\

1" l^iii

NBS TECHNICAL NOTE 1004

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901. The

Bureau's overall goal is to strengthen and advance the Nation's science and technology and

facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is performed by the National Measurement Laboratory, the National Engineering Laboratory,

and the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods of measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

Agencies; develops, produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

Absolute Physical Quantities^ — Radiation Research — Thermodynamics and

Molecular Science —Analytical Chemistry — Materials Science. THE NATIONAL ENGINEERING LABORATORY provides technology and technical

services to users in the public and private sectors to address national needs and to solve national problems in the public interest; conducts research in engineering and applied science

in support of objectives in these efforts; builds and maintains competence in the necessary

disciplines required to carry out this research and technical service; develops engineering data and measurement capabilities; provides engineering measurement traceability services; develops test methods and proposes engineering standards and code changes; develops and

proposes new engineering practices; and develops and improves mechanisms to transfer

results of its research to the utlimate user. The Laboratory consists of the following centers:

Applied Mathematics — Electronics and Electrical Engineering^ — Mechanical

Engineering and Process Technology^ — Building Technology — Fire Research —

Consumer Product Technology — Field Methods.

THE INSTITUTE FOR COMPUTER SOENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal Agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

economy in Government operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing the Federal Information Processing Standards Program, developing Federal ADP standards

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provides the technical foundation for computer-related policies of the Federal Government.

The Institute consists of the following divisions:

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'Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234. 'Some divisions within the center are located at Boulder, Colorado, 80303.

The National Bureau of Standards was reorganized, effective April 9, 1978.
NATIONAh BUKEAQ

OP STANDARDS

Calibrating Two 6-Port Reflectometers With^u^'Tms

Only One Impedance Standard

TecLKr^,cal A3(^Ve

no \c^^

Cletus A. Hoer

Electromagnetic Technology Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

'V*«o»*^*

5

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

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

i'ATlONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1978

BnH "'^^"""'«''—'
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1004

Nat. Bur, Stand. (U.S.), Tech. Note 1004, 46 pages (June 1978)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1978

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

Stock No. 003-003-01956-9 Price $1.60 (Add 25 percent additional for other than U.S. mailing)
CONTENTS

Page

List of Figures and Tables v

I. Introduction 1

II. Outline of Calibration Procedure 1

III. Review of 6-port Reflectometer Equations 5

A. a and b Parameters 5

B. V and i Notation 7

IV. Reference Planes Together 8

A. H in terms of H 9

B. Conditions on a„/a 10

V. Calibration Circuit 10

A. Measurement Sequence 11

B

.

Using P 11

C Relating H to H 12

D

.

Determining half of H 12

E

.

Exploiting redundancy 14

F. Five new constants 16

G

.

Six-Port //2 is also calibrated 20

H. Choosing the two terminations 21

VI. Adding the Standard 22

A. Standard Termination 22

B

.

Standard Line 23

1

.

Determining yZ of the line 23

2

.

Determining Z. of the line 25

3

.

Choosing the line length 25

VII. Filling in Calibration Matrices 27

VIII. Calibrating for Power, Voltage, and Current 28

IX. Evaluating the Calibration 29

X. Acknowledgment 30

References 31

iii
CONTENTS CContinued)

Page

APPENDIX (program outline) 32

A. Taking the Calibration Data 32

B

.

Data Sequence 33

C. Data Array 33

D. Calculating H and H2 35

iv
I

LIST OF FIGURES AND TABLES

Figure 1. Using two 6-port reflectometers to measure all

of the scattering parameters of a 2-port device

Page

Figure 2. The first part of the calibration consists of connecting the

two 6-port reference planes together (a) , and then connecting

each 6-port to a calibration circuit (b) . The output P of

the directional coupler is either measured or held constant

by leveling the generator. The two different terminations

e and f need not be known 2

Figure 3. A uniform length of transmission line or one known termination

is the only impedance standard needed. A power standard is

connected to one of the 6-port reflectometers to calibrate

them for making absolute power measurements 4

Figure 4. Defining the incident and reflected waves at the different ports 4

Figure 5. An example of choosing and numbering the 6-port sidearms so that I y V

I << 1. The left matrices apply when using detectors

3, 4, 5, and 6 (ignoring P-,)- The right matrices apply when

using detectors 3, 4, 7, and 6 (ignoring 7 ). Q is a quadrature

hybrid and D is an in-phase power divider as defined in figure 6. The parameters v, y, x, y, K, and K are calculated from

equations (81), (76), (74), (75), (80), and (73) 18

Figure 6. More examples of choosing and numbering the 6-port sidearms

so that |yv| << 1. Both 6-port designs have the same matrix

parameter if the sidearms are numbered as shown. The boxes

labeled Q and H are quadrature hybrids and 180° hybrids,

defined on the sides of the figure. The parameters v, y,

X, y, K, and K are calculated from equations (81), (76),

(74), (75), (80), and (73) 19

Figure 7. A calibration circuit with two terminations at e and f which have

a 90° phase difference. D is an in-phase power divider and Q is

a quadrature hybrid 21

Figure 8 . Defining the direction of Z . and Z 22
LIST OF FIGURES AND TABLES (Continued)

Figure 9. Effective phase shift through standard transmission lines as

a function of frequency where 41 is the phase shift through

the line and n = 0, 1, 2,.... In (b), n has been chosen so that |(j)-180n| is between and 90°. The line lengths are chosen so that they are multiples of a half wavelength at

f- + f-^» where f^ and f„ are the frequency limits of operation

of the 6-ports. For coaxial air lines.

Page

= 12 f „„ a , degrees. GHz cm "

^1 ^ £—+T~ ' ^^ ^°^ ^ ^° ^^^ ^^

Figure 10. One way of storing the sidearm power readings, P, so that

all readings taken at a given frequency are in one row (a)

.

After all the data is taken, the matrices P^^ , P^„, D^ , D„,

S^^ , and Sy are formed from any one row as indicated by the

example in (b) for row (frequency) //5 34

Table 1. Two possible sequences of measurement when using the calibration circuit. 11

vl
CALIBRATING TWO 6-PORT REFLECTOMETERS

WITH ONLY ONE IMPEDANCE STANDARD

Cletus A. Hoer

This paper describes a technique for calibrating a pair of 6-port reflectometers for measuring the reflection coefficient of 1-port devices, or the

scattering parameters of reciprocal 2-port devices. The operations in the

calibration consist of connecting the two 6-ports together, connecting each

6-port to a calibration circuit consisting of two terminations of unknown

impedance and a leveling loop, and then connecting the standard. The standard

can be one termination whose complex impedance is known, or a precision length

of transmission line whose cross-sectional dimensions are known. The length

and loss of the line are not required. The solution for the constants which

characterize each 6-port is closed, requiring no iteration.

Key Words: Calibration; current; impedance; network analyzer; power; reflection coefficient; scattering parameters; six-port; reflectometer; voltage.

I. Introduction

A 6-port reflectometer can be used to measure the impedance or reflection coefficient

of 1-port devices connected to its reference plane [1-8]. It will also measure voltage,

current, or power at that reference plane [9]. Two 6-port ref lectometers can be used to measure all of the scattering parameters of a 2-port device which is inserted between the

two reflectometers as shown in figure 1 [10]. This paper describes a technique for calibrating a pair of 6-port reflectometers for making all of these measurements.

Each 6-port reflectometer can be characterized by a square matrix containing 16 real

constants. The objective of the calibration is to determine these two matrices which will

be called H and H„ corresponding to 6-port #1 and 6-port #2.

The calibration procedure is first described without referring to any equations. This

is followed by a review of 6-port reflectometer equations which are expressed in matrix

notation. The solution for the constants contained in H and H„ is then given in detail

using matrix algebra whenever possible. An appendix outlines a computer program for

taking and processing the calibration data.

The calibration technique described in this paper is similar to that described by

Allred and Manney for calibrating two 4-port couplers [11]. The solution is a closed

solution, requiring no iteration.

II. Outline of Calibration Procedure

The calibration consists of two parts. The first part uses no standards and obtains

enough of the constants in H and H to make impedance ratio measurements at either 6-porf

reference plane. The second part of the calibration requires the use of one impedance
DIVIDER

FLEXIBLE CABLE

?) (P

^^FLEXIBLE CABLE

P) (P

Figure 1. Using two 6-port reflectometers to measure all

of the scattering parameters of a 2-port device.

i

6-PORT #1 OR #2

® ©

(a)

(b)

LEVELING

LOOP

^VVSr X ^yO-^VW

O-^VAAr

f

CALIBRATION CIRCUIT

Figure 2. The first part of the calibration consists of connecting the

two 6-port reference planes together (a) , and then connecting

each 6-port to a calibration circuit (b). The output P of

the directional coupler is either measured or held constant

by leveling the generator. The two different terminations at

e and f need not be known.

-2-
standard to find the remaining constants needed in measuring an absolute impedance or reflection coefficient. The first part of the calibration consists of the following three

operations which are illustrated in figure 2.

1. The two reference planes are connected together and all 6-port sidearm power

readings are recorded at four or more different settings of A , A and (j). The

values of attenuation A , A and the value of phase (ji do not need to be known.

This set of measurements is sufficient to determine H„ in terms of H , thereby

reducing the number of unknowns from 32 to 16. If 6-port #1 is already calibrated

so that H^ is known, then this set of measurements is sufficient to calibrate any

other 6-port.

2. Six-port #1 is connected to the calibration circuit shown in figure 2b and its

sidearm power readings are recorded at each of the two switch positions e and f within the calibration circuit. Power level, P , is either recorded for both

switch positions or held constant by leveling the generator.

3. Six-port #2 is connected to the calibration circuit and its sidearm power readings

are recorded as in step 2. P is again either recorded or held constant by

leveling the generator.

To make the connections in steps 1 to 3 requires that all connectors be of the same

type and sexless. The term "calibration circuit" is not meant to imply that this circuit

contains any standards, but only that it is a convenient circuit to use in the calibration

procedure. The impedance of the two terminations in the calibration circuit need not be

known. It is assumed only that the two values of impedance seen looking into the calibration circuit are the same when either 6-port is connected to it.

Steps 2 and 3 are sufficient to reduce the number of unknowns from 16 to 8. Half of

the constants in H.. are given in terms of the other half. The remaining 8 unknown constants

are reduced to just 3 unknowns by taking advantage of the dependent relationship between

the constants. Only 12 of the 16 constants in either H or H are independent. It will be

shown that these three remaining unknown constants are not needed for calculating the

impedance ratio of two terminations. Therefore the set of measurements described in steps

1 to 3 which uses no standards is sufficient to calibrate each 6-port for making impedance

ratio measurements.

The second part of the calibration uses one impedance standard to find two of the

three remaining unknown constants. These two constants are needed to calculate absolute

impedances or reflection coefficients from which S-parameters of 2-port devices are calculated. The standard can be a uniform length of transmission line which is inserted

between the two 6-ports as shown in figure 3. With the line inserted, all 6-port sidearm

power readings are recorded at two or more different settings of A , A„, and (j) . As in

step 1 above, the values of A , A , and it do not need to be known. This set of measurements is sufficient to determine the two needed constants and also yH of the line, where

Y is the complex propagation constant of the line and I is its length. The length and

loss of the line do not need to be known, only its cross-sectional dimensions need to be

known.

-3-
© ©

LENGTH OF LINE

f e^poRT #2

Ho

'1

OR .

! ^AArONE KNOWN TERMINATION

POWER STANDARD Ki)

Figure 3. A uniform length of transmission line or one known termination

is the only impedance standard needed. A power standard is

connected to one of the 6-port reflectometers to calibrate

them for making absolute power measurements.

4 ^ J^-1

6-PORT #1

ill ri c

^'^

V

2 "7 "8

1 t

6-PORT #2

h k k

"'2

Figure 4. Defining the incident and reflected waves at the different ports.

-4-
The standard can also be a termination (not a short or an open) whose complex reflection coefficient is known. With this standard connected to either 6-port, the sidearm

power readings for that 6-port are recorded. The two needed constants are calculated from

this set of readings and the previously known value of the reflection coefficient.

The last remaining unknown constant is determined only if each 6-port is to be used

for measuring voltage, current, or power as well as reflection coefficient and S-parameters.

To determine this last constant, a power standard is connected to either 6-port reference

plane as indicated in figure 3. The sidearm power readings of that 6-port as well as the

reading of the power standard are recorded. The last constant is calculated from this

set of readings. All constants in H and H„ are now known so that either 6-port can be

used to measure active as well as passive circuit parameters.

III. Review of 6-port Reflectometer Equations

A. a and b Parameters.

Using the notation shown in figure 4, let the emergent wave at sidearm //i of 6-port

#1 be [1,2]

b. = A!a, + B'.b, , i = 3---6 (1)

1 1 1 1 1

where Al and Bl are functions of the scattering parameters of the 6-port junction and also

of the reflection coefficient of each sidearm detector. The power, P., indicated by the

detector on sidearm #i is

P. = c. Ib.|2 = c. lA'.a^+B'.b, |2 (2)

1 I'l' x'll il'

where c . is a real constant that does not need to be known. It will be convenient to

1 combine c.111 with Al and B! so that eq . (2) becomes

P. = |A.a,+B.b, |2 , i = 3. .-6 (3) 1 ' 1 1 1 1

where A.11and B. are new constants which will be determined in the calibration procedure

described in this paper. Expanding eq. (3),

P. = |A.a,p + 2|a^B.| cos L^ la-b,! cos ii^ i 'il' 'il' i'll' 1

+ |B.bJ2 _ 2IA.B. I sin C. la.bJ sin \\i^, i = 3'"6 'il' '11' I'll' ^1 where

ii^ = arg (bj^/a^) , ?^ E arg(B^/A^)

In matrix notation, the four equations in eq. (4) can be written

-5-

(4)
^3

^6

I A3

1

2|A B I cos ?,

2 Ia.B, '44' cos C/

'

2|Ac.B^| cos ?c

2|AgBg| cos ?^

-2|a„B.| sin C„

-2 Ia.B, I sin z,, '44' 4 -2|A^B^| sin C^

-2|a^bJ sin ?^

[a^b^l cos ^^

|a-.b^ I sin ^^

(5)

or

^1 = ^^1 (6)

where (->) indicates a column vector and C) indicates a square matrix. From eq. (6),

t = "-^ " ^ ^ = ^1^ (7)

where

ei ^ ?i'

(^3 a^ a3 a^

3

"^4 "^5 "^6

^3 <^4 <^5 "6

(^ S3 S^ S^ Sg

J

(8)

Conditions for the existence of B are given in reference [4]. Since all elements in

eq. (5) are real, B and hence G are real matrices. Expanding eq. (7),

la-b^ I cos i). = y c.P. '11' 1

'^ 11

laib^ I sin ^. = T s.P. '11' 1 ^11

(9)

The ratio p = b /a^ can be written

Pi = 1-1^1

1 ^

\ 2 (cos il)^ + i sin \l)^) (10)

which from eq. (9) becomes

Pi =

E c.P. + i Z s.P. 11 "* 1 1

Z a. P.

1 1

i = 3- (11)

The symbol p is used instead of r because the ratio h /a is not a reflection coefficient in

the usual sense during much of the calibration process to be described. For a passive

termination, p is the usual reflection coefficient r.

-6-
B. V and i Notation.

The voltage v and the current i^ at the reference plane are related to the waves a and b^ by

v^ = a^ + b^

h^oi = ^1 - ^

(12)

(13)

where Z„^ is the characteristic impedance. Using eqs. (12) and (13), a matrix equation

similar to that for A can be written in terms of v and i Z_ ;

i, Z

1 01'

^""ihW """^ \

l^iH^oil "^^ ^ij

112

11-20

1-10

2

|a^b I cos ^^

|a^b^ I sin i(/^

(14)

V^ = K A^ as)

where

^01 (16)

If Z_^ is real, then =0 and 9^ is the phase angle of the impedance Z connected at the

Ui Zq^ 1 i reference plane. Using eq. (7) in eq. (15) gives

^1 = 5 ?1 ^1 = ?1 ^1 (17)

where

«i = 5 5i

^3 \ ^ ^6

^3 ^ ^ ^

^3 ^4 "^5 ^6

^3 ^ ^^5 ^6 }

(18)

Expanding eq. (17) yields a set of equations similar to eq. (9);

-7-
^=3---6

(^9^

|viP = I v.P.

The ratio Z^/Z„^ can be written

T^ =i // 1 ^ (cos e, + j sin e.) (20) % ih^oi' ^ -^

E q.P. + j E r.P. = ^^,.p. ^^ , 1 = 3-. 6 . (21)

1 1

A set of equations similar to eqs. (1) - (21) can be written for 6-port #2. Subscripts

1 or 2 on the matrices will indicate 6-port #1 or #2.

For converting between a and b parameters and v and i parameters, it is useful to know

that

k"-"- = IK . (22)

Then

G, = k"-"- H, = f K H, . (23) ~1 ~ ~1 4 ^ _1

The object of the calibration is to determine either G or H since one can easily be obtained

from the other. In the procedure to be described, it is easier to obtain H using v and i notation than it is to obtain G using a and b notation.

IV. Reference Planes Together

The first step in the calibration procedure is to connect the two reference planes

#1 and #2 together. Referring to figure 2a it is apparent that

a = b^ b = a- (24)

|aje ^1 = Ib^le ^2 |bje ^1 = \a^\e ^^ . (25)

Then

•^1 ^ *b^ -

-^a^ = ^a^ - ^2 = "'^2 • ^26)

The matrices A and A are then related by

-8-
X, = N X„

1 ^a 2 (27)

where

N E ~a

10

10

10

0-1

= N

(28)

The relation between V and V„ can be obtained by using eq. (15) in eq. (27).

-l-v -l-> K V, = N K V„

1 ~a^ 2 (29)

where

1 _v 2 (30)

N = KN K

-1

1 r

1 _I

-1 -I

1 -1

J

k

= N

-1

(31)

and where I is a 2 by 2 identity matrix and is a 2 by 2 zero matrix. The fact that N is

diagonal and can be so partitioned turns out to make eq. (30) a more useful relation to use

than eq. (27) later in the derivation.

A. H„ in terms of H^

.

To get an expression relating H„ and H^ , use (17) in (30) to obtain

h ^ = ^v »2 ^2 (32)

Four or more measurements are taken with the reference planes together by changing A., A„,

and ()) to different but unknown settings. Let these measurements be designated by subscripts

a, b, c,'*'. These measurements which each generate an equation like eq. (32) can be combined into the single matrix equation

^1 la lb Ic Id

] = N H„ [ P„ P„, P. P„,-

^v ,2 2a 2b 2c 2d ] (33)

or simply

»A = !?v»2^2 (34)

where underlining indicates a matrix that is formed from a set of column matrices, and is

usual]

gives

usually not square. If only four measurements are taken, P^ and P^„ are square and eq. (3A)

-9-
H„ = N H,

J

(35)

where

f

~= hh^ (36a)

When more than four measurements are taken, eq. (35) is a least squares solution of (34)

where J is then given by P^ times the pseudo- inverse of P^„ [12];

T T -1

P P (P P ) -1 -2 ^-2 -2^ (36b)

If H, is known by some previous calibration, H„ can be calculated from eq. (35).

B. Conditions on a„/a^

.

When taking four or more measurements with the reference planes together, the values

of a„/a at the common reference plane must not all have the same magnitude or all have the same phase. If so, the inverse in (36a) or (36b) will not exist. In the present NBS system,

4 to 6 measurements are taken with A , A„, and (}) set so that 3.^1 a.- at the common reference

plane is approximately equal to 1 at -90°, 1 at 180°, 1 at 0°, 0.3 at 90°, 1 at 90°, 0.3 at -90°.

V. Calibration Circuit

Two different terminations of unknown impedance can be used to complete the calibration

of each 6-port for impedance ratio measurements. Each termination is preceded by some circuit for measuring or controlling the relative power into the two terminations such as shown in figure 2b. Let r = r' + jT" be the reflection coefficient looking into this

"calibration circuit." When 6-port //I is connected to the calibration circuit, A^ defined

by (5) and (6) can be written

^1

= |a^b^|cos ^^

|a^b^|sin ^^ r"

|a,|^? (37)

where f is a new column vector which is a function only of the reflection coefficient F. Now add a subscript e or f to (37) to indicate which termination is being used;

le ' le' e

^If = 1^If I

(38)

(39)

-10-
A similar set of equations is obtained when 6-port #2 is connected to the calibration

circuit;

^,.= Ia,J^r *2e

2f

2ei

*2fl

(40)

(41)

Comparing (38) and (40) shows that when termination e is used,

U 12 ^le le

2e

^ 2e (42)

Likewise when termination f is used, (39) and (41) lead to

-> i^lfl'-.

^If = JI^ ^2f • (43)

A. Measurement Sequence.

Equation (42) assumes that F in (38) is the same as V in (40), and equation (43) assumes that T^ in (39) is equal to V in (41). To assure that this assumption is valid,

consider the two measurement sequences shown in Table 1. In the first sequence of measurements the switch must be repeatable so that V in measurement //I is the same as V in

e e measurement #4. The T^ in measurement #2 will be the same as T^ in measurement //3 because

the switch position is not changed. In the second sequence of measurements the switch need

not be repeatable. However, this sequence requires that 6-port //I be connected to the

calibration circuit twice instead of just once. In either sequence the connectors at each

6-port reference plane and at the calibration circuit reference plane must be repeatable.

TABLE 1.

Measurement

number

Switch must be repeatable Switch need not be repeatable ] 6-Port

Connected

Termination

used

6-Port

Connected

Termination

used

1

2

3

4

1

1

2

2

e

f

f

e

1

2

2

1

e

e

f

f

Table 1. Two possible sequences of measurement when using the calibration circuit.

B. Using P

For any one switch position, the calibration circuit can be considered as a 3-port

junction. It can be shown that for any such 3-port junction, the ratio of two incident

-11-
powers at the input port, say |a^| /|a„| , Is equal to the ratio of the two corresponding

values of P , P , /P ^^for a given termination at the third port. Therefore (42) becomes

C Cl Ci

P , -> cle -^ ^e=F7- ^2e (^*)

c2e

where a subscript e has been added to the power ratio to indicate which termination is used.

This equation shows the advantage of leveling the generator with the output from the power

detector measuring P so that the power ratio in (44) is unity and need not be measured.

C. Relating H to H^

.

Using (15) in (44) gives

P , -> cle -^ \.-TT ^2e («)

c2e

which can be expressed in terms of H^ and H„ using (17);

«1 ^le = pV ~2 ^2e ' (^6)

c2e

Starting with equation (43) leads to a similar expression for termination f

;

?1 ^If = pTT ?2 ^2f • (^7)

c2f

These last two equations can be combined into a single matrix equation

P P ^if^le^lf] = «2f pV ^2e pTT ^2f ^ (^8)

c2e c2f

or simply

!?1-1 = ?2^2 ('*9>

where D and JD are each 4 by 2 matrices containing power readings.

D. Determining half of H .

We now proceed to eliminate H„ from (49) and reduce the number of unknowns in H from

16 to 8. Using (35) in (49) gives

?1 -1 = ?v ?1 f -2 = ?v ?1 ^ (50)

where ^ is a 4 by 2 matrix

E = J D2 . (51)

-12-
Now partition H , ^ , N , and E^ into 2 by 2 matrices so that eq. (50) becomes

' h ; !!2

' ^1

r I 1 1 ~ 1 h ; !?2 ' !i

1

^ h 1 !}4 > • ^2

1 b ' !!4 - ' !2

(52)

Each symbol in eq. (52) indicates a 2 by 2 matrix. This equation expands into the two matrix equations

h d^ + h- d„ = h^ e^ + h- e„ b !i "• !?4 ^2 = -h !i - \ !2(53)

(54)

Solving eq. (53) for h„,

^2 ^^^2 ~ ^2^ "

'^l ^^1 ~ "^1^ (55)

or

h2 = h a (56)

where

a = (e^ - d^) (d2 - e^)

-1

(57)

Solving eq. (54) for h_,

N ^^1 "^ ^1^ " "\ ^^2 "^ ^2^ (58)

or h = ^

(59)

where

E - (e^ + d2)(d^ + e^) (60)

Using eqs. (56) and (59), matrix H^ can now be written

?1

' h ; h?

' h 1 " -1 , -

f I 1 a 1 ~ 1

^4^1 !?4 ~ 1 ~4 . I ? : ! J

(61)

Hj^ = h Y

(62)

-13-
where y is known but h is yet to be determined. Using eqs. (62) in (17), any measurement

with 6-port #1 can be written

\ = ^ih = ^Yl"! (63)

Since y and P are known, let

6,

t = yV^ = (64)

Then

^r^ = h t

where 6 is known. Expanding eq. (65) gives four equations

(65)

or

i,Z

1 01'

l^i^i^oil ^°" 'l

l^i^i^oil ^i^ \

V- V

3 4

1^3 ^

q^ q^

r^ r,

5 6

'^l"

^2S

/*.

'Veil

= v^&^ + v^62

= y36^ + y^62

'""ih^Oll "^ ®1 " 'l5'^3 + 'l6'^4

l^lH^Oll "^'^ ^1 = ^sS + ^6^

(66)

(67a)

(67b)

(67c)

(67d)

E. Exploiting redundancy.

It appears that this is as far as one can go in the derivation with simple matrix

manipulations. There are still 8 unknown constants in (67) to be determined. To reduce

the number of unknowns further, the nonlinear relationship between these 4 equations is

used. Note that the square of equation (67c) plus the square of equation (67d) must equal

the product of equations (67a) and (67b);

|v^i Zq |2 cos^e + |v,i,Z„J sin^e, = |vJ^|i,Z^J2 . (68)

1 1 01' ^1-^01

A similar but more useful relation between the remaining 8 constants will now be used

to determine 5 new constants, 3 of which allow each 6-port to make impedance ratio measurements with no further calibration.

-14-
Equations (67b) - (67d) can be used to get an expression for the impedance Z^ connected

to the reference plane of 6-port #1;

h IVi^oil

(q 6 +

q,6 J + j (r 6 + r.6 ) P3 6^ + y^ 62

(q5 + j r^) 6^ + (qg + j r^) 6^

^3 ^1 + ^ "^2

= K,

«3 + (x + j y) 6^

63^ + y 62

where

^0 = q5 + j r^

V5 + V5

(69)

(70)

(71)

(72)

(73)

(74)

(75)

li - Vi^lv^ (76)

Note that if the three real parameters x, y, and y can be determined, eq, (72) can be used

to measure the ratio of two impedances since K_Z. will cancel

Z^ can be obtained from the ratio of equation (67a) to (67b);

to measure the ratio of two impedances since K_Z. will cancel. Another expression involving

hi

2 ^1

4^01

^3 ^1 -^ ^ ^2 ^3 "^1 "^ ^4 ^2 (77)

The square of the magnitude of Z^/Z„^ calculated from (72) must equal that calculated from

(77);

^(«2 " ^ *1^

V^(&2. • ^ "^2^

I63+ (x+jy) 6J2

or

= I'^Ol" l6^ + y 62!^

(62 + V 6^)(6^ + y 62) = K [(63 + X 6^)2 + (y 6^)2]

(78)

(79)

-15-
where

IKqP U3

^3 \

(80)

V = v^/v^ (81)

F. Five new constants.

Equation (79) is a nonlinear equation with five new real constants to be determined:

V, y, X, y, and K. So far at least six measurements have been made with 6-port y/1, four

with the reference planes together plus two when connected to the calibration circuit.

These six measurements can be used to solve eq, (79) for the five constants. Expanding

eq. (79) leads to

or

(l+yv)6^62 = K 6^ + 2Kx&^&^ + K(x^+y'^)6^ - v5^ - ^Sj

6t 6„ = XT6f + x„6„6, + x_67 - x, 6? - x_6^ 12 13 234 34 41 52

(82)

(83)

where eq. (83) is linear in the new variables x^"*Xc. which are defined by

(l+yv)x^ = K

(l+yv)x = 2Kx

(H-yv)x = K(x2+y2)

(l+yv)x^ - V

(l+yv)x^ = y

(84a)

(84b)

(84c)

(84d)

(84e)

The six measurements on 6-port #1 each give an equation like eq. (83) which can be combined

into one matrix equation j

'3 '3\ 1 -'I -'2

6 rows or measurements

from 6-port #1

'^1

^

' ^1^2

X2

X3 = -

^4

. X3 J

(85)

/

-16-
or simply

^ = F . (86)

If only five of the six measurements are used, R is square and X is given by

X = r"-*-? . (87)

-v Using six or more measurements, X is calculated from

->. T _] T-> X = (R R) R F (88)

Finally the desired parameters v, y, x, y and K are calculated from the elements x •••x^. of

X using eq. (84)

.

The value of pv is first calculated from the product of eqs. (84d) and (84e) which

leads to

2x^X3

^^ =

1 - 2x,x, ± A-^x.x, • (^^^

4 5 4 5

It can be shown that 1/yv is given by this same equation. Therefore one of the two values

given by eq. (89) is yv and the other is l/(yv). To separate the two roots we need to know

only if I y V

I is greater than or less than 1. It is usually possible to choose and number

the 6-port sidearms so that ideally | y v

| << 1. For example, the choice and numbering of

the sidearms in figure 5 ideally gives y = v = 0. Then in the non-ideal case, |yv| < 1 and

the smaller of the two values given by eq. (89) is clearly the correct value of yv. Two more examples are shown in figure 6 which shows two circuits suggested by Engen [6]. If

the sidearms are numbered as shown, both circuits have the same G and H^ matrices, and

both have y = v = in the ideal case. With yv known, K, v, and y are calculated from eqs. (84a), (84d), and (84e) . The

values of x and y are independent of the value of yv;

X

X = ^^ (90)

x_ „ ^ - x^ • (91)

^1

The sign of y can be determined from a knowledge of what ideal components would give. For

the choice of sidearms in figure 6, y is negative.

Another way to determine or to check the value of yv and the sign of y is to calculate

the value of the ratio of the two impedances seen looking into the calibration circuit using

eq. (72). This ratio can be compared with that obtained by some other measuring device used

to measure the two impedances.

-17-
la+bl (Shx® a-bl

AAM^

la+jbl

e

Q

1 1 J 1

1 r—

D nirO 2 Ir &KS) la-jbl

r —

USING DETECTORS: 3 4 5 6

1

^1 = "6

2

-1

2

1

-1

-1 -1 2

OR a,

3 4 7 6 ~0 f

2 2 0-1

-110

11-20

ai'f

2

2

1 -1 1

^3 OR -y

1 -1 2

2

2

1 -1 1

1 1 -2

V = VI = X = y = -0.5, K = 1

K

V = y = X =

y = +0.5, K = 1 Kq = -j

Figure 5. An example of choosing and numbering the 6-port sidearms so that |yv| « 1. The left matrices apply when using detectors

The right matrices apply when

Q is a quadrature

3, 4, 5, and 6 (ignoring P ), using detectors 3, 4, 7, and 6 (ignoring Pq-) hybrid and D is an in-phase power divider as defined in figure

6. The parameters v, \i, x, y, K, and K are calculated from

equations (81), (76), (74), (75), (80),°and (73).

J

-18-
OJ CM

X3 -O + + o ^~•

•'—

)

hJ -r-) o 1 00 +—

to (T3

o

CO

1 LT) 1 CM

•<T> .— o CM O +

CM o ,— CM 1

CMO

1

o

(XI

1

1

CM

1 O

1

II o

LD

CM

3- 1 CM

II II o

^ UD 1— O I I O «* CM ^ O ^ CM >*

I O "* CM O I

C3S

II

tno

o

II

3.

II

w •H (0 p 4-1 X •K «t n) m o *• a. ,^v

(1) R Xi en vD

-a x) #« [-^ •H ti) cu •H ,> ^-^ M p x: u

cd H 42 en •* 4-1 tn >^ ^ ^'-N Ki X cu iH o Q) • 4J 00 p<^ rj OJ '^

1 J-l [5 e

vD 00 Cd cn

0) 4= rH (-1 3

(U > cn cd x: cfl 13 ex H

w x: to C 4J

nJ cfl (U Cd on en X 3

C C -a en H cr H M) 0) a cu

>J •H u •H

cu W <u i-i • e

Xi CU J3 XI OJ g -a B >> u n

3 3 x: 3 y-i c •u a M

n 0) •i-l T) -n o (U V4 M-J cu C Pu n 3 4J

nt 1 cd 4J CU Cd • VO td x: iH y-N Wl en M 4-1 3 ro c 42 e 13 CJ r-- •H 4-1 u to 14-1 .H N—'

en O rt 3 cd o FQ cu cr T3 o XI en C

x: •H cu cu cu CO o • CO ^-1 T3 M

.H tfl •H Cd ^

M-t 0) CO /--s O V x: ffi V 4-1 cu W 00

0) T) x: N-^

(U 14-1 d 4J 13 .H ? •H cd C »v O. 3. 3 cd •-*s fl H o- in m (» «N r^ X 4-1 4-1 T3 T3 i^ ^—

(U Cfl <U cu cu x: P. M 3 «\ #> OJ 4-1 cfl OJ •H >. ,.—

s

^ ^ X) 4-1 v3- o O C8 cd CU #t r-- a CO a-H 13 X

^"^

VO

(U^

3

W)

•H

Pn

|CM KM

-19-
After the five constants v, y, x, y, and K are determined, eq. (72) can be used to calculate complex impedance ratios, or eqs. (67a) and (67b) can be used to calculate voltage

and current ratios. Power ratios or reflection coefficient ratios cannot be calculated

without further calibration involving an impedance standard. Note that no standards have

been used in the calibration process so far.

G. Six-Port #2 is also calibrated.

When 6-port #1 is calibrated for making ratio measurements of voltage, current, and

impedance, so is 6-port #2. To show this, first note that for any measurement on 6-port #2

eq. (17) gives

^2 = «2 ^2

Using eq. (35) in eq. (92),

V- = N H, J P2 _v ^1 _ 2

(92)

(93)

But from eq. (62), H^ = h y so that eq. (93) becomes

V„ = N h T

2 ^v ^ (94)

where T is known;

-> -» T = Y J Po = (95)

Equation (94) for 6-port #2 is identical in form to eq. (65) which is for 6-port #1 except

for N which only has the effect c ~v

eq. (94) gives the four equations

for N which only has the effect of changing the sign of some of the equations. Expanding

|v2p = V3t^ + v^t2 = v^(t2+vt^)

IV02I' = Vl-' V2 = ^3<^V^'=2>

v„i„Z„J cos e„ = - q.t„ - q,t '2 2^02 5 3 ^6 4

v„t„Z.J sin e„ = - r^t_ - r,t

2 2 02 5 3 6 4

(96a)

(96b)

(96c)

(96d)

The last three equations give an expression for the impedance Z„ connected to the reference

plane of 6-port #2. Following the same steps as in eqs. (69) —(72) leads to

-20-
= - K,

^02

tj + (x+jy)t^

(97)

where K„, x, y and y are the same constants used in eq. (72). Therefore, when x, y, and y are determined, either 6-port can be used to make impedance ratio measurements. Also since

V as well as y is known, eqs. (96a) and (96b) can be used to calculate voltage or current

ratios with 6-port #2.

H. Choosing the two terminations.

One purpose of the measurements made when each 6-port is connected to the calibration

circuit is to define the measurement reference planes. For this reason, the magnitude of

the two values of reflection coefficient r and r, seen looking into the calibration

e f ^ circuit input port should be fairly large. Therefore, a short and an open were

initially chosen for the two terminations in the NBS system. However, the short and open

combination led to trouble at certain frequencies in inverting the matrices in eqs. (57)

and (60). When arg(r ) = and arg(r^) = 180° (or vice versa), the matrix being inverted

in (57) becomes singular. When arg(r ) = 90° and arg(r ) = -90° (or vice versa), the

matrix in (60) becomes singular. Either of these two sets of phase angles will occur at certain frequencies when using a short and an open as the two terminations. At angles

other than these two combinations, the solution appears to be well behaved even though

arg(r^) - arg(r^) = ±180°.

The optimum set of terminations has not been determined. However, it appears from

computer simulations that two terminations whose phase angles are 90° apart do not create

any singularities in the solution. One way of creating two terminations whose reflection

coefficients are 90° apart over a broad frequency range is shown in the calibration

circuit in figure 7. Assuming ideal components, r = 0.25 at 6 + 90°, and r^ = 0.5 at

e^ where B and 6 are unknown residual phase shifts in the lengths of line. The length

of line in switch position f is chosen so that The difference in phase is then

f e" 90° over the frequency range for which the quadrature hybrid and the power divider are designed to operate.

^e'^f

VNA/v-

<^e

SHORT

OR OPEN

H

Figure 7. A calibration circuit with two terminations at e and f which have

a 90° phase difference. D is an in-phase power divider and Q is a quadrature hybrid.

-21-
VI. Adding the Standard

The constant K_ in eqs. (72) and (97) has not yet been determined. To find K„ we can use a uniform length of line or a standard termination whose impedance Z or reflection

coefficient T is known. It is convenient at this time to rewrite eqs. (72) and (97) in

the form

^l^^Ol " ^0^1

^2^^02 ^0^2

(98)

(99)

where the small z's, z^ and z„, are the coefficients of K„ In eqs. (72) and (97). The

accuracies of the values of z^ and z_ are Independent of the accuracy of any standard used.

Their accuracies are functions only of the precision with which the previous measurements

have been made.

A. Standard Termination.

If a standard termination of known Impedance Z^ is measured on 6-port #1, eq. (98)

gives

^0^01 "

z Is

Is

(100)

so that 6-port //I is now calibrated for Impedance measurements. Since normally Z_^ = Z_„,

6-port #2 is calibrated also. Or using r ,

K = _i Is

° " ^is ^01

1 s

z- 1-r

Is s

(101)

Examination of (100) and (101) shows that the standard cannot be a short or an open since

K_ is finite and not zero. A value near Z„^ would be quite adequate.

LENGTH OF LINE

Figure 8. Defining the direction of Z. and Z .

-22-
B. Standard Line.

When using a uniform length of line as the standard, define the input impedance Z. and

the load impedance Z as shown in figure 8. The two impedances are related by [13]

Z. Z + Z_ tanh yl — = — (102)

Z- Z„ + Z tanh yl

where Z^. is the characteristic impedance of the line, y is the complex propagation constant,

and a is the length of the line. Connect 6-port //I to the input end of the line and 6-port #2 to the load end so that

Z, = Z. , Z„ = - Z^ . (103)

1 1 2 L

Then eq. (102) becomes

Z Z Z -i (1 - -r^ tanh y£) = - -^ + tanh ySL (104)

^0 ^0 ^0

Assuming that Z = Z = Z , eqs. (98) and (99) in eq. (104) give

Kq(z +Z2) = K^z z tanh yl + tanh yZ . (105)

This equation can be solved for K. if yi is known. If yi is not known, K_ can still be

determined but more measurements are required as discussed below.

1) Determining ySi of the line: By taking two or more measurements with the 6-port

reflectometers when the line is inserted it is possible to solve for K_ without knowing yl. In fact, Y^ can be determined along with K^^. First divide eq. (105) by K„ to get

z^ + Z2 = z^Z2Kq tanh yl + MBlLJri ^ (106)

This equation is linear in the two unknowns

u E K tanh yl (107)

V E tanlLYA ^ (108)

^0

Two or more measurements are taken with the line inserted by changing A^ , A , or (ji to

' different but unknown settings. In the present NBS system, two to four measurements are

' taken by changing (}> only. 4, is such that a /a == 1 at 90°, -90°, 0°, and 180°. Each

I measurement gives an equation like eq. (106) which can be combined into the single matrix

I equation

-23-

Hk^
' V2 1 ( u 1 z^ + Z2 — —

^ I ^ — — — V. ,* —

I

(109)

U V = w

If only two measurements are taken, IJ is a square 2 by 2 matrix and the solution is

-> -1 -> V = U W . (110a)

If three or more measurements are taken, a least squares solution is obtained using the

pseudoinverse for complex matrices [14].

V = (U*U) "'"U*W (110b)

where U is the transpose of the complex conjugate of U. Knowing u and v, K„ and tanh yl

are calculated from eqs. (107) and (108);

y V

tanh y2, = K_ V = u/K

o

(111)

(112)

One way of choosing the sign of K^. is by comparing the value from eq. (Ill) with an estimate

calculated from the 6-port circuit assuming all components are ideal. For example, in the

ideal circuit shown in figure 5, using detectors 3, 4, 5, and 6, the phase angle of K„

calculated from eq. (73) is

-1 ""5 arg (Kq) = tan -" ^ = 90° .

The sign of K„ from eq. (Ill) is chosen so that the phase angle is closest to 90°. The sign can also be chosen so that the impedances calculated from eq. (72) for the two terminations

in the calibration circuit are consistant with their estimated or previously calibrated

values. Or eqs. (107) and (108) can be used to write

tanh yH = ± /uv

K_ = — tanh yl V

(113)

(114)

in which case the sign is chosen so that tanh yZ is consistent with that calculated from an estimate of ySi determined from the dimensions of the line. In either case yi is determined

along with K^. Each 6-port is now calibrated for measuring impedance or reflection

coefficients relative to the characteristic impedance of the line.

-24-
2) Determining Z„ of the line: Since yH is known, it can be used in determining Z„ of

the line. For a coaxial transmission line [13]

Y = / (R+ju)L)(G+ja)C) (115)

Zn=/?f^ (116)

V G+j (dC

where R and L are the series resistance and inductance per unit length of the line, and G and C are the parallel conductance and capacitance per unit length of the line. For an air

line |g| << |a)C|. Then eqs. (115) and (116) combine to give

Z = —^— -

"^^ (117) G+juC jwCa • ^ ^

Since yl is known, only Ci needs to be determined. This can be calculated from the cross- sectional dimensions of the line or measured with a low frequency capacitance bridge.

Since C is independent of frequency, CZ is the total line capacitance at low frequency

and can be accurately measured at say 1 KHz. Thus yi and C£ and hence Z^ can all be

determined without knowing the length or series loss of the line when shunt losses are negligible.

3) Choosing the line length: When using a length of line as the standard, its length

must not be too near a multiple of half wavelengths long at the frequencies of operation,

or the solution will become ill-conditioned. One way to avoid this condition is to choose

the length as illustrated in figure 9a. If the 6-ports are designed to operate over the

frequency range f to f„, the length is chosen so that it is a half-wavelength long at the

frequency f^ * ^o* Then the effective minimum phase shift will be the same at both ends

of the band of operation, and will be 90° at the center of the band. If the effective

phase shift at the band edges is too small, or if the line length becomes physically too small, then multiples of this length can be used as illustrated in figure 9b. In figure

9b, |(j)-180n| has been plotted instead of <j), choosing n so that all values fall within

to 90°. For example, to operate over the 2 to 18 GHz range, f + f„ = 20 GHz. For a coaxial

system, a length of air line 0.75 cm long is a half wavelength at 20 GHz. So 0.75 cm would be a preferred length. This line would have a minimum effective phase shift of 18°

at both 2 and 18 GHz. If this phase shift is considered to be too small, a second line

three times as long as the first (2.25 cm) would increase the minimum effective phase shift

to 45° as shown by the shaded accent in figure 9b. The shorter line would be used from

5 to 15 GHz, and the longer one outside of this range.

-25-
•a: :r

FREQUENCY

(a)

21, 3J6| 4Jl,

CD

lUO

/ y.v.

10 12 FREQUENCY, GHz

(b)

Figure 9. Effective phase shift through standard transmission lines as

a function of frequency where ^ is the phase shift through

the line and n = 0, 1, 2,.... In (b), n has been chosen so that |({)-180n| is between and 90°. The line lengths are chosen so that they are multiples of a half wavelength at

f^ + f«, where f and f„ are the frequency limits of operation

of the 6-ports. For coaxial air lines.

= 12 f £ , degrees.

GHz cm

H- = 15

1 f, + f2 , cm for f in GHz,

-26-
VII. Filling in Calibration Matrices

Now that K„ has been determined, the calibration matrices H^ and H„ can be completed

except for one constant which is not required unless absolute measurements of power,

voltage, or current are desired. Let the components of K^^ be

K^ + j K^ (118)

Then from eq. (73)

,

^5 = ^3 ^1

^5 = ^3 ^2

(119)

(120)

From eqs. (74) and (75),

^6 = ^^5 - ^^5

= y3(xK^ - yK2) (121)

^6 = y^5 "*" ^5

= P3(yK^ + XK2) (122)

And from eq. (80)

\ = ^3 (123)

The elements of the 4 by 4 matrix h which was defined in eqs. (61) and (62) and expanded

in eq. (65) are now all known except for p . Factoring p. out of h it becomes

h

=

v_ V, 3 4 ^3^0

q3 q^

' '""s^e

^.^0

K^ xK^-yK^

K2 yK^+xK^

= v^n (124)

where H is the matrix h with the unknown parameter y factored out. The matrix H is calculated from eq. (62)

H^ = h Y = y3 H Y (125)

-27-
and H„ is calculated from eq. (35)

H„ = N H^ J = p„ N H Y J . (126)

The parameter y. will cancel in calculating impedance or ratios of voltage, current, or power. It is only needed to calculate absolute values of voltage, current, or power.

To calculate reflection coefficients or S-parameters , G^ and G„ are calculated from

H, and H„ using (23).

1 ^3

G, = r KH, =7^ KHy (127)

1 ^3

G^ = f- KH„ = -r^ KN HyJ . (128)

_2 4 ^^2 4 ^_v,J.^

The parameter y also cancels in calculating reflection coefficient or S-parameters.

VIII. Calibrating for Power, Voltage, and Current

To calibrate both 6-port reflectometers for measuring power, a power standard is con- nected to one of them, say #1. Then eq. (19c) gives*

q P Z„, = |v, i,Z„J c

s ^01 ^ ^ '^3 - l^l^l^Oll "°^ ^1 = ^3 ^ ^^i ' ^ = ^•••^ ^^^^^

where the parameters q./y- are known in Hy, and P is the net power into the standard power

meter. The constant y„/Z„^ can be calculated from (129). Then the net power into any

termination connected to reference plane #1 is

P-, =^I — P. . 1 = 3. 1 z_, y-, 1 ^01 ^3

I Note that Z_^ does not have to be known to calibrate the 6-port for making power measure- J

1 ments. However, Z_ does have to be known to find y„ which is needed for absolute measure- | ments of voltage or current. From eqs. (19a) and (19b) i

|vJ2 = y_ I^P^ (130)

3 ^ P3 ^i

i = 3.

^ j2 = 3 V i

1 1Zq3_i M3

|i, I" = i„- I V J — P, (131)

^ Mo 1

where the parameters v./y and Mj/Vo are known in Hy. When y and Z„ = Z_^ = Z„„ are known,

both 6-ports are completely calibrated.

*Equation (129) assumes that Z„^ is real so that 9^ = arg(v,/ip,

-28-
IX. Evaluating the Calibration

When taking more measurements during the calibration than are needed, the redundant

equations can be used to determine the consistency and precision of the measurements. When

the reference planes are together, each measurement made by the two 6-port relectometers

should satisfy

P^ = -P2 (132)

Z^ = -Z2 (133)

r^ = IIV^ (134)

^12 " ^21 " -^ " >^PiP2 • (135)

When connected to the calibration circuit, the 6-port reflectometer measurements should give

P^ = P2 (136)

Z^ = Z2 (137)

r^ = r2 . (138)

Lastly, when the standard line is inserted, each measurement should satisfy

r^ = e"^^^/r2 (139)

^12 " ^21 " ^~^^ " '^^fl' • ^^^°^

Another useful check on the calibration is to calculate the matrix B in (5) using (8);

B =

g"-^

. (141)

Note that the elements of B in (5) are not all independent since

4|a^|2|b^|2 = (2|a.B.| cos C.)^ + (2|A.B.| sin ?.)2 ^ (142)

In words, four times the product of the first two elements in any row of B should equal the

square of element 3 plus the square of element 4 in that row. The extent to which the calibration data satisfy equations (132) — (142) is a measure of

the quality of the calibration.

-29-
If a uniform length of transmission line is used as the standard, another check is to calculate aZ//F and &Z/f at each frequency f, where yl = aZ + j3^- Both of these parameters

should be constant with frequency.

X. Acknowledgment

The author wishes to acknowledge the earlier unpublished work of Charles Manney in

which he showed that self-calibration techniques can be applied to two 6-port reflectometers.

-30-
References

[I] Hoer, C. A., and Engen, C. A., Analysis of a six-port junction for measuring v, i, a, b, z, r, and phase. Presented at the Proc. IMEKO Sjmip. Acquisition and Proc. of Measurement Data for Automation, Dresden, Germany (June 17-23, 1973). Published in ACTA

IMEKO 1973, Vol. 1, pp. 213-222,

[2] Hoer, C. A., Using six-port and eight-port junctions to measure active and passive

circuit parameters, NBS Tech. Note 673, 23 pp. (Sept. 1975).

[3] Cronson, H. M. , and Susman, L. , Automated six-port microwave calibration system.

Sparry Research Center Final Technical Report No. SCRC-CR-76-32, 72 pp. (April 1977).

[4] Engen, G. F., The six-port ref lectometer : An alternative network analyzer, IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 1075-1080 (Dec. 1977).

[5] Cronson, H. M. , and Susman, L. , A six-port automatic network analyzer, IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 1086-1091 (Dec. 1977).

[6] Engen, G. F. , An improved circuit for implementing the six-port technique of microwave

measurements, IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 1080-1083 (Dec. 1977).

[7] Komarek, E. L. , An application of the six-port ref lectometer to precision measurement

of microwave one-port parameters, 1977 International Microwave Symposium, San Diego,

CA (June 21-23, 1977). Digest pp. 56-57.

[8] Weidman, M. P., A semi-automated six-port for measuring millimeterwave power and

complex reflection coefficient, IEEE Trans. Microwave Theory Tech., Vol. MTT-25,

pp. 1083-1085 (Dec. 1977).

[9] Engen, G. F. , and Hoer, C. A., Application of an arbitrary six-port junction to power

measurement problems, IEEE Trans. Instrum. Meas., Vol. IM-21, pp. 470-474 (Nov. 1972).

[10] Hoer, C. A., A network analyzer incorporating two six-port ref lectometers, IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 1070-1074 (Dec. 1977).

[II] Allred, C. M. , and Manney, C. H., The calibration and use of directional couplers

without standards, IEEE Trans. Instrum. Meas., Vol. IM-25, pp. 84-89 (March 1976).

[12] Greville, T. N. E. , The pseudoinverse of a rectangular or singular matrix and its

application to the solution of systems of linear equations, SIAM Review, Vol. 1, No. 1, pp. 38-43 (Jan. 1959).

[13] Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, New

York: Wiley, Table 1.23, p. 46 (1965).

[14] Penrose, R. , On best approximate solutions of linear matrix equations, Proc. Cambridge Philos. Soc, Vol. 52, pp. 17-19 (1956).

-31-
APPENDIX

This appendix gives an outline of a program for taking the calibration data, a description of how the data can be stored, and finally an outline of a second program for

calculating the matrices H^ and H„ which characterize the two 6-ports. Equations given in

the second outline refer to the equation numbers in the body of this paper,

A. Taking the Calibration Data.

This section contains an outline of a program for taking the calibration data using

a precision length of transmission line as the impedance standard. It is assumed that when

each 6-port is connected to the calibration circuit the generator is leveled with the output, P , of this circuit so that P is constafit and need not be measured,

c c

Connect reference planes together

For measurement #a, b, c, .. .corresponding to four or more settings of

aVa such as 1 at -90°, 1 at 180°, 1 at 0°, 0.3 at 90°, 1 at 90°, 0.3 at -90°.

For each frequency

For each sidearm on 6-port #1

Read sidearm power

Next sidearm

For each sidearm on 6-port #2

Read sidearm power

Next sidearm

Next frequency

Next measurement (next setting of a^/a,

)

Connect 6-port #1 to calibration circuit

Set calibration circuit switch to position "e" For each frequency

For each sidearm of 6-Port #1

Read sidearm power

Next sidearm

Next frequency

Set calibration circuit switch to position "f"

For each frequency

For each sidearm of 6-Port #1

Read sidearm power

Next sidearm

Next frequency

-32-
Connect 6-Port #2 to calibration circuit

For each frequency

For each sidearm of 6-Port #2

Read sidearm power

Next sidearm

Next frequency

Set calibration circuit switch to position "e" For each frequency

For each sidearm of 6-Port #2

Read sidearm power

Next sidearm

Next frequency

Insert standard line between the two 6-Ports.

For measurement if g, h, .. .corresponding to two or more settings of a„/a

such as 1 at 90°, 1 at -90°, 1 at 180°, 1 at 0°

.

For each frequency

For each sidearm of 6-Port #1

Read sidearm power

Next sidearm

For each sidearm of 6-Port #2

Read sidearm power

Next sidearm

Next frequency

Next measurement

End

B. Data Sequence.

In the above sequence for taking the data, all sidearm power readings are recorded

at all frequencies for any one setting of A , A„, and (f>. This sequence of taking the

data minimizes wear on the electromechanical switches in the components A.., A„, and ()).

C. Data Array.

All of the calibration data is taken before any computations are done. One way of

storing the calibration data is shown in figure 10. As the 6-port sidearm power readings

are taken, they are stored in one large data array called P^. Each element P in P^ represents

a power reading. Each row in JP contains all of the power readings taken at one frequency,

so that P^ contains as many rows as frequencies. The number of columns in P^ is at least 64,

but it can be larger depending on how many redundant measurements are made.

To save on computer memory, ^ can be broken into two equal parts as shown in figure 10

I and each part stored separately. All of the power readings for the first part of P^ with

the reference planes together are taken before any of the readings are taken for the

second half of P;. -33-
§

a:

REFERENCE PLANES TOGETHER

^1

6-PORT #1

l2

6-PORT #2

3

SIDEARM NUM]

4

3ER

5 6 7

SIDEARM NUMBER

8 9 10

MEASUREMENT

No. abed. .

MEASUREMENT

No. abed. .

MEASUREMENT

No. abed. .

MEASUREMENT

No. abed. .

MEASUREMENT

No. abed. .

MEASUREMENT

No. abed. .

MEASUREMENT

No. abed. .

MEASUREMENT

No. abed. .

1

2

3

4

5

P P P P P P

P P P P P . PP...

P . . .

PP...

1 2 3 4 5 6 7 8

P F P . . . .

G

Di

CONNECTED TO CALIBRATION CKT. STANDARD LINE INSERTED \ °2 ^1 ^2 m 6-PORT #1 6-PORT #2 6-PORT i\\ 6-PORT #2

f^ SIDEARM NUMBER SIDEARM NUMBER SIDEARM NUMBER SIDEARM NUMBER

bj 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10

bil

MEASURE!lENT NO. MEA!JURE!lENT NO. ^lEASUREMI;nt no. LEASUREMI:nt NO.

e f e f e f e f e f e f e f e f g h . . g h . . g h . . g h . . g h . . g h . . g h . . g h . .

1 P P P .

2 P P

3 P .

4

5 9 10~ 11 IT 13 14 15 16 17 18 19 20 21 22 23 24

P P P .

(«)

\-

— —

1

9 10

11

12

1 _J

13

14

1516

18

1920

21

2223

24

Figure 10. One way of storing the sidearm power readings, P, so that

all readings taken at a given frequency are in one row (a)

.

After all the data is taken, the matrices P^ , '?_^, D^, D2,

S^.| , and S^„ are formed from any one row as indicated by the

example in (b) for row (frequency) #5.

-34-
After all of the data is taken, the elements of P^ are used a row at a time to calculate

the matrices H^ and H„ for the frequency corresponding to that row. The elements in any

one row of P^ are first sent to six different smaller matrices as indicated in figure 10b

which shows an example using row #5. In general, the data in any one row is transfered

as follows:

1. The readings taken when the reference planes are together are sent to P^ and P^„

;

(33)

(34) P = fP P P P . . -1 ^ la lb Ic Id -2 '^^2a ^2b ^2c ^2d"-'

The readings taken when each 6-port is connected to the calibration circuit

are sent to D^ and D„

;

^1= f^le^lf]' °2 = fP2e' ^2f^

(48)

(49)

The readings taken when the standard line is inserted are sent to S^^ and S^~

;

^1= t^lg^lh-']' ^2= f^2gP2h--l

A subscript 1 or 2 refers to 6-port #1 or #2, and subscripts a, b, c,...h refer to the different measurement conditions. Each of the vectors in P^-. • • • S^o contains four power

readings. For example, D^ is

h = f^le ^If]

=

3e

•4e 5e

6e

3f

•4f

5f

6f

1 where subscript 3, 4, 5, or 6 on P refers to the sidearm number, and e and f refer to the

1 measurement condition where termination e or f was used in the calibration circuit.

D. Calculating H and H-.

This section contains an outline of a program for calculating the matrices H and

H„ which characterize the two 6-ports. The choice of roots and signs corresponds to the

6-port design shown in figure 5 using detectors 3, 4, 5, and 6. For each frequency

Fill in matrices P^ , P^ , ^ , ^ , S^ and S^„ from the £ array.

Calculate matrix J. If 4 measurements are taken with ref planes together.

£ = ?i li^

If more than 4 measurements are taken,

T T -1 £

= I1P2 ^^2^2 ^

-35-

(36a)

(36b)
Calculate E^ = J D„.

Partition D and E.

°1

= E =

[^ )

.%

(51)

(52)

Calculate a and 3 matrices,

a = (^1 - d^)(d2 - 62)

-1

= -(e2 + d2)(d^ + e^)

-1

(57)

(60)

Fill in Y matrix.

Y

=

I a (61)

(62)

Calculate 6 vectors for 6-port #1, store in &_. *

(64)

Set up and solve the matrix equation

R X = F

for X. Each column of £ is used to fill in one row of R and F. Calculate yv from elements 4 and 5 of X assuming | y v

| < 1.

(86)

yv =

2 x^X3

1 - 2x,x,- + /i-4x,x_

4 5 4 5

(89)

Calculate K, v, y, x, and y from the elements of X and yv.

K = x^(l + yv)

V = x^(l + yv)

y = X3(l + yv)

x ~ X2/2^1

V

!3 2 - X

(84a)

(84d)

(84e)

(90)

(91)

*See section C (Data Array) for definition of P and D . 6^ is a new matrix formed from

6 or more 6 vectors.

-36-
Using the standard line data, set up and solve

UV = W

Calculate

(109)

4 = !^,

4 = lih

For each measurement g, h,

z^ and z„ f

-53 + (x+jy)6^

calculate z,12and z„ for each column of 6 and T —s —

£

^1

=

63^ + V^2

(64)

(95)

(72)

(98)

tj + (x+jy)t^

^2 t^ + yt3

(97)

(99)

Fill In corresponding row of U and W.

U = "l"2 1 W = "1 + ^2 (109)

Next measurement

Solve UV = W for V =

If two measurements are taken with the standard line inserted,

V = u""HJ . (110a)

If more than two measurements are taken with the line inserted.

V = (U U) (U W) . (110b)

Calculate K and yZ from the elements of V.

o

o 1 - 2 \ / V

choose the sign of K such that

o

0° < arg(K ) < 180=

(111)

(112b)

-37-
tanh Y^ = '^r~ " x+JY (112)

ai = 0.25 Jin (1 +

4x

2 2 l-2x+x +y

?£ = 0.5 tan -^ { -^ p-^ ) 1-x -y

Fill in matrix H.

H = ^1 xK^-yK^

K2 yK^+xK2

(124)

Calculate H^ and H„, the desired calibration matrices, setting p = 1.

H^ = Hy (125)

^2 ^v^l^ (126)

where N is given by (31). (This completes the calibration. The remainder of

the program evaluates the calibration.

)

Evaluate the calibration.

Calculate G, = f-KH, _1 4^^1

?2 " 4TM2

^1 = ^1

^2 = ^2'

(127)

(128)

(8)

(8)

where K is given by equations (14) and (15).

Calculate inconsistency in calibration constants.

For each row in B calculate and print

A. =

1 B(i.l)B(i,2)

[B(i,3)^ + B(i,4)^]/4

-1 (5)

where i = row number.

tSee Dwight, "Tables of Integrals and Other Mathematical Data," Macmillian Co., 4th ed.,

p. 168, #722.2.

-38-
Next row in B

Repeat the calculation of A. using B„.

For data taken with reference planes together

calculate A = G P

^^ (7)

-2 " ?2-2 • ^^^

For each column of A^ and A„ corresponding to measurement

#a, b, c, .. .calculate and print

Ap = p^ - I/P2 C134)

AS21 = >V2 - ^ (135)

P +P

AP = ^ . (132)

Next column.

For data taken with standard line inserted

calculate A^ = G^S^^ (7)

-2 " ?2-2 • ^^^

For each column of A^ and A„ corresponding to measurement //g, h, .. .calculate and print

Ap = p^ - e'^^'^/p^ (139)

^^21 " ^^^1^2 " ®~^^ • ^^^°^

Next column.

End of evaluation.

Store H^ and H„ (or G^ and G„) on tape or disk.

Next frequency.

-39-
NBS-lUA (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS TN-1004

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

CALIBRATING TWO 6-PORT REFLECTOMETERS WITH ONLY ONE

IMPEDANCE STANDARD

5. Publication Date June 1978

6. Performing Organization Coc 724

7. AUTHOR(S) Cletus A. Hoer 8. Performing Organ. Report N

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No

7241403

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CCG 77-99

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Mr. Jerry L. Hayes

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14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

This paper describes a technique for calibrating a pair of 6-port reflectometers for measuring the reflection coefficient of 1-port devices, or the

scattering parameters of reciprocal 2-port devices. The operations in the

calibration consist of connecting the two 6-ports together, connecting each

6-port to a calibration circuit consisting of two terminations of unknown impedance and a leveling loop, and then connecting the standard. The standard

can be one termination whose complex impedance is known, or a precision length

of transmission line whose cross-sectional dimensions are known. The length

and loss of the line are not required. The solution for the constants which

characterize each 6-port is closed, requiring no iteration.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper J%

name; separated by semicolons)

Calibration; current; impedance; network analyzer; power; reflection coefficient;

scattering parameters; six-port; ref lectometer ; voltage.

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