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I" NBS TECHNICAL NOTE 1071

100

. U5753

No. 1071

1933
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

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JNATIONAL EUr.L Of st/,nd;ju f- UBRARy

fidDesign and Error Analysis for the WR10 u%?5£

Thermai Noise Standard '°^'

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 December 1983
National Bureau of Standards Technical Note 1071

Natl. Bur. Stand. (U.S.), Tech Note 1071, 45 pages (Dec. 1983)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1983

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

Page

1. Introduction

2. Antenna noise efficiency and construction of the horn.

3. Construction and operation of the cavity

4. Results, errors, and conclusions

5

.

Acknowl edgments

6. References

1

3

7

814

14

Appendix A:

Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G

The noise efficiency and noise temperature of an antenna

in a reflectionless, isothermal cavity

A simplified expression for the noise efficiency,

Evaluation of ri , Surface roughness

Higher-mode contami nation , Cavity-radiation error

Multiple reflections ,

1622

24

27

3133

39

m

Design and Error Analysis for the

WRIO Thermal Noise Standard

Will iam C. Daywitt

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

This note describes the design and error analysis of a WRIO thermal noise power

standard. The standard is designed to operate at the boiling point of liquid nitrogen

with a noise temperature accurate to + 1 K.

Key words: antenna efficiency; diffraction; error analysis; millimeter wave; noise

standard; plane-wave scattering matrix.

1. Introduction

Over the past twenty years, the Electromagnetic Fields Division of the National Bureau of

Standards (NBS) has built a number of coaxial and waveguide noise sources [1-3] consisting of

single-mode, uniform transmission lines terminated in reflectionless loads. The accuracy of their

calculated noise temperatures is typically 1% which tends to degrade as the operating frequency

increases. The basic design is illustrated in figure 1 where the termination and a portion of the

transmission line are immersed in a thermal reservoir at temperature 1^, with the remaining portion

of the line leading to the output connector at room temperature T^. The temperature distribution T^ of the line is also illustrated where the room-temperature portion of the line has a length Z. Radiation from the termination and the dissipative losses of the line result in a noise temperature

T, =T^+AT (1)

where, for the idealized distribution shown,

AT = (2a'Jl) (Tq - T^). (2)

The attenuation coefficient a' refers to the line at Tq, and the equations indicate that only that

portion of the line contributes to the excess (in excess of thermal equilibrium conditions) noise

temperature AT.

The largest source of error in calculating the noise temperature by eq (1) is the attenuation

2a'l, which is usually estimated to an error varying from lOX to 20X. With this large an error, it

is necessary to keep the attenuation small (to maintain the error in T^ less than 1%), implying

either a small attenuation coefficient a', a short transition length ^, or both. In the microwave

frequency range and below, the attenuation can be kept down with relatively simple engineering

designs; but as the frequency increases into the millimeter-wave range, this becomes more of a problem [3]. In this higher frequency range, surface roughness also plays a bigger role [4],

causing an additional increase in the attenuation.
Hot or Cold

Reservoir

Output

Connector-

-Ternninating | Load

Figure 1. Schematic diagram of a transmission-line type noise standard.

To circumvent the engineering difficulties, it was decided to abandon the transmission-line

type of noise standard in favor of a design incorporating a millimeter-wave horn antenna "looking"

at an absorber of known temperature. The resulting antenna noise temperature is close to the

measured temperature 1^ of the absorber, with additional noise contributions from the dissipative

antenna losses, and from the antenna side and back lobes. This type of noise source is not new, but

its use as a primary reference standard of high accuracy required the successful resolution of two previously unanswered questions: can a useful expression for the noise efficiency (defined in

Appendix A) of the antenna with sources in its radiating near field [5] be found; and can the error

caused by near-field excess radiation (in excess of thermal -equil ibri um conditions) entering the

side and back lobes of the antenna be estimated?

The antenna noise efficiency derivation described in the next section and in Appendix A is

based upon the plane-wave scattering-matrix (PWSM) theory of antennas [6]. This efficiency, because

of reciprocity, reduces to the antenna radiation efficiency. However, in contrast to the usual

definition [7] of the radiation efficiency, the PWSM description reveals enough of the structure of

the efficiency to permit a detailed calculation and error analysis of its magnitude. The PWSM

formalism is used solely to derive an expression for the antenna noise efficiency and excess noise

temperature with the cavity in the antenna radiating near field, and to show how the efficiency can

be evaluated in terms of antenna far-field quantities.

With the noise efficiency defined in Appendix A, insights gleaned from the Uniform Theory of

Diffraction (UTD) [8] are used in Appendix B to eliminate its angle-dependent components. The error
incurred from neglecting these contributions to the efficiency is then estimated. The residual,

angle-independent part of the efficiency is estimated in Appendix C by substituting the rectangular

waveguide expression for the absorption coefficient, changing the cross-sectional dimensions to confonm to the interior dimensions of the horn antenna. The waveguide expression is used because of

the nonexistence of pyramidal horn mode equations. An estimate of the error caused by this

substitution is also presented in Appendix C.

Effects of surface roughness on dissipative loss are reviewed in Appendix D where an attempt is

made to clarify some disagreements found in the literature [9,10]. A modified expression for the

corresponding noise efficiency is also presented.

Due to more intense fields in the waveguide portion of the horn, dissipative loss is greater

there than in the horn flare. Consequently, the horn waveguide lead is made as short as possible

without introducing a significant amount of higher-mode contamination in the antenna noise

temperature. The minimum allowable length is determined in Appendix E along with an estimate of the error due to these higher modes.

The UTD is used in Appendix F to estimate the noise temperature error due to the cavity wall

temperature being greater than the temperature of the absorber, and, in Appendix G, to estimate the

effect of multiple reflections between the horn and rear cavity wall. Appendix F sets an upper

limit to the magnitude of the excess cavity noise, while Appendix G justifies (by examining multiple

reflections between the horn aperture and rear cavity wall) separating the total excess noise into a sum of excess antenna and cavity noises.

The design equations and construction details will be described in a companion document.

2. Antenna noise efficiency and construction of the horn

The antenna noise temperature T^ is determined from eq (1) with a different excess noise

temperature AT than that given in eq (2) for transmission-line type noise standards. This new correction temperature has the form

AT - (1 - a) (Tq - T^^) (3)

for an antenna inside a reflectionless, isothermal cavity, a is the noise efficiency defined in

Appendix A, Tq is the physical temperature of the antenna, and T^ is the physical temperature of the

cavity. An equation for the efficiency, when the cavity is in the radiating near field of the

antenna, is derived by use of the antenna scattering matrix [6] and the Clausius statement of the

second law of thermodynamics [11]. The details of the derivation given in the appendix reveal that

the cavity walls need not be strictly reflectionless, only that no multiple reflections take place

between the cavity and the antenna (an important distinction that is utilized in constructing the

standard)
One interesting and useful feature that comes from the derivation is that the efficiency is the same whether the radiating sources (the cavity) are in the antenna near or far field. This result

allows the efficiency to be more easily evaluated with far-field quantities.

1 r '^ ^ cT- i

rE'(r) ^

rE(r) P^(Q)dQ (4]

and is obtained by combining eqs (A19), (A26), and (A27) from the' appendix--noting that the reflection coefficient of the antenna does not depend significantly (Sqq = Sqq) on the antenna loss. In eq

(4), Qg is the antenna solid angle, r_ is the radius vector from the antenna aperture to the farfield point (r is the magnitude), Pp(Q) is the normalized power pattern where Q stands for the

antenna pointing angles, dQ is the differential solid angle, and rE_(H is the E-field pattern. The

prime refers to the real (lossy) antenna, the unprimed quantities belonging to the same antenna with

no losses.

Ideally, the pattern ratio in eq (4) could be measured, or calculated from Maxwell's equations,

but the measurement is too inaccurate to be useful and the calculation is prohibitively difficult.

However, since most of the loss comes from the waveguide and flare portions of the horn (Appendix

B) , the efficiency is easily calculated (Appendix C) if losses from the remainder of the horn are neglected. Figure 2 shows an isometric view of the horn designed for the WRIO frequency band

(75 GHz to 110 GHz) to take advantage of this situation. Reflections from the waveguidfe-flare

junction, or throat, are minimized by joining the waveguide and flare with a circular arc of sufficiently large radius [12], beginning at Xj^ and ending tangentially on the flared walls. A quarterround (3 wavelength radius) matchiag section [13] at the aperture, in addition to a large aperture

cross section [14], minimize the aperture reflections. With multiple reflections interior to the

horn eliminated, a wave entering the horn flare from the waveguide maintains its TE-^Q-mode

configuration out to the aperture.

Before including the throat taper and the aperture matching section into the design, and after

choosing the E-plane aperture dimension, Braun's equations [15] were used to determine the other

aperture dimension and the flare angles and lengths to insure approximately equal E- and H-plane

beam widths and a simple butt joint at the waveguide-flare junction. The waveguide length x^ was chosen (Appendix E) to minimize the effects of higher modes (generated by radiation incident on the

antenna from the cavity) on the antenna noise temperature. Finally, a water jacket was included

around the waveguide-throat region to maintain this high-loss area at a constant known temperature.

The horn attenuation calculated in Appendix C neglects the effect of surface roughness on the

losses, predicting a loss that is less than the actual value. In Appendix D, it is argued that, for

an isotropically rough surface, the horn attenuation coefficient increases by a factor K, which is

independent of position inside the horn--depending only upon the roughness of the surface and the

operating frequency. Furthermore, it is reasonable to assume that K is independent of

temperature. These considerations imply that K can be determined by comparing the measured and

calculated (assuming no surface roughness) attenuation for a waveguide section with the same surface

roughness as the horn. A picture of the horn with the four waveguide sections and quarter-wave
Figure 2. Isometric view of the WRIO horn antenna.

Figure 3. WRIO horn antenna, waveguide sections, and quarter-wave short.

5
short (94.5 GHz) used to determine K is shown in figure 3. The horn was constructed by machining a stainless-steel mandril to the interior horn dimensions, flashing it with gold, and electroforming

copper on top of the gold. After electroforming and machining, the mandril was removed, leaving the

horn shown in the figure. The process was repeated to produce the four waveguide sections with the same interior surface roughness as the horn. Then, after determining K for the waveguide section,

the efficiency of the horn was calculated from

-2K/a'dz

(5)

where the attenuation coefficient a' (Appendix C) is a function of the dimensions and temperature of

the horn at position z along the horn axis. The integral is performed over the length of the horn.

Figures 4, 5, and 6 show the results of the calculations with K = 1. The attenuation

coefficient a' for 75 GHz and 110 GHz is plotted as a function of z along the horn axis, with the

waveguide flange of the horn at the origin. The waveguide portion of the horn extends from z = to

0.04

0.03

E

m

T3

g 0.02 O

co

o

CO <

0.01

-Waveguide

Flare

WR10 HORN/E-Plane Section

(scale 1/1)

Water

Jacket

Figure 4. Attenuation coefficient for the WRIO horn antenna.

6
90

Frequency (GHz)

Figure 5. Attenuation versus frequency for the waveguide, throat, and flare portions of the WRIO

horn antenna

z = 0.31, where the transition to the flare starts. The aperture and the beginning of the quarterround matching sections are at z = 7.0, and the water jacket extends from the flange to z = 3.6.

Figure 5 shows the total horn attenuation (the exponent of eq (5) with K = 1) as a function of

frequency, leading to the excess noise temperature shown in figure 6.

3. Construction and operation of the cavity

An assembly drawing of the WRIO noise standard is shown in figure 7. The horn antenna is

suspended at the top of the cavity by a yoke (items 18A and 18B) that attaches it to a flexible,

beryllium-copper membrane (17) which fits loosely in a holder (7) that allows approximately 8 mm

side motion. This motion, with the flexibility of the membrane, permits the horn waveguide flange

to be connected to a radiometer without supporting the weight of the entire standard, greatly reducing problems associated with flange misalignment. The cavity consists of the yoke and membrane, the

silicon-carbide absorber (13), and the load holder (14). The inside cavity walls are polished and

gold flashed to reduce thermal radiation from their surfaces. Two radiation shields (6C, 6E) help

reduce the amount of external radiation entering the cavity. The bottom portion of the cavity is

immersed in liquid nitrogen, allowing the liquid to leak through the bottom of the holder and be

absorbed by the silicon carbide. The liquid level is maintained between the maximum and minimum

levels shown in the figure. Millimeter-wave absorber (16) is inserted between the bottom of the

load holder and the vacuum flask (12) to absorb radiation entering the flask from outside the

standard. By using this absorber, the radiation temperature [16] of the radiation entering the

cavity from the flask area is reduced from approximately 300 K to 2 K above the boiling temperature

(77 K) of the liquid. Once the flask is filled to the maximum level, it takes approximately seven

7
'_>

^_

CC

^_<DQ.

E

(D

CO

CO

COo

o

X

LU

90

Frequency (GHz)

100 110

Figure 6. Correction noise temperature for the WRIO noise standard.

hours for the liquid to boil down to the minimum usable level. Figure 8 is a photograph of the

completed assembly.

The silicon-carbide absorber was manufactured from commercially available grinding-wheel stock

(grad P, grit C320, vitrified bond, silicon carbide) by cutting the stock into rectangular shapes

and sharpening the ends. The pieces were then fastened together and the assembly ground to form the

cylindrical shape shown in figure 9. When inserted in the holder (fig. 10), the cohesive action

between the silicon-carbide particles and the liquid nitrogen draws the liquid up into the wedges,

allowing the liquid to boil off on the wedge surfaces. With this technique, the temperature of the

radiating surface of the absorber is maintained at the boil-off temperature of the liquid nitrogen

to within 0.2 K, independent of the level of the liquid in the flask.

4. Results, errors, and conclusions

Equations (1), (3), and (5) are used to calculate the noise temperature Tp of the standard,

where T^j, is the measured temperature (liquid nitrogen boil -off temperature) of the absorber, Tq is

the temperature of the horn antenna (room temperature), a is the noise efficiency of the horn, K is

the roughness factor, and a' is the absorption coefficient at position z along the horn axis. Six

8
mftJi . tfi/gj ii<Siu.rJ> V2

/TJ/V. L£ i/£L LIQUID V^

Figure 7. Assembly drawing of the WRIO noise standard.

9
Figure 8. WRIO noise standard.

10
Figure 9. Silicon-carbide absorber.

possible sources of error accompany the use of these equations: (i) higher-mode contamination of

the noise temperature, (ii) multiple reflections between the horn and rear cavity wall, (iii) wall

temperatures greater than the absorber temperature, (iv) uncertainty in T^, (v) uncertainty in Tq,

and (vi) uncertainties in the noise efficiency. These sources of error are discussed below and sum- marized in table 1 along with the resulting errors they produce in the output noise temperature.

The length of the waveguide portion of the horn was chosen to adequately filter out the higher

waveguide modes generated in the horn flare by radiation from the cavity incident on the horn. The

calculations described in Appendix E show, that for 0.31 cm, the residual contamination amounts to

no more than + 0.12% of the noise temperature.

11
Multiple reflections between the back wall of the cavity and the horn affect both the validity

and magnitude of the correction noise temperature in eq (3). Calculations in Appendix G show the

discrepancy in the magnitude to be insignificant, indicating that the assumption of a reflectionless

cavity in the derivation of eq (3) leads to no larger an error than - 0.05%.

The temperature of the cavity back and side walls varies from room temperature at the top of

the cavity (fig. 7) to liquid-nitrogen temperature where the side wall meets the absorber. The

elevated wall temperatures cause the noise temperature to be in excess of the load temperature T^. The resulting error (Appendix F) is no larger than - O.IOX.

The temperature of the absorber wedges was found to be within 0.1 K (the experimental error) of

the liquid-nitrogen boil -off temperature (which was measured with the same thermocouple). When the

standard is in use, the absorber temperature is determined by reading the atmospheric pressure

(± 1 mm Hg) off a precision barometer and converting to the boil-off temperature (± 0.14 K) via

standard vapor-pressure equations [17]. Care was taken in constructing the standard to insure a positive flux of nitrogen boil -off gas from the enclosure, preventing contamination of the liquid

nitrogen by atmospheric gasses, and allowing the equations to be used to an assumed accuracy of

0.02 K. The total error (0.26 K) in measuring the absorber temperature is the sum of these three

errors. This causes an error of less than 0.341. in the noise temperature.

Table 1. Errors in the WRIO (75 GHz - 110 GHz) noise standard

Resulting Percentage

Source of Error Source Uncertainty Error in Tp

1. Higher modes —

+0

2. Multiple reflections between horn & cavity — - 0.05

+0

3. Elevated cavity-wall temperature • — - 0.10

4. Uncertainty in T^ + 0.26 K ± 0.34

5. Uncertainty in Tq + 2 K + 0.02

6. Uncertainties in a

+ 6.1 Neglecting losses beyond aperture — - 0.01

6.2 Using waveguide loss equation — ± 0.01

6.3 Dimensional uncertainties ± 0.0025 cm ± 0.06

6.4 Uncertainties in dc resistivity curve ± 5% + 0.01

6.5 Uncertainties in K ±5% + 0.06

Total error (linear sum) + 0.5 %

- 0.7 7o

12
The temperature of the horn, reduced slightly by cooling from the liquid-nitrogen boil-off gas,

is determined by circulating room-temperature water through the water jacket (fig. 2). The water

temperature is measured before entering the horn by use of a precision (+ 1 K) mercury thermometer,

and the temperature of the high-loss portions of the horn is depressed no more than 1 K by the boiloff gas. The horn temperature error is, therefore, no larger than 2 K, causing an error of less

than 0.02% in the noise temperature.

Errors in the noise efficiency fall into two classes: errors due to the approximations leading

to eq (5), and errors due to uncertainties in the parameters used in the equation. The first class

contains an error due to neglecting dissipative horn losses beyond the aperture (Appendix B),

resulting in a noise temperature error less than a negative 0.01%; and an error due to the nonexistence of equations describing pyramidal horn fields (Appendix C) resulting in an error no larger

than 0.01%. The parametric errors are due to uncertainties (+ 0.0025 cm) in the internal horn

dimensions, in the slope versus temperature curve for the dc resistivity of the horn walls (± 5%),

and in the roughness factor K. The first two result in noise-temperature errors of less than 0.01%

and 0.06%, respectively.

Figure 10. Top view of the cavity.

The roughness factor at 297 K was determined by comparing the measured and calculated values of

attenuation for the waveguide sections shown in figure 3. Errors in the roughness factor are due to errors (+ 0.0012 cm) in the measured dimensions of the waveguide sections, and uncertainties asso- ciated with the six-port millimeter-wave system [18] used to measure the attenuation. The roughness

factor was determined to be 1.14 (+ 0.05) at 94.5 GHz, with an error no larger than 0.06% in the

noise temperature.

13
The sum of errors in table 1 for the noise-temperature output of the noise standard is within

1%, showing that it is possible to construct a precision horn/absorber type of standard, and, thus,

eliminate the engineering problems mentioned in the introduction. Furthermore, this type of

standard can be easily duplicated at the higher millimeter-wave bands.

Calculations using the UTD show that, if the yoke and membrane (items 17, 18A, and 18B in fig.

7) are removed from the standard, the noise temperature increases by approximately 0.71.. If, in

addition, the aperture-matching quarter-rounds (the curved sections on the horn aperture in fig. 2) are removed from the horn, this figure increases from 0.7X to approximately 3%. Therefore, the yoke

and membrane are needed to insure an accurately known noise temperature.

5. Acknowledgments

The author takes great pleasure in acknowledging the many beneficial contributions from his

colleagues in the Electromagnetic Fields Division who unhesitatingly shared their expertise in a number of seminars and many private discussions.

6. References

[1] Arthur, M. G. ; Driver, L. D. A precision HF-noise power measurement system. ISA

Transactions, j^: 264 - 268; 1971.

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

microwave noise sources. IEEE Trans, on I&M, I M- 1: 17 - 28; 1964 March.

[3] Daywitt, W. C. WR15 thermal noise standard. Nat. Bur. Stand. (U.S.) Tech. Note 615; 1972

March.

[4] Tischer, F. J. Experimental attenuation of rectangular waveguides at millimeter

wavelengths. IEEE Trans, on MTT, MTT-27 , No. 1: 31 - 37; 1979 January.

[5] Hansen, R. C. "Aperture Theory," Microwave Scanning Antennas , Vol. 1., New York, NY: Academic Press; 1964.

[6] Kerns, D. M. Plane-wave scattering-matrix theory of antennas and antenna-antenna

interactions. Nat. Bur. Stand. (U.S.) Monogr. 162; 1981 June.

[7] IEEE standard definitions of terms for antennas. IEEE Std. 145-1973.

[8] The modern geometrical theory of diffraction. Vol. 1, Electro-Science Laboratory, The Ohio

State University; 1982.

[9] Benson, F. A. Waveguide attenuation and its correlation with surface roughness. Proc. lEE,

100 , Part III: 85 -90; 1953 March.

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

polished, electropol ished, electroplated and electroformed waveguides. Proc. lEE, 1Q2B : 251

- 259; 1955 Nbrch.

[11] Allis, W. P.; Herlin, M. A. Thermodynamics and statistical mechanics. New York, NY: McGrawHill Book Company, Inc.; 1952.

[12] Terzuoli, A. J., Jr.; Peters, L. , Jr. VSWR properties of E-plane dihedral corrugated horns.

IEEE Trans, on A & P, AP-26 , No. 2: 236 - 239; 1978 March.

[13] Burnside, W. D. ; Chuang, C. W. An aperture-matched horn design. IEEE Trans, on A & P, AP-30 , No. 4: 790 - 796; 1982 July.

14
Jull, E. V. Reflection from the aperture of a long E-plane sectoral horn. IEEE Trans, on A &

P, AP-20, No. 1: 62 - 68; 1972 January.

Braun, E. H. Some data for the design of electromagnetic horns. IEEE Trans, on A & P, AP4: 29 - 31; 1956 January.

Ko, H. C. Antenna temperature and the temperature of electromagnetic radiation. IEEE Trans,

on A & P, AP-12 : 126 - 127; 1964 January.

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 (10343 x 10^

N/m^). Nat. Bur. Stand. (U.S.) Tech. Note 648; 1973 December.

Weidman, M. WR-10 single six-port measurement system. Nat. Bur. Stand. (U.S.) NBSIR 81-1650;

1981 September.

Yaghjian, A. D. Near-field antenna measurements on a cylindrical surface. Nat. Bur. Stand.

(U.S.) Tech. Note 696; 1977 September.

f^orse, P. M.; Feshbach, H. F. Methods of theoretical physics. Part II. New York, NY: McGraw-Hill; 1953. p. 1764.

Middleton, D. An introduction to statistical communication theory. New York, NY: McGraw-Hill

Book Company, Inc.; 1960.

Burkhard, D. G. ; Lockhead, J. _V. S.; Penchina, C. M. On the validity of Kirchhoff's law in a nonequilibrium environment. Amer. J. Phys., Vol. 40: 1794 - 1798; 1972 December. Baltes, H.

P. On the validity of Kirchhoff's law of heat radiation for a body in a nonequilibirum

environment. Progress in Optics, XIII, North-Holland: 3 - 25; 1976.

Jull, E. V. Errors in the predicted gain of pyramidal horns. IEEE Trans, on A X P, AP-21 : 25 - 31; 1973 January.

Barrow, W. L. ; Chu, L. J. Theory of the electromagnetic horn. Proc, IRE: 51 - 64; 1939

January.

Lewin, L. Theory of waveguides. New York, NY: John Wiley & Sons; 1975.

Karbowiak, A. E. Theory of imperfect waveguides: The effect of wall impedance. Proc. IEEE,

102 , Part B: 698 - 708; 1955 September.

Silver, S.; ed. Microwave antenna theory and design. Radiation Laboratory Series, Vol. 12, Boston Technical Publishers, Inc.; 1964.

Bateman, H. Higher transcendental functions. Vol. II; New York, NY: McGraw-Hill Book

Company, Inc.; 1953.

Jackson, J. D. Classical electrodynamics. New York, NY: John Wiley & Sons, Inc.; 1962.

Korn, G. A.; Korn, T. M. Mathematical handbook for scientists and engineers. New York, NY: McGraw-Hill Book Company, Inc.; 1968.

Morgan, S. P. Effect of surface roughness on eddy current losses at microwave frequencies.

J. Appl. Phys., 20: 352 - 362; 1949.

Born, M.; Wolf, E. Principles of optics. New York, NY: Pergamon Press; 1959.

Stratton, J. A. Electromagnetic theory. New York, NY: McGraw-Hill Book Company, Inc.; 1941.

Siegel, R.; Howell, J. R. Thermal radiation heat transfer. New York, NY: McGraw-Hill Book

Company, Inc.; 1981.

Miller, C. K. S.; Daywitt, W. C; Arthur, M. G. Noise standards, measurements, and receiver

noise definitions. Proc. IEEE, Vol. 55, No. 6: 865 - 877; 1967 June.

15
Appendix A: The noise efficiency and noise temperature of an antenna in a reflectionless, isothermal cavity

The derivation for the antenna noise efficiency presented in this appendix employs the PWSM

theory [6] of antennas, and consists of two steps. First, a spectral -density function for the

cavity radiation is found by placing a lossless, reciprocal antenna in the cavity and applying

thermodynamic arguments. The density function is then used with a lossy antenna in the cavity to find an expression for the resulting antenna noise efficiency and noise temperature. The PWSM

theory and symbols are fully explained in the reference, and will not be reviewed here in order to save space.

Figure 11 and the use of plane-wave spectra to describe the cavity fields suggest that the

results are applicable only to an infinite, parallel -plate cavity. However, the results are more

general than indicated, and apply as well to a closed cavity, the PWSM formalism being used for

convenience because of its extensive development in the antenna metrology area. A rigorous

derivation for the closed cavity would proceed along the lines of Yaghijian's cylindrical surface

Figure 11. Schematic diagram of an antenna and termination in a reflectionless, isothermal cavity.

16
formulation [19] except for the use of spherical-wave expansions which [20], like their cylindricalwave counterparts, allow a scattering-matrix description of the antenna to be developed that

contains important orthogonality properties permitting a development similar to the one presented

here to be accomplished.

Cavity radiation density function

Figure 11 shows a lossless antenna (terminated by a refl ectionless load) in thermal equilibrium

(via radiation through the antenna) with the cavity, leaving the load and cavity at the same temperature T. Here da^ (m, _K_, _r) is a differential amplitude function for the radiation originating at position _r> where q

(= ± 1) denotes the front and back hemisphere surrounding the antenna; m is the

polarization index of the corresponding wave incident on the antenna; and K_ is the transverse (to

the z-direction) part of the propagation vector k_. The wave amplitude of the field in the antenna

waveguide lead produced by da^ is represented by db^, whose summation over both hemispheres is

denoted by b^. Sqq is the antenna reflection coefficient. Since there are no multiple reflections

between the antenna and the cavity walls [6]

db^ = ^0 ''^' -^ ^^ ^'^' -' -^ — ^^^^

where

t-=l J 1 (A2)

q K m

is a symbol used for convenience, and the integral is two dimensional, being performed over the

product dk^ dky symbolized by dj<. The integral limits are restricted to the region in k_-space where

K < k, implying that there is no evanescent-mode coupling between the antenna and cavity. Sq„ is

the receiving characteristic of the antenna. The total wave amplitude is

b^ = / db = ^ S^ (m, K) / da (m, K, r) dK (A3)

where the integral is over both hemispheres or H steradians.

The spectral density w^ of the available power [21] corresponding to b^ is

Tl u = ^ <b„ b*> (A4)

^ . ,3^^,2

where the bracket symbolizes an ensemble average, accounting for the fact that h^ is a stochastic

variable [21], and where the asterisk stands for the complex conjugate. tIq is the characteristic

admittance for the mode propagating in the waveguide. Combining eqs (A3) and (A4) leads to

to = — ° „ t p S (m, K) S* , (m' , K' ) w dK dK' (A5)

, 1^,2^^ oq'- oq '- c

' oo'

17
where

CO = J / <da^(m, K, r) da*,(m', K ' , r')> . (A6)

The power spectral density w of the load attached to the antenna is

w = kT (A7)

where k is BoHzmann's constant, and T is the load temperature. Since the load and cavity are in

thermal equilibrium, the second law of thermodynamics [11] implies that

Wq = u) . (A8)

Combining eqs (A5), (A7), and (A8) leads to

f f S^ (m, K) S* ,(m', K') o) dK dK' = kT . (A9)

1 i<;|2*'^ oq'- oq' '- c '^oo'

Since the antenna is lossless and reciprocal [6]

,2

t \ lv'\ dK = 1 (AlO]

1 - IS,00

1^ V^

where r]^(K) is the wave admittance for the incident wave with polarization index m and transverse

wave number K. The last equation can be rewritten in the form

n, S, (m, K) S* ,(m' , K') — ^f. t' -^ ~ ,°? ^-6 , 6 , 6(K-K') dK dK' - 1 (All)

1 . \sj^ \^ ^^ TM

where 6 and &^^< are Kronecker delta functions, and 6(1<^-K^') is a Di rac delta function.

Multiplying eq (All) by kT and combining it with eq (A9) leads to

(A12)

kT

%

[^r- Tiry 6,6. 6(K-K' )] dK dK ' =

'- c Ti^(K) qq mm -'

The quantity in the brackets of this expression is a property of the cavity and independent of the

antenna structure. Therefore, since eq (A12) must hold for any lossless, reciprocal antenna, the

quantity in the bracket must vanish. This yields the required spectral function for the cavity

radiation,

18
This function will be used in the next step to derive an expression for the noise efficiency of a lossy antenna in equilibrium with the cavity and termination.

Antenna noise efficiency and noise temperature

The antenna noise temperature T^ of a lossy antenna in the cavity is equal to the power

spectral density (divided by Boltzmann's constant) of the total available noise powers at the

antenna waveguide flange. The total power is the sum of the noise powers generated by the cavity

and the dissipative losses in the antenna. Hence, the spectral density is the sum of the separate

spectral densities of the antenna and cavity. The spectral density w^ for the cavity radiation

appearing at the waveguide flange of the antenna can be obtained by inserting eq (A13) into eq (A5)

and taking the cavity temperature T (since it is constant) out from under the integral sign. The

result is

u)^ = KTa (A14)

where

n„ IS' (m, K)|^

^ ^ ^1^

= ^

. t ^^5_z: dK . (A15)

(1 - iSiol^ ^0 ' ~

The second sum-integral symbol includes only the sum over q since the sum over m is absorbed into

the "complete" antenna receiving characteristic s' [6]. Yq is the wave admittance of free space,

and the primes in the equation signify that the corresponding parameters belong to a lossy

antenna . Note that, because of the loss, eq (A15) does not reduce to unity, as does eq (AlO) for a lossless antenna. Equation (A15) defines the antenna noise efficiency . For a reciprocal antenna,

it is equivalent to the antenna radiation efficiency [6],

The spectral density w of the total available power at the waveguide flange is

03 = OJi + OJo (A16)

where C02 is the spectral density for the antenna losses. Since the load terminating the antenna,

the antenna itself, and the cavity are at the same temperature, the second law of thermodynamics

implies that the w of eq (A16) is the same as that of eq (A7). Therefore, eqs (A7), (A14), and

(A16) can be solved for C02, yielding

19
^2 = kT(1 - a) (A17)

This important result is used to calculate the amount of noise power generated by the antenna

losses, and appearing at the antenna waveguide flange. When the temperature Tq of the antenna and

the temperature T^ of the cavity are different, eqs (A14), (A16), and (A17) combine to give the

antenna noise temperature Tp, where

Tp =

<"•' w = T^ a + Tq(1 - a) (A18)

This result depends on the validity of Kirchhoff's law of radiation [22] in nonequil ibri urn (T^, *

Tq)-

It is useful to rearrange eq (A15) to the form

where

Tl (K) E oq -

t "oq'y

"'

1 liq'y

1 - l^io'' ioq'i'

(A19)

:A20)

Tl k Is (K)r

da E ° '^q-' dK

^0^(1- l^ool^"

;a2i]

and Y is the z-component of the propagation vector. The antenna loss is now contained entirely in

definition (A20) with the result that'

^ da = 1 ;A22)

which is equivalent to eq (AlO). Since the antenna is reciprocal [6]

^0 ^ ^q^ii^ = ^0 ' V^"^^ :A23)

with a similar equation for the lossy or primed quantities, s is the "complete" transmitting

characteristic of the antenna. Equations (A20) and (A21) can now be expressed in terms of the

transmitting characteristic

%q(^^

IS 00

1 - IS' I 00

^lqo(-'<^

^Iqo^

:A24)

and

da (A25)

20
Equations (A24) and (A25) depend only upon the antenna structure, and apply when the noise sources

in the cavity walls are in either the radiating near field [5] or the far field of the antenna.

They may be conveniently evaluated in terms of the transmitted far field of the antenna [6] by using

and

%,'«

da

1 - IS 00

IS'00

2 r., \ 2 rE^

rE^lr)

P^(q, K) d^

;A26)

(A27:

where rE is the far field pattern, and Pp and Q^ are the normalized power pattern and antenna solid

angle, respectively. dQ is the differential solid angle referenced to the antenna aperture.

Combining eqs (A19), (A26), and (A27), and writing the integral in more conventional form.

1 - |S;j2 Q^

rE'(r)

rE(rl

P (e, (t)) dQ (A28)

where 9 and $ are the usual antenna angles, and where S^^' and the field pattern rE^' (r) contain the

antenna loss. The integral is over the entire An solid angle. The ratio in the integrand is the

ratio of the field pattern with loss to the field pattern without loss.

For convenience, the preceding development was done in the low-frequency approximation to the

radiation laws. This approximation results in a 3% error in the present situation, and, therefore,

the Planck quantum effects must be taken into account. This is done by replacing the low-frequency

radiation temperatures appearing in the equations by their Planck equivalents. For example

replaced by the expression

Tm i5

hv/K

hv/kT (A29)

where h is Planck's constant, v is the operating frequency, and K is Boltzmann's constant.

21
Appendix B: A simplified expression for the noise efficiency

For most horns, the first ratio in eq (A28) containing the reflection coefficient is not

significantly different from unity and can be discarded. The noise efficiency becomes

^/

rE ' ( r)

rE(r)

P dQ

n (Bl)

The horn in figure 2 was designed to minimize throat and aperture reflections so that a transmitted signal consists of a simple traveling wave that maintains the TEj^q spatial configuration

of the waveguide mode as it travels from the waveguide flange to the aperture. The expansion and

radiation of such a wave in the E plane of the horn is shown in figure 12. As the wave travels away

from the flange, it behaves like a TE-j^q mode until it arrives at the aperture A-A'. Considering the

9-direction, the principle wavefront continues to travel outward [8] along ray S. Two diffracted

waves [13] are set up at A and A' due to the abrupt change in curvature, portions of which travel

out along rays D-^ and Dj . Surface waves that are launched at A and A' travel along their respective

quarter-rounds until they encounter the edges at B and B' . Again, diffracted waves are set up with

Flange

Figure 12. E-plane section of the WRIO horn antenna showing the traveling-wave field.

22
components traveling along rays D^ and D2 in the Q-di recti on. To first order, the process stops

here. A similar explanation applies to the 9 '-direction, except there the principle ray S is

missing and a diffracted ray D3 representing leakage from the surface ray is added. To the

approximation considered, the field E_'(_r) in the 0-direction is the sum of the waves associated with

the rays S, D-^, D-^' , D2, and D2'. In the 9 '-di rection, the sum includes the rays D-^' , D2, D2', and

D3. The components of the field attenuation at r^ can be seen from this decomposition: attenuation

of the S, D]^ and D^ ' waves is a result of the " waveguide" attenuation [23] r] from the flange to the

aperture; while in addition to this attenuation, the D2 and D2' waves suffer a further attenuation

associated with the surface waves on the quarter-rounds. A similar interpretation applies to the

9 '-direction, from which it is clear that the difference in the ratio of eq (Bl) for the two directions 9 and 9' is due to a difference in the wave attenuations beyond the aperture A-A'

Therefore, if the attenuation beyond A-A' is neglected by setting the ratio equal to -n, then eq (Bl)

reduces to

a = Ti (82)

since r) is independent of the antenna angles and can be taken outside the integral and the integral

without the ratio is the antenna solid angle.

The relative error in the antenna noise temperature due to neglecting the attenuation beyond AA' can be obtained by combining eqs (A18), (Bl), and (B2), resulting in

AT T - T P dQ

n ^ (_ojT,^

T, / (1 - An(9, $)) -\- (B3)

n m a

where A-n denotes the efficiency associated with the additional attenuation and is equal to the ratio

in eq (Bl) divided by r). An upper bound is easily obtained by replacing A-p in eq (B3) by its

minimum value (corresponding to a maximum value of attenuation), reducing (B3) to

AT T - T

" =

{ \ ^ ] rid - An) . (B4)

n m

For a conductivity of 4.5 S/cm (gold at room temperature), the minimum value of Ati is estimated to

be no less than 0.99965. Then, using Tq = 300 K, 1^ = 77 K, and ti = 1 leads to a maximum error of

0.10% in the noise temperature.

23
Appendix C: Evaluation of r)

Precise evaluation of the antenna noise efficiency given by eq ( B2) is not possible for a pyramidal horn because mathematical expressions for the fields inside the horn do not exist. In sectoral horns where expressions do exist, the electric and magnetic fields do not propagate with

the same phase constant [24]. Even higher modes (other than TEj^q and TM^q) in a lossy, overmoded,

rectangular waveguide are not stable (i.e., retain the same field pattern along the guide)

[25,26]. Nevertheless, the attenuation of the wave described in Appendix B will be calculated as though it were the dominant waveguide mode in a waveguide of slowly increasing transverse dimensions

[23]. The error in using the dominant waveguide mode, instead of the nonexistent horn modes, is

assumed to be the difference in the attenuation calculated this way and calculated by using the

dominant mode fields for the E- and H-plane sectoral horns.

The attenuation coefficient a' for the fields is calculated from the expression [24]

a' = ^

^"^ (cn

where P is the power traveling longitudinally through the waveguide (or radially through the

sectoral horn with z - r) , and dP/dz is the corresponding power loss per unit length due to dissipative wall losses.

P = ^ Re // E X H* • ds (C2)

and

dz 2 "s ' ^ '"t

= iR. / IH j2 dT . (C3)

H^ is the magnetic field tangent to the wall, x is a transverse coordinate along the wall, and Re stands for the real part of the complex quantity of the arguement. The surface resistance R^ of the

wall is related to the wave number k, skin depth 6, and free-space impedance Zq by

R^ = ^ k & 1 . (C4)

s 2

The attenuation coefficient a^ for the waveguide mode is easily found in the literature [25] as

R ,2 2k^

a' ^ -^{^ + —^] (C5) w (jj^iP ^b a

'' g

where a and b are the cross-sectional dimensions of the waveguide, k^ (= n/a) is the cutoff

wavenumber, w is the radian frequency, ii is the magnetic permeability of the walls, and

p2 = k^ - k(,2 . (C6)

24
When the H-plane horn mode equations [27] are inserted into eq (CI), the resulting attenuation

coefficient a|!| for the geometry shown in figure 13 is (the H's in the following equation are Hankel

functions, and p = 71/20^)

R

H wiip '^c ^2re,

inj^^^kr)!^

4 Q^/^a^g

2re, H (2)', 2 (-^)

4 e^ Pg/uak'

2k.

f ^ V (C7)

E- Plane

y

X

A-

" b ,i ^^\

H-Plane

Figure 13. Diagram of an E-plane and a H-plane sectoral horn.

For the E-plane horn (fig. 13)

^E ' w^p. 2re 2/Tirp

(|H[(2),

2

.(2),2>

"27iFgF

2k.

,(2),

2

2/TtpgrJ (C8)

Using the proper asymptotic expansions of the Hankel functions [28], both eqs (C7) and (C8) can be

reduced to eq (C5) as the flare angles 0|_| and 9^ vanish.

Equations (C5), (C7), and (C8) were used to calculate the attenuation and noise efficiency of

the horn in figure 2 by varying the a and b dimensions to conform to the inner horn dimensions. The

efficiency and attenuation A are related by the equations

25
n = e"'^ C9)

A E 2 / a'dz . (CIO)

The integral in eq (CIO) is performed over the length of the horn. The error due to substituting

the waveguide mode fields for the nonexistent pyramidal horn mode equations was then assumed to be

no greater (see eqs (A18) and (B2)) than

AT T - T

n m

where ti^, ti^, ti^ are the efficiencies calculated by using the attenuation coefficients in eqs (C5),

(C7), and (C8), respectively. This error was no larger than O.OIX for the entire WRIO frequency

band (75 GHz - 110 GHz).

26
Appendix D: Surface roughness

The derivation outlined in this appendix was developed to help resolve a disagreement in the

literature [9,10] concerning the proper form of the equation describing the effect of surface

roughness on attenuation, and to extend these previous results into the range where the skin depth

is comparable to the rms surface roughness.

Figure 14 illustrates the E field of an electromagnetic wave that is constrained to move along

the z-direction of a rough, metallic surface where r\_ is the unit normal to the surface, and Z, is the

coordinate directed into the surface. H. is the magnetic field tangent to the surface, and d^P is a second-order differential representing the time-averaged Poynting vector directed into the surface

as a result of the surface losses. By neglecting the displacement current in the metal and

using Ampere's law [29] in the Poynting vector to eliminate electric field, the power absorbed by dA can be put in the form

h2p R 6 ^^t

-5-^ = -^ Re r H* • f - -^=1

( Dl

)

dA 2 L"^ I 55JJ lui;

where R^ is the surface resistance, and 6 is the skin depth.

Figure 15 shows a portion of the rough surface parallel to the x-z plane with surface

coordinates [30]|j. and u. The small (but nonvanishing) surface area AA is bounded by the z and z + Az planes, and the x and x + Ax planes. Ax • Az is the projection of this area on the x-y plane.

The power absorbed by AA is then found by integrating eq (Dl).

R 6 aH 5H

P =-^/ Re [- HJ-^- H^^] dA (D2)

AA

which can be put in a quasi -differential form resembling eq (Dl),

^2n R S 5H 5H .. dz • dT 2 '- ^i 55 V bZ Az • At

where the caret brackets stand for a surface average (the integral in eq (D2) divided by AA)

Equation (D3) applies either to a "y = constant" wall (x = x) or a "x = constant" wall (t = y). The

average should extend over at least 10 (a is the rms surface roughness) to be valid. In the

present case, the surface roughness of the internal horn surfaces is 10 |icm, easily small enough to make the quasi-differential useful even inside the WRIO waveguide lead of the horn.

27
Figure 14. Schematic diagram of an electromagnetic wave traveling along a rough surface.

For locally flat surfaces [29]

dH

Re [- H* -^]

5H

Re [- H* -^]

(D4)

D5)

The terms on the right of these equations are approximations associated with a first-order (in the

skin depth) solution for the fields at a metal surface where

fields. It is then convenient for the rough surface to define

skin depth) solution for the fields at a metal surface where H^ (H^^ or Hy) and H^ are the incident

28
* X

Figure 15. Schematic diagram for the coordinates of a rough surface.

<Re [- HJ 5H^/5C]>

IH J^/6

(D6)

<Re [- H* 5H /blh

K = V-^^ C -^ IHJV6

where

C = AA

Az • At

*

(D7)

(D8)

K^ is associated with the surface current containing a z component while the current associated with

K^ is contained entirely in the x-y plane. These two factors are expected to yield the same magnitude [9, 10] if the surface roughness in the z and transverse directions is the same. Furthermore, if the fields on the rough surface could be found to first-order in the skin depth, it

is not unreasonable to expect the partial derivatives in eqs (06) and (D7) to contain a factor 6 that would cancel the 6's in these equations. Therefore, K^ and K^ should depend on the frequency

and the surface geometry and be independent of the surface conductivity contained in the skin depth

(and thus the temperature).

For wavelengths much greater than the surface roughness, calculations [31] indicate that the K factors lie between unity and the constant C by which the real area is increased over the projection

of the area on a plane parallel to the real surface and that they approach C as the frequency

increases (or the skin depth decreases). Then, the ratios multiplying the constant C in the

equations lie between C"-*- and unity, approaching unity as the frequency increases. This behavior

suggests that the ratios are a measure of what fraction of the rough surface is available or "above

cutoff" to the incident fields. Therefore, since there is nothing in this description to suggest a

29
dependence on anything but the frequency and surface geometry (roughness), it is concluded that the

K factors are constant as long as the surface roughness and frequency remain constant. This

conclusion is the basic justification for determining the surface roughness factor for the horn by

the method described in section 2. Incidentally, this interpretation of the behavior of the ratios

explains why the effects of surface roughness increase with frequency (becoming important in the

millimeter-wave bands), an insight apparently missing from the literature.

Combining eqs (D2), (D6), and (D7) leads to

2 R

^ ^ - Mk IH 1^ + K IH J^) (D9) dz • dT 2 z T T z

K 4 (IH^I^ + IH J^) (DIO)

^ T Z

where eq (DIO) holds when the surface roughness is isotropic and K^ and K^ both equal the same constant K. Proceeding from eq (DIO) by the usual method [25] gives an efficiency

-2K / a'dT ,„, 1.

T) = e

•' (DID

in place of eqs (C9) and (CIO).

30
Appendix E: Higher-mode contamination

The noise temperature T^ of a lossless waveguide with gradually increasing transverse

dimensions that is terminated by a reflectionless absorber at its larger end can be expressed as

T = T + T y a..+ T y a.(El)

where Tj^ is the absorber temperature. The first term represents the noise power contained in the

dominant (TE-j^q) mode that propagates the entire length of the waveguide without attenuation. The

second and third terms represent noise in the higher TE and TM modes, respectively, which are attenuated in the smaller end of the guide. The efficiency factors are related to the increasing

transverse dimensions (a and b) of the waveguide and the wavelength \ [25] by

A. . -/^Ja..dz

a,, ^ e °

^J (E2)

where

TJ

- Q ri^ J. J^ 4U/2 ,c•^^ ^^

a b^ \^

The integral is taken from "o" at the small end of the waveguide to that point A-- where the

respective mode is propagating.

From eq (El), the relative uncertainty in the noise temperature due to the presence of

incompletely attenuated higher modes at the small end of the waveguide is

AT OO CO -J^= I a. +1 a . + Zll a.. . (E4) \ 2 ^° 1 °J 11 ''

This equation is applied to the horn in figure 2 from the waveguide flange to the aperture, where a and b are allowed to expand to conform to the transverse horn dimensions. The error depends on the

length x-^ of the waveguide portion of the horn as shown in figure 16, where it is seen, that for the

horn with an x-^ of 0.31 cm, the error is 0.12%. Thus, if a detector at the output of the standard

could respond to all of the modes, it would measure a noise temperature 0.12% higher than Tp^.

31
LU

0)>

CD DC

0.12

X-,(cm)

Figure 16. Higher-mode noise-temperature error at 110 GHz for the WRIO horn antenna.

32
Appendix F: Cavity-radiation error

The excess temperature (eq (3)) is derived on the assumption that the cavity walls and absorber

(fig. 7) are at the same temperature T,^, leaving just the antenna noise contribution. Therefore,

since the wall temperature varies from T^ up to room temperature, eq (3) predicts too small a correction and is in error. An upper bound to the magnitude of the error is estimated in this

appendix. The model of the cavity to be used in the estimate is shown in figure 17, which is an enlarged portion of the assembly drawing in figure 7. The heavy, superimposed lines indicate the

confines of the model in this profile view. The horn antenna is represented by lines 0-1 and 0-2,

where the matching quarter-rounds have been deleted to make the calculation easier and to- give a more conservative estimate of the error. The cavity walls consist of lines A-B-C-D and G-H-I-J.

The real absorber (the wedges) is replaced by the fictitious surface D-E-F-G from which the absorber

radiation is assumed to originate. The absorber temperature is T^^, while the walls vary in temperature from 1^ at D and G to room temperature at C and H. The rear portion of the walls, A-B-C and

H-I-J, are at room temperature. The absorber is reflectionless, while the walls and horn exterior

have emissivities [32,33] that vary with temperature.

The sectional view of the standard shown in figure 17 is in the E-plane of the horn, where

points 1 and 2 are at the center of the E-edges. In this plane, the sources of radiation in the

walls and absorber are included in the heavy lines of the figure. Since the walls of the cavity are metallic, a three-dimensional pattern of images of each source is set up, the source and its images

both radiating energy into the horn. To make the image pattern easier to construct, the cavity is

assumed to have a square (rather than circular) cross section, restricting the images of a particular source to the plane of the figure and to a plane perpendicular to the figure. The

calculations are simplified (and the error made more conservative) by rotating the images in the

perpendicular plane by 90° into the plane of the figure. A further simplification is gained by

translating the rotated images until each lies directly on top of its nearest neighbor. This last

operation doesn't have a great effect on the results, and permits the rotated images to be accounted

for by simply doubling the strengths of the images originally in the plane of the figure. The UTD

[8] is then applied to the resulting patterns, two examples of which are illustrated in figures 18 and 19. In figure 18 the source is an element of the absorber, and can be seen from the throat area

of the horn and the two horn edges, as indicated by the dashed lines connecting the source to these

points. The images, on the other hand, can only be seen by the edges. The number of images is

infinite, and appear to collect on the condensation line as the images get further away from the

edges. In figure 19, the source is in the upper wall behind the horn aperture and, with its images,

can only be seen by the upper E-edge. The images in both figures are drawn as though the horn were

not present to simplify the drawings and the subsequent calculations. This simplification is not expected to have a large effect on the results since, because of the rectangular horn geometry and

circular cavity geometry, no resonance structure is set up behind the horn.

For the source shown in figure 18, the UTD [8] approximation leads to a first-order magnetic

field H at point o of the form (for the source in figure 19, only the second term and the first

infinite series would be present)

33
sz:^srr^s=ssz3sz:^sz3SKZ^iz:ssc:ss:=^:=^cz^szi^s:r::^e

VJ/// //////// ///MI

I

l^^^^^v .^^'s^^v^^:^v

Ik ^vt- VS\< \ Vn\ V\V ^VnV VvV -^^V m vtvV\ ^VVV VVv\ ^w ^-\\ v\\ V

Figure 17. Enlarged view of the WRIO noise-standard assembly drawing showing the outline of the

cavity model

-jkr^ -jkr^ -jkr2

1 ^

-jkr^(i)

^2pii Hi(i) %,^( i ) (1>1)

+ 2 I I^. H^li)

-jkr2(i)

(2, i (Fi;

The first term is the field that comes directly from the source at distance r^. The absence of an

infinite series associated with the field Hq indicates that the images are not seen from point o.

Tlie second term is the portion of the source field incident on the first edge that is diffracted

34
i*^ image

Source

dA

Condensation

Line

Figure 18. Image pattern of a source in the absorber.

((1, o) is the diffraction coefficient) by the edge to the point o. The third term has a similar

interpretation, and the two infinite series (the factor of 2 comes from the image doubling mentioned

earlier) are the image fields diffracted to point o. The Z-symbols under the sum signs are the

35
compound reflection coefficients associated with the multiple reflections of the source in the

cavity walls that produce the images. For example,

^li = .\ ^li^J^ [F2)

is the compound reflection coefficient associated with the i^ image as seen by edge 1. The lowercase z's are the reflection coefficients [33] for the individual wall reflections which are n,- in

number. The symbol after the "equal" sign signifies that a product of the individual coefficients

is calculated. The various z's (indicated by the j in the argument) contain a small amount of wall

attenuation, and differ from each other because this attenuation is a function of the wall

temperature. The individual back and side wall coefficients differ because of the different angles

of incidence of the fields from the source.

The magnitudes Hq, H^(o), . . . , appearing in eq (Fl) are stochastic variables [21] which,

when multiplied together and averaged over an ensemble of systems identical to the actual system,

yield the source strengths of the radiation from the source or its images. This product takes the

general form

<H^(i) H^(j)> = C[e(Y^(i)) e{y^(3)) COSY^(i) COSY^(j)]^/^ kTB k^ dA K^^(i, j) (F3)

where m and n denote the edge (1 or 2), and i and j denote the source (o) or an image (1, 2,

. . . ). e is the emissivity which depends on the angle of incidence, k is Boltzmann's constant, T

is the temperature of the source, B is the frequency bandwidth of the system to which the noise

standard is attached, k is the wavenumber, and dA is the elemental area of the source. The y's are angles of incidence (fig. 18) , and C is the constant

?7

(F4)

where X is the wavelength, and Zq is the free-space impedance.

I 1, cavity walls

0, different sources and source images

(F5)

Equation (F3) was derived in the ray-optical approximation by using Kirchhoff's law of radiation

[34] and Poynting's vector. The emissivities are calculated from the reflectivity [32,33] of the

walls, and are equal to unity for the absorber.

36
The elemental noise temperature dip resulting from a particular source and its images is calculated from

dT =

n C'<H H*> kB (F6)

where C is a constant related to the transformation of the throat field H into the TE^q mode fields

in the waveguide lead of the horn. When eq (Fl) is inserted into (F6) and eq (F3) is applied to the

resulting products, there emerges an expression of the form

dT - C C T

n dA (F7)

where the bracket contains what is left of the product H H*.

i*^ image

Images

Condensation Line

Figure 19. Image pattern of a source in the side wall.

The process just described is relatively simple because it was initiated in the E-plane section

of the horn and cavity, the same calculation in any other plane being prohibitively complicated.

37
However, in other planes, eq (F7) overestimates the effects of the cavity walls on the error because

the wall contributions are strongest in the E-plane (since the diffraction is strongest in the

center of the E-edges). Thus, if eq (F7) is applied to all sections around the axis of the horn and

cavity (that is, if azimuthal symmetry is assumed), the error will be overestimated and

conservative. The resulting equation has the form

T^ = C C / T { } dA (F8)

where the integral is over the surface of the cavity. T and { } are independent of the azimuthal

angle, and dA can easily be expressed in spherical coordinates. The terms in the bracket are a function of the wall temperature since the conductivity of the walls depends on temperature. In thermal equilibrium, Tp and T must both equal the absorber temperature Tj^, allowing the product C C

to be easily calculated.

C C = (/ {m} dA)"^ (F9)

where the m in the brackets is a reminder that, in this case, the bracket is evaluated with the wall

temperature set equal to the absorber temperature. The antenna noise temperature now becomes

/ T { } dA

T„ = — . (FIO)

"

/ {m} dA

The nonisothermal wal 1 -radiation error corresponding to eq (FIO) is

AT T - T

n n

A computer program written to estimate this error produced the result

AT^

-^ = - 0.059 - 0.035 = - 0.094% (F12)

n

where the first component of the error (0.059) comes from the front portion (C-D and GH in figure

17) of the cavity wall and the second from the back portion (C-B-I-H).

Radiation can leak into the cavity around the radiation shields (6C and 6E in figure 7), and

possibly around the membrane (17). Because of the millimeter-wave absorber (16) underneath the load

holder (14), the radiation leaking around the shields has a radiation temperature [16] only 2 K greater than T^^, and causes an additional error no greater than - 0.001%. The radiation possibly

leaking around the membrane has a radiation temperature equal to room temperature, and causes an additional error no larger than - 0.01%. Thus, the total error due to the leakage and the walls is

- 0.10% (- 0.001 - 0.01 - 0.094).

38
m

Appendix G: Multiple reflections

Multiple reflections taking place between the horn aperture and the back wall of the cavity

cause an error in the calculated noise temperature since the derivation (Appendix A) of eqs (3) and

(5) assumes a reflectionless cavity. These reflections modify both the horn and cavity-wall

contributions to the noise temperature of the standard. The magnitude of the error due to the

changes in the horn-generated noise can be found by determining that portion of the reflection

coefficient of the standard that is a result of back-wall reflections. This was done by measuring

the change in the reflection coefficient of the horn waveguide port between the configuration shown

in figure 7, and the configuration with the horn, load holder (14), and load (13) in place and all other items in the figure removed. The resulting change in magnitude was 0.001. An expression for

the relative noise temperature error is derived by considering the horn as a two-port junction [35],

terminated by a load at temperature 1^ whose reflection coefficient changes from to 0.001.

(Gl)

where AT is the excess noise temperature and is estimated from eq (3) and figure 5 to be

approximately 2 K (T^^ and Tq are approximately 77 K and 300 K, respectively). Completing the

calculation in eq (Gl) shows this part of the error to be completely negligible.

An upper bound to the error in neglecting the excess noise generated by the cavity walls is

carried out in Appendix F, where the effects of multiple reflections are ignored. However, since

the rear cavity wall is so highly reflecting, it is reasonable to expect that the excess noise is

underestimated (a tighter coupling to the cavity can only increase the noise as the cavity walls

have a higher temperature than the absorber) by this zero-order approximation. A first-order

approximation can readily be obtained by considering the additional noise from the E-edge images.

Furthermore, after the first order, the effects of higher-order edge images of the edges fall off

rapidly. Figure 20 illustrates a pattern of source images with the addition of images from both E- edges, the crosses representing the upper edge and the asterisks the lower edge. The source and

each of its images excites either half or all of the edge pattern, depending on the location of the source or its images. Using the UTD, the edge images lead to an enhanced magnetic field H' in the

horn throat (the symbols are described in Appendix F).

- H,(o) ^"''^''l - Hjo) ^'-^'•'''2

H' = H + 2 I -J^_^^__ (1, 0, j) 1^.(1, j) + 2 I -^-^^ (2, 0, j) I^.(Z, j)

00 CO + 1

J

HJi) -^^'l^'^

2 l^ 2 _I^ Z^i-V %r^(i) <^' ^' J^ ^3j(^' J^

H (i) -J'^'^z'^^

2 I2 I 1^. ^^-^^.^ (2, i, j) Z,.(2, j) (G2)

OO GO + 1 j=l i=l

39
+ j*^ Edge

\ Image

DJth ^ Source

Image

!; Source

D

D

Source Images

+ Upper Edge Images

* Lower Edge Images

a o

D D

* *

Figure 20. Image pattern of a source in the absorber and the excited E-edges.

where H is the original field (eq (Fl)). The factor 2 dividing the source strengths is a result of

normalizing the previous equations for grazing incidence [8]. The only new symbols are the triplets

like the (1, i, j) in the four series of edge fields. This new diffraction coefficient describes

the excitation (fig. 20) at the j^" edge image by the source image at position "i." Similar

40
interpretations apply to the other coefficients. Proceeding as in Appendix F, the noise temperature

Tp that now includes first-order reflections has the form (c.f. eq (F8))

r=CC'/{}dA+4SCC'/T[]dA (Gs;

where S is a constant (close to unity) that is the result of the approximations made to arrive at

the equation. The symbol [ ] bears the same relationship to the additional terms in H' that the

symbol { } bore to H in eq (Fl). Again, using the second law of thermodynamics,

C C =

[ / {m} dA + / [m] dA)

-1

(G4]

where the m in the brackets indicates that they are evaluated at the absorber temperature T^^. Combining eq (G3) and (G4),

T' = T

n n

• 1 + 4 S / T [ ] dA/ / T { } dA

] 1 + 4 S / [m] dA/ / {m} dA

(G5)

With the help of a computer program.

AT T - T' n _ n n 0.054% (G6)

is the resulting error in the output noise temperature due to neglecting the next higher-order

(first-order multiple reflection) contribution of the excess noise generated by the cavity walls.

Comparing this with eq (F12) shows that the first-order multiple reflection is not insignificant

compared to the zero-order calculation.

41
NBS-n4A (REV. 2-ac) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1071

2. Performing Organ. Report No. 3. Publication Date

December 1983

4. TITLE AND SUBTITLE

Design and Error Analysis for the WRIO Thermal Noise Standard

5. AUTHOR(S)

William 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

[^J Document describes a computer program; SF-185, FlPS Software Summary, is attached.

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

bi bliography or literature survey, mention it here)

This note describes the design and error analysis of a WRIO thermal noise

power standard. The standard is designed to operate at the boiling point

of liquid nitrogen with a noise temperature output accurate to ± 1 K.

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

antenna efficiency; diffraction; error analysis; millimeter wave; noise

standard; plane-wave scattering matrix.

13. AVAILABILITY

[X] Unlimited

I I For Official Distribution. Do Not Release to NTIS

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

20402.

[2^ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

45

15. Price

USCOMM-DC 6043-P80

iiU.S. GOVERNMENT PRINTING OFFICE: 1984—777-494/124 REGION NO 8


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