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NIST Technical Note 1530

NIST Technical Note 1530

UNCERTAINTY ANALYSIS FOR NIST NOISE

PARAMETER MEASUREMENTS

James Randa

Electromagnetics Division

Electronics and Electical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, CO 80305

March 2008

\ TTT7 k %

^:^i^\#

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

National Institute of Standards and Technology

James M. Turner, Deputy Director
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to describe an experimental procedure or concept adequately. Such identification is not intended

to imply recommendation or endorsement by the National Institute of Standards and Technology,

nor is it Intended to imply that the entities, materials, or equipment are necessarily the best

available for the purpose.

National Institute of Standards U.S. Government Printing Office For Sale by the

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Technical Note 1530 U.S. Government Printing Office

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Tech. Note 1530 Washington, DC 20402-0001

40 pages (July 2008) Phone: (202) 51 2-1 800

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CONTENTS

INTRODUCTION

Page

2. THEORETICAL FRAMEWORK 2

2.1 Noise Parameters in the Wave Representation 2

2.2 Relation to IEEE Noise Parameters 4

2.3 Physical Bounds 6

3. MEASUREMENT METHOD 6

4. UNCERTAINTIES I: OFF WAFER, COAXIAL REFERENCE PLANES 8

4.1 Type_A Uncertainties 8

4.2 TypeB Uncertainties 9

4.3 Input Uncertainties 1

1

4.4 Sample Results 13

5. UNCERTAINTIES II: ON WAFER MEASUREMENTS 15

5.1 Differences from the Coaxial Case 15

5.2 Input Uncertainties for On^Wafer Measurements 15

5.3 Uncertainties in On_Wafer Measurements 17

5 .4 Occurrence and Treatment of Unphysical and Other "Bad" Results 20

5.5 Sample Results 22

6. SUMMARY 22

7. REFERENCES 23

APPENDIX A: JACOBIAN MATRIX FORX-> IEEE 25

APPENDIX B: OVERVIEW OF THE FORTRAN PROGRAM 28

B.l MAIN 28

B.2 Subroutine SETMEAS 29

B.3 Subroutine TMEAS 30

B.4 Subroutine FULLFIT 30

B.5 Subroutine FWDFIT 31

B.6 Subroutine FCN 31

B.7 Subroutines GAUSDEV and RECTDEV 3

1

111

UNCERTAINTY ANALYSIS FOR NIST NOISE-PARAMETER

MEASUREMENTS

James Randa

Electromagnetics Division

National Institute of Standards and Technology

Boulder, CO 80305

The uncertainty analysis is presented for NIST measurements of noise parameters

of amplifiers and transistors, in both connectorized (coaxial) and on-wafer

environments. We treat both the X-parameters, which are based on the wave

representation of the noise correlation matrix, and the traditional IEEE noise

parameters. An overview of the Monte Carlo program used to evaluate the type-B

uncertainties is included.

Keywords: amplifier noise, measurement uncertainty, Monte Carlo, noise

measurement, noisefigure, noise parameters, transistor noise, uncertainty

1. INTRODUCTION

Some time ago, what is now the Electromagnetics Division of the National Institute of

Standards and Technology (NIST) developed the capability to measure noise parameters of

amplifiers [1,2]. Recently, modified methods and analysis have been developed and have been

applied both to amplifiers [3] and to transistors [4], the latter in an on-wafer environment. For

such measurements to be meaningful, the results must be accompanied by corresponding

uncertainties. Many complications arise in the analysis for these measurements, and it is the

purpose of this paper to document and perhaps clarify the procedures used to estimate the

standard uncertainties [5] in the measured noise parameters.

The contributions to the standard uncertainty are divided into two groups. Generally,

type-A uncertainties are those that are determined by statistical means, and type-B are all others.

The standard or combined uncertainty (//,) in a quantity is the root sum of squares (RSS) of the

type-A and type-B uncertainties.

Noise parameters are computed by performing a least-squares fit to an over-detennined system

of equations obtained by measuring the output noise temperature (or power) for each of a number of different input terminations connected to the amplifier or transistor under test. The

type-A uncertainties can be computed from the covariance matrix of the fitted parameters, but

the type-B uncertainties require more effort. The uncertainties in the undcrKing or input

quantities, such as reflection coefficients, measured noise temperatures, etc., are known or can be

estimated; but the problem of propagating these underlying uncertainties to compute

1
uncertainties in the output noise parameters does not admit a simple analytical solution.

Therefore, a Monte Carlo approach is used for the type-B uncertainties. Matters are further

complicated by the fact that it is convenient to perform our analysis in terms of one set of noise

parameters (what we call the X parameters), but we need to express our results and uncertainties

in terms of a different set (the IEEE noise parameters) that is in near-universal use.

In the next section we present the theoretical framework used in the analysis. Section 3

reviews the measurement methods, both on-wafer and off-wafer. The uncertainty analysis for the

off-wafer, connectorized case is presented in Section 4, followed by the on-wafer case in Section

5. Section 6 contains a summary. There are two appendices: Appendix A presents the Jacobian

matrix for the transfonnation from our A^ parameters to the IEEE noise parameters, and Appendix

B gives an overview of the Fortran program used to evaluate the type-B uncertainties.

2. THEORETICAL FRAMEWORK

2.1 Noise Parameters in the Wave Representation

Our general theoretical framework was set out in Reference [6]. It follows the wave

representation formulated by Wedge and Rutledge [7]. For a linear two-port, such as an amplifier

or transistor, the amplitudes of the incident and outgoing waves are related by

b^Sa + c, (2)

where a and b are two-dimensional vectors whose elements are the incident and outgoing wave

amplitudes, as indicated in Fig. 1, 5 is the usual 2x2 scattering matrix, and c is a twodimensional vector whose elements are amplitudes of the waves emerging from the two ports even in the absence of any input waves, due to the intrinsic noise of the device. Although this

paper deals with noise parameters of both transistors and amplifiers, the figures will be drawn

with only the amplifier symbol, or with a generic box, and we will use DUT (device under test) to refer to either an amplifier or a transistor. We use a normalization in which the wave

amplitudes squared have dimensions of spectral power density, power per frequency interval.

When we refer to noise temperature, we take it to mean the available noise spectral power

density divided by Boltzmann's constant kg.

The noise properties of the two-port are characterized by the (intrinsic) noise correlation

matrix A^, defined by

Nn^Uc)), (3)
b\ ax a2 bi

Figure 1 . Reference planes for amplifier noise parameters.

Fr A Fr n

Tg^^g Ta^Fr,

(a)

2

(b)

Figure 2. Forward (a) and reverse (b) measurement configurations.

where the brackets indicate time or ensemble average (assumed to be equal). We find it convenient to work with temperature variables corresponding to scaled elements of the noise

correlation matrix. We define

^5^1 ^(hP)' ^B^2^[[^2/S2i\j, kBXy2^{cx{c2lS2i1 (4)

Dividing each C2 by Six effectively refers all four noise parameters (^i, Xj, ReXu, ImXn) to the

input plane 1. In fact, it can (and will) be seen that X2 is equal to the effective input noise

temperature for a reflectionless input termination, Xi = TeS) , sometimes also called the "50 Q

noise temperature." Xi is the intrinsic noise emerging from the input port (1); it is closely related

to the Trev of Wait and Engen [1],

T^ rev X,

\-\s11

(5)

Two different measurement configurations are used in the NIST noise-parameter

measurements: the "forward" configuration of Fig. 2(a) and the "reverse" configuration of Fig.

2(b). The forward configuration is the one usually used for noise-parameter measurements, but

the reverse configuration can also be used, and it provides a very good determination of the

parameter X\. The reverse configuration is particularly usefiil in the on-wafer case for poorly

matched transistors [8]. The output noise temperature for the two configurations can be written

in ternis of the scattering and noise parameters of the amplifier and the reflection coefficient T,, and noise temperature To of the source or generator. For the forward configuration, the equation

is
r,=

s^21 1-

A

i-Lr,G

l-TnS

Tn +

G^ll

/>

\-rnSG^ll

Xi + ^2 + 2 Re

/>7^ G^12

\-raSG^ll

(6)

and for the reverse configuration it takes the form

T^ = 1-r,

s,^\-\\-\r,G

Tn +

1 - /7^5 G^22

'^12'^21^G

1 - FnSG^22

X2+X1+2RQ

1 - FqS22

, (V)

where 71 is the reflection coefficient of the generator-amphfier combination at plane 2 in Fig.

2(a), and /I is the reflection coefficient of the generator-amplifier combination at plane 1 in Fig.

2(b). In performing the fits, the parameters to be determined are the Xs and the "reduced gain,"

defined by Go = {Siil^- All other S-parameters, including the phase of 5'2i, are taken from

measurements on a commercial vector network analyzer (VNA).

2.2 Relation to IEEE Noise Parameters

Although many different sets of noise parameters have been suggested, the set in most

widespread use is the IEEE set [9]. It is therefore desirable to present any measurement results in

terms of the IEEE set, in addition to the results for the Jf s. Even within the IEEE set, there are several variants; we use the one in which the effective input noise temperature Te is given by

^e ^ mm ^ ' ^opr - ^G

1 + /:opt \-\nG\

(8)

where Fg is the reflection coefficient of the input termination at plane 1 , and the four noise

parameters are Tmi„, t, and the complex Fopt. The parameter t is related to the noise resistance R„

by / = ARnTQlZa, where Tq = 290 K and Zq is the reference impedance, commonly taken to be 50 a.

To make the connection between the X-parameters and the IEEE noise parameters, we

force eq. (6) into the form

T2=G^,{T,+Tg), (9)

where G„v is the available gain.

Gnv =

>21 \-\Fr

1 -|/>;cl 1 |1 -

GS\ •^G'^11

(10)

This yields the expression for T^ in terms of the X-parameters,

4
T =

\rQ?- \\-rQSu\^

i-A \-\rr. l-/>

Re ^g(1-^G^1i)*^12 (11)

which can be compared with eq. (8) to derive the relationship between the A'-parameters and the

IEEE noise parameters. Happily, Wedge and Rutledge [7] have done the bulk of the algebra for

us, obtaining the relationship between the noise correlation matrix of eq. (3) and the IEEE noise

parameters. We then need only use the definitions of eq. (4) to obtain

X\1 —-^min Tmin Pll 5'i 1 — 1 +

^7 — -/min +

l-SuKopt

1 + /:opt

2 -' min

/:opt

1 + /:

|2

'

^12 -'^ll^min ~

opt\

f^optV'^W^opt)

1 + /:opt

(12)

The inverse relations are

t = X^ + \ + S^A '111 X^^2 -2 Re

t" ^

{\ + Su)*X,2

^opt - 1

2

X2

2 T-* =

min

ropt X^ +\SiA'- X2 -2Re[SyiXi2

1 + ropt

2\

(13)

where

fj = X2\\ + \Snlj + X^-2Re[SnXu

(^2'^11 -^12)

(14)

Thus, if the S-parameters are known, the IEEE parameters can be calculated from the Xparameters, and vice versa. Additional work is required to transform the uncertainties in the

noise parameters. That point is addressed below, in Section 4.
2.3 Physical Bounds

Basic physics and mathematics place certain constraints on the noise parameters. Some of

the constraints are obvious in either the IEEE representation or the X representation. For

example, r„„,„ t, X\, and X must all be positive, and 'j\X\j\ < Xj + X2 . Some are less obvious,

such as I/7I > 2. This last inequality can be proved by using eq. (4) to write the numerator and

denominator of eq. (14) in terms of the c wave amplitudes. The numerator can be put in the form

^2 1+S,ir +X,-2ReS,,X,2 + Cl^W

c\

>21 (15)

whereas the denominator is given by

XtiSi 1 — Xm —

Co

2^11 12 v^2i;

^2'^ll

V '^21

ci (16)

That \rj\>2 then follows from the Schwarz inequality.

In actual measurements, as well as in the simulations in the Monte Carlo program

discussed below, it is possible to obtain unphysical results that violate these bounds. This is rare when measuring well-matched amplifiers, but it is not uncommon when measuring poorly

matched, very low-noise transistors on a wafer. Small measurement errors can conspire to yield

fitted results for the noise parameters that are slightly (or even dramatically) outside the physical

bound rather than slightly within it. One approach to this problem would be to constrain the

fitdng procedure so that the fitting parameters range only over physically allowed values.

Instead, we choose to perform an unconstrained fit and to test whether the result satisfies the

physical bounds. This has the advantage of alerting us when the measurement resuhs prefer an

unphysical solution for the noise parameters, rather than just finding the best physical solution.

(It is also considerably easier than imposing this set of constraints in the fit.) We will return to

this point when we discuss the Monte Carlo program below.

3. MEASUREMENT METHOD

Basically, the measurement method for either off-wafer ("connectorized") amplifiers or on-wafer transistors (or amplifiers) consists of connecfing a series of different, known

terminations to one of the ports of the device under test and measuring the resulting noise

temperature at the other port. The noise parameters are then computed by performing a weighted

least-squares fit using eqs. (6) and (7). Most or all of the measurements are performed in the

forward configuration of Fig. 2(a), in which the known temiination (/g./, Tgj) is connected to the

input port, and the noise temperature (72,,) is measured at the output port. In the case of poorly

matched devices, typically transistors on a wafer, it is advantageous to also perform at least one

reverse measurement, as in Fig. 2(b). The tennination used for the reverse measurement is typically an ambient-temperature matched load. In principle, more than one reverse measurement
could be used; we are currently investigating the efficacy of that. For well-matched amplifiers,

the reverse measurement does not improve the noise-parameter uncertainties [6], and so we do

not use it in the fit for the noise parameters. In such cases, we often perform a reverse measurement and use it as a check, comparing the measured value for T\ to the value predicted

by eq. (7) with the values of the noise parameters that have been measured.

The set of input terminations typically comprises about eight ambient-temperature

terminations and one well-matched non-ambient noise source (usually hot). The input states are

discrete reflective or matched loads, connected manually to the appropriate port; we do not use a

tuner. We are currently developing an automated unit to switch automatically among the

different input terminations [10]. The ambient-temperature terminations include one matched

load and several reflective terminations chosen to provide adequate coverage of the complex-/'

unit circle. Although we have not made a great effort to optimize the choice of input reflection

coefficients, simulations have shown that we would gain little by using an optimized set, such as

is used by some commercial systems. The reflection coefficients of the input terminations and

the S-parameters of the device under test are measured on a VNA. As noted above, the quantity

Go = \S2i\ is treated as a free parameter in the fit; the fitted value usually agrees very well with

the value obtained from the VNA measurements. Since the same set of terminations is used

across the entire range of measurement frequencies, the reflection coefficients of the individual

terminations shift with frequency. The patterns of the reflection coefficients in the complex plane

at three representadve frequencies in the 8 GHz to 12 GHz band are shown in Fig. 3.

The noise temperatures of the ambient input terminations are taken to be the noise

temperature corresponding to 296. 1 5 K,

7>, =

hf

GJ [hfl{kB -296ASK) (17)

where 7c, / is the noise temperature of termination /, assumed to be one of the ambienttemperature terminations. The noise temperature of the non-ambient input termination is measured on the NIST coaxial radiometer NFRad [11], as are all output noise temperatures.

12 GHz

Figure 3. Reflection coefficients of input tcmiinations relative to the unit circle at three

frequencies.
Measurement of the output noise temperature for each input termination / yields a set of

measured values for 7?, which we call 72,,(meas), and possibly one or more measured values for

the reverse configuration, T\, which we call ri,/(meas). Each measured value is accompanied by

a standard uncertainty, u(T2j) or ii(T\j). For each measurement, we also have a corresponding

expression for the output noise temperature from eq. (6) or (7). The reflection coefficients and Sparameters in these expressions have been measured, and so the only remaining unknowns are

the noise parameters. We refer to the expressions as T2j{X) and T\j{X), where the Jfis understood

to include Go, as well as JTs. From measurements of the forward configuration, we obtain a set of equations of the form

T2j{meas) = T2j{X), (18)

for / = \,...,Nfi^J, where iV/wj is the number of forward measurements. If we include a measurement in the reverse configuration, we obtain an additional equation of the form

Tij{meas) = TyjiX). (19)

These equations can be used to determine the noise parameters and Go. To do so, we perform a weighted, least-squares fit, minimizing the function

^T2j{meas)-T2j(X)^

"(7^2,/)

2 r^ . X ^ ...x^2 + I

I, rev

Tij{meas)-Tij(X)

(20)

where the first sum runs over the forward measurements, and the second sum over the reverse measurements. For well-matched, packaged amplifiers, we do not typically use the reverse measurements in the determination of the noise parameters, and consequently the second sum is absent. In that case, the equations are all of the form of eq. (6), which can easily be put into

linear form by defining the variables Zi = GoX\, Z^ = GqXi, Z\i = GoA'12, Z5 = Go. In terms of the

Z's, the set of equations becomes linear. A commercial fitting program is used to minimize ;(^. If

only forward measurements are used, a linear fitting program would suffice, but if reverse measurements are present, the problem cannot be reduced to a set of linear equations. In practice,

we use the nonlinear fitting program in all cases, because it is the only one that provides the full covariance matrix, which we need in order to compute the uncertainties in the IEEE noise

parameters. The linear fitting program is used on the (sub)set of forward measurements to obtain an initial guess for input to the nonlinear fitting program. The nonlinear fitting program uses a modified Levenberg-Marquardt method (see, e.g., [12]) to solve a nonlinear regression model

using least squares.

4. UNCERTAINTIES I: OFF-WAFER, COAXIAL REFERENCE PLANES

4.1 Type-A Uncertainties

Type-A uncertainties are those that are evaluated by a statistical analysis of a series of

observations [5]. In our case, where the parameters are determined by a fit to an over-determined

8
system of equations, the packaged fitting program returns the covariance matrix F,y for the fitting

parameters, and the type-A uncertainties are given by the square roots of the diagonal elements,

w^(X,) = Vf,.,.(X), (21)

where A'/ represents any of the five fitting parameters (Jfs and Go).

Since the parameters in most common use are the IEEE parameters, we also need to evaluate the uncertainties in them. Because the fit is performed in terms of the A'-parameters, the

fitting program does not return values or a covariance matrix for the IEEE parameters. The IEEE

parameter values are computed from the A'-parameters by use of eqs. (13) and (14), and the

covariance matrix for the IEEE parameters must be computed from the A-parameter covariance

matrix by use of the Jacobian matrix for the transformation. If we use // to represent one of the

five IEEE parameters (including Go), then the type-A uncertainties in the IEEE parameters are given by

5 a/, dlj (22) V.{IEEE)= I —^ ^-Vi'i'iX).

/',/=! ^^/' SXj' '^

Calculation of the elements of the Jacobian matrix {dhldXi) is straightforward but tedious, and

the results are lengthy and unenlightening. They are relegated to Appendix A.

4.2 Type-B Uncertainties

Type-B uncertainties are those that are evaluated by any means other than the statistical

analysis of a series of observations. In our noise-parameter measurements, these uncertainties

arise due to possible errors in the input parameters, which include the amplifier S-parameters, all measured reflecfion coefficients, the temperature of the ambient-temperature terminations, the

noise temperature of the non-ambient termination(s), and the measured output noise

temperatures. We know or can esfimate all these input uncertainties, but the problem of how they

propagate into uncertainties in the noise-parameters does not admit a simple analytical solution.

We therefore adopt a Monte Carlo approach, a good description of which can be found in

Reference [13]. Our Monte Carlo procedure has been described previously in References [6] and

[14]. Here we present a somewhat more detailed and updated account.

The Monte Carlo uncertainty evaluation is built around the generation of simulated

results for the measurements that were actually performed. These simulated measurement results are analyzed in the same way as are the real data, yielding simulated values for the noise

parameters. This process is repeated a large number of times, and the distribution of \ alues

obtained for each noise parameter is used to compute its uncertainty. One full simulation of all the measurements that were actually performed will be referred to as a "sef' of measurements,

and the number of simulated measurement sets will be denoted M,w We choose M„„ to be large

enough that the computed uncertainties are approximately independent of A^..^,,. A value of

M//H= 10,000 is usually more than sufficient, but for poorly matched transistors, it is occasionally necessary to use larger values. In all cases, we use a large enough \alue that any
further increase in Nsi,,, changes the uncertainties by at most 10 % of their values. In most cases, the uncertainties are within a few percent of their asymptotic values.

Each simulated measurement is performed by choosing a number randomly from the

probability distribution for that physical quantity. The distribution is constrained to have an

average value equal to the true value and a standard deviation equal to the standard uncertainty

for that quantity. We use the real measurement results as the true values. Normal distributions are assumed for all quantities except for the ambient temperature, for which a rectangular

distribution is used, to better model the effect of a thermostatically controlled laboratory

temperature. (The program also allows the user to choose a normal distribution for the ambient

temperature, but we do not usually use this. It makes virtually no difference in practice.) For

each measurement set, we first simulate measurement of the input parameters; these include the

amplifier S-parameters, the reflection coefficients of all input terminations, the noise

temperatures of all terminations, and the output reflection coefficients, Fij in Fig. (2), if they are measured. (There is an option of measuring T?,, directly or of computing it by cascading Fgj with

the S-parameters.) To simulate the measurement of the output noise temperatures, we must first compute the true values from eq. (6) (for the forward configuration) or eq. (7) (for the reverse

configuration), using the true values for all quantities on the right hand side.

To simulate a measurement (of a quantity v, for example) we merely add a random error term S^' to the true value yo,

y{sim) = yQ+dy, (23)

where Sy is chosen randomly from the appropriate distribution (normal or rectangular), with

<dy> = and <^ > = u(y) , where u{y) is the standard uncertainty in y. In practice, we do this by

choosing a random number from the standard normal distribution (i.e., mean of zero, variance of

1 ) and multiplying by the standard uncertainty. For complex quantities, the real and imaginary

parts are simulated separately. In principle we could include correlation between errors in the

real and imaginary parts or work with uncertainties in magnitude and phase, but we have not yet

done so.

Because the input variables include sets of variables that are all measured in the same

manner, correlated errors can and do occur, and the correlations can cause significant effects in

the noise-parameter uncertainties [6,15]. We therefore include correlated errors in the

simulations. The correlations that we include are among all the measured reflection coefficients

and the S-parameters, among all the output noise temperatures, and among the temperatures of

the ambient-temperature terminations. If there is more than one nonambient input termination,

we include correlations between them. The way that correlated errors are simulated is to add an

additional term to eq. (23). If j^, and j^y are two physical quantities whose errors are correlated (at least in part), we write

yjisim) = V, + Svj{unc) + Svj(cor),

(24)

yj(sim) = yjQ + dyj{unc) + Syj{cor).

10
The correlated (cor) and uncorrelated (unc) error terms satisfy <Syi(cor)> = <Syi(unc)> = 0,

<dyicorf + dyi{uncf> = Ucoriyif' + Uunciyif = ufyif, and similarly for yj. The random errors for

the uncorrelated error terms are chosen independently, whereas the same random number (scaled

by the appropriate uncertainty in each case) is used for both correlated error terms. The

correlation coefficient is given by

Pij-

(Syjiimc) + ^i(cor)^Syj{unc) + Syj(cor)j

u{yi)ii{yj)

. (25)

dyi{cor)dyj{cor)) _ Ucoriyi>coriyj)

n{yi>{yj) niyiMvj)

Each set of simulated noise-temp measurements is analyzed in the same manner as a set of real measurements: we perform a least-squares fit to the measurement results and obtain a set of the noise parameters {X\, X2, Xu, Go). From these and the simulated measurement results for

the amplifier's S-parameters, we compute the set of IEEE noise parameters (T„,i„, t. Fop,, Go).

This is done for each of the Ns,m sets of simulated measurements. The average and standard

deviation of the measured values for each parameter {X and IEEE) are computed. The (type-B)

uncertainty in a single measurement of a parameter is then computed by combining the standard

deviation in quadrature with the difference between the average and the true value. This is just

the root-mean-square error (RMSE) of the sample,

ugiy) = RMSE(y) = yjvar(y) + {y - yf,,„f , (26)

where y is any of the noise parameters, and Var{y) is the variance of the sample of simulated

results for>'. For the complex quantities Jf12 and Fop,, statistics are done on the real and imaginary

parts separately. The fact that y is not equal to the true value, may be unsettling at first, but such

are the vagaries of nonlinear functions.

4.3 Input Uncertainties

The simulations require as input not just the true values of all parameters, but also the

standard uncertainties in the parameters that are directly measured, i.e., the amplifier Sparameters, the reflection coefficients of all terminations, the noise temperatures of all terminations, the output reflection coefficient, /:./ in Fig. (2a), if it is measured, and the output

noise temperatures. These input uncertainties (or the parameters that determine the uncertainties) are read into the Monte Carlo program along with the input data, so that they can be changed at any time, but we have a standard set of uncertainties that we usually use. This set is subject to change in the future, as we improve our knowledge of the uncertainties, and especially of the

correlated errors.

The reflection coefficients Fqj and /:., (if it is measured), are measured on a commercial

vector network analyzer (VNA). Since all reflection coefficients are measured with the same

calibration on the same VNA, there is significant correlation among the errors of the different

reflection coefficients. The connector used on the radiometer port is type PC-7. and so we

11
assume the packaged amplifier also has PC-7 comiectors. (If the amplifier has some other

connector, requiring use of adapters, the uncertainty analysis is similar in form to the on-wafer

case, which is treated in the next section.) The uncertainties in the reflection-coefficient

measurements can depend on the magnitude of the reflection coefficient; usually the

uncertainties are larger for larger reflection coefficients. Our program uses one value for small

|/] (less than or equal to 0.5) and another for large \r\ (greater than 0.5). In the program, we work

in terms of Uco,- and ?/„„,., defined after eq. (24) above, and then compute u and p from them. The

values typically used for small |/] are iico,- = 0.0025 and ih,„c = 0.001, corresponding to w(7~) «

0.002693 and /?« 0.8621. These uncertainties are somewhat larger than the manufacturer's

specifications, reflecting our own past experience with such measurements [15]. For large |/] we

use iicor = 0.004 and ii„„c = 0.001, corresponding to z/(7~) ~ 0.004123 and /?« 0.9412. The

uncorrelated part of the uncertainty, w„„c = 0.001, is due primarily to connector

(non)repeatability. The same ?/(/) is used for both the real and the imaginary parts of each

reflection coefficient, i/(Re/l = u{lmr) = z/(7~). The small S-parameters (i.e., ^n, ^'22, and S^)

have their errors treated the same way as the reflection coefficients, including the correlations.

For S21 we use u{RqS2\) = i/(ImS'2i) = 0.01. This value was obtained by an extrapolation of the

VNA specifications, which do not extend to 5'2i magnitudes as large as we encounter. The input

uncertainties in S21 are not very important because the magnitude of 5*21 is treated as a fitting

parameter, Go = |'S'2i|'.

The ambient temperature in the laboratory is set at 296.15 K and is kept within 0.5 K of

this value. We therefore use a rectangular distribution extending from 295.65 K to 296.65 K for

the ambient-temperature terminations. Because several hours intervene between measurements

on the amplifier with different terminations, there is essentially no correlation between the errors

in the temperatures of different ambient-temperature terminations.

In actual measurements of the output noise temperatures on NFRad [10], the

measurement software evaluates and writes the uncertainty in the noise-temperature

measurement. The uncertainty depends on a number of details, such as the reflection coefficient

of the DUT, the magnitude of the noise temperature being measured, statistics of the multiple

repeat measurements, and characteristics of the radiometer at the measurement frequency. Rather

than perform a full evaluation of the uncertainty for each simulated measurement, we use the

following approximate parameterization for the uncertainty in the noise-temperature

measurements,

i^iT^eas ) = 0.2K + 0.005{T^,,, - T, ), (27)

where Ta is the noise temperature corresponding to the ambient temperature of 296.15. This form

is adequate provided that the reflection coefficient of the DUT is less than about 0.2. For larger

reflection coefficients, further refinement is required, as described in Section 5.

There is significant correlation among the measurements of the output noise temperatures

for the different terminations. An examination of the uncertainty analysis that was performed for

noise-temperature measurements in coaxial lines [10,16] leads to the conclusion that most errors are correlated if all the reflection coefficients are measured with the same VNA calibration, as they typically are in the amplifier noise-parameter measurements. The major source of

uncorrelated error is the connector (non)repeatability. The magnitude of the connector error

12
relative to the correlated errors would lead to a value p ^ 0.98. Because correlated errors in the

noise-temperature measurements often lead to much smaller uncertainties [6], we are reluctant to use such a large correlation coefficient without being very sure of it. Until we have better

knowledge of these correlations, we will use p = 0.64, which corresponds to Ucorluunc = 4/3. This

may lead to a small overestimate of the uncertainty, but we prefer to err on the side of caution.

If two nonambient input terminations are used, there is a correlation between the errors in their noise temperatures if they are both measured on the same system, as is the case at NIST.

We have investigated the case of one hot and one cold input noise source, and we find that if both are measured on NFRad, the correlation coefficient is p^ -0.1 15. The negative sign occurs because the correlated errors have different sign for noise temperatures above and below ambient

temperature. In principle, there is also a correlation between errors in measuring the output noise

temperature from the amplifier and in measuring the nonambient input temperature(s), since both

types of measurements are performed on the same radiometer (NFRad) with the same standards.

In NIST measurements, this effect is usually small for the coaxial amplifier case because the

nonambient input tennination is usually measured with a different VNA calibration, often many

months before the noise-parameter measurements. We have therefore ignored this correlation in the Monte Carlo program for the type-B uncertainties in the noise parameters for coaxial

amplifiers. At many industrial laboratories, however, the same diode noise source that is used to calibrate the radiometer is also used as the hot input noise source. Furthermore, its noise

temperature is one of the major sources of uncertainty. In this case, the correlation between

errors in the hot input noise temperature and the measured output noise temperatures can be

important.

4.4 Sample Results

We have recently measured the noise parameters of a low-noise amplifier (LNA) from 8

GHz to 12 GHz. This was done to test an automated variable-termination unit (VTU) [10], which

can be used to automatically switch among a series of input terminations, greatly reducing the

time required for the measurements. For these measurements, 1 1 different input terminations were used in the forward configuration, of which one was hot (between 1000 K and 1 100 K,

depending on frequency), one cold (about 100 K to 110 K) and the remainder at room

temperature (about 296.15 K). In addition, one reverse measurement was done, with a matched

load on the output of the LNA.

Results for the X parameters are shown in Fig. 4. The quantity Xi is the equivalent input

noise temperature for a matched (reflecfionless) input termination, often called the 50 Q noise

temperature. There are two sets of uncertainties plotted for A2, to show the effect of an output

attenuator on the measurements. The actual measurements were performed with a 20 dB

attenuator on the amplifier's output (for the forward configuration), in order to keep the output

noise temperature in the measurement range of NFRad. Consequently, the noise at the

radiometer/DUT measurement plane is weighted 99 to 1 in favor of the attenuator over the

amplifier, and the uncertainty suffers significantly. The larger uncertainties on X2 in Fig. 4

correspond to the actual measurements, and the smaller uncertainties represent what could be

achieved if no attenuator were needed.

13
200 —

150

100 -

1^

50 -

-50

AAA ReX^2

V V V imz;.

H

m

"1 I I

I

I i I I

I I I I II I I I II

I I I I II I I r

9 10 11 /GHz)12 13

Figure 4. Results and uncertainties for X parameters. The smaller set of uncertainties for X2

applies to the case with no output attenuator in the measurements.

Another noteworthy point in Fig. 4 is the different sizes of the uncertainties for Xi at different frequencies; they are much larger at 10 GHz and 1 1 GHz than at the other frequencies.

A measurement of the reverse configuration gives the best determination ofXi, and one reverse measurement was made at each frequency. However, at some frequencies it is not possible to obtain a good least-squares fit when all the measured terminations are included. Presumably this

is due to one (or occasionally two) bad measurements (which contribute disproportionately to

2^). In those cases, one (or two, if necessary) terminations are deleted, and at 10 GHz and 11

GHz it was necessary to exclude the reverse measurement, resulting in larger uncertainties for

Xu

In Fig. 5(a-f) we show the results for Go and the IEEE noise parameters. Because it is in

common use, we have also shown the minimum noise figure in dB, which is related to T„,i„ by

F,^jn(t/5) = 101ogio(l + T'min /?o)- ^^ '^ P'§- '^' *h^ ^o ^^^^ of uncertainties correspond to the

actual measurements (larger uncertaindes) and the case with no output attenuator (smaller

uncertainties). For the phase of /"opt, Fig. 5(f), we show only the actual uncertainties, since they

are so small. As for the X parameters, the use of the attenuator causes a major increase in the

uncertainties. The approximate values for the actual uncertainties are: 0.25 dB for Go, 0.10 dB to 0.16 dB for F„,i„, 10 K to 17 K for /,„,„, 6 Q to 12 i^ for R„, 0.013 to 0.023 for |ropt|, and 0.8° to

1 .0° for (/>opi. If an attenuator had not been necessary, these uncertainties would have been 0.05

dB for Go, 0.03 dB for F,„„„ 2.5 K to 3.5 K for /,„,„, 0.08 Q to 0.16 Q for R„, 0.003 to 0.007 for

iroptl, and 0.2° to 0.5° for ^„/„.

14
35

34.5

34

33.5

33 d

ffl 3 32.5 H

32 -

31.5

31 —

30.5

30

03T3"c 1

I

I

I I I I

I

I I I I

I

I I I I

I

I I I I

I

I I I I

I 8 9 10 II 12 13 /(GHz)

I I I I

I

I I I I

I

I I I I

I

I I I I

I

I I I I II I I r

I 7 8 9 10 11 12 13 y(GH2)

Fig. 5(a) Results and uncertainties for Go. Fig. 5(b) Results and uncertainties for F„

fj 140

Bono 1 I I I

I

I I I I

I

I I I I

I

I I I I

I

I I I I

I

I I I I

I 7 8 9 10 11 12 13 ./(GHz)I I I

I

I I I I

I

I I I I

I

I I I I

I

I I I I

I 9 10 11 12 13 /(GHz)

Fig. 5(c) Results and uncertainties for r,„/„. Fig. 5(d) Results and uncertainties for /?„.

I

I I I

I

I I 1I

I

I ! I I

I

I I I I

I

I I

I I

I

I I I I

I 8 9 10 11 12 13 ./(GHz)

180 -n

150

120

90 —

60 -

30 -

§- 0-

9-

-30 -

-60

-90

-120

-150 H

-180

' ' '

I ' ' ' 'I

' ' ' '

I

' ' '

I 8 9 10 11 /(GHz)

I

' ' ' ' I

12 13

Fig. 5(e) Results and uncertainties ior\r„,„\. Fig. 5(f) Results and uncertainties fov^opt15
5. UNCERTAINTIES II: ON-WAFER MEASUREMENTS

5.1 Differences from the Coaxial Case

The measurement method for on-wafer noise-parameters is similar in principle to the

method for measuring coaxial amplifiers, but additional complications arise because the

measurement planes of interest are on-wafer, as indicated in Fig. 6. The reference planes for the

DUT noise parameters are planes 1 and 2. The locations of the reference planes are chosen to assure single-mode conditions in the calibration process. An expanded view of the on-wafer

structure for microstrip lines is shown in Fig. 7. We do not attempt to de-embed down to the

transistor reference planes Tl and T2, as is often done in industry.

In order to use eqs. (6), (7), (18), and (19) between planes 1 and 2, we need the reflection

coefficients, S-parameters, and noise temperatures at those planes. A two-tier calibration is performed using a multiline TRL set of calibration standards [17,18] that has been fabricated on

the wafer. The NIST MultiCal® software is used to perform the two-tier calibration. The

calibrated VNA is then used to measure the reflection coefficients and transistor S-parameters at planes 1 and 2, as in the amplifier case in the preceding section. The output noise temperatures

are measured by NFRad at plane 2' and corrected to plane 2, as described in Section 5.3 below.

Similarly, the non-ambient input noise temperature is measured by NFRad using the

configuration of Fig. 6, but with the on-wafer device replaced by a through line. Again, NFRad

measures at plane 2', and from that we compute the noise temperature at plane 1.

The uncertainty analysis for measurement of the noise parameters of on-wafer transistors

is thus similar to that for measurements on packaged amplifiers, but it presents a few additional

complications. Because the measurement planes of interest are on the wafer, we must

characterize and correct for the effects of the probes, which introduces additional uncertainty.

Also, the uncertainties in VNA and noise measurements are different on a wafer fi"om what they

are in coaxial lines, and therefore the input uncertainties are different for the on-wafer case. The

third complication is that the transistor may be very poorly matched, leading to relatively large

values of the reflection coefficient at its output. This requires that we refine our estimate of the

uncertainty in measuring the output noise temperature. It also requires us to adopt a prescription

for handling unphysical results in the simulations. We treat each of these complications in turn in

the following subsections.

5.2 Input Uncertainties for On-Wafer Measurements

The uncertainty in the noise temperatures of the ambient-temperature terminations is the

same as in the coaxial case above, and the uncertainties in the output noise temperatures and the

nonambient input noise temperature are treated in the next subsection. Reflection coefficients

and S-parameters measured at on-wafer reference planes have larger uncertainties than those

measured at reference planes in coaxial lines. The input uncertainties that we use for the on- wafer reflection coefficients and S-parameters (other than |^2i|) are Ui,„c = 0.004 and z/«„- = 0.003,

corresponding to uif) = 0.005 and p= 0.36. For on-wafer measurements, we (currently) use the

same uncertainties for large and small |/].

16
2 1

Figure 6. Reference planes for on-wafer measurements.

U

12 Urn

^ T2 T1

" 12 fim back from

center of Thru

Figure 7. Detail of reference planes for transistor in microstrip.

5.3 Uncertainties in On-Wafer Noise Temperatures

The measurement planes of interest are on the wafer, as indicated in Fig. 6, whereas the

noise-temperature measurements are performed by NFRad at plane 2'. We must therefore correct

for the effect of Probe 2 to get the noise temperatures at plane 2. The S-parameters of the probe

are determined in the two-tier calibration, as described in Section 5.1. To obtain the noise

temperature at plane 2 Tjj from the noise temperature at plane 2' Tyj , we treat the probe as an

adapter, characterized by its available-power ratio ayi.i ,

^2'2,i

Syih i-k2,/

(28)

where Sn is the S-parameter of probe 2 from plane 2' to plane 2, S22 is the reflection S-parameter

of probe 2 at plane 2 (what would normally be called ^n), /l., is the reflection coefficient at plane 2 (from the transistor), and Fyj is the reflection coefficient at plane 2' (from the probe).

17
Knowing arij , we compute the output noise temperature at plane 2 in terms of 72,, in the usual

manner,

T2i=^^—^ l.,U a

^ ^29)

where Ta is the noise temperature of the probe, which is assumed to be at ambient temperature.

From eq. (29) we see that in order to compute the uncertainty in the on-wafer output

noise temperature T2j, we need the uncertainties in Ta, Trj , and ayij The uncertainty in Ta is the same as in the case of the coaxial amplifier measurements, treated above. The probe effect, of

course, was absent from the treatment of the preceding section, and there is also a new

complication with the noise measurement at the coaxial plane T2'j. Because the on-wafer

transistor may have a relatively large value of \S22\, the reflection coefficient at the coaxial

measurement plane Fyj can also be large. For values of {Frj \ larger than about 0.2, the

uncertainty in T2'j increases, and the approximation of eq. (27) is no longer adequate. This occurs because as j/?'., | increases, the mismatch correction increases, and a larger mismatch correction

magnifies any error in the measurement of the reflection coefficient. Examination of the

radiometer equation for NFRad [11] reveals that (Ty,,- 7^) x (1 - {Frj I")"'. Because eq. (28)

indicates that a->n i x (1 - |/^v, |")"' as well, we can effect some simplification by regrouping eq. (29),

T2,i=-(T2',-Ta)+T, ^

(30)

1 ' a

where we have suppressed the subscripts on a for convenience, and pulled out and cancelled

factors of (I - {Frj f) by defining a ' = {I - \ry, |-)a and TyJ = (1 - \rrj p)x (^2',, - Ta). The

advantages of the form in eq. (30) are that it removes the necessity of dealing with important

correlations between the errors in a and Tyj, and that we can use eq. (27) for the uncertainty in

TyJ From the usual rules for propagation of uncertainty [5], the uncertainty in T2j can be written

as

ii'^{T2j) = ^u^{Tr-)+ ^'^^''~^'''^ u'^{a') + ir{T,), (31)

a'^ a'

where we have used Tyjlcc ' = Ti,, - Ta in the second term on the right side.

As already mentioned, ii{Tr,i) and ii{Ta) can be treated as in the coaxial amplifier case above. For a ' we use its definition and eq. (28), along with the propagation of uncertainty, to write
u^ia')

^ 2a' ^

Vl^2'2|;

" (|'^2'2|) +

2«F2,/

1-^2,/!

|2 "nk2,/)' (32)

where a small term has been discarded. The uncertainty for on-wafer reflection-coefficient

measurements was discussed in the preceding subsection. The relevant numbers for use in eq. (32) are iiQSril) = "(1^2,,!) = 0.005.

A final complication is that we sometimes need to use an attenuator at the output of probe

2 (plane 2') in order to keep the measured noise temperature within the dynamic range of the

radiometer NFRad. If an output attenuator is present, it can be lumped with the output probe, and

the combined available power ratio is given by the product of the available power ratios of the

attenuator and the probe, aiomi = cCaiiCCpmhe- In forming a\ T^v is to be taken at the attenuator

output, between the attenuator and the radiometer. From eq. (31) we can see that using an

attenuator degrades the uncertainty somewhat due to the smaller value for a '. There is no effect on the second and third terms on the right hand side because // (a') oc or' , but the first term

increases somewhat.

Pulling everything together, we use eq. (31) for the uncertainty u{TiJ) in noisetemperature measurements at the on-wafer reference plane 2. We get uijrj) from eq. (27) and

«(«) from eq. (32). Equation (31) applies not only to the measured noise temperatures at the

DUT output, but also to the measurement of the hot (or cold) input noise temperature. We have

checked that the uncertainties obtained in this manner are consistent with the uncertainties that were confirmed in our earlier work [19,20] on verification and consistency checks for on-wafer

noise-temperature measurements.

There is still the question of correlations among the errors in the measurements of the

output noise temperatures 72,, and also between the errors in 72,, and those in the noise

temperature of the hot or cold input termination. Of the three terms on the right side of eq. (31),

the first term corresponds to the highly correlated errors in coaxial noise-temperature

measurements {p = 0.64), considered in Section 4.3 above. The second term is comparable in size, but the errors contribufing to it are mostly uncorrelated, because the sign and magnitude of

a' vary randomly from one termination to another. The third term also corresponds to uncorrelated errors, but it is small relative to the other two terms. These considerations lead us to use pij = 0.36 for the correladon coefficient between errors in measurements of the on-wafer

noise temperatures 72,, and Tj,/. As for the correlation between measurements of the output noise

temperatures and of the on-wafer noise temperatures of the nonambient input tenninations, we

expect this correlation to be larger than in the coaxial case (Section 4.3) because the on-wafer

measurements are performed with the same VNA calibration, on the same NFRad port, w ith not

too long a time between. However, at present, we neglect this correlation, as we do in the coaxial

case. We hope to include this effect in the future.

For perspective, it should be remembered that //(T^,,) in eq. (31) is the uncertainty that is used for the simulated output noise-temperature measurement for a given tennination /. Each

output noise temperature in each set of simulated measurements has such an associated

uncertainty, and these uncertainties are used to detemiine the weighting iactors in the least-
squares fit in eq. (20). That fit then produces the simulated measured noise parameters for that

set, and there are Nsi,,, such simulated measurement sets.

5.4 Occurrence and Treatment of Unphysical and Other "Bad" Results

The final additional complication with which we must deal in on-wafer simulations is the occurrence of unphysical results, those that violate the physical bounds discussed in Section 2.3. For the CMOS transistors that we have measured [4,21], it is quite possible to obtain unphysical

results in the simulated measurements (as well as in real measurements). In a real measurement,

we would normally discard such results and repeat the measurements. In the uncertainty

program, we compute two sets of uncertainties, one with the unphysical results discarded, as we

would usually do in practice, and one with the unphysical results included. One reason for

evaluating the uncertainties with the unphysical results included is that if real measurements

produce an unphysical result, we will probably want to evaluate the uncertainties in those results,

in order to see whether they are consistent with physical results within the measurement

uncertainties. To evaluate the uncertainties in unphysical real measurement results, we need to include the simulated unphysical results in the calculation.

We impose two further cuts on the set of simulated results. If the best fit is not good

enough, we discard that set of simulated measurements, as we would with a set of real

measurements that did not admit a good enough fit. For this purpose we define "good enough"

by /'/ V < 1 , where v is the number of degrees of freedom in the fit. In several instances, we have

also evaluated the uncertainties with a cut of ^/v< 1.5, and it makes little difference. The other

cut is on the uncertainty in Fopr. It sometimes happens that an acceptable fit is obtained, but that

the variance of Fopt is enormous (corresponding to an extremely flat minimum in this variable). A

real measurement set with this property would be discarded, and we do so in the simulation as

well. The maximum standard deviafion allowed for either the real or imaginary part oi Fopt is 1.

100-

0-

O O O A-, D D D X AAA ReX|,

V V V \raX„

J5 ^

V

n 1

1 r

/(GHz)

Figure 8. Results and uncertainties for Xparameters of a transistor on a wafer.

20
O O O Noise Result + + + VNA Result

.= 0.5

n

'—

10

-

T Y

T <> ()

T <> o -'^ i

^

(>

1

'

' 1

AGHz) /GHz)

Fig. 9(a) Results and uncertainties for Go. Fig. 9(b) Results and uncertainties for

^/Mm(dB).

120 IIOH

lOO90

80

70 -E 60 g- 50-E

'^ 40-i s3 30- 20-

10- 0-

-10-

-20 -E -30 ^10- ~1

I I I

I I I 1 I I I 2 3 4 5 6 7 8 9 10 11 12 XGHz)

~l I I I I 0123456789 I I I i I I I 10 II 12 ./(GUzl

Fig. 9(c) Results and uncertainties for 7)„/„. Fig. 9(d) Results and uncertainties for R„.

1.1

0.9 c 1

0.8 -_

0.7

t=-0.5 \

$ I

I ?

$

$

i

5

0.4

0.3

0.2

0.1 ""i""i'i ll|ll 1 <

1

1 1 1

1

1 1

"1" ll|M M|M "1

) 1 2 4 5 r> 7 S 9 10 11 12 ^

/(GHz)

'l""|i"'l""l""l""l'iii|""l""l""l

3 4 5 6 7 8 O 10 II 12 rtGHz)

Fig. 9(e) Results and uncertainties for\r„p,\. Fig. 9(f) Results and uncertainties for^,,,,,.
5.5 Sample Results

The preceding analysis was used for measurements of the noise parameters of an on- wafer transistor fabricated in 0.13 fim CMOS technology [4]. The set of input terminations for

the forward configuration comprised one hot source (around 1 1 00 K) and eight ambienttemperature terminations (296.15 K). One reverse measurement was also made, with a matched

load connected at plane 1'. In the forward measurements, a 10 dB attenuator was connected

between probe 2 and the radiometer, to keep the measured noise within the linear range of the

radiometer. The results for the X parameters are shown in Fig. 8 and those for Go and the IEEE

noise parameters in Fig. 9 (a-f).

For Go we have shown a comparison to the value obtained from the VNA measurements. It is reassuring that the independent measurements generally agree within the uncertainties. The

approximate values for the uncertainties in the on-wafer noise parameters are: 0.22 dB to 0.30

dB for Go, 0.2 dB to 0.3 dB for F„„„, 15 K to 25 K for r„„„, 0.6 Q to 0.8 Q for i?„, 0.02 to 0.07 for

|/^pt|, and 1° to 6° for (popt- As expected, the uncertainties in the on-wafer case are generally

somewhat larger than for the connectorized amplifier. As in the amplifier case above, if the

measurements could be done without an output attenuator, the uncertainties would be smaller.

6. SUMMARY

We have set forth the uncertainty analysis for NIST measurements of the noise

parameters of amplifiers and transistors, where the reference planes can be either in coaxial

transmission lines or on a wafer. The standard uncertainty is the root sum of the squares of the

type A and type B uncertainties. The type A uncertainties are determined in the weighted least- squares fit to the overdetermined system of equations that result from the measurements of the

output noise temperatures for different configurations and terminations. The type B uncertainties

are evaluated by a Monte Carlo procedure. The program that is used to perform the Monte Carlo

computation is described in Appendix B. The uncertainty analysis described has been used for

both coaxial low-noise amplifiers [3] and very poorly matched on-wafer CMOS transistors

[4,20]. In the amplifier case [3], the results were subjected to checks and verificadon that

confirmed the general validity of the uncertainties, as well as the actual values of the parameters.

I am grateful to Dave Walker of NIST for many discussions and for performing the

measurements to which this analysis is applicable. I also thank Ken Wong of Agilent for

suggesting/requesting this paper. This work was supported in part by a contract from Agilent

Corp.

REFERENCES

[1] D. Wait and G. F. Engen, Application of radiometry to the accurate measurement of

amplifier noise, IEEE Trans. Instrum. Meas. 40: 433^37 (1991).

[2] D. F. Wait and J. Randa, Amplifier noise measurement at NIST, IEEE Trans.

Instrum. Meas. 46: 482-485 (1997).

22
[3] J. Randa and D. Walker, Amplifier noise-parameter measurement checks and

verification, Digest, 63rd ARFTG Conf, pp. 41-45 (2004).

[4] J. Randa and D.K. Walker, On-wafer measurement of transistor noise parameters at NIST, IEEE Trans. Instrum. Meas. 56(2): 551-554 (April 2007).

[5] ISO Guide to the Expression of Uncertainty in Measurement, International

Organization for Standardization; Geneva, Switzerland (1993).

[6] J. Randa, Noise-parameter uncertainties: A Monte Carlo simulation, J. Res. Natl. Inst. Stand. Technol. 107: 431-444 (2002).

[7] S. Wedge and D. Rutledge, Wave techniques for noise modeling and measurement, IEEE

Trans. Microwave Theory Tech. 40: 2004-2012 (November 1992).

[8] J. Randa, T. McKay, S. L. Sweeney, D. K. Walker, L. Wagner, D. R. Greenberg, J. Tao,

and G. A. Rezvani, Reverse noise measurement and use in device characterization, 2006

IEEE Radio Frequency Integrated Circuits (RFIC) Symposium Digest, pp. 345-348,

San Francisco (June 2006).

[9] H. Haus et al., IRE standards on methods of measuring noise in linear twoports 1959,

Proc. IRE, 48: 60-68 (1960).

[10] D. Gu, D. K. Walker, and J. Randa, Noise-parameter measurement with automated

variable terminations, submitted to 2008 Conference on Precision Electromagnetic

Measurements (CPEM 2008), Broomfield, CO (June 2008).

[II] C. A. Grosvenor, J. Randa, and R. L. Billinger, Design and testing of NFRad—A new

noise measurement system, NIST Tech. Note 1518 (March 2000).

[12] D. Luenberger, Linear and Nonlinear Programming, Ch. 7.8, Second ed. Reading, MA:

Addison-Wesley (1984).

[13] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes,

Chap. 14.5, Cambridge: Cambridge University Press (1986).

[14] J. Randa and W. Wiatr, Monte Carlo estimation of noise-parameter uncertainties, lEE

Proceedings—Science, Measurement and Technology, 149: 333-337 (2002).

[15] J. Randa, Simulations of noise-parameter uncertainties, 2002 IEEE MTT-S

International Microwave Symposium Digest, Seattle, WA, pp. 1845-1848 (2002).

[16] J. Randa, Uncertainties in NIST noise-temperature measurements,

NIST Tech. Note 1502 (March 1998).

[17] R. Marks, A multi-line method of network analyzer calibration, IEEE Trans.

Microwave Theory Tech., 39(7): 1205-1215 (July 1991).

23
[18] D. Williams and R. Marks, Transmission line capacitance measurement, IEEE

Microwave Guided Wave Lett., 1(9): 243-245 (September 1991).

[19] J. Randa, Noise temperature measurements on wafer, NIST Tech. Note 1390 (March

1997).

[20] J. Randa, R. L. Billinger, and J. L. Rice, On-wafer measurements of noise temperature,

IEEE Trans. Instrum. Meas., 48(6): 1259-1269 (December 1999).

[21] J. Randa, S. L. Sweeney, T. McKay, D. K. Walker, D. R. Greenberg, J. Tao, J. Mendez,

G. A. Rezvani, and J. J. Pekarik., Interlaboratory comparison of noise-parameter

measurements on CMOS devices with 0.12 \im gate length. Digest, 66th ARFTG Conf.,

pp. 77-81 (2005).

24
APPENDIX A: JACOBIAN MATRIX FOR X ^ IEEE

In order to compute the type-A uncertainties in the IEEE noise parameters from eq. (22),

we need the elements of the Jacobian matrix for the transformation from A'-parameters to the

IEEE parameters. It is a straightforward exercise to compute them, and we present them here.

The easiest elements are the partial derivatives of ? = AR^TqIZh, which in terms of the ^s

is given by

t = X^ + l + 5',,nY9-2Re (l + ^ll)*^12 (A.l)

The relevant partial derivatives are

dt = 1,

-^ = \\^Snf

dt

dX 5ReXi2

= -2(l + Re5n)'

dt

5ImJfi2 = -l\mSii. (A.2)

We consider Fopi next because it is used in writing T,„,„. The expression for Fopt is

^opt -

7 1- 1-

ri X2\\ + \Su\ ] + X,-2Kc[SuXu

(^2^11 - X\2)

(A.3)

We treat the real and imaginary parts of Fop, separately. The partial derivatives can all be written

in terms of partial derivatives of rj.

aRe/;„, 1 opt

dXi

aimr.., 1

1- 1- aRe// Re/7 d\ii\

opt

dX:

1- 1-

H

aim 7 Im rj d\?j\ dXj H\i- 4 dXi

(A.4)

25
The derivatives of rj are given by

5Re/7

where

= Re

dt]

dXi

dlmr]

dXj = Im

^ drj^

ai K

~dX~i = 2 Re 7

5Re/7 + Im//

ax,-

aim//

axI J

dri

axi ^:)5'ii -X12

8X2

1 V -^-"^Oa^^,

drj _ -2Re5n +^

a Re A 12 ^2"^11 -^12

a/7 -2Im»S'ii +7/7

aimX|2 Jl 2U1 ] ~ vl 12

(A.5)

(A.6)

Finally, we treat r„„„, which is given by

2f Xor • = nopt Xi+LSii X2-2Re[SuXi2J

1 + ropt

2\

(A.7)

It is understood that in eq. (A.7), occurrences of Fopt are just a shorthand way of writing the

expression on the right side of eq. (A. 3). The derivatives of T,„i„ are given by

26
ar,mm

dX1 1 + /:opt

nopt

I |2 i * - \ ^1+Pll| ^2 -2Re\jSiiA'i2/+^mm

-^^ nopt

J ^X^

dT,mm

dX2 1 + /:op?

^opt^X 1 2^

jr,I +|5nl^A'7-2Refc,*,A',7)+r. +P11 ^2 '11^12/^^min

^^opt

2"

dXi

>

•^^min

5 Re ^12 1 + ropt

nopt Re^ii- Xi+|Sii| ^2 -2Re^5'iiXi2J+7;^in

ropt

2 1

ar.mm

aimXi2 1 + ropt

ropt \mS11 Zi + 5ii X2-2Rekl^n)

JaReXl2

2 + r. n 5

mm

ropt

- 5Im A^|2

(A.8)

where the derivatives of |/^p,|" are given by

ropt

dXi = 2 Re/:

a Re/:o/?/

0/7/ dXj + 2Im/:

aim/:0/7/ opr dXi

(A.9)

and eqs. (A.4) through (A.6).

27
APPENDIX B: OVERVIEW OF THE FORTRAN PROGRAM

The type-B uncertainties in our noise-parameter measurements are evaluated with a Fortran Monte Carlo program called NISTNPUNC. In this appendix we give an overview of

the structure of the program, discussing each subprogram in turn.

B.l MAIN

The main program first initializes a few parameters, including the number of simulations

to be performed (NSET). It queries the user for the names of the output file (0UTFILE2), to which the final results are written, and the input file (INFILE), from which the input data are

read. It also defines OUTFILEl, naming it TRASH.TXT, to which it will write the noise

parameters from each of the simulated measurements sets. This file is usually discarded, but it enables one to examine some details of the individual measurement sets for diagnostic purposes,

should the occasion arise. The character variable QUANT(9), where 9 indicates the dimension,

contains the names of the nine quantities for which the program compiles statistics: X\, X2,

ReXu, ImZi2, Go, r,„,„, t, ReFopt, and ImToptAfter the initialization, MAIN reads the input data. The first line of data read from the

input file consists of the values for NMEAS, CHARl, AMBDIST, CHI_CUT, ALPHO_AT,

ALPHO_PR. NMEAS is the number of different measurements in each set, i.e., the number of

different forward-configuration terminations plus the number of reverse-configuration

measurements made in the actual measurement set from which the noise parameters were

determined. CHARl is a character variable that is either 'Y' if the output reflection coefficients are measured directly, or 'N' if they are determined by cascading the input reflection coefficient

with the amplifier or transistor S-parameters. AMBDIST is a character variable that specifies

the distribution for errors in the ambient temperature, either 'GAUS' for a normal distribution, or 'RECT' for a rectangular distribution. CHI_CUT is the maximum value of /^/ v retained in the

simulations, as discussed in Section 5.4. ALPHO_AT is the value of \S2\\'' for the attenuator (if it

is present; see Section 5.3). The other factors in the available power ratio a of the attenuator are computed during the course of the program. If no attenuator is present, a value of 1.0 is used.

ALPHO_PR is the value of \S2\\' for the output probe, probe 2 in Fig. 6. The second line of the

input file contains the "true" values for the noise parameters and Go, whose variable names are XI, X2, X12R, XI 21, G, where R and I refer to real and imaginary parts. The third line of the

input file consists of amplifier or transistor S-parameters, in the order SllR, Sill, S12R, SI 21,

S21R, S21I, S22R, S22I. The program then reads the information for the individual

measurements in each set, contained in lines 4 through 3+NMEAS of the input file, each line

corresponding to one of the measurements. Each line contains values for the number of the

measurement (1 through NMEAS), the configuration (1 for forward, 2 for reverse), the noise

temperature of the input termination, the real and imaginary parts of the reflection coefficient of

the input termination, and the real and imaginary parts of the output reflection coefficient (plane

2 in Fig. 2(a) or 6, or plane 1 in Fig. 2b). If the output reflection coefficients are not measured

directly (CHARl = 'N'), then the last two entries are absent. After reading the termination

information, the program computes the true values for the IEEE noise parameters and tests whether the true values of both the IEEE and the X parameters satisfy the physical constraints.

The final input quantities are the input uncertainties, which are read from the last five lines of the

input file. The first of these final lines contains the correlated and uncorrelated uncertainties in

28
reflection coefficients, first for small reflection coefficients, then for large reflection coefficients

(see Sections 4.2 and 5.3). The next line contains the correlated and uncorrelated uncertainties

for 5*2 1 and then the connector repeatability. We usually absorb the connector repeatability into

the input uncertainties in the reflection coefficients, so that this entry is 0, but the option is there

to enter it explicitly. The next line contains the correlated and uncorrelated uncertainties for

noise temperatures near ambient, and the following (next to last) line has the uncertainties for the

hot input terminadon, the cold input termination, and the correlation between the two (Secdon

4.3 or 5.3). The final line of the input file contains the correlated and uncorrelated uncertainties

for the measurements of the output noise temperatures (Section 4.3 or 5.3).

Once all of the input file has been read, the program assigns the appropriate uncertainty

to each of the quantities whose measurement will be simulated and inidalizes all the sums that

will be computed in the Monte Carlo process. It then enters a loop, each pass through of which

corresponds to one simulated measurement set. In the loop, the first step is to call the subroutine

SET_MEAS, which generates a complete simulated set of measurements, and returns the results

for S-parameters, reflection coefficients, input noise temperatures, and output noise

temperatures. It next calls the subroutine FULLFIT, which fits the simulated set of

measurement results to get the noise parameters, in both the Xand the IEEE representations. The

results are added to the set of running sums that include both good and bad results. The program

then checks the flags for "bad" results (Section 5.4), and, if the results are not bad, they are added to the running sums for good results. Thus, we have two sets of sums and averages, one

for all results and one for only good results. After the Monte Carlo loop is completed, the rest of

the main program just computes averages and standard deviations and writes the results to the

output file. The input data are also written to the output file, to ensure that the results are associated with the correct input.

B.2 Subroutine SET_MEAS

The subroutine SETMEAS simulates a complete set of noise-parameter measurements.

The true values and uncertainties of the variables to be measured are passed from the main

program through a common block. The number of different output noise temperatures to be

measured is NMEAS. The simulated measurement results are labeled by _M affixed to the

variable, except in the case of the output noise temperatures, which are designated TOUT. The

uncertainty corresponding to each simulated T_OUT is also computed and passed back to MAIN, for use in the fit of the simulated results.

The subroutine begins by generating the value for the complex deviate G_DEV. This one complex value is used for the correlated parts of the errors in all the reflecdon coefficients and Sparameters (except S2\) in this set of measurements. The deviates are all generated by the

function subprogram GAUSDEV, which returns a normally distributed deviate with zero mean

and unit variance. The program then generates simulated measurements for the S-parameters and

repeats the process for near-ambient noise temperatures and the reflection coefficients. Finally,

there is a loop that generates a simulated value for each of the output noise temperatures, along

with the associated uncertainty for each measurement. Each pass through the loop corresponds to one of the measurements. The measurement simulation is perfomied by calling the subroutine

T_MEAS, and the associated uncertainty is then computed.

29
B.3 Subroutine T_MEAS

This subroutine generates a single measurement of the output noise temperature from one

of the ports of the DUT. The input information comprises the true values of the noise

temperature and reflection coefficient of the source that is connected to the DUT, the true value

of the reflection coefficient of the DUT-source combination, the configuration (forward or

reverse), the true values or the S-parameters and noise parameters of the DUT, and the input

correlated and uncorrelated uncertainties for the measured noise temperatures. These input

uncertainties are not used directly as the standard deviations in the measured noise temperature,

but are used to set the scale and correlation coefficient for the standard deviations (Section 4.2).

The subroutine first computes the true value for the output noise temperature for the appropriate

configuration (forward or reverse). It then computes the uncertainty that is to be associated with

that measurement, taking into account the effects of potentially large reflection coefficients, the

probe, and the output attenuator, if it is present (Section 5.3). A simulated measurement result is then generated. The present version of the program allows inclusion of an output attenuator

either for all the measurements or for none at all. We plan to change this soon, to allow use of an

attenuator on some, but not all the measurements of the output noise temperature.

B.4 Subroutine FULL_FIT

This subroutine performs a weighted least-squares fit to minimize the function j^ defined

by eq. (20), thereby determining the noise parameters and Go of the DUT. This same subroutine

is used in other applications as well, and consequently it includes some features that are not

needed in this program. After some initial definitions, it calls the subroutine FWD_FIT, which

performs a linear fit to all the forward measurements in the set. The results of this linear fit are used to generate an initial guess for use in the full nonlinear fit. The nonlinear fit also allows

computation of the covariance matrix, which is used in other applications of FULL_FIT, but not

here. If there are no reverse measurements included in the measurement set, the linear fit produces the full, correct results. Nevertheless, we perform the nonlinear fit in all cases in this

subroutine because the covariance matrix is needed in the other applications, and the packaged

program used for the linear fit does not allow computation of the covariance matrix. An odd

feature of FULL_FIT is that we do not use the exact results of the linear fit as the initial guess for

the nonlinear fit, but rather we multiply the linear-fit results by 0.9. The reason for this is that if the starting point for the nonlinear fitting program corresponds to the minimum of the function

being minimized (as would be the case when the measurement set includes no reverse measurement), the program does not return the correct covariance matrix. (We have, of course, checked that the results of the nonlinear fit are independent of the initial guess and that it gives

the same results as the linear fit when there are only forward measurements present.)

After the nonlinear fit, the subroutine checks that ]^l v < CHI_CUT, computes the IEEE

noise parameters from the X parameters, and checks that the results satisfy the physical

constraints. It then computes the Jacobian matrix for the transformation from X parameters to IEEE parameters, which in this application is used only to check that the type-A uncertainty in

Fopi is not too large (Section 5.4).

30
B.5 Subroutine FWD_FIT

The subroutine FWDFIT performs a linear, weighted, least-squares fit to the output

noise temperatures of a series of forward measurements. The relevant S-parameters, reflection

coefficients, input noise temperatures, measured output noise temperatures, and uncertainties in

the measured output noise temperatures are all passed to the program via a common block. The X

parameters and Go are determined by minimizing the x defined in eq. (20). The minimization is performed by a packaged program that does a weighted, linear, least-squares fit.

B.6 Subroutine FCN

This subroutine defines the "error" in a given measurement for purposes of the fit, i.e., it computes the difference between the measured value of the output noise temperature and the

value computed from eq. (6) or (7) for the set of noise parameters X and Go. It is used in the

nonlinear fitting routine, which minimizes the weighted sum of the squares of the errors for all the measurements in the set, as given by eq. (20).

B.7 Subroutines GAUSDEV and RECTDEV

These two subroutines are taken fi"om and explained in Chapter 7 of Reference [13].

RECTDEV returns a random deviate between 0.0 and 1.0, using a system-supplied randomnumber generator. The point of RECTDEV is that it eliminates correlations in k-space, which

may be present in the system's random-number generator [13]. GAUSDEV generates and returns a normally distributed deviate with zero mean and unit variance, using RECTDEV as the source of uniform deviates.

31

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