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OFiMRSIRSTAND 78-869 & TECH

AlllDb T7flb75

tKKUN tuUATiOHS USED IN THE N8S EARTH TERMINAL

MEASOREMENT SYSTEM

Wiliiam C. Daywitt

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

December 1977

-QC

100

.U56

78-869

1977

NBSIR 78-869

MATIONAL BUKJEAU OF STANDABOB

LTBRART

OCT 5 1981

(00

ERROR EQUATIONS USED IN THE N8S EARTH TERMINAL

MEASUREMENT SYSTEM

William C. Daywitt

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

December 1977

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

CONTENTS

Page LIST OF FIGURES AND TABLES iv

ABSTKACT 1

I. INTRODUCTION > 1 II. THEORETICAL DEVELOPMENT 3

A. Step One - T^/G Calibration Curve 4

B. Step Two - Received Satellite Power Measurements————— 5 III. ERROR ANALYSIS = —— —~ 6 IV. CONCLUSIONS TM— 8 V. ACKNOWLEDGMENTS 8 VI. REFERENCES ~- —-~ 9 VII. APPENDIX: . MEASUREMENT AND ERROR EQUATIONS ^ ^ 11

A. T^/G Correction Factors - • 11

1. Atmospheric Transmission Factor - —— = —— —. , — n

1.1. Gaseous Absorption Factor - k -— —-_— n g 1.2. Refractive Spreading Factor - k^ — 12 1.2.1. Gaussian Main Beam With a Disk Source 12 1.2.2. Gaussian Main Beam With an Elliptical

Gaussian Source .

. 13 1.3. Tropospheric Scattering Factor - k^— —— 13

2. Star Shape Correction Factor - — ' '—— 1^

3. Bandwidth Factor - k^ —— 14

4. Differential System Noise Temperature Factor - k^ - - — -— 14

5. Antenna Pointing Factor - kg 14

6. Polarization Factor - k^ -- —•' " ' 15

7. System Response Factor - k^ - - ' 15

B. Radio Star Flux Density - S =-— ~ — 15

C. AY-Factor Ratios - Ay, AY ~— 15

D. Noise Bandwidth - Af 15

E. Space Loss - L , . ,. , , n . - ±(,

F. Aspect Angle Correction - A '

'

' 16

G. EIRP Correction Factors - e^^ 16

1. Atmospheric Transmission Factor - e^ - ' 16 1.1. Gaseous Absorption Factor - e^ — 16

1.2. Refractive Spreading Factor - e^ —~— =— 16 1.3. Tropospheric Scattering Factor - e^— =• — 16

2. Star Shape Factor - e, 16

3. Bandwidth Factor - e^ ' 17

4. Differential Operating Noise Temperature Factor - e^ ~— —~— 17

5. Antenna Pointing Factor - e^ — 17

6. Polarization Factor - e^ ,

, . 17

7. System Response Factor - e^ -,. , . . ..i. - . 17 iii
LIST OF FIGURES AND TABLES

Page

Figure 1. Interface between the earth terminal measurement system and

the earth-terminal receiving system 18 Figure 2.

''a^'^ earth-terminal calibration curve 19 Figure 3. Block diagram of a model representing the earth-terminal receiving

Figure 4. IF power detected by the earth terminal measurement system as a radio star traverses the earth-terminal antenna malnbeam 21 Figure 5. A portion of the RF power spectrum as it appears at the low-noise amplifier input 22 Figure 6. Parameters used in the definition of satellite ElKP and the resulting power flux at the earth-terminal antenna aperture 23 Figure 7. Total quadrature systematic EIRP error at 7.55 GHz versus earthterminal antenna elevation angle 24

TABLE I. ERROR IN MEASURING EIRP AT 7.55 GHZ 6. 12° ELEVATION ANGLE -~ 25

iv
ERROR EQUATIONS USED IN THE NBS EARTH TERMINAL MEASUREMENT SYSTEM

William C. Daywitt

An outline for the derivation of equations employed in a measurement

and error analysis of satellite EIRP using a calibrated radio star is presented. A table showing analysis results at 7.55 GHz using Cassiopeia A

for a satellite at 12" elevation angle is given. The quadrature sum of the

systematic errors appearing in the table is 10.1%. Also presented is a

ctirve of the systematic errors as a function of elevation angle showing a 7.3% minimum at high elevation angles. Included are working equations for the calculation of correction and errors.

Key Words: C/kT; EIRP; error analysis; G/T; precision measurements; radio

star, satellite communications.

I. INTRODUCTION

The National Bureau of Standards has constructed a system designed to measure satellite

EIRP using a calibrated radio star. The system operates off the i-f patch panel of the downlink earth terminal (ET) , and automatically controls the several measurement sequences. The method employed in the earth-terminal measurement system (ETMS) is a modified

version of the radio-star method [1] of measuring satellite EIRP (effective isotropic

radiated power), the modification [2] allowing ET receiver gain fluctuations to be removed

from the measurement, significantly reducing measurement error.

In addition to a digital voltmeter and accurately calibrated i-f attenuators, the ETMS features a very accurate power bridge [3], and an improved solid-state noise source [4] of high stability. Combined, the components can routinely measure power ratios to a few

hundreths of a decibel. The ETMS also incorporates well calibrated i-f bandpass filters

whose characteristics are combined with the observed ET passband characteristics by the

computer software in the ETMS to obtain the overall noise bandwidth needed for the calculation

of EIRP. A measurement and error analysis for the 1 to 10 GHz frequency range to be reviewed in

this paper is incorporated in the ETMS computer software. The analysis reveals additional

errors and a higher atmospheric correction than previously calculated.

The relationship of the ETMS to the ET receiving system is shown in figure 1. The

receiving system is represented by the ET' antenna, cross-guide coupler, low-noise amplifier

(LNA) , downconverter , and 1-f patch panel. The ETMS consists of the solid-state noise

source, and the power measurement and control console. The noise source is connected to the receiving system through the cross-guide coupler, and T^ is the noise temperature representing the noise power added to the system at the LNA when the noise source is turned on. The

gain of the system from the antenna far-field to the LNA is represented by G, and is the

product of the antenna gain and the waveguide loss from the antenna to the LNA. The noise source serves a double purpose in the EIRP measurement. The first is to eliminate gain fluctuations [2] from the LNA input to the power detector in the ETMS. This

is accomplished by performing all power measurements in pairs, one measurement with the
noise source turned off, followed innnediately by the same measurement with the noise source

turned on. The noise-source-of f measurement is then normalized by the difference between

the two measurements, thus eliminating the system gain (and consequently the possibility of gain fluctuations) between the LNA and the ETMS power detector from the power ratio. Besides

the immediate benefit of removing these short-term gain fluctuations, measurements may now

be accurately compared on a day-to-day basis. The second purpose is to provide a calibratable

source of power by which the magnitude of the received satellite signal power can be measured.

The calibration of this noise power is accomplished in the first step of the EIRP measurement by comparing the reduced noise temperature (T^/G) of the noise source against the power

from a calibrated radio star. The EIRP measurement method employed by the ETMS involves two steps: 1) A radio-star

calibration of the reduced noise temperature (T^/G) of the solid-state noise source as a function of ET antenna elevation angle; and 2) the measurement of the available powers at

the LNA input received from the satellite. The first step is performed by measuring the i-f power from the ET receiving system as the radio star drifts through the mainbeam of the antenna. These measurements are performed at various elevation angles as the star traverses

its path across the sky, and yield a set of T^/G data points as depicted in figure 2. The

least-squares curve fit to these data points represents a calibration curve for the "frontend" of the receiving system, and the solid-state noise source. The 6.5% value appearing in

the figure represents random scatter (lo) of the data points. The curve remains valid as long as T^ and G remain constant in time. Measurements performed on the noise source [4] show it to be stable to within ± 0.01 dB (la) over at least the time required to complete a set of EIRF measurements for a number of frequencies. The gain G is assumed to remain

constant for this same period, an assumption that appears to be bom out by experience. Once the calibration curve Is established, the value of T^/G for any satellite elevation

angle can be easily determined by reading the value of T^/G from the curve at that angle.

After obtaining the T^/G calibration curve, the ET antenna is pointed at the satellite

whose EIRP is to be measured, and a number of measurements of received satellite power are performed. The average of these measurements normalized by T^ yields a AY that can be used

with the proper (at the satellite elevation angle) T^/G value to determine the satellite

EIRP from the equation

k(T /G) Af L AY EIRP =- (1) Ae^-.e^

where k is Boltzmann's constant, Af is the combined ET-ETMS noise bandwidth, L is the space

loss (4irr/X)^ [5], r is the range from the ET antenna to the satellite transmit antenna, X is the measurement wavelength, and A is the aspect angle correction [1]. e^ through e^ are near-unity correction factors analogous to the G/T correction factors described in an earlier

publication [6]. Sequentially, these factors account for: 1) Atmospheric transmission

losses; 2) the angular extent of the received satellite signal as "seen" by the ET antenna;

3) frequency variations of the receiving system components across the receiver passband;

4) frequency variations of the downlink operating noise temperature across the receiver

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passband; 5) imperfect ET ancenna pointing; 6) polarization mismatch between satellite

signal and ET antenna; and 7) finite response time of the ETMS and instabilities in T^. The theoretical development leading to equation (1) is outlined in the next section.

II. THEORETICAL DEVELOPMENT

As mentioned earlier the EIRP measurement performed by the ETMS is accomplished in two steps, obtaining the T^/G calibration curve, and measuring the received satellite power. Before examining these two steps separately, basic equations used as a starting point for their theoretical descriptions will be presented. Although these equations and some of the approximations leading to the final working equations may be applicable over a wider frequency

range, they are designed primarily for the 1 to 10 GHz frequency range. A model of the ET receiving system is shown in the block diagram of figure 3. The

system (including the atmosphere) is outlined in solid blocks. The far-field source is the satellite or radio star, depending at which the ET antenna is pointed, and B(n,f) is the corresponding brightness distribution [7]. CI represents the pair of spherical polar angles

with origin at the ET antenna and z-axis along the center of the mainbeam, and f is the

frequency. a(n,f) is an operator representing the effects [3,6] of gaseous absorption,

refractive spreading, and tropospheric scattering on the incoming flux from the radio star

or satellite. M(f) is a mismatch factor [9] relating the polarization of the source radiation to the ET antenna polarization. A^(f) and G(f) are the effective area and gain respectively of the combined antenna, waveguide feed, and cross-guide coupler. is the noise

temperature of the solid-state noise source as it appears at the LNA input. When the noise

source is turned off, the added noise T^(f) is zero. T(f) is the noise temperature of the ET system and consists of cosmic background noise; atmospheric and man-made noise; ground

noise; noise from I^R losses in the antenna, waveguide feed, and cross-guide coupler; the effective input noise temperature of the LNA; and radiated satellite noise when the ET antenna is pointed at the satellite. w(f) is the spectral power (power per unit bandwidth)

corresponding to the sum of the spectral power from the far-field source, kT^(f), and kT(f).

g(f) is the ratio of the spectral power delivered to the power meter to the available

spectral power at the LNA input. P is the total power delivered to the power meter. The spectral power w(f) is related to other parameters appearing in figure 3 by the

equation

w(f) - k[T(f) + "T^(f)"] + Ag(f)M(f) /a(n,f)B(n,f)-P^(n-f2^,f) da. (2)

P^(n-Q^,f) is the normalized power pattern of the ET antenna, waveguide feed, cross-guide

coupler comibination, where the difference Ji-fl accounts for the fact that the antenna may

o not be pointing at the center of the far-field source. The quotation marks around T^(f) are a reminder that T (f) is zero when the solid-state noise source is turned off. dfl is a the infinitesimal solid angle sin 9 d9 d^.
The power P detected by the ETMS is related to the spectral power w(f) by the equation

P = / w(f)g(f)df . (3)

The form of equations (2) and (3) are a manifestation of the assumed linearity of the model

in figure 3, and form the theoretical basis for the following description of the two-step measurement process leading to the EIRP equation (1)

.

A. Step One - T /G Calibration Curve

a

The power measurements performed by the ETMS to obtain T^/G at a particular elevation

angle are indicated by the heavy dots in figure 4. As a calibrated radio star passes

through the center of the ET antenna mainbeam, the i-f power detected by the ETMS describes

a curve shaped somewhat as depicted in the figure. The slope of the baseline from t^ to t^ could be caused by a variation in the cosmic background surrounding the radio star, or a slow change in the ET receiver gain. Three pairs of power measurements are made, one at each time t^^, t^, and t^- At t^ a measurement P^ of the noise baseline is made, followed

immediately by the same measurement P^ + P^ with the solid-state noise source turned on. As previously described, the ratio y^ (=

^^.^^a) then calculated to eliminate the receiver

gain from consideration. Similar ratios (" ^2^^a^ ^3 ^3^^a} calculated from measurements at t2 and t^- The measurement at Z2 corresponds to the radio star being in the center of the ET antenna pattern. The measurement at t^ is another measurement of the noise

baseline and which, with y^, allows the value of the noise baseline at to be determined.

By subtracting the noise baseline from the measurement at the i-f power ratio difference

Ay corresponding to the radio star can be determined from the equation

Ay = y2~^^i ^3^^^

The values of the measured powers in figure 4 can be related to the quantities in the

block diagram of figure 3 through equations (2) and (3). For example,

P^ - /g(f)[kT(f) + A^(f)M(f) /B(n,f)a(fl,f)P^(n,f)dn]df (5)

and

- /g(f)kT^(f)df . (6)

The B(n,f) in equation (5) is the radio-star brightness distribution, and M(f) (approximately equal to 1/2) is the corresponding mismatch factor.

Equations (5) and (6) with similar equations for the measurements at and t^ can be

combined to give a theoretical expression for the Ay of equation (4) . When this is done and

equated to Ay the following equation can be derived

Z2. ,

^

. '^1"'^

(7) G

" Sir k* Ay

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X is the measurement wavelength, S » B(n,f)dJ^ is the radio-star flux density at frequency f

,

and k is again Boltzmann's constant, through k^ are correction factors defined to give

the functional form of equation (7), and are complicated expressions containing a number of integrals. Under certain circumstances these factors are close to unity, making accurate

radio-star measurements of G/T or EIRP possible. Reduced expressions for the k^'s have been

discussed in an earlier publication [6], and are presented in the Appendix.

The value for TjG calculated from equation (7) pertains to a particular elevation

angle. When this process is repeated for other elevation angles as the radio-star traverses

its path across the sky, the data points in figure 2 from which the calibration curve is derived, are generated. The T^/G value need to determining the satellite signal power is then obtained from the curve at the elevation angle of the satellite.

B. Step Two - Received Satellite Power Measurements

With the T^/G calibration completed the antenna is pointed at the satellite in order to measure the received signal power. Figure 5 illustrates the resulting rf power spectrum as it appears at the LNA input. In this example, the spectmm has a sloping noise baseline

with the satellite signal appearing at the center frequency f^. The shaded areas depict

those frequencies passed by the limiting i-f bandpass filter in the ETMS when the ET downconverter is tuned to f_, f^ or f_|_ respectively. The corresponding i-f powers P_, P^, and

P_l_ detected by the ETMS are related to this i-f power spectrum by the equation

- /w(f) g^(f) df (i - -,o,+) (8)

where i indicates that the downconverter has been tuned to f , f^ or f_^ respectively.

The procedure for normalizing these powers is different from the normalization procedure used in the T^/G calibration, but is still done for the same purpose, to eliminate

gain fluctuations. As previously discussed in the T^/G measurement with the radio-star, y^,

y^^, and y^ are calculated from pairs of power measurements with one member of each pair

having the solid-state noise source turned on. With reference to figure 4, the magnitude of

the added noise power P^ was large enough compared to the P^ (i = 1,2,3) that when P^ was subtracted from the measured power P^ + P^ to obtain P^ significant figures were not lost.

However, in the measurement of satellite power (figure 5), P^ and sometimes P_ and P_^, are sufficiently large compared to P^ to cause significant variation in the P^ determined from

subtracting P^(i » -,o,+) from P^ + P^. This problem is circumvented by pointing the ET antenna away (in azimuth) from the satellite to obtain P^(at f_, f^ and f_|_) by measuring a noise baseline comparable in magnitude to the noise baseline in figure 4, and then normalizing P.(i » -,o,+) with the resulting P 's. These P 's are related to T (f) and the

X a a a receiver gain g(f) by the equation

^ai "

/"^^a^^^ gi(f)df (i = -,o,+) (9)

where the subscript i signifies whether the ET downconverter has been tuned to

f ,f„, or f ,. - o +

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The ratio difference AY of the received satellite signal at the LNA input is calculated

from the equation

AY = - (y_ + y^)/2. (10)

With the Ay's of this equation replaced by their theoretical counterparts, equation (1) can

be derived. Like the k^'s of equation (7), the e^'s are defined in such a manner as to obtain the functional form of equation (1). As in the T^/G measurement, the utility of the radio-star method for measuring EIRP depends upon the e^'s being close to unity. The

working equations for the s^'a used in the ETMS are to be found in the Appendix.

In obtaining equation (1) from equation (10) a brief explanation of how the aspect

angle correction A is defined in this analysis is helpful. Figure 6 depicts a satellite

signal being radiated to an earth terminal located in the direction of the spherical angles

represented by u from the center of the mainbeam of the satellite antenna. The spectral

power p(f) of the signal available at the output port of the satellite transmitter (TXR) is fed to the satellite antenna whose gain in the u-direction is gg((ij,f). The resulting

intensity I(a),f) radiated into the infinitesimal solid angle dtu is

The aspect angle correction A is then defined in terms of this radiated intensity by the

equation

where 1(0, f) is the intensity radiated in the boresight direction of the satellite antenna,

and the integral^ are evaluated across a narrow bandwidth centered on the center frequency

f^ of the signal. The denominator of equation (12) is the EIRP. The ratio in equation (12)

reduces with negligible error to the normalized power pattern of the satellite antenna at

the center frequency f ^

.

III. ERROR ANALYSIS

Examination of equations (1) and (7) reveals that there are twenty-one possible sources

of systematic EIRP error. However, since Ay and AY, and the s^'s and k^'s are related, the

number of sources reduces to thirteen. These thirteen sources of error are listed in the

first column of Table I. This table was prepared for an earth terminal operating at 7.55 GHz with an 13 meter antenna looking at a synchronous satellite at 12° elevation above the horizon. The source magnitudes and uncertainties listed in column two reflect this choice.

The error in EIRP resulting from the various uncertainties are listed in the third column.

Typical random errors generated during T^/G and received satellite power measurements are given in rows 14 and 15. The Y-factor ratio uncertainty (0.029 dB) is calculated from the ratio of Ay appearing

in equation (4) to the AY appearing in equation (10) , although the 10 dB magnitude applies

to Y only. The uncertainty is mainly due to calibration errors in the i-f attenuators in

I(u,f) - p(f) ggdo.f) (11)

^ ^

' I(aj,f)df

/ l(0,f)df

(12)

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the ETMS, Che errors due to the power bridge being considerably smaller.

The Cassiopeia A flux-density uncertainty (26 fu) (fu stands for flux units, one flux

"26 2 unit being equivalent to 10 watts m /Hz/steradian) represents a careful but not quite

state-of-the-art flux density calibration [9]. This uncertainty and the corresponding 4.46%

EISP error appears difficult to reduce at the present time.

The 0.006 dB uncertainty in the space loss is due to range (r of figure 6) uncertainty

which was assumed to be the daily range variation of the satellite used in the example.

The composite ET receiving system-ETMS noise bandwidth is determined at each earth

terminal from previously measurement parameters of the ETMS filters and the gain slope of the ET passband.

The uncertainties listed in rows 5 through 11 were calculated from the k^/e^ ratios for

i equal to 1 to 7 respectively. Most of the ratios shown in column 2 are unity because like

correction factors tend to cancel each other. The exception is the star-shape correction

factor ratio (0.899) where ^2 always unity because the satellite subtends a vanishingly

small solid angle at the ET antenna, and the radio-star does not. The atmospheric transmission error (7.02%) varies with elevation angle, giving the

total EIRP error a characteristic shape shown in figure 7. Table I is a breakdown of this curve at 12° elevation angle. Uncertainties in the ^-^/^j. arise mainly from unpredictable variation in the atmospheric temperature pressure, and water-vapor density profiles

between the T^/G and received satellite power measurements. The large magnitude of this error (7.02%) is due in great part to a lack of measurement data and analysis concerning the

tropospheric scattering effect [6,8]. The error coiild probably be reduced with even a modest effort in this area. The basis for errors 6 and 7 have been previously discussed [6].

The error due to downlink noise temperature variations includes uncertainties arising

from cosmic background variation around the calibrated radio star, and from portions of the satellite signal (if the signal carrier is modulated) appearing in the f and f_^ frequency

bands (see figure 5) reserved for the baseline noise measurements. The satellite signal was assumed to be unmodulated for these calculations, accounting for the small 0.01% error. The 1.31% error from antenna pointing inaccuracies is essentially twice the value when

pointing at either the radio-star or satellite, and is calculated using an assumed pointing

uncertainty of 5% of the antenna half-power-beamwidth.

The polarization mismatch error was calculated assuming axial ratios for the ET and

satellite antennas of 0.5 dB and 0.05 dB respectively.

The time constant of the ETMS is sufficiently short and causes no appreciable EIRP

error. The 0.23% error in row 11 is due entirely to the T^ instability.

The normalized power pattern of the satellite transmit antenna is needed to determine

the aspect angle correction. The 5.29% EIRP error is the result of an aspect angle uncertainty obtained from the literature [1]. The application of emerging spherical near-field

scanning techniques [10,11] to satellite antenna measurements should significantly reduce

this error.

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The quadrature sum of the systematic errors in table I is 10.1%, and is dominated by errors resulting from uncertainties in the Cassiopeia A flux density, the atmospheric

transmission factor, and the aspect angle correction. Representative values for the random

errors generated during the measurement of T^/G and received satellite power increase this

quadrature error to 18.5%. The working equations used in the ETMS for measurement and error calculations are found

in the Appendix.

IV. CONCLUSIONS

The measurement of EIRP as performed by the ETMS has been described, and an outline of

the derivation of the equations leading to the error analysis presented. The results of this analysis are given in table I and figure 7. The results show a 10% quadrature systematic EIRP error for a 12° elevation angle,

increasing to 19% (3a) with addition of the random errors generated during the measurement.

At 12° elevation angle the total systematic error is dominated by errors resulting from

uncertainties in the radio-star flux density, the atmospheric transmission factors, and the aspect angle correction. It appears that the latter two uncertainties might be significantly

reduced with presently available knowledge and measurement techniques.

V. ACKNOWLEDGMENTS

The principle support for this work has come from the U.S. Army Communications Command, Fort Huachuca, Arizona 85613, and is gratefully acknowledged.

The Author would also like to thank Dan Bathker and Art Freiley of the Jet Propulsion

Laboratories, and Tom Bremer, Moto Kanda, Dave Wait, and John Wakefield of the NBS Laboratories

for a number of helpful discussions.

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VI. REFERENCES

[I] Steiabrecher , E. E. , and Gray, L. F., A computer controlled satellite signal monitoring

system, COMSAT Technical Review 1^, No. 1 (Fall 1971).

[2] Stelzried, C. T. , The JPL total power and noise adding radiometers, NBS High Frequency

and Microwave Noise Seminar, unpublished note (May 1970)

.

[3] Larsen, N. T. , and Clague, F. R. , The NBS type-II power measurement system, 1970 ISA Annual Conference Proceedings (Philadelphia, Pa.) Paper No. 712-70 25_, Part 3 (Oct. 1970).

[4] Xanda, M. , An improved solid state noise source, MTT Symposium Digest, Cherryhill, N-J. (June 1976).

[5] Satellite communications reference data handbook. Defence Communications Agency

(July 1972).

[6] Daywitt, W. C. , Error equations used in the NBS precision G/T measurement system, NBS Internal Note 76-842 (Sept. 1976).

[7] Kraus, J.D., Radio astronomy, (McGraw-Hill Book Co., New York, N.Y., 1966).

[8] Yokoi, H. , Yamada, M., and Satoh, T., Atmospheric attenuation and scintillation of microwaves from outer space. Astronomical Society of Japan 22 , No. 4, p. 511 (1970).

[9] Kanda, M. , An error analysis for absolute flux density measurements of Cassiopeia A, IEEE Trans. I&M, IM-25 , No. 3, p. 173 (Sept. 1976).

[10] Wacker, P. F., and Newell, A. H., Advantages and disadvantages of planar, circular

cylindrical, and spherical scanning and description of the NBS antenna scanning facilities, from a Workshop on Antenna Testing Techniques sponsored by the European Space

Research and Technology Center of the European Space Agency, Noordwijk, Netherlands

(June 6-8, 1977).

[II] Larsen, F. H., Probe correction of spherical near-field measurements of a satellite

model, from a Workshop on Antenna Testing Techniques sponsored by the European Space

Research and Technology Center of the European Space Agency, Noordwijk, Netherlands

(June 6-8, 1977).

[12] VanVleck, J. H., The absorption of microwaves by uncondensed water vapor, Phys. Rev.

71, pp. 425-433 (April 1947).

[13] VanVleck, J. H., Theory of absorption by uncondensed gases. Propagation of Short Radio

Waves, pp. 646-664 (McGraw-Hill Book Co., Inc., New York, N.Y., 1951).

[14] Bean, B. R. , and Dutton, E. J., Radio meteorology, NBS Monogr. 92, 271 pages

(March 1966).

[15] Bean, B. R. , Cahoon, B. A., Sampson, C. A., and Thayer, G. 0., A world atlas of

atmospheric radio refractivity , ESSA Monograph 1^ (1966).

[16] Valley, S. L. (Editor), Handbook of geophysics and space environments. Air Force Cambridge Research Laboratories, Office of Aerospace Research, United States Air Force

(1965).

[17] Daywitt, W. C, Atmospheric corrections used in the NBS earth terminal measurement

system, NBSIR (to be published. Fall 1977).

[18] Baars, J. W. M. , Mezger, P. G., and Wendker, H., The spectra of the strongest nonthermal radio sources in the centimeter-wavelength range, Astrophysical J. 142 , pp. 122-134 (1965).

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[19] Raghava Rao, R. , Medd, W. J., Higgs, L. A., and Broten, N. W. , Survey of 9.4 cm radio

emission in Cassiopeia and Cepheus, Royal Astronomical Society Monthly Notices 129 , No. 2, p. 159 (1965).

[20] Dent, W. A., Aller, H. D., and Olsen, E. T., The evolution of the radio spectrum of

Cassiopeia A, Astrophysical J. 183 , L 11 (Feb. 1974).

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VII. APPENDIX: MEASUREMENT AND ERROR EQUATIONS

The ETMS computer software contains a number of simplified working equations for calculating the parameters found in equations (1) and (7), and the various EIRP errors appearing in table I. These equations are collected here with an indication of the references upon which they are based, but with little or no further comment. The equations were

derived for the 1 to 10 GHz frequency range, and do not necessarily apply outside of this

range.

A. T /G Correction Factors - k. a 1

Although these correction factors are defined in precisely the same way as the G/T

correction factors with corresponding working equations previously appearing in the lit- erature [6], a number of the equations have undergone improvements since that time. Therefore, these correction factors will be listed here again. The corresponding errors E(k^) are included for completeness, and are used in determining the G/T or T^/G errors only, not

the EIRP errors. EIRP correction-factor errors are calculated from ^^/^^ ratios, and appear

in a later section of this Appendix.

1. Atmospheric Transmission Factor - k^

k - k k k,

1 g r d

1.1 Gaseous Absorption Factor - k

k - 10-^g

9 » ET antenna elevation angle ^3°

- 6.644 X 10"^P^/?g)^ &

2.75

(1+0. 02215 ZT /T)

o -10.52

0.9211 _^ 5.107(l+3596/f'^)

1+0.2912/f' (1-3596/f^)^

^1 - 5-1^5

\y-J 1-0. 02215 (11. 02-Z)

8.775 + 4.29 1-0. 02215 (11. 02-Z)y^

7.775>

+ (1. 047-0. 09502Z) 1-0. 02215 (11. 02-Z) 7.775

T

11-

(km)
-644/T

Y- = 1.451 X 10^ (p /p ) Z 0 3 1+0.022152

-5.262

1+493. 3/f

(1-493.

p(l+0.0046p) 1^

=. 2.09 + 0.27

1^1- -j-j (km) .

2 r T Y3 - 2.529 X 10"^(P^/Pg) 1^1+0.02215 Z ^l"^*^^^

•p(l+0.0046p) ^

= 2.17 (km) . ET antenna altitude above sea level (km) .

» 293 (kelvins).

Air temperature at EI antenna (kelvins)

.

• 1013.25 m bar

P^ = Reduced-to-sea-level value of atmospheric pressure at ET antenna in m bar.

p " Water-vapor density at ET antenna (gm/m^).

The error in k is

S E(k^) = 50(k"^-l) (%)

This error includes contributions from uncertainties in the constants appearing in

Yj^,Y2>V3»^]_'^2' '^3' ^'^^ earth-terminal altitude; atmospheric temperature and watervapor density profiles; the esc 3 approximation; termination of the calculation at 20 km

from ground level into the atmosphere; and ET antenna pointing errors.

•1.2 Refractive Spreading Factor - k

1.2.1 Gaussian Main Beam With a Disk Source

1 +

e -1

3t

3d

3t_ ^ TT (B -0.026 Ctn 9)Csc^e . ~

39 " 60 X 180

Star diameter (4. 6' for Cas A) ^ ' 1.2 (HPBW)

< 1 .

-12-
HPBW » Half-power-beamwidth (Same units as star diameter)

.

X

B(9>_3'') =- [0-9227(1 -0.022522)^'^^"^ + 0.0202p] Ctn 9.

E(k^) - 18 (k^^ -1) (%) .

1.2.2 Gaussian Main Beam With an Ellip tical Gaussian Source

3t

k^ " k^ factor for an elliptical gaussian source (equation appears later)

.

E(k^) equation is assumed to be identical with the previous E(k^) equation.

This error includes uncertainties in ET antenna pointing, and the refraction angle correction

for antenna pointing (see below)

.

The ET antenna elevation angle 9 differs from the true elevation angle to the extraterrestial source (satellite or radio-star) by a refraction correction x. The working

equation for t derived for the computer software is (9^3°)

T 0.952

T

P /P

° ^ 0.0219P

(1+0.0221SZT /T)^*^^^ ^ o

ctn 9

- 0.0135 ctn^ 9 (arcminutes)

.

E(t) =0.2 ctn 8 (arcminutes).

This error includes uncertainties from the approximate nature of the t working equation, the elevation angle 9, ET antenna altitude, and atmospheric temperature, pressure, and watervapor density profiles.

1.3 Tropospheric Scattering Factor - k^

k , = 1 +

a x^ . e -1

(lO^d C3C9/10_^^

- 0.0011 f2 (dB).

d

E(k,) = 75(k:^-l) (%). a a This error represents only an educated guess because of the present inavailability of a more

exact equation describing the energy lost by scattering of an extra terrestrial radio wave

as it traverses the atmosphere to the ET antenna.

The primary references upon which the preceeding equations are based are 6, 8, 9, 12,

13, 14, 15, 16, and 17.

-13-
2. Star Shape Correction Factor -

For a gaussiaa malnbeam,

^2 ^ —2?— (disk source)

31

\ VI (elliptical gaussian source)

.

= major half-power diameter of source.

= minor half-power diameter of source.

ECk.^) - Sd-k^) + 0.10 (%).

This error includes uncertainties from the approximate nature of the k^ equation, uncertainties in the Cas A disk diameter, and uncertainties in the ET antenna HPBW determined by the

ETMS. The error was derived for this disk source and is assumed to hold for the gaussian

source.The pertinent references for k^ are 6, 9, 17, and 18.

3. Bandwidth Factor - k^

k^ =• 1 ± 0.00% .

4. Differential System Noise Temperature Factor - k.

0.4 T 9

2 E(k^) g — (7,) .

T = Average noise temperature of the cosmic background surrounding the radio star

(= 1 kelvin for CAS A).

9„ = ET antenna HPBW (arcminutes)

.

a

S » Radio star flux density (f.u.).

The pertinent references for k^ are 6 and 19.

5. Antenna Pointing Factor - k^

kj-l.

ECkg) - 0.028 £2 (2).

e » ET antenna pointing error as a percent of the antenna HPBW. The pertinent references for k^ are 6 and 9.

-14-
6. Polarization Factor - kg

E(k,) - d Cos 2e (%).

o a

=• Ctn"-'-(AR^) . AR "• ET antenna axial ratio,

a

d Radio-scar polarization (%)

.

The pertinent references for k^ are 6 and 9.

7. System Response Factor - k.7

k^ - 1 ± 0.00%.

The pertinent references for ky is 6.

B. Radio-Star Flux Density - S

S = S e^*^ f °

o

S - 3154 f .u. ± 4.46%.

° -1

a - - 0.0097 ± 0.0005 (yr ). b - + 0.00126 i 0.00023 (yr"''')

.

o =• - 0.792 £ 0.006.

oAt « Epoch - 1965.0.

E(S) =- 4.46 + 0.050 At + 0.602 ln(£) + 0.023 At ln(f) (%) , The value used for S^ was obtained from an unpublished Jet Propulsion Laboratory measurement performed at 2.2735 GHz in 1972.6, and extrapolated to 1 GHz using the Dent- Aller-Olsen equation [20]. The error for S was derived from reference 20.

o

C. AY-Factor Ratio Differences - Ay. AY The equations for the ratio differences Ay and AY are given in the text. The error for

the ratio AY/ Ay is E(AY/Ay) - 0.66%.

D. Noise Bandwidth - Af

Af » Af^d + bNj^) (MHz) . Af^ " ETMS noise bandwidth (MHz). - 1.193* (MHz). » ETMS filter constant

= - 0.010 (MHz).

E(Af) " 0.38 (%). b * i-f gain slope of the ET spectrum around the signal frequency (MHz ^) . *These values differ from filter to filter and are given here only for illustration.

-15-
E. Space Loss - L

r = slant range (see figure 5) from ET antenna to satellite.

A = Signal wavelength.

The pertinent reference for L is 5.

F. Aspect Angle Correction - A A = Normalized power pattern of satellite antenna in the direction

of the earth terminal.

E(A) = 5.92 (%) (0.25 dB).

This error was obtained from reference 1.

G. EIRP Correction Factors -

1. Atmospheric Transmission Factor - e^

e, = e e e, . 1 g r d

1.1 Gaseous Absorption Factor - e

1 a

Same equation as that used for k^.

E(k /e ) - 60(e"-'--l) (%)

.

g S g

Note that in the EIRP measurement, correction factor errors are calculated from the ratios

k^/e^ because the k^ and e^ are related. In this and the next two errors only atmospheric

profile changes between the T^/G and satellite received power measurements cause significant

errors

.

1.2 Refractive Spreading Factor - e^

Same equation as that used for k^ with x » 0.

E(k^/e^) - 26(e^^-l)(%).

1.3 Tropospheric Scattering Factor - e^

Same equation as that for k, with k * 0. d

E(k^/e^) - 90(e^-'--l) (%)

.

2. Star Shape Factor -

" I ± 0.00%.

-16-
3. Bandwidth Factor - e

e3 = l.

ECk^/e^) =. 0.0058P (%)

.

p = Percentage of i-f passband occupied by received satellite signal.

4. Differential Operating Noise Temperature Factor - e^ % ' ^'

0.4 T P' + P' E(k^/e^) » ^ 2- + 25 ( ~

^

) (") •

P'= Signal power contained in the frequency band (see figure 5) reserved for the lower

frequency (f_) noise measurement.

P_J_ = Signal power contained in the frequency band (see figure 5) reserved for the upper frequency (f_|_) noise measurement.

T = Operating noise temperatures (see text).

5. Antenna Pointing Factor - e^

e^ = 1.

E(k5/e5) - 0.055 (%) .

6. Polarization Factor - e^

E(k,/e,) = d cos 2 e + 100 sin2(e-e ) + 100 cos 2e cos 2z (%) . o o a a a

- O.lld + 0.40 (%)

e - ctn~''"(Ail).

e =• ctn~^(AR ). a a

AH. * Satellite antenna axial ratio in the EX antenna direction (0.05 dB was used for AR,

and 0.5 dB was used for AS. ). a The pertinent reference for e^ is 9.

7. System Response Factor ~

e^ = l. AT E(k^/e^) i Eie^) - ^ = 0.23% . a

T^ » Noise temperature of the solid-state noise source as it appears at the LNA input.

AT^ " Maximum possible change in T^ between the T^/G and received satellite power measurements

.

-17-
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NBS.114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PIJRI iCATION OR REPORT MO

NBSIR 78-869

Z. tyrtv f A. r" f ^ ^ 'Z X f^n No.

4. TITLE AND SUBTITLE

ERROR EQUATIONS USED IN THE NBS EARTH TERMINAL

MEASUREMENT SYSTEM

5. Publication Date December 1977

6* Performing Organization Code

276.05

7. AUTHOR(S)

William C. Daywitt

8. Performing Organ. Report No.

g PFR FOR MING ORnANIZATTON MAMF AMFl ADnR F^"^ NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2765266

11. Contract/Grant No.

12* Sponsoring Organization Name and Complete Address (Street, City, Stste, ZIP) 13. Type of Report & Period

Covered

14, Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less (actual suwmary of most signiUcant information. If document includes a si^ificant

bibliography or literature survey, mention it here.)

An outline for the derivation of equations employed in a measurement and error analysis of satellite EIRP using a calibrated

radio star is presented. A table showing analysis results at 7.55 GHz

using Cassiopeia A for a satellite at 12" elevation angle is given.

The quadrature sum of the systematic errors appearing in the table is

10.1%. Also presented is a curve of the systematic errors as a function

of elevation angle showing a 7.3% minimum at high elevation angles.

Included are working equations for the calculation of correction and

errors.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

C/kT; EIRP; error analysis; G/T; precision measurements; radio star;

satellite communications

.

18. AVAILABILITY

^

Unlimited 19. SECURITY CLASS

(THIS REPORT)

21. NO. OF PAGES

1 1 For Official Distribution. Do Not Release to NTIS

UNCLASSIFIED 32

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Washington. D.C. 20402. SD Cat. No. CU

Office 20. SECURITY CLASS

(THIS PAGE)

22. Price

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Springfield, Virginia 22151

(NTIS)

UNCLASSIFIED $4.50

USCOMM.OC 29042-P74