REFERENCE

NBS

PUBLICATIONS

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O*

NAT'L INSTrOF^STAND &TECH NBS TECHNICAL NOTE 1083

M11D5 T77aHb

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

hoc

100

.U5753

No. 1083

1985
he National Bureau of Standards was established by an act of Congress on 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, the Institute for Computer Sciences and Technology, and the Center for Materials

Science.

The National Measurement Laboratory

Provides the national systern of physical and chemical 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, in- dustry, and commerce; provides advisory and research services to other Government agencies; conducts physical and chemical research; develops,

produces, and distributes Standard Reference 'Materials; and provides

calibration services. The Laboratory consists of the following centers:

• Basic Standards^

• Radiation Research

• Chemical Physics

• Analytical Chemistry

The National Engineering Laboratory

Provides technology and technical services to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains comp)etence 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 ultimate user. The

Laboratory consists of the following centers:

Applied Mathematics

Electronics and Electrical EngineeringManufacturing Engineering

Building Technology

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Chemical EngineeringThe Institute for Computer Sciences 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 guidelines, and managing Federal participation in ADP

voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and provides the technical foundation for computer-related policies of the Federal Government. The Institute consists of the following centers:

Programming Science and

Technology

Computer Systems

Engineering

The Center for Materials Science

Conducts research and provides measurements, data, standards, reference

materials, quantitative understanding and other technical information fundamental to the processing, structure, properties and performance of materials; addresses the scientific basis for new advanced materials technologies; plans

research around cross-country scientific themes such as nondestructive

evaluation and phase diagram development; oversees Bureau-wide technical programs in nuclear reactor radiation research and nondestructive evaluation; and broadly disseminates generic technical information resulting from

its programs. The Center consists of the following Divisions:

• Inorganic Materials

• Fracture and Deformation'

• Polymers

• Metallurgy

• Reactor Radiation

'Hcadquariers and Laboratories at Gaithcrsburg, MD, unless otherwise noted; mailing address

Cjaithersburg, MD 20899. -Some divisions within the center arc located at Boulder, CO 80303. 'Located at Boulder, CO, with some elements at Gaithcrsburg, MD.
RATIONAL BUREAU OF STANDARDS

LIEKARY A Radio-Frequency Power

Delivery System: Procedures for Error

Analysis and Self-Calibration

Motohisa Kanda

R. David Orr

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

\—'JZ^—7

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued August 1985
National Bureau of Standards Technical Note 1083

Natl. Bur. Stand. (U.S.), Tech Note 1083, 24 pages (Aug. 1985)

CODEN.NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

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

Page

1

.

Introduction 1

2. System S-Parameters 1

3. Expressions for Net Delivered Power 3

4. A Self-Calibrating System 7

5. Summary of Self-Calibration Procedure 9

6. Estimated Uncertainty in Computed Net Power 11

7. Conclusion 15

8. Acknowledgment 15

9. References 16

m

A RADIO-FREQUENCY POWER DELIVERY SYSTEM:

PROCEDURES FOR ERROR ANALYSIS AND SELF-CALIBRATION

Motohisa Kanda and R. David Orr

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

An expression is developed for net power delivered to a load in

terms of the indicated forward and reflected power and the system

S-parameters and reflection coefficients. The dual directional

coupler is treated as nonideal with power reflections assumed between all ports. The system itself is used to evaluate the major

S-parameter terms in net power computation, and uncertainty in the

computed power is derived from origins in the power meter readings

and incompletely known S-parameters.

Key words: errors; error analysis; radio-frequency power; scattering coefficients; S-parameters.

1. Introduction

Standard-gain horn antennas are employed to establish known electromagnetic (EM) fields in the National Bureau of Standards anechoic chamber.

Part of the uncertainty in our knowledge of these fields arises from the un- certainty in the net power delivered to the horn. In turn, this uncertainty

reflects our lack of knowledge of the amplitudes and phases of the various

reflection and transmission coefficients in the power delivery system, as well

as the uncertainty in measurements of the power incident upon and reflected

from the horn. To determine how the net power to an antenna depends on system

components, we now derive an expression for net power in terms of power meter

readings and the system S (scattering)-parameters.

2. System S-Parameters

We derive our net power expression from a set of simultaneous S-parameter

equations which describe voltage waves at an n-port microwave junction. In

figure 1, the voltage wave a,- is incident on port j when all other ports have

matched terminations. The voltage wave b^ emerging into the termination on port i is obtained from the incident wave a,- at port j by the relation defining the S-parameter S^ ^ [1]:
bi = S^j aj. (1)

The symbols b^ and a,- are complex and represent the magnitude and phase of the

waves at ports i and j; S^,-, therefore, is also complex. (For convenience in

writing the power equations that appear later, we let a and b represent voltage, i.e. E field, waves normalized to /Z^, the square root of the characteristic impedance of the system.) When i = j, the S-parameter is a reflection

coefficient; when i ^ j, the S-parameter is a transmission coefficient.

If all the ports are terminated in unmatched loads, there will be an incident a A at every j^ port contributing to the voltage wave b^- emerging from

the i^^ port:

b. = S.^a^ + 5-232 + ... S..,. + ... S.^a^. (2)

Because a similar equation can be written for the voltage wave emerging from

every i port, we have a set of n simultaneous equations in which all emerging waves are related to all incident waves through the S-parameters S^- ,• of

the junction.

In our system for establishing standard radio-frequency fields, we com- pute net power delivered to the transmitting antenna from measurements of incident and reflected power obtained with a dual directional coupler. Our

power delivery and measurement system can, therefore, be represented as shown

in figure 2 where the port terminations and numbering are:

1. Power meter to monitor forward power

2. Power meter to monitor power reflected from antenna

3. Rf generator

4. Antenna (load).

We now confine our attention to that part of the system in figure 2 within the dashed rectangle and, except for their action as "reflectors"

(imperfect matches), we exclude the four terminations. This four-port system,

including such components as dual directional coupler, switches, and cables is

described by four simultaneous S-parameter equations:
^2 " ^21^1 "• ^22^2 •" ^23^3 "^ ^24^4

^3 ~ ^31^1 "• ^32^2 "^ ^33^3 *" ^34^4 (^)

b i| — Sl).]^3]^ + ^42^2 "^ ^43^3 ''' Sm^S^

Again, each b^- represents the magnitude and phase of a voltage wave emerging

from the i^^ port, and a,- the magnitude and phase of a voltage wave incident

at the j^" port. Because the system (everything within the dashed-line rectangle in figure 2) described by these equations is reciprocal [1], we use the

reciprocity condition

Sij = Sj^ (4)

to simplify the simultaneous equations. (The reciprocity condition is S^ -Z- =

S^,-Z-: but we assume Z- = Z^.) Solving these equations for b-/b- is further

simplified by relating each b^ to a^- through the voltage reflection coefficient r^ of the termination on the i'''^ port:

ai = biTi; a2 = b2r2; an, = bi^v,,, (5)

3. Expressions for Net Delivered Power

An expression for net power absorbed by the load can be obtained from

inspection of figure 2 if we assume an ideal dual directional coupler. By

that, we mean a coupler in which there is complete isolation between ports 1 and 4, 2 and 3, and 1 and 2; i.e., S^ = S23 = 5^2 = 0« Thus, the power

emerging from port 1 is derived only from the generator power entering at port

3; and the power emerging from port 2 is derived only from power entering port

4 in the voltage wave reflected from the load on port 4. The power absorbed

by a load on port 4 is the difference between incident and reflected power at

the load: P^^^^ = P^-^^ - Ppgf ] • Then, where b^ and a^ are the emerging and

incident waves at port 4 and Yi^ is the voltage reflection coefficient of the

port 4 load.
Keeping in mind that the power transmission coefficient |S..|2 represents an attenuation and that P^- is the reading of a power meter whose sensor has a reflection coefficient r., we inspect figure 2 and write an expression for

the power incident on the port 4 load in terms of P^. The power |bj^

emerging from port 1 is P^ divided by 1 - |r^|2 (i.e., P^ = |bJ2

X (1 - |rj^)); |b^|2 divided by IS^sl^ is the power entering the coupler at

port 3; and this power multiplied by |S3^|^ is the power emerging from port

4. Using similar reasoning to relate Ib2|^ to the power reflected from the

load, we have

p s 2 p

1 nc SisPd - riP) refl izJ'd - iTzP)*

(7)

The power absorbed by the load antenna in the idealized system is now

1 S |2

net 13

31+

13

^ (1

2 rj^) S2^l' (1 - Ir^P)

(8)

1 - h\' S |2 ^21+ I 1 - r 1^ ^2l

This equation tells us which parameters of this ideal system we must know in

order to compute P^^^ from power measurements P^ and P2

We now derive an expression for P^^g^ for a real (i.e., nonideal) fourport system in which none of the S^ ^ =0.

P^-. = P^n^ - P_^T = lbJ2 - |aJ2; a^ = b.r

net refl

1+1 4 (9)

where ^^ is the reflection coefficient of the load. Because our power mea- surements are made at ports 1 and 2, we want to express b^ in terms of b-]^, and

a^ in terms of b2. We do this by solving eq (3) for b^/b-]^ and b4/b2:

b^ b^ a^

b7= 9(s,r); = h(s,r) (10)

where g(S,r) and h(S,r) are functions of the system S-parameters and the re- flection coefficients of terminations on ports 1, 2, and 4. Now

net

= |bi|2|g|2 - |b2|2|r,P|h|2. (11)
The quantities b^ and 62 can be expressed in terms of the power meter readings

at ports 1 and 2 by using the final form of eq (6):

Pi = Ibil^d - |ri|2); p^ = |b2|2(l . \T^\2) (12)

and so , ,„ Pi P2 IM'

P . =

-. irrlgl^-^ i—i-y IM^ • (13)

net 1 - |rJ2 iyi

X - Ir^p ' '

To obtain g and h, we modify eq (3) by using the reciprocity relation (eq

(4)) and eq (5) (a^ = b^- r^ ) . The voltage wave b3 is not involved in our com- putations for P^Qi and so we use only the first two and the fourth of eq (3).

These are now:

bi = SuFibi + Si2r2b2 + Sisaa + S^^r^b^

b2 = Si2ribi + S22r2b2 + 52393 + S24r4b4 ^^^

b^ = SmTibi + S24r2b2 + S34a3 + S4.,r4b4.

A convenient form for these equations can be obtained by grouping the two common b^- terms within each equation, then solving the first for a^ and substituting this for a^ in the second and third equations (this treatment is based

on [2]) . We then have

- Bbi = - Ab2 + Cb^

(15)

- Ebi = Fb2 - Dbi^

where

A = $13 (1 - S22r2) + 512523^2

B

= S23 (1 - SuTi) + Si2Si3ri

C

= (Si3S24 - SmS23)^ (16)

D

= S]^3 (1 - St^i^r^) + S]^^S3^^^

E = $34 (1 - SiiFi) + Si3Smri

^ = (513^24 - Si2S34)r2.

Equation (15) can now be solved for the ratios g(S,r) = b^/b-j^ and h(S,r) =

^/^/^2* 9^v''f^9 ^^^ following expressions for g and h in eq (13) for P^g^:
1 +

Si3 1 - Sm^Yi^ SmS^L^Ti^

^la^-'- " S44r4)

1 + [$23(1 - S^ri) + Si2Si3ri](Si3S2H - S^^^S^)^!

$13(1 - S22r2) + Si2S23r2j[S3^(l " S^^T^) + Si^Si^Ti\

^ " [Siad - S22r2) + Si2S23r2j[Si3(l - Shh^) + $m$3^r4j

^12523^^2

1 1 - $22^2 ^13(1 - $22^2)

h($,r) = -^— X = X =—^

^13^2tf

[$23(1 - SiiTi) + Si2Sl3ri][Sx3$2i+ - Si2S3H]r2

^ ^ lSx3(l - $22r2) + $i2S23r2j[S3^(l ' S^^T^) + S^SmrJ

^ [S23U - Surj + Si2Si3riJ[Si3(i - s^^t^) + s^^s^^v^]

[$34(1 - SiiTi) + $i3$ii+ri J[$i3$2i+- Sii+S23jr4

We see that the initial factors in g and h are $34/$i3 and l/$2tt> respectively. These factors are present in eq (8) for the P^-^ in an idealized

system. Thus, the rest of g and h corrects eq (8) for a system which is not

ideal. (Note that the second term in h contains l/Ti^ which removes the con- siderable dependence on Ti^ that P^g^ appears to have in eq (13)), $ection 6

further discusses the role of g and h in determining the uncertainty in P^^..

due to approximating eq (13) with eq (8).

If the S-parameters , reflection coefficients, and their uncertainites are

known for our four-port system, we can use eq (13) to compute P^g^ and its un- certainty from the power measurements P^^ and P2 and their uncertainties. The

amplitudes of some of the S^ • (see page 11) are so much less than unity that

Pj^g^ should be a weak function of their phases. To confirm this we computed g and h using these S^ • and r. magnitudes. We allowed for a complete lack of phase

information for these parameters by replacing each plus and minus sign in

eqs (17) and (18) with a ± sign. The uncertainty in P^g^ introduced by this phase
ambiguity is only +1.2 percent when |r^| = 0.05. For this reason, an order-ofmagnitude knowledge of some of the S^ amplitudes is sufficient for evaluating g and

h. Uncertainty components in P^^^ are discussed further in section 6.

4. A Self-Calibrating System

The functions g(S,r) and h(S,r) were written in the forms given in eqs (17) and

(18) to show which S-parameters have the most direct influence on P^ef These

parameters (S13, S24, and S3^) predominate in g and h in the forms S34/S13 and

1/S24, and in P^^^ as squares of magnitudes |S3^/Si3|2 and II/S24P which do not

involve the parameter phases. One of our intentions in developing an automated

system for establishing standard electromagnetic fields is to make the system

largely self-calibrating in the sense that the system itself is used to evaluate

|S34/Si3l2 and |1/S2^|^. This section presents the procedures for such a "self

analysi s".

In figure 3a, we have replaced the load on port 4 with a short (r^ = -1).

Using eq (12), the ratio of power measurements P2 and P-|^ is

(19)

t-1

2 1 - \^z'

1 - h\'

We solve eq (15) for b2/b^ and obtain

S2^S3^

'13

1 - S 11^ 1 jT~^~s:::j:j[T~^~s^^T^

1 -

1 +

s s •^m-^23

"~s^13-^24 s

s s r

^13 I-'- ' ^22T2J

1 +

1 + SmS3i^r4

S13II " Si+4r4j

(20)

[$23(1 - s^^rj + s^^Sis^iJlSuU - \,^,] + s,4S3,rJ

[Sg^^l - S^^Tj^J + S^gS^i^Tj^ J[Sj^^S23- S^3S2^J^^

'-^m^23 " ^13 24-''- 12 31+" ^13 24''^2^4

LSi3[l - s^.rj + s^^S3,rJts,3[i - S22r2] + 5,2^23^2]
in which \Vi^\ = 1. Writing this equation in this form (with jr^l = 1) shows

that Ibo/bip depends primarily on the first term containing the squares of

magnitudes of $24, S34, and S-^^- ^^^ ^ typical dual directional coupler, the

Sh ^- magnitudes are such that the numerators and denominators of the last four

terms can be written as unity with a small number added or subtracted. Because

we do not know the phases of the S-parameters, the value of each S^ .: may be

anywhere from + |S-jn| to - |S^,-| (similarly for the rs). Therefore, the last

four terms in eq (20) can each be put in the form |1 ± (a small number)]^ and

then the expression for |b2/b]^|'^ reduced to

S2hS3i+

'13

(1 ± A) (21)

where A is the uncertainty in |b2/b-]^| . The order of A can be obtained by

substituting nominal values for the S^ ^ and rs into eq (20^ we now have

S S ^21+•^31+

'13

P2 1 - \h\

P^(l + A) 1 - \T^\

From eqs (19,21)

(22)

where A and the uncertainty in P2/P1 "^^ke up the total uncertainty in

|S2i+S34/S 13I ^. The reflection coefficients r^^ and T2 are given in the manu- facturer's specifications for the power meter sensors on ports 1 and 2.

(Because our uncertainty estimates are derived from phase ambiguities, the

Ti and T2 magnitudes do not contribute to the total uncertainty in eq (22)).

For the next step in evaluating the predominant S-parameter terms in

^net' ^^^ system is in the configuration shown in figure 3b where the port 4

short has been replaced with the port 2 power meter, and port 2 is terminated

with a matched load, {T2 = 0.05). The ratio of the two power measurements is

(analogous to eq (19)

)

^ 1 - \h\

1 - |rj (23)

where we solve eq (15) (or take the reciprocal of eq (17)) to obtain
2 Sl3 2

X hn

s r hlh

1 +

s s r

Si3(l - Sn^rJ

1 +

s s r

^3^(1 - hlh)

(24)

1 - ^^^13^24 " ^12^34-'*'^ 13 24 " ^m^23>'^2^i

[S]^3(l - S22r2) + S^2S23^2il.^l3(^ " ^^l+^^) "*" ^li+^3»+^4i

^^^13^24' ^12^3'+''l-^23(--'- " ^ll^lJ "*" ^12^13^lJ^2

1 +

[Sj^3(l - S22r2) + Sl2^23^2i l-^34(-'- ' ^11^1' "*" ^U^li+^lJ

in which every r2 is then set equal to 0.05. Again, because many of the S^

magnitudes are small, and the phases not known, we reduce eq (24) to the form

bi

b^

2 Sl3

S34 (1 ± A) (25)

where A includes all that we do not know about the S^,-. From eqs (23,25) we ij now have

'13

'34

1 - |rJ2

P^d ± A) 1 - Ir^

(26)

which is the predominant S-parameter dependence in g(S,r) and, therefore, one

of the S-parameter multipliers we need for computing P^g^ from eq (13), Again,

A and the uncertainty in the ratio P1/P4 give the uncertainty in |Si3/S34p.

From eqs (22) and (26) we can now obtain a value for |l/S2it|^ which is the

predominant S-parameter dependence in h(S,r).

5. Summary of Self-Calibration Procedure

In this section we summarize the computation of system S-parameters from

readings of forward and reflected power at the side arms of a dual directional

coupler. System self-calibration proceeds as follows.

1. We must know which parameter groups are to be evaluated. For an ideal system, eq (8) shows the S-parameters that must be known to compute
the net power to the load antenna in figure 2. For a nonideal system, eqs

(13), (17), and (18) show how all the S-parameters of the system are involved in the net power measurement, and also that the S-parameters with

major influence in this measurement are the same as those identified in

eq (8).

2. Values for the S-parameter multipliers in eq (8) are obtained from

measurements of forward and reflected power with the system in the two

configurations shown in figure 3. For the system as in figure 3a, values

for power measurements P-|^ and P2 and their uncertainties are entered into

eq (22) to compute the quantity |S2^S3^/S 13 p. The uncertainty A is com- puted from the last four terms in eq (20). Then, for the system as in

figure 3b, power measurements P^ and P^ and their uncertainties are en- tered into eq (26) to compute the quantity |Si3/S34p, for which the un- certainty A is computed from the last three terms in eq (24). We can now compute 1 1/821+1^ and its uncertainty, and thus obtain values for the

leading terms in |gp and |h|2 of eq (13), where g and h are given by eqs

(17) and (18).

3. The computed values for iS34/Si3|^ and |1/S24|^ are substituted into

|g|^ and \h\^, respectively, in eq (13) for the net power to the load an- tenna in figure 2. The remaining terms in |gp and |h|2 cannot be com- pletely evaluated because the phases of the S-parameters and reflection

coefficients are not known. Therefore, each S and r is treated as a scalar quantity having the range of values +|S| and +|r|, and the terms

multiplying |S34/Si3p and |l/S2i+|^ in |g|^ and |h|2 are reduced to the

form 1+A in the same manner as in eqs (20,21) and (24,25).

4. The total uncertainty in the net power delivered to the port 4 load

has three parts which are entered into eq (13): 1) the uncertainty in the

S-parameter groups |S3^/Sl3|2 and II/S21+P in |gp and |hp, respectively;

2) the uncertainty, 1+A, which represents our lack of knowledge of the Sparameters and reflection coefficients in the remaining terms in \q\^ and

|h|^; and 3) the uncertainty in the power readings P-^ and P2.

10
6. Estimated Uncertainty in Computed Net Power

The uncertainty terms that we have represented by the symbol A are com- puted using nominal values for the magnitudes of S-parameters and reflection

coefficients. These values are obtained from the manufacturer's specifications for the dual directional coupler and for the power sensors. Conservative values for percent uncertainty in the power meter readings are also obtained from specifications for the meter and power sensors. We have computed

the estimated uncertainties using these magnitudes

Sill = IS22I = |S^^| = 0.05

Sisi = |S2^I = 0.1

SmI = IS23I = 10'^

S12I = 10"^

S3^| = 0.95

Til = |r2l = 0.05

r^l = 0.05 for a standard gain horn antenna (see page 14)

The power meter reading contains ±1 percent instrumentation uncertainty

arising in the metering circuits, +1.5 percent mismatch uncertainty in the

sensor calibration factor, and ±2 percent uncertainty in the power linearity

of the sensor. We therefore set a ±4.5 percent uncertainty on each power

meter reading. Using eq (13) for net power delivered to a load antenna on port 4, the system self-calibration procedure described in this report then

gives an estimated worst-case uncertainty of +16 percent (+ 0.64 dB, - 0.75

dB) in the computed value of the delivered power Ppg-(-.

The quoted uncertainties in the power readings P-^ and P2 are for a dual

channel power meter with two sensors. It is possible to reduce this uncer- tainty by using the meter in the single channel mode (or using a single channel meter) with a single power sensor. The instrumentation uncertainty in the

metering circuits is then quoted as +0.5 percent (instead of +1.0 percent).

In addition, the power sensor mismatch uncertainty (a function of frequency)

will cancel in the ratios P]^/P2 cind ^i/P^ because both readings in each ratio

are at the same frequency. However, the mismatch uncertainty remains in the

power readings P-|^ and P2 in eq (13). A single channel power meter with one sensor reduces the estimated worst-case uncertainty in P^^^ to +11.4 percent

(+ 0.47 dB, -0.53 dB), but at the expense of increased system complexity in

11
added circuits to switch the single power sensor between coupler ports 1, 2,

and 4.

Although the preceding computations give an estimate of the maximum un- certainty inherent in the self-calibration method, we emphasize that the Sparameter magnitudes on page 11 describe only the dual directional coupler for

which they are given. In practice, the procedures set forth in this report measure parameters for the entire system within the dashed rectangle in figure

2. For example, the S-parameter S21+ will be the transmission coefficient not

just for the path between ports 2 and 4 of the coupler, but for the path between a measurement plane on the rectangle outside port 2 and a measurement

plane on the rectangle outside port 4. Physically, these measurement planes

are at the power sensor for meter ?2 and at the input connector on the antenna

terminating port 4.

We have also used the S and r magnitudes on page 11 to determine the un- certainty component which appears in Pj^q^ as a consequence of considering the

system to be nonideal; that is, the uncertainty incurred in P^^^^ through using

eq (13) instead of eq (8) (the uncertainties in Pi,P2. |S3h/Si3|2, and II/S211P

being the same in each equation). To compare uncertainties in the computed

P^g^ for ideal and nonideal systems, the expressions |gp and \h\^ are inserted into eq (13) in the forms

2

2 -

'34

'13 [1 ± Ag); |h|2 =

1>2t

(1 + Ah) (27)

where the |l/r4J2 in |h|^ is omitted because it cancels the [T^l^ in eq (13).

For a nonideal system, P^^^ is then

net

'3»+

13 1 - h\'

(1 ± Ag) - '2H

^-^^-^(ItAh). (28)

Equation (8) for an ideal system is

net

'34

'13 1 - |rj '2^ 1 - Ir^P

(8)

12
Equations (28) and (8) now closely resemble one another except for the terms

(1 + Ag) and (1 + Ah) in eq (28). Therefore, neglecting the quantities Ag and

Ah is tantamount to assuming an ideal system and identifies Ag and Ah as un- certainties in P^g^ which are overlooked in this assumption.

To obtain the uncertainty in P^^^ in eq (28) due to the uncertainty Ag in

the forward power term and the uncertainty Ah in the reflected power term, we write P-^ and P2 in terms of Pq defined as the incident minus reflected power at port

3. Referring to figure 2, an approximation to the power Pi/(1 - Ir^p)

emerging from port 1 in a nonideal system is

P,

1 - 1^1

T= P0IS13I' ' PolS3^P I^P ISi.P (29)

^ Even for \Ti^\^ = 1, the second term is ~ 10 times smaller than the first

term, so we let

Pi

2 = PolSi3l' (30)

1 - |rj

The approximate power emerging from port 2 in a nonideal system is

P2

1 - |rJ = P0IS3.I' Irj' IS.J' + P0IS23P (31)

where, for jri+j^ = 0.05, the second term is about 25 times smaller than the

first term. Because the expression in eq (30) is then about 400 times larger

than that in eq (31), we let

1-T7FJT= PolS3^IMMMS2J2 (32)

with negligible effect on the computed uncertainty in Ppg^.

Equations (30) and (32) are good approximations for a nonideal system,

and are exact for an ideal system. Substituting eqs (30) and (32) into

eq (28), we have for a nonideal system

Ppet = l^3^P Pod t Ag) - IS3J2 |r,P Po(l ± Ah)

= |S3^I' Po[(l ± Ag) - 1^1^ (1 ± Ah)]

(33)

= IS3.P Po[(l - |rJ2) t (Ag + |rJ2 ^h)]

13
where the absolute uncertainties Ag and Ah are grouped to form

APnet = ± |S3^I^ Pq (Ag + \T^\^ Ah) (34)

which is the contribution to the total P^^^ uncertainty neglected in the

"ideal system" assumption. As a percent of Ppg-^, AP^g^ becomes

Ag + |r^|2 Ah

^^et(^°^

= ^ l-|rJ2 xlQO. (35)

Now putting eqs (30) and (32) into eq (8), we have for an ideal system

Pnet = |S3^P Po - lS3^P 1^^|2 Pq = |S3^I^ Pq (1 " l^^|2). (36)

Comparing eq (36) with the first term of eq (33) shows that the mathematical

descriptions of ideal and nonideal systems give the same expression for

^net* However, eq (33) also gives an additional uncertainty (beyond that common to P^^^ in eqs (8) and (13)) involving the terms Ag and Ah which characterize the nonideal nature of the power delivery system.

Because Ag, Ah, and the relative size of P^^ and P2 depend on r^, we com- puted AP^g^(%) for three values of \T^\, Numerical values for Ag and Ah were

obtained by substituting the S and r magnitudes into eqs (17) and (18) and

consolidating the second, third, and fourth terms in each of these equations

into one term of the form (1 ± A). For the following values for iTi+j, eq (35)

yields

. \T^\ = 0.05 (|r4|2 = 0.0025), AP^g^ = ± 1.2%

• Ir^l = 0.1 (|r4|2 = 0.01), AP^^^ = ± 2.1% (37)

. \T^\ = 0.224 (Ir^l^ = 0.05), AP^g^ = + 5.7%

(The maximum VSWR obtained by Slayton [3] for a pyramidal horn was 1.25, for

which I r| = 0.11. Our own measurements of |rp for a horn and an open-ended

waveguide were about 0.001 (|r| = 0.03) and 0.01 (|r| = 0.1), respectively.

These values suggested those we chose for | r^ | .

)

Referring to eq (28), the total uncertainty in P^^ comprises AP^^^ and

the uncertainties in the S-parameter multipliers and in P-^ and P2. For

14
smaller values of r^, AP^g^ is a smaller part of this uncertainty. In a power

delivery system using a dual channel power meter and having Ir^l = 0,05, AP^g^

= ± 1.2% out of a total P^g^. uncertainty of + 16%. If jr^l = 0.224, then

AP^g^ = + 5.7% out of a total uncertainty of ± 23%. Thus when r^ is small,

the assumption of an ideal system (i.e., the use of eq (8) instead of eq (13)

to determine P^et^ incurs only a small discrepancy in the estimated uncer- tainty i n Ppef

We are currently testing the system self-cali brat ion procedure by com- puting P^Qi with eq (13) and comparing that value with a direct power measurement at port 4. Results of this work will be given in a later report.

7. Conclusion

We derived an equation for computing the net power to a load antenna

using a power delivery system for which all of the S-parameters are assumed to

be nonzero. An S-parameter description of the system yields equations by

which the system can be self-calibrated using known terminations on port 4.

When the system uses a dual channel power meter and is calibrated by the procedure described, the maximum uncertainty in P^g^ obtained from eq (28) is + 16% (for |^^| = 0.05). This uncertainty can be reduced to 11.4 percent by

using a single channel power meter. We have shown that a P^^g^ computation

based on the "ideal system" assumption may be sufficient if the reflection

coefficient of the load on port 4 is small.

8. Acknowledgment

It is our pleasure to acknowledge the cheerful and expert assistance of

Wilbur Anson who wrote and ran a computer program which allowed us to observe

the behavior of g(S,r) and h(S,r) versus many combinations of the S^ • and r^

.

15
9. References

[1] Kerns, D. M.; Beatty, R.W, Basic theory of waveguide junctions and introductory microwave network analysis. New York: Pergamon Press; 1967.

150 p.

[2] Beatty, R. W.; Macpherson, A.C. Mismatch errors in microwave power mea- surements. Proc. I.R.E. 41(9): 1112-1119; 1953 September.

[3] Slayton, W. T. Design and calibration of microwave antenna gain standards. U.S. Naval Research Laboratory, Washington, D.C. Report 4433;

1954 November. 37 p.

16
Figure 1. An n-port microwave junction

17
System Boundary

f

Power Meters'

r

rf generator

©

i_n o

Dual Directional Coupler

Antenna

Figure 2. System for measuring rf power delivered to an antenna

18
Power Meters

a)

Short Circuit

D D

rf generator

b)

Matched Load

rf generator

Figure 3. Configurations for measuring system s-parameters.

19
NBS-n4A IREV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1, PUBLICATION OR

REPORT NO. NBS TN-1083

2. Performing Organ. Report No. 3. Publication Date

August 1985

4. TITLE AND SUBTITLE

A Radio-Frequency Power Delivery System: Procedures for Error Analysis and SelfCalibration

5. AUTHOR(S)

Motohisa Kanda and R. David Orr

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

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

13^ Document describes a computer program; SF-185, FlPS Software Summary, is attached.

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

bibliography or literature survey, mention it here) An expression is developed for net power delivered to a load in terms of the

indicated forward and reflected power and the system S-parameters and reflection

coefficients. The dual directional coupler is treated as nonideal with power

reflections assumed between all ports. The system itself is used to evaluate the

major S-parameter terms in net power computation, and uncertainty in the computed

power is derived from origins in the power meter readings and incompletely known

S-parameters.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons}

errors; error analysis; radio-frequency power; scattering coefficients, S-parameters

13. AVAILABILITY

|V] Unlimited

[~n Fo'' Official Distribution. Do Not Release to NTIS

|X I Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. Q^ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

24

15. Price

USCOMM-DC 6043-P80
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