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PUBLICATIONS

NIST TECHNICAL NOTE 1332

U.S. DEPARTMENT OF COMMERCE / National Institute of Standards and Technology

mtmimmmgitm
NATIOiN^- . [TUTE OF STANDARDS &

TECHNOLOGY

Research Information Center

nn Gaithersburg, MD 20899

m he National Institute of Standards and Technology^ was established by an act of Coingress on March 3, M 1901. The Institute's overall goal is to strengthen and advance the Nation's science and technology and

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* Located at Boulder, CO, with aome elements at Oaithenburg, MD.
Systematic Errors in

Power Measurements Made With

a Dual Six-Port ANA

Mis-rc

Cletus A. Hoer

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

^v^^*'^°^^o. \

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Raymond G. Kammer, Acting Director

Issued July 1989
National Institute of Standards and Technology Technical Note 1332

Natl. Inst. Stand. Technol., Tech. Note 1332, 56 pages (July 1989)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1989

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

Page

1

Introduction and Review 1

2. Measuring Power With a Six-Port Reflectometer 2

3. Connector Model and Reference Plane 5

3.1. Connector Model 5 3.2. Cascading Matrix Representation .... . . 5 3.3. Reference Plane 6

4. Mlcrocalorlmeter Reference Plane for Power . 6

5. Six- Port Reference Plane for Power 8

5.1. Calibrating the Six-Port for T Measurements With a Perfect

Line 9

5.2. Calibrating the Six-Port for Power Measurements Following

Calibration for T With a Perfect Line Standard 9

5.3. Calibrating the Six-Port for F Measurements With an Imperfect

Line. 10

5.4. Calibrating the Six-Port for Power Measurements Following

Calibration for T With an Imperfect Line Standard 12

5.5. Measuring »;^ of a DUT Power Detector 12

6. Efficiency of a 2 -Port 13

7. Efficiency of the Error 2 -Port 14

8. Approximation for e 15

9. Measured Power Constant 16

10. Total Systematic Error in tj^ . . 16

10.1. Error in dc Power Ratio 17

10.2. Error in rf Power Ratio at Six-Port Sidearm 22

10.2.1. Error Due to Detector Nonlinearity 23

10.3. Error in r)^ Due to Nonlinearity in power standard 24

10.4. Error in JG| 25

10.5. Error In M 26

10.6. Error Due to Imperfect Impedance Standards and Connectors ... 26

11. Total Systematic Error in tj^ for 7 mm Connectors 28

Table 2. Summary of systematic errors in dual six- port power measurements, 1-18 GHz. 27

12. Outline of Measurements and Calculations 28

13. Conclusions and Observations 30

14. References 31

ill
Al Appendix A

Bl Appendix B

Cl Appendix C

Appendix D

IV
Systematic Errors in Power Measurements

Made With a Dual Six- Port ANA

Cletus A. Hoer

The purpose of this report is to determine the systematic error in

measuring power with a dual 6 -port ANA. Most of the report

concentrates on developing equations for estimating systematic

errors due to imperfections in the test port connector,

imperfections in the connector on the power standard, and

imperfections in the impedance standards used to calibrate the 6-

port for measuring reflection coefficient. These are the largest

sources of error associated with the 6 -port. For 7 mm connectors,

all systematic errors which are associated with the 6 -port add up to a worst-case uncertainty of ±0.00084 in measuring the ratio of the

effective efficiency of a bolometric power sensor relative to that

of a standard power sensor.

Key Words: bolometric detectors; connector model; effective

efficiency; error analysis; microwave; power measurements; six-port.

1 . Introduction and Review

Microwave power measurements at NIST are based on the dc substitution

technique [1-5]. A bolometer unit, that is, a temperature-sensitive resistor

mounted in a transmission line, is biased to a specific resistance with a dc

current. The dc power to do this is measured by a bridge network [2] or a power meter [3]. Then, when rf power is applied to the input of the mount,

the dc power is reduced by the bridge or power meter to maintain the

resistance at its original value. The ratio of this change in the dc power

Pfjc to the net rf power P^-f app.lied to the input of the mount is defined as the effective efficiency rj , where,

r?=-^. (1.1)

rf

If all of the microwave power applied to the mount were dissipated in the

bolometer itself, the mount would be 100 percent efficient. However, because

of losses within the mount, the effective efficiency is less than 1. In

addition, there may be a small dc-rf substitution error due to the different

paths taken by the dc and rf currents. All of these imperfections in the

mount and the bolometer sensor are included in the measurement of r?

.

The effective efficiency of specially designed bolometric power detectors

is measured calorimetrically in NIST microcalorimeters which are the national

standards of microwave power. After calibration in a micro- calorimeter,

these "transfer standards" are used to obtain the effective efficiency of

other power detectors by means of a 6 -port reflectometer which measures the

ratio of two r/'s.
Source

lih^ iih2 lihi ^i^Q

6 -Port

^o ^ *^odc/'?o

Prf = Pdc/'?

a' b'

Mount

r = a'/b'

Power

Meter

Pdc

Figure 1. A 6 -port ref lectometer being used to measure the reflection

coefficient F and the net rf power Pj-f into a bolometer mount

whose effective efficiency is rj

.

2 . Measuring Power with a Six-Port Reflectometer

.

When a termination or power detector with a reflection coefficient F is

connected to the measurement port of a 6 -port ref lectometer as shown in figure

1, the net rf power into either device is given by^ [6]

rf

P^|l - wC/a|

Kp|l - BC/a|

d |r|^. (2.1)

where Pq is the emerging rf power from the 6 -port reference sidearm. This

sidearm is designed to have an output bg that is proportional mainly to b' at

the measurement plane. The complex quantity w is the ratio of one of the

remaining three sidearm emerging voltage waves (b^ , bj , or bg) to bQ and is

calculated from Pq ... Pg . The "error-box" parameters A, B, C are constants

that characterize the 6 -port and are independent of the reflection coefficient

of the generator and the load. These error-box parameters are determined by

the Thru-Reflect-Line (TRL) or Line-Ref lect-Line (LRL) calibration technique

[6,7].

The parameter Kp in (2.1) is also a constant ("power constant") of the 6-

port, and is determined by the effective efficiency rjg assigned to the

transfer standard by the microcalorimeter . For the 6 -port or any other

ref lectometer , F and the calibration parameters A, B, C and Kp all refer to

the same reference plane, so the reference plane for power is the same as the

reference plane for F.

The calibration parameters B and C/A appear in (2.1), but the parameters

A and C do not appear alone. Since B and C/A can be determined without the

"reflect" measurement in the TRL or LRL calibration, power as well as F

Kp in (2.1) above is written |r22|^ in reference 6.
measurements can be made independent of the "reflect" if the power detector is

measured on both 6 -ports to determine F^ on one 6 -port, and Fg on the other 6-

port. Then F calculated from VF^Fg is independent of the "reflect"

measurement as explained in Section 5.3 of Appendix G of reference [8].

For any ref lectometer , w and F are related by

AF + B „ w - B w or

CF + 1 A(l - wC/A)"

Using this result, (2.1) can be put in a more familiar form by writing

(2.2)

Then (2.1) becomes

CF + BC/A 1 - BC/A

1 - wC/A = 1 - cF^ 1 = 1 ^CF

p 1 - |fI o '

' ^^ Kp |1 -f CF|2*

(2.3)

(2.4)

The parameter C in (2.4) is sometimes written as -F„ or -S22 where Fp- or S22

represents the reflection coefficient looking back toward the generator from

the measurement port.

From (1.1) and (2.4) the effective efficiency r?^ of a detector connected

to the 6 -port can be written

where N^ is defined as Vn = Kp Nu>

"u-p

P^ ll + CF I dcu ' "'u' ou 1 |r I

' u'

2"

(2.5)

(2.6)

A subscript u has been added to parameters that refer to the device under

test.

The power constant Kp is determined by measuring a standard mount which

has a known effective efficiency tjg. From (2.5),

where

Kp = fys/Ns. (2.7)

P, ll + CF I dcs ' s

N = =

s P

OS 1 - |r

J

2' (2.8)

A subscript s has been added to parameters that refer to the standard,

Equations (2. 5) -(2. 8) combine to give

u

N

N

u dcu OS

1 + CFu

P^ P dcs ou 1 + CF

s

1 - |F I

' s

1 - |F I

' u'

(2.9)
CONNECTOR MODEL

TEST PORT DUT

1 ^ - "2 *^ t>6<yyxy^

r ^"7 '''

J

I

MEASUREMENT PLANE-*lA

Figure 2. Equivalent circuit for a pair of imperfect connectors

4
3. Connector Model and Reference Plane

When considering sources of systematic error it is necessary to take a careful look at the reference planes. The reference plane for power

measurements in the microcalorimeter may not be the same as the reference

plane for power in the 6-port. To investigate this possibility, we will first

look at the reference plane in the microcalorimeter where r?g of a transfer

standard is determined and then see how this rj^ is used in calibrating the 6-

port. The connector model developed by Hoer [11] which is described next will

be used to analyze all connections.

A typical connection of two 7 mm connectors is shown in the top of figure

2, where the shaded area represents unwanted modes generated by the

discontinuities present at the junction of the two connectors.

When more than one mode is present at the measurement plane, these modes

depend on the properties of both connectors. For this reason it is better to

define a model for a connector pair rather than for a single connector.

3.1. Connector Model

An exact equivalent circuit for the pair of connectors is shown in the

bottom of figure 2. In the derivation of this equivalent circuit, the

junction part of the connector is first represented by either a T-network or a ?r-network, either of which is sufficiently general to account for any

discontinuity at the junction. Either representation reduces to one series

and one shunt element as shown in figure 2 [11]. The T- or 7r-network is

inserted between two lengths of uniform lossy transmission line which

represent the rest of the connector. The lengths xj^ and X2 of these two lines are chosen long enough that all unwanted modes generated near the mating plane

are negligible by the time they reach the fictitious reference planes at aj^

and 32- Since only one mode is present at these fictitious reference planes,

everything between them can be , represented exactly by a set of network

parameters for a 2 -port. This 2 -port is therefore an exact representation of

a connector pair.

3.2. Cascading Matrix Representation

Let C be the cascading matrix of the 2 -port which represents the

connector pair. If C^i and Cx2 are the cascading matrices of the two lines of

length x^ and X2 , and C^, is the cascading matrix of the T-network, then

Q. = Sxi£c£x2 • (3.1)

The cascading matrices of the two lines are given by [9]

e'^i^i

e-'^i^i

C , —xi (3.2)

where 7^ is the complex propagation constant of line i = 1,2, whose

characteristic impedance is Zqj^.
The matrix C^, can be obtained by combining the cascading matrices for

the individual Z and Y elements of the T- or 7r-network [9] . The result is

1-y-z

Cc =

y-z

z-y

1+y+z

+ Az, (3.3)

where the dimensionless normalized admittance y and impedances z and AZq, are defined by

y=YZo/2, (3.4)

z = Z/2Zo, (3.5)

Azo = (Zoi - Zo2)/2Zo. (3.6)

The characteristic impedance Zq in (3. 4) -(3. 6) is the nominal value for the

transmission line. Zq^ and Zq2 are the characteristic impedances at the

reference planes a^ and a2 of the 2 -port representing the connector pair.

If the junction is represented by a T-network, Z in (3.5) is the sum of the

two series elements of the T-network. The same equations are obtained if the

junction is represented by a 7r-network. In that case Y in (3.4) is the sum of

the two shunt elements. In either case the results show that one series

impedance and one shunt admittance are adequate to represent the imperfections

at the junction. In deriving (3.3) we have assumed that

|2yz| « 1, |2yAZo| « 1, |2zAZo| « 1, and | 2AZo2 | « 1, (3.7)

which are true for all connectors in reasonably good condition.

3.3. Reference Plane

The choice of reference plane is not unique. For the application where a network analyzer is being calibrated with a length of line as the standard,

the reference plane is chosen as shown at the bottom of figure 2. This lumps

all of the discontinuity into the test port where it can be calibrated out

[11].

After calibration, if a termination with a connector that is identical to

the standard line connector is measured, the matrix C is the same as before

and none of the discontinuity is attributed to the device under test (DUT) . The discontinuity is calibrated out.

If a termination with a connector different from the line connector is

measured, the matrix C changes to a different matrix C'= C-AC. The matrix AC

contains this difference and appears as part of the DUT. The original

discontinuity remains calibrated out as part of the test port and creates no

error.

4. Microcalorimeter Reference Plane for Power

Let figures 3a,b,c represent the connection when the standard mount is

being calibrated in the microcalorimeter (MC) to determine its effective

efficiency r/g.
Perfect MC Test Port

Figure 3a.

^xi

Perfect Connector on Standard Mount

II n

1> iiX2

'^S

Pdc

Imperfect MC Test Port

Figure 3b. — Cxi'-^

c ' r '

iicm

Perfect Connector on Standard Mount

^1 Pdc

£x2

^S

Imperfect MC Test Port

Figure 3c.

Sxi

£m"

r ' — Ccm'-^

"'?cm~

Imperfect Connector on Standard Mount

»?c— AC

<J m

'7m-

£x2

l<- Range of possible

values for rjg

'dc

Figure 3. Equivalent circuit for the connector pair consisting of the

microcalorimeter test port and the connector on the standard mount for

different conditions.
In figure 3a, assume that the connector on the microcalorimeter test port

and the connector on the power standard are both perfect and lossless. The

cascading matrix Cj^ representing the connector pair is then simply C^-^ times

Cj^2 . ^^^ the measurement plane is clearly between the 2-ports represented by

these two matrices.

In figure 3b, assume that the connector on the microcalorimeter test port

is imperfect but the connector on the power standard is still perfect. Let

C^i' be the cascading matrix representing the connector pair, with component

matrices C^i '

, Qcm' > ^^"^ ^X2 • The matrix C^-^' represents all imperfections

near the junction. The measurement plane is chosen as plane 3 between C^.^' and Cx2 . so all of the imperfections in the connector pair are lumped into the

test port which, in this case, is the source of the imperfections.

In figure 3c, assume that the connector on the microcalorimeter test port

and the connector on the power standard are both imperfect. Let Cjj^" be the

cascading matrix representing the connector pair, with component matrices

C^i '

, Qcm" ' ^^*^ ^x2 '

• The matrix C^^jj^" includes the imperfections in both

connectors near the junction. Setting

^cm" ^ C(,jji'ACi^ (4.1)

makes reference plane 3 in figure 3c the same as reference plane 3 in figure

3b which corresponds to that for a mount with a perfect connector. Therefore,

3 is probably the best choice for the power reference plane because everything

to the left of this reference plane is associated with only the imperfect test

port. Let rjQ^ be the efficiency of the 2-port connector junction represented

by C(^jjj" , and let r;^,^! ^.nd r}.^ be the efficiencies corresponding to C^ni' ^^icl ACjj^, so

^c = 'Jcm^m- (4.2)

Then ^cxa. becomes part of the test port, and rj-^ becomes part of the mount.

However, it is usually not possible to separate r?^, into the components r^Q^xa. ^'^'^

»7jji. The efficiency r}^ of the 2-port connector junction causes a systematic

error in determining ryg because it is not clear how much of r/^, should be

assigned to the mount being calibrated. ry^, then becomes a systematic error associated with the microcalorimeter measurement of rjg.

5 . Six-Port Reference Plane for Power.

We next determine where the 6 -port reference plane is located for power

measurements. To do this, several different cases will be considered

depending on the quality of the connectors on the line standard. In all of

these cases, the connector on the test port of the 6 -port and the connector on

the power standard will be assumed to be imperfect.

Before the 6 -port can be calibrated for power measurements, it must be

calibrated for T measurements to obtain the "error box" parameters A,B,C.
5.1. Calibrating the Six-Port for T Measurements With a Perfect Line

In figure 4a, assume that the connector on the line standard is perfect.

This situation is similar to that shown in figure 3b. Let Cg be the cascading

matrix representing the connector pair, with component matrices C^i , Q.cs' ^''^^ £x2s- "^^^ matrix C^s includes all imperfections near the junction. The

measurement plane is chosen as plane 3 between C(,g and C-^2S' ^° ^^^ ^^ ^^^ imperfections in the connector pair are lumped into the test port which,

this case, is the source of the imperfections.

m

Since F and the error box parameters A, B, C refer to reference plane 3,

the 6-port reference plane for power must also be at 3.

6-port Test Port

Figure 4a. Cs

£xi I £cs

2

I

Perfect Connector on Line Standard

^X2S

3 F

P

6-port Test Port

Figure 4b

.

^xi Q-cm

£cs AC

Imperfect Connector on Standard Mount

ms ^x2m

Pd

Is

1 2 3 F

P Figure 4. Equivalent circuit for the connector pair consisting of the 6-port

test port and either the line connector or the connector on the standard

mount

.

5.2. Calibrating the Six-Port for Power Measurements Following Calibration for

F With a Perfect Line Standard.

In figure 4b, assume that the connector on the power standard is

imperfect. Let Cjj, be the cascading matrix representing the connector pair,

with component matrices C^j , C^m. ar^d Cx2m- Setting

£cm ~ £cs "^ms (5.1)
makes it possible to create reference plane 3 in figure 4b so that everything

to the left of 3 is the same as that in figure 4a. Since reference plane 3 is

the correct reference plane for measuring P and F, the effective efficiency tjq

of the transfer standard mount must be assigned to this reference plane. The

effective efficiency rj of other mounts will be measured at this reference

plane relative to fjg.

The matrix ACjjjg represents the difference between the perfect connector

on the line standard and the imperfect connector on the power standard. This

difference becomes part of the T of the power standard.

5.3. Calibrating the Six-Port for F Measurements With an Imperfect Line.

In this Section we consider the general case shown in figure 5 where the

connectors on the line standard and on the power standard are both imperfect . Let the cascading matrix C^ in figure 5a represent the connection when an imperfect line is used in the calibration, and let the components of this

matrix be Cxi, Q.cJi> ^^^ £x2iFor a line standard with perfect connectors such as shown in figure 4a,

the reference plane was chosen between the connector junction represented by

matrix C(.q and the uniform line represented by the matrix Cx2s- -"-^ ^^^ ^" port calibration, the line standard is always assumed to have perfect

connectors, and the reference plane is always taken to be between the matrix

representing the connector junction and the matrix representing the uniform

line. This is reference plane 4 in figure 5a, between the matrices C^,^ and

^x2i- This choice for the reference plane lumps all of the imperfections

associated with the test port and the line connector into the test port. The

imperfections in the line connector create errors that we need to analyze and

account for. Imperfections in the test port are still calibrated out.

Let P^ and F^ be the net power and reflection coefficient measured by the

6 -port at reference plane 4. If we let

Cci = Ccs-AC^s. (5.2)

then everything to the left of reference plane 3 in figure 5a is the same as everything to the left of reference plane 3 in figure 4a. P and F at

reference plane 3 are those values obtained after calibrating the 6 -port with

a line having perfect connectors. Therefore 3 is the correct reference plane

in figure 5a for measuring power and reflection coefficient. This means that

F^ is in error by AC^g, and P^ is in error by Arj , where Ar? is the efficiency

of the 2 -port whose cascading matrix is AC gg

:

An = P^/P. (5.3)

10
6-port Test Port

Figure 5a.

Imperfect Connector on Line Standard

^xi

£cs

Ci

Cci >

AC is

At, —

£x2i

3 r 4 r^

p p^

6-port Test Port

Figure 5b,

^xi

Imperfect Connector on Standard Mount

£cs

^m

~ ^cm

fici —

AC is <

ACmi

^x2m

Pd

Ar?s —

O

'7s

-

VJis

3 Fg 4 Fi,

Ps Pis

Figure 5. Equivalent circuit for the connector pair consisting of the 6-port

test port and either the line connector or the connector on the standard

mount .

6-port Test Port

I

Imperfect Connector on DUT Mount

iXl

^cs — Cu

^ Q-cu - £ci —^ ACui

ACis I <

^X2U

—

o

'?u

-

VJlu

3 r^ 4 Fiu

u Piu

Pdc

Figure 6. Equivalent circuit for the connector pair consisting of the 6-port

test port and the connector on the DUT mount.

11
5.4. Calibrating the Six-Port for Power Measurements Following Calibration for

r With an Imperfect Line Standard.

Figure 5b shows the situation when a power standard with an imperfect

connector is connected to the test port. This situation is the same as that

shown in figure 4b, where Cjjj is the cascading matrix representing the

connector pair, with component matrices Cj^^i , Qcra.' ^^"^ ^x2m- -'-^ figure 5b, C^,^

is first set equal to C,-. g

• ACg^g . and then C^^ is set equal to C^^g-AC^g to make

reference planes 3 and 4 in figure 5b the same as reference planes 3 and 4 in

figure 5a. The r/g of the power standard still refers to reference plane 3 as in figure 4b, since that is the correct plane of measurement, and P at

reference plane 3 is the correct power.

Let rjji be the effective efficiency measured by the 6 -port at reference

plane 4:

V,- ^. (5.4)

i

Then the correct effective efficiency rj at reference plane 3 can be written

P P P

dc dc i /c ex ^ = ^- = p7 p- = ''/^- ^^-^^

This equation says that the correct effective efficiency is equal to the

measured effective efficiency times an error term Ar? . For a line standard

with perfect connectors , Ar? = 1

Adding a subscript "s" in (5.5) to represent the power standard, we get

the relationship

'?s = ^Jls^^s (5-6)

between the effective efficiency, rjg , of the power standard at reference plane

3 and the measured effective efficiency, f?^s' ^^ reference plane 4.

Choosing reference plane 3 as the correct reference plane makes the

effective efficiency at that plane independent of the test port on the 6 -port,

independent of the connector on the line standard, and independent of the

connector on the power standard. The systematic error in the measured power

P^ and effective efficiency r?^ can be determined by estimating Arjg and Af?^ as discussed later.

5.5. Measuring r? ^ of a PUT Power Detector

Let figure 6 represent the general situation when the effective

efficiency ry^ of a power detector is measured following a 6 -port calibration

with an imperfect line and an imperfect standard mount.

The situation is the same as in figure 5b except that the connector on the

device under test is different from that on the standard mount, so subscripts

"m" and "s" are changed to "u" to represent the DUT . Adding a subscript "u"

in (5.5) to represent the device under test, we write

12
^u = VJlu^Vu- (5.7)

From (5.6), (5.7), and (2.9), the ratio of r/^ to rj^ Is

(5.8) u S.U. u iu u

s is s is s where N^^ and N^g ^^^ obtained from (2.6) and (2.8). They are repeated below

with the subscript i added to emphasize that they refer to reference plane 4

and are calculated from actual measured values of F^ at reference plane 4.

P, |l + c.r. 1^ _dcu _! 2 iu'

and

p |i + c r 1^

is P .

I

1

2

OS 1 - |r^J

We now need to determine Ajj^./^'/s which is the error in measuring ry^/r/g.

6. Efficiency of a 2 -Port.

The error term At? is the efficiency of a 2 -port whose cascading matrix is

AC ge; . The efficiency of any 2-port whose reference planes are designated 1 and 2, when energy is fed into arm 1, is given by [9]

\ |s2il^(i - Ir^l^

Va = p ^ o. (6.1)

1 |i - s^^rj - 1=11 - (hihi - =12=2i)'"lI

where Fl is the reflection coefficient of the termination on port 2, and the

Sj^A are the scattering parameters of the 2-port. In terms of the normalized

cascading parameters, the efficiency of this 2-port is

|d|'(i - |rj')

f}. = 5 ^, (6.2)

^ |1 -.cFj2 . |b + aFj2

where a, b, c, and d are the normalized cascading parameters defined by

a = 1^11/1^22 ~ "(S11S22 " S12S21) ~ "<i®t(S^)

,

^ - ^12/^22 =^11,

C = ^21/^22 ~ "^22'

d - l/r22 = S21, (6.3)

and the r^j are the cascading parameters of the 2-port.

13
7 . Efficiency of the Error 2 -port.

The efficiency, Ary , of the "error" 2 -port between reference planes 3 and

4 in figures 5 and 6 can be obtained from (6.2) using the cascading parameters

in the matrix AC gg in place of the a,b,c,d. From (5.2),

ACis = Scs'^ci (7.1)

Both C(,s and C^^ can be represented by the matrix in (3.3) if subscript "s" or

" £." is added to y, z, and AZq. Since the assumptions in (3.7) are satisfied,

substituting (3.3) into (7.1) leads to —cis

where

and

1-Ay-Az Az-Ay

Ay-Az 1+Ay+Az

+ Az,

Zi

r 1 -n

-1 1

AZq ^ AZq^ Az OS (Zos"^oi)/2Zo-

(7.2)

(7.3)

(7.4)

In (7.4), Zqs is the characteristic impedance of a perfectly uniform line

standard having diameters as specified in some standards document,^ and Zq^ is

the characteristic impedance of the actual line used to calibrate the 6 -port.

Comparing (7.2) with (6.3) defines the a,b,c,d parameters for the error

2-port. It will be more convenient to redefine the a,b,c,d parameters of the error 2-port as 1+Aa, Ab , Ac, 1+Ad. Then from (7.2) and (6.3),

a = 1+Aa = (1 - Ay - Az + AZo)/(l + Ay + Az + AZq) ^ 1 - 2Ay - 2Az

,

b = Ab = (Az - Ay - AZq)/(1 + Ay + Az + AZq) —Az-Ay - AZq,

c = Ac = (Ay - Az - AZq)/(1 + Ay + Az + AZq) -Ay-Az - AZq,

d = 1+Ad = 1/(1 + Ay + Az + AZq) = 1 - Ay - Az -AZ(

(7.5)

The approximations in (7.5) are valid when (3.7) is satisfied. Substituting

these parameters into (6.2) gives the desired expression for Ar?

|i+Adl2(i - |rj2)

Ar? |1 + AcrJ^ - |Ab + (l+Aa)rJ^

(7.6)

The outer and inner conductor diameters of a rigid precision 7 mm

coaxial air line standard are specified to be 7.000 and 3.040 mm respectively.

The conductivity of the material and the characteristic impedance of the line are not specified. See:

lEC Publication 457-2: Rigid precision coaxial lines and their associated

precision connectors, 1974, pp. 5 and 7. IEEE Standard for Precision Coaxial Connectors (IEEE Standards

Publication No. 287), IEEE Transactions on Instrumentation and Measurement, Vol

IM-17, Sept 1968, pp. 204-222.

14
For the ideal case, Aa = Ab = Ac = Ad = and At; = 1 . Typical values are given in the next Section.

If a second subscript s or u is added to designate the standard mount or

a DUT mount, (7.6) gives

^'?„ (1 - iTpJ^X |1 + Acr. |2 - |Ab + (1+Aa)r„ j2)

'u ^ ' iu' '^ ' ^s' ' ^ ' is

^''s (1 - |r. |2)( |i + Acr.j2 - lAb + (1+Aa)r,, |2)

(7.7)

is' ^^ ' iu' ' ^

' Jin' = 1 + €, (7.8)

where e is the error in the ratio of the effective efficiencies due to

imperfect impedance standards and connectors. Equation (7.7) shows that e = when F^^ = F^g or when Aa = Ab = Ac = . If Aa, Ab , and Ac are known, e becomes a correction to the ratio of the measured effective efficiencies.

However Aa, Ab , and Ac are usually unknown, and e is used to estimate the

systematic error in the ratio.

8 . Approximation for e

Let the real and imaginary components of Ay, Az , AZq, and F^ be

Ay = AG + jAB == jAB,

Az = Ar + jAx, (8.1)

AZq = Aro + JAXq,

Ti - Tr + jFx.

Then the real and imaginary components of Aa, Ab , Ac in (7.5) become

Aa = Aaj. + jAax =- (-2Ar) + j(-2AB-2Ax),

Ab = Abj. + jAbx = (Ar-Aro) + j (Ax-AB-AXq) , Ac = ACj. + jACx = (-Ar-Aro) + j (AB-Ax-AXq) . (8.2)

Appendix A shows that substituting these expressions into (7.7) leads to the

following approximate equation for e

€ - 4[AFrAr - AFxAXq - A|F|2Ar], (8.3)

where

Arr - r^ur - r^sr. (8-4)

A|rh - iTiuh - Irish- (8.6)

Estimates of Ar and AXq for the NIST 7 mm lines are shown in figure 7. These

estimates and the following expressions in Table 1 for Ay, Az , and AZq are taken from Table 6 of Part 4 of reference 8. When the values of Ay, Az , and

AZq at 18 GHz from Table 1 are substituted into (8.2), they lead to the

following worst-case errors for Aa, Ab , and Ac:

|Aa| - 0.0037, |Ab| - 0.0021, |Ac| - 0.0021. (8.10)

15
Table 1

1 GHz 18 GHz

Ay = AG + jAB; AG - , AB - ±0.000032f. ±0.00003 ±0.00058 (8.7)

Az = Ar + jAx; Ar - -0, +0.00007yf. 0.00007 0.00030

Ax - -0, +0.00007f. 0.00007 0.00126 (8.8)

Azq ^ Aro + jAXo; ArQ - ±0.00061 ± 0.00004/yf. ±0.00065 ±0.00062

AXq - ±0.00004/71. ±0.00004 ±0.00001 * '' where f is the frequency in GHz

9

Measured Power Constant

Substituting (7.8) into (5.8) leads to

is

Comparing this with (2.5) and (2.7) suggests that we write (9.1) as

where

%^^ ^^-'^

P

is

is the effective or measured power constant that refers to reference plane 4

in figure 6. K^p differs from Kp when F^g ^ ^s10

Total Systematic Error in n ^

Equation (9.1) or (9.2) is used to calculate r/^. Differentiating (9.1)

gives the following expression for the total systematic error dr/^^ in rj^:

d-u ' f^d-s - J7N^f2S;^<i(Niu/Nis) + ffd^, (10.1)

which reduces to

u s Jen Ms

The term dr]g is the total systematic error in determining fjg in the

microcalorimeter . For the second term on the right of (10.2) we need the

ratio N^^/N^s which from (5.9) and (5.10) is

Np P. P |1 + cF- 1^ 1 - |F. I^ xu. _ dcu OS J iu_| ' is ' .^ -. N~PP 9 9' \ ^^ ^

is ^dcs ou ll + cF„ I 1 - If „ I

' is' ' iu'

Differentiating this gives

16
where

and

^(^iu/^is) dcu dcs

d(

h —

:p /p ) diG 2 os' ou ' dM

^iu/^is P /P

dcv: dcs

P /P ' ^ 2 ^

os'^ ou G

_l_i_cru

1 + cFg '

>I 1

- Ts 2 " - 1

- Tu 2

•

(10.4)

(10.5)

(10.6)

Each of the error terms in (10.2) and (10.4) will now be evaluated to obtain

an estimate of the total systematic error in measuring power.

10.1. Error in dc Power Ratio

The error in the dc power ratio, the first term on the right in (10.4),

is given by

dcu dcs _ dcu dcs

(P, /P, ) ~ P , 'p.

dcu dcs dcu dcs

(10.7)

For the NIST dual 6 -port ANA, Pdcu. ^^^ ^dcs ^^^ both measured by the same NIST

type IV power meter which is connected through a scanner to a 5H digit DVM.

The power meter voltages can be read with or without a reference voltage

generator (RVG) in series with the DVM as shown in figure 8. The RVG enables

the DVM to operate on its optimum range for measuring small changes in

voltage, reducing the effects of the percent-of -fullscale error in the DVM.

Figure 9 shows the relative error-^ dP^^^/P^j^ expressed in percent when the

power is measured with a NIST type IV power meter and a HP3457 DVM, with and

without an RVG. These curves show that the error in P^^ due to the power

meter is negligible, and that most of the error is due to the DVM. If two

different DVMs are used to measure P^cu ^^^^ ^dcs > ^^^ ^^° terms on the right

of (10.7) can add, creating an error as large as 0.06% at 10 mW, and larger at

lower power levels.

In the NIST system, P^cu ^^"^ ^dcs ^^® read by the same power meter and

DVM, and P^cu ~ ^dcs- '^® systematic errors dP^j^u ^^'^ "^^dcs ^^® therefore

approximately the same, and the terms on the right of (10.7) cancel. Setting

dPdcu = dPdcs = dPdc (10.8)

reduces (10.7) to

2-'Obtained from a computer program written by Neil Larsen, based on the error analysis developed in his paper "A New Self -Balancing DC- Substitution RF

Power Meter," IEEE Transactions on Instrumentation and Measurement, vol. IM25, Dec. 1976, pp. 343-347.

17
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10

Figure 9. Relative error dP^c/Pdc expressed in percent when the dc power is

measured with an NIST type IV power meter and a 5H digit DVM, with

or without a Reference Voltage Generator (RVG)

20
.004 r=

3 .0035 ^^~~—

^/o live. ,003 — ~

^ .0025 — '

~

- R0: 200 c hms

j_ .002 —

VI : 2.500 \ o 1 ts

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~i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a - - 12 3 4 5 B 7 8 3 10

PoLue r , mi 1 1 i uat t s

Figure 10. Uncertainty dP^c in the dc power measured with an NIST type IV power

meter and a 5h digit DVM, with or without a Reference Voltage

Generator (RVG)

.

21
dcu^ dcs _ dcs dcu dc

(P /P ) ~ P P

dcu dcs dcu dcs

(10.9)

The uncertainty dP^j^ in the dc power is shown in figure 10. The error without

an RVG is given by

dPdc ^ ±(0.0035 - O.OOOOSPdc). mW. (10.10)

where dP^j^ and P^j^, are in milliwatts. This uncertainty assumes that the

power-off voltage V^ is = 2.5 V. As V^ varies between 2.4 V and 2.6 V, the

first constant in (10.10) varies from 0.00321 to 0.00375. Typically P^cs ^^^ ^dcu differ by no more than 0.5 mW in the NIST system. Assioming that

Pdcu ^ Pdcs =- 10 "iW, (10.11)

|Pdcs - Pdcul < 0-5 mW, (10.12)

and that dP,^^ is given by (10.10), (10.9) gives a worst-case uncertainty of

^^^dcu/^dcs^

= ±0.00002. (10.13)

^ dcu^ dcs^

This uncertainty is reduced by a factor of 6 to ±0.000003 if an RVG is used.

10.2. Error in rf Power Ratio at Six-Port Sidearm

The second term on the right side of (10.4) is the error in the ratio of

the rf power measured at the 6 -port reference sidearm. Each sidearm of the 6-

port is terminated with a thermistor mount which is used to measure the rf

power in terms of the dc substituted power. The net rf power Pq at the

reference port sidearm can be written

Po = PodcAo. (10.14)

where Podc ^^ ^^^ ^^ substituted power, and t^q is the effective efficiency of

the sidearm mount. Normally, t/q does not need to be known since it cancels in

the power ratios. For the NIST dual 6-port ANA, Podc ^^ measured by a NIST

type IV power meter which is connected through a scanner to the DVM. From

(10.14),

P »? P , _os^^u_odcs_ ^^Q^5^

ou OS odcu

A subscript has been added to f?Q to allow for nonlinearity in the sidearm

power sensor. Nonlinearity in a dual element thermistor mount can be caused

by differences in the two thermistor elements [10] . Taking derivatives of

(10.15) leads to

d(P /P ) d7? d77 dP ^ P ^ OS ou ou OS odes odcu ,.^ ,,, p /p ^~r ' ~r^ ~^~;~ ' f—• ^^^-^^^

OS ou ou OS odes odcu

22
Since Pqs == ^ou' ^^^ last two terms in (10.16) can be combined in the same way as the last two terms in (10.7) were combined to give

odes odcu odcu odes ,_, /ia it\

-p p = —^ p dg^. (10.17)

odes odcu odes odcu

where P^(, is given by (10.10). In the NIST dual 6-port design, Pq is about

half the power at the test port. With 10 mW at the test port, Pqs and Yq^^ are both about 5 mW. Assuming that (10.12) is true we will have

IPodcu - PodesI < 0-25 mW. (10.18)

Then (10.17) gives

odes odcu

P ' P

odes odcu

= ±0.00004. (10.19)

10.2.1. Error Due to Detector Nonlinearity

.

The net rf power P^-f into a bolometrie type power sensor can be written

Prf = bPdc + AP = Pdc/^?, (10.20)

where P^j^, is the de substituted power, AP is the de-rf substitution error at

Prf' '-/'-* ^'^ ^^^ effective efficiency of the mount at low rf power levels where

AP approaches zero, and rj is the effective efficiency at the rf power level

PrfExperimental data reported by Engen [10] indicate that the dual element

dc-rf substitution error has the form

where in the region from 5 to 10 mW where the 6-port sidearm thermistor mounts

are operated the exponent a is not constant but increases from 1 to almost 2.

1 < a < 2. (10.22)

At an rf power level of 10 mW, Engen' s tests on 10 pairs of elements in three

different mounts indicated a worst-case error of 1%, and a typical error of

0.2%. More recent tests by Neil Larsen^ at NIST on mounts similar to those

used on the NIST dual 6-port indicated a dual element error of only 0.02% or

less. Inconsistencies in the sidearm power readings of the NIST dual 6-port

Nell Larsen compared the output of a coaxial dual element mount with the

output from a single element waveguide mount, where both mounts were connected

to the arms of an X-band waveguide magic -T. Measurements were made from 1 to

10 mW at 10 GHz. Since the waveguide mount has no dual element error, the

difference in the responses was taken to be the dual element error of the

coaxial mount. Neil Larsen, NIST, 723.01, Boulder, Colo. 80303.

23
ANA'S indicate that the total nonlinearity in these power readings is

approximately only 0.01%. These percent errors give the following values of

Kg at 10 mW.

Engen worst case: K^ = IxlO"**

Engen typical case: K^ = 2x10"^

Larsen: K^ = 2x10"^ (10.23)

6-port inconsistencies: K^ = 1x10"^

A conservative estimate of K^ for the 6-port systems would be double the value

obtained by Larsen, or K^ = 4x10"^.

Using (10.21) with a = 2 for the worst case, the ry terms in (10.16)

become —

dr? — - —

dr? — = K (P

99- P ) = K (P + P )(P - P ). (10.24)

ri n e OS ou e os ou os ou

ou OS

When Podcs ^^"^ ^odcu ^^^ about 5 mW and are within 0.25 mW of each other, and

if we choose for Kg the value of 4x10"^, (10.24) gives

dfj dry —— - —— = ±0.00001. (10.25) ou OS

As< (10.16) indicates, the total uncertainty in the rf power ratio at the

6-port reference sidearm is the sum of (10.25) and (10.19).

d(P /P ) —p°^ ""^ = ±0.00005. (10.26)

OS ou

10.3. Error in t? ^ Due to Nonlinearity in Power Standard.

The value of r?g that is used in (9.1) is obtained from measurements made

in the microcalorimeter at a specific rf power level, say Pjnc > which is

usually near 10 mW. If the power level Pg into the standard mount when it is

connected to the 6-port is different from Pmc > ^^® value of r/g at Pg could

differ from the value of ryg at Pjjj^, because of nonlinearity in the standard

mount. Equations for calculating this error are the same as those derived in

the previous Section. Let

^mc ^ effective efficiency of the standard mount as measured in the

microcalorimeter at power level Pmc . ^^^ 1®^

f/ep = effective efficiency of this same standard mount at a power level Pg

which is the net power at into this mount when connected to the 6-

port.

The effective efficiency rj^^^ which is measured in the microcalorimeter at

a power level Pjn^ includes any nonlinearity such as the dual element

substitution error. When the standard mount is used at a different power

level such as Pg , it is only the change in the effective efficiency due to the

different power level that causes an error. An expression for this change in

effective efficiency will be derived next.

24
Replacing the subscript rf in (10.20) with mc to indicate power levels

used in the microcalorimeter , and with 6p to indicate power levels at the 6-

port test port, (10.20) gives

(10.27)

(10.28)

The difference is

This can be written

1 = b +

APmr,

Pdcmc

1 = b +

APfip

Pdcep

1 APmr AP ip_

'?mc ''ep Pdcmc Pdcep

(10.29)

^6p ^mc. ^mc''?6p

Pmr AP^p

PdcmcPmc

Pep APgp

PdcepPep

1 AP^p

'?mc Pmc

1 .APfip

'?ep Pep

fJep - Vmr. ^mc

'?f5p. APmr.

'Jmc Pmc

. APrp

Pep

• (10.30)

When Pmc ~ Pgp, then r/mc - rjgp, and (10.30) gives the following desired

expression for the dual element dc-rf substitution error:

A^mr. ^

AP^p _ APgp

^ (10.31)

^mc Pmc Pep

Substituting (10.21) into (10.31) with a = 2 gives

^ = Ke(Pn,c' - Pep') = Ke(Pr„c + Pep)(Pmc " Pep)- (10.32)

For the conditions stated in (10. 11) - (10. 12)

,

Pmc ^^ Pep - 10 mW, | Pj^c - PepI < 0.5 mW. (10.33)

If Kg = 4xl0-«, these values in (10.32) give

An^^ = ±0.00004. (10.34)

'?mc

10.4. Error in | g|

Equations for calculating d|G|2/|G|2 are derived in Appendix B. From

(B37) of Appendix B,

lAT^dCy - 2ArvdCv. (10.35)

1 + 2(Arj.Ci. - AFxCx)

'

To evaluate this expression we need to estimate dFj-, dF^, dCj-, and dC^. These errors are due to imperfections in the 6 -port and not in the standards used to

25
calibrate the 6 -port. They can be obtained from the Inconslstenqles In the 6-

port sldearm power readings [8]. Typical values are

\dTj.\ - IdFxl - |dCr| - |dCx| - 10-*. (10.36)

If we assume that

jr^l < 0.15, ITsI < 0.15, lc| < 0.3, (10.37)

the inequalities in (B24) give

lAFrl < 0.3, IaFxI < 0.3. (10.38)

For these values, (10.35) gives a worst-case error of

d|G|2/|Gh ^ ±0.00015. (10.39)

10.5. Error in M

Equations for calculating dM/M are derived in Appendix C. From (C14) of

Appendix C

,

dM/M - 2(Ari.drr + AFxdFx) . (10.40)

For the values assumed in (10 . 36) - (10 . 38) , (10.40) gives as a worst-case error

dM/M - ±0.00012. (10.41)

10.6. Error Due to Imperfect Impedance Standards and Connectors

The last term on the right of (10.2) is obtained by differentiating (8.3)

to get

de - ±4[|Arrd(Ar)| + |Arxd(AXo) | + | A|r| 2d(Ar) |

]

(10.42)

The d(Ar) and d(AxQ) in (10.42) are the uncertainties in estimating Ar and

AXq . If Ar and AXq were known, e is a correction and not an error. Then

d(Ar) and d(AxQ) would be zero, and so would de be 0. If Ar and AXq are error

limits, then d(Ar) and d(AxQ) in (10.42) are simply Ar and AXq .

For 7 mm connectors, Ar and Axq are given in figure 7. Over the 1 to 18

GHz frequency range this figure gives the limits

|Ar| < 0.0003, and [AXqI < 0.00004. (10.43)

Then for the values assumed in (10 . 36) - (10 . 38) , (10.42) gives as a worst-case

error

de

1+e

de - ±4[0.9 + 0.12 + 0.135]xl0-'' = ±0.00046. (10.44)

This is the largest of all the errors in measuring ratios of effective

efficiency.

26
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!l -^ - VI <J !m •H 01

O

4J c . . . o o CO

C

O rH rC c •l-l u o* o o o o o o o ^^^ v«^ 5 O CO 5m CO a 4-1 0) 0) CO w +1 4-1 +1 -hi +1 +1 +1 -I-l -I-l -I-l • B "d 5m o < "d ^ O g iJ s-l 13 Ph O VI— d) 4-1 CO MM C

<u S-lO II II II II II II II II II s in 73 4J -— P- to 3 O CO

>-l (U s CM

O cj) •• E 0) iJ -d

p CJ o tH o UJ

a in * (U- tU tM 4-1

OO 0)

u c 'd 5-1 s o s < CO CO < . O >^

C

C CO CM

a

(0

p < < < < < < < o , ^ c •H to (U E

a> IIV o •• ^ 0) ^ <u to -H S II II II II II II II II V c o 4J -H 4J j:: •H

y^S - (U Sm -P Sm 4J o u ,^<^ N u CO ^^ 3 > •d O O Q) ^ CO^rH

<a

au y~^ ^— sr- c- /•^ 3 OtM c P. 0) CO 4-1 "d

O 4-1 > )-l c o 3 o o < < u CO V) 3 cwPm bO to (4M -H C M CO o 0) T -0 c o e S \ sr c- c- sr S-l 4J Sm 0) > CO •H o v cm CLi P4 PL, sr \ \ d \ \ \ \ Pm ' to CO CO tu ^ 0) Sm

Q) \ •^ \ \ \ f< c^ s^ ^ 3 V 3 •H •d tU P, 4-1 (1) 5m 0) 4J e s (/ tn o ^^ ^ < < c- sr- c- sr 1 CO Sm 4-1 C C 01 4-1

>-l CO c o c O E o N.-^ v-x -d -d 0(^2 CO TJ to •rH t-I JZ

O o !-i T T3 p^ Pli c- -d -d COPLi > n C4M C .H —1 —

5 CO a CO

PL, Pm Pm -o

<! T)

Cfl

C4-H

5-1

— Dd

to

O CO cnrH 4-1

U - CO

X ^ M emC 4-1 bOtU CM "3

O 04= « mH o CO B^^ CO CO 1^ C 3 0) X O ^ n 4-> c- » •rM (]; 4-1 (4M <

ua

O \ 13 4-1 0) --I

)_l

O Sm •<-, 0) td -1 0) 3 e 3 s: 0) cO 0) + x: H g C sr- -V o 4-1 —1 r; P- O 4-J C 5 in ^ to '5 OJ 5m •o , oo

4-1O s 4-1

)1 -H •rM 3 '^

X CO

4-1 .4-1 < (4M

11 o c o o & O ^ (4M • CO ^ (4M O o ^ •o -d 4J rH CO to CM 0) c c 3 cd c CM 3 w t! CO 0) S:- !l d 0) o • 0) 7j (1) CM O 0) ^ CJ

4J "o to

O

to 0)

M "d

[_, rO X^ 0) 4-1

\rH

u c X S 5-1 •iH y^ Ct o CO f-i CO <u C4-I CO !l 3 4-1 =^ >

V4 T3 T5 Sh

O (4-1 COO o

V4 s - 5-1 3 S-l 3 CO <UM rH « s-l X T) CO CO i-i Pm O 0) C*M CO CO >^ t-H C -d CO c Pm S 5m CO C o <u u CO c <U (D !l 0) — \ p. Sm a^ E CO O-H

iH •d •I-l - CO £ -d

B Sm & M E

4J •H !-l CO >^ s-l J-i 4J s-l di 3 .. Q) e CO to^rH

a - bO tsj in , to 0)

v-'

O

rt 4J

w CO s-l 4J t-l E-i CO 4J CO s-l (4-14-15 •H III c e c 0) o •H T) •d 5-1 •d 3 S-< U O O « 5-1 32 ^ c s-l S-l Q) O c to Ph o a- tJO rH tU CO OJ > O 4J tj. Sm 4J N o; •p O s Sj •H s-l CO O o c; Pu CO CO Cu c < rC u i-i a) o ^ QJ <U 4-) 4J C 4J (U . . o •rM p

>i

U >-i p- c c CO ^ CO

B 4J (D T) -d VI III 0) CJ —- O ^ CO to 3 4J ^ 0) tn < 0) 5M

C4M

u Q) =5 1 o S •H <D Q) •d 5-1

O 3 O CO 03 «sD c > -H :i :3 0) c u C O C 0) ^M •rM

m tw CO

Q c '0 •d p, r-l 0) •r-l O- <U

X o p o o Q) 4-> s-l O £ S S-l 4-1 X (1> n, -C 3 T^ 4-1 C

.. -d ° -H

Sm

O

B CO o T3 c o s •H >^ o >^ 4-1 0) Q) CON >~ 0) •u c 4-> a, 4.) C (4-1 C P-

>^ o o O o CO TJ c p: P q

G Q) tU •!-!

O CO ^ CO rt (-1 5m o; -r -H 0) •1-4 •rH (U S-l •H •H g •H q •rH B 5-1 0) E 3 4-> P 4-) S-l CO CO •rH CO Q) OTS o o ^ o

CO

C 2 -^N

C

C CO "^^ tsl 0) > 0) E o CO CO 0) 0) Xl >. ;>, r^ 4J 4-1 5-1 4-1 -C c •rM C p w !-l !-l -0 u C u 4.) S-l >^ U 3 SM Pm CO P tM W <u CO c c 4J <U U 0) CO O X! 0) 4J _, 5m O —1

S-l

u e s 4-) •H •r-l (J

O

C o to P- {-. CO CO '"] T) 0) CO 9^ CD u W CO CO 0) C •H c 0) ' >i •^ m C 4M Q) rM > > CO s-l 4-> 4J <4-l 3 CO 3 B v^ 4J C 5m CO ?> to (4-1 Sm to o O (D a; S-l 5-1 Sh !-i 4-1 C •iM 0) 01 *^ tH 3 a P-T) 13 <u <D 0) d) r-H !-l rH 0) 01 'H 5 5m - -d 0) P

•r^

O s O

O &. CO (U CO CO ,c x: CO >-,O 0) J_> ^ o o <J-I w a^ o C c E 4J O 4-1 c 4J 4-J 4J 4J a- "d -C jM -d P CO -d M Ph P p M O C O o Sm C 0) 4-1 o 0) • • H P H •iH 4-1 4-1 0) •iH 0) N p. N CO OJ c 4-1 CO tO CJ CO ^ .H CO •H •rM x: o «-H CM ro <t in VO Q C iJ iJ rH •rH 4-1 r-M p

•H • . e 5-1 Sm 3 Sm CO CO CO c

4J o o O

O o O 3 (U 0) <D (4M E Sm 0) E X (4-1

O »-H rH rH »H ^H »-H CO 15 5 tu

OO 5m < 4-1 5m < O

Qi CO

OO rC

C

O

O

M <: Pm PM

H P Cm Z; 2i

27
1 1 . Total Systematic Error in n ^ for 7 mm Connectors

Table 2 summarizes all of the components of error for power measurements

in 7 mm connectors. All of the components of error in section 10 have been

substituted in (10.2) to obtain this result. All of these errors are independent of the type of connector used except de in (10.42) which is due to

imperfect impedance standards and connectors.

12 . Outline of Measurements and Calculations

This outline assumes that there is only one standard mount used in the

calibration of the 6-ports.

For the standard mount,

• Measure F^g, P^cs > ^ ^os f°^ ^ connections on each test port, where

k > 3 connections.

r^g = r of the standard mount as measured by the 6 -port,

^dcs ^ change in the dc power applied to the standard bolometer

mount when rf power is applied to the input of the mount.

Pqs = rf power at the 6 -port reference sidearm when rf power is

applied to the input of the standard mount. For thermistor

detectors we set Pqs = ^Podcs > where K is a constant and Podcs

* is the change in the dc power applied to the reference sidearm

mount when rf power is applied to the input of the standard

mount. When K is independent of power level, which we assume

is true for the 6 -port sidearm detectors, K drops out of the

following calculations and can be set equal to 1 . Then

Pos ^ ^odcs

• Calculate N^g from (5.10), and K^p from (9.3) for each connection.

^dcs 1^ -^ Vis'^ '^s N„^=-^^ ^-^, K = -^. (5. 10), (9. 3) '^ '-OS ^ - \r,f ' '^ ^.s

where ryg is the effective efficiency of the standard mount.

• Calculate K^p , the average K^p for each test port from the k values

of K^p for that test port.

• Calculate and save the average F^g, Pdcs > ^ ^os ^°^ 6 -port #1, and

the corresponding averages for 6 -port #2 for use in calculating

systematic errors.

For each mount under test,

• Measure F^^, P^cu' ^ ^ou f°^ ^ connections on each test port, where

'^ Jixx. = r of the mount under test measured by the 6 -port.

P(jcu ^ change in the dc power applied to the mount under test when

rf power is applied to the input of the mount.

Pqu = ^f power at the 6 -port reference sidearm when rf power is

applied to the input of the mount under test. As with Pqs- set

Pqu ^ ^odcu' where Podcu ^^ the change in the dc power applied to

the reference sidearm mount when rf power is applied to the input of

the mount under test.

28
For each connection calculate N^^^ from (5.9), and r?^ from (9.2)

using K^p, the average value of K^p for that test port.

"iu - rf , I, |2 "u-V^u- (5. 9), (9. 2)

iu' ou i - |r,

•

Calculate r/m , the average rj^ from the n values of rj^ on 6 -port #1,

and »7u2 . the corresponding average on 6 -port #2.

Calculate ^^ = (^ui + '7u2)/2-

• Calculate the long term random error S]-, (-Sjjgg) , the short term

random error s^, (=8^) , and the total random error Sy in 17^. following

a procedure similar to that described in Part 5 of reference [8]

.

• Calculate total systematic error A^ in Ifj^-j^ from (10.2) and related

equations listed in the right column of table 2 using average values

of r^s. Pdcs. Pqs. r^u. ^dcu. & ^ou fo^ 6-port #1. Calculate

corresponding total systematic error Aj in r?y^2 • • Calculate total systematic error A = (A^ + b.^)/!.

• Calculate total uncertainty U-p in 17^. U'p = A + 3sy.

The equations for calculating the components of the systematic error are collected below. When (10.4) is substituted into (10.2), there are 6 components of the total systematic error which we will write as

A = As + A(ic+ Arf + Aq + Aj^ + Ag (12.1)

where

Ag = dr/s/r?s is the uncertainty in the effective efficiency of the power

standard, presently given as

±As = 0.865 + 0.105yF, (12.2)

where f is in gigahertz.

Adc "" uncertainty in measuring the dc power ratio,

P - P dcs dcu

*S. - F P

,

''Sc ("•')

dcu dcs

where

dPdc ^ ±(0.0035 - O.OOOOSPdc). (10.10)

P<5c is the average value of P^cs ^^^ ^dcu' ^'^^ power is in milliwatts.

Aj-f = uncertainty in measuring the rf power ratio at the 6-port sidearm,

±Arf-2xlO-«(P^^+P^^)(P^^- P^^) (10.24)

^ !o|CHlIod=s^^^^ (10.17)

odes odcu

Aq = uncertainty in G due to dFj- , dP^, and dC which are obtained by

propagating the "power meter residuals" into T and C [8]. Since

this propagation is not presently being done, we will set each of

these errors equal to the power meter residual itself. This usually

29
results in values of dFj- =- dPx == dCj- = dC^ == lO"* .

±Ag = 2{|cl2Arrdrj- + IcI^Ar^drx + AFj-dCj- - ArxdCx) (10.35)

where in this equation we will use the absolute values of the

differences

Arr - Ir^sr- Tiurl. (8-^)

AFx = Ir^sx- r^uxl- (8.5)

Aj^ = uncertainty in M due to dr^- and dF^,

±Aj^ - -2(AT^dT^ + AFxdFx) . (10.40)

Ag = worst-case uncertainty in the position of the reference plane due to

imperfect standards and connectors,

Ag = de - ±4[|Arrd(Ar)| + |Arxd(AXo)l + | A|r| 2d(Ar) j ] (10.42)

-^-^^ A|r|2 = lr,3h - Ir^uP. (8-6)

d(AXo) = AXq - 0.00004/71, (figure 7)

d(Ar) = Ar - 0.0000771, (figure 7)

and where f is in gigahertz

.

The total systematic error excluding that due to the power standard is Agp

which is the 6 -port error in measuring the ratio of two effective

efficiencies

.

Asp = Adc + Arf + Ag + Am + A,. (12.3)

13 . Conclusions and Observations.

Equations have been derived for calculating estimates of the systematic

error in measuring ratios of effective efficiency of bolometric power sensors

on a dual 6 -port ANA. The largest source of error is imperfections in the

transmission line impedance standards used to calibrate the dual 6 -port for

making F measurements. This error, given by e in (8.3), is more than half the

total systematic error in measuring ratios of effective efficiency. This error is based on a model that accounts for connector imperfections as well as imperfections in the line standard itself.

One important observation is that the error e is usually considerably

less when calculated from (7.8) or (8.3) than it is when calculated directly

from (2.9) if worst-case errors are chosen for F and C. The reason for this

is because errors in F and C are correlated, and cannot be chosen independent

of one another. The correlation between errors in F and C is derived and

explained in Appendix D. For realistic situations, the error in C is nearly

equal to the error in F, and these errors then nearly cancel in (2.9),

30
For bolometric power sensors with 7 mm connectors, the total systematic

error in measuring ratios of effective efficiency on the NIST 1-18 GHz dual 6-

port ANA is estimated to be 0.00084 which occurs at 18 GHz. To this must be

added the systematic error in rjg of the power standard to get the total

systematic error in rj^j^ of the DUT power sensor. The systematic error in r^g of

the power standard is usually considerably larger than the 0.00084 contributed

by the 6 -port.

References

[I] Engen, G. F. , A Self-Balancing Direct-Current Bridge for Accurate

Bolometric Power Measurement. J. Res. Nat. Bur. Stand. (U.S.) 59(2):

101-105; 1957 August.

[2] Larsen, N. T.; CIague , F. R. The NBS Type II Power Measurement System.

Adv. Instrum, 25(3), 712-70; in Proc . 25th Annu. ISA Conf. (Philadelphia,

Pa.), 1970 October 26-29.

[3] Larsen, N. T. , A New Self -Balancing DC-Substitution RF Power Meter, IEEE

Trans. Instrum. Measurement, IM-25(4): 343-347; 1976 December.

[4] Runfelt, A. Y. ; Elwell, L. B. Radio Frequency Power Measurements. Proc.

IEEE., 55(6): 837-850; 1967 June. A review paper.

[5] Fantom, A. E., Power and Energy: National Standards. Proc. IEEE., 74(1):

94-101; 1986 January. A review paper.

[6] Engen, G. F. ; Hoer, C. A. Thru-Reflect-Line : An Improved Technique for

Calibrating the Dual Six-Port Automatic Network Analyzer. IEEE Trans.

Microwave Theory Tech., MTT-27(12): 987-993; 1979 December.

[7] Hoer, C. A.; Engen, G. F. On-Line Accuracy Assessment for the Dual

Six-Port ANA: Extension to Nonmating Connectors. IEEE Trans.

Instrumentation and Measurement . IM-36(2): 524-529; 1987 June.

[8] Hoer, C. A.; Engen, G. F. ; Juroshek, J. R. ; Judish, R. M. Theory,

Description, and Error Analysis for the NIST 2-18 GHz Dual Six-Port ANA.

To be published as an NIST Tech Note, 1989.

[9] Kerns, D. M. ; Beatty, R. W. Basic Theory of Waveguide Junctions and

Introductory Microwave Network Analysis. Pergamon Press, 1967. See page

77 for eq(3.2), page 80 for eq(3.3), and page 49 for eq(6.1).

[10] Engen, G. F. , A DC-RF Substitution Error in Dual-Element Bolometer

Mounts. IEEE Transaction on Instrumentation and Measurement, 58-64; 1964

June -Sept.

[II] Hoer, C. A., Equivalent Circuit for Imperfect Transmission Line

Connectors. Conference on Precision Electromagnetic Measurements,

Tsukuba, Japan, 1988, 264-265; 7-10 June. See also Appendix F of

reference [8] above for a more detailed derivation of the equivalent

circuit.

31

Appendix A. Approximate Expression for Ary^j^/ATyg

The purpose of this Appendix is to derive a simple approximate expression

for Arj^/Ar]g for use in estimating errors due to imperfections in the impedance

standard. From (7.7) the exact expression for Arj^/Ar/g is

% (1 - |r,J^( li+Acr^J^ - |Ab^ (UAa)r^J^ )

:= . . (Al ) ^''s (1 - lr^J^)( |i + Acr^^l^ - |Ab + (i+Aa)r^^l^

^ 1 + e. (A2)

Our purpose here is to get a simple expression for e which is zero for perfect

connectors and line standards. Let the real and Imaginary components of the A

terms be

Aa = La.-^ + jAa^,

Ab = Abj. + jAb^,

Ac = ACj- + jAc^. (A3)

Expanding the terms in (Al) with those in (A3) , the terms

|1 + Acr^s|2 and |i + Acr^^|2

become

|1 + Acr|2 = |i + [Ac^ + jAcxKFr + jrx)M

= |l + Ac^r^ - ACxFx + j[Acrrx + ACxFrll^

= 1 + 2[Lcy.Ty. - Ac^Fx]

+ [Ac^^r^ - ACxFxl^

+ [Acj-Fx + ACxFrl^

= 1 + 2[ACrrr - ACxFx]

+ Acj.2rj.2 . 2ACi-rj-ACxrx + ACx^Tx^

+ Acj-2rj^2 + 2Ac^rxACxrj. + ACx^Tj-^

= 1 + l[Lc^T^ - ACxFx] + Ac;^2|r|2 + Ac^^jrl^

= 1 + 2[ACrrj- - Ac^Fxl + lAc|2|r|2, (A4)

where T = T^j- + jTx represents either T^g or T^^. The terms

|Ab + (l+Aa)r^sP and JAb + (1+Aa)r^u|2

can be expanded to give

|Ab + (1+Aa)r|2 = |Abj.+jAbx + ( l+Aa^+j Aax) (Tj- + jrx)|2

= |Abj- + (1+Aaj.)rj- - AaxFx + j [Abx + (1+Aar)rx + AaxFj-lj^

= [Tj- + (Ab^+Aaj-r^^-Aaxrx) + j [Tx + (Abx+Aa^Fx+AaxFj.) ] | ^

Al
= rj-2 + 2rj-[Abj- + Aa^Fi- - Aa^Fx] + [Abj- + Aaj-Fj- - Aa^rx]^

+T^^ + 2rx[Abx + Aa^Fx + Aaxr^-j + [Abx + Aaj-Fx + AaxFj-j^

= |rl2 + 2rj-Abr + 2rxAbx + 2Aaj-|r|2 + Ab^^ + Aa^2r^2 + Aax^Tx^

+ 2Ab;f-Aaj-rj- - 2Abj-Aaxrx - 2Aaj-rj-Aaxrx + Abx^ + Aa^^Tx^ + Aax^T^-^ + 2AbxAaj-rx + 2AbxAaxri- + 2Aai-rxAaxrj-

= |r|2 + 2r;^Abr + 2rxAbx + 2Aaj^|r|2 + lAbp + |Aal2|r|2 + 2rj-(Abi-Aaj- + AbxAax)

+ 2rx(AbxAaj- - Ab^-Aax) , (A5)

where again T = Tj- + Tx represents either T^g or r^,j^. The first line of (A5)

contains the most significant error terms. Lines 2, 3, and 4 of (A5) contain

second order error terms (products or squares of A error terms)

Next, write (29) as

. — ry • (Ad)

(1 - |r,|^)

Substituting (A4) and (A5) into (A6) gives

|l+Ad|2/Aj7 = {1 - |r|2

2 + 2[AcJ.^;^- - ACxFx] + |Ac

- 2rj-Ab^ - 2rxAbx - 2Aav

- !Ab|2 - |Aal2|r|2

- 2rj-(Abj-Aaj- + AbxAax)

- 2rx(AbxAai. - AbxAax) }/(! - |r|2)

= 1 + {2[rr(ACj- - Abj.) - rx(ACx + Abx) " Aa^- 1 T | ^

] - |AbJ2 - |Aa|2|r|2 + |Ac|2|r|2

- 2rj-(Abj-Aaj- + AbxAax)

- 2rx(AbxAaj- - AbrAax)}/(l - |r|2). (A7)

Neglecting terms that contain squares or products of A error terms,

|l+Ad|2/Ar? - 1 + 2[rr(AC;^ - Ab^-) - rx(ACx + Abx) " Aa^lrp]. (A8)

The A terms are given in (8.2) as the following functions of the equivalent

circuit parameters.

Aa = Aar + jAax - (-2Ar) + j(-2AB-2Ax)

Ab = Abj. + jAbx ^ (Ar-Aro) + j (Ax-AB-AXq) , Ac = Acj- + jACx == (-Ar-Aro) + j (AB-Ax-AXq) . (8.2)

A2
Expanding the A terms in (A8) with these definitions reduces (A8) to

|l+Ad|VA»7 - 1 + 2[rr(-2Ar) - T^(-2/\x^) - (-2Ar)|r|2]

= 1 - U(T^Ar - TxAXq - Ar|r|2). (A9)

Substituting (A9) in (Al) and adding appropriate subscripts gives

This reduces further to

Arj

r'- {1 - 4[r- Ar-r. Axo-Arlr- I2]}{1 + 4[r._Ar-r„ Axo-Arlr. |2] + ...}, An '

^ isr isx ° ' is' ^ ^ iur iux ° ' iu'

s

where ... represents other terms in the series expansion of 1/(1 - 4[ ])

Discarding products or squares of error terms leads to

Ar? - 1 - 4[(r. -r. )Ar - (r. -r„ )ax - (|r. \ -\t. r)Ar] (aii)

At? ' isr iur isx iux o 'is' ' iu'

s

so that the error in the ratio of the efficiencies is approximately

e = 4[Arj-Ar - AF^AXo - AJrl^Ar] ' (A12)

where

AFr - r^ur " ^J>sr - (Al3)

A|rp - Ir^ul' - Irish- (ai5)

The approximate e from (A12) agrees with e calculated from (Al) to about 10%

for Ar = 0.01, AXq == 0.004, and |r| < 0.3. The approximation is more accurate

for smaller values of Ar , AXq , and |r|. Values of Ar and AXq for the NIST 7 mm lines are shown in figure 7

.

A3

Appendix B. Systematic Error in r/^ due to Systematic Error in C and r.

The purpose of this Appendix is to derive expressions for the error in G

due to errors in T and C, or due to errors in w and C/A.

1 . Exact solution.

Comparing (10.5) with (2.3) shows that G can be written as either

1 + r^c 1 - WgC/A

1 + r^c

or G =

1 - w^C/A

(Bl)

We will derive an expression for the error in G due to errors in C and T. The

resulting expression can also be used to calculate the error in G due to errors in w and C/A simply by replacing

r^ with Wg , Fg with w^, and C with -C/A.

Let

^u ~ '^ur • J^ux' "s " ^sr •" J^sx' and C - C^ + jC, (B2)

Then

G = 1 + (r,,^ + ir„^)(C^ + jC^)

1 + (Fgr + jrsxXCr + J^x)

and

_ l "^ ('^^nrCr ~ M^-yCy) + ] (r,,.yCy + r,,yC-p)

1

'' (^sr'-'r ^sx'-'x) "'' J (^sr^x + ^sx'-'r)

iG| 2 = [1 + (TurCr - ^TiyCy)] + (r^i-rCy + r,,yC-r)

[1 + (r sr'-'r - TsvCv)]^ + (r sr'-'x + TsxCr)'

(B3)

^ 1 + 2(r,]-t-c-,- - r,,yCy) + (r,,-rC-r - r,]yCy) + (r,,-rCy + r^iyC-,-) 1 + 2(rsr'-'r ^sx^^x) + (^sr^r ^sx'-'x) '' (•'^sr'-'x

''" ^sx^r)

1 + 2(r,,^c,

1 + 2(rsrCr - -^ sx^x

rnyCy) + r,]t- (Cy + Cy ) + T^^^ (C^ + Cy ^ ) X ) + ^sx (*^r + "^x' rc..C^) + rcv2(C^2 + c^2) + Tsx^CCr^ + Cy2)

1 + 2(r„,c, " r,,yCy) + r,! 2 C

2

1 + 2(rsrCr TsxCx) + Ts 2 c

2

•

Define the numerator of |g|2 as N and the denominator as D so that

f^l^-lThe systematic error in \g\^ can be written

d|G|2 sM!

ar dr ur

ur 5M

dr

-dr sr

sr

+ ^dc. . ^lac/^x

(B4)

(B5)

(B6)

(B7)

Bl
where from (B5) and (B6)

,

ar^r D ' ' N ^

^

a|G|^ ^ -2c^ + 2r,,^|cp ^ I |2 -2c^ + 2r,,^|ch srux D

I ' N

ilGji. _ N 2C^ + 2r.^|c|2

I ,, -2C^ - 2r,^|cp

arsj- D2 '"^l D

8\g\^ n -2c^ + 2r,^|c|2 _ |, 2c^ - 2r,^|c|^ srsx DAlso from (B5) we have

(B9)

(BIO)

(BID

d\G\^ _ D(2r,,^ + 2|r,j2C^) - N(2r^^ + 2|rJ^c^)

dCj. D2

2]G|2raW^ilJiC^ . (^sr + l ^sPc^)! (B12)

3^G|^ _ D(-2r,,^ + 2|rJ^c^) - N(-2r,^ + 2|rJ^c^)

5Cx D2

Substituting the partials from (B8)-(B13) into (B7) gives an exact equation

for the systematic error in |g|^.

2 . Approximate Solution.

An approximation for d|Gp can be obtained by assuming that the

systematic error in measuring F^ of the DUT mount is the same as the

systematic error in measuring Fg of the standard mount. That is,

dFur = dFsj. - dFr, (B14)

and

drux = dFsx - dFx- (B15)

Then (B7)-(B13) combine to give

d|Gp _ (D-N)C^ + |C|^(DF,,^ - NF.^) (N-D)C^ + Icl^CDr,,^ - NF.^)

2|G|2 DN

"^^^ DN

^

+ r (Dr,.^ - NF.^) + (D|F,J^ - N|Fj^)C^j

^^^

B2
When

|ruC| « 1, and JTsCl « 1, (B17)

both D and N in (B16) can be set equal to 1 except in the difference D-N where

second order terms must be retained. This reduces (B16) to

f|||4 - [|c|^(rur - Tsr) + (D-N)Cr]drr

+ [|ch(rux - Tsx) - (D-N)Cx]drx

+ [(iTuh - |rsl')Cr + (rur ' rsr)]dCr

+ [(iTuP - irsP)Cx - (rux - rsx)]dCx. (B18)

From (B6) we have

D-N = 2[Cx(rux - Tsx) - CrCrur " ^sr) 1 - |Cp(|ruM - [r^]^)

= 2[CxArx - Cj-Arj.] - |c|2A|r|2 (B19)

where

AFi- - r^r - Tsr, (B20a)

AFx - r^x - Tsx. (B20b)

A|rp - |r^P - iTgl^. (B20c)

Substituting (B19) and (B20) into (B18) gives

Ijlj^ = (ICpArj, + 2[CrCxArx - Cr^APr] - \c\^A\r\^C^)dT^

+ {ICpAFx - 2[Cx2Arx. - CrCxArj.] + | C | ^a] r| ^CxldFx

+ [A|r|2Cr + AFrldC^ + [A|r|2Cx - AFxldCx,

= [2CrCxArx + (Cx2-Cr2)Ari- -

| C | 2A|r| ^c^ldrr

+ [2CrCxAri. - (Cx2-Cr2)Arx + | C | ^AJ T | ^Cxl dPx

+ [A|r|2Cj. + Ar^ldCr + [A|r|2Cx - AFxldCx- (B21)

If we keep only the most significant terms, (B21) reduces to the following

approximate expression for the systematic error in |g|2.

fjlj^ - [2CrCxArx + (Cx2-Cr2)Arr]drr

+ [2Cj.CxAri. - (Cx2-Cj.2)Arx]drx

+ AFj-dCr - AFxdCx. (B22)

B3
3 . Approximate Upper Bound

An upper bound to (B22) can be obtained from the inequalities

lAFrl < Ir^l + iTsI, IaFxI < |r^| + ITsI. (B24)

and

|2CrCx ± (Cx'-Cr')l ^ y2|c|2. (B25)

Then (B22) can be written

M

1°! |||||< [y2|c|^(tdrr| + Idr^l) + IdCrl + |dCx|](|ru| + |rj). (B26)

4. Typical Values

Typical maximum values for the parameters are

ITuI < 0.15, ITsI < 0.15, |C| < 0.3. (B28)

Typical systematic errors in Tj-, r^, Cj-, and Cj^ due to sources other than the

impedance standards and the connectors are

\dTj.\ - IdFxl - |dCr| - |dCx| - 0.0001. (B29)

Subststuting (B28) and (B29) into (B26) gives

^^L < 2[0. 000025 + 0.00020](0.3) = 0.000135. (B30)

5Alternate Approximate Solution.

For the typical maximum values of F^, Fg and C given in (B28) we have

max|FuC|2 = max|FsC|2 = 0.002. (B31)

We can therefore make the assumption in (B5) that

|ruC|2 « 1, and |rsC|2 « 1. (B32)

This assumption reduces (B5) to

|g|2 =. 1 + 2[(Fur-rsr)Cr " (rux-rsx)Cx] . (B33) or

Ig|2 ^ 1 + 2(AFj.Cj- - AFxCx) . (B34)

Using this expression to obtain the partials in (B7) , (B7) becomes

d|Gl2 = 2Cj-dFur " 2CxdFux - ZCj^dFgj- + 2CxdFsx + 2AFrdCr - 2AFxdCx. (B35)

B4
When we can assume that the systematic error in measuring T^^ of the DUT mount

is the same as the systematic error in measuring Fg of the standard mount,

this equation reduces to

d|G|2 - 2Ari.dCj. - 2ArxdCx. (B36)

Then from (B36) and (B34)

,

d|G|^ ^ 2Ar^dC^ - 2ArvdCv.

|G|2 1 + 2(ArrCr - Ar^Cx)

^ ^

For the values given in (B28) and (B29) this expression gives a worst case error of 0.000146 which agrees fairly well with the value of 0.000135 obtained

from (B26)

B5

Appendix C. Systematic Error in ry^ Due to Systematic Error in |r| .

The purpose of this Appendix Is to derive expressions for the error in M

due to errors in |r|.

1 . Exact solution.

From (10.6), M is defined as the following function of |r|2.

M - f TT^T- (CI)

Let

^2u ^ ^ur + J^ux. r^s ^ Tgr + 2'^sx- (C2)

Then

r 2 . p 2 ^_ar L

" 2 . -p ur -^ ux Define the numerator of M as N and the denominator as D so that

M =

f 'f^, ^. (C3)

N

M - g. (C4)

The systematic error in M can be written

3M dK 3M 3M

d« - af^dr,, + ^dr,, . ^dr3, + ^dr3, (C5)

where from (C3)

,

dn 2r^ ^ -M^^^, (C6)

aPgr D N

arsx D N

* ^ ^

ar^r D D ^^ D

• ^ ^

ar^x D D

Substituting the partials from (C6)-(C9) into (C5) gives an exact expression

for the systematic error in M.

2. Approximate Solution.

An approximation for dM can be obtained by assuming that the systematic

tr in measuring r^ of the DUT mount is the same as the systematic error in

aring Ta of the standard mount. That is. error in measi. ^ _ j^ ._ _.. measuring Tg of the standard mount. That is.

CI
dFur = dFsr ^ drr' (CIO)

and

dr^x = dFsx - dr^. (Cll)

Then (C6)-(C9) combine to give

dM ^ -pr,^ + Nr,,^ -Pr.v + Nr

2M DN ^ DN

sx_±_i^i.irdr^ + '^^M ^^^dFx. (C12)

When

|r^|2 « 1, and |rs|2 « 1, (C13)

D and N defined by (C4) both reduce to approximately 1, and (C12) reduces to

^ = 2(Ari.drj- + Ar^dr^)

.

(ci4)

dM

M

where

AFr - r^ur " ^Jlsr> (C15)

Arx ^ r^ux - r^sx- (C16)

C2
Appendix D. Comparing Errors in r to Errors in C (or -Tg)

Dl . Introduction.

The purpose of this Appendix is to show that when there are imperfections

in the line standards used to calibrate the 6 -port, these imperfections create

errors in both the error-box parameter C (sometimes written -To-) and also in

the measured values of F of the DUT connected to the test port. The errors in

C and r are therefore correlated, and this correlation must be taken into

account when estimating errors in »?u/'?sThis correlation creates a problem if we attempt to calculate the error

in r?u due to errors on C and T from equation (2.9). It is possible to obtain

unrealistically large estimated errors for »7u/^s f^^oTM (2.9) by choosing errors

for C and T that give a worst case situation. The worst case usually comes about when the error in C has the opposite sign of the error in F, whereas in

real situations the error in C has the same sign as the error in F, and thes6 errors tend to cancel.

w —

c

> F —

c

>

A

C

B

1

1-1-Aa

Ac

Ab

1

error-box

parameters

Ti

parameters of error 2-port

4

w

error-box parameters

for reference plane 4

Ti

4

Pi

Figure Dl . Equivalent circuit for the 6 -port showing how the error box

parameters A,B,C, which refer to reference plane 3 of the 6-port, and the

parameters of the error 2-port combine to give a new set of error-box

parameters A^,B^,C^ that contain the imperfections of the line standard and

its connectors, and refer to reference plane 4.

Dl
D2. Error in F.

As shown in figure Dl , T and Tji are related by the parameters of the

error 2 -port which in turn are functions of the imperfections in the line

standard and its connectors

r =

(1+Aa)r + Ab

Acr + 1 (Dl)

When |Acr^|^«l this becomes

r = [(l+Aa)r^ + Ab][l - Acr^ + ...]. (D2)

If we discard terms that contain products or squares of A terms, (D2) becomes

r = r^ + Aar^ + Ab - Acr^2_ (D3)

From (D3) , the error in F due to imperfections in the connectors and the line

is

F-F^ - AaF^ + Ab - AcF^^

_

(D4)

For the NIST 7 mm air line standards, the estimated maximum values of Aa, Ab,

Ac at 18 GHz are given in (8.10) and are

|Aa| - 0.0037, |Ab| - 0.0021, |Ac| = 0.0021

Typical maximum values for the F are

|F^| < 0.15. IF3I < 0.15.

(D5)

(D6)

Substituting the maximum values from (D5)-(D6) into (D4) shows that the most

significant term in (D4) is Ab . Therefore

F^ == Ab. (D7)

D3. Error in C

We next want to get an expression similar to (D4) or (D7) for the error

in C. Returning to figure Dl , note that the error -box parameters A, B, C of

the 6 -port and the parameters of the error 2 -port combine to give a new set of

error-box parameters A^ , B^ , C^ that refer to reference plane 4. This new set

of error-box parameters contains all the imperfections in the connectors and

the line and is given by

^^ ^1 A B 1+Aa Ab

^^

1 C 1 Ac 1 (D8a)

' A(l+Aa) + BAc AAb + B

'

C(l+Aa) + Ac CAb + 1

D2
= (1+CAb)

' A(l+Aa) + BAc

1+CAb

C(l+Aa) + Ac

1+CAb

AAb + B

1+CAb

(D8b)

Comparing (D8a) and (D8b) shows that

C(l+Aa) + Ac

^^ ~ 1+CAb

When |CAb|2«l this becomes

C^ = [C(l+Aa) + Ac][l - CAb + ...]

(D9)

(DIO)

If we again discard terms that contain products or squares of A terms, (DIO)

becomes

Co - C + CAa + Ac - C^Ab. (DU)

From (Dll), the error in C due to imperfections in the connectors and the line

is

C-C^ - -(CAa + Ac - C^Ab). (D12)

The maximum value for C is typically

|C| < 0.30

Substituting this maximum value into (D12) shows that the most significant

term in (D12) is Ac. Therefore

C-C^ - -Ac (D13)

D4. Comparing Error in C to Error in V

To compare the error in F to the error in C for a given imperfect line

standard, expand Ab in (D7) and Ac in (D13) with their components from (7.5)

to get

and

F-F^ == Ab == Az - Ay - AZq

C-C^ - -Ac = Az - Ay + AZq.

(D14)

(D15)

These equations show that the error in F and the error in C are not

independent but are related approximately by

C-C^ - F-F^ + 2AZo. (D16)

Equation (D16) must be kept in mind when comparing the error in »?u/»7s

calculated from (7.7) or (8.3) with that calculated directly from (2.9). From

(7.7) we have

D3
A?7 u

(1 - |r.,J^( |i + Acr. |2 - lAb + (i+Aa)r. |2) iu' is

Aj7

(1 - |r«J^)( |i + Acr.j2 - |Ab + (1+Aa)r,,j2)

is iu' = 1 + e

iu'

(7.7),(D17)

(7.8),(D18)

Whereas from (5.8), (2.9), and (5. 9) -(5. 10) we have

2

Ary, ^u/^s

N /N u s ^^s '^iu/'^is ^iu/^is

1 + cru

1 + cr

Ir-J

K\

1 + "/iu

1 + =/is 1 - Ir,ul'

(D19)

D5 . An Example.

The problem is best illustrated by an example. Suppose we know that we have errors in |r| and |c| of about 0.005. If we were to use (D19) to

calculate the error A»?^/A?7g when

kul - iTsl - 0.2 and |c| - 0.1, (D20)

we would find that Ary^/Ar^g is a maximum of 0.0082 when the phase angles are such that

Tu = -jO.2, Tg = +J0.2, C = -0.1, (D21)

and

Tu-Tiu = Tg-r^s = jO.005, and C-C^ = -j0.005. (D22)

If the error in rj^/rj^ is calculated from (8.3) which is an approximation of

(D17) we get

e = 4[ArrAr - AF^AXo - AJrl^Ar] = 4ArxAXo = 4(0.4)AXo = 1

.

SAXq . (D23)

There is only one term in e because both AFj- and A|r|2 given by (8.4) and

(8.6) are for the values assumed in (D21). To satisfy (D16) and (D22)

requires that

Az = Ay = 0, and Azo = jAXo = -j 0.005, (D24)

so that Xq = -0.005. Substituting this value for Xq into (D23) gives e=0.0080

which agrees with the worst case value of 0.0082 calculated from (D19)

.

However, from (D14) and (D15) we can see that the errors in (D22) do not

represent a realistic situation. A more likely situation is where Xq is

negligible compared to Az and Ay, and where Az and Ay are mostly reactive.

Then the errors in F are nearly equal to the error in C and these errors nearly cancel in (D19) , which is what (7.7) or (8.3) also show.

D4
To continue this example where the error in |r| is known to be about

0.005, a more realistic approach would be to assume that the error in T is due

primarily to a gap or discontinuities at the connector junction. We might

assume that the normalized impedance Az and admittance Ay are

Az = Ar + jAx = 0.001 + jO.006, Ay = jO.OOl. (D25)

Then from (D14) , |Ab| and the error in |r| are 0.005, satisfying the original

observation. However, the error in measuring ratios of effective efficiency

as calculated from (8.3) is now zero for the values of F in (D21). If instead

of (D21) we assume T^ = -0.2, and Tg = +0 . 2 , e is a maximum of 0.0016. This

should be compared with the maximum worst case value of 0.0082 originally

obtained from unrealistic values of Az , Ay and Azq.

D6 . Another Example.

Equation (8.3) indicates that e can be 0.0080 also for the case when

r^ = -0.2, Tg = +0.2, C = any value

and

r^-r^u = Ts-r^s = c-c^ = 0.005.

From (D14)-(D16), this can happen when Az = Ar = 0.005, and Ay = AZq = 0.

Equation (8.3) then becomes

€ = 4[Arj.Ar - APxAXq - A|r|2Ar] = 4Arj.Ar = 1 . 6Ar = 0.0080.

However 0.005 is probably an unrealistically high value for Ar. From figure 7,

Ar is probably less than 0.0003. Then € = 0.00048.

D7 . Conclusion

If the error in measuring |r| is some known value, say 0.005, then rather

than calculate the maximiom possible (but unrealisltic) worst case error in

measuring ratios of effective efficiency from (2.9) or (D19) , it is better to

choose realistic values of Az , Ay, and AZq that make Ab given by (D14) equal

to 0.005. The error in measuring |r| will then be 0.005, but the error in

measuring ratios of effective efficiency as calculated from (8.3) will be

considerably less than the maximum possible worst case error.

D5

ft u s GOVERNMENT PRINTING OFFICE: 1989-Jacket 676-409/5113 Region 8

NBS-n4A (REV. 2-6C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NIST TN-1332

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

July 1989

4. TITLE AND SUBTITLE

Systematic Errors in Power Measurements Made With a Dual Six-Port ANA.

5. AUTHOR(S)

Cletus A. Hoer

6. PERFORMING ORGANIZATION (If joint or other thon N6S, see instructions)

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

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

10. SUPPLEMENTARY NOTES

[^2] Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual sun)mary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The purpose of this report is to determine the systematic error in

measuring power with a dual 6-port ANA. Most of the report concentrates

on developing equations for estimating systematic errors due to imperfections

in the test port connector, imperfections in the connector on the power standard,

and imperfections in the impedance standards used to calibrate the 6-port for

measuring reflection coefficient. These are the largest sources of error associated with the 6-port. For 7 mm connectors, all systematic errors which are associated with the 6-port add up to a worst-case uncertainty of ± 0.00084 in

measuring the ratio of the effective efficiency of a bolometric power sensor relative to that of a standard power sensor.

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

bolometric detectors; connector model; effective efficiency; error analysis;

microwave; power measurements; six-port.

13. AVAILABILITY

[^ Unlimited

I I For Official Distribution. Do Not Release to NTIS

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14. NO. OF

PRINTED PAGES

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15. Price

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