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United States Department of Commerce

Technology Administration

National Institute of Standards and Technology

N/ST Technical Note 1502

Uncertainties in NIST NoiseTemperature Measurements

J. Randa

1 QC

I 100

1 .U5753

NO. 1302

NIST Technical Note 1502

Uncertainties in NIST NoiseTemperature IVIeasurements

J. Randa

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

March 1998

^ATES O*

U.S. DEPARTMENT OF COMMERCE, William M. Daley, Secretary

TECHNOLOGY ADMINISTRATION, Gary R. Bachula, Acting Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Raymond G. Kannnner, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1502, 52 pages (March 1998)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1998

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
CONTENTS

Page

1. INTRODUCTION 2

2. BACKGROUND AND NOTATION 3

3. WAVEGUIDE SYSTEMS WITH SIX-PORT REFLECTOMETERS 6

3.1 Cryogenic Standard 6

3.2 Ambient Standard 7

3.3 Power Ratios 7

3.4 Mismatch Factors 8

3.5 Asymmetry 11

3.6 Connector 12

3.7 Isolation 13

3.8 Frequency Offset (Broadband Mismatch) 14

3.9 Nonlinearity 16

3.10 Type-A Uncertainty 17

3.11 Combined Uncertainty 21

4. COAXIAL SYSTEMS WITH SIX-PORT REFLECTOMETERS 21

4.1 Cryogenic Standard 21

4.2 Ambient Standard 24

4.3 Power Ratios 24

4.4 Mismatch Factors 24

4.5 Asymmetry 26

4.6 Connector 26

4.7 Isolation 28

4.8 Frequency Offset (Broadband Mismatch) 28

4.9 Nonlinearity 29

4.10 Type-A and Combined Uncertainties 29

5. OTHER SYSTEMS 30

5.1 WR-90 (8 GHz to 12.4 GHz) Switching Radiometer 30

5.2 Tuned Systems for 30 and 60 MHz 32

6. MEASUREMENTS THROUGH ADAPTERS 33

6.1 Background 33

6.2 Uncertainty Analysis 38

111
7. SUMMARY 41

8. REFERENCES 42

APPENDIX 44

IV
UNCERTAINTIES IN NIST

NOISE-TEMPERATURE MEASUREMENTS

J. Randa

Electromagnetic Fields Division

National Institute of Standards and Technology

325 Broadway

Boulder, CO 80303

Uncertainty analyses are presented for NIST measurements of noise

temperature. All systems currently used in NIST calibrations of thermal-noise

sources are treated. These include tuned systems for 30 and 60 MHz, coaxial

total-power radiometers for 1 to 12 GHz, a switching radiometer for the WR-90

(8.2 to 12.4 GHz) waveguide band, and total power waveguide radiometers for

the WR-62 (12.4 to 18 GHz), WR-42 (18 to 26.5 GHz), WR-28 (26.5 to 40

GHz), and WR-15 (50 to 75 GHz) bands. Measurements through adapters are

also analyzed. Typical expanded {k = 2) uncertainties for the measurements are

in the range 0.7 percent to 1.4 percent, depending on the particular system and

the frequency.

Keywords: noise; noise measurement; noise temperature; thermal noise;

uncertainty analysis
1. INTRODUCTION

The National Institute of Standards and Technology (NIST) offers noise-temperature

measurement services covering a wide range of frequencies and connectors. A tuned coaxial

radiometer is used for measurements at 30 MHz and 60 MHz [1]. For frequencies of 1 GHz

and above, almost all the measurement systems are total-power radiometers, with internal sixport reflectometers to measure the relevant reflection coefficients [2-4]. The systems currently

in use include coaxial radiometers covering 1 GHz to 12 GHz [2,3], as well as waveguide

radiometers for the WR-62 (12.4 GHz to 18 GHz), WR-42 (18 GHz to 26.5 GHz), and WR28 (26.5 GHz to 40 GHz) bands [2,4]. Coaxial sources are also measured from 12.4 GHz to

26.5 GHz using the waveguide radiometers and characterized adapters. One waveguide

radiometer of a different basic design is a switching, or Dicke, radiometer [5,6] for the WR90 (8 GHz to 12.4 GHz) waveguide band.

The many different systems and our understanding of them have evolved considerably

over the years, with concomitant changes in the uncertainty analyses. Even for systems

whose uncertainties have not changed, the method of reporting the uncertainties has changed.

Prior to about 1993 the common practice in the NIST Noise Project was to compute and

quote a worst-case or maximum possible error. This was done by estimating the maximum

possible value for each component of the uncertainty and then forming the linear sum of the

individual components. In 1992 NIST officially adopted the policy of reporting uncertainties

which conform to the ISO guidelines [7,8]. Accordingly, the measurement services offered by

the Noise Project now quote an expanded (2a) uncertainty, corresponding to a 95 percent

confidence level. This required a conversion from worst-case errors to standard uncertainties

for the individual components of uncertainty, as well as a change in the manner of combining

the components. As a result of all these changes—in the systems, the analyses, and the

method of reporting the uncertainty—the uncertainty analysis for a typical system is now

scattered in several different places, often rather inaccessible, and there are sometimes

conflicting forms for one analysis. The present paper addresses this problem; it assembles,

reconciles, improves (in some cases), and presents the uncertainty analysis for each of the
systems currently used in noise-temperature measurements.

The next section establishes notation and provides some background. Section 3

contains the uncertainty analysis for the waveguide systems with total-power radiometers and

built-in six-port reflectometers. It presents the estimates of the individual components of the

uncertainty as well as the method of combining them. The similar coaxial systems for 1 GHz

to 12 GHz are treated in Section 4. Section 5 deals with the WR-90 switching radiometer

and the tuned coaxial system. The uncertainty analysis for measurements made through

adapters is presented in Section 6. Section 7 contains a brief summary. There is one

appendix, devoted to the philosophy and practice of converting from worst-case errors to

standard uncertainties.

2. BACKGROUND AND NOTATION

In keeping with the notation of references [7] and [8], we will use Uj^ to denote the

standard uncertainty in the measurement of T^. The combined standard uncertainty is composed of type-A and type-B uncertainties. Type-A uncertainties are those that are

measured and determined by statistical methods, typically the standard deviation of the mean

of several independent measurements of the quantity of interest. Type-B uncertainties are

those determined by other means, such as estimates of systematic uncertainties. We shall deal

primarily with type-B uncertainties; the type-A uncertainty is treated near the end of each

section. We use ^to denote the fractional standard uncertainty in a parameter, for example,

^ry ~ "rcry/^ciy • Bccause we must deal with uncertainty estimates which predate the ISO

definition of standard uncertainty, we need notation for other types of uncertainty estimates.

For nonstandard uncertainties, such as worst-case errors, we use the symbol A with the

relevant variable or parameter. Thus Ar^^^ would be the worst-case error in the ambient

temperature. The relation between worst-case error and standard uncertainty is discussed in

detail in the Appendix. The symbol 5 will be used to denote arbitrary (small) variations in a

parameter. This is used in calculations of the propagation of uncertainties, to determine how

the uncertainty in the quantity of interest is related to the uncertainties in the various other
measured (or estimated) quantities. These variations in a parameter (z, for example) can be

related to the standard uncertainty in the parameter by

«, - <|6z|2> (1)

where the average is taken over the typical variations occurring in that parameter in a (very

large) set of independent measurements.

For the NIST total-power radiometers, which are dealt with in Sections 3 and 4, the

equation used to compute the noise temperature of the device under test (DUT) is

M,Ti, (y -1)

(2)

where a perfect isolator and a linear radiometer are assumed [2,4]. Equation (2) is referred to

as the radiometer equation. The Y 's are the ratios of the detected power from the cryogenic

standard or the DUT to the detected power from the ambient standard, Y^ = p^ /p^ and Y^ =

Px ^Pa- To define the relevant reference planes for the mismatch factors (A/) and efficiencies

(r|), we refer to fig. 1. The cryogenic standard, the ambient standard, and the DUT are

connected to three different ports of the front-end switch of the radiometer. The mismatch

factors in eq (2) are the ratios of delivered to available power at plane 3 for the cryogenic

standard {M^ and at plane 2 for the DUT (MJ. The efficiencies are ratios of delivered

powers at two different planes. For the cryogenic standard, r)^ = Po(S)/p3 , and for the DUT,

Figure 1. Basic setup for measurement of noise temperature.
"Hx = Po(^)^P2- The temperatures appearing in eq (2) are noise temperatures, which are related

to the available noise power hy P = k^BT, where kg is Boltzmann's constant, and B is the

bandwidth. For a passive load, the noise temperature and the physical temperature T^;,^.^ are

related by

k T '^B^ noise hf

(3)

where h is Planck's constant and /is the frequency. For low frequency or high temperature,

the two are approximately equal.

Uncertainties in T^ arise due to uncertainties in the determination of the quantities

appearing on the right side of eq (2) and due to departures from perfect isolation and

linearity. For quantities appearing in eq (2), the propagation of uncertainties is treated in the

usual manner. Thus, for example, the variation in T^ induced by a small variation in the

cryogenic-standard temperature T^ is given by

;(Cry)

=

( T \

1- '^

S a

hT.

(4)

and thus

uACry)

1 --^

a 3

'Cry

(5)

The next two sections evaluate the individual components of the uncertainty for the

waveguide and coaxial total-power radiometers used at 1 GHz and above. The bulk of each

section is devoted to the type-B uncertainties arising in a single measurement of the unknown
noise temperature T^ . The type-A uncertainties evaluated from multiple measurements of T^

are treated near the end of each section. The tuned coaxial system for 30 MHz and 60 MHz

and the WR-90 switching radiometer have different radiometer equations instead of eq (2)

[1,6]. The uncertainties for these two systems are treated in Section 5.

3. WAVEGUIDE SYSTEMS WITH SIX-PORT REFLECTOMETERS

3.1 Cryogenic Standard

The uncertainty in the noise temperature of the cryogenic standard contributes to the

uncertainty in T^ as

uACry)

T.

T

a o ^c^- («)

The fractional uncertainty in the noise temperature of the cryogenic standard, ^^ , depends on which specific standard is used. There is a different cryogenic primary standard for each

waveguide band.

The design and the analysis of the worst-case error for the WR-10 waveguide primary

standard are contained in reference [9]. The same basic uncertainty analysis applies to the

cryogenic standards in the other waveguide bands, with different numerical values for the

individual components of the uncertainty. A notebook kept in the calibration laboratory and

entitled "Thermal Noise Standards: Calculations and User's Manual" contains calculations of

the worst-case uncertainty for each of the NIST waveguide noise standards. It lists the

different components of the uncertainty for each standard, giving a worst-case value for each.

These values reflect an improved understanding of the uncertainties in the resistivity of the

gold-plated horns, but otherwise are based on the analysis of reference [9]. As discussed in

the Appendix, we convert each worst-case value to a standard uncertainty by dividing by y3.

The three principal components are the neglected excess radiation from the cavity walls, the
uncertainty in the load temperature, and the uncertainty in the noise efficiency. These three

are not correlated, and so we are justified in combining them incoherently using root sum of

squares (RSS). When we do so, the values obtained for the primary standards in the different

waveguide bands are

0,18% WR-9Q

0,22% WR-62

^CRY = <^-26% WR-42 .J. 0.17% F^-28

0.39% WR-22

0.48% WR-15.

We shall use the values of eq (7) in eq (6) for the contribution of the cryogenic standard

uncertainty to the standard uncertainty in T^.

3.2 Ambient Standard

The contribution of the uncertainty in the ambient standard temperature to the

uncertainty in the DUT noise temperature is given by

Uj, (amb)

TX

Ts-'T.

a s ^

a. . (8)

The ambient standard consists of a waveguide termination held at a constant temperature by a

water jacket with circulating room-temperature water. The water jacket is itself enveloped in

an insulating blanket. The physical temperature of the termination is measured with a

calibrated thermistor. A conservative estimate of the uncertainty in the temperature

measurement is Uj^ = 0.1 K, and thus ^^ = 0.034 percent.

3.3 Power Ratios

The measured powers enter the radiometer equation through the ratios Y^ = pjl\ and

^s - PjPz iri the factor (7^ - \)I{Y^ - 1), which we define as Y. An analysis of the uncertainty
in Y is outlined by S. Pucic in reference [10]. She parameterizes the effective efficiency of

the thermistor mount as ri^ = r\^Q + kp and then derives the expression for the uncertainty in Y

due to the uncertainty in k. Her expression can be simplified and put into the form

Uy = r\ eO y.-^

1 -Y.

\Px -Ps\

("^eO-^Pj

2 *' (9)

The resulting uncertainty in the DUT noise temperature is

uAY)

s a

X

(^eO-kPaf

(10)

\Px-Ps\ «*

Personnel of the NIST Microwave Power Project indicate that the best that they can do in

measuring linearity with a NIST Type-IV power meter is ±0.1 percent over a range of 10

mW. We assume that we can do half as well and take w^ = 0.02 percent per milliwatt. Since

rjgo " 1 and kp^« 1, and since we keep p^ and p^ less than about 2 mW in our

measurements, eq (10) then implies that

uAY) 1--^ X 0.04% , (11)

which is negligible unless T^ > 3 T^

3.4 Mismatch Factors

The mismatch ratio is one of the principal sources of uncertainty. It contributes to the

uncertainty in the DUT noise temperature according to

Ur(M/M)

^.MjM (12)
Because the ratio of mismatch factors is very near 1, ^^ = U/^^/(M/Af) « U/^^. The

mismatch-factor ratio itself is given by

^x li-TcFJ^ (1 - |rj') (1 - |r,J')

(13)

where the subscript S denotes the cryogenic standard, the subscript r,S denotes the radiometer

as seen from the standard's port, etc. If we compute the variation in M^IM^ due to small

variations in the F's, we get

6

(M,\

^ 2(j:, -xjibx^ - bxj H- 2(y^ ^yji^y, ^ 5v )

,

X r;x'^ X r^' ^x -T^-"- -^ x •' r^'

where x and y refer to real and imaginary parts of F, and where we have assumed that the

reflection coefficients are small. To obtain Uy^^ from eq (14) we need to know whether the

various 5x's and S/s are correlated. If they are all perfectly correlated (i.e., if all the 5jc's and

6ys are equal), then

This is the corrected version of the form in reference [2]. If the 5x's and 6ys are all entirely

uncorrected, eq (14) leads to

J (16)

where we have assumed u^^^= M,^f- . Consultation with the NIST Network Analysis and

Measurements Project indicates that we should expect the variations to be correlated (for

small |r|), which means that eq (15) would be appropriate. We are uncomfortable with eq

(15), however, particularly when the imaginary parts in eq (15) nearly cancel and we have to
rely on the equality of the variations in eq (14) to eliminate any effect of the real parts. This

is particularly unsettling since the mismatch factors are a major source of uncertainty. To be

safe, we shall use the maximum of eqs (15) and (16), thereby protecting ourselves from

artificially small uncertainties due to chance cancellations.

The remaining question is what is Ur^p? The values for u^^p measured with our

commercial vector network analyzer (VNA) are about 0.0013 for WR-62 and WR-42, and

about 0.001 for WR-28 with a precision calibration. In the checkout of each of the noisetemperature measurement system, we compared the results for reflection coefficients measured

with the system six-port to those measured with the VNA or by the NIST Network Analysis

and Measurements Project. The agreement required varied from system to system. For WR62 each reflection coefficient was measured many times on the system six-port, and we

required that the system six-port results agreed with the VNA results to within a standard

deviation of the six-port results. For WR-42 we required that the average of the six-port

measurements agreed with the VNA results within 0.007. For WR-28 we required agreement

within 0.014. For WR-28 we also did additional comparisons to the VNA and to the Network

Analysis and Measurements Project. Those agreed to within 0.002 for | F | and about 0.007

or 0.008 for Im Y. In the initial certification of the WR-15 system, the agreement was within

0.004, but the check was performed only at 63 GHz. Since the levels of agreement were

never exceeded in the tests, we take them to be expanded uncertainties, and we divide them

by 2 to obtain the standard uncertainties. We then have

«/ter = 0-^35 WR-62,WR-42

^^j^

0.007 WR-2S,WR-15

,

The WR-15 uncertainty was increased to the value of the WR-28 uncertainty because of the

lack of information across the entire band.

To summarize, in eq (12) we use ^^ « w^^ and

"a//m = Max{Uj^ji^(cor,), Uj^^j^(uncor.)]

,

(18)

10
with M^^(cor.) and Uf^^(uncoT.) given by eqs (15) and (16), and with w^^^ given by eq (17).

3.5 Asymmetry

The asymmetry is defined as the ratio of efficiencies ris/rix appearing in the radiometer

equation. Its contribution to the uncertainty in the measured noise temperature is

«j.(ti/ti)

L

1 --^ ^''^

where we have used ^/^ « u^/^. Reference [2] derived a value of 0.23 percent for the

worst-case error in the asymmetry. That was based on an analysis of the uncertainty in the

measurements of the power ratios which occur in the asymmetry measurement. However, the

basis for that analysis has been superseded by subsequent work [10], which found that the

power-ratio uncertainty is negligible. Consequently, we must reexamine the uncertainty in the

asymmetry.

The method for measuring the asymmetry is detailed in references [2] and [4]. Two

noise sources (xl and x2) are attached to the two ports {xl on the standard port and x2 on the

DUT port), and we measure the delivered power and mismatch factor of each. The two

sources are interchanged and the measurement is repeated. If we use primed quantities to

denote the second (interchanged) configuration, the expression for the asymmetry in terms of

measured quantities is [2,4]

Tl s ^.1 K (Y.2 -

1) (y^i -

1)

(20)

Ti. ^JM^,M;; (r;^ - 1) (>;; -

1)

Where the Y factors are the ratios of the delivered power from the designated source to the

delivered power from the ambient standard, 7^,' = p^^'/p^ , etc. The uncertainty in this

11
determination of the asymmetry is dominated by the uncertainty in the measurement of the

ratios of mismatch factors.

In principle, we could use the relevant Ps to evaluate the mismatch-ratio uncertainties using

eq (15) or (16), but the system software does not allow this. Instead we will use a

conservative estimate of the typical size of u^/^^. This is obtained by squaring and averaging

the expression in eq (14), and assuming the average of each reflection coefficient is 0.1. That

leads to 2^ 0.2 2 2 2 2 2 < «A<fA/> = ^<Xs ^Xr^ ^ys ^yr,S ^ X^ + ^r,r (22)

^yl^y%>uleT^

With the values of Mr^f from the preceding section (0.0035 for WR-62 and WR-42, 0.007 for

WR-28 and WR-15), we then have

«n/n " ^'^^^ WR-62, WR-42

^24)

0,56% WR-2S, WR-15.

These values are used in eq (19) to compute the asymmetry contribution to the uncertainty in

the DUT noise temperature. The values of eq (24) are quite large, particularly for WR-28 and

WR-15, and are not on a very firm footing. They are safe, but it may be possible to reduce

them significantly with some additional work.

3.6 Connector

Additional uncertainty arises from the variability from one waveguide flange to

another, even when both meet specifications. Such variations are already included in our

estimates of Mr^j- , and they require no further special treatment for their effects on mismatch

12
factors. They do, however, affect the measurement of the asymmetry, and this effect was not

included in Uj^^/r]). When the asymmetry is measured, noise sources are attached to the

ports of interest, and the delivered powers are measured. When this asymmetry is used in a

calibration, we assume that the connectors of the DUT and the primary standard are identical.

The error in that assumption can be written as Ar

1 (25)

where the last term contains the ratio of the efficiencies of the cryogenic-standard connector

and the DUT connector. This last term was estimated in reference [2]. Subsequent revisions

resulted in a value of ±0.003 dB x /''^(GHz) for the range of the term | rj/ri-l | , which we

will use as the standard uncertainty in that quantity. To convert from decibels to a natural

number we multiply by 0.23, thereby obtaining

Uj. (conn)

- 0.069% iL (26)

where /is in gigahertz. This is a sizable contribution to the uncertainty, roughly 0.4 percent

for a typical device at 36 GHz, and is probably an overestimate. An improved analysis could

reduce this uncertainty significantly.

3.7 Isolation

Reference [2] derived an equation for the worst-case error in T^ due to imperfect

isolation,

ATJisol)

0.01x<^0.47 ir. + 0.047 1 - —

|r I + 100 i^ (27)

where T, is in kelvins. This form assumes isolation of at least 50 dB. The WR-62, WR-42,

13
and WR-28 systems all have been measured to have greater than 50 dB isolation across their

entire bands. There was a minor arithmetic error in deriving eq (27) [2]; 0.47, 0.047, and 100

should be changed to 0.48, 0.048, and 108, respectively. Equation (27) represents a worstcase uncertainty. In this case, we are very confident (>95 percent) that the true error is less

than that given by eq (27), and so we divide by 2 (and assume a normal distribution) to get

the standard uncertainty,

Uj. (isol) = 0.01 x< 0.24 ir, + 0.024 + 54 —— (28)

for WR-62, WR-42, and WR-28. This is generally less than 0.03 percent. For WR-15 the

isolation was measured to be greater than 45 dB across the band. In that case,

Uj. (isol)

0.01x<^0.45 ir. + 0.045 1 --^ + 101-^ (29)

which is typically about 0.05 percent.

3.8 Frequency Offset (Broadband Mismatch)

This uncertainty arises from the fact that the reflection coefficients, and hence the

mismatch factors, are measured at the nominal measurement frequency, with negligible

bandwidth, whereas the power measurements are made across a bandwidth as large as tens of

megahertz, the center of which is offset from the nominal measurement frequency by the

intermediate frequency (IF). The uncertainty introduced by possible variation of the mismatch

factors across the measurement band is called the broadband mismatch uncertainty. The

uncertainty due to the center of the measurement band being at/i/p rather than at /is called

the frequency-offset uncertainty. Typical commercial noise sources have reflection

coefficients which are essentially constant for such frequency differences. The same is not

true for the radiometer itself, however. There is a significant length of transmission line

14
between the input ports (planes 2 and 3) and the isolator, which is the principal source of

reflections from the radiometer. Consequently, as the frequency changes, the phase of the

reflected wave observed at plane 2 or 3 will vary, even if the reflection from the isolator is

constant. The worst-case uncertainty due to this effect was calculated in reference [2].

(Slightly different forms were derived in references [11] and [12], but we shall use the more

conservative form of reference [2].) To convert from worst-case to standard uncertainty we

assume a uniform distribution and divide the result of reference [2] by \fi . This yields

u^jBBMM) 200%

COS

i^T^flFK^

30

InBl

sine

15 }

-1

(30)

x(|r5r,^h|r,r,J)

where /f^ is the intermediate frequency, B is the IF bandwidth, and sine x = (sin x)lx. The

electrical length 4 is related to the distance / from input port to isolator by

'. = ' N

/ 1--, (31)

where/, is the cutoff frequency of the waveguide. The values of the parameters for the

different NIST waveguide systems are given in Table 1. Although/^ is nominally 0, there is

Table 1: System parameters for use in eqs (30) and (31).

System /f (GHz) B (GHz) /c (GHz) /(cm)

WR-62 0.040 9.49 56.

WR-42 0.040 14.1 43.5

WR-28 0.040 21.1 50.

WR-15 0.040 39.9 36.

15
in fact a narrow notch in the frequency response at 0, due to the low-frequency response of

the mixer used in the down conversion. The shape of the frequency-response curve is

accounted for in the noise-temperature measurements. Its effect on the uncertainty estimate is

negligible, and we can use eq (30), which was derived with the assumption of a flat response

across the IF bandwidth.

3.9 Nonlinearity

The final contribution to the type-B uncertainty is due to the possibility of small

nonlinear behavior by the radiometer. Reference [2] derives an expression for this

uncertainty, but this result was subsequently subjected to major revisions. Rather than try to

resurrect the old analysis, we use our routine IF linearity checks [4] to place a limit on the

uncertainty due to nonlinear behavior.

Our routine linearity test consists of switching a (nominally) 3-dB pad in and out of

the measurement circuit, in front of the IF amplifier. The ratio of the two output powers is measured and plotted as a function of input power. The "linear" result is established by

averaging the low-power points, and we then require that the ratio not deviate from the linear

result by more than 0.005 dB (0.12 percent). The nonlinearity in the IF amplifier can be

parameterized by [2]

Si

out ^ „ ^m ag2

^^^Pin (32)

"<?2/'in(l-a/'«)'

where a = ag-^k. The ratio of output powers for the case of /?2,m ^ ^ P\,m is then given by

16
yifiut y\,in (33)

The linearity check assures us that op,^ < 0.12 percent. That is a worst-case error; we'll take

the standard uncertainty to be half that, 0.06 percent. This introduces a corresponding 0.06

percent uncertainty into the measurement of Y^ since the nonlinearity is negligible for

measurement of P^. The resulting uncertainty in T^ is given by

Uj, (lin)

TX

1 --^ \

(34) u-^ = 0.06% .

3.10 Type-A Uncertainty

In noise-temperature calibrations using our waveguide total-power radiometers, we

repeat measurements on several different levels. Typically we do three independent

calibrations of the entire system (six-port calibration plus asymmetry measurement). For each

of these three calibrations we perform one (for "mature" systems) or more (three for WR-28)

separate measurements of the reflection coefficients of the cryogenic standard and the DUT.

And for each of these separate measurements we do a number of readings (typically 20) of

the delivered powers from the two standards and from the DUT. A noise temperature of the

DUT is computed for each DUT power reading. Thus, in a typical WR-28 calibration, we

have three system calibrations times three measurements times twenty readings, for a total of

180 values of the noise temperature, which are combined to obtain the final value for the

measured DUT noise temperature. The question is how to evaluate the type-A uncertainty in

obtaining the average value from the multiple measurements.

17
We use Ty^ for the value of a single reading of the noise temperature. The index /, which denotes the number of the system calibration, runs from \ Xo N^, \ the index 7 denotes

the measurement number and runs from 1 to A'^ ; the index k denotes the reading number and

runs from 1 to Nj^. We assume that the number of readings N^ is the same for each

measurement and that the number of measurements N^ is the same for each calibration. We

use T^j to denote the mean of the N^ readings for each measurement and use a^ for the

associated standard deviation,

f^R ^iJ-

^'»^ t=i (35)

1 ^"

Similarly, we use T, and a, for the mean and standard deviation of the N/^ measurements for

calibration /,

(36)

2 Of =

1

and we use T and a for the mean and standard deviation of the N^ calibrations.

Nc

(37)

^CM

o^ = i y{T-Tf,

Reference [4] reported that we use the standard deviation of the mean of the A^^ values of T^

as the type-A uncertainty. It was recognized in reference [4] that this was not entirely

18
correct, but it was very close for the WR-62 and WR-42 systems and for some frequencies for

WR-28. In some recent cases in WR-28, however, it became clear that this estimate is

inadequate, and we must modify our computation of u^. In particular, the spread in the 20

readings for a single measurement is far from negligible and is not accounted for in the

previous evaluation.

To develop the more general expression for u^ we model [13] the variable T^^. as

TV, = X . C, ^ M.. ^ R.„ (38)

where x is the true value for the noise temperature of the device being measured, C, is a random variable representing variations from calibration to calibration, M,j is a random

variable representing variations from measurement to measurement (J), which also varies from

calibration to calibration (/), and Rjj^ is a random variable which varies with the reading k, the

measurement 7, and the calibration /. Our estimate of the true value is just the mean of all the

readings.

^C ^M N^

T « r^ —-—EEE^*-

^c^M^R «=i y=i *=i

(39)

The means of the three random variables are all 0, and their variances will be denoted by

<Cf> - Vc,

<Mf> = v^, (40)

<^k> = ^R,

where the averages are over all indices. The variances can be estimated from the measured

values Tjji, by

19
^ E(^.-7;-)^-v,,

' E(^(/-^)^«V^-^. (41)

The equalities are only approximate because we are dealing with a limited sample. We can

solve eq (41) for Vr , v^ , and Vc , and use eqs (35) through (37) to write

^R = <aj> >

Vm = <o]>- "« N/

Vc = O^

^M

Nm

v*

NmN,

(42)

where the averages are over all free indices and where it is understood that if a negative value

results for v^ or Vc it is taken to be 0. The type-A uncertainty in the determination of x « T

is then the square root of the variance in T,

^A "^ !^ ^ ^:«_ H. ^/? (43)

Comparison to the last line of eq (42) reveals that eq (43) is equivalent to u^^ = cj^/Nc , provided that eq (42) does not result in a negative value for v^. This is as it should be: if we

did enough independent system calibrations to determine a^ well, that would be sufficient,

since it includes the variations from one reading to another and from one measurement to

another. Since we only do three independent system calibrations, we supplement that

information with the measured a,^ and o,^. Equation (43) is now used to evaluate the type-A

uncertainties in the noise-temperature calibrations using our waveguide total-power

radiometers.

20
3.11 Combined Uncertainty

The type-B standard uncertainty for a single noise temperature measurement is

obtained by forming the square root of the sum of the squares of the individual contributions

eqs (6, 8, 11, 18, 19, 26, 28 or 29, 30, 34),

^B ^ ["r(^O') XX^ Uj-iamb) + Uj-iY) XX+ Uj-iM/Af) + Uj'(r\/r])

X (AA) + Uj{con) + Ujiisol) + Uj{BBMM) + «j.(/m)]'^.

In calibrating a customer's device, we make several measurements of its noise temperature.

Because the uncertainty defined by eq (44) depends on the measured noise temperature and

on various measured reflection coefficients, it is in principle different for each of the separate

measurements of the device's noise temperature. In practice, however, there is little difference

between the values, and we use the maximum.

The expanded {k = 2) combined uncertainty is computed from eqs (43) and (44),

u - 2^/;^T^. <«)

The expanded uncertainty varies with the device being tested, the frequency, and the

waveguide band. At present, typical values of the expanded uncertainty for a source with

high noise temperature (above a few thousand kelvins) are about 0.7 percent to 0.9 percent for

the WR-62 and WR-42 systems, and 1.0 percent to 1.4 percent for WR-28. The WR-15

system is currently undergoing tests. When it is back in service, we expect its expanded

uncertainties to be around 1.5 percent.

4. COAXIAL SYSTEMS WITH SIX-PORT REFLECTOMETERS

4.1 Cryogenic Standard

As was the case for the waveguide systems, the uncertainty in the noise temperature of

the cryogenic standard contributes to the uncertainty in T^ as

21
uACry)

a a

^.Cry

(46)

At present there are two coaxial, cryogenic, primary standards used in noise-temperature

measurements, called Standards C and D. A new standard. Standard E [14], is currently

under test. Standards C and E have GPC-7 connectors, whereas Standard D has a precision

type-N connector. Standards C and D are of nearly identical design, differing only in the

output connector. Their uncertainties were originally analyzed in reference [15]. The analysis

was later revised in an internal memorandum. The more recent result for the worst-case

uncertainty is AZ^ = [1.54 + (0.134 + 4.72Co3)/^ + QAA{f)]% , Cry

m =(CoiH-Co.+C)/^ + a11 (47)

1 +-H

f

where the frequency f is in gigahertz. The values of the parameters for the two standards can

be found in Table 2. To compute a "standard" uncertainty from this worst-case value, we

treat the separate contributions as independent and divide the root-sum-of-squares result by

yfh. That results in

^cry = [1.813 +(0.01013 +21. 174Co3)/ + 0.16^(/)2j^7\/3. (48)

Numerical evaluation of eq (48) yields the fractional uncertainties of Table 3 for the two

primary noise standards currently used in our coaxial calibrations. Equation (48) or Table 3

is then used in eq (46) to yield M;.^(Cry).

22
Table 2. Parameters for evaluation of uncertainties in coaxial primary standards, eq (47).

Cqi ^-^02 ^2 ^03 ^11 ^12

Std. C 0.0103 0.0060 0.0120 0.0245 0.0660 0.3654

Std. D 0.0092 0.0100 0.0080 0.0224 0.0450 0.3020

Table 3. Standard fractional uncertainty in noise temperatures of coaxial primary standards.

f(GHz) ^Std. C), % ^Std. D), %

1.0 0.782 0.782

2.0 0.787 0.786

3.0 0.792 0.791

4.0 0.797 0.795

5.0 0.802 0.800

6.0 0.807 0.804

7.0 0.812 0.808

8.0 0.816 0.813

9.0 0.821 0.817

10.0 0.826 0.821

11.0 0.830 0.825

12.0 0.835 0.830

23
4.2 Ambient Standard

The contribution of the uncertainty in the ambient standard temperature to the

uncertainty in the DUT noise temperature is given by

Uj, (amb) T-T

a s

a Qp (49)

The temperature of the ambient standard for the coaxial system is measured with a glassmercury thermometer rather than a thermistor as in the waveguide case. Comparison of the

thermometer to a thermistor calibrated by NIST (±0.002 K) indicated agreement to within

±0.02 K. The other contribution to the uncertainty in the temperature of the ambient standard

is the possibility of a small difference between the temperature measured by the thermometer

and the actual temperature of the load in the standard, due to imperfect thermal contact and

thermal gradients. This contribution should be similar to that in the waveguide ambient

standards. We therefore take the uncertainty in the temperature of the ambient standard to be

the same as in the waveguide case, Uj^ = 0.1 K, i^^ = 0.034 percent.

4.3 Power Ratios

In the waveguide section above, we concluded that this contribution is negligible.

Power measurements are done in the same way for coaxial calibrations as for waveguide

calibrations, and so Utx(Y) is again negligible.

4.4 Mismatch Factors

As in the waveguide case, Subsection 3.4,

Ur(M/M)

^.MjM

T. 1--^

*MIM

(50)

24
The mismatch uncertainty depends strongly on the poorly known correlation between

uncertainties in the measurements of different reflection coefficients and again we will use the

maximum of the uncertainties obtained by assuming complete correlation and no correlation

whatsoever.

As above, the correlated and uncorrected uncertainties are given by

(53)

where x and y refer to real and imaginary parts of the corresponding reflection coefficients.

The question then is what is WRer? References [2] and [4] use ±0.005 for the worst-case error

in the real or imaginary part. Comparison of reflection-coefficient measurements on our

system to those made by the NIST Network Analysis and Measurements Project on the same

devices indicated agreement within about ±0.002. Our year-to-year variations in

measurements on the same device are well within ±0.005. In the system calibration, we

require that the reflection coefficient of the check standards agree with historical values within

±0.0025. These facts lead to the conclusion that 0.005 represents a conservative estimate of

our worst-case error, and we therefore use half that for the standard uncertainty in real and

imaginary parts of reflection coefficients.

"jfer =«;TMr = 00025 (54)

for all our coaxial systems.

25
4.5 Asymmetry

The form for this uncertainty is the same as for the waveguide case, eqs (19) and (23)

above,

«t(ti/ti) 1--^ \h (55)

«n/n " ^'^^Rer- (56)

From the preceding subsection u^^^ ~ 0.0025, and so

«r(il/^)

1 -^xO.1% (57)

4.6 Connector

As in the waveguide case of Subsection 3.6 above, this uncertainty arises from

variability from one connector to another of the same type. This variability affects the

measurement of the asymmetry, which is measured by measuring the powers from two check

standards connected to the DUT port and the cryogenic-standard port, and then switching the

two check standards and comparing the measured powers [2,4]. For our coaxial calibrations,

the DUT reference plane is located between the DUT and the connector, and similarly the

cryogenic-standard reference plane is between the standard and the connector. Thus the

connectors are part of the two paths being compared. Any difference in the connection of

DUT to the test port and the connection of the check standard to the test port, or between the

connection of the cryogenic standard and the connection of the check standard, will result in a

difference between the measured asymmetry and the actual asymmetry occurring during the

measurement of the DUT.

26
To estimate the variability from one connector to another of the same type, we turn to

results of an Automated RF Techniques Group (ARFTG) comparison conducted by NIST

[16]. In this comparison several devices for each connector type were measured at NIST and

at a number of different industry laboratories. Each laboratory measured 5",, and ^,2 for each

device, and the standard deviations of the samples were computed. The variations observed

in this comparison are due to a number of factors, such as differences between the methods

and equipment at the laboratories, as well as differences between the connectors on the cables

used by the laboratories. Thus the standard deviations observed in reference [16] represent a

very conservative estimate of connector variability. The particular results we use are those

for the 20-dB attenuator for each connector, computed from the trimmed data (outliers

excluded). We expect the variability to increase roughly as/''. Taking into account the fact

that the results of reference [16] represent maximum values over the stated frequency ranges,

we infer the following parameterizations of the connector variabilities (in the square of

l'^,2l):

0.00053 v/? GPC-7

Oconn « 0.00066 v^ TypeN (58)

0.00062 v/f 3.5 mm,

where /is in GHz. Because the devices measured were two-port devices, the results of eq

(58) represent the variations from two connectors. Since there are two relevant connections in

the radiometer measurements, eq (58) can be used directly, resulting in

Uj, {Conn)

^x _

rp Conn X

(59)

Besides the connectors of eq (58), we also calibrate noise sources with 14-mm connectors,

which were not treated in reference [16]. For them we shall use the GPC-7 numbers.

27
4.7 Isolation

This contribution has the same general form as the waveguide case, provided that the

isolation of the radiometer is the same. All the NIST coaxial systems have isolation of at

least 40 dB. Measurement of the 8 GHz to 12 GHz system indicated that its isolation

exceeds 50 dB across its entire frequency range. We shall therefore use

Uj. (isot) = o.oix-to.24 ir

T.

l--^ + 0,024 1 - .54ii:^ (60)

for the 8GHz to 12 GHz system and

Uj^ {isot)

0.01 X

T, n

0.8 r. 1 -

" + 0.08 1 - "1 L ^.

+ 180 -i-^ (61)

for 1 GHz to 8 GHz, where T^ is in kelvins.

4.8 Frequency Offset (Broadband Mismatch)

The form of this uncertainty is the same as for the waveguide case,

u^jBBMM) ^ 200%

COS

30 J

. {tzBI\

sind

\ 15 j

-1

X (ir^r.J Hr.r, J)

T.

l--^

(62)

except that the electrical length ^ has reverted to the physical length /, and the parameters /p,

B, and / are different. The values of the system parameters are given in Table 4. The IF

frequency refers to the offset between the measurement frequency and the center of the pass

band. The bandwidth (for/p ~ 0) 'S the full detection bandwidth, which is twice the

bandwidth of the IF filter.

28
Table 4. Coaxial system parameters for use in eq (62).

System /p (GHz) B (GHz) / (cm)

1-2 GHz 0.010 116.

2-4 GHz 0.010 72.

4-8 GHz 0.010 76.

8-12 GHz 0.010 61.

4.9 Nonlinearity

The final type-B source of uncertainty in the DUT noise temperature is nonlinearity of

the system. For coaxial calibrations, system linearity is tested during each noise-temperature

measurement. Twenty five readings are taken with a 3-dB attenuator in the IF section of the

system, and 25 readings are taken without the attenuator. The two resulting noise

temperatures must agree within 0.2 percent. Using the same analysis as in Subsection 3.9

above, this translates into

u-r (lin)

-^ = 0.10% , (63)

TX

4.10 Type-A and Combined Uncertainties

For calibrations using the coaxial systems, the system six-port reflectometer is

calibrated just once, and the DUT is disconnected and remeasured several times on two

different measurement ports. The type-B standard uncertainty is essentially the same for each

of the measurements and is obtained by forming the square root of the sum of the squares of

the individual contributions eqs (46, 49, 50, 57, 59, 60 or 61, 62, 63),

29
"b "^ [«r(^0') "^ Uj{amb) + Uj{Y) + Uj(M/Af) + Uj(T]/r\)

(64) + Ujiisot) + Uj{con) + UjiBBMM) + Mj,(/m)]'^.

jr

The type-A standard uncertainty is just the standard deviation of the mean of the different

measurements.

^A

E (t - TJ' (65)

\ N(N -

1) '

and the expanded uncertainty is twice the combined standard uncertainty.

U^^=2J4^^4^. (66)

5. OTHER SYSTEMS

5.1 WR-90 (8 GHz to 12.4 GHz) Switching Radiometer

For the WR-90 band, a switching, or Dicke [5], radiometer is used [6]. The quantity

measured with this system is the effective noise temperature delivered to a reflectionless load

(r„J. It is related to the available noise temperature T, which is measured by all the other

systems, by

T^ = (l-lr^r, (67)

where F is the reflection coefficient of the noise source. The current uncertainty analysis uses

the original analysis [6] as a starting point. Five sources of uncertainty were identified

initially, and a sixth was added later. The five original components were due to the effective

noise temperature T^^^ of the working standard, the temperature T^ of the precision attenuator,

the attenuation difference D between the standard and unknown noise sources as measured

30
with the precision attenuator, the measurement of the reflection coefficients F using a tuned

reflectometer, and the determination of the mismatch factor M. The later addition was the

uncertainty in the loss in the waveguide flange. The original analysis took the maximum

values of the five separate uncertainties and added them linearly to obtain a worst-case error. We take each of the five maximum contributions and divide by 2 to obtain a standard

uncertainty, yielding

AT c u{Std) = 10^/10^^,

uiP) ^Q,23(T^-TJ^,

u(T.) - (1 - 10^/^0)

2

Ar.

«(r) - r,(io^/io|rjp-h|rj2)

AFe (68)

U{M) =1

(1^5,1 rj)^

(i-15„r,|)^

- 1 [Tne-T^l-K?)]

where the subscript S refers to the standard, and x refers to the DUT. The worst-case error in

the noise temperature of the working standard depends on frequency and the particular

standard used; it is typically around 100 K out of a noise temperature of the order of 10'^ K.

The worst-case error in D is AD - 0.006 or 0.008, depending on the settings of the attenuator.

Values of the other parameters appearing in eq (68) are

AT

l^iJ - 0.01, ^T^ = 2K, -=^ - 0.1. (69)

The contribution from the uncertainty in the flange loss is given approximately by

AF/

uiFi) - ^(10^° -l)(^,-^,)(l-|^5p)l0'^ (70)

31
with AFl - 0.005. The type-B uncertainty in the measured effective noise temperature is then

Ug = ^u(Stdf + «(D)2 + «(r^)2 + u(jf + u(Mf + u(Flf . (^1)

There are two type-A uncertainties. The noise temperatures of the working standards were measured several times against the primary (oven) standard [17]. The resulting

contribution to the uncertainty in T„^ of the DUT is

_D

10 .. (72)

where u^^ is the standard deviation of the mean of the measurements of the noise temperature

of the working standard. In measuring the noise temperature of an unknown device, five

separate measurements of its noise temperature are made and averaged. The standard

deviation of the mean of these five measurements is the other type-A uncertainty, u^j- The

full type-A uncertainty is then

and the expanded uncertainty quoted in calibration reports is

with u^ and u^ given by eqs (71) and (73).

5.2 Tuned Systems for 30 and 60 MHz

At 30 MHz and 60 MHz, noise temperature is measured on an unisolated radiometer,

using tunable standards [1]. The original estimation of the individual components of

uncertainty is still used, with two exceptions. The uncertainty in the temperature of the

ambient standard has been converted to a standard uncertainty by assuming a uniform

32
distribution and dividing by ^, and the uncertainty in the power ratio is now negligible, as

discussed in Subsection 3.3 above. For the other components—the cryogenic standard, the

mismatch factor, the nonlinearity, switch asymmetry, and the adapter (when present)—the

original values from reference [1] and the associated computer program are still used. The

type-A uncertainty is the standard deviation of the mean of the independent measurements,

and the individual components are added in quadrature to obtain the combined uncertainty.

6. MEASUREMENTS THROUGH ADAPTERS

6.1 Background

In certain frequency ranges and for certain connectors, the noise temperature

calibrations are performed through adapters. In such cases, we measure the noise temperature

of the device-adapter combination, and we must then correct for the effect of the adapter in

order to determine the noise temperature of the device alone. Referring to fig. 2, we can

write

T^ = T..n = a^i^r + (1 -ai,)^;, 2V ^a

^21

(75)

1

Figure 2. Reference planes for a noise source with an adapter.

33
where (Xj, is the available power ratio, a2i - P2 lp\ , and where the adapter is assumed

to be at ambient temperature, with noise temperature T^. Thus, if we can determine 0C2], we

can compute the device temperature T^ from the device-adapter temperature T^+^. To

determine 021, we follow Engen [18] in noting that oCj, = ri,2, where r|,2 is the efficiency,

''121 ~ P2 IPi ^^^ reciprocal devices r|,2 = r\2u ^^ so otj, = r|2i for reciprocal devices.

We then use Daywitt's method [19,20] to measure rij,.

We need not reproduce the algebra, but there are two key equations needed from

reference [20]. The first approximates the efficiency as the intrinsic efficiency for a low-loss,

well matched device,

^21 '^ Ti [i4-2/?e(xrp],

(76)

X « 522(1-Tl2i^),

where F, is the reflection coefficient of the load under "normal" use (the radiometer in our

case), and 1^21 is the intrinsic efficiency of the adapter, ri^^ =

|'^2iP/(^ ~ 1*^111^)- Since both

X and r, are small, eq (76) is used to justify the approximation riji " il2i . The other key

equation relates the intrinsic efficiency to the reflection coefficient r2 from the adapter at

plane 2 when the adapter is terminated with a low-loss reflective termination at plane 1,

\ \^2\ = ^2ljr«l - |X|C0S<J), (77)

where T^, is the reflection coefficient of the reflective termination (='!), and (]) is a phase angle

which varies (approximately) linearly and relatively rapidly (compared to tIji ) with

frequency. Equation (77) indicates that I Fj I consists of a rapidly varying piece, due to the

second term on the right side, superimposed on a more slowly varying piece due to the t]2i

term. The idea then is to measure I Fj I and smooth out the oscillations in frequency. That

34
should yield TI21 |r^|, which can be divided by I F^, I to yield the desired quantity. For a

good flush short I F^J can be taken to be 1, but for an offset short or offset open a correction

is usually required to account for the loss in the small length of line constituting the offset.

The process of smoothing the oscillations in frequency is facilitated by measuring

I F2 1 with two different reflective terminations, whose reflection coefficients differ in phase

by n. Typical choices would be a flush short and an offset short for a waveguide port, or an

offset open and an offset short for a coaxial port. As an example, we consider evaluation of

an adapter from 3.5-mm coaxial line to WR-42 waveguide. Referring to fig. 2, we identify

port 1 with the 3.5-mm port and port 2 with the WR-42 port. We measured I F2 1 , the

reflection coefficient from port 2, when port 1 was terminated with two different reflective

loads, in this case an offset open and an offset short. The offset refers to a short length of

transmission line before the open or short. The open for the coaxial line is achieved by

truncating the inner conductor, replacing it with a length of dielectric of equal radius, while

the outer conductor is continued to provide shielding. The results are shown in fig. 3. Each

curve is approximately what would be expected from eq (77), regular oscillations (x cos(()) on

a smooth overall frequency dependence (tjji |r^|). The smaller wiggles on the large waves

are attributable to VNA imperfections.

There could be some question of how we know that the larger waves, which we

attribute to Xcos(j), are not due instead to VNA errors. By comparing the respective

magnitudes of % and VNA uncertainties to the size of the oscillations, we can argue that the

oscillations are due primarily to the expected xcos(j) term; but there is also more direct,

compelling, experimental evidence. Figure 4 shows results of measurements of I F-, I made

first on an adapter with two different reflective terminations and then on the same adapter and

terminations with a 3.5-cm length of line inserted between the adapter and the termination.

The wavelength (in frequency) of the large oscillations becomes much shorter when the line is

introduced, indicating that these waves are due to the device rather than to the VNA. (We

will revisit this point in the uncertainty analysis section below.)

35
>

•ao

2

l-l

CO

N

o

(N

(N

O

<N

Os

m

00 o 00

ON On On On OOn

JU910IJJ903 U01P9|J9"^

36
"X ^^x+^*

oo

d a

o o • *-H • ^H M CO c d

u (U ^(U ^a> •5 -B

• ^H^ ?

p. t £i t;

d) o (U o

&- J3 a X

o CO o CO

•<-> -<-> -t-l -4-> (U D o (U

-t/1 -«5 .CO .CO s^

<+H ^<+H ^«+H ^<+-! O o O o

+ X o o

o

(N

e

N)

with

reflective

n

hne.

»n

om

adapter

(3.5

mm

to

typ

without

(upper

pair)

3.5

ci

/"^^N tb -O o

N ntsl )

an

1—

O

c^-i

Measured

reflection

coeffi

terminations,

with

(lower p

»n 2

o ^ s s s

oON ^ o o o o o

JU9piJJ903 U01P9]J9^

37
Returning to the determination of TI21 , at present we do the smoothing in fig. 3 by

hand, simply by drawing a smooth curve corresponding to an average of the sinusoids. As

pointed out in reference [20], it should be sufficient to use just one of the experimental

curves, but in this case the smoothed curves from the two sinusoids coincide. From eq (77)

we see that the smoothed curve corresponds to 1121 |r^| . Numerical values can be read off

the graph. To obtain the intrinsic efficiency we divide by I F^, I , the reflection coefficient of

the reflective termination. For the case shown here, I F^, I was calculated from data provided

by the manufacturer, with the result that | F,J = 0.9965 at 18 GHz and 0.996 at 26.5 GHz.

6.2 Uncertainty Analysis

Starting with the second form of eq (75) and taking small variations of both sides, we

obtain

bT = —bT - I — - 1 X x+a I a la

^bT+{T-T)^. (78)

. V . a^ ^

In principle, dT^+^ and dT^ are correlated since 5r^+^ contains a contribution from the

uncertainty in the ambient temperature. In practice, however, the dT^ term in eq (78) is

negligible (roughly 0.01 K), and we will use

^x =

\ a

The M^+^ is the standard uncertainty in the measurement of the device-plus-adapter, evaluated

as in Section 3 or 4 above. The problem is to evaluate u^ when a is determined by the

method described in the preceding subsection.

There are four major components to w^' the uncertainty associated with determining

Ti2i from the measured I Fj I , the VNA uncertainty associated with the measurement of

38
IFjI, the uncertainty introduced by the approximation 1121 ~

"Hzi ' ^"^ the uncertainty

introduced by connector repeatability (or lack thereof).

The determination of TI21 from the graph can contain both type-A and type-B

uncertainties. The type-A uncertainty arises from the fact that in effect we have two quasiindependent measurements of TI21 , arising from the measurements with the two different

reflective terminations. If we denote the two smoothed curves as r|, and r\2, this contribution

is given by

(80)

In principle, M;^ varies with frequency, but in practice we use the value at the point of

maximum separation between r|, and r\2. In the example considered above, this contribution

is negligible.

The type-B part of this contribution to the uncertainty arises from the uncertainty in

determining the appropriately smoothed curve for each of the measured I H I curves. The

uncertainty in drawing the smoothed curve is about 0.002 in typical cases, such as that of fig.

3, leading to

» StIi + 6TI2

Si,B- -^ (81)

,2 4 4

Combining the type-A and type-B uncertainties for the present example results in u, = u,^

0.0014.

39
The second source of uncertainty is the VNA measurement of i Fj I , for which we use

the uncertainties provided by the manufacturer. The manufacturer does not quote

uncertainties for one-port measurements, and so we use the values for the uncertainty in liS")] |

in two-port measurements. This should represent an overestimate of our actual one-port

uncertainties since the two-port uncertainties include the effect of cables, which are not used

in our measurements of the reflection coefficient. In addition, our smoothing procedure may

eliminate part of the VNA calibration error. However, since we do not have a good way of

estimating how much we have improved the VNA uncertainty, we will just use the full values

given by the manufacturer. These values depend strongly on the connector type and also

depend on the frequency and on the magnitude of the reflection coefficient being measured.

The relevant connector type for us is the connector at plane 2 (WR-42 in the example), since

that is where we are making the VNA measurement. The standard uncertainty for WR-42 is

«2 - 0.0025 . (82)

From the smoothing of the measured curves for | Fj I we obtain the intrinsic efficiency

T]2i , which we then use as an approximation to the efficiency r|2,, eq (76). The term we

neglect has a maximum magnitude of I 2xF, | . We do not know the relative phase of x and

F,, and so we assume the correction term has a uniform probability to assume any value

between plus and minus its maximum magnitude. We can estimate this magnitude by noting

that F, is the reflection coefficient of the normal load, the radiometer. For WR-42, 1 F, | <

O.L The magnitude of % can be estimated from the magnitude of the large oscillations in the

graphs of I Fj I , using eq (77). From fig. 3 we estimate that I x I is about .0035 for the

present adapter. Consequently, the neglected term has a maximum magnitude of about

0.00085. Dividing by \/3 to obtain the standard uncertainty, we have u^ = 0.005. As an

aside we note that we can also check the effect of neglecting the imaginary part of xF, in

obtaining eq (76). The effect is entirely negligible (-10"^), as expected.

Finally, there is the issue of connector repeatability. The uncertainty due to the

nonrepeatability of the connector at port 2 is included in the VNA uncertainty. The connector

40
at port 1 is 3.5-mm coaxial in our example. A recent comparison by the NIST Network

Analysis and Measurements Project achieved a repeatability of about 0.0003 with 3.5-mm

connectors. We will use a more conservative value of 0.001, which is also approximately

consistent with the experience of our technicians,

«4 = 0.001 (3.5-mm connector). (83)

The combined standard uncertainty for a is obtained by adding the four individual

components in quadrature, with the result

(84) - 0.003

This value is then used in eq (79) to compute the standard uncertainty in the determination of

T^. Typical values for u^^ range from 0.003 to 0.006. In a typical calibration, the use of an

adapter increases the expanded uncertainty by a few tenths of a percent or less for frequencies

up to 26 GHz.

7. SUMMARY

The uncertainty analyses for all the NIST noise-temperature calibration systems were

presented, including calibrations made through adapters. Although many of the individual

components had been evaluated previously, some had not been, and many were reevaluated,

corrected, or converted from worst-case errors to standard uncertainties. In some cases,

previously scattered treatments were integrated and, when necessary, reconciled. In all cases,

the uncertainty analysis is now consistent with the CIPM guidelines [7,8]. Typical expanded

{k - 2) uncertainties in the measured noise temperature are in the range 0.7 percent to 1.4

percent, depending on the particular system and the frequency.

41
Much of the content of this paper is the direct or indirect product of numerous

conversations with former members of the NIST Noise Project, in particular Bill Daywitt,

Jack Rice, and Dave Wait. I am also grateful to Bob Judish for discussions about statistical methods and to John Juroshek for discussions about VNA and six-port measurements and

connectors.

8. REFERENCES

[1] Counas, G.J., and Bremer, T.H. NBS 30/60 megahertz noise measurement system

operation and service manual. Nat. Bur. Stand. (U.S.) Interagency Report 81-1656;

1981 December.

[2] Daywitt, W.C. Radiometer equation and analysis of systematic errors for the NIST

automated radiometers. Natl. Inst. Stand. Technol. Tech. Note 1327; 1989 March.

[3] Counas, G.J. 2.0 GHz to 4.0 GHz automated radiometer operation and service

manual. Nat. Bur. Stand. (U.S.) Interagency Rep. 83-1697; 1984 January.

[4] Randa, J.; Terrell, L.A. NIST WR-28 noise-temperature measurement system. Natl.

Inst. Stand. Technol. Tech. Note 1395; 1997 August.

[5] Dicke, R.H. The measurement of thermal radiation at microwave frequencies. Rev.

Sci. Instrum. 17(7): 268-275; 1946 July.

[6] Miller, C.K.S.; Daywitt, W.C. The NBS switching radiometers. Nat. Bur. Stand.

(U.S.) Interagency Rep. 84-3004; 1984 May.

42
[7] ISO Guide to the Expression of Uncertainty in Measurement. International

Organization for Standardization; Geneva, Switzerland; 1993.

[8] Taylor, B.N.; Kuyatt, C.E. Guidelines for evaluating and expressing the uncertainty of

NIST measurement results. Natl. Inst. Stand. Technol. Tech. Note 1297, 1994 edition;

1994 September.

[9] Daywitt, W.C. A derivation for the noise temperature of a horn-type noise standard.

Metrologia21: 127-133; 1985.

[10] Pucic, S.P. The uncertainty in Y-factor measurements. URSI National Radio Science

Meeting Digest; Boulder, CO; 1995. p. 39.

[11] Pucic, S.P. Uncertainties of the NIST coaxial noise calibration system. Conference on

Precision Electromagnetic Measurements Digest; Boulder, CO; 1994. 254-255.

[12] Pucic, S.P. Broadband mismatch error in noise measurement systems. Conference on

Precision Electromagnetic Measurements Digest; Paris; 1992. 256-257.

[13] Graybill, F.A. Theory and Application of the Linear Model. Duxbury Press; Belmont,

CA; 1976.

[14] Daywitt, W.C. A new reference grade coaxial thermal noise standard for the 10 MHz

to 18 GHz frequency range. Submitted to IEEE Trans. Microwave Theory Tech, 1997.

[15] Daywitt, W.C. A coaxial noise standard for the 1 GHz to 12.4 GHz frequency range.

Nat. Bur. Stand. (U.S.) Tech. Note 1074; 1984 March.

[16] Juroshek, J; Ondrejka, C; Judish, R.; Vecchia, D; and Splett, J, Preliminary analysis

of ARFTG measurement comparison program. NIST-ARFTG Short Course; Boulder,

43
CO; 1994 November.

[17] Miller, C.K.S, and Daywitt, W.C. The NBS WR62 and WR90 reference noise

standards. Nat. Bur. Stand. (U.S.) Interagency Report 884-3005; 1984 May.

[18] Engen, G.F. A method of calibrating coaxial noise sources in terms of a waveguide

standard. IEEE Trans. MTT-16(9): 636-639; 1968 September.

[19] Daywitt, W.C. Determining adapter efficiency by envelope averaging swept frequency

reflection data. IEEE Trans. MTT-38(11): 1748-1752; 1990 November.

[20] Pucic, S.P., and Daywitt, W.C. Single-port technique for adapter efficiency evaluation.

45th ARFTG Conference Digest, Orlando, FL, May 1995, pp. 113-118; 1995.

APPENDIX

The question is how to estimate the type-B standard uncertainty when we already

know a "worst-case error." The reason this is a problem at all is that worst-case errors do not

translate directly into the CIPM/ISO scheme, which is based on standard deviations. To relate

the two we must inject additional information, either explicitly or implicitly (for example, by

the distribution used in whatever method we employ). In choosing a particular conversion

method, we shall be guided by two general principles (besides technical validity):

1. In providing the additional information required, it is better to use even qualitative

knowledge rather than to rely on uneducated guesses or arbitrary choices. If estimates

or best guesses are required, it is preferable to estimate quantities about which we

have some knowledge or intuition.

2. The bottom line is the expanded uncertainty, which should reflect a 95 percent

confidence level. That is what we provide to our measurement services customers.

44
We should choose a method for estimating type-B uncertainties which produces an expanded uncertainty that best reflects our knowledge of the measurement

uncertainties.

In its Section 4, reference [8] provides a number of ways to estimate type-B

uncertainties. Most of the methods consist of choosing a statistical distribution (normal,

rectangular, triangular) to model the quantity in question and then estimating the likelihood

that the true value of the quantity lies within some chosen interval of the estimated value of

the quantity (typically the mid-point of the interval, the average value). The standard

uncertainty is then obtained by determining the standard deviation for the chosen distribution

in terms of the estimated likelihood in the chosen interval. The case of estimating the

standard uncertainty when a worst-case error is known is addressed in Subsection 4.6. The

simplest course of action would be to blindly apply the method of 4.6, which does seem to be

in widespread use. There are some undesirable features of that method, however.

Suppose we are interested in a quantity A, and our best estimate of its value is a, and

the worst-case error is Aa, that is, for all practical purposes the probability is 100 percent that

the true value is between a - bsa and a + tsa. Then subsection 4.6 suggests that A be

modeled by a rectangular distribution (equally probable anywhere in the interval a ± tsd).

This leads to the relation that u^ = o = tsal^JZ. One immediate objection is that a rectangular

distribution seems a very poor choice. It means that the values at the very edge of the

interval are as likely as the values at the center, but that values just beyond the edge have

zero probability. That is usually unrealistic. A second, more practical, problem occurs when

we compute the expanded uncertainty (k = 2), which is supposed to correspond to a 95

percent level of confidence. This results in U^ = Iw^ = 2Aa/^ , which is greater than Aa.

That means that before the conversion we were 100% certain that A was within Aa of a,

whereas now we are only 95 percent confident that A is within 1.15xAa of a, which is a

bigger interval but less confidence! (It is true that u or ZW is typically just one contribution

to the overall uncertainty, but it may be the dominant one, and the other contributions are

45
subject to the same problem.) This inconsistency occurs in a particularly bad place for

measurement services, since the expanded uncertainty is the quantity of most interest; it is what we quote for the customers. Another relevant feature of this approach is the observation

made in reference [8] that if we had used a triangular distribution instead of a rectangular

distribution, then we would have obtained U3 = Aa/y/6, and U^ = 2Aa/i/6 < Aa. This is a

useful reminder that in this approach, the choice of the distribution is effectively a choice of

the uncertainty (for a given worst-case error).

The fact that the choice of the distribution determines the answer would not be

bothersome if we had some good way of choosing the distribution. But if we are quoting a

worst-case error we probably don't really know anything about the distribution anyway. A

more appropriate approach (for us) is given in Subsection 4.3 of reference [8]. It uses the

confidence interval associated with the known uncertainty and assumes a normal distribution

(unless there is information to the contrary) to compute a standard deviation and thus the

standard uncertainty. If Aa corresponds to a 95 percent confidence interval, then u^ =

Afl/1.96; if it is a 99 percent confidence interval, U3 = Aa/2.576. We prefer this approach

because we are in a better position to estimate a confidence level associated with the worstcase error than to estimate the distribution. Another advantage is that there is a clear, direct

link between the choice we make and the final quantity we are calculating, which is the 95

percent confidence-level uncertainty which we quote to the customer. In the previous method

(Subsection 4.6) that link was more obscure.

In practice then, we adopt the following approach. Rather than agonize over exactly

how confident we are, we shall use a two-state system. If we are very confident (>95

percent) of the worst-case error, we will use U3 = Aa/2. This results in an expanded

uncertainty Ug = Aa, that is, our uncertainty for 95 percent confidence level is equal to the

previous worst-case error. If we are less confident of the worst-case error, we shall continue

to use U3 = AalsfZ, which results from a 92 percent confidence level for the worst-case error.

46

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