A11102 bT?bfl3

NATL INST OF STANDARDS & TECH R.l

All 102697683

Daywltt. William C/Horn design equations QC100 U56 NO. 87-3073 1987 V19 C.1 NBS-P HUKN utblGN EQUATIONS FOR THE NBS

HORN-TYPE NOISE STANDARDS

William C. Daywitt

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303-3328

August 1987

“QC

100

. U56

#87-3073

1987

Research Information Center

National Bureau of Standards

Gaithersburg, Maryland 20899

NBSIR 87-3073

HORN DESIGN EQUATIONS FOR THE NBS

HORN-TYPE NOISE STANDARDS

William C. Daywitt

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303-3328

August 1987

Sponsored in part by

The Space Systems & Command, Control, & Communications

USAF, Washington, DC

and

The Naval Ocean Systems Center (NOSC)

San Diego, California

U.S. DEPARTMENT OF COMMERCE, Clarence J. Brown, Acting Secretary

C*

*

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

1. Introduction—Internal Reflections 1

2. Braun Equations 4

3. Sundry Equations 6

4. Results 9

5. Acknowledgments 9

6. References 12

Appendix A: Aperture Reflection 13

iii

Horn Design Equations for the NBS Horn—Type Noise Standards

W. C. Daywitt

Equations are given for calculating the interior dimensions of the horn pickup in the NBS millimeter wave noise standards. These dimensions insure negligible internal horn reflections, resulting in an

accurate calculation of the horn attenuation. Measurements in the WR10 horn show a reflection of magnitude less than 0.001.

Key words: electromagnetic horn; millimeter wave; noise source; reflection coefficient.

1. Introduction—Internal Reflections

The National Bureau of Standards (NBS) horn-type cryogenic reference noise standards [l] consist of a noisy

cavity and a pyramidal horn pickup. Their output noise temperatures are the sum of the cavity temperature

and the excess noise temperature generated by the dissipative horn loss. An accurate calculation of the horn

loss requires a knowledge of the magnetic field parallel to the internal horn surfaces, so the horn must be

structured to allow for an accurate prediction of that field.

Figure 1 shows an E-plane cross sectional view of one of the millimeter wave horns. The waveguide (wg)

and early flare portions are surrounded by a water jacket to maintain them at a uniform temperature close

to ambient. As is well known, no solutions for the field patterns inside the pyramidal flare exist. However,

by using a gradual (small angle) flare it is possible to maintain the TE\q mode field pattern generated in the waveguide portion of the horn far into the flare as long as the discontinuities at the wg-flare transition

and at the aperture are minimized.

The wg section of the horn in figure 1 is chosen just long enough to filter out higher-order modes

generated in the horn flare by the cavity radiation, but otherwise as short as possible to achieve a minimum

loss [ l] . In particular, the wg is chosen to keep the output noise temperature due to higher modes below

0.1% of the total output at the highest operating frequency of the noise source. Table 1 is a list of these minimum wg lengths for the wg bands from WR90 to WR10. The first column lists the various wg bands

by their WR designations with their corresponding frequency ranges listed in column two. The wg length

x\ required to produce a 0.1% higher mode error is listed in column three. Thus, as long as the wg portion

of the horn is x\ or longer, the output noise temperature of the source is contaminated by less than 0.1% of the higher-mode energy.

A schematic cross sectional view of a horn antenna is shown in figure 2, indicating the wg flange F, the wg-flare butt joint A, and the aperture edges B. The double arrows depict surface currents caused by fields entering the flange and exiting the aperture. Significant diffraction (single arrows) occurs at the joint and

aperture because of the abrupt edges at A and B, which in turn disturb the desired TE^q mode field pattern

and lead to errors in the calculated horn loss.

To reduce the diffraction effect at the butt joint a smooth circular arc from wg to flare is made Figure i shows this transition region for the El- and H-plane cross sections of the horn, where for convenien< < <mly i In1
Water

Jacket

Figure

WG Section

Matching

Section

Figure 1. A simplified E-plane cross section of the WR10 millimeter wave horn.

2. A schematic diagram of the interior horn surfaces indicating the surface currents and the diffraction

points.
upper horn walls are drawn. The dashed lines show the butt geometry before the transitions are included.

The narrow and wide dimensions of the wg are denoted by bo and ao respectively. The radius of the arc in

the E-plane is re and in the H-plane it is r^. The flare angles in these planes are <f) e and <f>h . Terzuoli &

Peters [2] found that the reflection at the butt joint due to the diffraction is markedly reduced if the length

of the arc re <f> e in the E-plane is 0.4A or greater, where A is the free space wavelength corresponding to the

lowest frequency in the WR band, i.e. the maximum free space wavelength. To be safe the following relation

is adhered to in the horn design (see figure 5 for zo):

zq = 0.4A. (1.1)

This leads to

r = zosin<j)/(l — cos<f>) ( 1 . 2 )

or

r<f> = 0.8A (1.3)

(twice the value suggested by Terzuoli & Peters) for both the E- and H-planes. Both re and r^ are determined

from equation (1.2) by inserting <p e and <f>h respectively. Up to this point only Xi, zo, and the relation in equation (1.2) have been determined. The narrow aperture dimension b is found next.

Table 1. MINIMUM WG LENGTH xx FOR 0.1% OUTPUT ERROR

WR Freq Range(GHz) xi (cm)

90 8.2—12.4 2.82

75 10—15 3.09

62 12.4—18 2.40

51 15—22 2.02

42 18-26.5 1.27 34 22—33 1.34 28 26.5—40 1.08 22 33—50 0.87

19 40—60 0.80

15 50—75 0.51

12 60—90 0.37

10 75—110 0.31

The reflection of the aperture as seen from the flare is a result of the edge diffraction at B in figure 2 and figure 4. The magnitude of the reflection is given by (see Appendix A)

Tap = 0.0912/B122

(1.4)

where

B = b/ A (1.5)

3
and b is the narrow (E-plane) dimension of the aperture. For the aperture reflection to be less that 0.01 equation (1.4) leads to the approximation

B = 6. (1.6)

This equation and equation (1.5) determine the narrow aperture dimension in figure 4. To further reduce

the aperture reflection quarter-round matching sections are attached to the H edges of the horn aperture

perpendicular to the page at B in figure 4 (see figure 1 also). Analysis [3] indicates that if the radius of the

quarter-rounds is 3A (or greater) the edge diffraction is greatly reduced.

2. Braun Equations

Braun [4] specifies a set of relationships which, starting with a given gain, lead to an optimum horn de- sign with approximately equal E- and H-plane beamwidths where the flare in both planes join the wg

portion of the horn in a simple butt joint. The starting point for the noise standard horn design is equations (1.4) and (1.5), however, since an optimum horn gain is less important than obtaining a small internal

aperture reflection. The Braun equations lead directly to

A = 4685/346 (2.1)

and

l e = 57600ABA/(346 x 468) (2.2)

where A is the normalized (a/ A) broad dimension of the aperture, and l e is the E-plane flare length in

figure 4. The corresponding H-plane flare length is

lh = l e [( 1 - 62 /4/e 2 )(( 1 - b0 /b)/(l - a0 /a)) 2 + a

2 /4Ze 2

]

1 / 2

. (2.3)

An approximate expression for the resulting gain g

is

g — 106 A5/(346 x 468). (2.4)

It is clear from figure 4 and the equivalent figure for the H-plane that the flare angles are given by the

equations

<t> e — sin 1 (6/2Ze ) (2.5)

and

<t>h = sin l [a/2lh ). ( 2 . 6)

The corresponding re and are then found from equation (1.2).

4
Figure 3. Cross sections of the upper halves of the El—plane (a) and H—plane (b) wg-flare transitions.

5
3. Sundry Equations

The fundamental equations relating the electrical requirements of the horn to the internal horn dimensions

have been presented in the previous two sections. Following are a number of derived equations that, although

easily obtained, are included here for completeness and their usefulness in calculating machine drawing

dimensions.

Figures 5 and 6 show cross sections of the El- and H-planes respectively in which the various dimension

designations are self explanatory from previous discussions.

E-Plane Dimensions (figure 5)

Pe — b0/2sin<j>e (3.1)

£2 = X 1 T z0 (3.2) Axe = re sin(f> e — zq (3.3) Aye = re (l - cos(t> e ) (3.4)

I3 — {J-e (3.5)

£5 = X2 + Axe (3.6)

Xq = X2 + I3 (3.7)

CC'J — jC (j 3 (3.8)

t/7 = 6/2 + 3Acos(f) e (3.9)

ye = C\ e + C 2 e(x £l) (3.10)

H-Plane Dimensions (figure 6)

ph = a0 / 2 sin<t> h (3.11)

Ax/, = rh sin<f> h - z0 (3.12) = rh (l - cos(f>h) (3.13)

I3 — (Z/i Ph)cOs4: h (3.14)

X^ — X2 + Axh (3.15)

yh = Clh T C 2 h[x £l) (3.16)

6
Figure 5. Cross section of the internal E-plane surface.

7
yh= c m+ c2h(x -x i> %

Figure 6. Cross section of the internal H-plane surface.

8
4. Results

Figure 7 is a photograph of the WRlO horn pickup that was designed and constructed according to the

equations presented which ensure low internal reflections and an accurately calculable attenuation. Figure

8 is an isometric drawing of the same horn. The total internal reflection including the wg-flare transition

and the aperture was measured to be less than 0.001. Analysis [l] shows the attenuation calculation to be

accurate to ±0.01%.

5. Acknowledgments

The Space Systems &: Command, Control, Sz Communications, USAF, Washington, D.C.; and The Naval

Ocean Systems Center (NOSC), San Diego, California funded much of the work leading to the millimeter wave horn pickup designs.

9
Figure 7. Photograph of the WR10 millimeter wave horn pickup.

10
Figure 8. Isometric view of the WRlO millimeter wave horn pickup.

11
6. References

[1] Daywitt, W. C. Design and error analysis for the WR10 thermal noise standard. Nat. Bur. Stand.

(U.S.) Tech Note 1071; 1983 December.

[2] Terzuoli, A. J., Jr.; Peters, L., Jr. VSWR properties of El-plane dihedral corrugated horns. IEEE

Trans., AP-26, No. 2: 236-239; 1978 March.

[3] Burnside, W. D.; Chuang, C. W. An aperture-matched horn design. IEEE Trans., AP-SO, No. 4: 790-796; 1982 July.

[4] Braun, E. H. Some data for the design of electromagnetic horns. IEEE Trans., AP~4'- 29-31; 1956

January.

[5] Jull, E. V. Reflection from the aperture of a long El-plane sectoral horn. IEEE Trans., AP-20, No.l:

62-86; 1972 January.

[6] Jull, E. V. Errors in the predicted gain of pyramidal horns. IEEE Trans., AP-21, 25-31; 1973 January.

12
Appendix A: Aperture Reflection

Jull [5] derives the following expression for the magnitude of the reflection seen from the flare portion of a

sectoral horn: rap = l/A4> e kle (Al)

where k is the free space wavenumber ( 2n/X ). Figure A1 shows a comparison of this formula with reflection data [6] from a pyramidal horn aperture as a function of 4<j) e kl e , where the formula is plotted as the solid curve in the figure. The dashed curve is a least-squares fit through the data as a function of (4 <f> e kle )~ c

, where the constant c was found to be 1.22 with a correlation coefficient of 0.98 for the fit. Recognizing that

2 <f> el e is the narrow aperture dimension b for small <j) e leads to equation (1.4), where B is the normalized

aperture dimension.

(4<f> e

k 4)

Figure Al. Internal aperture reflection for a pyramidal horn.

13
NBS-1 14A (REV. 2-80

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1 . PUBLICATION OR

REPORT NO. NBSIR 87-3073

2. Performing Organ. Report No. 3. Publ ication Date August 1987

4. TITLE AND SUBTITLE

HORN DESIGN EQUATIONS FOR THE NBS HORN-TYPE NOISE STANDARDS

5. AUTHOR(S)

William C. Daywitt

6. PERFORMING ORGANIZATION (If joint or other than NBS. see in struction s) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

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

The Space Systems & Command, Control, & Communications

USAF, Washington, DC

and

The Naval Ocean Systems Center (NOSC) , San Diego, California

10.

SUPPLEMENTARY NOTES

[ ] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

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

bi bliography or literature survey, mention it here)

Equations are given for calculating the interior dimensions of the horn

pickup in the NBS millimeter wave noise standards. These dimensions insure

negligible internal horn reflections, resulting in an accurate calculation

of the horn attenuation. Measurements in the WR10 horn show a reflection of

magnitude less than 0.001.

12.

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

electromagnetic horn; millimeter wave; noise source; reflection coefficient

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

[X[] Unlimited

[ _J For Official Distribution. Do Not Release to NTIS

[^3] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

20

15. Price

[X 1 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

USCOMM-DC 6043-P80

U.S GOVERNMENT PRINTING OFFICE: 1987-776-924/65,299 REGION NO. 8