = 2 reactive elements), for w=O.8 and r = 3, we obtain LA? = 0.639 db, which is too large. From Table 2 (which is for n = 4 reactive elements), for w = 0.8 and r = 3, we obtain LA? = 0.139 db, which is still somewhat too large. Finally, from Table 3 we find that for n= 6, Ldr=0.023 db, which is less than the 0.10 db required. Actually, in this case i t is clear that if the fractional bandwidth were reduced close to the w = 0.667 minimum required value, n = 4 reactive elements would be sufficient to give Ladr < O . 10 db.

Besides the fractional bandwidth w and the operat- ing-band-ripple LA? in Fig. 2, other aspects of the re- sponse of the normalized prototype circuit will at times also be of interest. The attenuation L.tdc a t zero fre- quency is given by

L A d c = 10 log10 - ( r + db 4r

where r is again the impedance or admittance transfor- mation ratio.

In some cases it will be desired to determine the at- tenuation accurately over a range of frequencies, pos- sibly for making use of the strong attenuation band of this type of structure above frequency w2. The attenu- ation characteristic in Fig. 2 can be predicted by map- ping the attenuation characteristic of a conventional Chebyshev low-pass filter (which has an equal-ripple pass band from w”=O to w” =@I”) by use of the map- ping function

J’ & - #0’2

A 01’’ -= (8)

where w“ is the frequency variable for the conventional Chebyshev filter,

A = - Wa‘’

2 (9)

94 1

is the frequency in the response in Fig. 2, which cor- responds to w “ = O for the corresponding conventional Chebyshev low-pass filter characteristic. hfaking use of (8) and (9) and the equations for the attenuation of a conventional Chebyshev low-pass filter (see, for ex- ample, Fig. 2 of Cohn5), we obtain

which applies in the “stop” bands O s w ’ sua‘ and wb’<w’< a. In ( l l ) ,

and n is the number of reactive elements in the im- pedance-transforming filter network. In the operating band w: 5 w‘ 5 Wb‘,

A conventional low-pass Chebyshev response for a fil- ter with n” reactive elements maps into a response of the form in Fig. 2 for a filter having n= 2n” reactive elements. The response in Fig. 2 corresponds to an n = 8 reactive-element design. However, this response could be obtained by mapping the response of a corresponding n” = 4 reactive-element conventional Chebyshev filter design, using the mapping in (8). This is because the mapping function in (8) doubles the degree of the trans- fer function polynomial.

TABLES OF PROTOTYPE ELEMENT VALUES Tables 6 to 10 (pp. 949-963) give element values for

prototype impedance-transforming networks for n = 2, 4, 6 , 8, and 10 reactive elements. After the designer has arrived at values for r , w, and n, the normalized element values can be obtained from the tables. However, since the networks under discussion are antimetric6 (;.e., half of the network is the inverse of the other half), only half of the element values for each network are tabulated, and it is necessary to compute the element values of the second half of the network from those of the first half.

45, pp. 187-196; February, 1957.

and Sons,. Inc., New York, N. Y . , ch. 11 : 1957.

5 S. B. Cohn, “Direct-coupled-resonator filters,” PROC. IRE, vol.

6 E. A. Guillemin, ‘Synthesis of Passive Setworks,” John Wiley

942 PROCEEDINGS OF THE IEEE August

This procedure will now be summarized for each value of n. For n=2 , go= 1, and gl is obtained from Table 6. Then

For n = 4, go = 1 ; and gl and g2 are obtained from Tables 7(a), (b). Then

g3 = g2r, g4 = - 7 and g6 = r. g1 r

For n= 6, go= 1; and gl, g?, and g3 are obtained from Tables 8(a)-(c). Then

g4 = 7 g3

g6 = gzr,

which results in a gradual loss of accuracy as the expan- sion progresses. Thus, values for gl could be obtained with great accuracy, values for g2 with somewhat less accuracy, etc. Tables 6 to 10 include all of the signifi- cant figures that were available in the computer print- out; however, in the case of the values for, say, ele- ments g4 and g5, some of the digits on the right end of each number are doubtless in error. In order to see how serious the errors might be, some responses were com- puted for several designs, including the rather extreme cases of w=O.8, and w = 1.0, having n = 10 and r = 50. The bandwidths were as predicted, and after checking the details of the pass band, the pass band Chebyshev ripples were found to have very nearly the peak values predicted, with a t most two or three units error in the third significant figure and better accuracy in most cases. Thus, though not all of the element values given in Tables 6 to 10 are as accurate as the number of sig- nificant figures shown indicates, the accuracy appears to be more than adequate for typical practical appli- cations.

For n= 8, go= 1; and gl to g4 are obtained from Tables SCALING OF THE NORMALIZED DESIGN 9(a)-(d). Then After a designer has selected a normalized design, the

element values required for his specific application are g6 = g4r, g6 = - j g7 = g21, easily determined by scaling. Let R be the desired resist-

Y ance level of one of the terminations, while R' is the cor- responding resistance of the normalized design. Simi- larly, let wm = (wa+m)/2 be the radian frequency of the

corresponding frequency for the normalized design.

g3

g1 gs = - 9 go = 1.

Y (17) center of the desired operating band, while Wm'= 1 is the

For n = 10, g = 1; and gl to g5 are obtained from Tables Then the scaled element values are computed using lO(a)-(e). Then

g9 = gzr, g1

g10 = - 9 Y

gll = r.

Note that for a design with n reactive elements, there are n+2 element values, go, gl, - . 1 gn, gn+l- Two POS-

sible interpretations of these element values are as indi- cated in Fig. 1. Observe that each of the terminating element values go and gn+l may be either a resistance or a conductance, depending on whether it is next to a shunt capacitance or a series inductance, respectively. This convention is used because it is convenient when applying the principle of duality to the element values. The two interpretations of the element values shown in Fig. 1 actually give the same final circuit for given ter- minations since the circuit shown at (b) could be ob- tained from that shown at (a) by simply turning the circuit a t (a) around and scaling its impedance level.

In carrying out the calculations of the element values in Tables 6 to 10, eight significant figures were carried by the computer. However, the computations were made using a continued-fraction-expansion proCessl6

Rk = Rk' (i) Ck = Ck' ($) R'

where Rk', Ck', and Lk' are for the normalized design and Rk, Ck, and Lk are for the scaled design.

DERIVATION OF THE TABLES The circuit designs given in Tables 6 to 10 were syn-

thesized by first starting with reflection coefficient functions for conventional Chebyshev low-pass filters that have pass bands from w" = 0 to w'' = w1", and have monotonically increasing attenuation above w1". Re- flection coefficient functions for such networks have the form'

problems in network synthesis,n J. Franklin Inst., vol. 249, pp. 189- 7 R. M. Fano, "A note on the solution of certain approximation

205; March, 1950.

1964 Mat thae i : Tables of Chebyshev Impedance- Transforming Networks 943

where 9” = d ’ + j w ’ ‘ is the’complex frequency variable We have not, as yet, discussed how the operating- and the P a k ” and P b k “ give the locations of the zeros band ripple enters into the synthesis problem. It can and poles of the function. Expressions giving the loca- be shown that tions of these poles and zeros for a given circuit com- plexity and Chebyshev ripple amplitude can be found in the literature.’ The poles and zeros of the reflection where coefficient function were then mapped to give the reflec- tion coefficient functions

LA^ 10 loglo (1 + e ) db ( 2 7)

( r - l)?

4r cosh2 [t cosh-l r;)] € = __ (28)

for the desired network. This was accomplished by use of (8) with w” replaced by p”/ j , and w‘ replaced by p ’ / j . The complex mapping function was then (with wl”= 1)

By properly adapting (24) it can be seen that for a given pole location P b k ” = @ , k ” + j W b k ” in the left half of the 9’‘ plane, the corresponding pole locations p b k ’ and a for the p’ plane are given by

~

pbk’, Pbk‘

where

Since ubk” is negative, the argument of the arctangent function is usually negative so that the arctangent is evaluated as between 0 and -a/2 radians. However, since Wbk’’ is sometimes positive and sometimes nega- tive, the argument of the arctangent can go positive in some cases. When the argument is positive, the arctan- gent is evaluated as having a value between -a/2 and -a radians. In this manner, every pole in the I’(p”) reflection coefficient function yields two conjugate poles in the I’(p’) function.

The mapping equations ( 2 5 ) and (26) also apply for mapping the zeros of the I’(p’’) function to give those of the I’(p’) function. The mapping of the zeros is some- what simplified, however, since they lie on the imagi- nary axis in both planes, ;.e., pal;“=jwak“ and Pak’, - P a k ’ = +jwak’.

8 Here p0l ) i s the complex conjugate of p.1’.

is the same E as appears in (11) to (13). Thus, LA^ (and e) are determined by wnt2/.4 (which controls the fractional bandwidth w ) , by r , and by n. Sext , we com- pu te

- 1

sinh-’ ,l/- e

a = (2 9) It - 2

and the parameter n is used in Fano’s’ equations for the pole and zero locations of ( 2 2 ) . By specifying n ” = n / 2 = [number of poles and number of zeros in ( 2 2 ) ] , w1”,

and a , the function in ( 2 2 ) is entirely determined, with the aid of Fano’s results.’

To summarize, the reflection coefficient function I ’ (p” ) , ( 2 2 ) , for a conventional Chebyshev filter is ob- tained from Fano’s paper, using ( 2 7 ) to (29) to fix the value of a to use in Fano’s equations. Sext, the poles and zeros of this function are mapped as discussed in connection with ( 2 5 ) and (26) in order to obtain the reflection coefficient function I’(p’) , ( 2 3 ) , for the desired form of network. This gives I’(p’) in factored form, and its factored polynomials must then be multiplied out so that the numerator and denominator are in unfactored form. Then, using standard methods of network syn- thesis,6 the input impedance function Z(p’) is formed, and the element values g k are obtained by expanding Z(p’) in a continued-fraction expansion.

ACKNOWLEDGMEXT

The mapping in (8) to (10) for use in the synthesis problem described in this paper was suggested to this writer by Prof. H. J. Carlin of the Polytechnic Institute of Brooklyn, N. Y., M. H. Lawton of Stanford Research Institute, Menlo Park, Calif., prepared the IBRl 7090 computer program used in making the calculations for the tables of impedance-matching network designs.

E

TAB

LE

1

LA

, in d

b vs

. w

and

r f

or n

3

2

W

c 1.5

2 -0

2.5

3.0

4.0

5.0

6.0

8.0

10.0

15 0

0

23.0

25.0

3l.

3

4C.O

50.0

-0 . 1

. 00 1800

. 0053

98

-009

712

a014

381

0242

40

0344

34

a044

782

,065

671

a086

637

a 136

917

1907

48

2420

36

-292759

0392

511

a 49

0055

__

_

0.2

. 0070

90

.021

235

-038

143

0563

97

0947

50

134143

1738

63

2532

33

0331

840

a523

445

e737

556

0864

410

1.05445

1 . 37

594

1.67535

0.3

0 15

549

. 0464

82

-083

313

1228

63

2053

59

a289

233

3729

48

a537

751

6978

55

1.07618

1 :425 30

1 74884

2.05014

2.59671

3.08228

0.4

.0266i37

a079

573

0 14

219.

7

2330

37

,347

127

a 48

5751

6224Q 7

ai336733

1.13735

la71

210

2.22374

2.67658

3.38935

3.~

~1

34

1

4.43

402

G -6

a054

487

a161

453

. 2 a64

2 9

417t280

5 63

492

0342

628

1. A9

134

1 65

534

2. G7

792

2 98

813

3.74

255

4.38

531

4. Y4

650

5.88881

6.bt291

0 -8

a065

225

a250

817

a441

537

a 639 1

li.

1.02961

1 . 40

Q3t

1 7

4776

2.37619

2. YZ

95 1

4.37

171

4.97

794

5.72825

5 36

8 22

7. 4

2049

80

2671

4

1.0

1142

95

a334

238

5'0426C

a839

801

1.33539

1 a

79

35

2

2.21849

2.96665

3.60972

409OO52

5 89

726

6.70

839

7.39

203

8 . 50

279

9.38680

TAB

LE

2

W

r

1.5

2.0

2.5

3.3

4.0

5.0

5.0

8.0

10.0

15.0

23.0

25

.0

30.0

43

03

53.0

0.1

0000

05

oOOO

O14

0000

24

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000061

0 00

0087

0001 13

.000166

. 0002

20

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0004

90

0006

2 5

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032

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72

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0003

9 1

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79

0009

77

. 00 1389

.Oi)18(19

002659

0035

16

0056

7C

0078

30

0099

93

-012155

0016479

020801

0.3

0003

66

oOO1098

0001976

.002928

. 0049

39

007023

0009142

- 0 13432

,017754

,028605

0039465

0503

12

oO61

140

C'82725

104210

3.4

.001154

0003452

0006229

009226

0015557

002213'9

0028765

-042

219

0557

46

oa9577

123258

156725

199958

25 570

4

-3 2 0493

0.6

0057

65

0 1727 3

o031C42

3459 lI

,

0077

193

. 1 i937R

141 885

2 G7

C'OI

I

.271701

.433265

. 5837

48

732

186

08

75

61

2

1.14959

1.43739

3.8

,017581

.052

530

.994

102

138633

e231536

.325719

.4 19483

ob03465

.78144(;

1. 19921

1.58151

1.93332

2 25897

2.04532

3.36201

1.0

039890

118586

0211117

-309

306

.5C9855

-7C8365

901453

1.26613

1 609r)l

2.36390

3. GO

880

3.57093

4.068~

4 0 92 088

5.633QY

946 PROCEEDINGS OF THE IEEE August

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LA

r in d

b vs

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and

r f

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0 0

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oOO

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31

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42

000053

0.4

0 0

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0 0

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6

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948 PROCEEDINGS OF THE I E E E August

In

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950 PROCEEDINGS O F T H E I E E E August

0 0 d

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m

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N

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LE

7(b)

ELEM

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VA

LUE

g2

bs.

wa

nd

r f

or n

=

4 . .~

W

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87

94

38

86

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80

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99

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1

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63

TAB

LE

8(a)

ELEM

ENT

VALU

E g

, vs.

wa

nd

r f

or n

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-

-

W

r

1.5

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S

3.0

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0

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49

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29

87

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38

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72

12

80

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71

56

5

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60

80

09

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35

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6

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98

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TAB

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VA

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r 1.5

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9(a)

ELEM

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VALU

E g

, vs

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an

d r

for

n

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~~

0.1

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13

23

83

23

32

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26

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65

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0

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87

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6

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59

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1

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5

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98

01

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1. 3

63

st,

1. 4

43

07

1.0

54

22

03

.rj4

42

73

7 1

36

92

. 767649

Oi3

53P

8E

91

56

41

.969

54cI

1,05

775

1.12

99?

1.27

253

1.3

b5

42

1.4d

139

1 .-

5662

3

1 7

13

87

1. P

41

95

TAB

LE

9(b)

ELEM

ENT

VALU

E g

2 v

s. w

and

r f

or

n 8

W

0.1

r

le 5

2.0

2- 5

3.0

4.0

5-0

6- 0

8.0

10.0

15-9

20.0

25

.0

33.3

43

.3

59.0

0.2

8448

24

-823

343

8 10

364

e799

795

e 77

9936

764694

0-3

e 88

2869

e 91

2871

,915

840

e913

547

-904

478

8944

60

e885

792

e869

951

8568

85

-831

899

,813

252

-798502

7862

99

7668 16

-751

629

0.4

-891

d26

e91

13

77

- 91 2 53

7

9389

4 1

warn

0887

791

8779

0 1

-860

801

8466

73

-819

587

e739

578

e793709

-770

575

-749

599

-733

139

0.6

e 86

9498

e904

2Cb

9Gl8

33

8314

61

-88Cb613

e 86

652 3

e 85

3892

.a32

512

e 8 E424

e 76

1 8

1 7

7 57

42 0

-738

122

-722

152

e695

624

e176535

0 -8

s 883

542

.a91146

8 8 33

2C

e872

552

8508

8.9

-831

597

-8 147

63

-786810

e764

262

e721

839

,691

015

6666

87

-546

623

e614

591

iE95

3.3

1.0

-872284

e 86

9198

e 85

3352

e836

225

8048

66

e 77

8309

e755

688

,718

858

5896

24

-635

568

5967

77

5665

79

-541

899

0 5 s39

7 3

4731

73

b

TAB

LE

9(c

)

ELEM

ENT

VALU

E g

3 v

s. w

an

d r

for

n

= 8

-.

~~

__

__

__

_.

W

~.

~ .

~..__

0.1

0.2

0.3

0.4

0.6

r

1.5

2.0

2.5

3. 3

4.0

5.0

6.0

8.0

1 . 25

93

4

1 4

56

45

1.58

652

1.68

459

1 0

56

05

1.9

97

83

2 1

09

95

2.30

736

1.2

96

12

1.4

65

26

1.5

91

25

l.bY

517

1.8

62

95

2,3

03

32

4.11831

2,3

17

04

10

.0

-2.4

7012

2.482

53

2

.51

53

3

15.3

2.71

634

2.

7933

0 2

ob

14

25

2.8

68

67

23.0

3

32

60

4

3

.G5

09

3

3.07750

3.1

51

54

3.7

22

55

3 . 77

99

3

3

.62

87

7

3.3

69

99

50.0

4.0026

1

4.0

51

98

4.1

12

34

4.2

63

06

___

0.8

1

.0

1 2

78

92

1,4

50

62

1.5

81

13

1.6

90

43

1.8

71

63

2. C

22

37

2.1

52

74

2.3

75

46

2.56429

2 .*

349 3

7

3.26

139

3.5

29

27

3.76730

4.1

wa

9

4.5

3P

70

1.2

64

86

1 4

4120

4

1 5

75

54

1 6

90

26

1.88

272

2.04

432

2.18

591

2.4

29

66

2.63

6544

3. C7037

3.42

553

3.73

421

4.G

lrJ6

4

4.49

736

4. 9

2336

x

TABLE 9(d)

ELEMENT VALUE g4 vs. w and r for n = 8

W 001 002 0.3 00 4 0.6 0 08 1.0 c

2.0 ,971922 0958291 9'36G7 0 908264 e875128

2.5 oYC.3029 0891491 rt368633 839925 805666

3.0 0852023 a 837866 08 15200 0785765 0751047

4.0 0769428 0758252 734992 0705234 670348

5.0 e 708982 0 730545 0677174 64743 1 0612734

5.3 -666138 -656 195 632 871 ob03277 m568883

800 0 60031 5 0591251 565250 0539351 505343

10.0 - 0 553993 0 545193 522369 493595 a 460 546

15.0 498537 ,478837 . 473935 0447747 0419926 a388285

20.0 - -438467 -431239 42 2 849 o401C32 373988 343462

25.0 -.. 407607 e397601 0399399 0366907 341613 0 3 12C03

50.0 .3n3524 372057 . 35 3998 3.i2952 .3171.10 0288273

45.3 e343258 334996 0327154 a306662' .28 1728 m254148

53.0 0315316 308895 30 1679 -281038 25682Y . 230252

TAB

LE

lO(a)

ELEM

ENT

VALU

E g,

vs.

w a

nd r

for

n

=

10

._ ~~

.

~~~

. ~

__

__

__

__

_~

W

.~ ._

~

.

~~

. ..__

0.1

0.2

0.3

0.4

0.6

O*A

1.

0 c 10s

0411731

431653

-451413

-480211

2.0

0452862

-488315

o5164YfS

5572

40

2.5

05

30

63

0

5242

48

-5

5821

6 607974

3.0

5249

50

. 5 507

95

05

8932

1 06

4653

1

4.0

05

58

05

2

5692

42

53519 7

7046

35

5.0

0531

406

.6

1738

0 66

922 5

0748

736

6.0

0632

333

ob

3961

3

0696

516

0784

761

8.0

63333.3

.47339

7 39

29 5

8424

69

10.0

0634

913

07Z0257

7 72

676

15.0

6953

35

743 24

6 0834819

. 8885

93

9771

72

23.3

0724

629

o7

8287

8

o88G

786

1 04

508

55

.0

a717

659

-7

4679

3 810293

9 1

79

03

1.10145

30

.0

0736

946

076510i)

0833

156

,9

4935

0 1

1503

3

40.0

7669

02

e7

9483

9 087C4277

loi30139 1

. 23350

50.0

7899

86

81

832

5 96

0G56

1

04409

1. 303

94

t-4 h

h

kl

TAB

LE

10(b

)

ELEM

ENT

VALU

E g

2 vs

. w

an

d r

for

n

= 10

c W

1.5

2.0

2.5

3.0

4.0

5.0

6.0

3.3

10.0

15.0

20

.0

25

. 0

so. 0

40

. 0

53.9

0.1

0.

2 0.

3

a 8

72

60

8

86

61

48

. 855

48

9

84

68

63

e8

5 1

50

4

08

81

63

0

89

44

02

-89

89

63

-90

12

33

n9

00

48

9

-89

84

23

08

93

31

2

08

88

54

5

o8

77

50

9

08

68

17

3

me6

0536

08

53

92

6

09

42

72

5

0833681

0.6

85

86

25

88

47

12

08

92

59

4

89

48

96

08

93

78

7

08

90

21

5

08

85

03

6

87

76

54

8 6

99

6 9

a 85

3825

8 4

39

50

83

02

58

08

21

09

2

.8i4

5899

. 793

53

3

08

60

76

7

08

R1

90

1

88

56

95

8.84

5 6

1

08

77

95

9

87

00

70

86

23

20

e3

48

23

2

8 3

60

18

0 1

14

67

07

92

44

9

07

76

87

5

76

36

44

07

41

88

9

07

24

33

3

85

97

01

87

23

99

86

94

71

86

28

17

84

74

90

08

32

82

4

a 8

19

52

6

07

96

64

5

77

75

67

-74

04

19

,71

23

55

.6@

97

06

06

70

66

6

06

39

71

2

6 1

49

90

0

“rl

b

e os

E x

b

Et 0.s f

TAB

LE

lO(c

)

ELEM

ENT

VA

LUE

g3

vs.

w and

r fo

r n

- 10

. .-

__

_

-

W

r

1.5

2.0

2. 5

3.0 4.3

5.0

6.0

8.3

10.0

15.0

20.0

25.0

33.0

40.0

53.3

0.1

0.2

Om 3

-

2.21549

2.31023

2.46527

2.59137

0.4

1.19672

1 30620

10 39716

1.46132

1 55648

1.52930

1.69879

1.60918

1.-89137

2.05722

2. 18713

2.29086

2 38

00'1

2.53177

2.55589

0.6

1.20715

1.32431

1.4068 1

1.47303

1. 57

754

1 66

064

1 73

040

1.64520

1 939GO

2.12274

2.26528

2.38381

2 48646

2.66020

2.8i15R3

0.8

1 2

04

87

1.32685

1.41462

1 48563

1 59944

1.69095

10 76

863

1.89779

20 004

63

2.21670

2.38365

2.52418

20 647

05

2.85726

3.33575

1.0

1.20163

1 33043

1.42546

1.50357

1 63084

1.73480

1. e2409

1 97463

2 0 10094

20 356

05

2056C95

2073599

2.69085

3. 15972

3.39173

TAB

LE

10(d

)

ELEM

ENT

VA

LUE

g4

vs.

w a

nd

r f

or n

=

10

__

__

W

r

1.5

2.0 2.5

3. 0

4.0

5.0

6.0

8.0

10.0

15.0

20.0

25. 3

39.3

40

.0

50

.0

0.1

0.2

0.3

a607

518

5780

24

5350

61

5042

94

0.4

1.09058

1.34576

0994

964

-956

031

0899

757

0857

803

81

972

0

7636

47

0725540

-657

517

61 1 571

e578

873

0553

219

0514

119

e 4

85

83

0

0.6

1.06762

1.01344

w963

926

0926

490

0865

317

8193

48

07ir

3020

o728

C82

,587

563

0618

468

-572

919

. 5 394

83

5133

1 1

,474

094

e445

368

0.8

1.04110

098

11

21

0930

146

0 888

42G

0824

186

0776307

a730

609

0681890

6402

51

5 697

2 3

5234

95

4897

1 I

46 34G 1

4241

40

3955

13

1.0

1.00815

0941

372

8864

68

8420

83

7744

6 1

0724

499

,685

385

6269

82

0584

402

5128

00

0466

326

04

32

56

5

4064

08

3676

51

,339

610

c,

TAB

LE

1 O(e)

ELEM

ENT

VALU

E g

5 vs

. w

ond

r f

or

n =

10

r

__

___

___

-

W

r

1.5

2.0

2.5

3.0

4.0

5.0

6.0

8.3

11.2

15

.0

20

.0

25.0

33.0

40.0

50.0

0.1

0

.2

0.3

5.5

24

26

6.0

82

52

7.0

54

45

7.

89

71

1

0.4

1.4

85

19

1.7

17

67

1.9

48

19

2.13122

2.4

32

18

2.6

85

09

2.9

43

50

3.38

62.3

3.7

40

99

4.5

19

53

5.1

85

05

5.7

53

20

6.25905

7.1

70

62

7.95889

C.p

1.4

03

56

1 7

26

45

1. 9

26

20

2.10366

2.4

11

18

2.07837

2 91608

3.3

33

51

3.6

97

25

4.4

64

68

5 1

37

47

50

37

01

4

60 1

75

21

7.0

71

31

7.8

5L

69

1.0

1 4

14

60

1 6

43

03

1 8

34

04

'

2.0

03

02

2 2

97

94

2.5

53

95

2.7

82

61

3.1

85

67

3.53800

4.2

81

96

4.9

05

76

5.4

52

42

5.9

44

95

6.81695

7.5

53

14