/ V \

V

•"•tAU 0« * NBS TECHNICAL NOTE 1012

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Calibrating A Six-Port Reflectometer

with Four Impedance Standards
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1

The National Bureau of Standards was reorganized, effective April 9, 1978.
jtettonar Boreaa of Standards APR 2 7 1971

Calibrating A Six-Port Reflectometer

with Four Impedance Standards

Cletus A. Hoer

Electromagnetic Technology Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

^iiii

f}Aj\I rcJ) »./(;

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

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued March 1979
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1012

Nat. Bur. Stand. (U.S.), Tech. Note 1012, 24 pages (Mar. 1979)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1979

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

Stock No. 003-003-02036-2 Price $1.20 (Add 25 percent additional for other than U.S. mailing)
CONTENTS

Page

I. INTRODUCTION 1

II. SIX-PORT REFLECTOMETER EQUATIONS 2

III. CALIBRATION EQUATIONS, a AND b PARAMETERS 3 A. Iterative Solution for the G-j 3 B. Initial Estimate of the G-j 5

IV. EQUATIONS FORT 6

I II A. Iterating to Find r and r : 6 B. Linear Solution for r , F , and |r|^ 7 C. Constant Matrix Solution for r', r", and |r|2 8

D. Comparison of Solutions for r 8

V. CALIBRATION EQUATIONS, v and i PARAMETERS 9

VI. SIMPLIFIED CALIBRATION EQUATIONS 10

A. Ideal Short 10

B. Ideal Open, Reactance, and Resistance 10

C. Initial Estimates of Ho H

VII. EQUATIONS FOR Z 12 A. Constant Matrix Solution for R, X, and |z|2 12

VIII. CONCLUSION 13

IX. REFERENCES 14

APPENDIX A 15

m

II IHimiBHUIliHHilllllllllH

Calibrating a Six-Port Reflectometer

with Four Impedance Standards

Cletus A. Hoer

This report is a theoretical study showing how four terminations of known

impedance can be used to calibrate a six-port reflectometer for measuring other

terminations of unknown impedance. The equations for obtaining the calibration

constants are exact but nonlinear, requiring an iterative solution. Several

ways are described for using these constants to calculate the impedance of

terminations being measured with the six-port reflectometer.

Key Words: Calibration; impedance; reflection coefficient; reflectometer; six-port.

I. INTRODUCTION

A six-port reflectometer is a circuit with six ports designed to measure power and

reflection coefficient at one of its ports in terms of power readings taken at four other

ports when a signal is applied at the remaining port. To measure reflection coefficients

with a six-port, it must first be calibrated to determine 11 constants which characterize

the six-port. This paper describes a technique for determining these constants with four

different terminations whose reflection coefficients are known.

Several techniques have already been published for calibrating a six-port

reflectometer. Engen [1,2] has described a technique using a sliding short and a sliding

load. However, at frequencies below about 100 MHz, it becomes impractical to use sliding

components because their length becomes too long. It then becomes more practical to use fixed terminations which have known values of impedance. A technique for calibrating a pair

of six-port reflectometers is described by Hoer [3]. This technique also becomes

impractical below about 100 MHz because the standard transmission line used in the

calibration procedure becomes too long. Additional work by Engen [2] has shown that it is possible to calibrate a six-port reflectometer with only three known terminations if one first reduces the six-port to an equivalent four-port. This reduction requires that the

six-port sidearm power readings be recorded when a minimum of five but preferably nine

different unknown terminations are connected to the reference plane. Three of this set of

five to nine terminations can be the three known standard terminations. Cronson and Susman

[4] have published equations for use in calibrating a six-port reflectometer with four known

terminations. Their solution is a closed linear solution, but it assumes that the wave

incident upon each of the four terminations is the same when they are connected to the

reference plane. This assumption is not usually satisfied. The calibration procedure

described below in this paper also uses four known terminations, but the solution for the

calibration constants does not require that the incident wave be held constant. However,

the equations then becomes nonlinear, requiring a solution which is an iteration of an approximate linear solution.

This report should be viewed as a preliminary theoretical study. Further work needs to

be done to determine the relative merits of the different proposed solutions.
II. SIX-PORT REFLECTOMETER EQUATIONS

Let the six-port power detectors be labeled as shown in figure 1. It has been shown

[3] that the power P-j measured at port i = 3,4,5,6 is given by

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

where a,b are the waves incident upon, and reflected from, the termination at the

measurement reference plane, and A-j , B-j are functions of the scattering parameters of

the six-port and the reflection coefficients of its four power detectors. Engen [5,6] has

rewritten eq (1) in the form

Pi = iB^al^ |r- q.|2, i = 3. ..6, (2)

where r is the reflection coefficient of the termination at the reference plane, and the

q-j are new constants describing the six-port;

r = b/a, q.= -A./B. . (3)

The four complex q-j and three of the four |B-jl are the 11 constants needed to calculate

an unknown r from the four power readings.

For this paper it will be more convenient to write eq (1) in the form

P. = |A. a|2|l + G.r|2, i = 3. ..6, (4)

where

G. = B./A. = -1/q.

\AA/rorZ

Figure 1. Six port reflectometer.
III. CALIBRATION EQUATIONS, a AND b PARAMETERS

The calibration is performed by recording each of the four P-j when four different

terminations having known reflection coefficients are connected to the reference plane

Add a subscript SL = 1,2,3,4 to P-j, a, and r in eq (4) to indicate these four known

terminations.

Let the six-port be designed and the sidearms numbered so that P3 is primarily a function

of the incident wave a. Then P3 will never be zero and we can divide eq (5) by P^g to

get

!lt, INl' l^"°i 'f 1 = 4.5.6;

: P3, IA3I2 II + G3 rf £= 1...4. («'

This ratio eliminates la^l so that no assumption about the incident wave needs to be

made in the calibration process.

For the first termination, eq (6) gives

"n IM^ i = 4,5,6. (7)

|l + G,ri|2

"31 \hl' ll.GjFjl^

Choose the first termination so that its reflection coefficient r^ , is such that none of the P-ji are zero. For most existing six-port reflectometer designs, if ^1 = 0, none of the P-ji will be zero. Then dividing eq (6) by eq (7) gives

^U-'-AL^l l ^^'^i^^ ' ' S ^ll' ^- = 4,5,6;

(3)

'3,'n " il^^S^. • l^^i '\\

' ^ = 2,3,4,

where the ^^^ are calculated from the observed power readings. There are nine equations

like eq (8) which can be solved for the four complex constants G3...G5 (8 real

unknowns). The equations are nonlinear and in general an iteration procedure must be used

to solve for the unknowns.

When r^^O, the remaining three terminations should be highly reflecting and have

values of r whose phase angles are perferably in different quadrants.

A. Iterative Solution for the G-j

To simplify the solution of eq (8) for the four G^, define

g^, ^|i + G. rj^ i = 3.. .6, (9)

and let the components of G-j and Fjj^ be

G. =G.' + j G." , i = 3. ..6, (10)
Expanding eq (9) with eq (10) and eq (11) leads to Su= ll^S^j'-'^^S'--,' -2G,"r^"- |G,|' Iff . (12)

Using eq (9), eq (8) can be rearranged and written as

'iz^2i ^n = 9i£93i' I = 1:1:1. (13)

Next define f-jj^ as Ui = '^l^3l 9ii - 9^,931 = 0.

^ = ^^^^^^ ^^^

and expand f-j£ in a Taylor series [7] about the best estimates of G3 and each G-j

.

Keeping only linear terms, eq (14) becomes

where foi£ is the value of f-jj^ calculated from eq (14) using best estimates of

G3 and Gi . The partial deriatives in eq (15) are obtained from eq (14);

^ ^ »93£ 9931

aGj' = ^ii ^G^ ^n " ^ii "96^

^Ui 993£ 9931

"36^ ~ ^ii 9G3" 9ii -

9iji

-^

8G.' " ^il ^31 3G.' " 8G.' ^Zl * " '^''''^' ^ " ^iA 93JI -^ - ^

931

where from eq (12)

,

3g. i = 3... 6; '' 2 0=1. (17) 9G." = -2r^" + 2 G."|r^p A = 1...4.

These partial derivatives are calculated using the same best estimate of G3 and G, used

in calculating foi)i«
ifie nine equations represented by eq (15) are solved for the eight unknown aG-j ' and

AG-j". Since the equations are linear in these unknowns, the solution is straightforward.

One solution is outlined in Appendix A. The aG-j ' and aG-j" are then used to improve the

previous estimate of the G-j ' and G-j".

new G^' = old GV + aG^.

'

new G." = old G." + AG."

i = 3 ...6.(18)

These new estimates of G^ ' and G^-" are used to compute new fg-jj^ and partial

derivatives. The solution of eq (15) is then repeated for a new set of aG^ ' and aG-j".

The iteration is continued until the AG-j' and aG-j" become insignificant.

B. Initial Estimate of the G.

Since the solution of eq (8) for the G-j is an iteration process, estimates of the

G-j are needed to begin the iteration. If the six-port is designed so that P3 is

primarily a function of the incident wave, then G3 will be small. Also, we have assumed

that ri is small. Under these conditions, estimates of the G-j may be obtained by

letting

S= h= 0- (19)

Then eq (8) reduces to

'u= Ii*«i'-.i 'U

i = 4,5,6;

i = 2,3,4

(20)

which has been expanded in eq (12). Let the three remaining terminations have a large

constant reflection coefficient magnitude such as for three offset shorts. When

ITj^I = constant for z = 2,3,4, the three equations in eq (20) combine to give

^i2 ^i3=2G,-' (^2'-^3') - 2Gi"(r2"-r3")

«12 -«i4 = 2G^' ^^z'-h'^ - 2Gi"(r2"-r4")

(21)

(22)

which can be solved for G-j' and G-j", i = 4,5,6;

_ ^^2-^3^^V-V^-^^2-^4^^V-V^

^^ (r '-r ') (r "-r,") - (r '-r') (r '' -r ") (23)

2G " =

i ^v-v^^vy-^v-v^^v^3^

(r,'-r3') (V-V) - (r^'-r^') ir^"-Tp (24)

Since all quantities on the right of eq (23) and eq (24) are known, estimates of G-j (i= 4,5,6) can be calculated from these equations.

When one of the terminations is a flat short, and another termination is an open,

eqs (23) and (24) can be simplified. Let
r^ = 1 (open), ^2~ "1 (short).

Then ^2 -^3 = 0» ^nd eq (23) and eq (24) reduce to

Gi' = '% ' (25)

,.. 2(6.^-6.^) -(1-I^4-) (^,,-^,3) . G. ^ (26)

Note that the value of G-j is independent of r^. it is possible to obtain G-j"

independent of r4 also. For the short (r3= -1), eq (20) gives

5.3 = (1-G.')2 + G."2 (27)

from which

G/'2=«,3-(l-G.')2

,28)

where Gi ' is obtained from eq (25). This gives G-j" independent of ^4 except for a choice of sign. The sign of G-j" can be chosen to agree with that obtained for an ideal

six-port corresponding to the design being used.

Equations (23) and (24), or eqs (25) and (26) or eq (28) give estimates of the G-j (i= 4,5,6) to begin the iterative solution of eq (8) for the actual value of G-j.

IV. EQUATIONS FOR T

Once the G-j are determined, the ratios |A.j|2/A3|2 can be calculated from eq (7) . Call these ratios K-j

;

lA^I^ Pil 11 ^ Vll^

K. =—^-=^^ ^-^ , i = 4,5,6. (29)

1^3! P3I 11 ^'^i^ll

Then eq (7) can be used to obtain the reflection coefficient ^^^ of any other termination

conn connected to the reference plane;

p. |1 + G. r 1^ p^=K. ^-^ i = 4,5,6. (30)

3 '11+ G3rj^

The three equations given by eq (30) yield r^^.

With the G-j and K-j known, there are several ways to calcualteT^j. The most accurate

value of Tjj is obtained by iterating the three equations in eq (30) to solve for the two

unknown components ^^' and r^" of ^^^. A less accurate but closed linear solution may

be obtained from the three equations in eq (30) by considering ry', T^^" , \T^^\^ to be

three independent unknowns. A different linear solution can be obtained from eq (4). These

three different ways of calculating r^ are outlined below.

A. Iterating to Find T' and T".

The solution of eq (30) for ^^ is similar to the solution of eq (8) for the G-j.

Using eq (9) in eq (30), it can be rewritten
Pi 93 = Qi' i = 4,5,6, (31)

where the second subscript on g has been dropped and p-j is defined as

P.

3 i

Define fi as

fi =

Pi 93 - g^ = 0. i = 4.5,6, (33)

and expand fi in a Taylor series about the best estimate of r^^;

u u

where foi is the value of fi calculated from eq (33) using the best estimate of T^^. The partial derivatives

afi ago ag.

^ ^ , , , af^ = Pi aA Tr^ = ^(p. 103^ - |G. r) r^ + 2(p.G3 - G. ) (35)

u u u

*fi «g3 _ agi 2 2"

-^pr - p^ yp-tr -jf-r = 2(p.|G3| " |G^ | ) F^ - 2(p.G3 - G. ) (36)

for i = 4,5,6 are also calculated using the same best estimate of r^^. The three equations

represented by eq (34) are solved for the two unknowns ATl,' and AFu" which are used to improve the previous estimate of r^. The iteration is repeated until ATy' and aPu"

become insignificant.

An initial estimate of ^^J to begin the iteration with can be obtained from either of

the two linear solutions given below.

B. Linear Solution of r' , F", and |rl^.

A closed linear solution of eq (30) can be obtained by writing eq (30) in the form of

eq (31) and expanding with eq (12). This leads to

p.[U2G3T; -2G3"r;' -h IG3I2 lrj2] = i.2G.'r; -2G."r;' . 1G.|2 \rf (37)

or

l-P-2(p,G3'-G,') r; -2(p,G3"-G,")r;'MpJG3|2-|G,l')|rj2, (38)

1 = 4,0,0.

The three equations represented by eq (38) are linear in the three quantities ^^^' , Tl,", ITliI^. The solution for these quantities is therefore straightforward. Note that the

coefficients of r^j', r^', and |F^j|2 are the same terms needed to calculate the

partials in eqs (35) and (36).
C. Constant Matrix Solution for r, r", and |r|^ .

A closed matrix solution for r', r", and Irj^ can be obtained directly from eqs. (1),

(2), or (4). A derivation for these quantities beginning with eq (1) is given by Hoer [3], A similar derivation beginning with eq (4) leads to the matrix equation

iar

1 h'

Irl^ ^

r = G '5

II r !e\

(39)

where G is a real four-by-four matrix which is a function only of the calibration constants

IA3I, Gi, and Ky,

G

=

K4IG4I

K5IG5I

2G,

2K^G^

2K5G5

^hh

2G,

2K4G4

2K5G5"

- ^KgGg

(40)

The reflection coefficient is obtained from elements 1,3,- and 4 of the left column vector

in eq (39), i.e., from |a|2, |a|2p', and lal^r";

r =

2 2

r' | a | + j r"l , a | (41)

Since |A3|2 in eq (40) cancels when calculating r with eq (41), it can be set equal to one.

D. Comparison of Solutions for.r

The iterative method in A is more accurate than either of the closed linear solutions

described in B or C. This is becauseiall three equations in eq (30) are used to find only

two instead of three unknowns. When r', r", and |r|2 are considered to be three

independent unknowns, these three values will not usually satisfy the relation

.2 . 1.2 _ I ,2 r + r - Irl •

However, the loss in accuracy may not be significant or worth the extra time required to iterate.

One advantage of solution C over the other two solutions is that the matrix G is

independent of rand therefore needs to be calculated only once at a given frequency for any

number of measurements.
V. CALIBRATION EQUATIONS, v and i PARAMETERS

Instead of expressing the equations in terms of the reflection coefficient at the

reference plane, it is sometimes more convenient to write the equations in terms of

impedance. This is especially true when a high quality flat short is one of the four

standard terminations used in calibrating the six-port. If in eq (1) we let

V = a+b (42)

ii

then eq (1) becomes

iZ = a-b (43)

Pi = iC^v + D.iZj^ , i = 3.. .6, (44)

where v,i, and Zg are the voltage, current, and characteristic impedance at the reference

plane, and

A. + B. A. - B.

C. = -^2 -'

^i = "4 ~

^ " ^"•^' ^^^^

Now add a subscript £ as before to indicate the termination being used, and factor out

D-jiZo from eq (44) to obtain an expression similar to eq (4);

"u- i"iuZoi' ii*"i^j' i:?:::': (46)

where Zj^ is the impedance Z^ of the termination at the reference plane divided by Zq,

and H-j is a new constant describing the six-port;

z, - _^- A (47)

" \^o " ^0

H. ^ ^ * ^ (48)

Again assume that P3 is primarily a function of the incident wave, a, so that P3

will never be zero. Dividing eq (46) by P^i gives

PuJ^iij^V/ ''''''''(49)

P3, \0/ 11 + Hjzj^ , = I...4.

For the first termination, eq (49) gives — -^- -^ i = 4,5.6. (50)

P3I = \^3\ 11 " Vll

Finally, assume again that zj of the first termination is such that none of the P-j]^, are zero. Then dividing eq (49) by eq (50) yields
«i, _ 'n P31

_

I 1 ^ ^ih 1 ' hh

'2z'u~ I ''hh' i^^^i

2

i = 4,5,6

(5^)

£ = 2,3,4.

In this solution of this equation for the H-j is identical to the solution of eq (8) for

the G-j . Initial estimates of the H-j can be obtained from initial estimates of the G-j through eq (48), or from equation given below.

VI. SIMPLIFIED CALIBRATION EQUATIONS

A. Ideal Short

Choose the first termination to be a high quality flat short so that it can be assumed

that li=0. Then eq (51) becomes

^, . li^V,!' I : I'lf (52)

Since this equation assumes that none of the P-ji are zero, none of the sidearms can be

proportional to v = a+b since v is zero when a short is at the reference plane. Although

eq (52) is also a nonlinear equation, it is somewhat easier to solve for the calibration

constants than in eq (51) or eq (8).

No assumption about the remaining three terminations are made in eq (52). The value of

Z^ for these three remaining known terminations could be an open, a reactance, and a resistance. If these three terminations are sufficiently ideal, the solution can be

simplified as described below.

B. Ideal Open, Reactance, and Resistance

If the second termination is an open circuit, z^ becomes infinite so eq (52) gives

,2 6 IH., i2 = -"-^y i = 4,5,6. (53)

Let the third termination be a pure reactance, and the fourth termination be a pure

resistance. Then if we define

Hi = H.' + jH." (54)

i

and assume that Zq is real, eq (52) yields

i2

6 l^^^-^^3l'

i3"

|1 .H3JX3I2 ' i= 4,5,6, (56)

"5.4 = '-^ , i = 4,5.6. (57) |1+HR|^

Using eqs (53) and (54) in eqs (56) and (57) leads to

10
i3 = 1^ I-2H3 X3. 2 2 l"3l V

i = 4,5,6, (58)

^.„ 1 4 i2 ' 3' 4 T4 = ;

1 + 2 H, R +

3 4

2 2

i = 4,5,6 (59)

Solving eq (58) for Hj and eq (59) for H-j yields

Hi .i_l3,^3H3 -. X3 ( 12

^ 13) ^3]' (60)

^1

6 14-1 , ^4^3' ^^^4 ('14- '12) \hy

2R, (61)

for i = 4,5,6. Finally, adding the square of eq (60) to the square of eq (61) and

substituting into eq (53) gives

(H.'y + (H.')2 = |H. |2 =

<S IH 121"; ^+ 6 H

" + X-^^i^-^ 1^^3i' 2 Xo 13 "3 ^3 2 -^

2 R. " '14 ^3 " h

1 = 4,5,6. (62)

The three equations represented by eq (62) can be solved for H3 and H3 . The

remaining H-j and H-j for 1 = 4,5,6 are then obtained from eqs (60) and (61).

Although eq (62) is a nonlinear equation and must be solved by Iteration, the iteration is

faster than when solving eqs (8), (51) and (52) because only two unknowns are being solved

for.

C. Initial Estimates of H3

We have already noted that G3 is small for a six-port designed so that P3 is primarily a function of the incident wave. For this design eq (48) gives

H3 = 1 + jO.

This value of H3 is a good estimate to use in beginning the iterative solution of eq (62).

11
VII. EQUATIONS FOR Z

Once the Hi are determined, the ratios |D-j|2/|D3|2 can be calculated from

eq (50). Call these ratios small k-j

iD^i^ Pil 11 ^"3^1 ^

i = 4,5,6. (64;

ID. P3I 11 ' ^ h

Then eq (50) can be used to obtain the normalized impedance z^^ of any other termination

connected to the reference plane;

k. I^^^^i^ul' i = 4,5,6. (65;

These three equations can be solved for z^ in the same manner as eq (30) was solved for

Y^^^ One can iterate to find R^J and X^, as outlined in section IV A, or derive a linear

solution for R^, X^, and |zul as outlined in section IV B. A linear solution for

|v|, MZqI, Ru and Xy is similar to that out-lined in section IV C and is given

below.

A. Constant Matrix Solution for R, X, and \z\^ ,

A closed matrix solution R, X, and |z|2 at the reference plane can be obtained from

eq (46). Dropping the subscript z and expanding eq (46) gives

P.- = I D,i ZJ^ 1 + 2H.'r - 2H." X + |H.|^|z|^

1 ' 1

Using eq (64) in eq (66),

]

Pi = iDjI^ |iZj^ k. + 2K.H.'R - 2K.H."X + K.|H.|2|z|a

The four equations represented by eq (67) lead to the matrix equation

Iizj2

(66)

i = 3. ..6. (67)

z|^ '3

1 P4

R = H ^^5

X /6J

(68]

where]! is a constant matrix similar to G and is a function only of the calibration

constants ID3I, Hi, and ki

12
(69)

k4 |H4i'

The impedance is obtained from elements 2, 3, and 4 of the left column vector in eq (68):

Z

2H3' -2 H3"

2 k^ H4 - 2 k^ H4

'h^ - 2 k^ H5

^hh - 2 kg Hg

|1Zj2 R+ J |iZ I^X

Since |D3|2 in eq (69) cancels when calculating z^ with eq (70), it can be set equal

to one.

It can be shown that H and G are related by [3]

H = K G where

K

=

(70)

1 1 2

1 1 -2

1 -1

2

(71)

(72)

VIII. CONCLUSION

Equations have been derived for determining the calibration constant of a six-port

reflectometer and also for calculating r or z once these constants are known. Further work

needs to be done to determine how much accuracy is lost when using the closed linear

solutions for r or z instead of the nonlinear iterative solution.

The equations given in this report can be extended to the case of more than 6 detectors

or when more than four known terminations are available by letting i = 3,4,5,6,7,... and/or

£ = 1,2,3,4,5

13
IX. REFERENCES

[1] Engen, G. F., Calibration of an arbitraty six-port junction for measurement of active

and passive circuit parameters, IEEE Trans. Instrum. Meas. IM-22 , 295-299,

(Dec. 1973).

[2] Engen, G.F., Calibration the six-port reflectometer, 1978 International Microwave

Symposium Digest, 182-183, Ottawa, Canada, June 27-28 1978.

[3] Hoer, C. A., Calibrating two six-port reflectometers with only one impedance standard,

NBS Tech. Note 1004 , 399 (June 1978).

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

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

[5] Engen, G. F., The six-port reflectometer: An alternative network analyzer, IEEE Trans. MTT 25, 1075-1079 (Dec. 1977).

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

measurements, IEEE Trans. MTT 25 , 1080-1083 (Dec. 1977).

[7] Daniel, C. and Wood, F. S., Fitting Equations to Data (Wiley, p. 276 1971).

14
APPENDIX A

This appendix outlines a solution of eq (15) for the eight AG, and also gives

useful some approximations to simplify the solution.

If the first standard termination has a reflection coefficient Ti- 0, then eqs (12)

and (17) become

9il

~- 1'

8G.' 3G." "

which can be used to simplify the partial derivative in eq (16) to

9f.„ 6.„ 3g,„

303' 3G3'

'Uz 'u '93£

363" ^^3

''u ^9i£

3G.- 3G.'

'U, '^H

3G." 3G."

To simplify writing, define

K 3g

u 3G.

3g

U = 2r^' + 2G.'lrJ'

LU^ ;^=-2V-2G."|rJ

3G."

i = 3. ..6 (Al)

i = 3. ..6, (A2)

i = 4,5,6

£ = 2,3,4.

(A3)

i = 3... 6,

£ = 1...4.

(A4)

where eq (12) was used to obtain the partials. Substituting eqs (A3) and (A4) into eq (15)

gives

f •„ + -S-o K,„ AG-' + 6.„ L-„ AG-" = K. AG.' + L.„ AG." OU U 3£ 3 U 3£ 3 U 1 U 1

i = 4,5,6;

£ = 2,3,4.

(A5)

15
One way of solving this equation for each AG is to first eliminate aG^- ' and AG^" for

i = 4,5,6, and then solve for AG3' and AG3". The two equations obtained from eq {A5)

for measurements with termination numbers two and three can be written as one matrix

equation;

foi2

^oi3

or simply

6-2 K32 6^2 "-SZ

^i3 '^33 i3 43

F. + M. AG3 = N . AG.

AG3'

AG3"

=

K.2 1-2

^•3 ^3

AG."

II AG,

i = 4,5,6. (A6)

i = 4,5,6 (A7)

where the matrix symbols in (A7) are defined by the corresponding matrices in eq (A6).

Solving eq (A7) for AG^

,

AG. = N"-^ F. + N."-^ M. AG., i = 4,5,6. (A8)

Equations (A5) for the measurement with termination number four can also be written as a matrix equation;

f . + S. AG, = T. AG. oi4 —I 3 —I 1

where S^ and J-j are l-by-2 matrices defined by

S = ffi K -S L 1

-i [ i4 34 i4 34

J

T = Tk L 1

-i [i4 i4j.

Substituting eq (A8) into eq (A9) yields

f + S AG = T (N -1 F + N-^ M AG ) oi4 -i 3 -i ~i i ~i ~i 3

i = 4,5,6 (A9)

(AlO)

(All)

i = 4,5,6 (A12)

or

f + (S -T N"'^ M ) AG = T N--^ F oi4 i i ~i ~i 3 i~i i

i = 4,5,6. (A13)

The three equations represented by eq (A13) can be combined into one matrix equation;

16
^o44

'^4 -1

- T N M -4 ~4 ~4 AG

3

-1 _

T N F -4 ~4 4

fo54 + s-5

-1

- T N M -5 -^ ~5

= -1 ^

T N F -5 ~5 5

f

o64

s-6

-1

- T N M -6 ~6 ~6

T N " T -6~6 6

or Simply

? + VAG3 = w".

(A14)

(A15)

The solution for AG3 is _ T -1 T, ^^ AG ={ V V) V (W-U) (A16)

With AG3 known, the remaining aG^ for i = 4,5,6 can be obtained from eq (A8),

17
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

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1. PUBLICATION OR REPORT NO.

NBS TN-1012

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Calibrating a Six-Port R,ef lectometer \yrith. Four

Impedance Standards

5. Publication Date March 1979

6. Performing Organization Code 724.01

7. AUTHOR(S)

Cletus A. Hoer

8. Performing Organ. Report No.

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NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

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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 report is a theoretical study showing how four terminations of known

impedance can be used to calibrate a six-port reflectometer for measuring other

terminations of unknown impedance. The equations for obtaining the calibration

constants are exact but nonlinear, requiring an iterative solution. Several

ways are described for using these constants to calculate the impedance of

terminations being measured with the six-port reflectometer.

ii

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Calibration; impedance; reflection coefficient; reflectometer; six-port.

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