1E13ETRANSACTIONSON MfCROWAVETIilZORYAND TECHNIQUm, VOl,. MTT-19, NO. i 1, NOV13v15ER197i 843

A Precision Compact Rotary Vane Attenuator

TOM Y. OTOSHI, iW3MBER, IEEE, AND CHARLES T. STELZRIED, MEMBER, IEEE

Absfracf—The accurate attenuation range of many precision

rotary vane attenuators is fimited to about 40 dB because of a trans-mission error term that is not accounted for in the familiar COS2

@attenuation law. This paper presents a modified law that makes it

possible to extend the useful dynamic attenuation range. The samemodified law also makes it practical to reduce the length of the rotor

section and, therefore, to develop compact rotary vane attenuators

that are accurate over reduced dynamic attenuation ranges. Themodified law requires the additional calibrations of the incremental

attenuation and incremental phase change at the 90° vane anglesetting.

To verify the modified law, a precision compact WR 112 rotaryvane attenuator was fabricated and tested. The attenuator has a

totaf dynamic attenuation range of about 30 dB and a rotor section

length approximately one-third that of a conventional WR 112attenuator. Application of the modified law resulted in good agree-

ment between theoretical and measured incremental attenuations

over the total dynamic attenuation range.

1. INTRODUCTION

I

N RECENT years, a great deal of attention has

been focused on the development of high-precision

rotary vane attenuators (RVAS) for use as primary

and interlaboratory attenuation standards, An ideal

RVA possesses the desirable characteristics of a variable

attenuation standard. Among these characteristics are

broad-band performance, low residual loss, negligible

phase variations with changes in attenuation, and elec-

trical performance that can be predicted by a mathe-

matical law. I t is well known that, in the idealized case,

an RVA will obey the COS2 0 law. However, in practice,

there are several major types of RVA errors that cause

deviations from this mathematical law. These are as

follows :

1)

2)

3)

4)

5)

vane angle readout error, which includes errors

due to initial center vane misalignment, gear drive

imperfections, and dial scale inaccuracies;

error due to mutual misalignment of the stator

vanes;

transmission error, which is also referred to as the

error due to insufficient attenuation of the vane in

the rotor;

error due to internal reflections;

error due to external leakage from rotary joints.

lVIost of these error types have been investigated by a

number of authors [1 ]– [7 ]. Other sources of errors are

vane warpage, perturbations of the TEII mode by the

Manuscript received September 24, 1970; revised April 29, 1971.This paper prese~ts the results of on: ph?se of ~esearch earned outat the Jet Propulsion Laboratory, Cahforma Institute of Technology,and sponsored by NASA under Contract NAS 7-100.

The authors are with the Jet Propulsion Laboratory, CaliforniaInstitute of Technology, Pasadena, Calif.

center vane, and eccentricity of the center vane. These

latter error sources have not been fully analyzed but,

with the exception of vane warpage, are presently be-

lieved to produce minor types of errors.

At the Jet Propulsion Laboratory it has been rec-

ognized for several years that, for purposes of antenna

gain and noise temperature calibrations, it is desirable

to incorporate a precision RVA in the receiving system.

The RVA would enable power ratio measurements to be

made by RF substitution methods and therefore would

reduce the present requirements for amplifiers with a

high degree of linearity over a large dynamic range.

The “front end” of the deep space communication an-

tenna systems operating at 2.3 GHz utilizes WR 430

waveguide components that are assembled inside a

Cassegrainian cone housing [8]. Due to the need to keep

the waveguide losses to a minimum, the installation of a

conventional WR 430 RVA (4 or 5 ft in length) was not

considered practical. The requirements for a shorter unit

promoted the development of the compact RVA.

One method of achieving a compact RVA size is to

physically shorten the lengths of the transition and rotor

sections. Shortening the transition secticms should not

noticeably affect RVA performance if the transitions

are well matched. A shortening of the rotor section, how-

ever, generally results in 1) a corresponding reduction in

the useful dynamic attenuation range, 2) an increase in

phase-shift variation as a function of vane angle setting,

and 3) an increase in frequency sensitivity of attenua-

tion and phase-shift characteristics at the higher vane

angle settings.

Such a compact device, therefore, would not be useful

in a swept frequency microwave calibration system

where flat frequency responses of amplitude and phase

characteristics are important. However, a type of sys-

tem where a compact RVA can be useful k a deep space

communication antenna system which operates at spec-

ified fixed frequencies. In the calibrations of the operat-

ing noise temperatures of a low-noise antenna system

by the Y-factor method [9], the accurate dynamic

range of the attenuator does not have to exceed 20 dB

and the relative phase shift does not have to be con-

stant with attenuation setting or with frequency.

The most serious result of shortening thle rotor section

can be the increased effect of the transmission error

signal if it is not properly treated. Transmission error

occurs because the familiar COS28 law assumes complete

attenuation of the E-field component that is tangential

to the rotor resistance card. In this paper, a modified

law that corrects for the effect of the transmission error

signal is derived from basic considerations, The present

844 1SSSTRANSACTIONSON MICROWAVSTHEORYAND TSCPINIQIJE$,NOVSMSSR1971

treatment is similar to that of James [3], but differs in

that the transmission error signal effect is no longer as-

sumed to be of unknown magnitude or phase. Since the

modified law does not require the transmission error

signal to be zero, the attenuator is not required to be

physically long. Therefore, it becomes practical to de-

velop compact RVAS with short rotor sections. The

same modified law can also be used to extend the useful

dynamic attenuation range of high-precision conven-

tional RVAS.

Section II of this paper presents a derivation of the

modified law. Section I I I gives a description of a com-

pact RVA that was fabricated and tested to verify the

modified law. Experimental results and a discussion of

errors are given in Sections IV and V, respectively.

II. THEORY

An RVA consists of three sections of waveguide

mounted in tandem. Across the center of each wave-

guide section is a thin dielectric card that has been de-

posited with 10SSY resistive films. The two end sections,

referred to as stators, are usually rectangular-to-circular

waveguide transitions. The center section, referred to as

the rotor, is a circular waveguide that is free to rotate

between the two stator end sections.

The rectangular waveguide TE1O mode is launched

at the input stator to establish a vertical electric field

~. at the input to the rotor as shown in Fig. 1. This is

decomposed into electric field vectors ZO cos 8 and ~0

sin 6. These two components propagate through the

rotor section of length 1. Defining -y. and ~~ to be the

propagation constants for the electric field vectors nor-

mal and tangential, respectively, to the rotor card, the

vertical component of the electric field at the output of

the rotor can be expressed as

7?U(0) = I?o(e-@ cosz 0 + e–~’Z sin~ 0). (1)

Only the vertical component is considered because the

horizontal component will be attenuated by the output

stator card.

The electrical properties of an RVA that are generally

of interest are incremental attenuation and incremental

phase change. Incremental attenuation is defined as the

change in attenuation of the output wave that results

from rotating the center vane from the 0° setting to the

angle O [10]. Incremental phase change shall be sim-

ilarly defined in this article as the change in phase of the

output wave that results from rotating the center vane

from the 0° setting to angle 0. By definition, and from

(1), the expressions for incremental attenuation and

incremental phase change are

E.(O) ‘~d~ = -– 10 log10 —

EV(0)

IFig. 1. Rotary vane attenuator voltage vectors

at the input to the rotor.

——

where

and

a

( — e–”z sin @ sin2 etan–l

)(3)

cos2 0 + e–aZ cos /31 sin2 0

cY+j(3=yt-yn (4)

cliff erence between the attenuation con-

stants of the tangential and normal elec-

tric field components in the rotor, nepers

per unit length;

difference between the phase shift con-

stants of the tangential and normal elec-

tric field components in the rotor, radians

per unit length;

phase angles of EV(6) and E“(O), respec-

tively, radians.

It is now convenient to introduce two new parameters

that can be experimentally evaluated. These are

& = 1010g~~t?2a’

C$=-fil.

Substitutions into (2) and (3), respectively, result in

modified attenuation law

z4dB = — 10 ]Oglo [COS4 6 + 10–LdB/20

(5)

(6)

the

(7).— 10 Ioglo I COS20 + e–(~+i~)z sin2012 (2) .(2 cos@COS2Osin’0) + 10-~’B/’O sin’0]

OTOSl$lANO STELZRI~D: ROTARYVANE ATTENUATOR 845

and the incremental phase change relationship

[

10–LdB/20 sin @ sinz OA+ = tan–l

1(8)

COS26 + io–LdB/20 cos ~ sin2 6 d

Note that when O= 90° is substituted in (7) and (8),

A dB becomes .&j and A* becomes ~. Therefore, these

two parameters can be experimentally determined by

measuring the incremental attenuation and incremental

phase change when the vane angle is set at 90°. Since the

values of LdB and @ will generally vary with frequency,

the incremental attenuation and phase change relation-

ships given by (7) and (8) will also be functions of fre-

quency. Although (7) and (8) were derived in terms of

the wave at the rotor output, the same equations also

apply to the total attenuator since only incremental

values are of interest. It is assumed that the addition of

the output stator section does not introduce mismatch

or misalignment errors.

It is of interest to examine some special cases of the

modified attenuation law. Note that when LdB ap-

proaches infinity, (7) will reduce to the familiar unmod-

ified law. For cases of finite LdB, analysis of (7) reveals

that, when cos @ < 10–LdB/20, the incremental attenua-

tion will become a maximum at a vane angle setting less

than 90° and have a maximum value greater than Lm.

The following relationships apply for cos @ < 10–~dB/20

(~dB)rnax = ~dB 10=&n = LdB

(L–2dZcos4+l+ 10 logl(l

L sin’4 )(9)

where

L = 10~dB/10 (lo)

‘i

l–dzcosrp6’M = COS–l

L–2~~cos4+l”(11)

If cos @ z 10–LdB/20, the maximum incremental attenua-

tion will always be equal to Lm and will occur at 9 = 90°.

For most RVAS, there will be some deviations be-

tween the indicated and true vane angle due to bore-

sight and other readout errors. The true vane angle can

be expressed as

6’ = & + Cil + a2(eI) (12)

where

& indicated vane angle;

al boresight error (it is the angular misalignment

of the vane in the rotor with respect to the vanes

in the stators when&= 0°;

a7.(61) angle runout error calibrated relative to 61= 0°

setting (it is a function of Or and is due to gearing

errors, bearing runout, eccentricities, etc.).

The vane angle errors al and ~2(&) should be calibrated

to ensure that the RVA follows the law given by (7).

Fig. 2. Compact RVA shown with interchangeable transitions.

Fig, 3. Compact and conventional WR 112 rotary vane attenuators,

With proper mechanical design and use of high-pre-

cision components, the angular errors al and c2Z(01) can

often be made negligibly small.

III. DESCRIPTION OF A TEST

MODEL COMPACT RVA

The test model compact RVA may be seen in Fig. 2.

A pair of stepped transitions, shown installed on the

attenuator, can be interchanged with the other pair of

tapered transitions shown to the left anc[ right of the

attenuator. The attenuator was tested in both the

stepped and tapered transition configurations in order

to determine if any differences would result from use of

shortened transitions as well as a shortened rotor

section.

A comparison of the physical length of the compact

RVA and a conventional WR 112 RVA may be seen in

Fig. 3. (Both attenuators were fabricated by the Mea-

surement Specialties Laboratory of Van Nuys, Calif. )

The rotor length of the compact attenuator is approxi-

mately 3.3 in as compared to a rotor length of about

10.9 in for the conventional attenuator. When the

stepped transitions are installed on the compact attenu-

ator, the overall length is 8.1 in as compared to an over-

all length of 21.6 in for the conventional attenuator.

The compact RVA is fabricated to have a very precise

vane angle readout; the resolution of the readout of the

vane angle position is O.OOO1° and backlash appears to

be less than O.OO1°.

The rotor section of the compact RVA is WC 125

waveguide and contains a vane made from 0.003 -in-

thick mica sheet having a thin layer of metallized resis-

tive film. The resistive film, which has a resistivity of

846

,.

INPUT SIDETRANSITION

IEEE TRANSACTTONS ON MICROWAVE THEORY AND TECHNIQUES, NOVSMSER 1971

9 !

Fig. 4. Rotary vane attenuator represented as the cascade of three multiports. (Numbers refer to measurement portswhile arrows refer to polarization directions.)

90–0+5 Q/square, has been deposited on the mica sheet

in an inverted taper pattern so as to minimize reflec-

tions.

IV. EXPERIMENTAL RESULTS

The experimental evaluation of the compact RVA

consisted mainly of calibrating reflection coefficients,

boresight error, incremental attenuations, and incre-

mental phase change characteristics. The residual loss

of the compact RVA in either the tapered or stepped

transition configuration was measured to be 0.1 dB at

the test frequency of 8448 MHz.

A. Reflection Coejicient

To facilitate the study of internal mismatch errors,

measurements were made of the reflection coefficients

of the individual transitions and rotor secticm. Reflec-

tion coefficients were measured through the use of

WR 112 and WC 125 reflectometer systems tuned at

8448 MHz. The WC 125 reflectometer system was made

to respond only to a single polarization component by

causing the reflected wave of the orthogonal component

to be absorbed in a matched resistive card,

Table I gives a summary of voltage reflection coeffi-

cients measured at various ports of the transitions and

rotor section. As indicated in Fig. 4, an individual port

is identified by a number and is defined for an electric

field that is linearly polarized in the direction of the

arrow. The general symbol S~i denotes the complex

voltage reflection coefficient as seen looking into port i

when all other ports are terminated in nonreflecting

loads. Using a coordinate transformation method similar

to that given in [11], the rotor reflection coefficient

relationships, for the unprimed ports in terms of the

primed ports, are derived as

SS6 = S5,51 cos2 $ + Sb, ~, sin~ $ (13)

S77 = S77V Cosz O + S8t~t sinz 0. (14)

From (13) it can be seen that S~~, and Sb,B, can be deter-

mined by measuring the reflection coefficient at port 5

TABLE I

MEASURED 8448 MHz REFLECTION COEFFICIENTS OFCOMPACT RVA COMPONENTS.

Descriptionof Component

WR 112-t. -WC 125 taperedtransition m inputside of attenuator

WR 1 lZ. to. WC 125 taperedtransition .n outputside .af attematm

WR 112- to-WC 125 steppedtr.nsitmn .n inpntside of attwmator

WR 112. to. WC 125 steppedt=ans,ti.n .n outputside of attenuator

WC 125 rotor section

VO1tage Reflection C.effic lent

Syrnbc.la

‘11

533

544

SZ2

’99

‘%0, 10

%1

S33

544

S22

’99

Slo, 10

‘5,5<

%6$

s ,,, <

S8E8,

Ma@tude

O. 0063

0.0057

0.0102

0.0082

0.0077

0,0101

0.0335

0.0342

0.0063

0.0302

0.0307

0.0014

0.0004

0.0447

0.0045

0.0490

Phase,deg

19.7

-73.5

137.0

18. z

-29.0

156.0

135.1

.87.0

86.8

141.4

-88.5

-173.0

148.1

132.5

138.8

126.6

‘For c.mflgnr.tmn of measurement port t c.rrespmdmg t. S,,, refer mFlg 4.

when 6= O and 90°, respectively. In a similar manner,

S7, ~, and SS ~, can be measured at port 7. The experi-

mentally determined values are given in Table 1. At

other vane angle settings, the measured reflection co-

efficient magnitudes of S65 and S11 agreed with the

values predicted by (13) and (14) to within + 0.001.

The values in Table I were obtained from direct

measurement and are useful for analysis of internal

reflections. When only the overall RVA reflection coeffi-

cient characteristics are known, additional insight into

internal reflections can be obtained from use of a method

described by Helm et al. [12]. The reflection coefficient

characteristic curves of the overall compact RVA and a

conventional RVA are shown in Fig. 5 for comparison

purposes.

847OTOSHI AND STELZRIED : ROTARY VANE ATTENUATOR

0.070 [ 1 I 1 I I I I

(o) TAPERED TRANSITION CONFIGURATION0.060

1 ~Isll[0.050 D--D IS221

~ Islll

0.040G-a IS*21

s 0.030~z

:0.020‘1

COMPACT RVA 1

~ae

CONVENTIONALRVA

R’,n-

,n’

/n’

,ti

/u’

/d

2=

Q 0.010t

g-o-++ .e.Q-~+.&-@-o-+u<)

ozoEu 0.C605.%Lll 0.050

$ <5

$? 0.040

0.030

0.020

0.010

0.

I I I I I I I 1

(b) STEPPED TRANSITION CONFIGURATION

‘Q

‘Q

t I I I 1 t I I10 20 30 40536070S09

1

81, deg

Fig.5. Compact RVAreflection coefficient versus 81 setting.

B. Foresight Error

Boresight error is the misalignment angle term al

defined in (12). A simple procedure for calibrating this

error for conventional RVAS has been described by

Larson [4]. His method is equivalent to substituting an

experimentally measured value of attenuation into an

expression for al derived from the COS2O attenuation law.

The final calibrated al value is obtained by averaging

the al values determined from experimental attenuation

data at several vane angle settings, Larson showed that

the average al value results in a good fit between mea-

sured and theoretical attenuation values over a large

range of vane angle settings for a conventional RVA.

The procedure used to calibrate al of the compact

RVA is identical to the one described for conventional

RVAS except that measured attenuation values are sub-

stituted into a more general al expression derived from

the modified law rather than from the COS2 6’ law. This

general al expression is given in Appendix 1. Due to the

fact that attenuations of a compact RVA deviate sig-

nificantly from the COS2O law even at vane angle settings

as low as 20°, the use of the more general expression is

recommended for accurate calibration of al. A beneficial

outcome of the use of the described calibration proce-

dure is that, if the stator vanes were misaligned with

respect to each other, the effect of this misalignment

would tend to reduce to a type B [6] error which is gen-

erally negligible. This result is also discussed in Ap-

pendix 1.

Since al is a mechanical misalignment angle, its value

can be determined from RF calibrations at ons fre-

quency if internal reflection errors are small. Using 8448

MHz calibrated values of .L~~ and @ and measured at-

tenuation values in the general cu expression (Appendix

I), average values equal to (0.0064 f 10.001 8s1)0 and

(– 0.178 t 0.004si)0 were calibrated for the compact

attenuator in the tapered and stepped transition con-

figurations, respectively. The symbol S? denotes the

calculated standard error based on the number of mea-

surements. The average al value for each configuration

was based on measured attenuations at 27 different vane

angle settings (between 19 and 80°). Thle differences of

the al values for the two transition configurations are

believed to be due mainly to differences in the mechani-

cal alignments of the stator vanes.

C. Incremental A tterwation and Incremental Phase

Change

For conventional RVAS, the incremental phase

change characteristics are not usually measured because

phase changes are negligibly small at attenuation set-

tings below 40 dB. Phase changes are negligible because

LdB is usually very large (90 dB or greater) for most

conventional RVAS. For example, when ~dB is at least

90 dB, the maximum incremental phase change as calcu-

lated from (8) will be less than + 0.25° at vane angle

settings below 85°.

Since the value of LdB is not large for compact RVAS,

the incremental phase change will not be negligibly

small (unless the value of @ is fortuitously equal to zero

or an integer multiple of r rad). Therefore, in the verifi-

cation of the modified attenuation law, it was of interest

to calibrate the incremental phase change characteristics

as well as incremental attenuations. Incremental atten-

uations were calibrated at various indicated vane angles

through the use of an ac ratio transformer test set and

measurement technique described in [13]. (Incremental

phase change measurements were made by the Micro-

wave Standards Laboratory, Hughes Aircraft Com-

pany, Culver City, Calif.)

The results of incremental attenuation and phase

measurements at 8448 MHz are given in Tables I I and

II 1, respectively, for the compact RVA in the tapered

and stepped transition configurations. Theoretical

values are also shown for comparison purposes. I t can

be seen that good agreement between theoretical and

measured values was obtained over abcmt a 20-d B at-

tenuation range. The overall accuracies of the incre-

mental attenuation and incremental phase change

measurements are estimated to be better than f (0.0005

+0.001 XAdB) and f 0.5°, respectively. Theoretical

values were computed through the use c,f (7), (8), (12),

and the following experimentally determined parameter

values.

848 IEEE TRANsACTIONS ON MICROWAVE THEoRY AND T13CHNIQUES, NOVEMRER 1971

TAB LE 11

INCREMENTAL ATTENUATION AND INCREMENTAL PHASE CHANGE FORCOMPACT RVA IN TAPERED TRANSITION CONFIGURATION AT 8.448 GHz.

‘I’ deg

2.787

3.937

4.819

5.562

6.142

6.216

8.779

12.248

12.388

15.142

17.451

19.255

19.474

20.000

27.284

32.712

33.109

37.880

40.000

41.964

44, 932

45.550

Incrementalattenuation, dB

Theo-

reticalvalue

0.0202

0.0403

0.0604

0.0805

0.0982

0.1005

0.201

0.392

0.401

0.601

0.802

0,979

1.002

1.058

z. 003

2. 92S

3.003

4.004

4.506

5.005

5.824

6.006

Measuredminus

theoreticalvalue

-0.0001

-0.0005

-0.0008

-0.0011

.0. 0014

-0.0013

-0.002

-0.001

-0.001

-0.001

0.000

0.001

0.001

0.001

0.000

-0.003

-0.002

-0,001

0.001

0, 002

0.002

0.001

2ncrem,ental phasechange, deg

Theo-

reticalvalue

-0.2

-1.0

Measuredminus

theoreticalvalue

-0.1

-0.1

81, deg

48.752

51.646

54.286

55.782

56.710

59.9.21

60.000

61.028

64.779

68.089

71.051

71.565

73.749

76.259

78.677

80.000

81.146

84.019

85.000

86.012

88.497

89.994

Incrementalattenuatlcm, dB

Theo-

reticalvalue

7.006

8.007

9.008

9.615

10.008

11.468

11.506

12.010

14.012

16.014

18.016

18.384

20.019

22.021

24.025

25.114

26.027

28.030

28.569

29.029

29.714

29. 825a

Measuredminus

theoreticalvalue

0.002

0.004

-0.002

-0.004

-0.005

-0.002

-0.003

-0.003

-0.008

-0.003

0.002

0.003

-0.007

-0.006

0.009

0.013

0.013

0.003

0.000

-0.003

-0.003

—

Incremental phasechange,. deg

Theo.reticalvalue

-4.0

-25.9

-41.6

-50. 7a

Measuredminus

theoreticalvalue

-0.1

0, 0

-0.2

—

Tapered Transition Configuration:

Lm = 29.825 dB

@ = – 50.?0

al = + 0.0064°.

Stepped Transition Con.guration:

Ld, = 29.800 dB

@ = – 50.1”

al = — 0.178°.

The runout error term az(Or) in (12) was assumed to be

zero at all vane angle settings.

As will be discussed further, the small differences in

the measured values of LdB and @ given above for the

two transition configurations resulted from experimen-

tal procedure. Since LdB and ~ are defined to be param-

al%ese are measured values used in the theoretical (7) and (8).

eters of the rotor section only, their values should be

calibrated with the transitions removed. However, for

convenience, calibrations were performed on the overall

RVA configuration. The values of incremental attenua-

tion and phase change (measured when rotating the

center vane from the 0° to 90° setting) were assumed to

be equal to L~, and @, respectively. This experimental

procedure resulted in measured values of&~ and 4 that

were dependent on the mismatch interactions between

the rotor and transition sections. It may be seen from

Table I that the reflection coefficients of the stepped and

tapered transitions are significantly different.

Although the described experimental procedure

causes small errors to occur in the measured values of

&B and @, it is generally more convenient to perform

calibrations in rectangular waveguide rather than in

circular waveguide. It can be shown from an error

analysis study that, in general, the measurements of ~d~

and @ do not have to be made to a high degree of ac-

OTOSHI AND STELZRIED: ROTARY VANE ATTENUATOR

TAB LE III

INCREMENTAL ATTENUATION AND INCREMENTAL PHASE CHANGE FORCOMPACT RVA IN STEPPED TRANSITION CONFIGURATION AT 8.448 GHz.

81, deg

2.787

3.937

4.819

5.562

6.142

6.216

8.779

12.248

12. 38I3

15.142

17.451

19.255

19.474

20.000

27.284

32.712

33.109

37.880

40.000

41.964

44.932

45.550

Incrementalattenuation, dB

Theo-

reticalvalue

O. 0176

0.0366

0.0559

0.0752

0.0923

0.0946

0.192

0.380

0.389

0.586

0.784

0.960

0.982

1.038

1.974

2.892

2.967

3.961

4.459

4.954

5.768

5.948

Measuredminue

theoreticalvalue

0.0005

0.0009

0.0011

0.0010

0.0011

0.0006

0.002

0.002

0.001

0.000

-0.001

0.002

0.002

0.003

0.000

-0.003

-0.003

0.000

0.002

0.004

0.004

0.006

Incremental phase

change, deg

Theo-

reticalvalue

-0.2

-1.0

Measuredminus

theoreticalvalue

-0.2

-0.5

f+, deg

48.752

51.646

54.286

55.782

56.710

59.921

60.000 ‘

61.028

64.779

68.089

71.051

71.565

73.749

76.259

78.677

80.000

81.146

84.019

85.000

86.012

88.497

90.000

90.18

Incrementalattenuation, dB

Theo-

reticalvalue

6.942

7.936

8.930

9.533

9.923

11.372

11.410

11.910

13.896

15.882

17.868

18.233

19.855

21.844

23.840

24.930

25.848

27.883

28.441

28.922

29.659

29.798

29. 800=

Measuredminus

thee.reticalvalue

-0.003

0.009

0.005

0.001

0.004

0.007

0.009

0.005

-0.004

-0.003

-0.004

-0.004

-0.020

-0.034

-0.037

-0.043

-0.052

-0.073

-0.076

-0.075

-0.041

-0.009

—

Incremental phanechange, deg

Theo-reticalvalue

-4.0

-25.1

-40.6

-50. la

Measuredminus

theoreticalvalue

-0.6

-1.2

-0.9

—

849

curacy if the useful dynamic attenuation range of the

RVA is restricted to approximately 2/3 the value of LdB.

D. Broad-Band Frequency Response Data

Incremental attenuation and phase characteristics of

the test model compact RVA over the frequency range

8.0 to 10.0 GHz were measured by means of a computer-

controlled Hewlett-Packard network analyzer system

operated in a phase-locked mode [14]. In this system,

the major systematic errors (such as those due to mis-

match and imperfect couplers) are calibrated so that

corrections can be applied to the measurement data by

a digital computer. (The calibration work was per-

formed by the Computer Metrics facility, Palo Alto,

CaIif.)

Fig. 6 shows the measured frequency responses of LdB

and @ of the compact RVA in the tapered transition

aThese are measured values used in the theoretical (7) and (8).

confkuration. From the accuracy m-aphs for the auto-

mati~ network analyzer [14], it- is- es~imated that the

LdB and @ data are accurate to within ,+ 0.10 dB and

~ 0.5°, respectively. The variation of @ vvith frequency

as indicated in Fig. 6 was greater than anticipated, A

separate experimental study [15 ] showed that the main

cause of the large frequency variation was the resistive

film on the center vane and not the dielectric (mica)

material. However, the dependence of f r{equency sensi-

tivity on resistivity tolerances is not clearly understood.

Later tests made on another $10-,+5-fl rotor card showed

that the total variation over the same frequency range

was a fadOr Of 2 leSS fOr &B and a factOr Of s leSS

for 4.

lVhen calibrated values of LdB and ~ and al= 0.0064°

were applied to the modified theoretical law at each test

frequency (50 MHz steps between 8.0 to 10.0 GHz), the

850 IEEE TRANSACTIONS ON MLCROWAVE THSORY AND TECHNIQUES, NOVEMBER 1971

I I I I I

4

2

-2% 10~.

oz.n%6n

;.

4

2

10-3

-101 I I I

(b) FREQuENCY RESPONSE OF+-15 –

-20 –

-25 –

U -30 –

8z -35 –

EF ~o _.

:$. -45 –

-50 –

-55 –

-60( /

/’///

//,/

I [ I I5 10 15 20 25

FREQUENCY, GHz THEORETICAL ATTENUATION, dB

Fig. 7. Standard error curves for compact RVA.

additional requirement for obtaining broad-band per-

formance of the compact RVA is to calibrate the fre-

quency responses of&B and ~.

V. DISCUSSION OF ERRORS

Fig. 6. Measured frequency response of compact RVAin the tapered transition configuration.

TAB LE IV

THEORETICAL AND MEASURED ATTENUATIONS FOR COMPACT RVAIN TAPERED TRANSITION CONFIGURATION OVER FREQUENCY

RANGE 8.o TO 10.0 GHz.

Theoretical IncrementalAttenbati. ns ,a dB

81.deg

10

20

30

40

50

60

70

80

90

Meamred Minus TheoreticalValues,b dB

8.oGIiz

8.0CR.

-0.01

-0.01

-0.01

0.01

0.01

0.02

0.05

-0.01

—

Error analysis equations for the compact RVA are

derived in Appendix II. Fig. 7 shows error contributions

from various sources of random type errors for the test

model compact RVA. It can be seen that for incremental

attenuations below 10 dB, the principal contributor is

the vane angle readout error. As the incremental attenu-

ation approaches the maximum value, the principal

contributor is the uncertainty in the value of Lm.

The standard error curves in Fig. 7 were derived from

random error equations given in Appendix I I and the

following input parameter values:

10.0GM.

8.5GHz

9.0GHz

9.5GH z

10.0GHz

0,00

0.01

0.00

0.01

0.03

0.03

-0.01

-0. 0.?

O. 262

1.061

2.449

4.522

7.458

11.565

17.379

25.038

28.93

0.261

1.058

2.442

4.508

7.432

11.524

17.354

25.581

31.39

0.263

1.067

2.464

4.557

7.529

11.730

17.872

27.517

38.04

0.00

-0.01

-0.01

-0.01

0.00

0.02

0.02

0.00

-0.02

-0.02

-0.02

0.00

0.00

-0.02

-0.02

0.00

-0.02

-0.02

-0.01

0.00

0.01

-0.01

0.00

-0.04

‘The. ret,cal valuee shown at spot frequencies t. illustrate frequencysensitivity.

LdB = (29.83 i 0.04uj)dB

q = (–50.7 + o.2cr,)0bMeasured values from automatic network analyzer system were given t.two decimaL places.

al = (0.0064 ~ 0.0020-j)0

a2(er) = (0.00 + o.015@)0agreement between theoretical and experimental values

was typically as good as shown in Table IV. The results

appear to indicate that the compact RVA will obey the

modified law over a broad band of frequencies. The where uoI is the standard error of 01. The values of UZ,

OTOSm AND 8TELZRIETJ: ROTARYVANE ATTSNLTATOR

I I I I I

8448 h!tiz CALIBRATION DATA #

● MEASURED DEVIATION, STEPPEDTRANSITION CONFIGURATION

O MEASURED DEVIATION, TAPERED ●

TRANSITION CONFIGURATION●

— THEORETICAL STANDARD ERROR● ,

●✏

/

❑ 0

● ❑

•1●

❑

❑

/

❑ ’@

● 00

●la

I I I I Io 5 10 15 20 25

❑ C

THEORETICAL ATTENUATION, dB

Fig. 8. Plot of measured deviations from theoretical attenuations.

which denotes the standard error, were based on calcu-

lated standard errors or best estimates. Although the

above input parameters are applicable to the compact

RVA in the tapered transition configuration, no signifi-

cant differences in the plotted error curves were found

when these input parameter values were replaced by

those of the stepped transition configuration.

The standard error curve for A dB and absolute values

of the measured deviations from theoretical attenua-

tions at 8448 MHz (Tables II and III) are shown in

Fig. 8. It can be seen that the measured deviations for

the attenuator in the tapered transition configuration

tend to fall under the standard error curve. In the case

of the stepped transition configuration, however, many

of the measured deviation data points are above the

standard error curve. The larger deviations in the case

of the stepped transition configuration are attributed to

increased contributions from systematic errors. Large

systematic errors can result from increased interactions

between the rotor and transition fields due to junction

discontinuity effectsl and internal reflections.

1 At the j unction of the rotor and the stepped transition, the wave-

guide cross section of the stepped transition k somewhat octagonal

rather than circular.

851

An error study indicates that, in the absence of junc-

tion discontinuity effects, the main effect of internal

reflections would be to cause an error to occur in the

calibration of the boresight error angle GUIfor the com-

pact RVA. The analysis of the error due to internal

reflections is presented in Appendix II. It should be

pointed out that the calibration of al can also be affected

by the combined effects of internal reflections, stator

vane misalignment, and rotor vane waqpage.

It has

valid for

VI. CONCLUSIONS

been demonstrated that the modified law is

compact RVAS. When the transitions on the

attenuator were reasonably well matched (VSWRS

E1 .02), the agreement between theoretical and mea-

sured values at 8448 MHz was typically + (0.001

+0.0003 XA dB) over a 30-dB dynamic range. The agree-

ment was degraded to ~ (0.001 +0.001 XA dB) over a

20-dB dynamic range when the stepped transitions

having VSWRS of 1.06 were used. Although these types

of agreement were verified only at a single test fre-

quency, they are representative of those obtainable at

other frequencies if calibrations are performed with a

high-precision insertion loss test set.

The validity of the modified law over a broad band of

frequencies was essentially verified by network analyzer

measurements. Although the values of the parameters

LdB and@ will generally be frequency sensitive and will

vary from unit to unit, they can be very accurately and

economically calibrated over the entire frequency range

by means of an automatic network analyzer. The values

of LdB for a compact RVA will typically be 30 or 40 dB

and therefore be within the accurate measurement range

of the network analyzer. Once the broad-band responses

of LdE and ~ are measured, the data can be used in the

modified attenuation law. With the availability of high-

speed digital computers and programmable desk calcu-

lators, specialized attenuation tables at many frequen-

cies can be supplied for each unit at low cost and with

little additional effort.

The main advantage of the compact RVA is its

reduced size and low residual loss. The disadvantages of

the compact RVA are 1) a reduction in the total dy-

namic attenuation range, 2) the phase-shift variations

that occur as a function of attenuation setting, and 3)

frequency sensitivity. The compact RVA is not intended

to replace the high-precision laboratory-type unit which

offers a large dynamic attenuation range and essentially

constant incremental attenuations and phase change

characteristics with frequency. The compact RVA, how-

ever, would be useful in some systems applications

where a large dynamic attenuation range is not needed

(e.g., operating noise temperature calibrations). Fur-

thermore, a given amount of phase-shift variation can

be tolerable for some types of systems application if the

phase-shift change can be predicted, as is possible in the

case of the compact RVA.

852 IESSTRANSACTIONSON MICROWAVB‘IH50RY AND TSCHNIQUSS,NOV13MB13R1971

APPENDIX 1

ANALYSIS OF BORESIGHT ERROR

CALIBRATION PROCEDURE

A. Bo~esight Error Calibration Equations

From algebraic manipulations of (7) and (12), the

explicit relationship for al is obtained as

al = * arccos v’; — 82 — az(dr) (15)

where

–B+4B2–4ACx=

2A

A = 1 – 10-L’’/20 (2 COS ~)

(15a)

+ 10-L’B”” (15b)

B = 2(10–LdB/20 COS @ – lf)-L’’/lO) (15C)

C = 10–LdB/10 _ lo–AdB/l”. (15d)

The plus sign shown in (15) is chosen if 191has a positive

value and the minus sign is chosen if flI has a negative

value. For (15a) to yield the correct result, it is required

that vane angle settings be restricted to the region

IO I <&.X, where Ore.. has a positive value and is the

vane angle at which maximum attenuation occurs.

The al calibration procedure is to measure the incre-

mental attenuation at a 01 setting and then substitute

the measured value for A dB in (15d) and compute m

from (15). It is assumed that LdB, ~, and cYz(O1) are

known or were previously calibrated. After computing

al values based on measured attenuations at several Or

settings, an average value of al is computed. For best

accuracy, use of data obtained at vane angle settings

close to minimum and maximum attenuation regions

should be avoided.

B. Effect of Stator Vane l.misalignment

The purpose of this analysis is to show that if the

stator vanes were misaligned with respect to each other,

the al calibration procedure will cause the actual rotor

index plane to be established at a plane located approxi-

mately midway between the two stator vanes, By es-

tablishing the rotor index at this plane, a good fit will

result between measured attenuations and the modified

law.

Fig. 9 depicts the geometry of a general stator vane

misalignment case. An arrow at the end of an arc indi-

cates the plane to which the angle is measured with

respect to the reference plane located at the beginning

of the arc. When the arrow points in a counterclockwise

direction, the angle has a positive value in the equations

presented in this analysis. For the general stator vane

misalignment geometry of Fig. 9, the equation for

theoretical attenuations (relative to minimum attenua-

tion) can be derived as

A ‘dB = – 20 ]Oglo

ei+ .

“ Cos “ Cos ‘e’ + “) + Vz “n ‘“ ‘in ’00+ “) ’16)

Fig. 9. Geometry for a general rotorand stator vane misalignment case.

where

ev=eI+&,(eI) +8=8–al+a. (17)

8 angle between the output stator card and the

indicated rotor index plane, rad;

O’ angle of misalignment between stator vanes, rad;

and L was defined previously by (10). Other angles used

in this analysis were previously defined by (12) or can

be defined from Fig. 9.

In the absence of internal reflections, the measured

attenuation values will closely follow those given by

(16). Substitution of (16) for Ad~ in (15d) and compu-

tations of al from use of (15) at many vane angle settings

will result in an average al value of

e’E1E6+ T+;

where the angles are expressed in radians and

( [e’~’

and

(18)

(Cos (#l )sin2 0~— COS2Oi + —

2 4Z L1+

1

(19)Cos + 1

COS2 0< — — cos 20~ — — sin2 0~4Z L,

(20)

The derivation of (19) is involved and cannot be

discussed adequately here. Details of the derivation can

be found elsewhere [16]. The approximate formula

OTOSHI AND STELZRIED : ROTARY VANE ATTENUATOR

given by (18) k useful for showing the relationship be-

tween 0’ and the calibrated ctI value. For most compact

RVA cases likely to be encountered in practice, the

accuracy of the approximate formula for El will typically

be better than 0.001 percent. The approximate formula

becomes inaccurate when vane angle settings approach

e = O, T/2, and fl~, which was defined by (11).

From the geometry of Fig. 9 and substitution of (18)

we obtain

e’Cto=e’+a-ffl =--;.

2(21)

The last expression shows that the new rotor index

plane, established by the boresight error calibration

procedure, will be located approximately midway be-

tween the two stator vanes. If the rotor index were lo-

cated exactly midway between the stator vanes, the

associated attenuation error would be called a type B

error [6] whose magnitude is very small when # is small.

APPENDIX 11

ERROR ANALYSIS

A. Random Errors

For analysis of the accuracy of calibration results it is

usually necessary to know not only systematic error, but

also the individual random error contributions. These

random errors include 1) the random instability of the

measurement system and calibration procedure used,

2) the random instability of the device being calibrated,

and 3) the random error associated with operator per-

formance. In stating magnitudes of random errors, it is

convenient to use the standard error (defined as stan-

dard deviation of the mean value) as a measure of

dispersion.

For the general case, let x be the error source and u.

be the standard error of x. Then the standard

AdB due to CTZ iS

dAdp,rTAdB/x = Uz — .

dx

The individual standard error contributions of

error of

(22)

AdB are

/uAdB d standard error of .4 cIB due to the standard

error of O, decibels;

/uAdB LdB standard error of A dB due to the standard

error of L~B, decibels;

/uAdB 0 standard error of A dB due to the standard

error of 0, decibels.

Application of (22) to the modified law given by (7)

results in

/ r ~$$I (10 log,, e)lo-LdB’20&AdB 4 = —180

.(2 sin @ COS20 sin’0) loA”B1’O I (23)

/~AdB .LdB = aLdB x I(lo-LdB20cos@COS20 sin2 0

+ l&LdBj10 sin4 o)l@dB/10 ] (2A)

853

/fJAdB 8 = ---0 ff~ I (10 loglo e) (4 sin 0 cos 6)

. [(~ - ~@@20 cos ($) (,0S2 @

_ (~@LdB/lo – l@ LdB/20 co, ~)

. sinz O]loAdBi10 (25)

where

u+ standard error of O, degrees;

a~dB standard error of LdB, decibels;

and from (12)

fJ8 = (U012 + Ualz + ua2 )2 112 (26)

where

U@I standard error of Or, degrees;

(ral standard error of al, degrees;

~a2 standard error of cq (Or), degrees.

Treating the above individual sources of errors as if

they were normally distributed, we can compute the

standard error of .4 dB from [17]

rAdB = [(~AdB/d2 + (UAdB/LdB)2 + (ffAC,B/d)2]1/2. (27)

B. Error Due to Internal Reelections

If the effect of internal reflections is accounted for,

the expression of the incremental attenuation for the

compact RVA can be derived as’

()!&E sin’ o2Ad~ = – 20 loglo cos e + ~7,5,

. 20 loglo I 1 + I? sin’0 ~ (28)

where if I Sg, o,/S7,5, ] ~0.032, and neglecting higher order

terms,

R = (S,6 – SWW)SS3 + (SW,, – S,IV)SW -- (SV,)2SWSW

+ (s,,5?)’ COS20(s44s9,+ S33S,O,1O– s33S39)

‘[l+&Fi[(s’’’’-s)s44)+ (s8)8, – 57)7} )510,10 – (s7,5)2s44510,10Sin2g]. (29)

The last term of (28) is the effect due to mismatched

transitions. For example, using measured values of

(SVV) = 0.995 exp (j39.4°)

(s&&) = 0.032 exp (–jl 1.3°)

and the reflection coefficient data for the compact RVA

in the stepped transition configuration in Table 1, the

effect of the transitions is calculated to be – 0.011 dB

at 0=45° and —0.020dB at (3=90°.

z The derivation of this equation was based on scattering parame:

ter coordinate transformations [1 1 ] and a mu ltiport interconnectionmethod [18].

854 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTr-19, NO. 11, NOVEMBER 1971

ACKNOWLEDGMENT

The author wishes to thank T. Mukaihata and P.

Roberts of the Hughes Aircraft Company for their

cooperation in making incremental phase change cali-

brations on the compact RVA. The work of R. B, Lyon

of Jet Propulsion Laboratory in calibrating reflection

coefficient phase angles is acknowledged. informative

technical discussions with Dr. R. W. Beatty, Dr. G. F.

Engen, B. J, Kinder, W. Larson, and W. E. Little, all

of the National Bureau of Standards, are also acknowl-

edged and appreciated.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

REFERENCES

B. P. Hand, “Broadband rotary waveguide attenuator, ” Elec-tronics, vol. 27, Jan. 1954, pp. 184-185.P. F. Mariner, “An absolute microwave attenuator, ” Proc.Inst. Elec. Eng., vol. 109, Sept. 1962, pp. 415-419.A. V. James, “A high-accuracy microwave-attenuation stan-dard for use in primary calibration laboratories, ” IRE T’nwzs.Instrum., vol. I-11, Dec. 1962, pp. 285-290.W. Larson, “Analysis of rotation errors of a waveguide rotaryvane attenuator, ” IEEE Trans. Instrurn. Mess., vol. IM- 12,Sept. 1963, pp. 50-55.—, “Gearing errors as related to alignment techniques of therotary-vane attenuator, ” IEEE Trans. Instrum. Meas., vol.IM-14, Sept. 1965, pp. 117-123.— “Analysis of rotationally misaligned stators in the rotary-vane’ attenuator, ” IEEE Trans. Instrusn. Meas., vol. IM-16,Sept. 1967, pp. 225-231.W. E. Little, W. Larson, and B. J, Kinder, “Rotary-vane at-tenuator with an optical readout, ” J. Res. Nat. Bur. Stand.,

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

vol. 75C, Jan.–Mar. 1971, pp. 4148.G. S. Levy, D. A. Bathker, W. Higa, and C. T. Stelzried, “Theultra con;: an ultra-low noise s~ace communication groundradio-frequency system, ” IEEE Trans. Microwave TheoryTech., vol. MTT-16, Sept. 1968, pp. 596-602.C. T. Stelzried, “Operating noise-temperature calibrations oflow-noise receiving systems, ” Microwave J., vol. 14, June 1971,pp. 4148.R. W. Beatty, “Insertion loss concepts, ” PYOC. IEEE, vol. 52,June 1964, pp. 663-671.C. G. Montgomery, R. H. Dicke, and E. M. Purcel!, Principlesof Mzcrowave Czwxiis (Radiation Laboratory Series, VOI. 8).New York: McGraw-Hill, 1948, p. 351.J. D. Helm, D. L. Johnson, and K. S. Champlin, “Reflectionsfrom rotary-vane precision attenuators, ” IEEE Trans. Micro-wave Theory Tech. (Corresm ), vol. MTT- 15, Feb. 1967, DD.

123-124. - ‘ “ ‘. .

C. J. Finnie, D. Schuster, and T. Y. Otoshi, “AC ratio trans-former technique for precision insertion loss measurements, ”~ F“~~lsion Lab., Pasadena, Calif., Tech. Rep. 32-690, Nov.

S. ‘F. Adam, “A new precision automatic microwave measurementsystem, ” IEEE T?ans. Instrum. Meas., vol. IM-17, Dec. 1968,pp. 308-313.T. Y. Otoshi, “Improved RF calibration techniques: a precisioncompact rotary vane attenuator, ” Jet Propulsion Lab., Pasa-dena, Calif., Space Programs Summary 37-64, vol. II, Aug. 31,1970, pp. 67-69.T, Y. Otoshi, “Analvsis of the boresizht error calibration cmo-cedure for compact “rotary vane atte-nuators, ” Jet Propul~ionLab., Pasadena, Calif., Tech. Rep. 32-1526, vol. III, June 15,1971, pp. 126–132.A. Worthing and J. Geff ner, Treatment of Experimental Data.New York: Wiley, 1960, p. 213.Thomas-Alfred Abele, “Uber die Streumatr~x Allgemein Zusam-mengeschalteter Mehrpole, ” J“The scattering matrix of a gen-eral mterconnectlon of multlpoles”), Ad. Etek. Ubert~agung,vol. 14, pt. 6, 1960, pp. 262–268 (Transl, m NASA ReportNASA-CR-1014O4).

A Group Theoretic Investigation of the

Single-Wire Helix

JEFFREY B. KNORR, MEMBER, IEEE, AND PAUL R. McISAAC, MEMBER, IXEE

Abstract—Thk$ paper discusses the way in which symmetrygroups may be utilized in the analysis of periodic microwave struc-

tures. The theory of group representations is introduced, and the

relationship of these representations to the vector electromagnetic

fields which are solutions to the Hehnholtz equation (subject to the

boundary conditions imposed by the microwave structure) is briefly

explained, Also ezplained is the concept of time reversal. Symmetry

analysis involves collecting all of the symmetry operations of a

microwave structure into a group, and then finding the irreducible

representations of that group. Each solution of the Hehnholtz equa-

tion must belong to an irreducible representation of the space group,

and by examining the irreducible representations it is possible to

determine the symmetries and degeneracies of the waves. Symmetry

analysis is employed to describe some of the characteristics of the

Manuscript received October 21, 1970; revised April 19, 1971.J. B. Knorr was with the School of Electrical Engineering, Cornell

University: Ithaca, N. Y. He is now with the Department of Elec-trical Engineering, Naval Postgraduate School, Monterey, Calif.

P. R. McIsaac is with the School of Electrical Engineering,Cornell University, Ithaca, N. Y.

waves of the unsupported wire helix and of the single-wire helix

supported symmetrically by three dielectric rods. In particular, the

conditions for the occurrence of branch crossings on the k-b diagram

are discussed.

I. INTRODUCTION

SYMMETRY analysis has been extensively applied

in chemistry and physics to provide information

concerning the solutions of compIex problems aris-

ing in those fields. Since most microwave structures of

practical interest exhibit some degree of symmetry, it

would appear that the application of symmetry analysis

to microwave theory would also be fruitful. However,

with the exception of the analysis of symmetrical micro-

wave junctions [1 ], [2], symmetry analysis has received

relatively little attention from microwave engineers.

There appear to be several reasons for this neglect of

symmetry analysis. Firstly, the abstract group theory