A UNITEO STATES DEPARTMENT OF COMMERCE

PUBLICATION NBS TECHNICAL NOTE 640

U.S. APARTMENT

OF

COMMERCE

National

Bureau

QC ,f ^ s loo

JL57 53

Considerations

for the Precise Measurement

of Amplifier Noise
NATIONAL BUREAU OF STANDARDS

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research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

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< Located at Boulder. Colorado 80302.

* Part of the Center for Building Technology.
wo

(LI OO

53

»*#

Considerations

for the Precise Measurement

of Amplifier Noise

David F. Wait

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

<*»

4 \

-*£&

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued August 1973
National Bureau of Standards Technical Note 640

Nat Bur. Stand. (VS.), Tech. Note 640, 129 pages (August 1973) CODEN: NBTNAE

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

(Order by SD Catalogue No. 03.46:640). 81.25
CONTENTS

Page

1. Introduction 1 1.1 The Basic Measurement 2 1.2 Accuracy Considerations 3 2. Hot and Cold Standards 4 3. Mismatch Ambiguity 5 4. Mismatch Error 8 5. The Bandwidth Problem-- 9 6. The State-of-the-Art 10

7. An Example 17

7.1 Measurement of the Post Amplifier Noise 20

8. Conclusions 22

9. References 24

Appendix A. Measurement Errors that Depend on

Thot and Tcold 26

Appendix B. Mismatch 49

B.l Mismatch Uncertainty 49

B.2 Mismatch Error 50

B.3 Derivations 52

Appendix C. Effect of Front End Loss on Noise Figure

and Effective Input Noise Temperature 83

Appendix D. Computer Programs 86

D.l Figures 1 Through 4 87

D.2 Figures Bl Through B7 95

D.3 Table 3 99

D.4 Tables AI-AXXI 101

D.5 Tables BIII-BXXII 105

Appendix E. Reprints of Useful Papers 109

[1] Wait, "The Measurement of Amplifier

Noise" 110

[3] Engen, "A New Method of Characterizing Amplifier Noise

Performance" 115

[8] Wait, "Thermal Noise from a Passive

Linear Multiport" 121

in
List of Figures

Figure Page

1-4 The state-of-the-art errors in measuring

noise figure 11-14

B1-B7 Mismatch uncertainty 55-61

CI Alternate input reference planes for an amplifier 84

List of Tables

Table Page

Thermal noise sources 4 Errors for various T, . , T , •, combinations- 6 hot' cold

Translation between effective input noise

temperature T (K) , and noise figure, F, R 16

1

2

3

A-I - A-•XXI

B-I

B-•II

B-•III

B-•XXII

D-•I

D--II

D-•III

D-IV

D-V

D-VI

D--VII

asurement errors that depend on T, ot Me and

cold

Key to mismatch uncertainty figures 50

Key to mismatch error tables 62

Maximum mismatch error 63-82

Computer program for figures 1-4 89

Print out of D-I program 93

Computer program for figures B1-B7 96

Print out of D-III program 98

Computer program for table 3 100

Computer program for tables A-I through A-XXI 102

Computer program for tables B-III through

B-XXII 106

IV
CONSIDERATIONS FOR THE PRECISE MEASUREMENT

OF AMPLIFIER NOISE

For the best accuracy in measuring noise figure,

attention needs to be given to the choice of the hot

and cold noise standards and to mismatch problems.

Tables and graphs are presented to aid in choosing

the proper measurement conditions, and an example is

given to demonstrate their use. This paper essentially

supplements a previous paper (included in an appendix)

,

treating in more detail topics that become important

when state-of-the-art measurements are required.

Key words: Amplifier noise; effective input noise

temperature; mismatch error; mismatch uncertainty;

noise figure.

1. Introduction

The problem of measuring amplifier noise is extensively

documented in the literature [1,2]. For the reader's con- venience, a general overview of this problem is reprinted in

Appendix E (along with other related papers) . This technical

note provides additional detailed information concerning the

accuracy of measuring amplifier noise with the various noise

standards presently available. This additional information

should be helpful in evaluating or designing measurements of

amplifier noise as accurate as ± 0.3 dB in noise figure or

± 1% in effective input noise temperature.
With the present state-of-the-art, the least inaccuracy

to which amplifier noise figures can' be measured is approximately

0.1 dB in noise figure or 2% in effective input noise temperature. This paper is especially addressed to those who intend to make such a state-of-the-art noise measurements.

1 . 1 The Basic Measurement

The most accurate measurements of amplifier noise use

some version of the so-called Y-factor method. In this method,

two noise standards with noise temperatures of T, and T , , hot cold

are sequentially connected to the input of the unknown amplifier*, and the ratio of the noise powers, Y, out of the unknown

amplifier is measured. This parameter Y may be written in terms

of the Effective Input Noise Temperature** of the unknown

amplifier, T , as follows:

Y =

C Thot

+ V^coid + V- CD

Because T, and T , j are known, and Y is measured, then T hot cold '

' e is known:

T

e

= (Thot " Y T cold)/(Y - 1). (2)

Also in common use as a measure of amplifier noise is

noise figure, F, R. To avoid the ambiguities and difficulties

of the IEEE definition of F, R discussed in [1] , the definition

of F, R used in [1] is used here, namely

F dR

= 10 log[l + T

e /290]. (3)

*In this paper, the term "unknown amplifier" is used to refer

to the amplifier whose Effective Input Noise Temperature, or Noise Figure we want to know.

**The definition of T and F Jr> are discussed in [1] which is

e dB L J reprinted in Appendix E.

2
1 . 2 Accuracy Considerations

In crude terms, measurements of T within 2% or F, R within

0.1 dB require nT, . >> T >> T , , , and require that the re- hot e cold n flection coefficients of the noise standards closely match the

reflection coefficient of the "antenna"* in both magnitude

and phase angle.

There are at least seven important sources of error in a typical measurement of T : (1) uncertainty in the value of

T, . , (2) uncertainty in the value of T .. , , (3) uncertainty hot' ' cold' ^

* 7 in the measurement of Y, (4) amplifier -gain instability during

the time required to measure the ratio Y, (5) inequality of

the reflection coefficients of the "antenna" and the two standard noise sources, (6) uncertainties in input connector

losses with the "antenna" and each of the two standard noise

sources connected to the unknown amplifier, and (7) errors in

correcting for the noise originating in the measuring system

(cascade error). As we shall see, the magnitude of these

various errors depend r on the choice of T, . and T , ,. hot cold

To minimize the measurement error requires two steps.

First the hot and cold standards are selected, then the other

measurement conditions are investigated. Some major factors

that influence measurement accuracy are discussed first, then

the state-of-the-art is discussed, and last a measurement

example is given to clarify the use of this technical note.

*The term "antenna" in this paper refers to all of the components that will be attached to the input of the unknown amplifier when it is being used in its intended application.

(If the end use is not known, then it is usual to assume that

the "antenna" is reflect ionless . However, the concept of

an unknown "antenna" is not in the spirit of a precision amplifier noise measurement.)

3
lo avoid breaking up the text, the long series of tables or graphs are collected in the appendices.

2 . Hot and Cold Standards

Some of the choices for the hot and cold standard noise

temperatures that are available are shown in Table 1.

Table 1. Thermal Noise Sources

1.

2.

3.

4.

5.

6.

7.

9.

10

11

12

Source*

Neon gas -discharge

Argon gas -discharge

WR15, WR62, WR90--

NBS Standards

WR284--NBS'-Standard

Commercial coaxial hot

standard

14 mm coaxial --NBS

Standard

Unregulated ambient

temperature load

Regulated ambient

temperature load

Commercial LN„

z 14 mm coaxial,

LN

2 Standards

Commercial LH

e WR90--NBS LH

load

WR90- NBS

standard

standard

Typical Effective Temperature

and state-of-the-art uncertainty

18,000 ± 270 K (1.50%)

11,000 ± 165 (1.50%)

1250 ± 3 692 ± 0.9

373 ± 0.5

373 ± 0.15

300 ± 1

300 ± 0.1

80 ± 1

(0.24%)

(0.13%)

(0.13%)

(0.04%)

(0.33%)

(0.03%)

(1.25%)

10 ± 0.2 (0.25%)

4 ± 0.5 (12.5%)

4 ± 0.1 (2.50%)

Four of the error contributions to T or F JT,, namely the

e dB ' J uncertaintyJ in the values of T, . , T „,,, Y, and gain insta- hot cold' to bility,7 depend on the choice of T, . and T , ,. Appendix A ' r hot cold rr

*Frequency coverage for the NBS 14 mm coaxial source is d.c

to 1.2 GHz, for WR284 is 2.60 to 3.95 GHz, for WR90 is 8.2

to 12.4 GHz, for WR62 is 11.9 to 18.0 GHz, and for WR15 is

50 to 75 GHz.
contains tables that list these dependent measurement error contributions for various values of T or F JT) for combinations

e dB

of T, and T , , selected from Table 1. Table 2 summarizes hot cold

some of the results given in Appendix A. The error listed

under F,_ in Table 2 is the magnitude of the uncertainty of

F, R. Thus by F, = 8 ± 0.1 dB, we mean the true value of F,

is between 7.9 and 8.1 dB. The uncertainty in Y, and variations in amplifier gain used in Table 2 and Appendix A approximate the present state-of-the-art. For different uncertainties

in the parameters used, one needs to refer back to Appendix A

and make the modifications explained there.

3 . Mismatch Ambiguity

Mismatch ambiguity (or uncertainty as it is called in [1])

is the ambiguity in effective input noise temperature because

of an ambiguity in specifying the reflection coefficient of

the "antenna," r ... It is best to measure the mismatch ' ant

ambiguity (see [1] in Appendix E) , but to give an idea of

the magnitude of the mismatch ambiguity, it is examined below.

The simplest assumption for amplifier noise is that the

amplifier is linear (that is, its output voltage y is related

to its input voltage x in the form y = ax + b , where a and b

are constants). If in addition to being linear, we assume

isolation such that an impedance change at the amplifier's
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output -termination does not alter T , then the most general

dependence of T with input reflection coefficient is [3]

t r ,, V1+b ' r !nt- g l 2

) T

e (ant) =

_ (4 )

' ant '

where the parameters r T , b, |r -3 I and |r are chosen to

a ' ' an t ' ' ant 1 be terminal invariant (i.e., their value does not depend on

the location of the input or output terminals provided the

choice is limited to lossless regions) . The parameter T is

a the amplifier's characteristic noise temperature, bT is the

magnitude of reverse radiation (i.e., the noise temperature

of the radiation from the amplifier as seen by the "antenna"),

3 can be thought of as a measure of the correlation of the reverse radiation with the internal noise or alternately it

may be thought of as a measure of the difference in conditions

for maximum power transfer and minimum noise figure, and

*

r' = ant " amp

ant -

1

_ C5J

ant amp

where the asterisk implies the complex conjugate of the

t "amplifier's" reflection coefficient. Note that r = for

maximum power transfer, and has magnitude unity when

|r fc = 1. For the case that r _ and r „ are small, then 1 ant

ant amp

it is convenient to use the approximate form of equation (5)

,

i *

r ~ r - r cfi~i an t an t amp

*- D > In Appendix B are graphs which can be used to estimate

mismatch ambiguity.
Frankly, it is a bit of a problem to decide what value

of b, and 3 to use for any given amplifier because so little

is known about what are typical values of b and 3. Engen [3]

has measured |@| = 0.13, b = 0.65, T = 496 K for an X-band

crystal mixer amplifier. For an X-band tunnel-diode amplifier

|3| = 0.03, b = 0.35, T = 825 K was measured. For a 30 MHz

vacuum tube amplifier, |3| = 0.22, b = 0.59, T = 161 K was measured.

* For any7 case,' and a typical J * value of r -r = 0.1, ' ant amp ' ' the mismatch ambiguity is less than about 2% of T (0.1 dB of

F,,,) if the magnitude of the uncertainty of r is less than dB' & ' ant

0.02. This means that precise measurements of amplifier noise

require n fairly} accurate knowledge& of r ant

4 . Mismatch Error

Mismatch error is the error resulting when the "antenna,"

the hot standard, and the cold standard do not have identical

reflection coefficients. The maximum mismatch error is listed

in tables B-III to B-XXII for ideal linear amplifiers.

As pointed out in [1] (see Appendix E) , estimates of mismatch errors using the tables in Appendix B are hazardous.

When it is practical, the measurements suggested in [1] are preferable

.
5 . The Bandwidth Problem

One problem that complicates the measurement of amplifier noise figure is caused by the dependence of the amplifier's noise on the variations in the "antenna" reflection

coefficient versus frequency. For example, if local oscillator power in the superheterodyne amplifier leaks out the

signal input port, then the mixer diode bias depends in part

on the amplitude and phase of the reflection coefficient of

the "antenna." This causes the noise figure and other amplifier parameters to vary depending on the phase of the "antenna"

reflection coefficient at the local oscillator frequency . In

addition, the noise figure depends on the phase of the "antenna"

reflection coefficient at the signal frequency . The resulting

combination effect modulates the amplifier noise figure and

other parameters with a period characteristic of the IF amplifier frequency. In addition, the noise figure in theory can

depend on the reflection coefficients of any pair of frequencies within the passband of the amplifier so that to

check out reflection coefficient dependent effects, one needs

to consider an unwieldly range of possible frequency dependences. These difficulties can be eliminated or reduced

if the reflection coefficient of the standard noise sources

can be adjusted to equal that of the "antenna" at all frequencies. There is evidence in the literature that significant

measurement errors can occur by not using the correct frequency

dependent reflection coefficient [9]

.

9
6. The State-of-the-Art

Several factors which affect the accuracy of measuring

amplifier noise have been mentioned, and details of their

effects examined in the appendicies. In figures 1-4, the

resulting& error caused byJ the uncertainties of T\ . , T ,,, hot' cold'

Y, gain, connector loss, and mismatch are shown for four com- binations of hot and cold noise standards. For these figures

it is assumed that Y is measured within 0.01 dB, gain is

stable within 0.1% over the measurement time, and the worst

case uncertainty in loss in the connectors joining the hot

or cold standards to the amplifier during the measurement is

0.01 dB (see Appendix C for the effect of connector loss on measurement accuracy). The mismatch error selected corresponds

to |r -r ., | < 0.01 and |r .

I < 0.05. For the amplifier

ant s t u. ant

properties, b = 3 = 0, were selected not because these are typical, but because the error of this assumption is within a factor of 2 of the probable situation for the various amplifiers represented in the range of amplifier noise indicated.

In figures 1-2, the measurement error is expressed in decibels.

As for Table 2, a 0.1 dB error say at 8 dB means the amplifier

noise is between 7.9 and 8.1 dB. For values of F,

R where the

error is a sizable fraction of F, R

, the figures no longer have

a simple interpretation other than "this is no way to be

measuring amplifiers with such small noise figures."

10
0.20

COT3O

HOT

n 1 r

10,000 +150K T

~i 1

= 300 ±0,5K

0.18

0.16

0.14

0.12

COLD

QUADRATURE SUM

MISMATCH

CONNECTOR LOSS

GAIN

Y-FACT0R

T COLD

LINEAR SUM

0.10

0.08

0.06

0.04

0.02

Ufflnmm iZ^MW®^

r dB

Figure 1. The state-of-the-art errors in measuring noise

figure, F,y,, using an argon gas discharge noise

source (about 10,000 K) as the hot standard, T, ^

,

k > j > not' and a room temperature resistive termination (about

300 K)' for the cold standard, T ,,. The uncer- cold

tainty assumed for Y-factor, gain, connector loss,

and mismatch are ± 0.01 dB, ± 0.11, 0.01 dB, and

(for v mismatch)' r -r ^,\ ' ant std 1 as discussed in the text.

< 0.1 and

ant

< 0.05

11
0.20

0.18 -

0.16 -

0.14 -

COT3 0.12 - o

or

en

LU 0.10 -

0.08 -

0.06 -

0.04 I

0.02

0.00

dB

Figure 2. The state-of-the-art errors in measuring noise

figure using the National Bureau of Standards' WR15,

WR62 or WR90 "black-body" standard as the hot standard, and a room temperature resistive termination

as the cold standard. Other parameters as noted in

figure 1.

12
eno

5.0

4.5 -

4.0

3.5

3.0

2.5

2.0

1 .5

1 .0

0.5

T

H0T

= 373 ±0.5K TC0LD = 80 ±1K

QUADRATURE SUM

MISMATCH

i CONNECTOR LOSS

GAIN

Y-FACTOR

T,

T

e (K)

Figure 3 The state-of-the-art errors in measuring the effective input noise temperature, T

e

, using a commercial

hot standard near boiling water temperature and a cold standard near liquid nitrogen boiling temperature. Other parameters as noted in figure 1.

13
5.0

o

en

en

UJ

en

IS)

4.0

3.0

1 .0

2.o m

T (K)

Figure 4. The state-of-the-art errors in measuring T

e using

a room temperature resistive termination as the hot

standard, and a liquid helium cooled resistive termi

nation as the cold standard. The other parameters

as noted in figure 1.

14
The inaccuracy of hot and cold standards used is indicated in each figure. In general, the measurement situation

depicted is typical of the best measurement conditions, and

at some frequencies (e.g., where no calibration service for

noise sources exists) somewhat better than the best conditions

The four combinations of hot and cold noise standards

selected are 10,000 K and 300 K, 1270 K and 300 K, 373 K and

80 K, and 300 K and 4 K. The combination 1270 K and 300 K differs from the others in that it assumes the use of the NBS

primary standards, or if you will -- the best NBS can measure

noise figure. The results from these figures can be converted

to either noise figure or effective input noise temperature

using table 3. In figures 1-4, the dashed line is the quadrature sum (root mean square addition) of the errors. As we

see, the quadrature sum is typically half of the linear addition of errors. The quadrature sum is the appropriate sum

if the errors meet three criteria: (1) the error sources are independent, (2) the errors are equally likely to be positive

or negative, and (3) the errors are more likely to be small

rather than large. In other words, quadrature addition is

appropriate if the error distributions are something like a gaussian distribution about the value used in the measurement

calculation.

Unfortunately, three corrections, namely clipping, con- nector loss, and mismatch correction, seldom satisfy the

15
Table 3. Translation between effective input noise temperature , T (K) , and noise figure , F The asymmetry in the

e

- dB

error statements, because of the logarithmic non- linearity of F, , is avoided by using the slope of F dB

at the corresponding T

dB

T

e

:k; = FdB FdB

- T (K)

e

v J

10

15

20

+

+

+

i%

1%

1%

= 15

.22

29

+

+

+

.0014

.0021

.0028

dB

dB

dB

1

23

+

+

+

.1

.1

.1

dB

dB

dB

= 75

169

238

+

+

+

11

6

4

22%

24%

62%

30

50

70

+

+

+

1%

1%

n _

43

.69

.94

+

+

+

.0041

.0064

.0084

dB

dB

dB

4

56

+

+

+

.1

.1

.1

dB

dB

dB

= 433

627

864

+

+

+

33

3

831

371

.07%

100

150

200

+

+

+

1%

1%

-L o =

11

2

29

.81

28

+

+

+

.0111

.0143

.0177

dB

dB

dB

78

9

+

+

+

.1

.1

.1

dB

dB

dB

= 1163

1539

2013

+

+

+

22

2

83%

.74%

.63%

300

500

700

+

+

+

1 s- ±

1 9-

1

9- 1 a

=

3

4

5

.08

35

33

+

++

.0221

.0275

.0307

dB

dB

dB

10

11

12

+

+

+

.1

.1

.1

dB

dB

dB

= 2609

3360

4306

+

+

+

2

2

2

.56%

.50%

46%

1000

1500

2000

+

+

+

1 9-

1

9-

1

=

6

7

8

.48

90

97

+

+

+

.0337

.0364

.0379

dB

dB

dB

13

14

15

+

+

+

.1

.1

.1

dB

dB

dB

= 5496

6994

8880

+

++

22

2

42%

.40%

33%

3000

5000

7000

+

++

1%

1%

1%

= 10

12

14

.55

61

00

+++

.0396

.0410

.0417

dB

dB

dB

16

17

18

+

+

+

. 1 .1

.1

dB

dB

dB

= 11255

14244

18007

+

+

+

2

2

2

36%

35%

34%

10000

15000

20000

+

+

+ 11

= 15

17

18

50

22

45

+

++

.0422

.0426

.0428

dB

dB

dB

19

221

+

+

+

.1

.1

.1

dB

dB

dB

= 22745

28709

36218

++

+

2

2

2

.33%

33%

32%

30000

5 0000

70000

+

++ n

li = 20

22

23

19

39

84

++

+

.0430

.0432

.0433

dB

dB

dB

22

23

24

+

+

+

.1

.1

.1

dB

dB

dB

= 45671

57572

72554

+

+

+

2

22

32%

.31%

.31%

16
quadrature conditions. In principle this can be arranged but

it is more practical to keep the level of these errors low,

and then add their contributions linearly to the quadratic

sum of the remaining errors

.

Frequently the errors due to the uncertainty in the hot

noise standard, the cold noise standard, the measurement of

Y, and the amplifier gain instability add in quadrature. In

this case, the contributions from these four sources will be

about half of their linear accumulation of error.

7 . An Example

To illustrate a measurement problem we begin with the

following measurement specification provided by a "buyer".

WR15 mixer-preamplifier specifications:

Local Oscillator: 60 GHz

I.F. Bandpass: 10-110 MHz

Maximum Noise Figure: < 8 dB (1500 K) for

Ir1 J < o.i

ant

Input | r|

< .1

Output | r|

< .1

Gain: > 10 dB

Gain Instability: < 0.005 dB/min

Measurement Accuracy Goal: 0.1 dB

First a comment on the specified accuracy goal. From

reference [1] (see Appendix E) we note that a calibration from

17
NBS exists so figures 1-4 can reasonably represent the stateof-the-art. Using figure 1, we see that 0.1 dB error is near

the state-of-the-art using an argon gas discharge noise standard and a room temperature standard providing some of the

error contributions can be placed in quadrature. Further, from

figure 2, it appears NBS should be able to provide a measurement verification if needed. However, it is clear that the

specified accuracy goal is going to be difficult to obtain.

To compare the specified accuracy goal with the estimated

mismatch ambiguity, I would use figure B4 (it is my guess that

bT - 300 K so that b = 0.2, and 3 = 0.2 to be consistent with

* Engen's & measurement [1]) L i with r - T - 0.2, and ' ' ant amp ' '

|r ^.j 1 - T . I - 0.1. The mismatch ambiguity indicated is std ant

& J about 1.1%. This is several times the accuracy goal specified.

At this point one should stop and reevaluate whether it is

economically more effective to lower the amplifier noise

figure to allow for the mismatch ambiguity, or to specify the

antenna impedance more precisely. If the specified accuracy

goal is maintained, a difficult measurement is being undertaken

with a real risk that the "buyer" cannot make use of it. In

a situation like this there must be a very clear mutual understanding with the "manufacturer" of the conditions under which

the amplifier will be utilized. Otherwise the measurement can unintentionally become invalid.

18
As a final consideration of the accuracy goal, we note

from [10] or from table 2 in [1] (see Appendix E) that

connector losses as low as 0.01 dB have been measured in WR15.

But from experience at NBS, a 0.1 dB loss can occur if con- nectors from different manufacturers are mated without careful inspection to see that the flanges close properly. Thus,

flanges must be examined and handled with "laboratory" care,

right through the time the amplifier is finally installed.

The gain of the WR15 mixer-preamplifier in this example

is not great enough so that the cascade noise contribution

due to a post amplifier can be ignored. This contribution is

approximately equal to T (post) divided by the power gain of

the WR15 mixer-nreamplif ier . For a F Jr> (r>ost) of 5 dB (627 K) , the post amplifier contribution of 63 K represents a 0.1 dB

increase of F, R (spec + post) over F, R (spec). We need to know

T (post) and the gain of the specified amplifier within about

10%.

At this point, we can list a set of conditions which will

make it probable that a measurement of F_, R is within .1 dB.

1. A quadrature error situation using an argon-gas discharge

noise standard and room temperature standard. For example

a. An argon-standard adequately calibrated by a standards laboratory.

b. A Y- factor measurement system accurate to within

0.01 dB (Including resolution limitations).

l'J
c. A thermometer accurate to within about 0.1 K

to measure the room temperature standard.

2. Mismatch ambiguity measured for |r ,

| = 0.1 (with the

most unfavorable phase) preferably to better than 1%.

(Note: to meet the specification, the measured F„

must be lower than that specified by this mismatch

ambiguity.

)

3. Connector losses should be less than about 0.02 dB.

4. Cascade correction

a. 60 GHz amplifier gain measured within 10%.

b. Post amplifier noise temperature measured within 10%

7 . 1 Measurement of the Post Amplifier Noise

The example of measuring the WR15 mixer -preamplifier is

completed. But the problem of measuring the post amplifier

noise temperature within 10% (± 0.3 dB at 5 dB noise figure)

is a sufficiently different problem that it merits further

discussion. To be specific, assume the manufacturers specifications of the amplifier chosen for this task are:

Bandpass 10-110 MHz

Noise Figure < 5 dB (627 K)

Input |r| < 0.05

Output | r

| < .1

Gain > 70 dB

Gain instability < 0.005 dB/min

1 dB gain compression at 100 mw or greater.

20
The gain selected permits one to operate a bolometer power

bridge with 0.1 milliwatt output when a room temperature noise

source is connected to the WR15 mixer-preamplifier. The gain

compression level selected is such that less than 0.1% of the

noise power will be clipped when the hot noise source is

connected to the WR15 mixer-preamplifier.*

To estimate the mismatch uncertainty for the post amplifier (due to the low frequency noise standard's reflection

coefficient being different than the output reflection coefficient of the mixer-preamplifier) , I would use figure B7 (no

hard information for this choice) with

l r TMi- " r omJ ~

l r a n1-

1 " r^ . , I a 0.1. The result is off the ant amp 1

' ant std 1 graph but extrapolates to be near 131 (or .4 dB via table 3).

A 101 overall accuracy is our goal so we would like to tune

the standards (which decreases their accuracy) to equal the

output impedance of the mixer-preamplifier. But to match the

output impedance from 10-110 MHz is probably impractical so like it or not, we may have to accept the 131.

For the post amplifier noise measurement, a solid-state

noise source with T, near 10,000 K, and a room temperature hot

cold standard are reasonable choices. Scaling the errors from

*From Cohn [4] average amplitude/saturation amplitude is approximately equal to 0.07 for 0.11 clipping in a square law

detector. Bolometer-power bridges work accurately up to 10 mw

of power so this suggests > 147 mw saturation level. This hard

limiting model is not easy to interpret for real amplifiers.

From our experience, 1 dB compression at 100 mw is adequate to

keep clipping correction to less than 0.1%. For other discussions of the effects of clipping see Deutsch and Hance [5],

Bell [6], Van Vleck and Middleton [7].

21
table A-IV for T = 700 K, T, + 3% contributes 4.401,

e not

T Qld i 10 K contributes 1.6%, Y ± 0.1 dB contributes 3.6%,

and gain stability o£ 1% over the measurement time contributes

1.6%, connector loss of 0.05 dB contributes .05 dB to F Jr> etc (see appendix C) or 1.69% via table 3. If the noise standards

both have |r J < 0.02, then from table B-IX, 3 = .3,

bT /T = 1 (this is just a guess but in line with the measa e

J b

urements in [1]) and using F(dB) = 6 (because 5 not listed),

for |r1 > .1 a mismatch error of 0.068 dB (2.1% using

ant ' & table 3) needs to be added to the 13% mismatch uncertainty

for a total mismatch error of about 15%. The grand total

measurement error expected is 27.9%. If the errors due to

the hot source (4.42%), the cold source (1.6%), the Y-factor

(3.6%) and the gain instability (1.6%) are in quadrature,

then the grand total error reduces to 22.8%, or still uncomfortably large compared with the 10% goal.

8. Conclusions

Under favorable measuring situations, noise figure measurements within 0.1 dB or effective input noise temperature

within 2.5% is about the best that can be done. One of the

greatest problems to accurate measurements is mismatch error.

This problem is compounded by the general ignorance of the

magnitude and variation of the pertinent amplifier parameters.

22
The accuracy achieved in a particular measurement of

amplifier noise depends not only on whether a commercial automatic noise figure meter is being used, or whether a refined

Y-factor measurement scheme is utilized, but also on whether

a national reference noise source exists at the frequency of

interest. It also depends on the stability of the amplifier

properties during the measurement. But in addition, the

accuracy depends on the experience and skill of the metrologist

This experience and skill needs to be learned, preserved, and

shared if accurate amplifier noise measurements are to become

a reality. This paper provides information on the accuracy

of the noise figure measurement in terms of the uncertainties

of various measurement parameters without stating how to

estimate the uncertainties of these measurement parameters.

For these important estimates we anticipate that the average

metrologist will utilize the information provided by the

manufacturers of the test equipment he chooses to use. If

the metrologist hopes to have individual error contributions

combined in quadrature, then an even greater skill and understanding is required. We hope that for some of the amplifiers

whose noise figure he has measured, he will have the National

Bureau of Standards verify the measurement. This provides

an opportunity to assure the metrologist that his methods and

his equipment are adequate, and to give him the confidence to

share his ability with others.

2 3
9 . References

[1]* D. F. Wait, "The Measurement of Amplifier Noise," Microwave

Journal, pp. 25-29, Jan. 1973.

[2] M.G. Arthur, "A Guide to the Measurement of Noise Performance Factors," NBS Monograph _, 1973.

[3]* a. Glenn F. Engen, "A New Method of Characterizing

Amplifier Noise Performance," IEEE Trans. Instr. § Meas., Vol. IM-19, No. 4, pp. 344-349, Nov. 1970.

b. Glenn F. Engen, "Mismatch Considerations in Evaluating

Amplifier Noise Performance," IEEE Trans. Instr. § Meas., Vol. IM-22, Sept. 1973.

[4] Charles Erwin Cohn, "Errors in Noise Measurements Due to

the Finite Amplitude Range of the Measuring Instrument,"

Rev. Sci. Instr., Vol. 35, pp. 701-703, June 1964.

[5] Ralph Deutsch § Harold V. Hance , "A Note on Receivers for

Use in Studies of Signal Statistics," IRE Convention

Record, Part I, Information Theory, Vol. 1, pp. 7-13, 1953.

[6] R. L. Bell, "Linearity Range of Noise-Measuring Amplifiers,"

Wireless Engineer, pp. 119-122, April 1947.

[7] J. H. Van Vleck $ David Middleton, "The Spectrum of

Clipped Noise," Proc. IEEE, Vol. 54, No. 1, pp. 2-19,

Jan. 1966.

[8]* D.F. Wait, "Thermal Noise from a Passive Linear Multiport,"

IEEE Trans. Microwave Theory § Techniques, Vol. MTT-16,

No. 9, pp. 687-691, Sept. 1968.

*Copies of these papers are in Appendix E

24
9] C.K. Mayer, "Improved Noise Measurements Using Ferrites,"

IEEE Trans. Microwave Theory § Techniques, Vol. MTT-4,

pp. 24-28, Jan. 1956.

10] B.C. Yates and C.J. Counas , "Summary of WR15 Flange

Evaluation at 60 GHz," NBS Technical Note 642, Oct. 1973.

25
Appendix A. Measurement Errors that Depend on T and T hot cold

The tables in this appendix are summarized in part in Table

2 on page 6. These tables are computer print outs using the

program noted in Appendix D. The symbols have the following

meanings (see equations (1) , (2) , and (3) in text) : Symbol Meaning

TM

Thot (A-l)

TC T cQld (A-2)

DY = .01 dB Y-factor measurement inaccuracy of

± 0.01 dB (A-3)

DG = .101 Amplifier gain instability of 0.11 (A-4)

TE(K) T expressed in degrees Kelvin (A-5)

F(DB) F dB (A-6)

Y(DB) Y expressed in decibels, i.e.

Y dB

= 10 log Y (A-7)

ETH Error to T due to uncertainty in T, ^ (A-8)

e

' hot J ETC Error to T due to uncertainty in T ,, (A- 91

e

J cold v * EY Error to T due to uncertainty in Y (A-10)

EG Error to T due to uncertainty in

e

' amplifier gain (A-ll)

+ ± (A-12)

The error listed next to value of T under TE(K) is the sum of the errors listed under ETH, ETC, EY and EG. Then

this error is converted into the corresponding error in F,,,

and listed next to the value of the corresponding F, R

.

As an example of using Table A-l, note that an amplifier with an effective input noise temperature near 7000 K

(noise figure of 14.0 dB) can be measured to an inaccuracy of

26
± 2.1% (+ 0.087 dB in noise figure) if the hot standard is

18000 ± 270 K, the cold standard is 300 ± 1 K, the Y-factor

(near 5.35 dB) is measured with an inaccuracy of 0.01 dB,

and the gain is unstable within ± 0.1%. Of the 2.1% inaccuracy,

1.59% is caused by the ± 270 K uncertainty in the hot noise

standard, 0.02% is caused by the ± 1 K uncertainty in the

cold noise standard, 0.34% is caused by the 0.01 dB in measuring the Y-factor, and the remaining 0.15% is caused by the

gain instability of 0.1%. Mismatch error, connector loss

error, and cascade error are neglected in these tables. If

instead of the ± 270 K, ± 1 K, ± 0.01 dB , and ± 0.1% inaccuracies assumed in Table A-l for T, . , T , , , Y(dB) , and gain hot ' cold >»-'» & respectively we had ± 270K, ± 2 K, ± 0.005 dB , and ± 0.5%,

then the corresponding error contributions to T in our example would expand or contract in proportion to 1.59%,

0.04%, 0.17%, and 0.75% for a total error of ± 2.55%. Similarly for any other measurement uncertainty situation the

tables can be modified to obtain the appropriate error contributions to T .

27
Table AI

TH=18000 + 270.00 K (1.50%)

TC= 300 + 1.00 K (0.337.)

DY=.01 DB DG=.10%

TE(K) F(DB)

ERROR CONTRIBUTIONS TO TE

Y(DB) ETH ETC EY EG

13 +67.9% 0.15 + .093 17.64 47.29% 10.13% 7.26% 3.15%

15 +45.9% 0.22 + .098 17.57 32.33% 6 . 79 % 4.92% 2.14%

20 +34.9% 0.29 + .098 17.51 24.41% 5.39% 3.75% 1.63%

30 +23.9% 0.43 + .097 17.37 16.73% 3.40% 2.53% 1.12%

50 +15.1% 0.69 + .096 17.12 10.63% 2.04% 1.64% 3.71%

70 + 11 .3% 3.94 + .095 16.39 8.06% 1.46% 1.24% 3.54%

100 + 8.5% 1.29 + .094 16.56 6.10% 1.02% 0.94% 0.41%

150 + 6.3% 1.31 + .093 16.36 4.58% 0.68% 3.71% 0.31%

200 + 5.2% 2. 23 + .392 15.61 3.31% 3.51% 0.59% 3.26%

32)0 + 4.1% 3.08 + .090 14.84 3.05% 0.34% 0.43% 3.21%

530 + 3.2% 4.35 + .033 13.64 2.44% 0.21% 0.39% 0.17%

700 + 2.3% 5.33 + .037 12.72 2.13% 3.15% 3.35% 3.15%

1000 + 2.6% 6.43 + .036 1 1 .65 1 .93% 3.11% 0.32% 0.14%

1530 + 2.3% 7.9Z + .085 10.35 1.33% 2.37% 3.30% 3.13%

2330 + 2.2% 3.97 + .335 9.39 1.75% 0.06% 0.30% 3.13%

3000 + 2.1% 10.55 + .085 3.34 1.68% 0.04% 0.30% 3.13%

50 00 + 2.1% 12.61 + .036 6.37 1.62% . 03 % 3.32% 0.14%

70 00 + 2.1% 14.30 + .08 7 5.35 1.59% 0.02% 0.34% 0.15%

10030 + 2.1% 15.53 + .09 3 4.34 1.57% 0.02% 0.33% 0.16%

15000 + 2.2% 17.22 + .394 3.34 1 .56% 0.01% 3.44% 0.19%

23330 + 2.3% 13.45 + .093 2.72 1.55% 0.01% 3.50% 0.22%

33033 + 2.5% 20 . 1 9 + .136 2.33 1 .54% 0.01% 3 . S3 7, 3.27%

51'? 30 + 2.3% 22.39 + .122 1 .31 1.53% 0.01% 3.39% 0.39%

70330 + 3.2% 23.34 + .133 3.93 1.53% 0.31% 1.15% 0.53%
Table A-II

TH=13339 + 273.00 K (1.50%)

TC= 30 + 1 .30 K (1 .257.)

DY = .31 DB DG=.10%

TE(K) F(DE) Y(DB)

ERROR CONTRIBUTIONS TO TE

ETH ETC EY EG

10 +26.6%

15 +18.3%

23 +14.2%

33 +10.1%

53 + 6.8%

73 + 5.4%

133 + 4.3%

153 + 3.5%

230 + 3.1%

330 + 2.7%

533 + 2.3%

703 + 2.2%

1303 + 2.1%

1530 + 2.0%

2000 + 2.3%

3303 + 2.3%

5033 + 2.3%

7030 +2.3%

13303 + 2.1%

15303 + 2.1%

20333 + 2.2%

30303 + 2.4%

53333 + 2.3%

70303 + 3.1%

3.15 +.039

3.22 +.039

0.29 +.04 3 3.43 +.341

0.69 +.343

3.94 +.045

1 .29 +.043

1.81 +.052

2.28 +.055

3.08 +.359

4.35 +.065

5.33 +.368

6.48 +.071

7.90 +.074

3.97 +.376

13.55 +.379

12.61 +.382

14.33 +.084

15.53 +.337

17.22 +.091

13.45 +.395

2 0.19 +.134

22.39 +.120

23.34 +.136

23.31 13.56% 10.35% 2.08% 0.93%

22.78 9.54% 6.73% 1 .47% 3.64%

22.56 7.53% 5.33% 1.16% 0.53%

22.15 5.52% 3.35% 3.85% 0.37%

21.43 3.92% 2.31% 3.63% 3.26%

20.31 3.23% 1 .44% 3.53% 0.22%

20.32 2.71% 1 .01% 3.42% 0.18%

13.97 2.31% 0.63% 0.36% 0.16%

13.13 2.11% 0.51% 0.33% 0.14%

16.33 1 .91% 3.34% 0.30% 0.13%

15.34 1 .75% 3.21% 3.28% 3.12%

13.83 1.68% 3.15% 3.27% 3.12%

12.45 1 .63% 0.11% 3.26% 3.11%

13.91 1 .59% 3.0 7% 0.26% 3.11%

9.83 1 .57% 3.36% 3.27% 3.12%

3.34 1.55% 0.34% 3.28% 3.12%

6.56 1 .53% 3.33% 3.33% 3.13%

5.43 1.52% 0.32% 3.32% 3.14%

4.44 1 .52% 0.32% 3.36% 0.16%

3.43 1.51% 0.01% 3.43% 3.19%

2.77 1.51% 3.31% 0.49% 3.21%

2.03 1.51% 3.31% 3.62% 3.27%

1.33 1.51% 0.01% 0.33% 3.38%

3.99 1.51% 0.31% 1.13% 3.49%

29
Table A-III

TH=1S300 + 270.23 K (1.53%)

TC= 4 + 2.50 K (12.50%)

DY = .01 DB DG=.10%

TE(K) F(DB)

ERROR CONTRIBUTIONS

Y(DB) ETH ETC EY TO TE

EG

10 + 7 ,S% 3.15 + .311

15 + 5 .7% 3.22 + .312

20 + 4 .7% 3.29 + .013

30 + 3 .7% 0.43 + .015

50 + 3 ,3% 0.S9 + .019

70 + 2 ,7% 3.94 + .022

130 + 2 .4% 1.29 + .027

153 + ;> ,2% 1.81 + .033

230 + 2 . 1% 2.23 + .338

300 + 2 ,0% 3.38 + .045

503 + 2 ,3% 4.35 + .354

700 + 1 .9% 5.33 + .059

1 300 + 1 .9% 6.48 + .0 64

1500 + 1 .9% 7.92 + .069

2000 + 1 ,9% 8.97 + .3 72

3000 + 1 .9% 10.55 + .275

5330 + 1 .9% 12.61 + .330

7000 + 2 ,0% 14.00 + .232

10003 + 2 , 0% 15.53 + .335

15 300 + 2 , 1% 17.22 + .393

22300 + 2 , 9 % 13.45 + .394

30003 + 2 ,4% 23.19 + .133

50003 + 2 .8% 22.39 + .1 19 70330 + 3 .17. 23.34 + .135

31.39

29.77

23.76

27.25

25.24

23.33

22.41

2 3.71

19.53

17.30

15.65

14.24

12.77

11.13

9.99

3.45

6.52

5.53

4.47

3.42

2.79

2.34

1 .34

0.99

2.10%

1 .90%

1 .80%

1 .70%

1 .62%

1 .59%

1 .56%

1 .54%

1 .53%

1 .52%

1.5 1%

1.51%

1 .51%

1 .50%

1 .50%

1 .5 3%

1 .50%

1 .5 3%

1 .53%

1.50%

1 .50%

1 .5 3%

1 .53%

1 .50%

5.33%

3.34%

2.50%

1 .67%

1 .00%

3.72%

3.53%

3.34%

3.25%

3.17%

3.10%

2.3 7%

3.05%

0.04%

3.03%

0.32%

3.31%

0.01%

3.0 1%

0.31%

0.31%

0.33%

3.00%

3.33%

3.32%

3.29%

3.28%

0.26%

3.25%

3.24%

3.24%

3.24%

3.24%

3.24%

2.24%

3.24%

0.24%

3.25%

0.25%

3.27%

3.29%

0.32%

2.3S%

0.42%

3.49%

0.51%

0.37%

1.13%

3.14%

0.13%

3.12%

1 1%

1 1%

3.11%

3.10%

3.10%

3.13%

3.13%

3.12%

3.13%

3.11%

3.11%

3.11%

3.12%

0.13%

0.14%

0.16%

0.13%

0.21%

3.27%

3.33%

0.49%

30
Table A-IV

TH=11330 +

TC= 333 +

DY=.01 DB

165.33

1 .30

(1.53%)

(3.33%)

DG=.13%

TE(K) F(DB)

ERROR CONTRIBUTIONS TO TE

Y(DB) ETH ETC EY EG

13 +63.67.

15 +45.4%

23 +35.3%

30 +24.1%

53 +15.2%

70 +11.4%

100 + S.6%

150 + 6.4%

200 + 5.2%

303 + 4.1%

530 + 3.3%

700 + 2.9%

1300 +2.6%

15 03 + 2.4%

2030 + 2.3%

3333 + 2.2%

5033 + 2.2%

70 03 + 2.2%

10339 + 2.3%

15003 + 2.4%

23033 + 2.6%

303 03 + 2.8%

53300 + 3.5%

733 30 + 4.1 %

3.15 +.399

0.22 +.399

3.29 +.099

0.43 +.093

3.69 +.097

0.9 4 +.39 7

1 .29 +.096

1.81 +.094

2.23 +.093

3.33 +.39 1 4.3 5 +.389

5.33 +.338

6.48 +.337

7.^3 +.03 7 8.97 +.037

13.55 +.333

12.61 +.393

14.33 +.092

15.53 +.096

17.22 +.133

13.45 +.109

23.1.9 +.123

22.39 +.149

23.34 +.176

15.53 4 7.33% 13.29 % 7.34% 3.19%

15.44 32.33% 6.86% 4.93% 2.16%

15.37 24.67% 5.15% 3.79% 1 .55%

15.24 16.96% 3.44% 2.61% 1 .13%

14.99 12.79% 2.37% 1.66% 0.72%

14.76 3.15% 1 .43% 1 .26% 0.55%

14.43 6.17% 1 .24% 3.96% 3.41%

13.94 4.63% 3.69% 3.72% 3.31%

13.50 3.86% 0.52% 0.63% 0.26%

12.75 3.03% 0.35% 3.49% 2.21%

11.53 2.4 7% 0.21% 2.40% 3.17%

13.63 2.23% 0.15% •J'A . \J^^7'0 /o 3.15%

9.65 2.30% 0.11% 3.34% 0.15%

8.42 1.85% 3.03% 3.32% 0.14%

7.52 1 .77% 3.36% 0.32% 3.14%

6.23 1.73% 3.34% 3.33% 2.14%

4.80 1 . 63 % 0.03% 3.36% 3.16%

3.92 1 .61% 0.02% 0.43% 3.11%

3.39 1 .59% 0.02% 3.4 7% 3.23%

2.33 1.57% 0.02% 3.57% 3.25%

1 .34 1.57% 2.01% 3.6*% 3.29%

1.31 1.56% 0.01% 2.39% 0.39%

3.34 1 .55% 0.31% 1 .32% 3.57%

0.62 1.55% 3 .31% 1 .75% 0.76%

31
Table A-V

TH = 1 1000 + 155.30 K (1 .50%)

TC= 80 + 1.03 K (1 .25%)

DY = .01 DB DG=.10%

TE(K) F(D3)

ERROR CONTRIBUTIONS TO TE

Y(DB) ETH ETC EY EC

10 +26.7%

15 +18.4%

23 +14.3%

30 +10.1%

53 + 6.3%

73 + 5.47.

103 + 4.3%

150 + 3.5%

203 + 3.!%

333 + 2.7%

530 + 2.4%

703 + 2.2%

103 + 2.1%

1503 + 2.1%

2003 + 2.3%

3333 + 2.0%

5030 + 2. !%

7400 + 2.1%

10303 + 2.2%

15333 + 2.3%

20033 + 2.5%

30333 + 2.8%

50303 + 3.4%

70002 + 4.0%

0.15 +.039

0.22 +.339

0.29 +.043

0.43 +.041

0.69 +.044

0.94 +.346

1.29 +.3 43

1.31 +.052

2.23 +.355

3.33 +.059

4.35 +.065

5.33 +.069

5.43 +.072

7.90 +.075

3.97 +.077

13.55 +.033

12.61 +.334

14.03 +.383

15.53 +.092

17.22 +.099

13.45 +.106

23.19 +.1 19 22.39 +.145

23.34 +.172

20.38 13.60% 13.03% 2.39% 3.91%

23.64 9.57% 6.72% 1 .47% 0.64%

20.42 7.55% 5.05% 1.16% 0.50%

20.01 5.54% 3.5 7% 0.35% 3.37%

19.29 3.93% 2.32% 0.61% 3.26%

13.68 3.24% 1 .4 5% 3.50% 0.22%

17.93 2.72% 1 .32% 0.42% 0.!3%

1 S.36 2.32% 0.63% 0.36% 0.15%

1 5.32 2.12% 3.51% 3.33% 0.14%

14.73 1 .91% 0.3 4% 0.30% 3.13%

12.97 1 .75% 0.21% 3.23% 0.12%

11.76 1 .58% 0.15% 3.27% 3.12%

13.46 1 .63% 3.11% 3.27% 3.12%

8.98 1 .59% 0.03% 0.23% 0.12%

7.96 1.57% 0.36% 3.29% 0.12%

6.58 1 .55% 0.34% 3.33% 3.13%

4.98 1 .54% 0.03% 0.34% 0.15%

4.35 1 .53% 0.02% 3.38% 0.17%

3.19 1 .52% 0.32% 0.45% 0.19%

2.37 1 .52% 0.02% 3.55% 0.24%

1 .89 1 .52% 0.01% 3.66% 0.29%

1.35 1 .52% 3.01% 0.37% 0.33%

0.36 1.51% . 1 % 1 .29% 3.56%

3.63 1 .51% 3.01% 1.71% 3.74%

32
Table A-VI

TH=11090 + 165.03 K (1.50%)

TC= 4 + 0.50 K (12.53%)

DY = .31 DB DG:.10%

TE(K) F(D3)

ERROR CONTRIBUTIONS TO TE

Y(DE) ETH ETC EY EG

13 + 7.67. 3.15 + .011 28.96 2.13% 5.31% 0.32% 3.14%

15 + 5.7% 3.22 + .012 27.63 1 .9 3% 3.34% 0.29% 3.13%

20 + 4.77. 0.29 + .013 26.62 1 .83% 2.51% 0.28% 0.12%

30 + 3.7% 0.43 + .015 25.11 1.70% 1.67% 3.26% 0.1 1%

50 + 3.0% 3.69 + .319 23.11 1 .52% 1.33% 0.25% 3.11%

73 + 2.7% 0.94 + .022 21.75 1.59% 0.72% 0.25% 0.11%

100 + 2.4% 1.29 + .027 23.28 1 .5 6% 0.53% 0.24% 0.13%

150 + 2.2% 1.81 + .333 18.63 1 .54% 0.34% 0.24% 3.13%

200 + 2.1% 2.28 + .038 17.40 1.53% 3.25% 0.24% 0.137.

330 + 2.0% 3.33 + .045 15.73 1.52% 0.17% 0.24% 0.13%

500 + 2.3% 4.35 + .054 13.58 1.51% 0.13% 3.24% 0.11%

700 + 1.9% 5.33 + .060 12.21 1 .51% 0.33% 0.25% 3.11%

1000 + 1.9% 6.48 + .365 10.77 1 .51% 0.35% 0.25% 0.11%

1500 + 1.9% 7.90 + .070 9.20 1 .50% 0.04% 0.26% 0.11%

2003 + 1.9% 8.97 + .073 8.12 1.50% 0.03% 0.27% 0.12%

3300 + 1.9% 10.55 + .077 6.58 1.53% 0.02% 3.29% 0.13%

5030 + 2.3% 12.61 + .082 5.05 1 .53% 0.01% 3.34% 0.15%

7003 + 2.1% 14.00 + .336 4.10 1.53% 0.31% 3.33% 0.16%

10033 + 2.1% 15.53 + .090 3.22 1.53% 0.31% 0.44% 3.197.

15333 + 2.3% 17.22 + .393 2.39 1.53% 0.01% 0.54% 0.24%

20033 + 2.4% 18.45 + .134 1.93 1 .53% 0.01% 3.65% 0.23%

30333 + 2.7% 20.19 + .113 1.36 1.53% 0.31% 0.36% 0.37%

50033 + 3.3% 22.59 + .144 0.86 1 .50% 0.01% 1.23% 3.55%

70303 + 3.9% 23.84 + .170 0.63 1.53% 0.31% 1.73% 0.747.

33
Table A-VII

TH=

TC = 1253 +303 +

DY=.31 DB

3.30 K (3.24 7.) 0.13 K (3.93%)

DG=.12%

TE(K) F(DB) Y(DB)

ERROR CONTRIBUTIONS TO TE

ETH ETC EY EC

13 +24.7%

15 +15.8%

20 +12.87.

30 + 8.3%

50 + 5.67.

70 + 4.3%

100 + 3.3%

150 + 2.5%

200 + 2.17.

300 + 1 .8%

500 + 1 .5%

700 + 1 .4%

1000 + 1 .5%

1500 + 1.57. 2300 + 1.7%

3030 + 2.0%

5000 + 2.77.

7000 + 3.37.

10300 + 4.4%

15030 + 6.1%

20033 + 7.8%

33300 +11 .3%

53003 +18.57.

70000 +25.8%

0.15 +.33 6 0.22 +.03 6 0.29 +.336

0.43 +.335

3.69 +.336

0.94 +.036

1 .29 +.037

1 .31 +.037

2.28 +.038

3.33 +.339

4.35 +.042

5.33 +.044

6.48 +.049

7.9 3 +.05 6 S.97 +.364

10.55 +.079

12.61 +.109

14.00 +.139

15.50 +.184

17.22 +.260

18.45 +.336

20.19 +.488

22.39 +.797

23.34 +1.1 15

6.39 9. 79% 1.33% 9.47% 4.11%

6.04 6.63% 0.39% 6.44% 2.337.

5.99 5.35 7. 0.677. 4.93% 2.14%

5.89 3.477. 0.457. 3.41% 1 .48%

5.73 2.217. 0.277. 2.21% 0.96%

5.52 1.67% 0.23% 1.69% 3.73%

5.23 1.26% 3.14% 1.31% 0.57%

4.93 0.95% 0.13% 1.02% 3.44%

4.62 0.79% 3.38% 3.33% 0.33%

4.12 0.63% 0.05% 0.757. 0.33%

3.40 0.51% 0.04% 0.63% 0.29%

2.93 0.45% 0.33% 0.68% 0.29%

2.33 0.417. 3.02% 0.71% 3.31%

1.34 0.38% 3.02% 0.30% 0.357.

1 .53 0.36% 0.32% 0.91% 3.39%

1 .13 3.35% 0.317. 1.13% 0.49%

0.72 3.33% 0.01% 1 .61% 0.73%

0.53 3.33% 0.317. 2.09% 0.91%

3.33 0.33% 3.31% 2.81% 1 .22%

3.25 0.32% 3.017. 4.02% 1.75%

0.20 3.32% 3.317. 5.24% 2.27%

0.13 3.32% 0.31% 7.69% 3.33%

0.03 0.32% 3.01% 1 2 . 69 % 5.44%

3.06 0.32% 3.01% 17.38% 7.57%

34
Table A-VIII

TH= 1250 + 3.03 K (3.24 7.) TC= 321 + 0.20 K (0.25%)

DY=.01 DB DG=.10%

TE(K) F(DB)

ERROR CONTRIBUTIONS TO TE

Y(DB) ETH ETC EY EG

13 + 7.7%

15 + 5.3%

23 + 4.2%

33 + 3.3%

53 + 2.1%

70 + 1 .7%

133 + 1 .4%

150+1 .2%

203 + 1.1%

300 + 1.3%

533 + 0.9%

700 + 0.9%

1333 + I .0%

1500 + 1.1%

2303 + 1 .'2%

3333 + 1 .5%

5030 + 2.1%

7033 + 2.6%

13033 +3.5%

15330 + 4.9%

23333 + 6.3%

33300 + 9.1 % 53003 +14.9%

73330 +20.7%

3.15 +.01 1 3.22 +.011

0.29 +.012

0.43 +.012

0.69 +.013

3.94 +.314

1 .29 +.315

1 .31 +.017

2.28 +.019

3.08 +.321

4.35 +.026

5.33 +.329

6.48 +.334

7.93 +.341

3.97 +.047

13.55 +.063

12.61 +.385

14.00 +.110

15.53 +.147

17.22 +.238

18.45 +.270

23.19 +.393

22.39 +.64 2 23.34 +.89 6

11.46

11.24

11.34

10.66

10.33

9.44

8.75

7.34

7.14

6.11

4.33

3.98

3.19

2.41

1 .94

1.40

0.93

0.66

0.43

0.32

0.25

0.17

0.13

0.37

2.31% 2.15% 2.23% 0.97%

1.62% 1.44% 1.53% 3.63%

1 .23% 1.09% 1.25% 0.54%

3.94% 3 . 73 % 3.92% 0.40%

0.67% 0.44% 0.67% 3.29%

0.55% 0.32% 0.56% 0.24%

3.46% 3.23% 3.43% 0.21%

3.39% 3.16% 3.42% 0.13%

3.36% 0.12% 0.43% 0.17%

3.32% 0.39% 3.39% 0.17%

3.30% 0.06% 3.40% 3.17%

0.29% 3.05% 3.43% 0.19%

3.28% 0.04% 3.43% 3.21%

0.27% 3.33% 0.57% 3.25%

3.27% . 33 % 3.67% 3.29%

3.26% 0.32% 0.36% 3.37%

0.26% 3.02% 1 .25% 3.54%

0.26% 0.32% 1.64% 3.71%

0.26% 3.32% 2.23% 3.97%

0.26% 0.32% 3.22% 1 .43%

0.26% 0.02% 4.21% 1.82%

3.26% 0.02% 6.19% 2 . 63 % 0.26% 3.32% 10.21% 4.43%

3.26% 0.02% 14.32% 6.12%

.3 5
Table A-IX

TH= 1250 + TC= 4 +

DY = .01 DB

3.00 K (0.247.)

0.10 K (2.50%)

DG=.10%

TE(K) F(DB) Y(D3)

ERROR CONTRIBUTIONS TO TE

ETH ETC EY EG

10 + 1 .37. 0.15 + .003

15 + 1 .4 7. 0.22 + .003

23 + 1 ,27. 0.29 + .003

30 + 1 .07. 0.43 + .004

50 + .87. 0.69 + .005

70 + 2 .87. 0.94 + .007

100 + 3 ,77. 1.29 + .008

150 + ,77. 1.81 + .010

200 + .77. 2.28 + .012

300 + ,17. 3.08 + .015

500 + ,11 4.35 + .023

730 + .37. 5.33 + .024

1000 + 0,,9% 6.43 + .029

1503 + 1 ,07. 7.90 + .036

2000 + 1 ,17. 8.97 + .042

3 030 + 1 .47. 10.55 + .055

5303 + 1 .9% 12.61 + .078

7030 + 2 .47. 14.03 + .102

10000 + 3 .27. 15.50 + .136

15300 + 4 ,67. 17.22 + .194

23300 + 5 ,9 7. 13.45 + .252

33033 + 3 .67. 20.19 + .368

50330 + 13 .9 7. 22.39 + .601

70030 + 19 .4 7. 23.34 + .333

19.54

13.23

17.24

15.76

13.82

12.51

11.13

9.59

3.52

7.07

5.41

4.42

3.50

2.62

2.10

1.51

3.97

3.71

3.51

0.35

3.26

0.18

3.1 1 0.03

0.347

0.307.

0.29%

0.277.

0.26%

3.25 7.

0.257.

0.257.

0.257.

0.24 7. 0.24%

0.247.

0.247.

0.24%

0.24%

2.24%

0.24%

3.247.

3.24 7. 0.24%

0.24 7. 0.24%

0.247.

3.24%

1 .017. 0.63%

0.517.

0.347.

0.217.

0.157.

0.117.

0.077.

0.36%

0.047.

0.33%

0.027.

0.027.

0.017.

0.01%

2.017.

0.017.

0.017.

3.01%

3.017.

3.31%

2.01%

0.017.

3.317.

0.33%

0.307.

0.23%

0.277.

0.26%

0.26%

0.26%

0.277.

0.277.

2.29%

0.337.

0.3 67.

0.427.

0.51%

3.60%

0.79%

1.167.

1 .537. 2.28%

3.01%

3.937.

5.30%

9.56%

13.42%

0.14%

0.137.

0.12%

0.127.

0.117.

0.117.

0.1 1%

3.12%

0.122

3.137.

0.14%

0.167.

0.187.

0.22%

0.267.

2.347.

0.507.

0.667.

3.9 3%

1 .337.

1 .71%

2.517.

4.127.

5.747.

36
Table A-X

TH = 692 + 3.9.3 K (3.13?.)

TC = 3 33 + 2.10 K (3.037.)

DY=.31 DB DG=.10 7.

TE(K) F(F)B)

ERROR CONTRIBUTIONS TO TE

Y(DB) ETH ETC EY EG

13 +2 7 ,2% 3.15 + .039 3.55 7.127. 1.797. 12.73% 5.55%

15 + ] 3..5% 0.22 + .343 3.51 4.32% 1.237. 8.72% 3.79%

23 + 14 ,2% 3.29 + .043 3.47 3.677. 3.917. 6.69% 2.91%

33 + 9 .3% 3.43 + .343 3.40 2.537. 3.61% 4.67% 2.33%

53 + S .4% 0.69 + .041 3.26 1.617. 0.38% 3.35% 1 .337. 73 + 4 ,9% 0.94 + .041 3.14 1.217. 3.28% 2.37% 1 .037.

133 + 3,,8% 1.29 + .34 2 2.97 3.927. 3.23% 1 .86% 0.31%

15?) + 3 ,3% 1 .31 + .344 2.72 0.697. 3.14% 1 .4S7. 3.647.

203 + 2 ,6% 2.23 + .045 2.51 0.5 77. 0.11% 1 .317. 0.577.

3 '3 3 + 2 ,2% 3.33 + .349 2.13 0.4 67. 3.33% 1.17% 3.51%

533 + 2 ,0% 4.35 + .056 1.73 0.3 77. 3.0 6% 1.12% 0.49%

733 +

r> ,17. 5.33 + .363 1.44 3.33% 0.35% 1 .17% 3.517.

1000 + 2 ,2% 6.4 3 + .374 1.14 3.3 37. 3.04% 1 .29% 3.5 5%

1503 + 2 .5% 7.93 + .392 0.36 0.23% 0.047. 1.55% 3.67%

20 3 2 + 2 ,9% 3.97 + .1 10 . 63 3.26% 3.33% 1.32% 3.79%

3033 + 3,,7% 10.55 + .147 3.49 0.25% 0.03% 2.39% 1 .04%

530 3 + 5 ,4% 12.61 + .223 0.31 3.24 7. 0.03% 3.55% 1.547.

7303 + 7 ,37. 14.03 + .293 0.23 3.24% 3.03 7. 4.727. 2.35%

10003 + 9 ,6% 15.53 + .404 0.16 3.24% 0.037. 6.49% 2.31%

15333 + 3 ,37. 17.22 + .539 3.11 3.23% 3.33% 9.43% 4.09%

23000 + 18,,27. 13.45 + .777 3.03 3.237. 3.33 7. 12.52% 5.37%

33033 + 27 ,07. 23.19 +1.163 0.0S 3 . 23 7. . 03 % 13.327. 7.96%

53 330 + 46 .4'. 22.39 +2.331 3.33 0.23% 0.037. 32.3 77. 13.23%

70333 + 69 ,4 7. 23.84 +3.00 0.32 0.23% . 33 7. 50.397. 13.71%

37
Table A-XI

TH= 692 + 3.90 K (3.137.)

TC: 80 + 0.20 K (3.257.)

DY = .01 DB DG=.10Z

TE(K) FCDB)

ERROR CONTRIBUTIONS TO TE

Y(DB) ETH ETC EY EG

10 + 7 ,3% 3.15 + .013 3.92 1 .32% 2.29% 2.33% 1 .33%

15 + 4 ,9% 3.22 + .013 8.72 0.937. 1 .54% 1 .63% 3.73%

20 + 3 ,3% 3.29 + .311 3.52 3.74 7, 1.15% 1.347. 3.58%

30 + 2 ,37. 3.43 + .01 1 3.17 0.547, 3.79 7. 1 .33% 3.437.

53 + 1 .9 7. 3.69 + .012 7.56 3.387. 0.43% 0.73% 3.32%

73 + 1 ,6% 0.94 + .013 7.36 0.3 2% 3.36% 0.61% 3.27%

100 + 1 ,37. 1.29 + .314 6.43 0.26% 3.25% 3.54% 3.23%

150 + 1 .1% 1.31 + .316 5.64 . 23 % 3.13% 0.49% 0.21%

20 3 + 1 ,0% 2.28 + .318 5.03 3.21 % 8.15% 3.4 7% 0.20%

300 + 1 ,3 7. 3.38 + .821 4.17 3.197. 3.117. 3.47% 0.21%

5:53 + 1 ,3 7. 4.3 5 + .327 3.13 0.177. 3.33% 0.5 27. 3.23%

733 + 1 . 1% 5.33 + .353 2.52 0.16% 3.3 6% 3.58% 3.257.

1000 + 1 ,27. 6. 48 + .043 1.95 3.16% 0.3 6% 3.69% 0.3 3%

1500 + 1 ,4% 7.90 + .353 1.42 3.15% 3.0 5% 0.87% 3.38%

2000 + 1 .77. 8.97 + .3 65 1.12 0.15% 0.34% 1 .057. 3.4 6%

3000 + 2 ,27. 13.55 + .33" 0.79 3.15% 0.04% 1 .43% 3.62%

5 303 + 3 .3% 12.61 + .136 0.49 0.15% 0.04% 2.18% 3.95%

7300 + 4 .4% 14.00 + .183 0.36 0.15% 0.34% 2.937. 1 .27%

10000 + s ,3 7. 15.53 + .253 3.26 0.15% 0.03% 4.067. 1 .76%

15030 + q ,77. 17.22 + .371 3.17 3.15% 3.03% 5.9S% 2.53%

20302 + 11 .47. 18.45 + .49^ 0.13 0.15% 0.03% 7.36% 3.43%

33303 + 17 .07. 20.19 + .729 0.39 2.15% 0.03% 1 1 .737. 5.34%

50003 +28 .4 7. 22.39 +1.224 3.35 0.15% 3 . 33 % 19.82% 3.35%

73303 + 40 .57. 23.34 + 1 .754 0.34 3.15% 0.33% 23.65% 11.72%

38
Table A-XII

TH= 692 + TC = 4 + 3.90 K (0.13%)

0.10 K (.2.57)7.) DYr.31 DB DG=.13%

TE(K) F(DB) Y(OR)

P.RROR CONTRIBUTIONS

ETH ETC FY TO TE

10 + 1 ,77 3.15 + .00 2 17.00 0.13% 1.02% 0.33% 3.14%

15 + 1 .3% 3.22 + .033 15.71 0.177. 3.69% 0.30% 3.13%

20 + 1 ,1% 0.29 + .303 14.72 0.167 3.52 7. 3.29% 3.12%

33 + . 9 % 3.43 + .004 13.27 0.157, 0.35% 3.27% 3.127.

50 + .7% 0.69 + .005 11 .30 3.147 0.227. 0.27% 3.12%

70 + p, .7% 0.94 + .006 1 3 . 1

3

0.14% 0.16% 0.277.

100 + g ,6% 1 .29 + .00 7 3.82 3.147. 0. 127. . 20 7 0.127

150 + 3 ,67. 1.01 + .039 7.33 3.13 7. 0.08% 0.29% 0.13%

2 30 + ,6% + .311 6.41 0.13 % 0.36% . 3 * 7. 3.13%

300 + .7 7. 3.00 + .015 5.14 0.13/1 3. 05 % 0.34% 0.15%

530 +

r/ .7% 4.3 5 + .020 3.74 3.137 .33 7 3.437 0.17%

700 + Q ,3% 5.3 3 + .026 2.96 3.13 % 3.03 7 0.47% 3 . ?.£ %

10-20 + l .0% 6.43 + .033 2.27 7

. 1 3 7 0. 3?% 3.57% 25%

1530 + l .2/5 7.90 + .344 1 .64 0.137 0. 32% 0.74% o!32%

?030 + l .4 7. 3.9 7 + .05 5 1 .25 3.13% 3.02% 3.90% 0.39%

3300 + l ,9? 1 . 5 5 + .375 0.90 .13% 3.327. l.°47 3.54%

5003 +

o .9% 12.61 + .110 3.5 5 3.13% 0.02 7, 1.917. 0.03%

70 30 + 3 ,0% 14.00 + .160 3.4 1 3.137 0.22% 2.53% 1 .12%

10030 + 5 .3% 15.53 + .223 0.29 0.13 % 0.32% 3.53% 1 .5-7

15003 + 7 ,7% 17.22 + .320 . 1 <3 3.13% . ? % 5.27 7 2.23%

20003 + 10 . 1 7. 13.45 + .453 0.15 3.13% 3.327 6.967 3.317

3000? + 15 . 3% 20.19 + .64 5 Z. 13 0.13 1 0.31 7. 10.33% 4.47%

50300 +25 .3% 02.39 + 1 .300 0.0 6 .13 7 3 . 1 7. 17.4 7% 7.41%

70000 +35 .6% 23.04 +1.539 0.04 0.137. 3.31 % 25.36% 13.33%

39
Table A-XIII

TE(K)

10 +13.5%

15 +12.7%

20 + 9.8%

33 + 6.9%

53 + 4.6%

73 + 3.6%

133 + 2.9%

153 + 2.4%

230 + 2.1%

333 + 1 .9%

530 + 1.9%

730 + 2.1%

F(P3)

3.15 +.027

0.22 +.027

3.29 +.327

0.43 +.023

0.69 +.329

0.94 +.333

1.29 +.332

1.81 +.335

2.23 +.033

3.33 +.343

4.35 +.053

5.33 +.053

TH= 3 73 + TC= 30 +

DY = .01 DB

Y(DB)

6.29

6.11

5.94

5.64

5.12

4.73

4.23

3.57

3.11

2.48

1.78

1.39

0.50 K (3.13%)

1 .00 K (1.25%)

DGr.13%

ERROR CONTRIBUTIONS TO TE

ETK ETC EY EG

1 .54% 13.37% 2.71 % 1.13%

1.33% 3.33% 1.33% 3.34%

3.35% 6.71% 1.54% 0.67%

0.63% 4.53% 1.16% 0.53%

0.44% 2.39% 0.86% 0.33%

0.3 7% 2.16% 0.75% 0.32%

0.31% 1.61% 0.67% 3.29%

0.26% 1.19% 0.63% 0.27%

3.24% 3.98% 0.63% 3.27%

0.22% 3.77% 3.67% 3.29%

0.20% 3.60% 0.33% a. 35%

0.19% 0.52% 0.94% 3.41%

1330 + 2.3% 6.43 +.373 1.04 0.13% 0.47%

1533 + 2.8% 7.90 +.103 0.74 0.13% 3.43%

2303 + 3.4% 8.97 +.12S 3.57 3.13% 3.43%

3303 + 4.5%

5033 + 6.7%

7303 + 9.3%

13 303 +12.4%

15 300 +13.1%

20033 +24.0%

33033 +36.4%

53300 +55.3%

73000 +135.3%

10.55 +.177

12.51 +.275

14.33 +.3 74

15.53 +.522

1 7.22 +.773

18.45 +1 .329

23.19 +1 .566

22.39 +2.318

23.14 +4.575

1.17% 0.51%

1.55% 3.67%

1.94% 3.34%

3.39 0.13% 0.33% 2.72% 1.13%

3.24 3.17% 3.37% 4.30% 1.36%

0.13 0.17% 3.36% 5.83% 2.55%

3.12 3.17% 0.35% 3.27% 3.57%

0.08 3.17% 0.35% 12.33% 5.29%

3.36 3.17% 3.35% 16.53% 7.31%

0.04 0.17% 3.3 5%

3.03 3.17% 3.34%

0.32 3.17% 3.34%

25.33% 13.53%

4 7.32% 17.74%

79.76% 25.53%

4

()
Table A-XIV

TH= 373 +

TC= 4 + 0.53 K (0.13%)

0.53 K (12.507.)

DY=.01 DB DG=.137.

ERROR CONTRIBUTIONS TO TE

TE(K) F(DB) YCDB) ETH ETC EY EG

13 + 5.9% 0.15 + .333 14.37 0.19% 5.19% 0.337. 0.15%

15 + 4.1% 0.22 + .009 13.13 0.17% 3.50% 0.31% 0.13%

20 + 3.2% 0.29 + .339 1 2 . 1 4 3.16% 2.66% 0.297. 0.137.

30 + 2.4 7. 0.43 + .313 13.74 0.157. 1.82% 0.29% 3.127.

53 + 1.7% 3.69 + .011 8.94 3.15% 1.15% 0.29% 0.12%

73 + 1 .4% 0.94 + .012 7.77 0.14% 0.36% 0.29% 0.15%

133 + 1.2% 1.29 + .314 6.53 0.14% 3.64% 3.31% 0.13%

153 + 1.17. 1.31 + .316 5.31 0.14% 0.47% 3.34% 0.15%

200 + 1.07. 2.28 + .319 4.49 0.14% 0.39% 0.36% 3.15%

330 + 1.17. 3.33 + .323 3.45 0.147. 0.30% 0.43% 3.13%

500 + 1.27. 4.35 + .332 2.39 0.14% 3.24% 0.55 7. 0.247.

700 + 1.37. 5.33 + .043 1 .33 0.14% 0.21% 0.67% 3.297.

10 33 + 1 .57. 6.43 + .352 1.36 3.14% 0.19% 0.367. 0.37%

1533 + 2.3% 7.90 + .372 0.95 3.147. 0.17% 1.17% 0.517.

20 03 + 2.4 7. 8.97 + .392 0.73 0.14% 0.16% 1.48% 0.64%

3033 + 3.3% 13.55 + .131 0.53 0.147. 0.157. 2.11% 0.92%

5033 + 5.1% 12.61 + .209 3.31 0.14% 0.15% 3.36% 1.46%

7330 + 6.9% 14.00 + .2S7 0.22 3.14% 0.14% 4.61% 2.30%

10003 + 9.67. 15.53 + .435 0.1S 0.147. 0.147. 6.53% 2.31%

15033 + 14.17. 17.22 + .532 0.1 1 0.14% 0.14% 9.63% 4.17%

20303 +13.7% 13.45 + .302 0.03 3.14% 0.14% 12.92% 5.547.

30333 +:23.2% 23.19 +1 .214 0.05 0.14% 0.14% 19.66% 8.297.

53033 + 49.17. 22.39 +2.113 0.03 0.14% 0.14% 34.33% 13.91%

70333 + 74 . 5 % 23.34 +3.221 0.32 0.147. 0.14% 54.41% 19.73%

41
Table A-XV

TH= 373 + 0.15 K (0.04%)

TC= 303 + 0.10 K (0.03%)

DY=.01 OB DG: .10%

TEC K) FCDB)

ERROR CONTRIBUTIONS TO TE

YCDB) ETH ETC EY EG

171 +65.3% 0.15 +.095 0.92 6.37% 5.25%

15 +44.7% 0.22 + .096 0.91 4.32% 3.54%

20 +34.4% 3.29 +.096 0.89 3.29% 2.69%

37.46% 16.26%

25.70% 11.16%

19.34% 3.61%

30 +24.2%

50 +16.0%

70 +12.5%

100 +13.0%

150 + 3.2%

200 + 7.4%

0.4 3 +.098

0.69 +. 102

0.94 +.136

1 .29 +.112

1.31 +.121

2. 23 +.131

0.37

0.82

0.78

0.73

3.65

0.59

2.26% 1.34% 13.93% 6.07%

1 .44% 1.16% 9.34% 4.06%

1 .09% 3.37% 7.39% 3.21%

0.32% 0.65% 5.97% 2.59%

0.62% 0.43% 4.95% 2.15%

0.51% 3.39% 4.52% 1.96%

300 + 6.8% 3.03 +.153 3.53 3.41% 0.31%

503 + 6.9% 4.35 +.139 3.33 0.33% 3.24%

730 + 7.4% 5.33 +.229 0.31 3.29% 0.21%

4.25% 1.34%

4.41% 1.91%

4.84% 2.13%

1003 +3.5% 6.48 +.238 0.24 0.27% 0.19%

1503 +10.6% 7.95 +.336 0.17 3.25% 0.17%

2030 +12.3% 3.97 +.435 3.14 0.24% 3.16%

5.64% 2.45%

7.11% 3 . 33 % 3.66% 3.74%

3003 +17.3%

5030 +26.7%

7303 +36.6%

10.55 +.686 3.10

12.61 +1 .096 0.06

14.03 +1 .523 0.04

3.23% 3.15% 11.34% 5.39%

3.22% 0.15% 13.50% 7.34%

3.21% 3.14% 25.64% 10.64%

13000 +53.0%

15 030 +8 7.5%

20333 +142.2%

15.53 +2.233 0.03

17.22 +3.727 3.32

18.45 +6.335 0.02

3.21% 3.14% 37.74% 14.9-3%

3.21% 0.14% 64.66% 22.47%

3.21% 3.14% 111.11% 30.72

42
Table A-XVI

TH = 3 73 + 0.15 K (3.34%)

TC = 80 + 0.20 K (0.25%)

DY = .01 D8 DGr.13 7.

TEC K) F(DB)

ERROR CONTRIBUTIONS TO TE

Y(D3) ETH ETC EY EG

10 + 7.0%

15 + 4.9%

23 + 3.3%

0.15 +.013

3.22 +.313

3.29 +.011

6.29

S.ll

5.94

0.4 6%

0.32%

3.26%

2.61%

1 .77%

1.3 4%

2.71%

1 .93%

1 .54%

1 .13%

0.34%

3.67%

33 + 2.8%

53 + 2.3%

73 + 1.6%

0.43 +.311

3.69 +.012

3.94 +.314

5.64

5.12

4.70

0.19%

0.13%

3.11%

3.92%

3.53%

0.43%

1.16%

0.36%

0.75%

3.53%

3.33%

3.32%

133 + 1.4% 1.29 +.315 4.20 3.39% 0.32% 0.67% 3.29%

15-3 + 1.2% 1.31 +.313 3.57 3.33% 3.24% 0.63% 3.27%

230+ 1.2% 2.23 +.321 3.11 0.07% 3.20% 0.63% 3.27%

33 3 + 530 +1 .2%

1.3%

733 + 1 .5%

3.33 +.326

4.35 +.336

5.33 +.04 6

2.43

1.73

1.39

3.36% 3.15% 3.67% 3.29%

0.36% 0.12% 0.33% 0.35%

3.36% 3.13% 3.94% 3.41%

1333 + 1 .3%

1533 + 2.4%

2333 + 2.9%

6.43 +.361

7.90 +.336

^.97 +.1 11

1.34

0.74

3.57

3.0 6%

0.05%

0.05%

3.09% 1.17%

3.39% 1.55%

3.33% 1.94%

3.51%

3.67%

3.34%

3333 + 4.3% 13.55 +.160 3.39 3.35% 3.03% 2.72% 1.13%

5000 + 6.3% 12.61 +.253 3.24 3.35% 0.37% 4.30% 1.36%

7300 + 3.5% 14.00 +.357 3.13 3.35% 0.37% 5.83% 2.55%

10333 +12.3%

15333 +17.7%

2333 3 +2 3.6%

15.53 +.505 3.12 3.05% 3.37% 3.27% 3.57%

17.22 +.756 3.03 0.35% 0.07% 12.33% 5.29%

13.45+1.012 0.36 3.35% 0.37% 16.53% 7.31%

33333 +36.3%

53333 +64.9%

73333 +135.4%

23.19 +1 .549 3.34 3.05%

22.39 +2.3>J1 3.33 3.35%

23. S4 +4.553 0.02 3.35%

3.37% 25.33% 10.53%

3.37% 47.32% 17.74%

3.37% 79.76% 25.50%

43
Table A- XVI

I

TH = TC=373

4 DY=.01 D3

3.15

3.10

(3.04%)

(2.50%)

DG=.10%

TECK) F(DB)

ERROR CONTRIBUTIONS TO TE

Y(D3) ETH ETC EY EG

10 + 1.6%

15+1 .2%

23 + 1.0%

33 + 2.87.

53 + 3.7%

70 + 3.6%

130 + 3.6%

153 + 3.6%

230 + 3.6%

333 + 3.7%

533 + 3.9%

703 + 1.3%

1030 + 1 .3%

15 33 + 1 .*%

2333 + 2.2%

3333 +3.1%

5303 + 4.9%

7333 + 6.7%

13333 + 9.4%

15303 +13.9%

23303 +1P.5%

333 3'! +23.0%

50333 +48.9%

70333 +74.3%

3.15 +.002

3.22 +.333

3.29 +.033

0.43 +.333

3.69 +.334

3.94 +.335

1 .29 +.30 7

1 .31 +.339

2.23 +.31 1 3.33 +.31

S

4.35 +.324

5.33 +.332

6.43 +.344

7.93 +.364

3.97 +.333

13.55 +.123

12.61 +.201

14.03 +.279

15.53 +.396

17.22 +.593

IS. 45 +.793

23.19 +1 .205

22.3S +2.1 13

23.34 +3.212

14.37 3.36% 1.34% 3.33% 3.15%

13.13 3.35% 0.73% 3.31% 3.13%

12.14 3.35% 3.53% 3 . 29 % 3.13%

10.74 3.35% 3.36% 3.29% 3.12%

3.94 3.34% 3.23% 3.29% 0.12%

7.77 3.04% 3.17% 3.29% 3.13%

6.53 3.34% 0.13 % 3.31% 3.13%

5.31 3.34% 3.39% 3.34% 3.15%

4.49 3 . 34 % 3.03% 3.36% 3.16%

3.45 0.34% 3.36% 3.43% 3.13%

2.3^ 3.34% 0.05% 3.55% S.24%

1.33 3.04% 3.34% 3.67% 3.29%

1 .36 3.34% 3.34% 3.36% 3.37%

3.95 3.34% 3.33% 1.17% 3.51%

3.73 3.04% 3 . 33 % 1.43% 1 . 64 %

3.53 3.34 7. 3.33% 2.1 1% 3.92 % 0.31 3.34% 3.33% 3 ,3S% 1 .4 6%

3.22 0.04% 3.03% 4.61% 2 .

9| 3 % 3.16 3.34% 3.33% 6.5?% 2.31%

0.1 1 3.34% 3.03% 9.63% 4.17%

3.33 3.34% 3 . 33 1 12.92% 5.54%

3.35 0.34% 3.33% 19.66% 3.29%

3.33 3.04% 3.33% 34.33% 13.91%

D.32 3.34% 3.33% 54.41% 19.73%

44
Table A-XVIII

TH = TC:

30 +

3 +

DY = .01 DB

1.03 K (0.33%)

1.33 K (1.25 7.) DG=.10%

TE(K) F(D3) Y(DB)

ERROR CONTRIBUTIONS TO TE

ETH ETC EY EG

10 +22.4%

15 +15.4%

2 5 +1 1 .9%

33 + 8.5%

53 + 5.7%

70 + 4.67.

133 + 3.77.

153 + 3.1%

200 + 2.3%

323 + 2.6%

5 00 + 2.6%

700 + 2.8%

1303 + 3.2%

1530 + 3.97.

2000 + 4.6%

3000 + 6.17.

5 300 + 9.0%

7033 +12.1%

10000 +16.6%

15030 +24.57.

20000 +32.6%

30033 +50.37.

50030 +93.0%

73030 +197.17.

0.15 +.032

0.22 +.033

0.29 +.033

0.43 +.035

0.69 +.037

0.94 +.039

1 .29 +.041

1 .81 +.046

2.28 +.050

3.08 +.058

4.35 +.073

5.33 +.037

6.43 +.107

7.90 +.141

3.97 +.174

10.55 +.243

12.61 +.371

14.00 +.503

5.37

5.21

5.05

4.77

4.3

3.92

3.47

2.91

2.52

1.93

1 .40

1 .03

0.31

0.57

0.44

0.33

0.13

0.13

4.39% 14.09%

2.33% 9.55%

2.27% 7.27%

2.92% 1.277.

2.09% 3.917.

1.67% 0.73%

15.53 +.703 0.09

17.22 +1 .042 0.06

13.45 +1.396 0.35

23.19 +2.165 0.03

22.39 +4.232 0.02

23.34 +3.526 0.01

1.67%

1.187.

0.97%

0.82%

0.737.

3 . 64 % 0.53%

0.53%

0.51%

0.497.

0.437.

3.4 77. 0.47%

3.4S7.

0.46%

5.03%

3.137.

2.42%

1.82%

1.367.

1.14%

0.91%

0.73%

0.65%

0.59%

0.55 7. 0.52%

0.50%

0.43%

0.47%

1.27%

0.957.

3.837.

0.75 7. 0.72%

. 73 7. 3.33%

0.977.

1.17%

1.47%

1.99%

2.537.

3.557.

5.65%

7.77%

0.55 7. 0.41%

0.36%

0.33%

0.31%

0.32%

0.35%

3.42%

0.51%

0.64%

0.36%

1.09%

1 .54%

2.457.

3.35%

J. 46% 0.47% 10.997. 4.73%

0.4 57. 0.4 6% 16.52% 7.0 2%

0.46% 0.467. 22.34% 9.34%

0.4 6% 0.4 6%

0.45% 0.4S%

0.46% 0.46%

35.35% 14.07%

72.957. 24.157.

160.617. 35.6

45
T£(K)

10 + 7.4%

15 + 5.2%

20 + 4.17.

33 + 3.0%

50 + 2.1%

70 + 1.8%

133 + 1.5%

150 + 1.4%

2 33 + 1.3%

300 + 1 .4%

5 30 + 1.6%

700 + 1 .9%

1000 + 2.3%

1500 + 3.0%

2000 + 3.7%

3000 + 5.2%

5300 + 3.27.

7003 +11 .3%

Table A-XIX

TH= 303 + 3.13 K (3.03%)

TC = S3 + 0.20 K (0.25%)

DY=.31 DB DG=.10%

F(DB)

0.15 +.011

3.22 +.311

0.29 +.01

1

0.4 3 +.012

3.69 +.014

2.94 +.215

1 .29 +.017

1.81 +.020

2.28 +.024

3.38 +.033

4.35 +.044

5.33 +.357

6.48 +.077

7.90 +.139

3.97 +.142

10.55 +.207

12.61 +.333

14.30 +.4 70

Y(DB)

5.37

5.21

5.35

4.77

4.33

3.92

3.47

2.91

2.52

1 .93

1 .40

1 .08

3.81

0.57

0.44

0.30

0.18

0.13

ERROR CONTRIBUTIONS TO TE

ETH ETC EY EG

0.41% Cm • <* /• 2.92% 1.27%

0.29% 1.91% 2.39% 3.91%

3 . 23 % 1.45% 1 .67% 3.73%

0.17% 1.03% 1.27% 0.55%

0.12% 0.64% 3.95% 0.41%

0.10% 0.43% 0.33% 3.36%

0.08% 0.36% 3.75% 3.33%

0.37% 0.27% 0.72% 3.31%

0.06% 0.23% 0.75% 0.32%

0.36% 3.13% 3.30% 0.35%

0.35% 0.15% 3.97% 3.42%

0.35% 3.13% 1.17% 0.51%

3.35% 0.12% 1.4 7% 3 . 64 % 0.05% 3.11% 1.99% 0.36%

0.05% 3.13% 2.53% 1.09%

3.05% 3.10% 3.55% 1.54%

0.35% 0.13% 5.65% 2.45%

0.35% 0.09% 7.77% 3.3 6%

10300 +15.9%

15030 +23.7%

20 030 +31 .3%

15.50 +.673 3.39

17.22 +1 .009 3.06

18.45 +1.362 3.05

0.35% 0.39% 10.99% 4.73%

0.05% 0.09% 16.52% 7.32%

0.05% 0.09% 22.34% 9.34%

30330 +49.6%

53030 +97.2%

73030 +196.4%

20.19 +2.132

22.39 +4. 199

23.34 +3.492

3.33

0.02

0.31

0.05% 0.09% 35.35% 14.37%

0.05% 3.09% 72.95% 24.15%

0.05% 0.09% 163.61% 35.62

46
Table A-XX

TH= 390 +

TC= 4 +

DY=.01 DB

1 .33 K (3.33%)

0.50 K (12.50%)

DG=.13%

TE(K)

10 + 6.2%

15 + 4.4%

23 + 3.5%

33 + 2.7%

50 + 2.3%

70 + 1.7%

130 + 1.5%

153 + 1.4%

200 + 1.3%

333 + 1.4%

533 + 1.5%

700 + 1.7%

1033 + 2.0%

1533 + 2.5%

2033 + 3.1%

3333 + 4.2%

5 333 + 6.4%

7 303 + 3.7%

10333 +12.1%

15333 +17.3%

23033 +23.6%

33333 +35.8%

50030 +64.2%

7 3 333 +1^3.7%

F(DB)

0.15 +.339

3.22 +.309

0.29 +.313

3.43 +.311

3.69 +.313

3.94 +.314

1.29 +.317

1 .31 +.323

2.28 +.024

3. 33 +.333

4.35 +.342

5.33 +.352

6.43 +.363

7.93 +.393

3.97 +.1 13

13.55 +.167

12.61 +.264

14.33 +.352

15.53 + .5 ^9 17.22 +.75 7 13.45 +1.313

23.19 +1 .543

22.39 +2.772

23.34 +4.435

ERROR CONTRIBUTIONS TO TE

Y(DB) ETH ETC EY EG

13.45 3.4 7% 5.24% 0.34% 0.15%

12.23 0.43% 3.55% 3.31% 0.13%

I 1.25 0.41% 2.73% 0.30% 0.13%

9.37 0.33% 1.86% 3.29% 3.13%

3.12 0.3 6% 1.13% 0.29% 3.13%

6.99 0.36% 0.39% 3.33% 3.13%

5.35 0.35% 3.53% 3.3 2% 3.14%

4.66 0.35% 3.51% 3.36% 0.16%

3.39 3.34% 3.4 2% 3.4 3% 3.17%

2.95 3.34% 0.34% 3.47% 3.21%

2.31 0.34% 3.27% 3.63% 3.27%

1.52 3.34% 0.24% 3.73% 3.34%

1.12 3.34% 3.22% 1 .02% 3.44%

3.73 0.34% 3.23% 1 .43% 0.61%

0.63 0.34% 3.19% 1 .79% 3.73%

3.4 1 3.34% 3.19% 2.57% 1 .12%

3.25 3.34% 0.13% 4.13% 1.79%

3.13 3.34% 3.13% 5.73% 2.4 7%

3.13 3.34% 3.17% 8.3 7% 3.49%

3.33 3.34% 0.17% 12.33% 5.13%

3.36 3.34% 0.17% 16.23% 5.39%

0.34 3.34% 0.17% 24.9 6% 13.34%

3.33 3.34% 3.17% 45.21% 17.49%

3.32 0.34% 3.17% 73.32% 25.16%

47
TECK)

13 + 1 .6%

15 + 1.2%

23 + 1 .07.

33 + 3.3%

50 + 0.7%

73 + 0.7%

133 + 0.6%

153 + 0.7%

233 + 0.7%

300 + 0.3%

530 + 1 .0%

730 + 1 .2%

1303 + 1 .5%

1533 + 2.1%

2:333 + 2.6%

3333 + 3.3%

5300 + 6.3%

7333 + 8.2%

133 33 +11.6%

15030 +17.3%

23333 +23.2%

30303 +35.4%

53330 +63.3%

70330 +103.2%

Table A-XXI

TK=

TCr

333 +

4 +

F(DB)

3.15 +.332

0.22 +.033

0.29 +.333

3.43 +.033

3.69 +.334

0.94 +.305

1.29 +.007

1 .81 +.013

2.23 +.012

3.38 +.017

4.35 +.027

5.33 +.33 7

6.48 +.052

7.90 +.076

3.97 +.100

13.55 +. 149

12.61 +.246

14.03 +.344

DY=.01 DB

YOB)

13.45

12.20

11 .25

9.37

8.12

6.99

5.35

4.66

3. 39

2.95

2.01

1.52

1.12

3.78

0.60

3.41

3.25

0.18

0.13 K (0.33%)

3.13 X (2.53%)

DG=.10%

ERROR CONTRIBUTIONS TO TE

ETH ETC EY EG

3.05% 1.05% 3.34% 3.15%

3.34% 3.71% 0.31% 3.13%

0.34% 3.54% 0.33% 0.13%

0.34% 0.37% 3.29% 3.13%

0.04% 3.?4% 3.29% 0.13%

0.34% 3.13% 3.30% 3.13%

0.04% 0.14% 3.32% 0.14%

0.03% 0.10% 3.36% 0.16%

0.03% 3.33% 3.43% 3.17%

0.33% 3.37% 3.47% 3.21%

0.33% 0.35% 3 . 53 % 0.27%

0.03% 0.35% 3.78% 0.34%

0.03% 0.34% 1.32% 0.44%

0.03% 3.34% 1.43% 3.61%

3.03% 3.34% 1.79% 3.78%

0.03% 3.34% 2.57% 1.12%

. 33 % 0.34% 4.13% 1 .79%

3 . 33 % 0.04% 5.73% 2.47%

15.53 +.493 3.13 0.33% 3.03% 3.07% 3.49%

17.22 +.733 3.23 0.03% 3.33% 12.33% 5.18%

13.45 +.991 0.06 3.33% 3.33% 16.23% 6.39%

23.19 +1 .521 3.04

22.39 +2.753 3.33

23.34 +4.466 3.02

0.33% 3.03% 24.96% 10.34%

3.33% 3.03% 4 6.21% 17.49%

0.33% 3.33% 73.32% 25.16%

48
Appendix B. Mismatch

In this appendix, different mismatch errors are expressed

in a set of figures and tables. The derivations of the equations used in the computer program are presented last. The

computer program actually used is contained in Appendix D.

B.l Mismatch Uncertainty

The following figures, referred to as the "Mismatch

Uncertainty" figures, are based on Eq. (4) in the text. This

set can be used to estimate the mismatch ambiguity discussed

in Section 3 or they can be used to estimate the error caused

by using noise standards that have a different reflection

coefficient than the "antenna". The maximum uncertainty in

* T versus r - r is graphed for various values of (3, e ' ant amp ' & r

b, and |r ., - r I. For estimating mismatch ambiguity ' ' std ant ' &

|r , - r is the magnitude of the uncertainty of r

1 std ant

to ant

An upper bound for the mismatch ambiguity is obtained using

the maximum value for |r - T consistent with uncer- 1 ant amp

tainty of r 1 ant

The values of 3 and b for the various mismatch uncertainty

figures is listed in Table BI

.

4 9
Table B-I. Key to mismatch uncertainty figures

F igure B ti Bl

B2 .1 . 2 B3 .1 1 B4 o 2 B5 .2 1 B6 .3 . 2 B7 .3 1

As an example of how to use these figures, consider an amplifier where 8 = 0.2 and b = 0.2. From Table B-I we see

that figure B4 is appropriate. For this amplifier if

Ir^J 1 < 0.01, |r I < 0.05, and |r J < 0.1, then std 1 — ' ' amp 1 — ' ' ant 1 — '

I r ,.

1 - r I < 0.15 and I T^ , - V J < 0.11. Thus using

ant amp 1 — ' std ant 1 — 6

figure B4 the maximum mismatch error caused by using a standard with a reflection coefficient different from the "antenna"

is about ± 7%. For this same example, to estimate the mismatch ambiguity (because r can have any amplitude a phase

* restricted onlv by r < 0.1) we use r - T < 0.15 '

J

' ant ' — ' ant u.mp ' —

and |r ^.j - r < 0.1. The mismatch ambiguity is about 'std ant 1 — & ' ±6%.

B.2. Mismatch Error

Mismatch error is the error in measuring& either T or

e

Fjti caused when the reflection coefficients of the "antenna",

the hot standard, and the cold standard differ from each

other. Part of the mismatch error, namely the error caused

because the "antenna" reflection coefficient differs from

50
the reflection coefficient of the two standards, is already

given in the mismatch uncertainty figures, and this contribution is NOT included in the mismatch error tables. In other

words, the tables only include the error because the impedance

of the hot and cold standards differ from each other. The

magnitude & of the error depends f on the magnitudes b of T, , hot'

T .,, 3i and bT /T . A key to the 19 mismatch error tables cold a e

J

is provided in Table B-II. The errors listed in the tables

are decibel errors to F, R. The meaning of the symbols used in

the mismatch error tables compared with the symbols used in

Eqs. (1), (3), and (4) are as follows:

Symbo Is Meaning

T(HOT) Thot

T(COLD) Tcold1 A BETA 3 BTA/TE bT /T

a e

a b

ERR 1 r hot r cold'

ANT 1 r' 1 ant '

' ant

F(DB) F dB

*

r amp

As an example of how to use these tables, consider an am- plifier where 3=0.2 and b = 0.2, and where a hot standard of

10,000 K and a cold standard of 300 K will be used for the meas- urement of noise figure. Then from Table B-II (page 62) we note

that Table B-VI is appropriate for use with these given conditions

If the amplifier has a noise figure near 6 dB, and I\ , T -.,

51
and r _ < 0.05 and r . < 0.15 so that "ERR** < I I\ -r , ,1 amp — ant — — > hot cold 1 * r = 0.2, the maximum mismatch error is

amp

0.1 , "ANT" = r <_ 1 ant

± 0.281 dB.

B . 3 . Derivations

The output power, P , out o£ a linear amplifier when a standard with temperature r T , and reflection coefficient r -, std std

is attached to it is [3]

P , = kGBM «,(T ^,+T ) rB.ll out std *• std e

J lu,iJ

where k is Boltzmann's constant, B is the appropriate bandwidth

G is the appropriate gain, and M , is the mismatch factor

( M

std

= X T ^ I 2

) where r ^j defined in eq. (5) except std 1 '* std n*.; r with the subscript "ant" replaced by "std"). Using eq. (4)

except with the subscript "ant" replaced by "std":

T (std) = e ^

'

T (1+bl r ^ , - B

I

2

) a

y

' std ' '

r 1 21 std 1 (B.2)

then

(B.3)

P /(GBk) = T „, + T - [T ^J 2 (T «,-bT ) out c -* std a ' std 1 v std a

J

+ bT

a [|3| 2 -2Re(3*r^

td )]

where Re ( ) implies the real part of the vector in parenthesis,

and the asterisk indicates the complex conjugate. If we let

std ant std

then using eq. (B.3) and eq. (B.4)

P „/CGBk) = M „(T ,,+T -AT _) out v ant ^ std e std^

(B.4)

(B.5)
where M„. and T (ant) are the mismatch factor and effective an l e input noise temperature when the "antenna" is connected to the

ampl if ier

,

M ^AT , = T ,(l-bT /T „,) [|e\ J

2 +2Re(r' V* )] ant std std^ a std^ L

' std 1 v ant std^ J

+ 2bT

a [Re(,3V

td )]. (B .6)

If G and B do not change6 as V changes, then

ant 6 '

T, + + T - AT, „ v hot e hot Y = , CB.7)

T . , + T - AT _ , cold e cold

where the subscript "std" is changed to "hot" or "cold" as appropriate. Solving for T ,

(T, , -AT, «J - Y(T . ,-AT , ,) „ ^ hot aot J y cold cold^

T

e

= • (B.8) e Y - 1 For the computer calculations, eq. (B.6) is modified

so that

ReClWe std) " ± r ant e std'

and

Re

^

std) - ± &e'std .

For mismatch error we assume |e, | = I e ro i ^ I anc*

'*" e

~"~ n ^

(B.8) is computed for the eight sign combinations of the parat i i meters T , ± c, ^ , ± c , -, , and 6, and the greatest difference

ant ' hot cold

"

! 1 from the value of T with e, «_ = e , , = is used as the mise not cold

match error.

For mismatch uncertainty e, = e , = e

1 so that using — hot cold & eqs. (B.2) and (B.4)

5 3
T

e (std)

T [1+blr .-3| 2 +b|

e

1

I 2 +2bRe(e'*(r «.-S))l ,,> a

L

' ant ' '

' v

^ ant ' J J —t— H5

- |

e

1

| 2

- 2Re(r_c') ant ant

Using eq. (B.2) and eq. (B.9),

CB.9)

i * i T (ant)-T (std) L+Lblr + -3

I

2 +b I e

'

I

2 +2bRe [e (r -3)

e

v ^ e

v J ' ant ' '

' L *- ant

where

! I 2 L

=

d-|r ant | 2 )(i-L)

2ReCr;;

t e')

(B.10)

r J

2 ant '

Equation (B.10) is the mismatch uncertainty and unlike the

mismatch error calculated from eq. (B.8), the mismatch uncer

tainty1 is independent v of the values for TV . and T , -, . For hot cold

the computer calculation, eq. (B.10) is modified so that

ant" Re(T .e") + T _e a n t" an t and

Re[e'*(r' L v -6)] -»• e» (T -3) ant J } v ant J

Then eq. (B.10) is computed for the four different sign combina

tions of the parameters rr ^ , ± c', and ± 3', and the greatest

ant

value is used as the mismatch uncertainty.

54
6 = b =

|r ant amp

Fig. Bl Mismatch Uncertainty

Explanation on p. 49

55
3 = 0.1

b = .2

i—

UJO

o

I—

<:

00

x

<

ant amp

Fig. B2. Mismatch Uncertainty.

Explanation on p . 49

56
3 = 0.1

b = 1

10

cc

UJ

=>

jz

t—

g - */////// f / / / / / // // //

?c7- / / / / / / /

1 Y ****'// / / / / 8 ~

1 1 1 1 Y

<=>' r-S- °> / / // // //

1 1 1 / t F> JSP / / / /

1 1 1 1 Y ^ / / / /

////// * / / /

6 /////// / / /

/////// & / /

5 ////// &' / /

1 1 / / / / / & /

/// / / / ^ /

4 // / / / / y

& ' /

r -

3 7 // / / / &

// / / X S

-

2 /y/^^^^ ^^^^

*^

1

n ^-— l r std '

[ ^

1 1 1 0.2 0.3

I !' r

di!l P Fig. B3 Mismatch Uncertainty.

13 xp Ian at ion on p. 49.

57
3 = 0.2

b = 0.2

0.1 0.2 0.3

ant

r amp

l ; ig. B4 . Mismatch Uncertainty

lixpl an at i on on p . 4 9
3 = 0.2

b = 1

o /

i—q;

LlJ <_>

o

X

|r ant amp

Fig. B5 Mismatch Uncertainty

Explanation on p. 49

59
0.3

0.2

0.1 0.2 0.3

r ant

- r * amp

Fig. B6. Mismatch Uncertainty

Explanation on p . 49

60
g = 0.3

b = 1

10u

9 -III 1

' ~r ' / / / / // //

/

1 /? d° // // // //

-

8 / / ^ / / / /

I / / & / / / / / i? / / /

/ / / ** / / /

/ -

"7/ L / / & / / /

/ / / </ / /

-

6 I / / f / /

/ / / & /

/ / / /' /

-

5 / —

4 / / / & /

/ / / jP' /

3 -

2 ^^^ ^^o-^

-

1

n 1 1 1 0.2 0.3

ant amp

I

; ij4. B7. Mismatch Uncertainty,

lixplanat i on on p . 49 . 61
Table B-II. Key to mismatch error tables

Table Thot T cold 3 bT /T

a e III 10,000 300

IV 10,000 300 .1 .2

V 10,000 300 .1 1 VI 10,000 300 . 2 . 2 VII 10,000 300 .2 1 VIII 10,000 300 . 3 .2

IX 10,000 300 .3 1 X 373 80

XI 373 80 .1 .2

XII 373 80 .1 1 XIII 373 80 .2 .2

XIV 373 80 .2 1 XV 373 80 .3 .2

XVI 373 80 .3 1 XVII 1,270 300

XVIII 1,270 300 .2 1 XIX 1,270 80

XX 1,270 80 .2 1 XXI 300 4 XXII 300 4 .2 1

62
Table B-III

MAXIMUM MISMATCH EP.ROF. CDB)

TCHOT)= 100312 K T(CCLD)= 333 X 3ETA= 3 STA/TE= 3

--GAMMA 1 --

ERP. AMT -FC

4

,335

.32

.05

, 332

,3 34

,332

,3 34

,331 333,301 ,333

» 33 1 .333

3.331 .333

. 1

. 2.35

.338

.3 17 .033

,338

,316

,33

3 37 .314 ,326

.3 36 .212 .323.33 5 .311

.321

.335

.31 1 ,32

.31 3.32 .333

, £33

,333

.333

3 33 J03 536 .333

. 336

3.302

.335

. 1 .2.35

.3 17 ,335

,066

315

3 32 36

,313

, 327

,052

.312

.324

.04 6

.311 ,323

.34 3

.311 .322

.341

.32 3.32

.05

3.007

.017

,331

. 307

,016

.031 ,336

,314

.331 .336

.313

,031 .336 ,312.332

.306

.312

. 1 .2.35.334

.37

. 1 33

,331 ,364

121

, 327

,355

, 134

.025

.35 ,094

.023

.346

.387

.322

.044

.333

.02

.35

,0'3 2

,3 19 .044

.034 ,319

,342

.337

,3 19 .339

,333

,02 .337

.339

.02 ,336

,3 1 ,32 ,335

. 1

• £ .35

336

, 175 334

.33 .161 ,336

,372

141 ,265

,367

, 1 29

,24

, 363

,121 ,224

,361

,116

.214

3.32

.35

3'39

, 34 2 ,392

.316

.346 .392

, 326

, 352

.091

.333 ,056

,391

.3 37

.353

,39

.339

.059

.09

. 1 .2.35

,177 .355

,673

. 169

.33 1 .62

, 157 297 545. 15 .275 .498

. 145

.261 .463

. 142

.252

.4 5

63
Table B-IV

MAXIMUM MISMATCH ERROR CDB)

TCHOT)= 13302 K T(COLD)= 333 K BETA= .

1

BTA/TE= .2

--GAMMA' -- ERR AMT 1 2 F(DB)-

4 6 8 13

.335 3 .001 .33 1 .001 .301

.32 .332 .332 .332 .332 .002 .332

.05 .034 .034 .334 .333 .003 .333

. 1 .003 .338 . 307 .306 .036 .306

.2 .017 .315 .313 .3 11 .31 .31 1 .35 .333 .029 .024 .321 .019 .02

.31 3 3 .331 .032 .332 .302 .333

.32 . 034 .304 .3 34 .004 . 304 .035

.35 .009 .033 .307 .007 .307 .037

. 1 .317 .315 .313 .312 .311 .312

.2 .034 .331 .026 .0 23 .321 .022

.35 .•065 .053 .348 .042 .037 .34

.02 3 .031 .302 .304 035 .036 .037

.32 .338 .003 .303 309 .009 .01 .05 .017 .017 .015 .3 15 .315 .015

. 1 .334 .33 1 .027 .025 .023 .324

.2 .369 .062 .353 .346 .34 2 .344

.35 .131 .117 .398 .385 .376 .331

.05 .335 .303 .0 14 . JI8 .021 .324

.32 .321 .323 .326 .023 .33 .332

.35 .345 .344 .343 .343 .043 .043

. 1 .087 .38 1 .373 .063 .365 .367

.2 . 174 . 159 . 137 . 1 22 .112 .118

.35 . 3 29 .296 .25 .219 . 197 .239

. 1 3 .315 .326 .343 .355 .364 .072

.02 .047 .355 .367 .075 .381 .337

.05 .096 .093 . 132 . 1 35 .138 .1 1

. 1 . 179 .172 . 162 . 156 . 152 .158

.2 .353 .327 .239 .265 .243 .261 .35 .664 .634 .517 .46 .419 .445

64
Table B-V

MAXIMUM MISMATCH EHROn (DQ)

TC!!OT)= 13300 : T(C0LD)= 330 K 3ETA= .

1

3TA/TE= 1

—GAMMA' —

ERR AMT 1 2 FCD5)-4 6 3 13

.005 3 .001 .032 .333 .334 .335 .036

.32 .332 .303 .304 .005 .335 .337

.35 .005 .305 .005 .037 .303 .308

. 1 .338 .333 .333 .31 .311 .012

.2 .316 .314 .014 .317 .319 .321

.35 .33 .025 .324 .03 .034 .336

.01 .33 2 . 334 .307 .039 .311 .313

.32 .335 .336 .333 .31 .311 .314

.05 .039 .31 .311 .3 14 .015 .316

. 1 .3 17 .316 .316 .32 .323 It Q /, .2 .333 .323 .323 .33 5 .039 .342

.35 .361 .351 .348 .36 .367 .072

.32 .005 .0 33 ,314 .319 .323 .323

.32 .3 1 .313 .317 .321 .323 .329

.05 .319 .02 .323 .329 .332 .334

. 1 .334 .332 .334 .042 .347 .351

.2 .366 .357 .357 .37 .079 .335

.35 . 1 22 . 102 .398 . 122 . 137 . 146

.05 .3 1 4 .326 .343 .357 .069 .082

.32 .329 .037 .35 .362 .369 .384

.35 . 05 1 .355 .366 .032 .092 .093

. 1 .038 .334 .392 .115 . 129 . 133

.2 . 163 . 147 . 15 .136 .209 .224

.35 .339 .26 .253 .315 .354 .379

. 1 .337 .367 .113 .146 . 174 .203

.32 .367 .39 . 126 . 156 . 1 75 .233

.35 .111 . 125 . 157 . 196 .22 .235

. 1 . 136 . 185 .211 .263 .295 .316

.2 .345 .3 11 .327 .437 .457 .469

.35 .6 23 . 533 .536 .667 .75 .302

65
Table B-VI

MAXIMUM MISMATCH EP.F.OP: (D3)

T(H0T)= 13000 K T(C0LD)= 333 K 3ETA= .2 3TA/TE= .2

--GAMMA* -- ERR ANT 1 2 FCD3)-

4 6 8 13

.005 3 3 .031 .331 .332 .03 2 .003

.02 . 032 .002 .002 .033 .303 .003

.35 .334 .004 .304 .004 .304 .335

. 1 .039 .003 .007 .007 .007 .307

.2 .017 .016 .014 .012 .011 .312

.35 .033 .03 .025 .322 .32 .321

.31 .30 1 .332 .3 33 . 04 .304 .005

.02 .304 .334 .335 .005 .005 .337

.35 .009 .039 .039 .339 .009 .339

. 1 .317 .316 .315 .014 .313 .013

. 2 .035 .032 .327 .3 24 .023 -.023 .35 .066 .0 59 .35 .344 .34 .342

.02 .032 .034 .306 .3 08 .31 .312

.32 .333 .309 .311 .312 .013 .315

.05 .018 .318 .018 .318 .319 .02

. 1 .035 .033 .03 .328 .027 .327

.2 . 37 .364 .055 .35 .346 .34 3 .35 . 1 32 .119 . 1 .038 .33 .384

.35 3 .007 .312 .02 .026 .331 .037

.02 .023 .326 .332 .336 .34 .344

.05 .0/17 .34 8 . 24 9 .351 .053 .256

. 1 .339 .335 .379 .376 .375 .376

.2 . 176 . 162 . 143 .131 . 123 . 126

.35 .331 .3 .256 .223 .209 .213

. 1 3 .0 18 .033 .055 . 37 1 .334 .39' .32 .05 1 .362 . 379 .39 1 . 1 32 .112

.05 . 1 . 136 .114 .121 . 123 . 136

. 1 . 133 . 179 . 174 .172 . 1 73 . 175

. p . 357 .334 .332 .23 1 .269 .277

.35 .663 .612 .531 .478 .442 .463

66
Table B-VII

TAX I MUM MISMATCH EP.P.OP. CD3)

T(H0T)= 13300 K T(C0LD)= 330 K 3ETA= .2 3TA/TE= 1

—GAMMA'

.005

.31

.02

.05

AT IT

A.32

.35

. 1 .2.35

.32

.35

. 1 .2.35

.32

.35

. 1 .2.35

.32

.05

. 1

. 2 .35

.02

.35

. 1 .2.35

, 332

.303 .036

.039

,017 .032

.004

.037

,311

,019

.335

,063

,308

,014

.023

. 333

,07

, 1 27

,024

338

36 1

,098

,173

,32

,356

,036

, 13

, 235

,365

,65

.034

,33 5 ,337

, 339

,016

,027

.337

.31 .313

,319

.332

.355

.015

.02 .027

.339

.3 64

. 1 1

.343 .355

,372

, 1 32

. 166

,23

102 126

161

,348

,573

FCD3J4 6

336 . 033 007 .309

3 33 .31

311 .313

316 .32

027 .334

313 .317

314 .017

316 .32

322 . 327

333 .34 1 354 .367

326 .335 029 .036

334 .042

344 .355

363 .334

1 1 . 136

073 .097

38 . 1 092 . 1 14

1 19 . 143

177 .22

233 .352

173 .226

137 .233

21 .261

264 .323

331 .474

596 .74 1

.31

.31 .311

.315

.023

.033

.321 ,021 .323

,03 ,046

,076

.043

.344 .047

.062

.394

. 153

.119

. 12

, 123

, 166

,247

,396

.275 .276

.293

.369

.533

.833

13

.313

.313

.0 13

.016

.325

.34

.326

.326

.327

.032

.05

.031

.353

.354 .356

.366

.101 -164

.145

. 147

.151

. 173

.265

.423

.328

.333

,341

.395

,57391

67
Table B-VIII

MAXIMu.i MISMATCH ERROR CDB)

TCHOT)= 13330 K T(COLD)= 333 X

BETA= .

3

BTA/TE= .2

--GAMMA' -- ERR ANT 1 2 FCD3>-

4 6 3 10

.035 .031 .33 1 .002 .033 .303 .024

.32 .332 .033 .303 .004 .334 .035

.05 .035 .005 .005 .005 .305 .036

. 1 .039 .338 .038 .033 .038 .038

.2 .013 .316 .3 14 .013 .312 .012

.35 .033 .03 .026 .323 .321 .322

.01 .031 .332 .004 .335 .336 .033

.02 .305 .335 .006 .337 .333 .01 .35 .339 .31 .01 .01 .311 .012

. 1 .013 .017 -016 .315 .315 .316

.2 .035 .032 .029 .026 .025 .325

.35 .066 .36 .051 .046 .042 .043

.02 .033 .005 .003 .011 .3 14 .017

.02 .009 .011 .013 .315 .317 .02

.05 .319 .019 .02 .021 .323 .325

. 1 .036 .334 .032 .331 .032 .033

.2 .37 .065 .358 .353 .351 .051 .35 . 133 . 12 . 103 .392 .385 .338

.05 '3 .003 .015 .026 .034 .041 .349

.02 .025 .03 • .033 .044 .05 .357

.05 .049 .351 .055 .059 .063 .069

. 1 . 39 .033 .035 .035 .035 .088

.2 . 173 . 166 . 149 . 139 . 133 .134

.35 .333 .304 .263 .237 .22 .227

. 1 3 .022 .34 • 3 6L .037 . 134 .122

.32 .355 .069 .39 1 . 137 . 122 .137

.35 . 104 .113 . 126 . 137 . 143 . 161

. 1 . 186 . 136 . 187 . 189 . 193 .2

.2 .361 .342 .315 .293 .29 .293

.35 .67 3 .62 .544 .496 .465 .481

68
Table B-IX

MAXIMUM MISMATCH ERROR (DB)

TCKOT)= 13033 X T(COLD)= 333 K 3ETA= .3 BTA/TE= 1

--GAM' ERR1A'--

ANT 1 2 FCDS)-4 6 8 13

.335 3 .033 .33 5 .039 .312 .315 .019

.02 .034 .337 .31 .313 .315 .31 9 .35 .037 .003 .311 .313, .215 .32

. 1 .01 .311 .013 .017 .319 .32

.2 .018 .018 .319 .3 24 .327 .329

.35 .033 .329 .33 .037 .34 2 .345

.01 3 .036 .311 .019 .325 .331 .333

.32 .339 .313 .02 .026 .331 .339

.35 .313 .317 .022 .027 .331 .34

. 1 .021 .323 .3 27 .334 .333 .341

.2 .037 .335 .039 .348 .354 .353

.35 .065 .059 .36 .075 .384 .39

.32 3 .312 .322 .338 .051 .363 .378

.32 .013 .027 .34 1 .052 .364 .379

.35 .327 .334 .045 .355 364 .331

.1 .042 .046 .355 .368 .377 .334

.2 .074 .07 2 .078 .098 . 1 1 .117

.35 .131 .113 . 122 .151 . 1 7 . 1 82

.35 3 .033 .361 . 133 . 137 . 17 .207

.02 .343 . 37 2 . 1 1 . 141 . 17 .21 .05 .37 .39 .121 .147 .171 .2 14

. 1 . 133 . 12 . 145 .131 .23 3 .221

.2 . 188 . 1 34 .234 .254 .236 .335

.35 .331 . 3 .313 .339 .437 .468

. 1 3 .376 . 133 .233 .307 .376 .453

.02 . 135 .161 .247 .3 14 .377 .4 53

.-^5 . 149 . 196 .263 .326 .33 .466

. 1 .225 .256 .317 .394 .44 3 .43 2 .2 .33 5 .334 .436 .542 .63 9 .651 .35 .672 .619 .655 .8 15 .916 .93

69
Table B-X

MAXIMUM MISMATCH ERROR CDE)

T(HOT)= 373 K T(COLD)= 30 K BETA= .2 BTA/TE= .2

—GAMMA' -- ERR ANT 1 2 FCDE)-

4 6 3 10

.005 .001 .002 .005 .01 .013 .03 .02 .001 .002 .035 .01 .018 .032

.05 .002 .0 04 .036 .01 .019 .035

. 1 .004 .035 .3 08 .311 .021 .04

.2 .008 .009 .011 .312 .024 .05

.35 .315 .316 .017 .017 .031 .37

.01 .00 2 .004 .01 .02 .036 .361 .02 .003 .335 .311 .02 .337 .065

.05 .005 .037 .313 .021 .038 .07

. 1 .009 .311 .316 .022 .34 1 .379

.2 .016 .019 .022 .325 .043 . 1 .35 .029 .032 .334 . 3 3b .362 . 14

.02 3 .333 .008 .32 .34 .072 . 123 .02 .006 .011 .023 .04 1 .374 . 13 .05 .31 .015 .026 .342 .073 .141

. 1 .017 .022 .332 .345 .384 . 16

.2 .032 . 337 .345 .05 .098 .231 .35 .059 .065 .369 .07 . 125 .28

.05 .01 .022 .055 .107 . 187 .314

.32 .317 .029 .361 . 139 . 193 .332

.35 .3 27 .04 .37 .112 .23 1 .359

. 1 .045 .353 .03 5 .117 .216 .436

.2 .03 2 .097 .113 .131 .251 .511

.35 .143 . 166 . 178 . 1 32 .32 .739

. 1 3 .325 .354 . 126 .233 .396 .652

.32 .339 .363 . 133 . 237 .437 .63 3 .35 .06 .039 . 1 56 .243 .424 .742

. 1 .09" . 126 . 136 .254 .454 • 3 ° 6 .2 . 17 .233 .252 . 232 .524 1 .346

.35 .303 .343 .374 .335 .665 1 .444

70
Table B-XI

MAXIMUM .MISMATCH EP.P.OP. CD3)

T(HOT)= 373 X T(C0LD)= 83 X BETA= .

1

3TA/TE= .2

--GAr-!MA'-- FCDB)-

ER?. AMT 1 2 4 6 8 10

.035 3 .301 .302 .005 .039 ."3 15 .02 . 031 .332 .333 .035 .039 .017

.35 .032 .003 .004 .305 .01 .32

. 1 .334 .304 .33 5 .006 .312 .324

.2 .008 .338 .339 .339 .015 .334

.35 . 31 A .315 .314 .316 .32 1 .352

.01 3 .301 .032 .005 .31 .313 .331 .32 .332 .333 .036 .31 .319 .334

.05 .334 .00 5 .333 .311 .321 .34

. 1 .003 .039 .011 .012 .024 .049

.2 .015 .317 .317 .3 18 .33 .369

.35 .328 .33 .329 .333 .342 . 105

.32 .032 .034 .011 . 021 .337 .062

.32 .335 .007 .313 .322 .339 .069

.35 .309 .01 1 .317 .023 .042 .38

. 1 .016 .319 .023 .025 .04 3 .399

.2 .331 .034 .035 .037 . 361 . 138 .35 .357 .061 .353 .067 .385 .211

.35 .336 .013 .332 .058 .39 9 . 163 .02 .313 .321 .337 .06 . 134 .181

.05 .324 .331 .046 .063 .113 .238

. 1 .341 .349 .061 .363 . 127 .253

12 .37 3 .038 .093 .093 . 159 .353 .35 . 144 . 156 .151 . 174 .22 .536

. 1 .313 .037 .379 .136 .22 .35 .02 .332 .351 .39 . 14 .231 .385

.35 .053 .372 . 108 . 146 .243 .439

. 1 .338 . 133 . 133 . 156 .276 .531

.2 . 162 . 185 .20 2 .217 .341 .73 .35 .295 .324 .32 .371 .465 1 .399

71
Table B-XII

MAXIMUM MISMATCH ERROR (DB)

TCHOT)= 37 3 K TCCOLD)= 80 K BETA= .

1

BTA/TE= 1

—GAMMA'

FCDB)-

ERR ANT 1 2 4 6 8 10

.035 .032 .034 .012 .3 24 .344 .376

.02 .002 .034 .313 .323 .35 2 .09

.05 .003 .034 .315 .034 .363 .111

. 1 .034 .305 .319 .043 .033 . 147

.2 .336 .338 .026 .0 64 . 125 .223

.35 .01 .015 .04 .131 .202 .3 63

.01 .334 .0 39 .024 .049 .389 . 152

.02 .035 .039 .027 .35 6 . 134 . 18 .35 .306 .039 .031 .363 . 127 .222

. 1 .338 .31 • 3 3 3 .337 . 166 .294

.2 .312 .317 .353 . 1 23 .251 .447

.35 .02 .33 .33 .233 .434 .726

.02 .003 .319 .35 .099 . 179 .306

.02 .01 .319 .055 .114 .21 .361

.05 .312 .319 .363 . 137 .256 .446

. 1 .016 .021 .377 . 175 .334 .589

.2 .325 .035 . 107 .257 .503 .396

.35 .041 .06 . 162 .437 .839 1 .455

.05 .023 .J 51 . 132 .258 .459 .776

.02 .327 .352 . 145 .295 .535 .915

.35 .033 .352 . 165 .351 .65 1 . 126

. 1 .043 .357 .2 .447 .847 1 .434

.2 .365 .392 .27 5 .653 1 .268 2.253

.35 .134 . 155 .414 1 .329 2,3^6 3.65

. 1 3 .355 .119 .29 .549 .954 1 .592

.02 . .36 3 .119 .316 .623 1 . 136 1 .37

.05 .374 . 12 .357 .735 1 .337 2.291

. 1 .095 . 13 .426 .927 1 .73 3.308

. P . 139 .2 . 577 1 .34 2.574 4.547

.35 .219 .329 .857 2.396 4.113 7.345

72
Table B-XIII

MAXIMUM MISMATCH ERROR <DE>

TCHOT)= 373 X TCCOLD)= 30 X 3ETA= .2 BTA/TE= .2

--GA-:'MA'-- FCDE)- ERR AMT 1 2 4 6 3 10

.005 3 .301 .302 .335 .31 .018 .03

.32 .031 .302 .335 .01 .018 .332

.05 .032 .334 .036 .31 .019 .035

. 1 .304 .035 .333 .311 .021 .34

.2 .338 .309 .011 .312 .024 .35

.35 .015 .316 .017 .317 .031 .37

.31 3 .33 2 .3 34 .3 1 .32 .336 .361 .32 .033 .335 .311 .02 .337 .065

.35 .305 .337 .3 13 .021 .333 .37

. 1 .039 .311 .316 .022 .34 1 .379

.2 .316 .319 .022 .325 .043 . 1 .35 .029 .332 .334 .335 .362 . 14

.02 3 .333 .308 .32 .34 .372 .123

.32 .006 .011 .023 .041 .374 . 13 .05 .31 .015 .026 .342 .078 .141

. 1 .017 .3 22 .03 2 .045 .384 . 16 .2 .032 .337 .045 .05 .098 .231 .35 .059 .3 65 . "",9 .37 . 125 .23

.05 3 .31 .022 .355 . 137 . 137 .314

.32 .317 .329 .361 . 139 . 193 .332

.35 .327 .04 .37 .112 .23 1 .359

. 1 .045 .353 .035 .117 .216 .436

;2 .082 .397 .113 .131 .251 .511

.35 . IAS . 166 . 173 . 132 32 .739

. 1 3 .325 . 354 . 126 .233 .396 .652

.32 .339 .363 . 1 33 . 237 .437 .633

.35 .36 .339 . 156 . 243 .424 .742

. 1 .395 . 126 . 136 .254 .454 .336

.2 . 17 .233 .252 .23 2 .524 1 .346

.35 .333 .343 .374 .335 .665 1 .444

73
Table B-XIV

MAXIMUM MISMATCH ERROR CDB)

T(HOT)= 373 K T(COLD>= 33 K BETA= .2 BTA/TE= 1

--GAMMA" -- ERR AMT 1 2 F<DB>-

4 6 8 13

.035 3 .334 .009 .324 .249 .388 '. 152 .32 .334 .039 .025 .05 2 .396 .165

.35 .035 .339 .327 .053 . 108 .187

. 1 . 336 .30 9 .331 .068 . 128 .223

.2 .038 .31 .039 .039 . 171 .332

.35 .012 .317 .054 . 129 .252 .449

.01 3 .003 . 01 8 .348 .3 98 . 177 .303

.02 .333 .313 .351 .135 . 192 .331

.35 .31 .313 .355 .116 .215 .374

. 1 .012 .013 .362 . 136 .255 .447

.2 .015 .32 .373 . 173 .342 . 63 j .35 .024 .333 . 137 .258 .5 34 .899

.32 3 .315 .336 .097 . 196 .355 .639

.02 .317 .336 . 102 .211 .336 .664

.35 .319 .036 .111 .234 .432 .749

. 1 .324 .337 . 125 .273 .512 .395

.2 .032 .341 . 156 .358 .686 1.212

.35 .349 .067 .216 .518 1.3 1 1 .3

.35 3 .34 1 .295 .25 .50 1 .9 1 .533

.32 .345 .395 .264 .533 .976 1 .672

.05 .351 .395 .234 .594 1 .092 1 .384

. 1 .362 .396 .32 .692 1 .292 2.24 9 .2 .334 . 109 .398 .935 1 .727 3.341

.35 . 125 . 174 .549 1 .335 2.535 4.512

. 1 . 39 1 .205 . 527 1 .234 1 .836 3.135

.02 .399 .23 6 .554 1 . 103 1 .933 3.334

.05 .111 .237 . 594 1 .221 2.22 3.333

. 1 . 132 .239 .666 1.416 2.62 4.537

. 2 . 177 .234 .824 1 .345 3.492 6. 1 23 .35 .261 .365 1 . 1 £8 2.64 6 5.117 9.369

74
Table B-XV

JAX I

>

MISMATCH i?.?.OS <C3)

tchot>= 373 :< tccold)= 33 :-: 3ETA= .3 3TA/TE= .2

—3A"MA' -- ANT 1 O FCDE)-

4 6 3 13

.335 3 .331 .333 .337 .315 .327 .346

.02 .R32 .333 .333 .315 . 3 27 .347

.35 . 333 .334 .339 .315 .323 . 35

. 1 .335 .336 .31 .316 .33 .355

.2 .33 3 .31 .314 .317 . 333 . J66

.35 .315 .3 17 .32 . 32 . 34 1 .33"'

.31 3 .332 .3 35 .3 15 . 329 .353 .39 1 .32 .334 .3.37 .316 .33 . 354 . 395

.35 .336 .339 .3 17 .33 .356 . 1

. 1 .3-39 .313 .32 1 .332 .35 9 . 1 1 .2 .3 17 .32 . 327 .335 .357 . 1 32

.35 .33 . 334 . 34 . 34 1 .352 . 1 74

.02 3 .335 .311 .33 . 36 • 13d . 1 34 .32 .333 .3 14 .33 2 .36 . 1 1 .191

.35 .312 .313 .336 .362 .113 .2 32

. 1 .319 .326 .342 . 364 .119 .221

.2 .33 A .341 .355 . 37 . 134 . 264

.35 .36 .359 • 36 .333 . 165 .349

.35 3 .3 14 .331 .379 . 155 .275 .466

.32 .321 .338 .335 .157 .23 1 .4 34 .35 .331 .34 9 .394 . 16 .239 .5 11

. 1 .34 9 .367 .139 . 1 66 .335 .559

.2 .33 6 .13 6 . 14 2 .131 .34 3 .663

.35 . 153 . 176 .235 .215 .421 .33 1

. 1 3 .333 .37 1 . 174 .33 .573 .95 5 .^2 .347 .035 . 1 35 .334 . 534 .991 .05 .367 . 1 37 .2 33 . 34 . 531 1 .346

. 1 . 133 .14 3 .2 34 .352 .532 1.142

.2 . 177 .222 .331 .33 3 .73 3 1.361

.35 .311 .363 .423 .451 .866 1..733

75
Table B-XVI

IAXIMUM MISMATCH E?.r.O?- (33)

TCHOT)= 373 '< T(COLD)= 93 K BETA= .3 3T A/TE= 1

--GAM' 1A'-- A? JT 1 2 FCDB)-

4 6 3 1 3

.3-35 3 .336 .213 .33 6 .37 3 . 132 .227

.32 .336 .313 . 337 .377 . 14 . 24 1 .35 .337 .313 .33 9 .332 . 152 .2 63

. 1 .333 .313 . 343 .39 2 . 172 .3 .2 .31 .3 14 .351 .114 .217 .331

.35 .914 .3 13 .367 . 156 .332 .535

.31 3 .311 .326 .37 2 . 146 .265 .455

.32 .312 .326 .374 . 153 .23 .433

.35 .113 .326 . 379 . 165 .334 .525

. 1 .315 .3 27 .336 . 135 .344 .6

.2 .32 .023 . 1 32 .229 .434 .763

.35 .323 .337 . 134 .313 .635 I .37 1

.02 3 .323 .353 . 144 .293 .532 .911 .32 .3 24 .353 . 15 .333 .562 .967

.35 .327 .0 54 . 153 .331 .639 1 .353

. 1 .331 .354 . 173 .37 1 .69 1 .23 1 .2 .34 .356 .236 .459 .37 1 .527

.35 .3 57 .37 5 .27 .623 1.211 2. 145

.35 3 • .36 . 133 .369 .743 1 .34 2.29

.32 .364 . 133 .332 .73 1 .417 2.43

.35 .37 . 139 .4 33 .337 1.534 2. 643

. 1 .33 . 14 .439 .936 1 .737 3.313

.2 . 133 . 145 .522 1 . 153 2. 136 3.33

.35 . 146 .192 .634 1.531 3.34 5.375

. 1 3 . 1 23 .292 .764 1.513 2.7 17 4.613

.32 . 136 .292 .79 1 1 .592 2.37 4. 393

. 35 .143 .293 .332 1 .737 3 . 1 34 5.325

. 1 . 169 .296 .9 35 1 .936 3.51 6.356

.2 .215 .337 1 .371 2.349 4.4 1 7.7

.35 .333 .432 1.393 3.2 i. 122 10 .794

76
Table B-XVII

"lAXrtUM MISMATCH EI'.riOH CDB)

TCHOT)= 1273 K TCCOLD)= 333 K 3£TA= 3 3TA/TE= 3

-GAM'-TA'--

;RR AMI 1 2 FCD3)-

4 6 3 13

005 3 3 3 3 3 3 .32 .332 .332 .332 .332 .332 .332

.35 .335 .335 .335 .334 . 334 .334

. 1 .311 .31 .039 .333 .308 .338

g 2 .323 .321 .319 .3 17 .316 .316

.35 .343 .34 .336 .333 .331 .33

31 3 3 3 3 3 3 .32 .334 .334 .334 .334 . 333 .0 33 .05 .31 1 .31 .339 .33 9 .333 .338

. 1 .322 • 3 2 .3 13 .317 .316 .316

.2 .34 5 .342 .337 .33 5 .333 .332

.35 .336 .2? .37 1 .366 .362 .36

32 3 3 .33 1 .33 1 .331 .20 1 .332

.32 .339 .339 .038 .333 .333 .338

.35 .322 .321 .319 .313 .317 .317

. 1 .344 . 34 f .337 .335 .033 .032

.2 .39 .384 .375 .37 .366 .3 64 .35 . 1^3 .161 . 143 . 133 . 126 .121

35 3 .332 .304 .337 .303 .339 .31 .32 . 324 .324 .324 . 325 . 325 .325

.35 .356 .354 .351 .34 9 . 34 3 .34 7

. 1 .111 .13 5 .397 . 39 1 .33 3 . 3 3 6 .2 .227 .213 .193 . 13 . 172 . 166

.35 .433 .4 34 .363 .337 .321 .31

1 3 . •? 3 9 .316 .3 25 .3 33 .037 .23 9 .32 .352 .356 .062 . 365 .063 .269

.35 .117 .116 .116 .115 .115 .114

. 1 .227 .219 .236 .199 .194 .191

.2 .4 59 .43-1 .399 .376 .362 .353

.35 .87 1 .3 17 .741 .693 .662 .643

11
Table B-XVIII

MAXIM'iv MISMATCH ERF.O . CDB)

T(HOT)= 127 3 :< T<COLD)= 333 K

-3ETA= ' .2 3TA/TE= I

iA'TMA

ETIR

.335

.31

.92

.35

V* --

A'.'T 1 2 FCD3)-

4 6 8 1 3

r> .333 .336 .313 .322 .335 .355

• 3 2 .335 .333 .314 .3 23 .337 .359

.35 .333 .31 .3 15 .3 24 .34 .365

. 1 .313 .313 .316 .2125 .345 .37 6 .2 .323 .32 1 .319 .329 .357 .399

.35 .34 .335 .327 .037 .379 . 142

3 . 337 .313 . 327 .34 5 .37 1 . 1 1 .'3 2 .31 .3 16 .023 .046 .375 .118

.35 .316 .32 .329 .343 .081 . 13

. 1 . 325 .327 .332 .351 . 391 .151

.2 .345 . 342 .333 . 359 .114 . 198 .35 .331 .069 . 354 . 37 4 . 15S .234

3 .313 .3 27 .355 .091 . 143 .222

.32 .321 .332 .357 .393 .151 .233

.35 .33 2 .34 .36 .397 . 163 .262

. 1 .351 .355 .365 . 133 . 134 .334

.2 .391 .335 .37 6 .119 .23 .397

.35 . 162 . 14 . 1 1 . 149 .313 .57

3 .336 .37 1 . 145 .237 .369 .567

.32 .35 5 .385 . 15 .243 .333 . 637

.35 .332 . 1 36 . 1 57 .252 .413 . 663

. 1 . 13 .14 1 . 17 .269 .471 .773

.2 .229 .217 .19 9 .337 .537 1 .3 34 .35 . 433 .35 5 .23 3 . 384 .3 39 1 .433

3 .33 1 . 153 .316 .53 7 .774 1.173

.32 .118 . 1 86 . 325 .519 .813 1 .253

.35 . 174 .227 .34 .537 .873 1 .375

. 1 .263 .29 3 .3 66 .57 . 9 7 3 1.586

n .4 63 .45 1 .426 . 647 1.212 2. 349

. 35 .827 .728 . 596 .8 55 1 .659 2. 92

78
Table B-XIX

[AMIMUM MISMATCH EP.30R CD3)

TCH0T)= 1273 X TCCOLD)= 33 X

3ETA= 3 BTA/TE=

--GAMMA' --

TP.?- AMT 1 2 F(DE)4 6 3 13

.335 3 3 3 3 3 3 3 .32 .03 1 • 3 3 1 .03 1 .031 .33 1 .03 1

. 3 5 .332 .332 .032 .305 . 332 .33 2

. 1 .333 .334 . 3 34 .334 .335 . .335

.2 .336 .337 . 3 C 3 .3 39 . 31 .3 1 .35 .312 .314 .316 .317 .313 .319

.31 3 •3 3 3 3 3 .32 .3'3 1 .332 .332 .335 .332 .332

.35 .033 .334 .334 .305 .3 35 .035

. 1 .33 6 .3 37 .3 33 . 2H .31 .01 .2 .313 .314 .317 .3 13 .319 .32 .35 • 3 24 .323 .032 .335 .337 .333

.32 3 .33 1 .331 .33 1 .331 .3 32 .32 .(533 .333 .334 .33 5 .33 5 .335

.35 .336 .033 .339 .31 .311 .311

. 1 .313 .3 15 .317 .3 19 .32 .021

.2 .326 .329 .334 . 337 .339 .34 1 .35 . 34 9 .35 5 .365 . 371 .374 .077

.35 3 .332 .334 .037 .3 33 . 039 .31 .32 • 33 3 .311 .315 .317 .313 .319

.35 .317 .021 .3 27 . 33 .332 .333

. 1 .333 .339 .347 .352 .355 .357

.2 .366 .276 .39 .399 . 134 . 136

.35 . 124 .14 1 . 1 66 .135 . 192 . 193

. 1 3 .3 39 .316 .326 .333 .33 7 .339

.32 .351 .33 .042 .35 .35 5 .058

. 35 .33 9 .35 .366 .37 6 . .3 3 2 . 33 6

. 1 .37 1 .736 .13 7 . 12 .129 . 134

. 2 . 136 . 1 6 . 193 .214 .223 .236

.35 .253 .292 .343 .333 .435 .419

79
Table B-XX

MAXIMUM MISMATCH EP.SOP. CD3)

TCK0T)= 1273 :• TCCCLD)= 33 K

3ETA= .2 3TA/TE = 1

--GAMMA'-

AM! 1 2 FCD3)-

4 6 3 13

.305 ') .332 .035 .01 .3 17 .327 .34 3 .32 .333 .335 .3 1 .313 .329 .347

.35 .333 .0 35 .31 .319 .332 .352

. 1 .334 .337 .311 .321 .337 .361

• C .306 .31 .3 15 .3 26 .347 .33 .35 .31 .317 .327 .336 .067 .117

.31 3 .035 .039 .32 .334 .35 5 .367

.32 .33 5 .31 .32 .336 .359 .394

.35 .337 .311 .221 .333 .364 . 1 34

. 1 .339 .313 .322 . 343 .37 4 . 122

.2 .313 .32 .333 .353 .395 .161 .35 .021 .333 .354 .372 . 135 ,234

.32 .31 .319 .34 1 .37 .112 . 176

' .32 .311 .32 .342 .373 .119 . 189

.35 .314 .322 .343 .373 . 13 .21

. 1 .3 13 .327 .345 . 337 . 149 .246

. 2 .325 .341 .363 . 137 . 192 .324

.35 .342 .3 67 . 1 1 . 145 .27 2 .47

.35 3 .327 .3 54 .111 . 135 .29 .451

. 32 .33 1 .356 .113 . 1 93 .3 29 .485

.35 .337 .359 .116 .235 .336 .537

. 1 .347 .373 .121 .223 .385 .626

m o .363 . 139 . 177 .277 .491 .322

.35 . 137 . 174 .233 .373 .692 1.133

. 1 3 .363 . 123 . 243 .432 .617 .94. .32 .37 1 . 128 .252 .413 .654 1.339

.35 .332 . 135 .253 .444 .71 1.113

. 1 . 132 . 162 .269 .433 .636 1 .292

. o . 145 .234 .331 .533 1 .32 1 .634

.35 .225 .365 .596 .733 1 .426 2.42

80
Table B-XXI

MAXIMUM MISMATCH E?.?.0?. CDE)

TCKOT)= 300 K T(COLD)= 4 K BETA= 3TA/TE=

--GAMMA' -- FCDB)-

EP.R ANT 1 2 4 6 3 10

.005 3 3 3 3 3 3 .02 3 .031 .301 .301 .03 1' .05 .30 1 .031 .331 .032 .23 2 .332

. 1 .331 .002 . 303 .303 .234 .334

.2 .33 2 .0 04 .336 .037 .033 .303

.35 .304 .007 .3 11 .014 .315 .3 16

.01 3 3 .32 .001 .001 .002 .33 2 .33 2 .05 ..33 1 .032 .303 .3 34 .3 24 .334

. 1 .33 2 .304 .336 .037 .338 .339

.2 .304 .007 .312 .314 .316 .31 7 .35 .333 .314 .022 .327 .33 1 .333

.02 3 .30 1 .331 .33 1 .331 .302

.32 .031 .002 .033 .3 34 .334 .005

.05 .332 .004 .037 .003 .339 .31

. 1 .0 04 .0 33 .012 .3 15 .317 .313

.2 .039 .0 15 .0 24 .029 .033 .335

.35 .017 .323 .045 .355 .362 .366

.05 .032 .004 .007 .003 .309 .01 .02 .034 .007 .012 .015 .3 17 .313

.05 .337 .013 .32 .025 .323 .33

. 1 .312 .321 .3 34 .342 .347 .05 1 .2 .323 .34 .364 .07 9 .333 .3 94 .35 .043 . 373 .116 . 144 .161 . 1 72

. 1 .009 .3 16 .326 .03 3 .037 .339

.02 .313 .0 23 .3 37 .04 6 .352 . 355

.05 .3 19 .333 .354 .366 . 374 .03

. 1 .03 .051 .3 32 .101 .113 .121

.2 .052 .0 33 .141 . 174 .195 .203

.35 .091 . 1 56 .24? .306 .343 .366

81
Table B-XXII

MAXIMUM MISMATCH ERROR <CB)

TCHOT)= 333 K TCCOLD)= 4 K 3ETA= .2 3TA/TE= I

--GAM"!A'-- FCD3)- ERR AMT 1 2 4 6 3 13

.335 3 .333 .037 .321 . 345 .334 . 146 .32 .333 .337 .323 .349 .391 . 16 .05 .333 .333 .025 .354 . 103 .131

. 1 .334 .339 .329 .364 . 123 .217

.2 .336 .0 11 .033 .356 . 166 .296

.35 .339 .0 17 . 355 . 127 .247 .44 1

.31 . 336 .3 14 .342 . 09 . 163 .293

.32 .336 .3 15 .345 .093 . 133 .32 .35 .33 6 .316 .35 . 109 .236 .362

. i .337 .313 .353 . 129 .246 .435

.2 .311 .322 .376 . 173 .333 . 592

.35 .319 .033 . 1 1 .254 .494 .383

.32 3 .312 .329 .336 .131 .337 .537

.32 .312 .33 1 .392 . 197 .363 .642

.05 .012 .033 .131 .22 .414 .726

. 1 .315 .337 .117 .26 .494 .871

.2 .323 .345 . 153 .347 • 66 3 1.185

.35 .037 .3 67 .22 .539 .991 1 .768

.05 3 .332 .07 3 .222 .463 .354 1 .473

.32 .333 . 33 1 .237 .531 .93 1.617

.35 .334 .037 .261 .559 1 .346 1 . 323

.- 1 .341 .396 .3 32 .6 59 1 .246 2. 139

.2 .36 .118 .391 .877 1 .631 2. 576

.35 . 397 . 174 .56 1 . 234 2.4 39 4.434

. 1 3 .373 .171 .469 .953 1 .744 2.996

. 32 . 374 .173 .5 1 . 335 1 .3 97 3.273

• ^5 .376 . 1 3 9 . 54 3 1.151 2. 129 3.694

. 1 .39 .239 .63 1.351 2.529 4 . 4 1 S

• 2 . 13 .252 . 3 1 1.739 3.4 5. 992

. 35 . 234 .3 65 1.149 2.635 5.319 3.912

82
Appendix C. Effect of Front End Loss on Noise Figure and

Effective Input Noise Temperature

If the reference plane used to define the input of the

amplifier is changed, then the noise figure, F, R

, and the

effective input noise temperature, T changes if there is any

loss between these reference planes. These changes in amplifier noise are calculated in this appendix. One application

of the results of this appendix is to estimate the uncer- tainty in amplifier noise because of the uncertainties in

the connector loss.

Using figure C-l, the effective input noise temperature

at reference plane r 1 is T , and at reference plane 2 is T . e e The two-port between 1 and 2 has the physical temperature T

and an absorption loss of A so that the thermal radiation from

the two-port expressed relative to port 1 is A T [8]. The

effective input noise temperature at reference plane 2, T ,

is the amplifier noise and the two-port noise expressed relative to port 2, namely

T* = (A T +T )/(l-A) (C.l)

e

v conn e

J K J y J or

T* - T = —-— (T +T ). (C.2)

e e

i a conn e

Expressing the absorption loss in decibels,

A

dB

= -10 log

1() (l-A) , (C3)

83
u

o

o

84
then

A /10

T - T = (10 -1) fT +T ) e e

*• conn e

J (C4)

For small losses (i.e., A-, R << 10)

T = 0.2303 (T +T ) A, D. e

* conn e

J dB CC.5)

To express the difference in noise figure at reference

plane 2 from plane 1, recall that F-, R

= 10 log,. F, where

F = 1 + (T /T ) , and where T

Q

= 290 K. Because

dFi

R

= (10 log,

n e)dF/F, then for small losses

?ah " ¥ ah ~ 4.343 (T -T )/(T +T ) dB dB * e e

J v o eJ

or using eq . (C . 5)

F dB FdB

fj + T conn e T + T *• o e

v dB'

(C.6)

(C7)

If the tivo-port's physical r J temperature, v T , is nearly ' conn J equal n T = 290 K, then

o

F dB F dB A

dB' (C.8)

Thus an amplifier which includes a two-port with 0.1 dB

absorption loss at 290 K (16.8 C or 62.3 F) has a noise figure

0.1 dB greater than the same amplitude excluding this two-port

85
Appendix D. Computer Programs

This appendix contains the computer programs used for the

figures and tables in this paper. The programs are in BASIC

language

.

86
D.l. Figures 1 Through 4 For figures 1-4, the program DW11 in table D-I was used.

A typical output of this program is shown in table D-II.

The basic relationships used for this table are as follows

using eq. (1) of reference 1 and changing the notation to

correspond with the remark statements in Table D-I, and adding

the gain function G [cf step 4400 in Table D-I]

Y = G*(TH+TE)/(TC+TE)

,

(D.l)

or rearranging [cf steps 4300, 4440, 4450, 4480, 4490, 4520,

4530, 4560, 4570]

TE = (G*TH-Y*TC)/(Y-G) . (D.2)

This eq. (D.2) is used to calculate the error contributions

from TH , TC , Y and G. Equation (2) of reference 1 (cf steps

4340 and 4330) is used to convert from F(DB) to TE

,

F(DB) = 10*CLG(l+TE/290)

,

(D.3)

and the derivative of eq. (D.3) is used to convert errors in

F(DB) to errors in TE (cf step 4427 which is also divided by

100 to convert to percent)

DF(DB) = 10[(F-l)/(F*Log(10))] DTE/TE, (D.4)

where the symbol D denotes the derivative, and F is defined via

(cf step 4425)

F(DB) = 10 CLG(F)

.

(D.5)

The error caused by loss in the connector uses the following relationships. First converting loss in decibels to

a ratio (step 4435) uses CONNECTOR LOSS (DB) = -10 CLG(A). (D.6)
The changed noise temperature TH (or similar for TC) caused

by a loss at 300 K is given by [8] (cf step 4640)

TH + DTH = A*TH + (1-A)*300. (D.7)

For mismatch error, eq. (A. 6) is used as explained in

appendix A (of steps 1000 through 1150)

.
Table D-I. Computer program for figures 1-4.

DW11 11:03 CSS FRI. 11/24/72

5 REM MODIFIED STEPS 74,95, 96, 43 15, 4597-4633, 1033+, FROM TNES 5391

10 REM USES NBS 5S53 FOR MISMATCH

20 N=3

30 M = 3 43 C=62

42 V=0

45 REM 1=1 FOR TE,I=2 FOR F(DB),I=3 FOR TE RECYCLE, 1=4 FOR FCDB) RECYCLE

53 PRINT

63 PRINT

65 REM B2=BETA

73 PRINT

71 REM T = TE

72 REM TlrTH

73 REM T2=TC

74 REM T3=BTA

75 REM T4=TA

76 REM

77 REM D1=DTH

73 REM D2 = DTC

79 REM D3 = DY(D3)

B0 REM D4=DG(Z)

81 REM D5=CONNECTOR LOSS (DB)

82 REM D6=CLIPPING (7.) 83 REM

84 REM EUETH

85 REM E2 = ETC

86 REM E3 = EY 87 REM E4=E3

88 REM E5=E(MISMATCH)

89 REM E6=E(C0NMECTOR>

90 REM E7=E(CLIPPING)

91 REM

92 REM F=F

93 REM F1 = F(DB)

94 REM

95 REM G1=GAMMA'(ANT)

96 REM G2=GAMMA'(H0T)-GAMMA'CANT)

97 REM G5=GAMMA'(COLD)-GAMMA'(ANT)

93 REM

103 REM Y=Y

102 REM Y2=DY

104 REM

105 REM

106 REM T5=PERTURBED TH

108 REM T6=PERTURBED TC

150 GOSUB 4333

193 READ G1,G2,T3,D3,D4,D5,D6

195 B2=3

200 READ Tl ,D1,T2,D2

205 READ I,K1,H2,H3

245 GOSUB 5033

247 ON I GOTO 253,810,253,813

250 PRINT USING 5550

263 PRINT

89
Table D-I (Continued)

300 FOR T=H1 TO H2 STEP H3

333 G0SU3 4343

343 PRINT USING 5553, T, El , E1+E2, E1 + E2+E3 , E3, E3+E6, ES+E6+E7, Eg, P 353 GOSUB 3000

770 NEXT T 775 IF 1:3 GOTO 205

800 GOSUB 3330

802 GOTO 190

310 PRINT USING 5570

815 PRINT

820 M=M+2

830 FOR F1=H1 TO H2 STEP H3

840 GOSUB 4330

890 PRINT USING 5530, Fl ,R 1 ,R 1+R2,R 1+R2+R3,R8, RS+R6, RS+R6+R7,R9, P*B

895 GOSUB 3000

900 NEXT Fl 931 IF 1=4 GOTO 205

905 GOSUB 3033

910 GOTO 193

1000 K=0

1005 G5=G2

1010 FOR J:l TO 8 1320 B2=-B2

1033 X=K+l

1043 IF K=2 GOTO 1060

1345 K=l 1050 G5:-G5

1060 IF J<>5 GOTO 1083

1070 G2=-G2

1083 T5=T1+FNM(T1,G2,B2)

1090 T6=T2+FNM(T2,G5,B2)

1100 W=ABS(FNT(T5,T6,Y,1)-T)

1110 IF J<>1 GOTO 1130

1120 X=W

1130 IF X>W GOTO 1150

1143 X=W

1153 NEXT J 1160 RETURN

3003 M = fH-l 3310 V=Vfl

3020 IF V<>3 THEN 3370

3030 V=3

3040 PRINT

3350 PRINT

3060 M=fH-2 3070 IF M<C-3 THEN 3123

3330 FOR Q=M TO C+S

3093 PRINT

3100 NEXT Q

3110 M=3

3120 RETURN

4300 DEF FNT(H,C,Y,G)=(G*H-Y*C)/(Y-G>

4310 DEF FNE(W,X,Z)=50*ABS((W-X>/Z)

4315 DEF FNM(X,Y,Z)=UX-T*T3)*(Y*Y+2*G1*Y)+2*T*T3*Z*Y>/C1-G1*G1)

4320 RETURN

90
Table D-I (Continued)

4333 T=290*(10T(F1/10)-1)

4343 Fl=10*CLG(l+T/290)

4430 Y=CT1+T)/CT2+T)

4410 Yl =13*CLG(Y)

4415 Y2 : Y*L0G(13)* D3/10

4425 F=10T(F1/10)

4427 B=3.I*CF-1)/(F* LOGC10))

4430 T4=T-G1t2*CT+T3)

4435 A:13t(-D5/13)

4440 W=FNTCT1+D1,T2,Y, i: 4453 X=FNTCT1-D1,T2,Y, 1 4460 Z=T

4473 E1=FNECW,X,Z)

4475 R1=E1*B

4483 W=FNTCTl,T2+D2,Y,i:

4493 X=FNT(T1,T2-D2,Y, l: 4510 E2=FNECW,X,Z)

4515 R2=E2*B

4520 W=FNT(T1,T2,Y+Y2, I! 4533 X=FNTCT1,T2,Y-Y2,i:

4550 E3 = FNE(W,X,Z)

4555 R3=E3*B

4563 W=FNT(T1,T2,Y, 1+D4/130)

4573 X=FNT(T1,T2,Y,1-D4/103)

4590 E4=FNE(W,X,Z>

4595 R4=E4*B

4633 GOSUB 1303

4613 E5=103*X/T

4635 R5=E5*B

4640 WrFNT(A*Tl+(l-A)*330,A*T2+Cl-A)*333,Y, 1) 4650 X=T

4660 Z=T

4673 ES=2*FNE(W,X,Z)

4675 R6=E6*3

4630 W=((1-D6/100)*T1-Y*T2)/(Y-1)

4713 E7:2*Ft)E(W,X,Z)

4715 R7=E7*B

4717 E9=E1 + E2+E3+E4+E5+E6+E7

4713 R9=E9*3

4720 E8:E1 + E2+E3 + E4 4725 R8=B*S3

4727 PrSQRCEl T2+E2 T2+E3 T2+E4T2+E5t2+E6r2+E7t2 ) 4733 RETURN

5303 PRINT USING 5533, Tl , Dl , 1 03*D1 /Tl

5035 PRINT USING 55 13, T2, D2, 1 00*D2/T2

5010 PRINT

5023 PRINT USING 5523, T3 , G2, Gl ,B2

5053 PRINT USING 5533, D4, D3 , D6, D5 5060 PRINT

5065 MrM+8

5370 RETURN

5333 PRINT USING 5550

5093 PRINT

5100 PRINT

5113 RETURN

5533: TH=##### + fit.* K (#.##%)

5513: 1Cz44444 + 4.4 K (4.447.) 5523: BTA/TEr##.# G( STD- ANT) =#.### GCANT):#.### 3ETA:.##

5533: T>Gz44.47. D\z. 44 DB CLIPPING:*** .47. LOSZz. 444 DB

5553: TEC K) ETH ETC EY EG LOSS CLIP MATCH QUAD-E

5563:#/«l #i? 444.44 444.44 444.44 444.44 444.44 444.44 444.44 444.44

5573: FCDB) ETH ETC EY EG LOSS CLIP MATCH QUAD-E

5580:##.# .4444 .4444 ,44ft .4444 .4444 .4444 .4444 .4444

91
Table D-I (Continued)

8000 DATA .05, .01,3, .01, .1, .01,0

8313 DATA 1 0030, 1 50,303, .5 8015 DATA 2,2, 16.5, .5 8020 DATA .05, .01, .3,. 01,. 1, .01,.

8033 DATA 373, .5,80,1

8040 DATA 3,100,330,20

8053 DATA 1,330,803,50

8063 DATA .05, .31 ,0, .31 , . 1 , .31 ,0

8070 DATA 1270,3,330, .2 8072 DATA 4, .1,1,.

I

8074 DATA 2,1,14, .5 8075 END READY

92
Table D-II. Print out of D-I program

READY

RUN

DW11 13:44 CSS FRI. 11/24/72

TH:10030 + 150.0 K (1 .537.) TC = 300 + 0.5 K (0.1 7Z) BTA/TE: 0.0 G(STD- ANT) =0.0 1 DG= 0.17. DY = .01 DB G(ANT)=0.050

CLIPPING: 0.07. BETA:. 33 LOSS:. 010 DB F(DB) ETH ETC EY EG LOSS CLIP MATCH QUAD-E

2.0

2.5

3.0

.0686

.0685

.0683

.0736

.0729

.0723

.0843

.0836

.0831

.0389

.0883

.0878

.0991 .0985

.09 79

.3991 .0985

.0979

.1359

.1359

.1051

.0709

.0707

.0735

3.5

4.0

4.5

.0682

.0681

.0680

.0713

.0713

.0709

.0826

.0822

.0818

.03 73 .0869

.3866

.0975

.3971 .0967

.0975

.0971 .0967

.1044

. 1333 .1033

.3704

.0702

.3701

5.0

5.5

6.0

.0679

.3678

.0677

.0705

.0701 .0698

.0816

.0813

.081 1

.0864

.0362

.0361

.0965

.0963

.0961

.0965

.09 63 .3961

.1029

.1025

.1022

.0700

.3700

.0699

6.5 7.0

7.5

.0677

.0676

.0676

.0696

.0693

.0691

.0810

.0839

.0360

.0360

.03 60

.0963

.0963

.3960

.39 63 .3963

.39 60

.1323

.1319

.1019

.0698

• 36S3

.0698

8.0

8.5

9.0

.0675

.0675

.0675

.0689

.0688

.0686

.0809

.0813

.0311

.0361

.08 62 .0865

.0961 .0963

.0965

.0961

.09 63 .0965

.1019

.1019

.1321

.3698

.0698

.0698

9.5

10.0

10.5

.0674

.0674

.0674

.0685

.0634

.0633

.0312

.0814

.0817

.0367

.0871 .0375

.0968

.3971 .0975

.0968

.3971

.09 75

.1023

.1326

.1329

.0693

.3698

.9699

11 .0 11.5

12.0

.0673

.0673

.0673

.0682

.0681 .0680

.0820

.0823

.0828

.0880

.0335

.0892

.0983

.0986

.0992

.0933

.0986

.0992

.1334

.1039

.1345

.0599

.3700

.0701

12.5 13.0

13.5

.0673

.0673

.0673

.0679

.0679

.0678

.0833

.0839

.0846

.0900

.0908

.0918

. !000

. 1008 .1013

.1003

.1003

.1313

.1053

.1361

.1371

.0703

.0734

.0706

14.0

14.5 15.0

.0673

.0672

.0672

.0678

.0677

.0677

.3353

.0362

.0872

.0929

.0942

.0956

.1329

.1042

.1056

.1329

.1042

.1056

. 1082 .1394 .1133

.0708

.071 1 .0714

15.5 .0672 .0677 .3883 .39 73 .1073 .1373 .1124 .0718

93
Table D-II (Continued]

TH= 373 + 0.5 K (0.13%)

TC= 80 + 1.0 K (1 .257.)

BTA/TE: 0.,3 G(STD-ANT): 0.010 G(AHT) =0.050 BETA:. 00

DG : 0.1% DY :.0! DB CLIPPING: 0. 0% LOSS:. 010 DB

TECK) ETH ETC EY EG LOSS CLIP MATCH QUAD-E

100 0.31 1.92 2.59 2.88 3.80 3.80 4.17 2.05

120 0.28 1 .69 2.33 2.61 3.42 3.42 3.75 1.82

140 0.27 1.52 2.15 2.43 3.15 3.15 3.45 1.65

160 0.26 1,39 2.02 2.29 2.96 2.96 3.25 1 .53

180 0.25 1.30 1 .92 2.20 2.81 2.81 3.09 1 .44

200 0.24 1.22 1.85 2.12 2.70 2.70 2.96 1 .37

220 0.23 1.15 1.79 2.06 2.61 2.61 2.87 1.32

240 0.23 1 .10 1.74 2.02 2.54 2.54 2.79 1.28

260 0.22 1.05 1 .70 1.99 2.48 2.48 2.73 1.24

280 0.22 1 .02 1.68 1 .96 2.44 2.44 2.68 1 .22

300 0.22 0.98 !.65 1.94 2.40 2.40 2.64 1.20

TH: 373 + 0.5 K (0.137.)

TC: 80 + 1.0 K (1 .25%)

BTA/TE: 0,,0 G(STD-ANT): -.010 G(ANT) :0.050 BETA:. 00

DG : 0.1% DY :.01 DB CLIPPING: 0. 0% LOSS: .010 DB

!(K) ETH ETC EY EG LOSS ' ' CLIP MATCH QUAD-E

300 0.22 0.98 1 .65 1 .94 2.40 2.40 2.64 1.20

350 0.21 0.91 1.61 1.92 2.34 2.34 2.57 1.16

400 0.20 0.86 1.59 1.91 2.31 2.31 2.53 1.15

450 0.20 0.83 1.59 1 .92 2.30 2.30 2.51 1.14

500 0.20 0.79 1.59 1 .94 2.30 2.30 2.51 1.15

550 0.20 0.77 1.60 1.96 2.32 2.32 2.52 1 .16

600 0.19 0.75 1 .61 1.99 2.33 2.33 2.53 1.18

650 0.19 0.73 1.63 2.02 2.36 2.36 2.56 1.20

700 0.19 0.71 1 .65 2.06 2.39 2.39 2.58 1.23

750 0.19 0.70 1.68 2.10 2.42 2.42 2.62 1.25

800 0.19 0.69 1 .70 2.14 2.46 2.46 2.65 1.28

94
D.2. Figures Bl through B7

For figures Bl through B7 , the program DW21.3 listed in

table D-III was used. A minicomputer was used for this calculation and call 11 in step 1052 reads a switch position on the

front panel for a diagnostic routine. A typical output from

this program is listed in table D-IV. For the calculation,

eq. (B.9) is used as explained in appendix B.

95
Table D-III. Computer program for figures B1-B7.

1 p^TMT *"Drf21.3 (23 NOV 72)"

o ^w MI "WATCH UNCERTAINTY; SWITCH .3 UP FOR DIAGNOSTICS

13 REM 'FROM NBS PAGE 5927, PROGRAM MODIFICATION FO 533S

12 REM B1=3ETA

14 REM GUGAMMA' (ANT)-GAMMA' (STD)

16 REM G2 = GAMMA* (AWT)

1^ REM 3=REVERSE RADIATION PARAMETER

<J3 DEF FNA(C) = INT ( 1 2l(233*C+.5>/l 33

133 DIM D[25]

11'? FOR 1=1 TO 23

120 READ D[ I]

133 NEXT I 143 READ 31

20:? PRINT

213 PRINT

223 PRINT

233 PRINT TAB (13);

?43 PRINT "MAXIMUM MISMATCH UNCERTAINTY a OF TF) M 253 PRINT

253 PRINT

273 PRINT ,,"FOR BETA=";B1

233 PRINT

23 3 PRINT

3 -3 3 PRINT

313 PRINT

323 PRINT " --GAMMA'--"'

325 PRINT " -j

343 PRINT ""fr""""t: ^

VER ^ RADIATI0N PARAMETERS

35* FOR K=l TO S 3S3 ?3INT TA3 ( D[ 1 7+K ]+ 1

)

;D[ 1 1+X]

373 NEXT K 33 3 PRINT

3

7 5 PRINT

39 3 PRINT

433 F03. 1=1 TO 5 435 FOR J=l TO 6 410 LET G 1 = D [ I ] 423 LET G2 = D[ 5+J ] 433 IF J<>1 GOTO 45^

44 3 PRINT Glj

453 prist TAB (7),G2;

453 FOR K=l TO 6 47 ^ LET B=D[I1+K]

4'7 3 GOSUS 1303

HI LET X= FNA(E) + .0001

«S0 IF X> 3 GOTO 513

52? ^JZf'-P?

(Dti7+"- LOG (X)/ LOG (13)), FNA(E);

96
Table D-III (Continued)

53'? NEXT K 54 3 PRINT

553 LET V=VH

5S3 IF V=3 GOTO 61:3

57"' IF "v/=6 GOTO 553

539 GOTO 623

5?3 LET V: S03 PRINT

si g PRINT

520 NEXT J 533 NEXT I 64 3 FOR M = l TO 17

65 "1 PRINT

653 N EX T [1 673 GOTO 143

1330 LET E= 3 1335 FOR Q = l TO A 1010 LET GU-G1

1323 IF Q<>3 GOTO 134 3 133 3 LET BU-31

1343 L ET L = ( G 1 * G 1+2* G2 * G 1 ) / ( 1 -32* G2 ) 1341 LET E2=L*B*(G2-B1)*(G2-B1)

1^42 LET E3 = B*G1*G1

1343 LET E4=2*3*G1*(G2-B1)

1344 LET E5=(1-G2*G2)*(1-L)

1353 LET E1=(L+E2+E3+E4)/E5

1352 CALL 11, 3,P

1353 IF P= 3 GOTO 1363

13 54 PRIi'JT 1355 PRINT "E=" Ej " ETa;"="El ; "G1 = "G1 ; "G2 = "G?

1356 PRINT

1357 PRINT " E2 =

" E2;"E3 = "E3 ; " E4 = "E4 • " '3- "E5

13 63 IF AB5 (E1)<E 30TO 1333

1 373 LET E= ABS (El)

imn N E

v T Q

139 3 RETURN

53 33 RET" G1=GAMMA' (AMT)^GAMMA' (STD) FOLLO

5310 DATA .335, .01, .32, .05, .1 5323 R EM 32 = G AM M A " ( A N T ) FOLLO WS

5333 DATA 3, .32, .35, .1, .2, .35

5343 REi* REVERSE RADIATION PARAMETE ? FOL

5353 DATA 3, .2, .5, 1, 2, 5 53 73 REM TAB POSITIONS FOLLOW

5375 DATA 17, 27, 36, 45, 55, 64

53 3 3 REM BURET A FOLLOWS

53P3 DATA 3, .1, .2, .3, .4, .5

"31="31;"3:"B:"L="L:

97
Table D-IV. Print out of D-III program.

RUN

D\'2 1 .3 (30 MOV 72)

MBS PAGE 5940, RECHECK 5928

VJ w U i

,05

. 1 .2.35

,04

.02

• 35

. 1 .2.35

12 3.32

.35

. 1

.35

1 3 3.02

.05

. 1 .2.35

.14 3.32

.35

. 1

. 2 .35

13 5.32 °?

MAXIMUM MISMATCH UNCERTAINTY (% OF TE)

FOR BETA=

--GAMMA'-- P.EVE^.SE RADIATION PARAMETERS

ERR ANT 3 .2 .5 1 2

,33

.32

.09 % . 1 1 % . 14 /o . 18 C7 .27 % .54 % .21 % .25 Z .32 % .42 «7 .63 % 1 .26 ? .39 «7 .47 % .59 % .79 % 1.13 <*t 2.36 %

.71 % .85 % 1.3 6 % 1 .42 % 2. 13 or 4.25 J 1.42 °»

1 .7 :5 2. 13 % 2.34 % 4.26 % 8.51 % 2.92 % 3.5 % 4.33 % 5.83 Z 3.75 V 17.5 Z

. 16 % . 19 % .24 % .32 ? .48 Z .96 7 .32 7= .39 % .43 % .64 °7 .96 z 1 .93 % .57 % .63 °7 .85 «? 1.13 % 1.7% 3.4 %

.99 % 1.19 % 1 .43 % 1.98 % 2.97 % 5.93 %

1 .95 •7 2.33 % 2.92 % 3.89 z 5.84 z 1 1 .67 % 3.93 % 4.77 7 5.97 % 7.96 % 1 1 .93 % 23.87 7

1 .46 % 1 .75 % 2. 19 % 2.92 =7 4.33 % 3.77 ^

1 .96 % 2.35 % 2.94 % 3.92 % 5.83 7 1 1 .76 % 2.73 " 3.27 % 4.39 % 5.45 % 8. IS Z 16.35 % 4.33 % 4.89 % 6. 1 1 07 3. 15 % 12.23 Z 24.46 % 7. 24 % 3.69 g" '10.36 % 14.43 % 21.72 % 43.45 % 14.39 % 17. 27 Z 21 . 59 % 23.79 % 4 3. 18 <v 36.36 "

1.72 V 2.3 6 j> 2. 53 % 3.44 " 5.16 * 10.31 7 2.26 z 2.7 1 % 3.39 Z 4. 52 % 6. 79 % 13.57 % 3. 1 ' 3.72 % 4.65 Z 6.2 3 9.29 % 18.59 %

4. 53 z 5.49 % 6.36 % 9. 15 % 13.73 % 27.4 5 fo 3.35 % 9.67 w 12.03 % 16.11 " 24.16 J 43 .33 j7 15.93 % 19.17 23.97 % 31.96 % 47.93 % 9 5.37 %

2 Z 2.4 f 3 % 4 % 6 Z 12 % 2.59 % 3. 1 5 3.3 3 % 5. 17 % 7. 76 " 15.52 % 3.49 % 4.19 % 5.24 z 6. 99 fo 10.43 % 2 . 9 7 %

5.1 % 6.12 =7 7 .65 z 10.2 % 15.3 1 z 33.61 » 3.9 Z 13. 69 TM 13.36 7, 17.31 z 26.71 % 53.43 % 17. 64 ?.. 21.16 f. 26.4 5 % 35.27 % 52. 31 %

98
D.3. Table 3 For table 3, the program DW13 listed in table D-V was

used. Essentially eq. (D.4) is used in step 320.

9 9
Table D-V. Computer program for table 3.

LIST

DW13 15:02 CSS THU. 05/04/72

53 N=l

S3 V=3

103 PRINT

113 PRIST

123 PRINT

133 PRINT USING 143

140: TECK) = F(DB) FCD8) = T£(K)

153 PRINT

150 FOR J=l TO 4 173 FOR K=l TO S 133 ON K GOTO 233,213,220,23*3,243,250

203 T:13TJ

235 GOTO 333

213 T=1.5*13TJ

215 GOTO 333

220 T=2*13tJ

225 GOTO 333

233 T=3*13tJ

235 GOTO 333

240 T=5*l*Tj

245 GOTO 333

253 T=7*13tJ

255 GOTO 333

303 Fl=13*CLG(T/293+l)

313 F=l+T/290

323 B=.1*(F-1)/(F* LOGU3))

333 T5=290*(13f< . l*r.: )-l)

34 3 F5=l+T5/29 3 353 TS: r5*LOG(13)/CF5-l)

3S3 PRINT USING 3 73, T, Fl ,B, N, T5, T5

37i3:##### + 1% ##.## + .#### DB *# + .1 DB ##### + ##.##%

33 5 V- "HI 393 VzJ+l

433 IF V<>3 THEf! 433

41 J V:^

423 PRINT

42 c> RINT

43-* NEXT K 44? M EXT J

5'>3 FOR 1:1 TO 13

51J PKINT

52 d \!EXT I 999 END

R E A DY

5.i: FOR 1=1 TO 2' 100
D.4. Tables AI--AXXI

For table A-I through A-XXI the program DW10 listed in

table D-VI was used. This program is very similar to parts of

DW11 listed in table D-I.

101
Table D-VI. Computei- program for tables A-I through A-XXI,

LIST

DW10 08:50 CSS THU. 04/20/72

20 N = 3 30 M=3

40 C=62

42 V:0

45 1=1 50 PRINT

60 PRINT

70 PRINT

71 REM T = TE

72 REM T1=TH

73 REM T2 = TC

74 REM T3 = BTA

75 REM T4 = TA

76 REM

77 REM DUDTH

78 REM D2 = DTC

79 REM D3 = DY(DB)

S3 REM D4 = DG(7.)

81 REM D5=C0NNECT0R LOSS (DB)

82 REM D6=CLIPPING (Z) 83 REM

84 REM EUETH

85 REM E2=ETC

86 REM E3 = EY

87 REM E4=EG

88 REM E5=E(MISMATCH)

89 REM E6= EC CONK ECTOR)

90 REM E7=E(CLIPPING)

91 REM

92 REM F=F

93 REM F1=F(DB>

94 REM

95 REM GUGAMMA (AMP)

96 REM G2=GAMMA (STDS)

97 REM Y=Y

98 REM Y1=Y(DB)

99 REM Y2 = DY

100 DEF FNTCH, C, Y, G) = (G*H-Y*C) /CY-G)

110 DEF FNE(W,X,Z)=53*ABSUW-X)/Z)

120 DEF FNF(T)=10*CLG(l+T/290)

190 READ G1,G2,T3,D3,D4,D5,D6

200 READ T1,D1,T2,D2

245 G0SU5 5030

300 FOR J=l TO 4 310 FOR K:l TO 6 320 ON K GOTO 1000,1023,1043,1060,1383,1103

330 G0SU3 4433

343 PRINT USING 55 63, T, ES, Fl ,R8, Yl , El , E2, E3 , E4 350 M=M+1

355 V:V+1

360 IF V<>5 THEN 760

370 V=0

380 PRINT

390 PRINT

102
Table D-VI (Continued)

400 MrPH-2 760 IF M<= C-3 THEN 77<3 763 IF M> C-3 THEN 765

765 GOSUB 3300

770 NEXT K 7S0 NEXT J 800 GOSUB 3000

802 GOTO 200

803 GOTO 200

804 GOSUB 5000

1000 T=10tJ

1010 GOTO 330

1020 T=1.5*10TJ

1030 GO TO 330

1043 T=2*10tJ

1050 GOTO 330

1060 T=3*10TJ

1070 GOTO 330

1080 T=5*10fJ

1090 GOTO 330

1100 T=7*10tJ

1110 GOTO 55<3 3000 FOR Q=M TO C+6

3010 PRINT

3020 NEXT Q

3025 M=3

3040 RETURN

4430 Y=(T1+T)/(T2+T>

4410 Yl =13*CLGCY)

4415 Y2 = Y*LOG(13)*D3/13

4420 F1 = FNF(T)

4425 F=10t(Fl/10)

4427 B=0.1*(F-1)/(F*LOG(10))

4430 T4:(1-G1 T2)*T-T3*G1

4435 A=l0t(-D5/10)

4440 W=FNT(T1+D1,T2,Y,1)

4450 X = FNT(T1-D1,T2,Y, 1) 4460 Z=T

4470 E1=FNE(W,X,Z)

4480 W=FNT(T1,T2+D2,Y,1)

4490 X = FNT(T1,T2-D'2,Y, 1) 4510 E2=FNE(W,X,Z)

4520 W=FNT(T1,T2,Y+Y2, 1) 4530 X=FNT«T1,T2,Y-Y2,1)

4550 E3 =FNE(W,X,Z)

4560 W=FNT(T1,T2,Y, 1+D4/103)

4570 X=FNT(T1,T2,Y, 1-D4/130)

4590 E4=FNE(W,X,Z)

4600 W=(T4+T3*((Gl+G2)/Cl-Gl*G2))>/<l-<<Gl+G2)/(l-Gl*G2))t2)

4610 X=(T4+T3*ASS(G1-G2))/(1-(A3S(G1-G2))T2)

4630 E5=FNE(W,X,Z)

4640 V=FNT(A*T1+<1-A>*300,A*T2+(1-A)*300,Y,1>

4650 X=T

4663 Z-T

4673 ES=2*FNE(W,X,Z>

4683 W=((1-D6/133)*T1-Y*T2)/(Y-1>

4710 E7r2*FNE(W,X,Z)

4723 E8=E1 + E2+E3 + E4 4725 R3:B*E3

4733 RETURN

103
Table D-vi (Continued)

5030 PRINT USING 5500, Tl , Dl , 1 00*D1 /Tl

5005 PRINT USING 5510, T2, D2, 100* D2/T2

5010 PRINT

5020 PRINT USING 5520,D3,D4

5050 PRINT

5060 PRINT

5065 M=Mf8

5070 PRINT USING 5540

5080 PRINT USING 5550

5090 PRINT

5100 PRINT

5110 RETURN

5500: TH=##### + ### K (#.##7.)

5510: TC:##### + #.# K (#./#Z)

5520: DY=.## D3 DG=.##%

5543; ERROR CONTRIBUTIONS TO TE

5550: TE(X) FCDB) Y(DB) ETH ETC EY B 5560:##### +##.#% ##.# + .## ##.# ##.##% ##.##% ##.##% /|

8010 DATA .1 ,.1,300, .01, .1,. 02, .1 8020 DATA 18000,270,300,1

8030 DATA 18030,270,80, .5 8040 DATA 18000,270,4, .5 8050 DATA 10000,150,300,1

8060 DATA 10030, 153,80, .5 8070 DATA 10300, 153,4, .5 8080 DATA 1270,3,333,1

8090 DATA 1270,3,83, .5 8100 DATA 1270,3,4, .5

81 10 DATA 373, .1 ,290, .1 8120 DATA 373, .1,80, .5 8130 DATA 373, .1,4, .5 8143 DATA 300, 1,80, .5 8150 DATA 300, 1,4,.

5

9999 END

READY

RUN

104
D.5. Tables BIII--BXXII

For tables Bill -- BXXI I , the program DW22 listed in

table D-VII was used. This uses the same mismatch error calculation from appendix B that was used in DW11 listed in table

D-I.

105
Table D-VII. Computer program for tables B-III through B-XXII

LIST

1 PP.IMT "D' : 22(3 DEC 723"

PRIMT "-PAGE 5856, CK 5391"

REM :MISMATCH ERROR..! SMITCi

2

51012

1416

17

1823

2122

2426

23

33

3343

53

93

103

1 13 123

133

143 233 213

223

233

243 253

263

273 272

274

276

273 233 293

333

313

3 23 325 333

343

353 363

373 333

335 393

P.EM REM REM

REM

P.EM PEM

REM REM

REM REM

REM

P.EM P.EM

pr-,« REM

DEEDIM FOR

Ri

MOD !E 5935),

5 u? fo; DV 2 1.3

:riA" (and

D'.'l 1 (PA

B1=BETA

Gl =GAMX,V (HOT) - GA G2=GAMMA' (AMT 3 G3=GAMMA* (COLD) - GAMMA' (AMT)

B= REVERSE RADIATION PARAMETER

T = T3 T1=T(H0T)

T2=T(C9LD3

T5=PERT'.;P.3ED T6=?ERTUR3ED

T7=?ERTUREED

DIAGMOSTICS

(PAGE 5 97J

)

T(HOT)

T(COLD)

TE

PI =F(D33

FMA(X3 IMT C1323*> DC25]

1=1 TO 23 AD D C I

+.53/1303

Mr XT I READ

PR IMT

PP.IMT

PP.IMT PRIMT

PP.IMT " PRIMT

PP.IMT PRIMT , PRIMT , PRIMT

PRIMT ,

P.P. I JIT , PRIMT PRIMT

PRIMT PRIMT PRIMT " PRIMT " PRIMT " PRIMT " FOR K=l PRIMT

ME XT !< PRIMT

PRIMT

PRIMT

1 ,T2,31 ,3

TAB (133 MAXIMUM 1ISMATCH ERROR CDS)'

TCiIOT) = "TU

t(cold3=-t;

0ETA="31

3TA/TE="!

--GAMMA' --'

AMT";

(DC i7+:<] + i 3;dc i '+:<]: ro 6 TAB

106
Table D-VII (Continued)

430 FOR 1=1 TO 5 435 fop J=l TO 6 4 13 LET Gl=Dd]

423 LET G2 = DC5+.J1 433 IF J"l GOTO 453 443 PP.IMT Gl ; 453 PP.IHT TAP C7>,G2J

463 FOP X=l TO 6 473 LET F1=DC 1 t+KJ 472 LET T=293*C 1 3' CFI/13}- 1

473 LET F=13' CF1/1 ~> 474 LET Y=CT1+T)/<T2+T)

476 LET E4=l 3*(F-1 )/(F* LOG C 13 )

)

483 GOSUO 1300 435 LET X= FMACE)+.3C0I

4 93 IF X> 3 GOTO 510 530 LET :<= . 3 3 1 513 PP.IMT TA3 CDC17+K3), FNACE);

533 NEXT X 540 PRINT

550 LET V=V+1

560 IF V=3 GOTO 613

573 IF V=6 GOTO 593

580 GOTO 623

593 LET V= 3 633 PP. I MT

613 PP.IMT 623 TJEXT J 633 r;EXT i 64 3 FOP. '1=1 TO 17 653 pr.niT 663 r.'EXT r; 673 GOTO 143

15103 LET 3= 3 1332 LET E3= 3 1305 LET G3=G1

1306 DEE FME<X)=CX*X+2*32*X)

1307 LET E1=1-G2*G2

1013 FOR P.= l TO 3 1023 LET 1 =-31

1033 LET S = S + 1 1043 IF S=2 GOTO 1 363

104 5 LET S=l 1350 LET G3=-G3 . 1353 IF ?= 3 GOTO 1053 1354 PP. I NT

1056 PP.IMT

10 63 IF*P.<>5 GOTO 1 133 1073 LET G 1 = -G1

1 133 DEF FXFCX)=2*3*T*X*31

1 123 LET T5 = C<T1 -3*T>* E"ECG1>+ FNFCG1))/E1

1 125 LET T5=T1-T5

I 130 LET T6=C<72-3*7)* FNE<G3)+ F?JFCG3))/E1

1 135 LET T6=T2-T6

1 150 LET T7=CT5-Y*T6)/(Y-1

)

1 160 LET 3 2 = A3S C(T7-T)/T)

1173 CALL 11, 3 , ? 1232 IF P= 3 GC x O 1233 123? PPIT

1213 pp.imt "f= ,- f; ,, fi ="fi ;"T="T;"Ti="Ti ;"T2="T2

1215 PP.IMT 'T5="T5;"T6 = "T6;"T7 = "T7;-'Y="Y

1223 PP.INT "Gl = "G1 ;"G3 = "G3;"G2 = "G2; ,- D1="31 ;"3="3

1225 pp.imt "e<"~.;")="e;"ei ="Ei ; ,, e2 = "E2; ,, e2 = "E2;"E4 =

' •E4 1233 IF E2<E3 GOTO 1253

124 3 LET E3=E2

1250 LET E=E3*E4

107
Table D-VII (Continued)

1261

1271

5035!

5015! 5020

5030 5351

5 37 5075

508 3 5-39 3 5103

5113 5123

5133 5143

5150

5160 5173

5 130 5193 5230

5210

5220 5233

5240 5333

5310

5433

54 10

NEMT

?.etu: "EM DATA

pem DATA

"EM : DATA

t.em DATA

HEM : DATA

DATA

DATA

DATA DATA

DATA

DATA DATA

DATA DATA

DATA

DATA

DATA DATA

DATA DATA

DATA

DATA DATA

DATA

GI=GAMMA* f AMD-GAMMA* C5TD3 FOLLOWS

.305, .31, .02, .05, .1 g2=ga;:ma**cahtj follows

0, .32, .35, .1, .2, .35 l=F<DE) FOLLOWS

1 , 2, 4, 6, 3, 13 TAB POSITIONS FOLLOW

17, 27, 36, 45, 55, 64 SETS OF Tl ,T2,0ETA,BTA/TE FOLLOW

13093, 300, 0, 10303, 300, 2 10000, 3 30,

10003, 330,

13333, 300,

130 30, 33-3, 10003, 330,

. 1 ,

. 1 , .2,

.2,

.3,

.3, 373, 83,

373, S3,

373, 80,

373, 33,

373, 30,

373, 30,

373,

0, ;

. i,

. i , .2,

• 2, .3, 33, .3,

127-3, 3 33, 3, 3 1273, 300, .2, 1 1273, S3, 3, 1270, 33, .2, 1 303, 4, 3, 3 303, 4, .2, 1

10 8
Appendix E. Reprints of Useful Papers

References [1, 3, and 8] are reprinted here.

109
Reference [1] Reprinted with Permission of the Publisher

Technical Feature

Measurement of Amplifier Noise

mismatch errors and amplifier noise measurements are interrelated -- other errors are exposed

D. F. WAIT National Bureau of Standards. Boulder, CO

Measurement of amplifier noise, in particular a state-of-the-art measurement using hot-cold noise standards of 10,000 K - 300 K

or 373 K - 80 K, is discussed. An error that is especially difficult to estimate is the mismatch error. Measurements are suggested to help with this problem. Also, the traditional NBS thermal noise

services are briefly sketched.

In its RF and microwave thermal-noise services, the National Bureau of Standards has in the past concentrated on developing means for transferring the fundamental

standards of noise power to the scientific and engineering community of the Nation. With substantial capability for doing this now on hand, NBS has turned its attention to the application of these standards, and of related techni- ques, to measurements of more direct concern to the "ultimate user." One of the key objectives of these more

recent efforts is to transfer to the user the ability to make

his own measurements and/or to verify that these are in- deed valid. In special instances NBS may simply make

measurements for the user; in others it may repeat the customer's measurement, if necessary at the customer's own laboratory, to validate his technique and results.

This paper outlines and discusses the amplifier noise

figure that has been used by NBS in the validation process and that, most important, can be used by others. It is typical of the new directions in NBS noise services. The emphasis in this paper is on how to make amplifier

noise measurements and on sources of measurement error and uncertainty. But because these and so many other

similar measurements critically depend on the availability of noise standards, the standards that can be made avail- able through the more traditional NBS calibration services

are first briefly sketched.

TRADITIONAt CAPABItlTIES IN NOISE

The National Bureau of Standards maintains reference

noise standards and calibration radiometers, and has been

active in developing the theory to support these areas. The most utilized noise service is that of noise-source cali- bration, although more and more frequently NBS is asked

to analyze a radiometer rather than to calibrate a noise

source.

Reference Standards

All NBS noise standards are resistive elements maintained at a uniform and known temperature T

1 ' 3

. The

theoretical power P

a available from these blackbody radi- ators is P

a = kTB where k is Boltzmann's constant, and 8

is the bandwidth being viewed. The cold standards maintained by NBS are liquid-nitrogen cooled and have noise temperature near 80 K. The temperatures of the hot

standards vary: the NBS 14mm coaxial hot standard

operates at 373 K; the WR284 standard (2.60 - 3.95 GHzi at*9lK;and the WR90 (8.2 - 12.5 GHz), WR62 (11.9-

18.0 GHz), and WR15 (50 - 75 GHz) standards near 1250 K. Figure 1 indicates rreauencies at which NBS

maintains the hot and cold primary standards.

Calibration Services

The comparison radiometer, used to calibrate an un- known noise source relative to the primary standard, is 10 °T -«GH2

| *&* 15

tz I '• = ROUTINE SERVICE WR 62

\VR 90

|WR 28 4|

18 GHz ,

8.2 GHz

3.95 GHz

2.60 GHz \*fi

60 MHz '

30 MHz

3 MHz * COI.D CALIBRATION

SERVICES

This paper is a contribution of the National Bureau of Standards

not subject to copyright. COLD STANDARDS „ T STANDARDS

Fig. 1 NBS calibration services and standards.

60 MHz

30 MHz

3 MHz •"HOT

CALIBRATION

SERVICES

January — 1973 110
the most elaborate part of any calibration service, and the need to understand its operation spawns most of the NBS

experimental and theoretical noise studies''"' 6

. The fre- quencies at which both reference standards and comparison radiometers exist are the frequencies of the available noise calibration services shown in Figure 1. Typically

noise temperatures between 1000 to 300,000 K are cali- brated to within 2% total uncertainty (excess noise ratios between 4 and 30 dB to within 0.1 dB) ' 7

. The calibration uncertainty is degraded somewhat if the connector of the source to be calibrated differs from that of the corresponding NBS standard. For frequencies at which cold-source

calibrations are indicated in Figure 1, noise sources are typically calibrated to within 1% at temperature above 75 K. MEASUREMENT OF AMPLIFIER NOISE

At present, the National Bureau of Standards can measure effective input noise temperature only in WR15

waveguide but, in principle, the noise figure service can be

readily implemented at any frequency where an NBS pri- mary noise source exists (see Figure 1 ). Thus, the potential frequency coverage of NBS noise-figure-measurement

capability is greater than the coverage of the noise-source

calibration services, which are limited by the frequency

ranges of the comparison radiometers.

An Overview

The most crisply defined measure oi amplifier noise is Te

, the effective input noise temperature.* If two noise standards with corresponding noise temperatures T hot and Tco| d are sequentially connected to the input of an amplifier, then the change in output power of the amplifier, measured as a ratio Y, is Y = <T hot *Te )/(Tcold+ Te

) (1)

We see that Teis essentially a measure of the amplifier's

internal noise expressed in terms of the available** power

density of a noise generator at the input o.f the amplifier. Another popular measure of amplifier noise is noise

figure' 8

, but the ambiguities and difficulties 19 " 26 with

the IEEE definition of noise figure limits the usefulness of noise figure. The most serious ambiguity is related to superheterodyne amplifiers for which two different input channels contribute to the same output channel. The

noise figure for the same amplifier differs depending on

whether the user expects useful information through only

* "Effective Input Noise Temperature, Average T (of a multi- port transducer with one port designated as the output port). The noise temperature m degrees Kelvin which, assigned si- multaneously to the specified impedance terminations at all frequencies at all accessible ports except ttie designated output

port of a noisefree equivalent of the transducer, would yield the same total noise power in a specified output bond delivered to the output termination as that of the actual transducer con- nected to noisefree equivalents of the terminations at all ports except the output port."

•* The noise temperature of a noise source is defined to be the

physical temperature of a blackbodv radiator which has the lame available power density as the noise source (at the fre- quency in question).

Te <K)

10 ±1% = 1G ±1% = 20 ± 1 %

30 ±1% = 50 ±1%'

70 ± 1 % =

100 ±1% = 150 ri%

200 ± 1 %

300 ±1% = 500 ± 1 %

700 ± 1 % =

1000 ±1% = 1500 ±1% = 2000 ±1% =

0.15 ±.0014 dB 1 ±.1 dB

0.22 ±.0021 dB 2 ±.1 dB

0.29 ±.0028 dB 3 ±.1 dB =

0.43 ±.0041 dB 4 ±.1 dB = 0.69 ±.0064 dB 5 ±.1 dB " 0.94 ±.0084 dB 6 ±.1 dB =

= 1.29 +.0111 dB 7 ±.1 dB = = 1.81 ±.0148 dB 8±.1dB = = 2.28 ±.0177 dB 9 ±.1 dB =

3.03 ±.0221 dB 10 ±.1 dB = 4.35 ±.0275dB 11 ±.1 dB = 5.33 ±.0307 dB 12 ±.1 dB =

6.48 ±.0337 dB 13 ±.1 dB = 7.90 ±.0364 dB 14±.1dB = 8.97 ±.0379 dB 15±.1dB =

Te IK)

75 ±11.20%

169 ± 6.24%

283 ± 4.62%

438 ± 3.83% 627 ± 3.37% 864 ± 3.07%

1163 ± 2.88% 1539 ± 2.74% 2013 ± 2.63%

2609 ± 2.56% 3360 ± 2.50% 4306 ± 2.46%

549G ± 2.42% 6394 z 2.40% 8880 ± 2.33%

3000 ±1%= 10.55 ±.0393dB 16 ±.1 dB = 1 1255 ± 2.36% 5000 ±1%- 12.61 ±.0410 dB 17 ±.1 dB = 14244 ± 2.35% 7000 ±1%= 14.00 ±.0417 dB 18 ±.1 dB = 18007 ± 2.34%

10000 ±1%= 15.50 ±.0422 dB 19 ±.1 dB = 22745 ± 2.33% 15000 ±1% = 17.22 ±.0426 dB 20 ±.1 d8 = 23709 ± 2.33% 20000 31%= 18.45 ±.0423 dB 21 ±.1 dB = 36218 ± 2.32%

30000 ±1% = 20.19 ±.0430 dB 22 ±.1 dB = 45671 ± 2.32% SO0OO ±1%= 22.39 ±.0432 dB 23 -A dB = 57572 ± 2.31% 70000 ±1%= 23.84 ±.0433 dB 24 r.1 d8 = 72554 ± 2.31%

Conversion between effective input noise temperature, T (K), and noise figure, F dB. The asymmetry in the error statements re- sulting from the logarithmic non-linearity of F^,- is avoided by

using the slope of F

d~ at the corresponding T .

the signal channel, or through both channels. Nevertheless,

noise figure is still used as a measure of amplifier noise. To bypass the past difficulties an unorthodox definition

is used here, namely

FdB = 10 log [1 +Te /290]. (2)

This definition corresponds approximately to double- channel noise figure as discussed by Mumford and

Sheibe 24

, and is -equivalent to the IEEE definition of noise figure for a ingle-response amplifier. Defined this way, F dB can be .

: ewed as a cnange in scale of Te

. The

conversions bet'.-.esn Te and FdS are given in Table 1.

The Accuracy G-.al

The most accurate and easiest measurements of Te are based on Equation 1. The effective input noise temperature T. is known rrom a measurement of Y because

(T hot -YTcold )/(Y-1) (3)

There are seven important sources of error in a measure- ment of Te

: uncertainty in the value of Thot ; uncertaint

in the value of T^^, uncertainty in the measurement of

Y; the amplifier-cjin instability during the time required to meas".-? tH» ratio Y; the inequality of the three reflec111 the microwave jo urn.
tion coefficients, vis., the hot and cold noise standards, and the reflection coefficient "seen" by the amplifier in- put port when it is in actual use (mismatch error); differ- ence in losses associated with the connector of the hot and cold standards and the amplifier under operating conditions are either not reproducible or are unknown; uncertainty in the noise generated in the measuring system

is not accurately known (cascade error).

200 300 500 700 Ik 1.5 k 2k 3k 5k 7k 10k

fllSMATCH ( | rANT-rsTD | <0.01; irAM2.-raMP '

I < 0.05)

do.o; s

s

TCOLD = 30° ±0S K

THriT = 10.000 ±150 K

CONNECTOR LOSS -(o.Ol drj) gain -cp.i^; Y-KACTOR -(p.01 dB)

rCOLD<017 °

!

>.)

THOT <1.57»)

Fig. 2 Physical parameter contributions to noise-figure measure- ment limit of error.

, = 373 ±0.5 K

80 ±1 K

TCOLD •C1 - 25"") ^

T„ OT -(0.13^J

^ 100 200 300 -100 500 600 700 800 Te (K)

Fig. 3 Physical parameter contributions to effective input noise- temperature measurement limit of error>

In setting an accuracy goal for the measurement of amplifier noise, it is useful to keep in mind the state of

the art, the magnitude of mismatch uncertainty that can

be expected, and the quality of available connectors. The State of the Art In establishing an accuracy goal, one of the best guides is an awareness of how closely one

wishes to approach the state-of-the-art. Figures 2 and 3 show the accumulated errors for state- of-the-art measurements of F dB and Te

, using two fre- quently employed hot-cold standard combinations; namely, 10,000 K and 300 K, or 373 K and 80 K. With one of

these combinations, measurement uncertainties of about

0.1 dB in noise figure or effective temperature uncertainties of about 2.5% are state of the art. Mismatch Uncertainty The effective noise temperature Te

is a function of the reflection coefficient (i.e., impedance) of the "antenna" (components attached to the input

of the amplifier). If the reflection coefficient of this "antenna" is not known exactly, then the Te that the

amplifier will have when in use cannot be known exactly. TABLE 2 (PRELIMINARY! CONNECTOR ABSORPTION LOSS IN dB AND

ABSORPTION COEFFICIENT A IN PERCENT

There is very little evaluated connector loss literature. The

values in this table should be treated as preliminary data suggesting

the general magnitude of losses to be expected. FREQ. TYPE (GHz) LOSS (dB] Al%]

1. COMMERCIAL TYPE Nib 24 0.01. <0.1 0.23, < 2.3

2. PRECISION TYPE N° 0.1 0.01 0.23

2 0.02 0.46

0.4 0.04 0.9

1.0 0.04 0.9 20 008 1.8 4.0 0.11 2.6 8.0 0.20 46

3. COMMERCIAL WR90' 8-12 <0 01 <0.23 COMMERCIAL WR9ud 9 4 0.003 0.07 NBSTYPE WR90*

(dirtyl 9 0.003 0.07

(lappedl 9 0.001 0.023 CMR WR90*

(lapped or not! 9 0.0004 0.009

4. SPECIAL CPR WR112' 0.0002 00046

5. SPECIAL WR430' 23 ^OOOOOl 0.00023

6. PRECISION COAX1

• 14 mm 0.01-8.5 0.003 v-f 0.07^1

7 mm 001 17 007^/f O.IBx/f

7. COMMERCIAL WR15' GO 0.007 to 0.045 0.16 to 1.04

• in CHi

a. Charlei T. Stllcrftid IJPU pn.ateci>mmonrcaoo

Q Robert T Ada.r (NBSI pn.are commonicanon

e. R W B-atty. Proc IEEE Vol SI PP 642 642 1965

d R W Beany. G F Enqen. and W Amon, IRE Tranr. on Iminffr—r, eon.

Vol. 1-9. pp 219 226, 1960.

«, W E. Utile IIVBS) private commont^tion.

f. IEEE Standard lor Predion Coa-ia- -onnecion IEEE Irani, on Imtr. and Meat.. Vol 1M- 1 7. pp. 204 222. 1 968

g. B Yetei 1NBS). Pr.,a:e common, car :r-. This uncertainty in effec;:ve input noise temperature is referred to here as the .—.ismatch uncertainty. One should

be hesitant to set an accuracy goal better than this mis- match uncertainty.

In Figure 4, the maxmum mismatch uncertainty is plotted for an ideal am" ! :fier as described by Engen . It

is clear from Figure 4 t"Jt the mismatch uncertainty be- comes very important ' the magnitude of the difference between the "antenna's'' and standard's reflection coeffi- cient is expected to di'J :r by more than 0.01

.

Connector Quality Table 2 is a list of connector loss measurement results. Tbs change in F^g due to the connector loss is equal to the decibel loss of the connector when it is at the temper;;' ',, 290 K. Experience in waveguide has shown that a g'-.'J! deal of care is required to duplicate these values. A carelessly marred flange, or a

112
poorly mating pair can easily be worse by a factor of ten.

It is good practice to remeasure the noise figure after a break and a remake of some of the connectors. Bad connectors can sometimes be spotted in this way.

Measurement Technique

In principle the measurement of Te

is simple. The hot

standard is connected, then the cold standard is connected

and Y, the ratio of output powers, is measured. The details of changing from hot to cold (e.g., by changing current through a solid-state noise source), of measuring Y, and

displaying the results depend on the equipment used. The

manufacturer's literature and the prior literature 23 " 33are very helpful but often the information needed to establish

error limits is difficult to obtain, especially for the auto- matic noise-figure meters. Often the straightforward (con- nect a hot standard and measure the output power, then connect a cold standard and measure the output power)

technique is best. One precaution however: it has been found 34

at NBS that, for 1-dB gain compression in the power measuring system, the power level needs to be about a factor of 10 greater than the mean noise power if the error from clipping the noise is to be less than 0.1%.

This is also true for the commercial noise-figure measuring

systems. Systems operating with too much gain can pro- duce serious errors 35 " 37

The most insidious measurement hazard is "mismatch"

error,23,38 " 40

. The problem of using standards having a different reflection coefficient from the "antenna" has already been discussed as mismatch uncertainty. The additional problem of the hot-standards and the cold standards having different reflection coefficients from each other is the mismatch error displayed in Figure 2 and 3. Here the two standards reflection coefficients have arbitrary ampli- tude and phase within the limits indicated. When the re- flection coefficients are different many ugly things can

lOr

Fig. 4 Moxiit

coeffic

OO 5

0.2 r1 V ' I ANT l AMP ' natch uncertainty — "antenna" reflection amplifier input reflection complex conjugate.

happen as these standards are exchanged for each other: changes in gain, in gain stability, in bandwidth, and in linearity. In Figure 2 and 3 these ugly possibilities were

ignored, and only the changes for the best possible ideal amplifier described by Engen"' were considered. It is safer to estimate mismatch error from the measurements

suggested below. The change of Te with a change of reflection coefficient can be measured by inserting an isolator and a slide- screw tuner (or the appropriate equivalent) between the

iniiimjiiimiiiim i m 3^^i

Fig. 5 Block diagram for measurement of mismatch uncertainty

as related to reflection coefficient.

noise standards and the amplifier as shown in Figure 5. From this variation of Te the mismatch uncertainty can

be estimated. This measurement is not exact because the

losses change in the tuner-isolator combination as the re- flection coefficient changes, but this loss change is usually

small. For example, for commercial X-band slide-screw

tuners, measured changes in absorption loss are less than 0.015 dB as reflection coefficient is changed from zero to

0.5 (any phase). Because of the definition of the noise

figure (Equation 2), a 0.015-dB increase in absorption loss

(at 290 K) increases the noise figure by 0.015 dB. One estimate of mismatch error is obtained by measuring Te with and without the isolator-tuner shown in Figure 5. If the isolator-tuner is adjusted to the "antenna"

impedance, then the change in noise figure should be

equal to the absorption loss or the isolator-tuner. If this absorption loss can be estimated (insertion loss for lowreflection elements is nearly equal to the absorption loss), then any serious mismatch errors will be spotted. A better method, if it can be implemented, is to attach a slide-screw tuner to the hot standard only, and to observe the change

in Te with the change of reflection coefficient. This pro- cedure is then repeated with the slide-screw tuner at- tached to the cold standard only.

Estimating Measurement Error

Figures 2 and 3 are useful for estimating measurement

errors because the error contributions to FdB or Te are

essentially in direct proportion to the magnitudes of the

respective error source parameters. Therefore, if Y is measured to within ± 0.02 dB. the Y-factor contribution error bands in Figures 2 and 3 double. However, this proportionality is not a good approximation of the mismatcherror estimate, but direct measurement of Te versus rstd should be used'for estimates of the mismatch errors. Thus,

Figures 2 and 3 can be used to translate from uncertainties

in the various measurement parameters into the error in- duced in Te

. The cascade error is not shown in Figures 2 or 3. The

desired Te can be deduced 25 ' 41 from the measured T |otal when the effective input noise temperature of the measuring system is Te2 from

113
T. =T. Te2 /G,

where G, is the available power gain of the amplifier being measured. The cascade error depends on how accurately Te2 and G, are known. CONCLUSIONS

An accurate measurement of amplifier noise requires

that particular attention be paid to the mismatch error. This error depends on several amplifier characteristics and

their changes with the different terminations on the amplifier. It is therefore important to make measurements of the

the change in amplifier noise with a change in terminating

reflection coefficient, in order to estimate this mismatch

error. From a discussion of the state-of-the-art measurements

for hot-cold standards of 10,000 K - 300 K or 373 K - 80 Kr it is seen that measurements of noise figure to within about 0.1 dB, or effective input noise temperature to within about 2.5%, is the best that can be done. Such

small uncertainties are possible only at those frequencies

at which NBS calibration of noise sources exist. REFERENCES

1. Daywitt. W. C. W. J. Foote, and E. Campbell, "WR15

Thermal Noise Standard," NBS Tech Note 615. March 1972.

2. Trembath, C. L., D. F. Wait, G. F. Engen, and W. J. Foote. "A Low-Temperature Microwave Noise Standard," IEEE

Trans MTT, Vol 16 No 9, pp 704-714. Sept 1963.

3. Trembath, C. L.. W. J. Foote. and D. F. Wait, "A Liquid Nitrogen Cooled Microwave Noise Standard," Rev Sci fnstr. Vol 42 No8, PP 1261-1262, Aug 1971.

4. Wells, J. S.. W. C. DaYwitt, and C. K. S. Miller, "Measure- ment of Effective Temperature of Microwave Noise Sources," IEEE Trans Inst Meas, IM-13, pp 17-28, March 1964.

5. Wait, D. F.', Seminar Coordinator, Notes for High- Frequency and Microwave Noise Seminar, NBS, Boulder, Colo., 21 Apr -

1 May, 1970. (Unpublished).

6. Miller, C. K. S., W. C. Davwitt.and M. G. Arthur, "Noise

Standards, Measurements, and Receiver Noise Definitions," Proc IEEE, Vol 55 No 6, pp 865-877, June 1967.

7. Miller, C. K. S.. W. C. Daywitt, and E. Campbell, "A Waveguide Noise-tube Mount for Use as an interlaboratory Noise Standard," Acta Imeko III, pp 371-381, 1964. . 8. Wait, D. F., "Sensitivity of the Dicke Radiometer," J Res NBS, 71C (Engand Instrl, No.»2, pp 127-152, April-June 1967.

9. Wait, D. F. and T. Nemoto. "Measurement of the Noise Temperature of a Mismatched Noise Source," IEEE Trans MTT, Vol 16 No 9. pp 670-675, Sept. 196S.

10. Wait, D. F., "The Precision Measurement of Noise Temperature of Mismatched Noise Generators," IEEE Trans MTT,

Vol 18, pp 715-724, Oct 1970.

11. Wait, D. F., "Thermal Noise from a Passive Linear Multiport." IEEE Trans MTT. Vol 18 No 9, pp 637-691 , Sept 1968.

12. Nemoto T. and D. F. W ait, "Microwave C.rcuit Analysis

using the Equivalent Generator Concept," IEEE Trans MTT,

Vol 16 No 10. pp 866-873, Oct 1963.

13. AMred, C. M., "A Precision N"cise Spectral Density Comparator." J Res NBS. S6C I Engand Instrl, No 4, pp 323-330. Oct - Dec 1962.

14. Arthur, M. G.. C. M. Allred, and M. K. Cannon, "A Precision Noise-Power Comparator," IEEE Trans Instr and Meas, IM13. No 4. pp 301-305. Dec 1964.

15. Arthur, M. G., "A Precision HF-Noise Power Measurement

System." ISA Trans. Vol 10 No 3. pp 264-268, 1971.

16. Engen, G. F., "A Method of Calibrating Coaxial Noise Sources

in Terms of a Waveguide Standard." IEEE Trans MTT, Vol 16 No 9, pp 636-640. Sept 1969.

17. Mason, H.. Ed., "Calibration and Test Services of the National

18

19.

21

Bureau of Standards," NBS Spec Publ 250. "IRE Standards on Electron Tubes Definitions of Terms,

1962," (62 IRE 7.S2). Proc IEEE, Vol 51. pp 436-442, March

1963. Cohn, Seymour B., "The Noise Figure Muddle," microwave

journal. Vol 2. pp 7-1 1 , March 1959.

20. Greene, J. C, "Noisemanship - The Art of Measuring Noise

Figures Nc3rly Independent of Device Performance," Proc

IRE. Vol 49 No 7, pp 1223-1224.

Hendler, A. J., "Noise-Figure Measurements Definitions, Techniques, and Pitfalls," Topics in Noise. Airborne Inst Lab,

Deer Park, NY. pp 1-4, Aug 1960.

22. Rhodes, D. R.. "On the Definition of Noise Performance,"

Proc IRE. Vol 49 No 1 , p 376, Jan 1 961

.

23. Engen, Glenn F., "A New Method of Characterizing Amplifier Noise Performance," IEEE Trans Instr and Meas. Vol IM-19 No 4, pp 344-349, Nov 1970. Mumford. W. W. and E. H. Scheibe, Noise Performance Factors in Communication Systems, Horizon House-Microwave,

Inc., Dedham, Mass.. 1968.

Sleven, Robert L , "A Guide to Accurate Noise Measurement,"

Microwaves, pp 14-21, July 1964.

Letzter, S. and N. Webster, "Noise in Amplifiers," IEEE Spectrum, Tech Notes T-243, pp 67-75, Aug 1970.

27. Topics in Noise Figure Measurements, Airborne Instruments

Lab.. 1963.

Noise Figure Primer," Application Note 57, June 22, 1962,

Hewlett-Packard. Sweeney, J. J. and G. A. Deschamps, "A Wave Amplitude Approach to Noise Analysis with Application to a New

Method of Noise Measurement," Tech Report AFAL-TR-70-

20, Air Force Avionics Lab. AFSC, Wright-Patterson AFB,

Ohir-. Hildner, E. G., "A Note on Noise Factor Measurement," ESSA Tech Report ERL133-WPL 9. 31. Haus, H. A., et al, "Standards on Methods of Measuring

Noise in Linear Twoports," Proc IRE, Vol 48, pp 60-68, Jan I960.

32. Haus, i1. A., et al, "Representation of Noise in Linear Twoports," Proc IRE, Vol 43, op 69-74, Jan 1960. Lebenbaum, M., "Measurement of Noise Figure," Handbook

of Electronic Measurements, Vol 2 Chap 1 1 , pp 1-51, Poly- technic Inst of Brooklyn, 1956.

Reeve, G. R.. (Nat Bur Stand. Boulder). Private communication.

Bell, R. L., "Linearity Range of Noise-Measuring Amplifiers,"

Wireless Engineer, pp 1 19-122. Apr 1947. Cohn, C.-E., "Errors in Noise Measurements due to the Finite Amplitude Range of the Measuring Instrument," Rev Sci

Instr. Vol 35, pp 701-703, June 1964.

37. Deutsch, R. and H. V. Hance, "A Note on Receivers for Use

in Studies of Signal Statistics." IRE Convention Record.

Part I. Information Theory, Vol 1 . po 7-13, 1953.

38. Imboldi. E.. "Transmission-Line Mismatches and System

Noise Figure," Electrical Communication, Vol 40 No 2. 1965.

39. Brady, M. M., "The Influence of Mismatch Error in Noise Performance Measurements," Proc IEEE. Vol 52 No 9, pp

1075-1076. Sept 1967.

Pastori. W. E., "New Equation Analyzes Mismatch Errors in Noise Measurements," Microwaves. Vol 7 No 4, pp 53-53,

April 1968.

Stelzried, C. T., "Post-Amplifier Noise Temoerature Contri- bution in a Low-Noise Receiving Svstem," Proc IEEE. Vol

52. pp 76-77. Jan 1964. David F. Wait received the Ph.D. degree in Physics from

the University of Michigan in 1962. In 1963 he was a senior

scientist with Lear Siegler, Inc., Ann Arbor, Michigan. From

1964 to the present he has been with the National Bureau

of Standards. Boulder, Colorado, and was the coordinator of the 1970 "High Frequency and

Microwave Noise Seminar."

28.

29.

30.

33.

34.

35.

36.

40

41 r~\

I \

'- v /

in

Dr. Wait is a member of tJ'e IEEE. MTT Group (Chairman 1970. Vice-chairman 1969 of Cwnvcr-Boulder Chapter).

January — 1973 114 29
Reference [ 5

]

TrTM TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-19, NO. 4, NOVEMBER l970 A New Method of Characterizing Amplifier

Noise Performance

GLENN F. ENGEN

Abstract—Although the use of noise figure or noise temperature

to characterize amplifier performance is a well-established practice,

It is also recognized that this parameter provides only a partial description of the amplifier noise properties. In general, the noise

figure (or temperature) depends upon the generator impedance and

Is thus a function of the signal-source and amplifier combination.

Typically, the noise figure is measured by the Y-factor method

using hot and cold noise sources that are nominally matched (reflectionless). The result of this measurement is of value as a figure of merit; however, if optimum performance is to be realized, the appli- cations engineer must know whether to adjust the signal source im- pedance for maximum power transfer, minimum noise figure, or some other criterion, and he must know the deterioration in perfor- mance that results if this is not done. It is the purpose of this paper to present an alternative method of characterizing amplifier noise per- formance in terms of parameters that provide ready answers to these

questions. In addition, the measurement of these parameters via a simple extension of the Y-factor method will be described.

I. Introduction ALTHOUGH the use of noise figure or noise temperature to describe amplifier performance is a well-established practice, it is also recognized

that this parameter provides only a partial description

of the amplifier noise properties. More recently, the use

of noise measure [1] has been proposed, but again, this provides only a partial description. As usually defined [2], the noise figure depends upon the generator impedance and thus is a function of the signal-source and amplifier combination.

. If optimum use is to be made of a given amplifier,

its performance characteristics must be known in sufficient

detail, to predict its operation in arbitrary systems, and

the options available in attempting to optimize the

overall system performance. It is quite possible, for example, for two amplifiers to hi.ve the same noise

figures and yet have markedly different sensitivities to changes in source impedance. In order to account for

this and other phenomena, it is necessary to begin with

a "complete" description. The remaining problem is one of putting this description in the most convenient

form. This is certainly not to suggest that this point of view

is new, and indeed a number of useful descriptions have

been given (for example [3], [4]). It is the purpose of

this paper to put this description in an alternative form

that provides additional insights for certain problem

areas.

II. General Theory

Undoubtedly it would be possible to obtain the material

that follows as an extension or reinterpretation of a number of earlier results. It will prove more instructive, however, to give its derivation from an elementary

model; this provides the additional benefit that the

presentation is "self-contained." The model chosen for the amplifier is a two-port

that is linear and active (both in that it provides gain and includes noise sources). This device, shown in Fig. 1,

is conveniently described by the equations

61 = SllOl + <Sl2<*3 + &1.

62 = S2iai + S72<h + &2„

(1)

(2)

where 6i, 63 are the emergent wave amplitudes at ports

1 and 2, respectively, and au a? are the corresponding

incident waves. The S„„ are the scattering coefficients of the two-port. Finally, the internal noise sources are

represented by the terms 6 ln , 62 „, such that the power

spectral density (per cycle of bandwidth) delivered to

passive and matched (reflcctionless) terminations on

ports 1 and 2 is given by |&i„|

2 and |&j„|

r

, respectively.

It will prove convenient to express i 3 „ au~the sum of two components: the first aft,, is completely correlated with & ln, the second bi0 is uncorrected. Thus,

bin — a&i. + b3 (3)

In keeping with the usual design objectives, it will be assumed that |Si 2 | is negligibly small. In addition, it

is also convenient1 for the present discussion to assume S„ = 0. The boundary condition imposed on b^ and at by the

generator is conveniently expressed in the form:

a, = 6„ + 6,r„. (4)

Manuscript received June 4, 1970. The author is with the National Bureau of Standards, Boulder,

Colo.

1 If Sn = 0, the term Sn is only of interest in attempting to optimize the coupling between the amplifier and the system that

follows it.

115
ENOEN: CHARACTERIZING AMPLIFIER NOISE PERFORMANCE 345

ro ! sn

b, o,

5, 1 r<

I "r

IB

Fig. 1. Basic circuit for discussion of amplifier noise.

Here, 6, is the source term and r, is the generator re- flection coefficient. In order to examine the dependence

of the amplifier noise upon the source impedance, it is convenient initially to assume that be = 0. These equations and assumed conditions may now

be combined to yield the following expression for the

noise power delivered to a (matched) load at the amplifier output terminals:

16,1' <* + i - saf,\ I6..I' + |6,.r*. (5)

This result explicitly displays the dependence upon

source impedance V, and this dependence is a rather complicated one. A substantial simplification in this expression may be

effected by introducing "terminal invariant" parameters. (Tor an earlier discussion see [5].) The desired trans- formation begins with the substitution^

rj = (r. - S*)/(i - Sur.) (6)

so that (5) can be written3

T. = |&»|' = 277(1 + b |rj - 0fj (7).

where T„ is the noise power output (to the matched load),

G =

i«..r = amplifier power gain,

T. = i>..r G

minimum amplifier noise temperature (as a function of source impedance) referred to amplifier input,

, 1_ / |6,.|* \ _ ratio of the noise temperature of T. VI - |S„|V theinpu it port* to T.,

P = -s?i gfl - |M). /Sji The physical significance of may be interpreted as expressing to what extent the conditions for maximum

power transfer and minimum amplifier noise differ. In (7), the parameters T„ G, b, and |/3| are dependent on the amplifier properties only, and moreover their values are "terminal invariant" (i.e., their values do

not change if a "lossless" tuner is added to the amplifier

input). The entire dependence upon the generator im- pedance is found in the term TJ.

III. Evaluation of Parameters The terms Ta , b, and |/S| may be evaluated by an

extension of the F-factor method.

Let a temperature T, be assigned to the generator of

Fig. 1. Then (7) becomes5

t. = T,G(i - |r;|

2

) + TjG{\ + b |r; f). (8)

As in the ^-factor method, the measurement procedure

calls for the use of "hot" and "cold" loads (noise sources). These will be designated T,h , T,„ respectively, and

?"•», T.c the corresponding outputs when the noise sources are connected to the amplifier input. As a further condition it is assumed that the impedances of these loads

is such that T'a = (see the Appendix). After the observation of T,K and Tac , a sliding short is connected to the amplifier input and the maximum and minimum

values of T„ observed in response to motion of the short. These will de designated T, u and T.m, respectively. These four observations provide a set of simultaneous

equations, which can be solved to yield

101 = v.* - r.J(r.» - t„)

2|(r„„ + T„ - 2T.t)(T,k - TJ + 2T«(T.> - T.c )\

* This transformation has the following properties: 1) TQ

' —• when the conditions for maximum power transfer are satisfied (rfl = Sii*), 2) jr/l = 1 when |r„| = 1, 3) the value of !l7! is "invariant" to the choice of terminal surface between generator and

amplifier, provided th:it this choice is limited to a lossless region. For a more complete discussion, see |G|.

1 Although the functional form of this equation is equivalent to that obtained by a number of earlier authors, it is important to tecognize the following distinctions. I) The emphasis is on noise temperatures instead of noi.se figures. 2) In addition Ut "characterizing the rapidity with which Ta increases ab^ve Ta as r o

' departs from 0," the term 6 is also the ratio between the noise temperature of the input port and T . '.i) The complete dependence upon source impedance is contained in the single term IV- A) The

behavior with either power matching or noise matching is im- tnedu t'v evident from inspection.

T.b

T,

(T.„ - r.„)(r„ - t.<) 4 |/3 1 (r.„ - TJ

T., - {TJT„)T„ _ , , {TJT„) - 1 |P| T£.

0)

(10)

(ID

4 This is the available noise power (per cycle of bandwidth)

from the input terminals expressed as a temperature.

6 In obtaining this result, it is helpful to note that

(i - |r,'|»)

(i ir.i'Xi - is„p

|i -s„r,|=

116
IEEE TRANSACTIONS ON INSTEUMENTATION AND MEASUREMENT, NOVEMBER 1970 The last two equations nan also be solved explicitly

for T„ and 6 :n terras of the observed quantities; however,

the suggested computation follows the indicated stepj. [Note that (10) requires the value of \ff\, and (11) the value of both |/3| and 2'„6.] Finally, 6 is determined by

taking the ratio of (10) and (11). It is of interest to note that the first term on the right in (11) is just the usual one for determining amplifier

noise temperature via the F-factor method. In addition

it may be recognized that these equations call (implicitly) only for ratios between the different output temperatures

(e.g., Tok /T.c , etc.). If the absolute value of these terms

is measured, it is also possible to obtain the amplifier

gain. The only remaining quantity to complete the description is the argument of 0. This may be obtained by

computing the argument of T't that provides r„„. This argument may, in turn, be obtained from (6) provided

that Sn and the corresponding argument of r, are known.

IV. Power Matching Versus Noise Matching

Assuming that the applications engineer wishes to obtain the best overall performance, (i.e., the maximum

signal-to-noise ratio at the amplifier output) it is necessary

to know whether to adjust the generator impedance

for maximum power transfer, minimum amplifier noise, or according to some other criterion, and the penalty

that results if this is not done.8 These topics will now

be investigated. Returning to Fig. 1, it will be assumed that the signal source is of variable impedance and that its available power includes a signal component S„ and a noise component kT,B. The signal power S at the amplifier output

is given by

S. = S,G(1 - \T'f) (12)

while the noise power output 7V„ is given by

N. = kT,BG(l - |rj|

2

) + kT.BG{\ +b\T',- /S| 2

). (13)

The signal-to-noise ratio at the amplifier output is &

N kT,B(l

s.(i - lr;i

2

) |r;i

2 ) + fcr.S(i + b |r; - /3|

2

) (14)

By inspection, for maximum signal-to-noise ratio, the argument of T'a should equal that of jS. The remaining

problem is that of maximizing

(i - |r;!

2 )/[i + 6(|r;| - |/3|)

2

].

8 The point of view adopted here is that there are many situations

in which thfi applications engineer might hi willing to adjust the

impoditr.ee inarching bar. unwilling to tackle the more ambitious problem of also adding inverse feedback. For a discussion of this more general problrm, see (H, see also, J. S. Engberc, "Simultaneous

input power match and noise optimization using feedback," M.S.

thesis, Syracuse Universltv, Syracuse, N. Y., July 1960.

By use of the calculus, this maximum occurs for

|r;| = (\/D)[i - Vi - d'} (is)

where D = (26 |0|)/[1 + 6(1 + W)}. (16)

It can be shown that D < 1 and (15) can be expanded

to yield

|r:i = AM.

1 + 6(1 + |/

(6 M)1

[1 + 6(1 + \fi\>)]> + •I (17)

These results can now be substituted back into (14) to obtain an expression for the optimum signal-to-noise

ratio. Because of its complexity, this result will not be

given. If |/3| is small, however, the following approximation

is useful:

S.N.S. kB[T, + r.(l + 6 |017(1 + 6))] (18)

It is of interest to evaluate the potential improvement

over matching oa a maximum power transfer or minimum

noise basis. The required expressions may be obtained,

in turn, from (11) by letting rj = and r; = 0. If the

subscripts p (for power) and n (for noise) are used to

identify these two cases, a little manipulation leads to the following approximate expressions:

and

(SJN,)\ (SJN.)\m

(S./N.)\. (S„/iV„)L

It is worth noting that the potential improvement is small if |/3| is small, but increases rapidly as |0| increases.

V. .Experimental Results

In an effort to explore the method further, the measure- ment procedures described in Section III were applied

to a number of different amplifiers. The first item to be evaluated was an X-band tunnel- diode amplifier. In this case the measurement was straight- forward and yielded the results: \&\ = 0.03, 6 = 0.35,

!T„ = 825 K. It should be noted that if the circulator, which is required in such devices, is "ideal," this would

lead to |/3| = and bT„ = ambient temperature. The

above results agree with the theoretical expectations. The evaluation of an A'-band crystal mixer gave

|/5| = 0.13, 6 = 0.65, T„ = 496 K. Again the results appear plausible enough; it may be noted that 67", is only slightly above ambient while the value for T, is

117
ngen: characterizing amplifier noise performance 347

half the usually quoted value since the measurement method utilized both the signal and image channels. In spite of this plausibility, however, these results are subject to some question in that a substantial shift in crystal bias was observed in response to the short motion. The source of this problem was due to a substantial coupling of local-oscillator power into the input arm, and when this was reflected back into the system, a

shift in bias resulted. This, in turn, gives rise to some

question as to whether or not the device was behaving

in a linear manner. An application of the method to a 30-MHz vacuumtube amplifier gave |/3| = 0.22, b = 0.59, T„ = 161 K.

Initially these results were believed to be in error in that

this implies an input-port temperature of 94 K. Some

further effort was expended in attempting to refine the impedance matching, but this proved to be a difficult broad-band problem in that the bandwidth of the dewith "available" noise power. If the second interpretation

is used, no change in (7) is required. With regard to the <S, 2 = postulate, it can be shown

that the functional dependence of output power upon

TJ is still in the form of (7), even when S13 9^ 0. There

is, however, a dependence of the different parameters

(including T',) upon IV

Finally, it is possible to prescribe a more general method for evaluating Ta , b, and |/3| than that outlined

in Section III. Returning to Fig. 1, it is convenient to assume for the generator a noise source of temperature T„ whose impedance can be adjusted in such a way

as to vary the argument of T't while keeping its magnitude

constant. Inspection of (S) shows that the first term on the right will remain constant while the second will go through maxima and minima. An observation of this response can be substituted for the sliding-short operation

described in Section III. The appropriate equations are w

\t:\ (t.„ - r.„)(r. t - t„)

2((r.„ + r.„ - 2r„)(r„t - r„) + 2[t„ - t..(i - \v'.\

2)](T^ - t.,)\

tection system was a significant fraction of the operating

frequency. An indirect confirmation of this result was

finally obtained by an evaluation of the input-port temperature using a narrow-bandwidth radiometer. This gave the value 120 K, which may be regarded as a "confirmation" even if the agreement is not very good. It is unfortunate that it has not, as yet, been possible to repeat the entire evaluation with this narrow-band

radiometer. Although these results represent only a cursory eval- uation of the method, they do call attention to certain

practical problems that must be considered in its application.

VI. Extension to More General

Operating Conditions

In the preceding analysis, a number of simplifying approximations were introduced. These will now be examined in greater detail. The theory can be easily generalized to account for arbitrary impedance conditions (£22 r 0, etc.) at the output port. In order to retain the terminal-invariant

character of G, it is redefined as the ratio of the net power delivered (to the amplifier input) to the available power at the output port. (That is, the definition that

follows (7) should contain the factor (1 - l^iV)-

The right-hand side of (12) and (13) should be multiplied by (1 — |r;|

2 ) where

in I st

|i - Sn r,\ (21)

and T, is the reflection coefficient of the terminating

load. The same is true of (7), provided that T„ is as- sociated with the "delivered" noise power as contrasted

T.b =

T. =

(T.„ - T,„)(T, h - T„)

4 | r; I |/3 1 (T.h - T„)

T. h - (T.JT.JT,. (T.JT„) - 1 W T.b.

(22)

(23)

(24) By a proper choice of \T'S

\ it should be possible to avoid

the nonlinear operating region noted earlier. On the other hand, the realization of the required noise source,

for which the argument of TJ is adjustable without changing its magnitude, is somewhat more of a problem. One possible solution is to add a tuning transformer

to the amplifier input terminals and adjust for an im- pedance match. To the extent that it may be considered

lossless, the presence of this transformer will not affect the values of T„ b, and |/3|. For the tuned amplifier, Sn = and V, = r„. An adjustable generator (at ambient

temperature) is now conveniently provided by a matched

attenuator followed by a sliding short. At this point it is possible to consider the tuner either

as part of the amplifier or part of the adjustable noise generator. The former interpretation is convenient when the tuner losses are negligible since the resulting impedance requirement on the "hot" and "cold" loads

(required to complete the evaluation of T„ b, |j}|) is now that these are also matched. In the projected evaluation of cryogenic systems, however, these losses may not be negligible and further study is indicated.

VII. Summary An alternative method of characterizing amplifier noise has been presented, which (assuming linearity) completely accounts for variations in source impedance

and in which all the parameters have a simple physical meaning. With the exception of the argument of J, these

118
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, NOVEMBER 1970

parameters are "terminal invariant." In addition, methods

for measuring these terms have been described and an

analysis, leading to optimum source impedance, presented. The preliminary experimental evaluation suggests

that the methods should prove of value in system optimization, although confirmation of this awaits further work. In particular it appears that the method may be of

real value in the evaluation of parametric- and masertype amplifiers, since here there is some reason to anticipate larger values of b and |/3| than those reported

above. On the other hand, in the case of the parametric

amplifier at least, this expectation is seriously qualified by recognition that these devices typically exhibit a

substantial sensitivity to changes in source impedance, and the region of linear operation may be difficult to

establish. The possible use of a moving short alone for the eval- uation of maser noise has been previously suggested. From (7) it appears that it is indeed possible thus to determine an upper limit to Ta , provided that G is known. At the present, the most serious limitation to this description appears to be in the assumed amplifier

linearity (in respect to changes in source impedance,

not output versus input). However, if this method puts

this aspect in better focus, even this may prove to be a

contribution.

Appendix The measurement procedure outlined in Section III

calls for the adjustment of the hot and cold load im- pedances to equal the complex conjugate of the amplifier input reflection coefficient r„. In principle, this can be

effected by measuring this impedance (by use of a slotted

line, for example) and then using the impedance meter

to recognize when the hot and cold load impedances

are the conjugate of this value. An alternative procedure

follows. Referring to Fig. 2 it is convenient to assume a system

comprised of a directional coupler, tuning transformers (T„ Ty ), load isolators, generator, and detector as shown.

Scattering equations can be written for this system as

follows:

6, = S„a, + S„a, + S13 a3 (25)

62 = S21 a, + »S22 a2 + S23 a3

. (26)

The boundary conditions imposed by the different terminations are

a, = fc,rd

Oi = &2 Tj

a3 = b, + b3 T,.

(27)

(28)

(29)

€H^EH—

Fig. 2. Measurement system, for recognizing conjugate impedance

match.

either rd or Tg

; thus there is no loss in generality

7 in assuming Td = V = 0. Combining these results with (25) and (26) leads to

&i = K

(iS 12 02 £13022)!^ ~f~ Q 13 1 - s22 r.

(30)

Next it is assumed that Tx has been adjusted such that

6, vanishes when port 2 is terminated by the load of

interest (r, = r„), and T, has been adjusted such that

|6,| is constant in response to a moving short at port 2. The conditions imposed on the scattering parameters by these adjustments are, respectively,

and*

S„ + (S, 2 S:3 - S13 S12)V. =

"13 "T" (Sl2 S23 — <S l »lS22)lS*2 —

i° By inspection, s„ = r*.

Equation (26) may now be written

bt = S23 6„ + T*a,

(3D

(32)

(33)

(34)

In the operation to follow, there is no occasion to change

or, stated in words, the equivalent generator that obtains

at port 2 has the source reflection coefficient r*. To

obtain a termination with this reflection, it is only neces- sary to replace the generator on arm 3 by a passive load

of the same impedance. The purpose of the isolator on arm 3 is to minimize the requirement that this im- pedance be the same as for the generator and to avoid

possible pulling effects with the short motion. The isolator on arm 1 avoids the requirement of a constant detector impedance with changes in power level. Finally, the adjustment, of other impedances (e.g., the hot and cold loads) to equal this value may be effected by anjr of the procedures for recognizing the equality

of two impedances.

7 For a discussion of this point, see [7].

8 See, for example, [6].

119
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-19, NO. 4, NOVEMBER 1970 349 Acknowledgment

The author expresses his appreciation to O. L. Patty and M. G. Arthur for providing the experimental results described. References

[1] H. A. Haus and R. B. Adler, Circuit Theory of Linear Noisy

Networks. New York: Wiley, 1959.

[2] "IRE Standards on electron tubes: Definitions of terms, 1957"

(57 IRE 7.S2), Proc. IRE, vol. 43, pp. 933-1010, Julv 1957.

[3] H. Rothe and W. Dahlke, "Theorv of noisy fourpoles,1

' Proc. IRE, vol. 44, pp. 811-818, June 1956.

[4j H. A. Hans, W. R. Atkinson. G. M. Branch, W. B. Davenport,

Jr., W. H. Fonger, W. A. Harris, 8. W. Harrison, W. W. McLeod.

E. K. Stodola, and T. E. Talpey, "Representation of noise in linear twoports," Proc. IRE, vol. 43, pp. 69-74, January I960.

[5] J. Lange, "Noise characterization of linear twoDorts in terms of invariant parameters," IEEE J. Solid-State Circuits, vol. SC-2,

pp. 37^40, June 1967.

[6] G. F. Engen, "An introduction to the description and evaluation

of microwave systems using terminal invariant parameters," NB3 Mono. 112.

[7] G. F. Engen, "A variable impedance power meter and adjustable

reflection coefficient standard," J. Res. Nat. Bur. Stand., Sect. C,

vol. 68, pp. 7-24, January-March 1964.

Reprinted bv permission from IEEE TRANSACTIONS ON

INSTRUMENTATION AND MEASUREMENT

Vol. IM-19, No. 4, November 1970

Copyright <© 1970, by the Institute of Electrical an.] Electronics Engineers Inc PRINTED IN THE U.S.A. NRC 11S48

120
Reference [8]

Thermal Noise from a Passive Linear Multiport

DAVID F. WAIT

Abstract—The thermal noise from passive multiports is discussed from fundamentals so that it can be understood, measured, or calculated by a microwave engineer. The multiports are assumed to have a uniform

temperature, but with no restrictions on reciprocity or mismatch. The

noise temperature, TV, contributed by such a multiport is 7\ = AT where T is its physical temperature and A is its absorption coefficient. An approximate method of measuring A, and a method of measuring an A as small as 0.008 within 5 percent, are pointed out. Also, exact and approximate

expressions for A in terms of scattering matrix elements and termination

reflection coefficients are derived. Finally, the crosscorrelation of the noise from different ports is briefly considered. M:

I. Introduction ORE THAN 40 percent of the noise in low-noise communication systems and 95 percent of the noise in low-noise radio astronomy receivers is due

to "black body" radiation from passive linear elements.

Likewise, this type of noise accounts for most of the noise

in a Blum radiometer [I

J, [2] used at the National Bureau of Standards to compare cryogenic noise sources [3]. The literature on the subject of noise is very extensive. Some of it applies to the problem of describing the noise from multiports like those used in modern low-noise communications and measuring systems [4]-[7]. But if one

wants to determine how much of the noise power a three- port switchable circulator,' or a four-port hybrid tee, or

other multiport element contributes to the system noise, the

literature is sometimes inconvenient to use, especially if the elements used to terminate the multiport are mismatched.

This inconvenience may be due to unfamiliar mathematical

language, but more likely, it is because a conversion is re- quired to reexpress the results in more useful terms. What is useful depends on the details of the system involved. Thus, a

laboratory measurement of the scattering matrix parameters might prove helpful in estimating the noise contributed from

a circulator, but relatively useless in estimating the noise contribution from a hybrid tee. This paper will discuss the thermal noise from passive

linear multiports in various terms that might be useful to a microwave engineer. The results are derived from funda- mentals to make them more understandable. An exception

is the discussion of the correlation of noise from dill'erent ports where the results can be obtained from the literature with little difficulty. Since a tractable extension to mis- matched multiports is not known, this is not belabored.

Manuscript received Ft.bru.uy 2, 196S; revised May n. Ilffi. This re- search was supported by the Advanced Research Projecls Agency of

the Department of Helcnsc. under I'rojccl 1)1 I T.NDI.R. The author is with the National Uureau of Standards. Boulder, Colo. S0302

1 The elFect of the time dependence of the noise on system sensitivity

is treated by [8 J.

In Section Injunctions with uniform physical temperature

are considered. For them, the noise contribution is directly proportional to their physical temperature. The constant of

proportionality is referred to as the absorption coefficient which may either be measured (Section III) or calculated (Section IV). The calculated form is stated in terms of scatter- ing matrix elements and reflection coefficients of the port

terminations. In Section V, the correlation of noise from

different ports is discussed. Multiports with almost reflec- tionless terminations are treated in Section VI. In Section

VII, the absorption coefficient is related to loss concepts known for two-ports. Conclusions are given in Section VIII.

II. The Noise Temperature of a Passive Multiport

Neglecting quantum effects, the power P available in each waveguide mode from a thermal source at a uniform temperature T is P=k TB. where k is Boltzman's constant and B is the bandwidth. Thus, the available power per unit bandwidth (available power spectral density) may be expressed as a temperature, and conversely, a temperature can be used to express an available power spectral density. In a somewhat

similar way the available noise from a "passive"' multiport is described. Specifically, if ail ports of a multiport are termi- nated with loads at the absolute zero of temperature, while

the multiport itself is at a different temperature, then the radiation emerging from any port is due only to thermal

sources within the multiport. The acaitable noise power

spectral density- at, say. port i expressed as a temperature is known as the noise temperature of the multiport at port /. .Thus defined, the value of the noise temperature depends on

which port is being considered and also upon the reflection coefficients of the other port terminations, i.e., the reflection coefficient "'looking*' out of the multiport. A multiport with independent port generators is shown in Fig. I. Because the sources are independent (thus uncorre- cted) they each make an additive contribution to the total available output of port j, so £ aijTj + r.v., (i)

where T, is the available power spectral density of the y'th generator at its own output port and TV , is the remaining

contribution due to the thermal sources within the- multiport

itself. The proportionality factor a u

is the ratio of the power

spectral density available out of port i to the power spectral density available from the generator on port_/ when only the

2 The available noise power spectral density is detined as the maxi- mum noise power spectral density that could be delivered to a toad. The circuit will deliver the maximum power when the load has the proper rejection coelticient. No manipulation of the rellection eoelli- cient of any of the port generators is implied.

121
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, SEPTEMBER 1968 O

s it,)

T, r. ©

1

j Tj

r, T

T(

Fig. 1. A multiporl with independent thermal noise generators.

generator^' has nonzero amplitude. The proportionality fac- tors ay depend on the reflection coefficients of the other port

terminations. It should be intuitively obvious that a„ is also equal to the ratio of available power out of port /to the avail- able power from the generator on porty when a CW source

is used on porty. 3 Thus a,; will be referred to as the available power ratio. The noise temperature of the multiport TXni is a characteristic of the multiport and the reflection coefficients of its terminations but independent of the amplitudes of the port generators. To determine 7",v.„ consider the multiport when it is in thermal equilibrium with all its port generators at temperature T. Then from the second law of thermodynamics,

7"totai.. as well as all the T„'s must equal T. Thus

T,v., = A,T

where

A,=-{'-£4-

(2)

(3)

The constant A, is here called the absorption coefficient for port i of the multiport. For a two-port. (1) reduces to the form given by Daywitt [10]: 7".ot3l = a7s+(l— a)T, where a

is the ratio of available power at the output to the available power at the input, and Ts is the temperature of the source.

HI. Measurement of the Absorption Coefficient The losses within a junction can change the power avail- able at the output port, and the amount of the change can be

used to determine the absorption loss. For example, if the junction is lossless and each port of the junction is termi- nated by a noise generator with the noise temperature T„. then the output from the junction 7", ora i., would be T . But

if the junction has a nonzero absorption coefficient Ai, then from Section II the difference between T„ and 7"tma i., is

where Tis the physical temperature of the junction. Thus to measure A„ one needs to adjust a set of noise generators4 to have the same known noise temperature T„, attach them to the ports of the junction to be measured, and then measure

the available power spectral density [11], TM*\,i from the output port. Because the absorption coefficient depends upon

the reflection coefficients of the port terminations, the noise generators used in the measurements must have the appropriate reflection coefficients.

If the reflection coefficients are less than one tenth, then

the small difference of (Ta — rtota i,) can be measured within 2°K when T„ is 10 800°K [8] and (Ta -T) can be known to

2.5 percent [12]. Thus A, as small as 0.008 (a two-port loss of 0.03 dB) may be measured within 5 percent. In case only an approximate measure of absorption coeffi- cient is needed, a simpler measurement will suffice. For ex- ample, consider the problem of determining the approxi- mate thermal contribution of a two-port placed between an

antenna and a low-noise receiver as shown in Fig. 2. The

two-port might be a directional coupler terminated with a gas tube which, when fired, adds a known amount of noise

to the system for calibration purposes. When the tube is off, the terminated directional coupler may be treated as a twoport at uniform temperature. For small Ai and 7" »r, (4) becomes A.~0.230APdB (l + T/T„) (5)

T„ = A.(T, - T) (4)

where iPdB is the relative change in available power ex- pressed in decibels when the two-port is inserted into the system. Ignoring the receiver noise, APd B can be determined by measuring the relative change in power when a noise source

feeds the receiver directly (from reference plane 1 in Fig. 2) compared with when a source with the same noise temperature feeds the receiver through the two-port (from reference plane 2). The noise sources used must have the same reflec- tion coefficients, amplitude, and phase as the circuit they

replace 5 (f i and I\, respectively) so that the same fraction of the available power is delivered to the receiver for both

parts of the measurement. The precision of measuring 1P.:b depends on the precision of the readout of the receiver, on the stability and resolution of the receiver, on the noise temperature of the receiver, and on the relative accuracy of the noise sources. If a single noise source with adjustable reflection coefficient, like that shown

in Fig. 3, is used for both parts of the measurement, the change in noise temperature due to the change in tuner losses

is typically less than 0.15 percent at A"-band for reflection coefficients in the range of zero to 0.3 [14]. The error due to neglecting the receiver noise is on the order of (.4, Tr„/T„), where rrec is the receiver noise temperature.

1 The rigorous mathematical proof that the value a,-> is the same for noise as the CW signals is beyond the scope of this paper. The foundations for such a proof are in Middleton [')], however, the route to the proof may be more obvious from the appendices of the paper by the author [8J.

1 An "internal" attenuator is the usual way to modify the noise temperature of a noise source.

1 A simple procedure for adiusting the reflection coefficient of one

circuit to be equal in amplitude and phase to another is described by Engen [13].

122
WAIT: THERMAL NOISE FROM A PASSIVE LINEAR MULTIPORT 689

Fig. 2. A receiving system with a two-port between

the antenna and the low-noise receiver.

^=^

CIS OISCWKCE nose sounce

Fig. 3. An adjustable reflection coefficient noise source. If necessary, the noise temperature of the source is reduced by the level set attenu- ator so that the source does not saturate the receiving system.

IV. Calculation of the Absorption Coefficient At times it is useful to know the relationship between the absorption coefficient and the parameters that describe the multiport and its terminations. To do this, the available power ratios a,, are first expressed in terms of the scattering matrix elements of the multiport and the reflection coeffi- cients of the port terminations, and then the absorption coefficient is obtained using (3). The available power ratio will be calculated as the ratio of

the CW power available at port i, /

>b,

ivai ', to the CW power

available from the generator on port/, /

> a/""1 ' 1

.

P*r"'/Pa. (6)

The power available from port i expressed in terms of the equivalent "generator" wave6 amplitude 6, is [16] iV = fcVU r«h (7)

where f , is the reflection coefficient "looking" into port ;'. Similarly, the power available from the generator in tern" -> r

its "generator" wave amplitude a, is Nva- |r,h (8)

where I\ is the reflection coefficient of the generator. The

relationship between <3, and a, can be established by matrix

techniques (see the Appendix). A compact and convenient

expression of this relationship is

S,= \D liSi) /D ( (9)

where Dusfi is the determinant of the matrix (1 — Sr) when

its ith column is replaced by the /th column of scattering matrix S, r is the diagonal matrix with elements (T),, equal

to the reflection coefficient of the load on port /. and Duo is the determinant of (1 — Sr) when its ith row and column are

• The "generator" wave amplitude is denned as the amplitude the generator would deliver to a reflectionless load [15], [16]. In this paper, a basis has been chosen so that the characteristic impedance Z. }i associ- ated with a port / will not be expressed explicitly. For those who wish an 'explicit statement, each voltage such as m should be replaced by

at /^/2ot , and each scattering matrix element S,, should be replaced by

deleted. Thus for is*/(1- | rj|») |Z>r.-«, " (1- if.

I

2

) \D,

so that At-l-Z (l- |r*h|z> (

.- s S (i- |r.h|D,.o|'

For example, for the three-port, 7

\Sij —SxkTk I \sts (i-s«r*)l

(1 - SuTi) -SitTt I -skj v,- (i-Swrol

and

\Sa -5„r, -S*Tt f, = [/)„„]-' |5„ (l-5„r; ) -SjtTt

(10)

(11)

(12)

(13)

(H)

V. Crosscorrelations for the Matched Multiport

In some applications it is interesting to know if the internal junction noise emitted from one port is crosscorrelated to the noise emitted at another port. A measure of crosscorrelation is the time average of the product of the time dependent voltages. However, the analysis of this paper is entirely

in the frequency domain. In the frequency domain, the Fourier transform of the crosscorrelation is appropriate, namely, the cross-spectral density (•K,, [17]. Wa= *{(n,V)av + (n*rij)* (15)

where ( )., denotes the ensemble average and the asterisk denotes the complex conjugate. If//, and //> are the voltages front port :' and port/, respectively, of the generator waves

due to the noise sources within a matched multiport, then

[5] <n,V>.v = NUT (16)

where T is the physical temperature of the multiport and A

r

,v

is an element of N= 1 - SSj (17)

where the dagger indicates the Hermitian conjugate (trans- pose, complex conjugate). Bosma [5] refers to TV as the noise

distribution matrix. For the matched multiport, a "gener- ator" wave is the same as the emergent wave. The special case of /=/ may also be obtained from (2) and

(11) by setting all T's equal to zero, that is

(n,«.*>.v = (l-|f,|=)r,v.,= {l -Zw.j T. (18)

' When actually writing out Du s/). one should take advantage ofthe

properly that Dasfl is unchanged when the y'th row and column of the determinant are deleted. This property is expressed by D,,s,i = DdSiiu/).

123
690 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, SEPTEMBER 1968

An interesting special case is the two-port

(nxn,*),T = - (SiiS21* + SI2S2 o*)2'. (19)

Thus, if the two-port is matched to rellectionless loads so

that Sn and S22 are zero, then there is no correlation 5 in the thermal noise emitted out the two ports.

VI. Multiports with Low-Reflection Port Terminations

In many situations the port terminations have small reflec- tion coefficients. For this situation, the results of Section 111 simplify. In particular, neglecting the squares of the reflec- tion coefficients [15],

D(iSj)/D(.ii) — Sij + 2-i Sm T„S„ (20)

If in addition |f,-| 2 is negligible, then the absorption coeffi- cient simplifies to Ai^Af+ti (21)

where A? is the absorption coefficient of the junction when

all T's = and

22 (Sim rmN„, + s,m*ra*Nmi*) (22)

where Nml are defined in (17). If the elements Nmi are not known, then it is of interest to determine an upper limit for the magnitude of the N,fs. From the definition of Nt, in (17)

it is clear that

/V- = ;V--

: (23)

Because N=(l — SSt) is a positive semidefinite matrix for a

passive multiport [5], [16], then [18] \Nv \*^Ntjri,. (24)

This in turn implies Na + Nn < Nu + Njs. (25) The diagonal elements of N are directly related to the absorption coefficients of the multiport, viz., Nu = (1 - IS,,!'-).!, (20) where A," is defined in (21). Thus, because 2Zm |Sim j 2 <l

and all |r„| 2 and A," are less than some maximums |rmax | and Amax ", then

I €i| < 2(n - l)Um«°| rm«| (27) where n is the number of ports.

VII. The Aksorption Coefficient as a Loss Many types of losses have been distinguished in the

literature. Among those described by Bculty [19], the absorption coefficient is related most nearly to dissipative

3 One should remember that zero correlation does not imply that the signals arc unrelated. For example, lor the related signals iii = v+r

and n~ = x —v, where x and v are uncorrelated, C/'i"-.«v=-''.v.v* vav- ~~ fyy*'** ' s zero when the power spectral density in v equals v. II' the

spectral densities of x and v are equal at all frequencies, then n t and /rj are uncorrelated. This principle is used in the IJIum radiometer [1 ], [2].

loss Ld= — 10 logic n (in decibels), where v is the efficiency. The absorption coefficient expressed in decibels, i.e., absorptive loss, would be (/4,-,)dB= — 10 logn a,-,-. Efficiency is a

ratio of delivered powers in the same way that a,, is the ratio of available powers. The efficiency and the available power

ratio are related in the following way.

aii = (Mj/Mdn (28) where M, and M, are the mismatch factors of the /th andy'th

ports, respectively.* Thus, if both ports are conjugate matched [16], dissipative loss and absorptive loss are indis- tinguishable. Otherwise, dissipative loss is a loss that de- pends on the output load impedance, but independent of the input generator impedance, while absorptive loss is a loss that depends on the input generator impedance, but is inde- pendent of the output load impedance.

VIII. Conclusions The noise from the thermal sources within a multiport ele- ment is equal to the absorption coefficient of the element, times the physical temperature of the element. The absorption coefficient may be measured or calculated. The calculated absorption coefficient is simply the sum of the avail- able power ratios from all ports to the output port. In turn, the available power ratios are expressible in terms of scat- tering matrix elements and reflection coefficients. Only

linearity was assumed; no restrictions on reciprocity, symmetry, loss, or condition of match are needed. The correlation of the junction noise emitted from differ- ent ports when the junction has refleetionless terminations depends on the appropriate matrix elements of N=l — SSt

and the junction temperature. When all ports of the junction have nearly refleetionless terminations, the expression

for the absorption coefficient greatly simplifies, and a limit of error can be established if the port reflections are merely

ignored. Appendix

Equivalent Wave Generators The mathematics and the notation in this appendix have

been influenced by N'emoto. Further details and interpreta- tion of the concepts introduced here are discussed else- where [15]. A multiport is described by a scattering matrix that relates the emersinu waves b, to the incident waves a,-. b =Sa. (29) The boundary conditions imposed by terminating each port with a load or a generator is described bv a + rb (30) where d, is a generator wave, that is, a wave a generator would deliver to a refleetionless load, and r is a diagonal matrix whose diagonal elements are the reflection coefficients

9 Mismatch factor is used here in accordance with Miller el nt. [10],

A/, -(I — ! r,l-)(l-| P.;-)/ j l-r,f',|-, where 1\ is the reflection coelli- cient "looking" out of the multiport and 1\ is the rellection coellicient "looking" into the multiport at reference plane /'.

124
WAIT: THERMAL NOISE FROM A PASSIVE LLNEAR MULTIPORT

looking out of the multiport. The generator wave a, characterizes the generator on port i in the sense that its amplitude depends only on the output of the generator and that it is independent of the load that terminates the generator. Elimi- nating a between (29) and (30),

Using (3S) we obtain

where

fa = Sa

Se(J- Sr)-'S.

(31)

(32)

Because a matrix element of a product C = A~'B, where A and B are square and have the same dimension, is C,, = det A(xBj, /det A, where det Auh,) is the determinant of A after the ith column of A has been replaced by they'th col- umn of B;

Sn = Diis»/D (33) where D stands for the det (1 — Sr).

In the same way that the inputs a, were each divided into a generator wave plus a reflected wave, the outputs 6, can

be divided into an equivalent generator wave plus a reflected wave.

fa = fa + ta (34)

where f is a diagonal matrix whose elements are the reflec- tion coefficients looking into the multiport. Eliminating fa from (31) using (30) and (34),

fa = Ga (oo)

where one form for G is G = |(1 - rf)s - rj. (36) From the nature of the definition of fa and a, the diagonal elements of G must be zero. Setting the diagonal elements of

(36) equal to zero and solving for f , we obtain

Pi = Dusa/Dwt (37)

where D<«) is the determinant of (/ — ST) after the j'th row

and /th column have been deleted. Again looking at a diagonal element of (36) and solving

for (1 — r,f ,), replacing the remaining l\ by (37), we obtain

(38)

Gn = D(lSi) /D( (40)

D = (i - r.-roD,w. An ofT diagonal element of (36) is G„ = (1 - TitdDusn/D. (39)

To obtain (9), i, is obtained from (40) and (35) for the case when only they'th generator, and thus only they'th generator wave dj is nonzero. Acknowledgment

The author is indebted to J. J. Fari?, W. Daywitt, G. F. Engen, and T. Nemoto for helpful discussions and to M. B. Hall for his support and encouragement.

References

[1] E. J. Blum. "Sensibilite des radioteUscopes and recepteurs a correlation," Ann. Asiwphys., vol. 22. pp. 140-163. 1959.

[2] J. J. Faris, "Sensivitiy of a correlation radiometer." J. Res. NBS.

vol. 71C, pp. 153-170. April-June 1967.

[3] D. F. Wait and C. L. Trembath. "A comparison between a switched and a correlation radiometer for cryogenic noise source measurements," 1965 Proc. 20th Ann. ISA Con/., pt. 1, vol. 20, no.

14.4-3-65.

[4] A. E. Siegman. "Thermal noise in microwave systems—Part I: Fundamentals," Microwave J., vol. 4, pp. 81-90, March 1961.

[5] H. Bosma, "On the theory of linear noisy systems," Philips Res.

Repis. Suppi, no. 10, 1967.

[6] W. R.Bennett, Electrical Noise. New York: McGraw-Hill, 1960.

[7] H. A. Has and R. B. Adler, Circuit Theory of Linear Noisy Networks. New York: Wiley, 1959.

[8] D. F. Wait, "Sensitivitv of the Dicke radiometer," J. Res. NBS,

vol. 71C, pp. 127-152,' April-June 1967.

[9] D. Middleton. An Introduction to Statistical Communication

Theory. New York: McGraw-Hill, 1960.

[10] C. K. S. Miller, W. C. Daywitt. and M. G. Arthur. "Noise stan- dards, measurements, and receiver noise definitions," Proc. IEEE

vol. 55, pp. 865-877, June 1967.

[11] D. F. Wait and T. Nemoto, "Measurement of the noise temper;

ture of a mismatched noise source," this issue, page 670.

[12] J. S. Wells, W. C. Daywitt, and C. K. S. Miller, "Measurement

effective temperatures of microwave noise sources," IEEE Tra

Instrumentation and Measurement, vol. IM-13, pp. 17-28, Mai

1964.

[13] G. F. Engen, "A variable impedance power meter, and adjust;

reflection coefficient standard," J. Res. NBS, vol. 68 C, pp. 7

1964.

[14] G. F. Engen, private communication.

[15] T. Nemoto and D. F. Wait, "Microwave circuit analysis the equivalent generator concept," lEEETrans. Microwave T

and Techniques {to be published).

[16] D. M. Kerns and R. W. Bcatty, Basic Theory of Waveguide

lions and Introductory Microwave Network Analysis. New

Pergamon, 1967. ("'Generator" waves are denoted as bo f discussed on p. 50.) [17] G. M. Korn and T. M. Korn, Mathematical Handbook fa

lists and Engineers. New York: McGraw-Hill, 1961, sec. [18] Ibid., sec. 13.5-6. [19] R. W. Beattv, "Insertion loss concepts," Proc. IEEE, vc 663-671, June 1964.

Reprinted from IEEE TRANSACTIONS ON MICROWAVE THEORY AM) TECHMQCES

Volume MTT-16, Number u, September, 1P6S

pp. 6S7-691 Copyright © 106S—Tin: Institute or Klhtuic.m. and Electronics Engineers, Inc. PRINTED IN THE U.S.A.

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4. TITLE AND SUBTITLE

Considerations for the Precise Measurement of

Amplifier Noise

5. Publication Date August 1973

6. Performing Organization Code

7. AUTHOR(S)

David F . Wait

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16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) For the best accuracy in measuring noise figure, attention needs to be

given to the choice of the hot and the cold noise standards and to mismatch

problems. Tables and graphs are presented to aid in choosing the proper

measurement conditions, and an example is given to demonstrate their use. This paper essentially supplements a previous paper (included in an appendix)

,

treating in more detail topics that become important when state-of-the-art

measurements are required.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Amplifier noise; effective input noise temperature; mismatch error; mismatch

uncertainty: noise figure

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