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NAT'L INST. OF STAND & TECH

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REFERENCE

NBS

Publications

NBS TECHNICAL NOTE 1074

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

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the Institute for Computer Sciences and Technology.

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'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. ^Some divisions within the center are located at Boulder, CO 80303.
IJJATIONAL BUREAU OF STANDARDS

LIBRARY QC

A Coaxial Noise Standard for the 1 GHz Ig^^

to 12.4 GHz Frequency Range No /oi

William C. Daywitt

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued March 1984
National Bureau of Standards Technical Note 1074

Natl. Bur. Stand. (U.S.), Tech Note 1074, 44 pages (Mar. 1984)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1984

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
CONTENTS

1. Introduction and Description,

2. Noise Temperature and Errors.

3. Conclusions >

4. Acknowledgments

5. References

Appendix A

Appendix B

Appendix C

Appendix D

Bead Support Design Equations

PI ati ng Depth

Noi se Temperature Deri vati on

Noise-Temperature Computer Code (Exact)

Page

1

8

14

14

16

18

24

28

35

m

A Coaxial Noise Standard for the 1 GHz to 12.4 GHz Frequency Range

William C. Daywitt

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

This note describes the design and construction of a coaxial

thermal noise standard. The standard is designed to operate at

the boiling point of liquid nitrogen with a noise temperature

accurate to ± 1 K in the frequency range from 1 GHz to 12.4 GHz.

Key words: coaxial; error analysis; microwave; noise standard.

1. Introduction and Description

The block diagram shown in figure 1 depicts two of the steps involved in

calibrating the spectral power output [1] of a noise source by use of a noise

standard and a noiseless radiometer, where the ratio R of the net spectral

power p^ from the noise source under test to the net power p^ from the

standard is measured by the radiometer. In terms of the reflection

coefficients shown in the figure and the noise temperatures [1] T^ and Tg of

the source under test and the standard.

-mr:

IM'

l^sl

1 - r^ r^

r r r X

1)

In the microwave and millimeter-wave calibration systems presently in use at

the National Bureau of Standards (NBS) [2,3], only the magnitudes of the

reflection coefficients are measured, and a determination of T^ from eq (1) is

not exact since the corresponding phases must be known to calculate the last

factor in the equation. Consequently, in addition to requiring low-reflection

standards, the front ends of the present systems must be tuned to reduce r^

and the resulting errors associated with these unknown phases. However, a 6-

port reflectometer [4] has been incorporated into a new, automated system
standard

Noise

Source

Ts

H

Noise Source

Under Test

Tx

H

^P

INoiseless

Radiometer

Px

Figure 1. Block diagram for a noise calibration,
under development [5] for the frequency range from 2 GHz to 12 GHz, allowing

the phases of the reflection coefficients to be measured in addition to their

magnitudes. This capability obviates the need to tune the system front ends

or the need to build broadband, reflectionless noise standards. A description

of the design, construction, and calculation of the noise temperature for the

standard to be used with this new system is presented in this note.

Although not reflectionless, the noise standard is nominally 50 fi, and

consists of a modified commercial termination immersed in a liquid-nitrogen

bath. The termination is connected to a precision type N (male) connector at room temperature via a coaxial transmission line whose center conductor is

supported by two low-loss beads. Two unique features are incorporated into

the design: to eliminate the strain on the connector when the standard is

attached to the radiometer, the termination and line have been loosely

connected to the cryogenic bath; and the bead supports have been manufactured

from a ceramic material of high thermal conductivity to positively lock the

center conductor to specified temperatures at the bead positions, eliminating

significant heat flow into the termination and condensation on the center

conductor of the output connector. Figure 2 shows an assembly drawing of the

standard and a subassembly drawing of the termination, center, and outer con- ductors. Figure 3 is a photograph of the assembled standard. The line and

termination (items 1, 4, and 6 in figure 2) are fixed to a set of legs (3)

that rest on, but which are not attached to, the cover (15) of the enclosure

(27) housing a vacuum Dewar (25A). The line and termination are thus free to move within the confines of the heating ring (9A) through which water is

circulated to maintain the enclosure cover (15) at room temperature. The

upper, or room-tennperature, bead (34) and line are maintained at room temperature by circulating water through the housing (IB) attached to the

upper part of the line. The environment of the stainless steel, temperaturetransition portion (4) of the line is kept at a reduced temperature by the

copper sleave (21), the height of which is adjusted by rotating it with

respect to the lower barrel (22). This barrel rests on a funnel (25B) which,

by adjusting the legs (23), permit the entire section (21 and 22) to be

centered in the Dewar. The funnel collects boil -off gas and directs it to the

bubble collector (38) which forces the gas up past the lower, fan-shaped bead

(35) into the cavity between the center and outer conductors of the
ft

®^_(2

-a

s-SO

s: +j

(/)

<u

(/) o

c:

<u

CDc

•I—

S- •o

0) <

CVJ

0)

C7>
Figure 3. Photograph of the assembled standard,
temperature-transition part of the line. The lower part of the terminating

element (6) below the bubble collector remains bathed in the liquid nitrogen. Keeping the liquid out of the transition region and away from the top of

the terminating element considerably reduces the reflection from these regions

that would otherwise be present. The standard is usable, and the noisetemperature output equation described in the next section is valid, when the

liquid-nitrogen level resides between the top of the Dewar and the minimum

level shown in the drawing, the boil -off time between these levels being

approximately 5 hours.

A large number of possible bead materials were considered, the primary

requirements being a relatively low dielectric constant (6 or less), a low

loss tangent (0.001 or less), and a relatively high thermal conductivity. The

most suitable materials satisfying the first two requirements were cross- linked polystyrene, fused silica, and the ceramic beryllium oxide. The thermal conductivities of these materials is shown in table 1 along with those of

copper and nitrogen gas [6,7]. The most striking feature of the table is the

high thermal conductivity of beryllium oxide, which even exceeds that of

copper in between the temperatures shown. The effect of these various conductivities on the temperature of the center conductor is illustrated by estimating the temperature T2 of the center conductor at the connector and the

temperature T;^ just below the liquid-nitrogen bead (see figure 2). The

estimates are shown in the last two columns of table 1. Since the temperature

(271 K) of the line above an upper polystyrene bead is less than the dew point

(282 K) in the laboratory, this material was immediately eliminated from con- sideration. With a fused silica bead, the temperature (85 K) below the lower

bead is 9 K greater than the boiling temperature of the liquid and the base of

the terminating element. Thus, there will be a 9 K temperature gradient from

that point to the base of the element, causing a significant error in estimating the noise temperature of the standard. Beryl Ilium oxide was chosen to

eliminate this gradient. Although otherwise very tough and stable, one drawback from using beryllium oxide is that the dust generated while grinding

the material is toxic [8], with the resulting inconvenience of having the

beads manufactured in a grinding shop that handles this type of material. The

equations used to design the beads are found in Appendix A.
(XI

(Ti

CM

CM

CM

r-l I CM I

1*^ 0^

CM

•T3C

to i^ . ^ i^ ^ V r^ t—

t—

•M

«o

(/) f-^(0

•F"I.o;

!-> -t-> W re E >*—«*. V) ixi 3 "^ i>!i o e

•r- ^i-- >

«*- >> o +-> •1— >» > M "r~

+-> > o

3 M -a U c 3 o

•O o :i^ C

o ,— l£) o re r^ E

p_ s_ « a; E j=

l- 1—

o; x:

•M

OJ x:

CO

1^

CT»

CM S O ^ O

<X)

00 <—

o o f^ o

O CD Ooo

re

re o

E=(U

-M

O

CL X re "o 00 o o oj re

'r- j^ cri

•I— 00

S- I—

OJ r— -o

Q. >> CD Q. $_ t/) O <U rj

C_> CO U.

1

O)

1 CD

</) O

(/) s_ o M SC_) 5
The stainless-steel center and outer conductors were gold plated to

reduce their dissipative losses, and thereby increase the accuracy of the

calculated noise temperature of the standard. The plating thickness was determined from an equation derived in Appendix B giving the relative error in

the attenuation coefficient versus depth of plating.

2. Noise Temperature and Errors

An expression for the noise temperature of the standard is derived in

Appendix C, and takes the approximate form

T^ ^ dT^ + AT) (2)

where c is the approximate quantum mechanical correction discussed at the end

of appendix C, T,„ is the temperature of the termination, and AT is a correction noise temperature arising from excess noise contributed to the output by

dissipative losses in the transmission line and room- temperature bead. From

Appendix C

AT = 2.64 X 10"^ f^/^ |r^|^ (T^ - T^) + [0.511 - 2.55 x lO"^ (T^^^ - 76)

+ 3.82 X 10'^ (T - 297)] (0.6 f^^^ + 0.011 f)/0.611 (3)

a

where T^ is the temperature (kelvins) of the water flowing around the upper

bead and line and is close to room temperature; Tg is the measured reflection

coefficient of the standard; and f is the measurement frequency in GHz. T^ is

calculated {± 0.01 K) from the atmospheric pressure P via the vapor pressure

equation [9] (T,^ = T + 0.04, where the 0.04 is due to the average hydrostatic

pressure of the liquid above the termination)

1 Q*! 3

inP = N^/T + N2 + N3T + N^(T^ - T)^-'"'' + H^r

+ HJ^ + N^T^ + NoT^ + N„ ilnT (^^

6 7 8 9
where the pressure is expressed in atmospheres, T is in kelvins, and where T^

is the critical temperature (126.20 K). The coefficients are given in table

2, while a convenient table of the results with the pressure in millimeters of

mercury is given in table 3. The first term in eq (3) involving the reflection coefficient is a correction due to that noise generated by the portion of

the transmission line physically above the room- temperature bead which is

propagated downward and reflected back by the combined reflection of the two

beads and the termination. The upper curve in figure 13 is a plot of eq (3)

with Tg = 0, Tn, = 76 K, and T^ = 297 K.

Table 2. Coefficients for the nitrogen vapor- pressure equation [9]

Coefficient

N4

Numerical value

0.8394409444 x 10 ^

- 0.1890045259 x 10 ^

Coefficient

- 0.7282229165 x 10 1

0.1022850966 x 10"!

0.5556063825 x 10'^

N7

Ng

Numerical valu e

- 0.5944544662 x 10"^

0.2715433932 x

10"'^

- 0.4879535904 x 10"^^

0.5095360824 x 10 ^

'coefficients are for temperature in kelvin and pressure in atmospheres.

Seven sources of error in using eqs (2) and (3) to calculate the noise

temperature arise from: uncertainties in T^, T^, and the dc resistivity of

the conductor walls; uncertainty in the temperature distribution of the con- ductors between the two beads; surface roughness of the conductor walls;

uncertainty in the loss tangent for the room- temperature bead; reflections by

the termination and the beads. The error due to T^ is a result of: uncer- tainties {± 0.01 K) in the vapor-pressure eq (4); uncertainty (± 1 mm Hg) in
determining the atmospheric pressure; variation (+ 3 cm) in the liauidnitrogen head about its average level above the teniination; and an uncertainty in the noise output of the terminating element due to a slight temperature gradient (0.1 K) across its surface. The resulting noise-tenperature

error {+ 0.21%, - 0.07%) is displayed along with the uncertainties in the sources of the error under item 1 in table 4. The uncertainty (+ 1 K) in the

water temperature resulting in a 1 0.02% error in the noise temperature is

displayed as item 2.

Table 3. Liquid-m'trogen boiling temperature in kelvins verus pressure in millimeters of mercury

Temperature in kelvins

Pressure

mm of mercury 1, 2 3 4 5^ 6 1_ 8 9

610 75 .53 75 .54 75 .56 75 .57 75 .58 75 .60 75 .61 75 .62 75 .63 75 .65

620 75,.66 75,.67 75,.69 75 .70 75 .71 75 .73 75 .74 75 .75 75 .76 75 .78

630 75,.79 75 .80 75 .82 75 .83 75 .84 75 .85 75 .87 75 .88 75 .89 75 .91

640 75 .92 75,.93 75,.94 75 .96 75,.97 75 .98 75 .99 76 .01 76 .02 76 .03

650 76 .04 76 .06 76 .07 76 .08 76 .09 76 .11 76 .12 76 .13 76 .14 76 .16

660 76 .17 76,.18 76,.19 76 .21 76 .22 76 .23 76 .24 76 .26 76 .27 76 .28

670 76 .29 76 .31 76 .32 76 .33 76 .34 76 .35 76 .37 76 .38 76 .39 76 .40

680 76 .42 76,.43 76,.44 76 .45 76 .46 76 .18 76 .49 76 .50 76 .51 76 .52

690 76 .54 76 .55 76 .56 76 .57 76 .58 76 .60 76 .61 76 .62 76 .63 76 .64

700 76 .66 76 .67 76,.68 76 .69 76 .70 76 .72 76 .73 76 .74 76,.75 76 .76

710 76 .77 76 .79 76 .80 76 .81 76 .82 76 .83 76 .84 76 .86 76 .87 76 .88

720 76 .89 76 .90 76 .91 76 .93 76 .94 76 .95 76 .96 76 .97 76 .98 77 .00

730 77 .01 77 .02 77 .03 77 .04 77 .05 77 .06 77 .08 77 .09 77 .10 77 .11

740 77 .12 77 .13 77 .14 77 .16 77,.17 77 .18 77 .19 77 .20 77 .21 77 .22

750 77 .23 77 .25 77 .26 77 .27 77,.28 77 .29 77 .30 77 .31 77 .32 77 .34

760 77 .35 77 .36 77 .37 77 .38 77 .39 77 .10 77 .41 77 .42 77 .44 77 .45

770 77 .46 77 .47 77 .48 77 .49 77 .50 77 .51 77 .52 77 .54 77 .55 77 .56

10
Table 4. Noise temperature error components (1 GHz to 12.4 GHz)

Source error Magnitude % error in T,

1. T,V-P equation

Pressure

Head

Gradient

± 0.01 K

± 1 iim Hg

± 3 cm

+ 0.1 K

+ 0.21

- 0.07

2. T. ± 1 K ± 0.02

3. pdc value

Plating depth

± 5%

± 5.6%

r+ 0.06 ^ .1/2

^- 0.026^ ^

4. Surface roughness + 2% + 0.01 f 1/2

5. Loss tangent + 10^ ± 0.0036 f

6. Temperature distribution

7. Internal reflections < 0.1

0.073 f^/2

[[:o:°«)to.o54|r^iifi«

Total error (linear sum) (^ ^'^^] + [(^ q'J^)

± 0.05 |r^|] f^^^ ± 0.0036 f

An uncertainty of ± 5% is assumed for the resistivity, where the error

due to an insufficient depth of gold plating is included. This latter uncer- tainty arises because the attenuation coefficient of the line is calculated

11
assuming solid gold conductors. The resulting uncertainty is estimated in

Appendix B to be ± 6.6%, and combined in the resistivity error leads to the

noise-teinperature error shown under item 3 in the table.

Since the surface of the conductors are not perfectly smooth, there is a greater surface area available to absorb the energy traveling down the transmission line than that assumed for an ideally smooth surface. The effect of

this surface roughness is not great in the microwave frequency region [10],

and is assumed in the present case to cause an attenuation increase of less

than 2%.

The attenuation in the room-temperature bead support is calculated from

the measured loss tangent of the beryllium-oxide bead material. The measure- ment uncertainty is estimated to be no greater than ± 10% of the loss tangent,

and results in the noise- temperature error shown under item 5.

As the liquid-nitrogen level in the standard drops (due to boil off) from

near the top of the Dewar to the minimum point shown in figure 2, the temperature gradient along the inner and outer conductors vary, causing a change in

the excess noise from that portion of the line between the two bead supports. Therefore, since the corresponding noise temperature is calculated

(see eq (C2)) for the average gradient, an error results. The magnitude of

the error is shown under item 6.

Internal reflections due to the two beads and the termination also result

in an error in the calculated noise temperature. By keeping the reflection

coefficient magnitude of the lower bead and termination, and each of the two

upper bead faces below a 0.1, it is possible to keep the resulting noisetemperature error less than the amount shown in the table under internal

reflections.

The total error is defined as the linear sum of the right-hand column in

table 4, and is reproduced in graphical form in figure 4. For example, the

systematic error at 10 GHz for a total 0.3 reflection coefficient is + 0.91%

and - 0.56% of the calculated noise temperature.

12
1.0

1 1 1 1 1 1 1 1 1 1 1 1 1 ^0.3-

0.9

y

y/^y

0.8

+ >/X

0.7 — ^

>-«v g.

1- 0.6 — ___^0,3_

o^

. ^ f\ k. ^ zJr ^0

/> ^

LU //

^^ ^

TsL

<D 0.5 ~

_>

// .^^^^ / y^.^ 1 '*-• '/ y

^^y 05

0.4

1 yyy^

CC

X^ /y^

0.3

y

y^ 1 Tm = 76K

0.2

Ta = 297Kl

1 0.1 —

1

1

\

1 1 5 10

Frequency (GHz)

Figure 4. Relative noise temperature error versus frequency,

13
3. Conclusions

A cryogenic noise standard (fig. 3) has been described whose noise temperature is calculable to an error less than + 1% in the frequency range from

1 GHz to 12.4 GHz.

Dielectric constant/loss tangent measurements have been performed (fig.

5) for the ceramic material beryllium oxide in the temperature range from

100 K to 300 K that had previously been unavailable, and an equation was derived (Appendix B) to estimate the plating depth for a required accuracy in

calculating the attenuation coefficient for a wave constrained to propagate

along a metallic surface.

4. Acknowledgments

The author gratefully acknowledges the support of the following individuals and organizations: Drs. Howard Bussey and Robert Beatty for making some unpublished work on calculating TEj^l ^^^^ resonances available; Dr. Bussey for

providing a computer program (and some patient coaching) and a silver cavity

for determining the dielectric properties of the beryllium oxide; Dan Briggs

of the Coors Spectro-Chemical Laboratory for performing the dielectric

measurements; Mike Kuettner of the NBS shops for some yery fine machining;

Newark Air Force Station for support in manufacturing the standard; and the

Calibration Coordination Group of the Defense Department for support of some early (1972-73), preliminary investigations.

14
6.6

6.5

CC

m 6.4

0-

LU

>

LU

CC

6.3

6.2

F

/

//

Relative

Permittivity'

A

Loss

'/^ Tangent /

o

X

Iz

LU

O

z

CO

C/D

Q

100 200 300

TEMPERATURE (K)

Figure 5. Dielectric properties of BeO versus temperature at 9.3 GHz.

15
5 . References

[I] IEEE Standard Dictionary of Electrical and Electronic Terms. New York,

N. Y.: Wiley-Interscience, a division of John Wiley and Sons, Inc.;

1972.

[2] Wells, J. S.; Daywitt, W. C; Miller, C. K. S. Measurement of effective

noise temperature of microwave noise sources. IEEE Trans, on I & M, IM2: 17-28; 1964 March.

[3] Daywitt, W. C. A reference noise standard for millimeter waves. IEEE

Trans, on MT & T, MTT-21, No. 12: 845-847; 1973 December.

[4] Engen, G. F. An improved circuit for implementing the six-port

technique of microwave measurements. IEEE Trans, on MT & T, MTT-25, No.

12: 1080-1083; 1977 December.

[5] Counas, G. J. 2.0 Griz to 4.0 GHz automated radiometer -- operation and

service manual. Nat. Bur. Stand. (U.S.) NBSIR 83-1697; 1984 January.

[6] Chi Ids, G. E.; Ericks, L. J.; Powell, R. L. Thermal conductivity of

solids at room temperature and below. Nat. Bur. Stand. (U.S.) Monogr.

131, 1973 Septeidber.

[7] Rose-Innes, A. C. Low temperature techniques. D. Van Norstrand

Company, Inc.; 1964.

[8] Ferreira, L. E. Recommended practices for safe handling of beryllium

oxide ceramics. Symposium on materials and electron device processing.

Special Technical Publication No. 300, American Society for Testing and

Materials: 145-155; 1961.

[9] Jacobsen, R. T.; Stewart, R. B.; McCarty, R. D.; Hanley, H. J. M.

Thermophysical properties of nitrogen from the fusion line to 3500 R (1944 K) for pressures to 150,000 PSIA (10342 x 10^ N/M^). Nat. Bur.

Stand. (U.S.) Tech. Note 648; 1973 December.

[10] Allison, J.; Benson, F. A. Surface roughness and attenuation of

precision-drawn, chemically polished, electropol ished, electroplated,

and electroformed waveguides. Proc. IEEE, 102B: 251-259; 1955 March.

[II] Moreno, T. Microwave transmission design data. New York, N.Y.: Dover

Publications; 1958.

[12] Cruz, J. E. Experimental verification of an equation for determining

the effective dielectric constant of center conductor supports in

coaxial transmission lines. Nat. Bur. Stand. (U.S.) Report 9195, 1966

March.

[13] Daywitt, W. C. A broadband coaxial noise source -- preliininary investigation. Nat. Bur. Stand. (U.S.) NBSIR 73-334; 1973 October.

16
[14] Bussey, H. E.; Beatty, R. W. Higher mode resonances of dielectric

support beads in coaxial lines. Private communication,

[15] Jackson, J. D. Classical electrodynamics. Nev/ York, London: John

Wiley and Sons, Inc.; 1962.

[16] Miller, C. K. S.; Daywitt, W. C; Arthur, M. C. Noise standards, mea- surements, and receiver noise definitions. Proc. IEEE, Vol. 55,

No. 6: 865-877; 1967 June.

[17] Ragan, G. L. , ed. Microwave transmission circuits. Radiation Laboratory Series, Vol. 9; 1948.

17
Appendix A: Bead Support Design Equations

The equations collected in this appendix were used to design the two bead

supports for the center conductor. The upper bead {fig. 2) is solid in cross section while the lower bead has a fan-shaped geometry to permit passage of

boil -off gas up into the space between the center and outer conductors. Both were constructed from beryllium oxide whose dielectric properties were carefully measured at 9.3 GHz in the temperature range from 100 K to 300 K, the

results of which are presented in figure 5. The relative dielectirc constants

for designing the room- temperature (297 K) bead and the liquid-nitrogen (76 K) were determined from the figure to be 6.524 and 6.284, respectively.

Room-Temperature Bead

The diameter of the center conductor inside the bead was reduced to maintain a 50-fi characteristic impedance via the equation [11]

where Dq (0.7000 cm) is the inner diameter of the outer conductor, D^ is the

reduced diameter of the center conductor, e^g (6.375) is the effective

dielectric constant of the bead with a slight gap between its inner diameter

and the inner conductor diameter, and e^ and y^ are the free-space electric

and magnetic permeabilities, respectively. The effective dielectric constant

was calculated from the equation [12,13]

^ _ &n(b/a)

iln(a^/a) + — Jln(a2/a^) + - inib^/a^^ + )ln(b/b^)

(A2)

where

n E (1 - N0/2Tr) + (Ne/2ti)e^ . (A3)

The parameters are defined in figure 6, where e^ is the dielectric constant

for the bad material (6.524 at room temperature). For the solid bead Ne is

2tt, h^ is the same as b (0.7 cm) since the bead was press fit into the outer

18
N = 3 = # of Fan Blades = Blade Angle

Figure 6. Geometry of a bead support.

19
conductor, and the inner diameter a^^ of the bead is 0.0008 cm larger than the

inner-conductor diameter (a = D^) to allow for a slip fit. Equations (Al) and

(A2) were solved for a^^ and e^g to obtain 0.0861 cm and 6.375, respectively.

The maximum bead length is determined by the appearance of the first TE-ii

waveguide mode in the bead region for the operating frequency range of the

standard. This length i was determined from the equation [14]

, ^

. 1/2 / \^/\i^ - 1 y/^

tan[|l(e^^-x2/4) .]= (A4)

VVe - 'HJ

where X is the free-space wavelength corresponding to the maximum operating

frequency (12.4 GHz), and X^,^ and \^^ are the cutoff wavelengths [11] for the

TE-j^j^ mode in the bead region (with no bead material present) and the conductor

region, respectively. For the room- temperature bead, the first root of this

equation produced a length of 0.8 cm. The length of bead actually used was 0.45 cm.

The stray step capacitance at each end of the bead where the centerconductor diameter is reduced was eliminated (ideally) by removing some of the

bead material in the immediate vicinity. The length d (cm) of material

removed was calculated from [13]

1.5 C (e' )^/2

d = -—

1

r-^;^^ (A5)

1 - ^re/^e

where C is the step capacitance [11] with the dielectric removed (Mori no's

formula with his e^ replaced by unity), and where the positions of the effective dielectric constants e' and e^ (both calculated from eq (A2) to be

2.022 and 6.375, respectively) are indicated in figure 7. The calculation

produced a length of 0.0866 cm. Figure 8 shows a cutaway drawing of the bead

where the previously discussed dimensions are collected.

20
I Outer Conductor ' !

Figure 7. Equivalent circuit of a bead support.

21
(Dimension in Centimeters)

-0.2738—1

^0.0866

Room-Temperature Bead

0.6985

Liquid-Nitrogen Bead (Room-Temperature Dimensions)

Figure 8. Dimensions for the room- temperature and liquid-nitrogen bead

supports.

22
Liquid-Nitrogen Bead

The process just described was repeated for the lower, fan-shaped bead,

with the results shown in figure 8.

23
Appendix B: Plating Depth

The conductors between the beads were constructed from stainless steel to

reduce heat transfer into the cryogen and terminating element. To eliminate

the corresponding increased loss caused by a lower electrical conductivity,

the conductors were then plated with gold whose conductivity is some ten to 20

times that of stainless steel. Since the plating depth is relatively shallow,

a portion of the fields propagating along the conductor penetrate through the

gold and into the stainless steel. Thus, the loss will be greater than that

calculated assuming solid gold conductors. The purpose of this appendix is to

determine the resulting error as a function of the plating thickness.

Figure 9 shows a portion of a semi-infinite base metal (e.g., stainless

steel) that has been plated with another metal to a thickness of ci to reduce

the attenuation coefficient of the wave traveling along the surface. The magnetic field in free space that is parallel to the surface is denoted by H,

with H^, and H^ being the magnetic fields in the two metals that are induced

by _H. To a first approximation [15], the fields in the metals obey the equations shown in the figure, where n is the unit normal to the surface and c is

the inwardly-directed coordinate. The skin depths (MKS units) are given by

(i = 1,2)

6, =

( ? )^/2 (Bl)

where y^- and o^ are the magnetic permeabilities and electrical conductivities,

respectively, for the two metals, and w is the radian frequency. The fields

have the solutions (a harmonic time variation of e~^^^ is assumed)

(l-j)c/6i -(l-j)^/6,

H^ = A e ^ + B e ^ (B2)

-(l-j)(^-?,)/6^

H2 = C e

^ "^ (B3)

24
Metal

Free Space —H Plating

n-*

e=o ^ = C.

Figure 9. Schematic diagrdin of a plated conductor.

25
where the constants A, B, C are evaluated by satisfying the boundary conditions at the surfaces ? = and ? = ci*

The power loss per unit area is calculated from

f

=lRe (n .E^xH;) (B4)

at the c = surface which, after inserting the fields, reduces to the form

'' - O^ «e 1,1 - J) dA

^ , -2(l-j)Ci/5;

(35)

where

(B6)

The first factor on the right of eq (35) is the power loss assuming the

plating thickness to be infinite. Thus, if the power loss is calculated

assuming a finite thickness, the relative error e is (z = ^i/<Si)

e ^ |Re {{1 - j) [ ]} - 1| = 2(^-^) e"^^ |cos 2z + sin 2z| (37)

= 1.3 e"^^ Icos 2z + sin 2z| . (37')

The "1.3" factor is for gold plating (7.6 x 10' mhos) on 304 stainless steel

(0.38 X 10' mhos) at 186 kelvin. Figure 10 shows a graph of eq (37') which,

for z = 2 (the plating thickness ratio used in the standard), indicates a 3.3%

error in the calculation of the power loss.

It is convenient to include this error as an error in the resistivity,

and, since the power loss is proportional to the square root of the

resistivity, the relative error in the resistivity is 2 e or 6.6%.

26
01

10-2--

10-3 -

10-4

Figure 10. Relative power-loss error versus the ratio of plating depth to

skin depth.

27
Appendix C: Noise Temperature Derivation

The essential features of the line and termination shown in figure 2 are depicted in figure 11a. The lower bead and termination are immersed in liquid

nitrogen, while the upper bead and surrounding conductor are maintained at room temperature by circulating water. The inner and outer conductors have

distinct temperature distributions which vary as the liquid nitrogen level

varies, the inner (outer) conductor distribution being assumed to lie in the

horizontally (vertically) cross-hatched region in figure lib. Above (to the

right in figure Ua) the room- temperature bead, both conductors are at room temperature T^. Figure He is an electrical representation of the standard,

where the lower bead and termination are lumped together in the impedance Zq

(with reflection coefficient r ) since they are both at the same temperature. The compensation regions at the faces of the upper bead are represented

by the T networks shown, while the portion of the bead between these regions

is represented by a length of line. If the step capacitances at both faces were perfectly compensated, then both Bj^ and B2 would be infinite, leaving

only the conductor and dielectric losses included in the resistances of the T

networks. The line representing the bead between these networks includes both

losses, also. The a's appearing in the figure represent noise efficiencies

(available power ratios) [16], a^ [3] being the efficiency of the length of

the temperature-transition portion of the line from point x to the bead at

X]^. There is a separate T^ and o^ for the inner and outer conductor since the

conductors generally have different temperature distributions.

The noise temperature is conveniently written in the form

T, = T^ + ATi + ... + aTc (CI)

s m 1 5

where aT-j^ through aT^ represent the excess noise contributions from to Xj^,

Xj^ to X2, X2 to X3, X3 to X4, and x^ to X5, respectively.

^1

AT^ = [/ T^ (1 - \) dx] 02 03 a^ a^ (C2)

28
MAXIMUM

MINIMUM

LIQUID NITROGEN

'-^y^'- LIQUID NITROGEN RoomTemp

TERMINATION >'^^ I LEVEL Bead

I

Figure 11. Schematic and electrical diagrams for estimating the noise temperature.

29
''h

'- '^a -

^ra' " - "2' «3 % ^ <")

^^^3 - (T^ - TJ (1 - aj) a^ aj (C4)

'^4 = '^a - TJ (1 - "4' % <«)

'^Ts = ^\ - TJ (1 - -5) (C6)

v/here T^ is the temperature gradient corresponding to T^. The term in the

bracket of eq {C2) represents the sum of tv/o similar terms, one each for the

inner and outer conductor, respectively. All of the noise efficiencies except

ag were estimated under the assumption that there were no internal reflections, an approximation that leads to no significant error. After the preceeding equations were combined and reduced to a convenient nonintegral form

by approximating the "exact" computer calculation shown in Appendix D, there

resulted

AT^ + ... + ATg = 2.64 x 10"^ (Tg - T^) |r^l^ f^^^

+ [0.611 - 2.55 X

10'"^ (Tj^ - 76) {C7)

+ 3.82 X 10"^ (T - 297)] (0.6 f^^^ + 0.011 f)/0.611

a

with an accompanying error of

for the 1 GHz to 12.4 GHz frequency range. In evaluating eq (C2), the averages of the extreme limits of the shaded area in figure lib were used for the

inner and outer temperature distributions.

Attenuation coefficients for the various sections shown in figure lie were evaluated according to the following equations: for the conductor losses

[11]

30
I

1.44865 X 10'^ (f e^^ p^.)^''^ a. = ___-^_^^_^^_jj. (C9)

1.44856 X 10 ^ (f e p )^/^

a"^0 = ^' ':|__q (no)

D JlnTDTD.T ^^ '

where a^ and ag are the attenuation coefficients (expressed in dB/cm) for the

inner and outer conductors, respectively, f is the frequency in GHz, e^^ is

the effective dielectric constant, p^- and Pq are resistivities in micro-ohn

centimeters, and Dq and D^ are the outer and inner conductor diameters,

respectively. Pq and p^. are different only when the temperatures of the outer

and inner conductor are different. For dielectric losses, the attenuation

coefficient is (dB/cm)

27.288 e^^ tan6 (An(D,/D,)) (Jln(D /D.))^^^

a

= ^ [ l^J -—-737-2-] ^^1'^^

(Xn(D^/D.) + e^ S.n{D^/D^)y^^

where e^ is the dielectric constant of the material, tanS is the loss tangent

of the material, and \ is the free-space wavelength in centimeters. The

remaining symbols in the equation are indicated in figure 12. The last equation was derived using a static approximation [17], and reduces to the equation given by (loreno [11] when the dielectric completely fills (D^ = Dq) the

space between the conductors.

The magnitudes of the various excess noise temperatures, eqs (C2) through

(C6), are plotted as a function of frequency in figure 13 along with the total

(AT) at the top of the figure. The dashed curves represent the effect of the error in (C8) on the calculation.

In the preceeding development, the temperatures T^ and T^ appearing

explicitly in eqs (CI) through (C6) were, for convenience, assumed to be the

thermodynamic or physical temperatures of the reservoir and transmission line.

31
Outer Conductor

Dielectric

Sleave

Center

Conductor

Figure 12. Schematic diagram of a center and outer conductor, and a dielectric sleave.

32
LU

DC

<

DC

LU

CL

LU

O

LU

DC

DC

OO

FREQUENCY (GHz)

Figure 13. Various components of the noise temperature versus frequency

33
respectively. To be exact, both of these temperatures should be replaced by

their corresponding radiation temperatures, where the radiation temperature

corresponding to a physical temperature T is calculated from the equation

radiation hf/kT

e - 1

where h is Planck's constant, and k is Boltzmann's constant. An adequate

approximation to eq (CI) for the 1 GHz to 12 GHz range where the thermodynamic

temperatures are retained is given by

Tn = ^(T„ + AT, + ... + AT.) {C13)

n m 1 o

where

e^ - 1

X E -Jl . {C15) kTm Neglect of the correction factor c in eq (C13) leads to an error varying from

0.05% to 0.4% in the frequency range under consideration.

34
I

Appendix D: Noise-Temperature Computer Code (Exact)

Following is a copy of the computer program used to perform an "exact"

calculation of the noise temperature from which the approximation given by eq

(07) was derived.

35
83/08/26. 13.27.52.

PriOCRAivj TCAl

1 REM TCXl. NOISE TEMP FOR COAX

2 REM 9/10/82

4 DIM A(2000)«B(2000)>C(2000)

5 DIM V(100)>V.(100)>A(2000>«1(2000)#U(2000)

7 GOSUB1000

9 GOSUB7000

10 GOSUB6000

20 GOSUB3000

48 PRINT

50 PRINT ••Ll*L2*L3"

52 PRINT "LT*T2*13"

bA PRINT "Ul»U2*D3*DA"

56 PRINT "ER£»EREP#L4#L5"

60 PRINT L1*L2*L3

62 PRINT G#12#T3

6A PRINT Dl«02>D3>iJ4

66 PRINT E1*E2*L4*L5

6 9 PRINT

70 PRINT "Al <DB)*DTr'

72 PRINT ••A2L*A2U*A2S*D12"

74 PRINl ••A3L*A3D*D13"

76 PRINT "A^L^A^D^A^ii^DT^"

76 PRINT ••A5>DT5"

60 PRINT -10*L( T (Al )#H1

82 PRINT Bl#B3*B2*h2

84 PRINT B4*B5*H3

86 PRINT Bl#33*d2*H4

88 PRINT -10*LGT<A5)>h5

89 PRINT

90 PRINT ••XMs"T0*"TA="T3*"DT="H

99 PRINT

100 tj)5=T3-T0

102 W6=H/(T3-T0)

105 PRINT ••TA-TMs"u5*"<TN-TM)/<TA-TM)="bl6

999SI0P

1000 REM CONSTS

1005 N5=2000

1010 C0=1 .44866E-4

1020 F=12.4

1030 Ol=.304

1040 D2-.7

1060 C2>LOC(10>/10

1070 N6aN5-l

1080 O3-.08534

1082 D4=.7

1086 £1*6.375

1088 £2-2.022

1090 L4S.2738

36
1090 L4S.2738

1092 L5=.0866

1094 L2=L4+2*Lb

1096 G=. 000353

1098 L6=<Ul-D3)/2

1099 D5=(Dl+D3)/2

1 1 10 L3=4.7

1499 «E1URN

2000 HEM RESISTVY CALC

2010 Rs-. 17*.008051*T

2099 RETURN

2100 REM All.^ AND EFF

2110 Cl=C0*SuR<F)/LUG(D2/Dl)

2130 FOR 11=1 TO N6

2140 T=(T<I1 )"^T<Il + l))/2

2 150 t-OSUB2000

2160 ACIl )=Cl*StiR(R)/Dl

2170 T=<U<I1 )+U(Il+l ))/2

2180 GOSUB2000

2190 8(11 )=C1*SUR<R)/D2

2200 NEXT II

2210 Q1=X(N5)/N6

2220 Q2=(A(N6)*B<N6>)*Ql/2

2222 C(N6) = l/10r (£.2/10)

2229 N7=N5-2

2230 FOR 11=1 TO N7

2240 y2 = (A(N5-Il ) +8 (N5- 1 1 ) ) b(l/2

22 50 y2 = t-2+<A(N6-Il ) 3(N6- 1 1 ) ) +^1/2

2260 y2=l/10T(u2/10)

2270 C<N6-I1 )=C(N5-I1)*U2

2280 NEAT II

2290 A1=C(1 > 2300 REM A2

2305 T=T2

2307 GOSUB2O00

2310 C1=C0*SQR(F*R*E2)

2312 B1=C1*(1/D3+1/D4)*L5/L0G(D4/D3)

2314 B2=C1*L6/D5/LOG<04/D5)

2316 B3=27.268*S&R<E1 )*G*L5*F/29. 9793

2318 B3=B3*L0G(Dl/D3)*SyR(L0G<D4/D3))

2320 B3=B3/<<L0G(01/D3)+El*L0t(U4/Dl))r 1 .5)

2324 A2=B1+B2+B3

2325 A2=l/10t(A2/10)

2328 A4=A2

2330 REM A3

2332 Cl=C0*Sb>R<F*R*El)

2334 B4=C1*(1/U3*1/D4)*L4/L0G<D4/D3)

2336 85=27. 288*G*Sb(R(El ) L4*F/29. 9793

2336 A3=B4*B5

2340 A3=l/10t<A3/10)

2350 REM A5

2352 T=T3

37
2352 T=T3

2354 tOSUB2000

2356 Cl=C0*SaR<F*R)*L3

2 358 A5 = C 1 d/Dl + 1/02) /LOG <D2/D1 ) 2360 A5=l/10t(A5/10)

2499 RETURN

3000 REM T.OUl

3005 GOSUB2100

3010 S3-0

3030 FOR 15=1 TO N6

3040 S3=S3*(T<I5)*A(I5)*U<I5)*8(I5))*C(15)

3050 NEAT 15

3060 S3=S3*A<N5)/N6

3065 S3=C2*S3

3070 H1=S3*A2*A3*A4*A5

3072 H1=H1-T0*(1-A1 )*A2*A3*A4*A5

3080 H2=(T2-T0)*(1-A2)*A3*A4*A5

3090 H3=<12-T0)*<1-A3)*A4*A5

3100 H4=(T2-T0)*(1-A4)*A5

3110 H5=(T3-T0)*C1-A5)

3120 Hs:hH-H2"t-H3*h4+H5

3199 RETURN

6000 REM INPUT DATA READOUT

6010 RESTORE

6020 READ N9

6030 PRINT ••FREQ = "F

6032 PRINT

6070 PRINT ••a"*"T"

6071 PRINT •• INNER"

6080 FOR 17=1 TO N9

6085 READ \J»V.

6090 PRINT y/#W

6100 NEXT 17

6110 PRINT •• OUTER"

6118 READ N9

6120 FOR 17=1 TO N9

6130 READ V#W

6140 PRINT V#W

6150 NEXT 17

6299 RETURN

7000 REM DIST CALC

7005 RESTORE

7010 GOSUB7500

7020 &=V<N9)/N6

7025 X<1)=0

7030 FOR J9=2 TO N5

7040 X<J9)=X< J9-1 )(»(

7050 NEAT J9

7060 18=2

7 064 S9=(V^(2)-WC1))/(VC2)-V<1)*1E-10)

7068 T<1)=U<1)

7070 FOR 19=2 TO N5

38
7070 FOR 19=2 TO N5

7080 IF X<I9)<V(I8) THEN 7090

7082 18=18+1

7083 IF X<I9)>V(I8) THEN 7082

7084 S9=<W(I8)-W<I8-1 ))/<V(I8)-V<I8-l)+lE-I0)

7086 T<I9)=U<I8-1)+S9*(X<I9)-V(I8-1)>

7088 C-OTO7I00

7090 T(I9)=1(I9-1)+S9*U

7092 T2=T(N5)

7094 r3=T2

7100 NEXI 19

7 110 COSUB7500

7120 18=2

7 130 S9=(U(2)-V.<1))/(V(2)-V<1)'«'1E-10)

7 140 U(1)=V.(1 ) 7150 FOR 19=2 TO N5

7160 IF X<I9)<V(I8) THEN 7190

7162 18=18+1

7163 IF X<I9)>V(I8) THEN 7162

7164 S9=(KI8)-kCI8-l ))/<V(I8)-V<I8-l) + lE-10)

7166 U<I9)=W<I8-l)+S9*(XCI9)-V<I8-n)

7188 GOTO7200

7190 U<I9)=U<I9-l)+S9*tl

7200 NEXT 19

7499 RETURN

7500 REM READER

7510 READ N9

7520 FOR J9=l TO N9

7530 READ V<J9)

7540 READ V.(J9)

7560 NEXI J9

7562 L1=V(N9)

7564 T0=Vi<l ) 7599 RETURN

9000 REM DATA FORMAT: INNER THEN OUTER COND AS N*X1*Y1*

9002 REM XI =0 & LAST X SAME FOR IN & OUl CUND

9003 REM X*S IN CM

9010 DATA 3

9012 DATA 0«76«1.6#76

9014 DATA 10.6

9016 DATA 297

9100 DATA 4

9104 DATA 0#76«3.2«76

9106 DATA 10.U297«10.6#297

9999 END

39
83/08/26. 13.43

PROGRAM TCXl

18.

FREQs 12.4

INNER

76

1 .6 76

10.6 297

OUTER

76

3.2 76

10. 1 297

10.6 297

L 1 * L2 * L3

LT*T2>T3

D\,D2»D3»bA

EHE*EREP*L4i L5

10.6 .447 4.7

.000353 297. 297.

.304 .7 .08534

6.375 2.022 .2738

Al <DB)>D1 1 A2L*A2D*A2S* DT2

A3L*A3L>>DT3

A4L*A4U>A4Sj DT4

A5#DT5

3.19245E-2 .80176

5.84851E-4 9.49576E-•5 4.74420E-4

3.28330E-3 2.75436E-3 .305519

5.84851E-4 9.49576E-•5 4.74420E-4

2.02131E-2 1.0262

.7

.0866

5.83582E-2

5.84549t-2

TM= 76 TA= 297. DT3 2.25029

TA-TM= 221. (TN-TM)/<TA-TM)= 1.01823E-2

40
NBS-n4A IREV. 2-8CI U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS TN-1074

2. Performing Organ. Report No 3. Publ Icacion Date March 1984

4. TITLE AND SUBTITLE

A Coaxial Noise Standard for the 1 GHz to 12.4 GHz Frequency Range

5. AUTHOR(S)

W. C. Daywitt

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as #6.

10. SUPPLEMENTARY NOTES

1^^ Document describes a computer program; SF-I8S, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gnificant

bibliography or literature survey, mention it here)

This note describes the design and construction of a coaxial

thermal noise standard. The standard is designed to operate at

the boiling point of liquid nitrogen with a noise temperature

accurate to ± 1 K in the frequency range from 1 GHz to 12.4 GHz.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

coaxial; error analysis; microwave; noise standard

13. AVAILABILITY

Qj Unl imited

I I For Official Distribution. Do Not Release to NTIS

[Xj Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

1

] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

44

15. Price

USCOMM-DC 5043-Pe'^ * U.S. Government Printing Office 1984 - 778-586/9228 Reg. 8



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