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Stimulating America's Progress 1913-1988

REFERENCE

NISI

PUBLICATIONS

NBS TECHNICAL NOTE 1319

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards
Tm he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The Bureau's overall ^_^^ go^ 's ^o strengthen and advance the nation's science and technology and facilitate dieir effective application for public benefit. To this end, the Bureau conducts research to assure international competitiveness and leadership of U.S.

industry, science arid technology. NBS work involves development and transfer of measurements, standards and related science and technology, in support of continually improving U.S. productivity, product quality and reliability, innovation

and underlying science and engineering. 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 Institute for Materials Science and Engineering.

The National Measurement Laboratory

Provides the national system 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,

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produces, and distributes Standard Reference Materials; provides

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System. The Laboratory consists of the following centers:

• Basic Standards^

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• Analytical Chemistry

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Provides technology and technical services to the public and private sectors

to address national needs and to solve national problems; conducts research

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capabilities; provides engineering measurement traceability services; develops test methods and proposes engineering standards and code

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and improves mechanisms to transfer results of its research to the ultimate

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• Applied Mathematics

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The 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

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participation in ADP voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and

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Systems and Software

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Architecture Advanced Computer Systems

The Institute for Materials Science and Engineering

Conducts research and provides measurements, data, standards, reference

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• Ceramics

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'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address

Gaithersburg, MD 20899. 'Some divisions within the center are located at Boulder, CO 80303.

'Located at Boulder, CO, with some elements at Gaithersburg, MD
Generation of

Standard Electromagnetic Fields

in a TEM Cell

Motohisa Kanda

R. David Orr

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303-3328

^ fc^ ^ ^

x>-

U.S. DEPARTMENT OF COMMERCE, C William Verity, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued August 1988
National Bureau of Standards Technical Note 1319

Natl. Bur. Stand. (U.S.), Tech Note 1319, 246 pages (Aug. 1988)

CODEN:NBTAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1988

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

Page

1. Introduction 1 2. Advantages and Limitations 2 3. Physical Characteristics 3 4. Electrical Characteristics 4

4.

1

Impedance 4

4.2 Field Uniformity 5 4.3 Higher Order Modes 6

5. Field Measurement Procedures . 8

6

.

Setup and Measurement Procedure 10

6.1 Setup 10

6.2 Measurement 12

7. Standard Field Uncertainty 13

8. Statistical Control 13

9

.

Summary 15

10. Acknowledgements 16

11. References 16

Appendix A. Larsen, E. B. "Techniques for producing standard EM . . 23

fields from 10 kHz to 10 GHz for evaluating radiation . . monitors"

Appendix B. Donaldson, E. E.; Free, W. R. ; Robertson, D. W. ; Woody, . 43

J. A. "Field measurements made in an enclosure"

Appendix C. Crawford, M. L. "Generation of standard EM fields .... 55

using TEM transmission cells"

Appendix D. Tippet, J. C; Chang, D. C. "Radiation characteristics . 65

of dipole sources located inside a rectangular,

coaxial transmission line"

Appendix E. Kanda, M. "Electromagnetic field distortion due to . . . 105

a conducting rectangular cylinder in a transverse ....

electromagnetic cell"

Appendix F. Spiegel, R. J.; Joines, W. T. ; Blackman, C. F.; 115

Wood, A. W. "A method for calculating electric and . . . magnetic fields in TEM cells at ELF"

Appendix G. Gruner, L. "Higher order modes in rectangular 125

coaxial waveguides"

Appendix H. Tippet, J. C; Chang, D. C; Crawford, M. L. "An .... 131

analytical and experimental determination of the ....

cutoff frequencies of higher order TE modes in a ....

TEM cell"

Appendix I. Tippet, J. C; Chang, D. C. "Higher order modes .... 165

in rectangular coaxial line with infinitely thin ....

conductor"

Appendix J. Weil, C; Gruner, L. "High-order mode cutoff in .... 209

rectangular striplines"

Appendix K. Hill, D. A. "Bandwidth limitations of TEM cells .... 215

due to resonances"

Appendix L. Wilson, P. F. ; Ma, M. T. "Simple approximate 233

expressions for higher order mode cutoff and

resonant frequencies in TEM cells"

1.1 X

Generation of Standard Electromagnetic Fields

in a TEM Cell

Motohisa Kanda

R. David Orr

This paper documents the facilities and procedures used by the

National Bureau of Standards to calibrate radio frequency electric

field probes using a transverse electromagnetic (TEM) cell. The

advantages, limitations, and physical characteristics of TEM cells

are presented. Impedance, field uniformity, and mode structure,

critical aspects of a cell as a standard field enclosure, are discussed. The paper concludes with sections on setup and measure- ment procedures for users, uncertainty in the standard field, and

statistical control of the calibration system. Copies of key

references are included to provide ready access to the details of

topics summarized in the text.

Key words: antenna; antenna calibration; calibration; electromagnetic field; radio frequency; standard field; TEM cell;

transverse electromagnetic cell.

1 , Introduction

The National Bureau of Standards (NBS) uses three different methods for

generating radio frequency (rf) electromagnetic (EM) fields of known

frequency, field strength, and polarization [1]. The facilities and their

approximate frequency ranges are (1) transverse electromagnetic (TEM) cells

(10 kHz-150 MHz); (2) a 30 m x 60 m open-site ground screen (25 MHz-1000

MHz, and (3)a5mx6.7mx8.5m anechoic chamber (500 MHz-40 GHz). This

report summarizes the physical and electrical properties of TEM cells and

the NBS measurement methods for exploiting their well characterized EM

field.

The error budget is presented to quantify just how well the field is

characterized, and references include the details of each topic or procedure

summarized. This report, therefore, serves to document the procedures and

equipment used by NBS to generate and evaluate standard EM fields within TEM

cells. Donaldson et al. [2] discuss TEM cells and other enclosures used for

generating known radio frequency fields. Their paper provides a useful

background for the present report. Except for reference [3], which contains

TEM cell construction details and engineering drawings, copies of all cited

references are provided with this report.
2 . Advantages and Limitations

Two of the three facilities mentioned in the previous section are containers (the TEM cell and anechoic chamber), and the third (the open

site) is a half space. Each is an environment into which a signal source projects an EM field. For the open site and anechoic chamber, the source is an antenna, and thus the performance of these two facilities as environments

for known EM fields is subject to the limitations of the antennas

themselves. Because antennas are bandwidth limited, field measurements over

a given frequency range may require more than one antenna. The finite

extent of an open site or anechoic chamber may force the measurement point

into the near field of the antenna where the radiated field is not planar

and where, therefore, the uncertainty in computed field values is greater.

In open site and anechoic chamber measurements, an antenna is the

transducer between EM waves on a transmission line and the radiated EM

field. However, the TEM cell (fig. 1) is just an enlarged portion of a transmission line and thus eliminates the need for a transducer. As an rf

field enclosure, the cell is operated from dc to the frequency at which

higher order modes of propagation appear. Over this frequency range the

cell supports only TEM plane waves. Thus, in eliminating the source

antenna, the TEM cell avoids near- field problems and has only an upper

frequency limit. The exterior of the cell is a grounded metal enclosure

which shields the internal plane wave field from the external electromagnetic environment and, conversely, contains that field which otherwise

would radiate into the surroundings and perhaps interfere with associated

equipment. The primary shortcomings of the TEM cell are the upper frequency

limit and the restricted working space within the cell compared with an open

site or anechoic chamber. The latter is particularly severe in the smaller,

higher frequency cells.

In sum, the TEM cell is a shielded enclosure for rf plane wave fields.

It is of simple construction and, in its smaller versions, is portable. The

field is well characterized and is sufficiently uniform to be useful in the

calibration of EM field probes and in susceptibility measurements. Higher

frequency operation is attained with a smaller cell but at the expense of

working volume within the cell.
3. Physical Characteristics

Details for the construction of a TEM cell are given in [3] and the

configuration of a typical cell is shown in figure 1.

The test portion of the cell (ericlosing the uniform, well characterized

plane wave field) is the center section which is an expanded rectangular

"coaxial" transmission line (RCTL) in which the center conductor (the

septum) is a thin sheet of metal supported parallel to two opposite walls of

the cell. (Some authors refer to the TEM cell as a rectangular- strip

transmission line and use the term "rectangular coaxial transmission line"

for a line with a thicker center conductor. We use the latter notation to

be consistent with the references summarized in this paper) . The shell of

the cell can be square or rectangular in cross section with the septum

either midway between opposite walls (a symmetric cell) or offset from the

center plane of the cell (an asymmetric cell) [4] . The trade-off between

test volume and field uniformity determines which cell configuration is best

for a given application.

The NBS cell that is the subject of this report is used for creating a known EM field in which radio -frequency field probes are calibrated for use

as transfer standards, hazard level probes, or general purpose field probes.

Because such probes are usually small and their calibration requires a very

well defined plane wave field, NBS uses a cell that is rectangular and

symmetric with a 2:1 cross section aspect ratio (a/b in fig. 1) at the test

volume. The septum is parallel to the large walls of the cell and is

supported by dielectric rods. The cell is constructed of 3.2 mm aluminum

sheet which is sufficiently rigid to give the cell dimensional stability.

The central region of the cell (the "box" between the two tapered sections)

is 1.2 mx 0.6 m in cross section and 1.2 m in length. The tapers at each

end of the cell electrically match the central region to a 50 fi input

transmission line and a 50 f] termination at the output end of the cell.

Each tapered section is terminated with a coaxial connector whose center

conductor is attached to the septum. Access to the test volume is through

either of two doors, one in each of two adjoining walls at the test volume.
4. Electrical Characteristics

As a device whose purpose is to provide NBS- traceable calibrations of EM

field probes, the TEM cell discussed in this report should have (1) minimum

impedance mismatch with its input transmission line and the load terminating

its output; (2) a known uniform field throughout the test volume (the region

in which the device to be calibrated is placed), and (3) a known upper

frequency limit.

4 . 1 Impedance

The characteristic impedance of an RCTL can be expressed in terms of C^

,

the capacitance per unit length of transmission line [5] as

o vC 1 , C

^^^

O -^2^;^;;^ C O

o o where r)^ == 1207r is the intrinsic impedance of free space, e^ - 10"^/367r

F/m is the free space permittivity, /x^ = 47r x 10"'' H/m is the free space

permeability, and v = (n^e^)'^^^ = 3 x 10^ m/s . C^ is given in terms of the

cell geometry by the approximate expression

O / r^ 2 , , — . TTg. , AC ,^. ==4 [^ - - In (sin^f)] - - (2) o

°

where the dimensions a, b, and g are shown in figure 1. AC accounts for the

interaction of the edges of the septum with one another. Equation (2) is

applicable and AC is negligible when

g

> 1, and ^

> 1/2. (3)

These conditions are satisfied by the NBS TEM calibration cell. For several

values of a/w (cell width with respect to septum width) , the dependence of

cell capacitance per unit length on a/b is shown in figure 2. Subject to

the conditions in eq (3), eqs (1) and (2) can thus be used to determine the

septum width necessary to give a 50 Q characteristic impedance to a cell of

known relative dimension, a/b. By means of laboratory models and a time
domain ref lectometer , the septum width can be adjusted to minimize

reflections along the RCTL and determine the optimum septum shape required

to produce the desired cell characteristic impedance.

4.2 Field Uniformity

The unperturbed electric field within the central portion of the cell,

as shown in figure 3, has been well characterized by measurements of

electric field vs. position in the cell [4] and by theoretical expressions

for the cell electric field vs. position [5]. These data are relative

values giving the field variation with respect to the field at the center

point of the cell between the septum and side wall. When P watts is

transmitted through the cell, the absolute electric field in volts per meter

at the center of the cell test volume is computed from the expression

E = ^

(4)

where R is the real part of the cell characteristic impedance in ohms and d

is the distance in meters between the septum and side wall that bound the

test volume (in figure 1, d=b) . This absolute electric field point serves

to "calibrate" the relative field data from which the absolute electric

field can then be obtained at any point within the test volume.

The theoretical expression for field distribution was derived for an empty cell. The measured field distribution was obtained with a calibrated

dipole probe small enough to cause negligible distortion of the field within

the cell. However, if a probe to be calibrated occupies one -fifth to one- third of the septum- to-wall distance, there would be some field distortion

(in the form of field enhancement) due to the probe shorting out part of the

field between septum and wall. In other words, because of the presence of

the probe the physical configuration of the cell would differ significantly

from the configuration on which the standard field determination was based.

Therefore, the EM field within the cell would deviate from the theoretically

and experimentally determined distribution and would not have the value at

the center point given by eq. (4). To preserve the integrity of the

environment giving rise to the standard field, the probe to be calibrated

should occupy less than one- third of the septum- to-wall distance.
Theoretical and experimental analyses [4, 6] have been done on field

distortion due to a rectangular metal object in a TEM cell. Spiegel et al.

[7] have used a quasi-static approximation and numerical integration to

derive expressions for the electric field in a TEM cell. Their results are

in close agreement with [5]. They have also obtained analytical expressions

for the X and y components of the cell magnetic field.

4.3 Higher Order Modes

A very useful feature of the TEM cell as a source of plane wave rf

fields is its ability to support fields down to zero frequency. As the name

of the cell suggests, these fields are in the TEM mode. The maximvim

frequency at which the cell provides such a well characterized electromagnetic environment is determined by the low- frequency cutoff of the first

higher order mode. This frequency can be interpreted equivalently as the

lowest frequency at which the first higher order mode can exist in a given

cell, or as the highest frequency at which the TEM mode exists alone. The

cutoff frequencies of the higher order modes depend purely on the cell

geometry and have been determined theoretically and experimentally by a number of observers [8, 9, 10]. Figure 4 shows this dependence on geometry for several TEM cell modes whose cutoff wavelengths are computed

from Gruner's analysis [8]. Weil and Gruner [11] give an alternative

presentation of the first eleven higher order mode cutoff frequencies by

plotting them as a function of cell aspect ratio (a/b) for several values of

cell characteristic impedance.

As the frequency is increased and some of the power entering the cell

couples into higher order modes, those modes are superimposed upon and thus

degrade the field configuration of the pure TEM mode. When a higher order

mode first appears, it behaves as a traveling wave that reflects back and

forth within the cell and which, if not replenished by incoming power, would

vanish due to wall losses. Reflection occurs within the cell tapers where

the cross section of each taper has narrowed to that of a waveguide whose

cutoff frequency is lower than the field frequency within the cell. This

traveling wave condition is the first type of perturbation inflicted on the

TEM mode by a newly appearing higher order mode.

6
As the frequency is further increased, the guide wavelength A of the

field in the cell decreases until it is twice the distance between the two

reflection planes (one in each taper). That is, the effective length of the

cell is then A /2 and a cavity resonance that enhances the fields of the

higher order mode is established. Thus, these modes perturb the TEM mode

through their presence either as traveling waves or standing waves. Theoretical and experimental analyses of mode cutoff and resonance

frequencies are given in [12, 13].

In his measurements of the E fields of higher order modes. Hill [12]

observed that a given mode may not be detectable between its cutoff

frequency and its first resonance frequency. (He speculates that within

this frequency range the guide wavelength may be too long with respect to

the cell length for the wave to develop.) For this reason, the frequency

response of a cell contains windows within which the cell field is primarily

the TEM mode even though the field frequency is above cutoff for some higher

order modes. These windows in the cell's mode structure vs. frequency may

be usable for purposes such as investigating the electromagnetic

susceptibility of or emissions from an electronic component. In general,

however, the field configuration within the cell after the onset of higher

order modes is sufficiently complex and uncertain that NBS uses the

calibration cell only in the pure TEM mode region below the cutoff frequency

of the first higher order mode. When a cell is used as a standard field

source, the analyses of higher order modes are useful only for predicting

the lowest cutoff frequency; that is, the highest frequency at which the

cell field is reliably pure TEM. For the present cell this limit is

approximately 150 MHz. Table 1 gives the dimensions and TE^^ q cutoff

frequencies for several NBS TEM ceils.
Table 1. Dimensions and TEj^q cutoff frequencies (fj, ) for some NBS TEM

cells. When used for calibrating electric field probes, the cell must be

operated at less than f^

.

TEjo Cutoff Frequency

(MHz)

Cell Dimensions Cell Form Factor

2a 2b

(cm)

2w a/b

300 300 249 1.0

120 120 100 1.0

100 60 72 1.67

120 60 82. 8 2.0

50 30 36 1.67

30 20 22. 8 1.5

12 12 10 1.0

29

72

128

125

256

408

718

5. Field Measurement Procedures

The creation of a known rf field within a TEM cell is accomplished by

applying a voltage, V, between two parallel plates (septum and cell wall) a distance, d, apart. The electric field at the center of the region between

the plates is then E = V/d volts per meter. The geometry and construction

of the cell are basic (fig. 1). As a consequence of this conceptual and

practical simplicity, the procedure for establishing and computing a standard rf field is also straightforward.

As rf pov7er flows through the cell, the field within the cell can be

determined by either of two measurement methods. In figure 5a the output of

a signal generator is amplified, cleansed by a low pass filter, passed

through the cell, and almost completely absorbed by the 50 Q termination on

the output port of the cell. An rf voltmeter reads the septum- to-wall

voltage by means of the voltmeter tee connector at the cell input port.

In figure 5b, the voltmeter and tee are replaced by a dual directional

coupler whose side arms are terminated by power meters that read the

incident and reflected power at the cell input port. From the net power, P

watts, entering the cell and the real part, R ohms, of the cell complex

impedance, eq (4) gives the rf electric field, E volts per meter, at the
center point of the test volume between septum and wall,

E = ^

(4)

At lower frequencies where the length of the propagating wave is

several times the cell length, the methods are equivalent. Regardless of

the voltage standing wave ratio (VSWR) , the voltage measured at the input

port will not differ significantly from the value at the cell center.

However, at higher frequencies the voltage standing wave could be short

enough to cause noticeable voltage variation between the cell center and

either end. For example, suppose a measurement of incident and reflected

power at the cell input indicates a VSWR of 1.2:1. This means that, due to

a standing voltage wave on the transmission line (including the cell), the

maximum line voltage is 20% greater than the minimum line voltage. The

distance between the maximum and minimum voltage points is one quarter of

the line wavelength A. Therefore, if the distance from the cell input to

the cell center is much less than A/4 (that is, the cell is electrically

small) , a voltage measurement at the cell input will be a good approximation

to the voltage at the cell center. If the cell is electrically large, the

input line voltage could be 20% greater than the line voltage at the cell

center. Thus, the operator must consider the VSWR on the line and the

electrical length of the cell in deciding if a voltage measurement at the

cell input gives a sufficiently accurate value of the TEM electric field at

the cell center. If not, a better value for this field can be obtained by

using a time domain ref lectometer to measure the line impedance at the cell

center. The electric field is then computed from eq (4). In another

procedure, a vector voltmeter is used to measure the incident and reflected

complex voltages at the cell input. The field at the cell center is

obtained by using transmission line equations (or the Smith chart) to

translate the line voltage (or line impedance) to the cell center. The

frequency and power limitations of the system components such as directional

coupler, voltmeter, and power sensors may also determine which measurement

method to use. The power handling capacity of the present components sets an upper limit of about 200 V/m to the cell field.
6 . Setup and Measurement Procedure

6.1 Setup

1. Check the interior of the cell to be sure that it is clean, that

nothing was left in it from a previous measurement, and that the

septum's supports and electrical connections are sound.

2. If the absence of a dedicated calibration setup requires gathering

and interconnecting components, examine all connectors for obvious

defects such as bent center pins, broken or bent spring- loaded

receptacles, and dirty or worn mating surfaces. A dimensional mismatch

between connectors should be corrected because it may result in either

poor electrical contact or the need for extreme closing force which may

damage the connectors.

3. Use only the highest quality terminations (that is, lowest VSWR)

when outfitting a TEM cell as a calibration facility, and do not exceed

their quoted frequency and power specifications.

In Section 4 . 3 we discussed how higher order modes deteriorate the TEM

field configuration within the cell. Further distortion is caused by waves reflected back into the cell from a poorly matched termination on

the cell output port. While higher order modes are so intractable that

they limit the frequency range over which the cell serves as a standard

field enclosure, reflections from the output port can be greatly

reduced by employing high quality terminations. The endpoints of the

cell are 50 Q type N connectors and therefore the output termination

must be as close as possible to 50 Q. For example, 150 W terminations

are commercially available with a maximum VSWR of 1.1 (reflection

coefficient magnitude = |r| - 0.05) from dc to 1.5 GHz;

5 W and 10 W terminations can be obtained with a maximum VSWR of 1.03

(|r| - 0.01).

10
4. Regularly recalibrate voltmeters and power meters required for the

calibration. This will verify the measurement uncertainty which is

assigned to them in deriving the net uncertainty quoted for the

calibration service. These well .characterized meters should be dedicated

to the measurement setup and thus made part of the documentation history

of the calibration.

5. Calibrate directional couplers and attenuators over the frequency

range of the TEM cell. This provides a more accurate value of coupling

factors and attenuation at each measurement frequency than the nominal

values furnished by the manufacturer.

6. Interconnect the system components as shown in figure 5a or 5b. If

power rather than voltage is being measured, use eq (4) to compute the

net power needed to give the maximum electric field required for the

calibration. From this value and the forward coupling factor of the

directional coupler, compute the power that will be incident on the

forward power sensor. Be sure that the sensor is rated to handle this

power; if necessary, insert an attenuator between the sensor and coupler

sidearm. (The cell characteristic impedance is near enough to 50 Q that,

for the present purpose, the computed net power to the cell may be used

to approximate the incident power at the coupler)

In general, be aware of power, voltage, and frequency ratings of all

components in the setup. Exceeding these ratings may damage some components or cause them to function outside their optimum operating

range

If the amplifier is driven near its rated output or if its

performance deteriorates, a low pass filter at its output will attenuate

signal frequency harmonics that may be introduced during amplification.

As the frequency is varied, other filters may have to be used in order

that the filter cutoff frequency is always less than the second harmonic

of the signal frequency.

7. Orient the probe in the cell according to the type of measurement

specified by the customer. The probe handle, fixed or mounted in a rotator, enters the cell through small doors in the cell walls. The

11
field sensor is positioned at the length and width center of the septiam

and midway between the septxam and wall. Battery-powered probes have a switch to check the battery condition. This should be done before and

after each calibration.

8. Prepare a well organized data sheet. It should contain not only

columns for all parameters to be varied and measured (and perhaps

quantities to be computed), but also spaces for the date, time of day, names of the observers, description of the measurement, and

identification of all the equipment used (for example, serial numbers

as well as model number so that specific individual components are identified) . Such information is essential if the measurement has to

be repeated or the data and setup referred to at a later date when the

observer's detailed recollections have faded.

6 . 2 Measurement

1. Turn on all instruments and allow them time to stabilize.

2. a. Calibrate the power sensors against the power meters' internal

reference sources.

b. If a voltmeter is being used to set the field within the cell,

calibrate the voltmeter at each frequency and voltage at which a measurement is to be made .

3. Set the signal generator to the first measurement frequency and

adjust the signal level until the rf voltmeter (or power meters)

corresponds to the desired electric field in the cell.

4. Record frequency, rf voltage on (or power through) the cell, and the

probe response as displayed by the probe metering circuit,

5. Because setup impedances and the signal generator output vary with

frequency, decrease the signal level before changing to the next

measurement frequency. This reduces the risk of exceeding the power

rating of some components at the new frequency.

6

.

Perform the measurements at the remaining frequencies and record

all data.

12
7. Repeat the measurement at the first frequency and compare the two

probe readings at this frequency to verify the stability of system

parameters during the calibration. In the course of the calibration

(or previous calibrations) the observers will have acquired a sense of

system repeatability. If the original and comparison probe readings

differ by more than this expected repeatability, the system should be

checked to find the cause of the discrepancy and then the calibration

repeated.

7. Standard Field Uncertainty

Following the analysis in [4] , the expression for the percent

uncertainty in the standard field is obtained from eq (4) and has the form

AE(%) . I 1/2 (^ + ^) +^+ ej X 100 (5)

where AP, AR and Ad are absolute uncertainties in the net cell power, P, the

real part of the cell complex impedance, R, and the septum- to -wall distance,

d. In deriving eq (5) higher order error terms were neglected. The

quantity eg is the fractional field nonuniformity due to a test object that

occupies no more than approximately one -fifth of the septum to wall

distance; eg was evaluated through field mapping with a small probe. The

fractional uncertainties in eq (5) are:

• net power to cell: ±0.03 (power at coupler side arms: ±0.01;

coupling factors: ±0.02),

• real part of cell impedance: ±0.03,

• septum- to-wall distance: ±0.01; and

• field nonuniformity: ±0.06.

The total percent uncertainty in the computed standard electric field is

then

AE(%) « |l/2 (0.03 + 0.03) + 0.01 + 0.06| x 100 - 10%, or « 0.8 dB.

8. Statistical Control

Before a measurement is begun, the operator must verify that the TEM

cell system is operating properly. We must therefore define what we mean by

"proper operation" and specify which cell parameter (and its value or range

of values) will be used to indicate that condition.

13
Because of the mechanical and electrical simplicity of the system, the

reflection coefficient of the cell (as a measure of cell impedance)

characterizes the operating condition of the cell. The impedance of the

cell is affected by deteriorated or broken electrical contacts, a change in

septum orientation within the cell (due to loose or broken dielectric

supports), objects in the cell, and the impedance of the nominally 50 Q

termination on the output port of the cell (see item 3 in Section 6.1). The

influences of these factors on the cell impedance can be detected by

measuring the power or voltage reflection coefficient at the input to the

cell. In this role as a revealer of things amiss, the reflection

coefficient (through its dependence on the cell impedance) not only monitors

the condition of the system but also gauges the quality of the system's

performance. This function is often assigned to any suitable device (called

a check standard) whose response to the system in which it is placed

indicates the operating condition of that system. In the present case, a check standard might be a small antenna located within the cell to act as an indicator of the cell electric field. Then, if the antenna did not give the

expected response to a reference level of rf power entering the cell, this

discrepancy would alert the user to a possible fault in the cell system.

However, any check standard will introduce some perturbation into the system

it is monitoring. This concern does not arise here because impedance, while

providing the same surveillance as a check standard, is an intrinsic

property of the cell.

A record of the cell's performance history will be maintained in the

form of a statistical data base from which control charts for the reflection

coefficient will be developed and used. The data for these charts are obtained by measuring the cell's reflection coefficient each time the cell

is used. When adequate data have been obtained on the system, the proper

control chart methodology will be determined. Typical control charts

monitor not only the mean value of a measurement process but also the

spread, or dispersion, in that process. The control limits for each chart

are determined from the statistical properties of the underlying measurement

data. The magnitude of the cell voltage reflection coefficient approaches

zero as the cell impedance is matched more closely to the transmission line

and to the cell termination. Therefore, the control chart for the mean

14
voltage reflection coefficient magnitude will not have a lower control

limit.

The cell reflection coefficient was measured at 100 MHz for each of

three commercially available terminations on the cell output port. The

magnitude of the voltage reflection coefficient for the empty cell ranged

from 0.027 to 0.029. This very low value indicated an excellent impedance

match at the cell input and output ports and that the cell itself was operating properly. Therefore, the cell operating point on the voltage

reflection coefficient control chart will be approximately 0.028.

The cell operating condition may also be monitored by an electric field

probe as a check standard to sense the cell field as the indicator quantity.

In principle this is a workable method and has the advantage of monitoring

the cell field directly. The probe would be positioned in the cell at a predetermined point and orientation, and the rf power into the cell set to some reference level. The probe response would then be the parameter used

for statistical control. A deficiency in this scheme is that it depends on

the repeatibility of power settings and the operator's ability to reproduce

the position and orientation of the probe in the cell. The uncertainty in

these experimental factors (power settings and probe positioning) will

introduce additional components of error into the data used for the control

charts. Thus the effectiveness of this method for evaluating the operating

condition of the cell may be reduced. Monitoring the cell performance by

reflection coefficient measurements does not require critically positioning

a detector in the cell or setting an absolute power level; only power ratios

are used. For any such control scheme, the statistical properties of

experimental data on the TEM cell must be determined before the proper

control chart methodology can be developed. Here we are suggesting an approach that is reasonable from both a statistical and an engineering

perspective.

9 . Summary

A TEM cell is employed for creating standard electromagnetic fields.

The most important electrical properties of the cell in establishing a known

electromagnetic field are the cell characteristic impedance, the highest

frequency at which the cell can operate in the TEM mode, and the field

uniformity between septum and side wall. All three properties have been

15
documented by theoretical analyses supported by measurements. Field

nonuniformity introduced by the presence of a test object in the cell has

been evaluated by field mapping in the vicinity of the object.

Electromagnetic fields up to about 150 MHz and 200 V/m can be produced with

an uncertainty of ±0.8 dB,

10. Acknowledgements

The authors thank Dennis L. Friday for his review of and contributions

to Section 8 of this paper and M. L. Crawford for providing original figure

artwork.

11. References

[1] Larsen, E. B. Techniques for producing standard EM fields from 10

kHz to 10 GHz for evaluating radiation monitors. Proc . Symp. on Electromag. Fields in Biolog. Systems; 1978 June; Ottawa, Canada.

96-112.

[2] Donaldson, E. E. ; Free, W. R. ; Robertson, D. W. ; Woody, J. A. Field

measurements made in an enclosure. Proc. IEEE 66(4): 464-472; 1978

April.

[3] Decker, W. F. ; Wilson, W.A. ; Crawford, M. L. Construction of a large transverse electromagnetic cell. Nat. Bur. Stand. (U.S.)

Tech. Note 1011; 1979 February.

[4] Crawford, M. L. Generation of standard EM fields using TEM transmission cells. IEEE Trans. Electromagn. Compat. EMC-16(4): 189-195;

1974 November.

[5] Tippet, J. C. ; Chang, D. C. Radiation characteristics of dipole sources located inside a rectangular, coaxial transmission line.

Nat. Bur. Stand. (U.S.) NBSIR 75-829; 1976 January.

[6] Kanda, M. Electromagnetic field distortion due to a conducting

rectangular cylinder in a transverse electromagnetic cell. IEEE

Trans. Electromagn. Compat. EMC-24(3): 294-301; 1982 August.

[7] Spiegel, R. J.; Joines, W. T.; Blackman, C. F. ; Wood, A. W. A

method for calculating electric and magnetic fields in TEM cells at

ELF. IEEE Trans. Electromagn. Compat. EMC-29(4) : 265-272 ; 1987

November.

[8] Gruner, L. Higher order modes in rectangular coaxial waveguides.

IEEE Trans. Microwave Theory Tech. MTT-15(8): 483-485; 1967 August.

16
9] Tippet, J. C; Chang, D.C.; Crawford, M. L. An analytical and

experimental determination of the cutoff frequencies of higher order

TE modes in a TEM cell. Nat. Bur. Stand. (U.S.) NBSIR 76-841; 1976

June

10] Tippet, J. C; Chang, D. C. Higher order modes in rectangular

coaxial line with infinitely thin conductor. Nat. Bur. Stand.

(U.S.) NBSIR 78-873; 1978 March.

11] Weil, C; Gruner, L. High-order mode cutoff in rectangular

striplines. IEEE Trans. Microwave Theory Tech. MTT- 32 (6) : 638 -641

1984 June.

12] Hill, D. A. Bandwidth limitations of TEM cells due to resonances.

J. Microwave Power. 18(2): 181-195; 1983.

13] Wilson, P. F. ; Ma, M. T. Simple approximate expressions for higher

order mode cutoff and resonant frequencies in TEM cells IEEE Trans.

Electromagn. Compat. EMC-28(3): 125-130; 1986 August.

17
COAXIAL LOAD

SEPTUM (INNER CONDUCTOR)

OUTER SHIELD

DIELECTRIC

SEPTUM SUPPORTS

ACCESS DOOR

COAXIAL CONNECTORS

(BOTH ENDS)

2b

outer conductor

f—

upper chamber

^^

-g-H

Inner conductor

lover chamber

kg-.

side vail

/

2a

Figure 1. A typical NBS TEM Cell,

18 I
35.0

32.

S

30.0

27.5

^ 25.0 o^ 22.5

t-o

z

<^ 20.0

1.0

—J r

APPROXIMATE

T 1—I—r—

9-^1 KflU2w-*J — 2a

I I I I I 10.0

a/b

Figure 2. Capacitance per unit length of a TEM cell as a function of cell geometry [5],

19
•HC

£

0) •H

0)

-1.0 -.8 -.6 -.4 -.2

x/a

.4 .8 1.0

1.6

1.4 .

0) 1.2 -

0)>

•H•P

el

0)

1.0 -

0.8 -

0.6

-1.0

x/«

Fractional distance from center of cell.

Figure 3. Computed and measured values of the total electric

field magnitude, normalized to the TEM cell central

electric field V/d volts per meter [4, 5].

20
2.4

2.2

2.0

1.8

1.6

1.4^

1.2 —

1.0

0.8 I0.6

0.4

1 1 1 1 1 1 1 1 .a/b = 2.0 TMii,a/b = 1.0

^^,.^^'T'E2o . ALL a/b*V^

-? \

—

::

CM

2w

1

2a

1 1 1 1 1 1 1 0.1 0.2 0.3 0.4 0.5

w/a

0.6 0.7 0.8 0.9

Figure 4. Cutoff wavelengths of some higher order modes as a function of TEM cell geometry [8].

21
AMPLIFIER LOW PASS VOLTMETER

SIQNAL QENERATOR

TERMINATION

AMPLIFIER

SIQNAL GENERATOR

LOW PASS

FILTER

DIRECTIONAL COUPLER

Figure 5. Setup for determining the TEM cell standard field from a measurement of: a) septum- to -wall voltage; b) net power

into the cell.

22
Appendix A

Larsen, E. B. "Techniques for producing standard EM fields from 10 kHz to

10 GHz for evaluating radiation monitors."

Reprinted from the Proceedings of the 1978 Symposium on Electromagnetic

Fields in Biological Systems, 1978 June; Ottawa, Canada.

23

TECHNIQUES FOR PRODUCING STANDARD EM FIELDS

FROM 10 kHz to 10 GHz FOR EVALUATING RADIATION MONITORS

E. B. Larsen

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

1. Introduction

The approach used at the National Bureau of Standards (NBS) for evaluating

and calibrating rf radiation monitors is to generate a calculable or "standard"

field and then immerse the probe of the monitor being tested in this known

field. The optimum type of field-generating instrumentation depends on the

frequency band and desired accuracy. A convenient device at frequencies up to

300 MHz is a rectangular "coaxial" transmission line known as a transverse electromagnetic mode (TEM) cell. In the frequency range of 300 to 1000 MHz, a series

of rectangular waveguide transmission lines can be used to generate a field with

approximately known intensity. Also, in the frequency range of 100 to 500 MHz,

high-level fields extending over a larger volume can be produced at an outdoor

field site in the near zone of open-end waveguide radiators. Calibrating fields

above 500 MHz are produced in an anechoic chamber by a series of standard-gain

pyramidal horns. In all cases it is possible to calculate the electric and

magnetic field strengths (and equivalent free-space power density) in terms of

the measured power flow through the cell or waveguide, or the power delivered to

the open-end waveguide or horn antenna. Standard fields can be established over

the entire frequency range up to 10 GHz with an uncertainty less than ± Oj5 dB.

The purpose of this paper is to describe the above techniques, suggesting a preferred approach for each frequency band in the total 10-kHz to 10-GHz range.

2. Description of techniques for generating known field intensities

for calibrating radiation hazard monitors

2. 1 TEM cells, 10 kHz to 300 MHz

A TEM cell consists of a large 50-ohm transmission line in which the

center conductor is a flat metal strip and the outer (grounded) conductor has

a rectangular cross section. Two TEM cells are used at NBS to cover the lower

frequency bands. The larger one has a 0.6 m x 1 m cross section and a length

Reprinted from Proceedings of the 1978 Syitposium on Electromagnetic Fields in

Biological Systems, 1978 June; Ottawa, Canada.

25
of 2 meters. It is accurate for frequencies up to about 150 MHz, but above this

frequency higher-order waveguide modes may cause errors. A smaller cell, with a smaller calibrating volume, is used to cover the 150- to 300-MHz frequency band.

At frequencies sufficiently low so that only the principal wave (TEM mode)

propagates through the cell, a uniform and calculable electromagnetic field is

established [1,2, and 3]. The electric field at these frequencies, midway

between the center conductor and the bottom of the cell, is given by the equation;

v^v/^oIp

*^ D D ^^

where:

E = Electric field strength, volts/meter,

V = Voltage between the center conductor and the outer cell wall, volts,

D = Distance from the center conductor to the bottom of the cell, meters,

P = Power transmitted through the cell, watts,

50 = Characteristic Impedance of the transmission cell,

= Resistance of the terminating load, ohms.

A block diagram of the instrumentation used to produce a standard field by means of a TEM cell is given in figure 1. The field intensity, as calculated

from equation (1), can then be used to calibrate a radiation hazard meter or a small transfer probe with an uncertainty of ±0.5 dB [3 and 4]. An electrically

short dipole 5 cm long is used at NBS as a transfer probe since its response is

essentially flat vs. signal frequency [3, 5, and 6].

2.2 Waveguide chambers, 300 to 1000 MHz

For the frequency range of 300 to 1000 MHz, NBS makes use of a set of three

waveguide sections, each having a rectangular cross section with width-to-height

(aspect) ratio of two to one. A block diagram of the instrumentation is given

in figure 2. The length of each guide "cell" exceeds two wavelengths over the

specified frequency band, in order to create a fairly uniform field within the

guide. Electromagnetic power is transmitted through the guide to a matched

resistive load, and the upper frequency in each guide is limited to that in

which operation is in the dominant TE.„ mode. In this well-known case, the

direction of the electric field vector is across the narrow face of the guide, as shown in the following sketch.

26
SIGNAL

GENERATOR

0.01-300 M Hz

LINEAR

AMPLIFIER

LOW -PASS

FILTER

JEM CELL

TRANSFER DIPOLE OR

HAZARD METER PROBE

RESISTANCE LINE

700 Q/CM

ISOLATED

DC VOLTMETER

Z IN =100M n

50 Q LOAD

OR PAD AND

POWER METER

Figure 1. Instrumentation for generating a known electromagnetic field in^

a TEM cell to calibrate a dipole transfer probe or an rf radiation

monitor

27
RF POWER

OSCILLATOR

300-lOOOMHz

POWER

METER

(INCIDENT)

LOW- PASS

FILTER

20 db

ATTENUATOR

PAD

DUAL

DIRECTIONAL

COUPLER

POWER

METER

(REFLECTED)

WAVEGUIDE CHAMBER

TRANSFER

ISOLATED

DC VOLTMETER

Z IN=100Mn

PROBE

RESISTANCE LINE

700 Q/CM

50 fi LOAD

OR PAD AND

POWER METER

Figure 2. Instrumentation for generating an electromagnetic field in a waveguide chamber.

28
The E-field amplitude (E ) is constant in the y direction and varies sinusoidally

in the x direction. Propagation is in the z direction. Assuming good conductivity

of the waveguide walls, air dielectric, and sinusoidal excitation, the lowest

"cutoff" frequency which will propagate in the guide (f ) is given by:

1.5 X 10

CO

where f is in Hertz and a is the guide width in meters.

CO

The largest or "cutoff" wavelength is given by X = 2a. The wavelength

inside the guide (A ) for these operating conditions is given by:

1wg

1

I o J

1CO

(2)

where X is the free-space wavelength.

The transverse impedance (Z ) of the wave traveling in the guide is: w

-E

Z = -rr-^ = Z w H

wg

ll - (X^/2a)

"2^ (3)

where;

Z = Wave impedance in the guide, ohms, and

Z = Intrinsic impedance of free space = 377 ohms.

The E and H fields at the center of a rectangular waveguide can be obtained

from the equations:

E,=i

^ Z P w z ab and H =

X

2 P

z Z ab w (M

29
where:

E = RMS magnitude of the E field in the center of the waveguide, V/m

H = RMS magnitude of the H field in the guide center, A/m

P = Total power flow in the guide, watts, and

^

2 ab = Cross section area of the waveguide in square meters = 0.5 a . The above equations can also be used to calculate the power required to

produce a desired E or H field in the center of the guide. The value of an

"equivalei

given by:

2 "equivalent" free-space power density at the center of the guide, in W/m , is

2P

ab

f \ Zw

o Z Z o

However, the uncertainty of the calculated field in a waveguide is quite

large (2 dB or more) due to standing waves which may occur in the guide at

certain frequencies. It is thus necessary to clheck the field uniformity in a waveguide calibration chamber by measuring the field strength with a short

dipole probe which has been checked and is known to be essentially flat over the

frequency range in question [3]. A typical calibration curve for an electrically

short dipole is shown in figure 3. The amplitude range covered is 1 to 1000 V/m

2 which corresponds to a free-space power density range of about 0.25 yW/cm to

2

250 mW/cm . By this procedure the uncertainty of the calibrating field within a waveguide cell can be established with an uncertainty of ± 0.5 dB [7 and 8].

2.3 Open-end waveguide radiators, outdoor field site, 100 to 500 MHz

Another approach now being evaluated to cover the 100 to 500 MHz frequency

range involves the use of large waveguide launchers [1 and 3] . This technique

is especially useful for occasional calibrations of larger antennas and probes

which would cause perturbation of the field inside a TEM cell or waveguide

chamber. The open-end guide (OEG) is placed at ground level at an outdoor field

site, with the beam launched vertically upward toward the open sky. A carriage

mounted on a utility pole is used to position the radiation monitor probe or antenna under test (AUT) in the center of this beam. A set of three rectangular

open-end waveguides is available to cover 100 to 500 MHz, each having a two-toone aspect ratio and a height of 2 meters. The AUT or hazard meter probe is

placed in the beam center of the OEG launcher. The distance between the

30
10K r

F.lcrtrli Kli Id

Figure 3. Detected output of a 5 cm dipole transfer probe as a function of field

intensity and signal frequency.

31
+ 90

PLANE OR

H-PLANE

PATTERN

ABSORBING \RESPONSE

MATERIAL

SENSOR

SPHERE

TRANSMITTING

PYRAMIDAL

HORN

ANTENNA

0.5-10 G Hz

DIRECTION

OF

PROPAGATION

S

-90°

RADIATION

MONITOR

Figure 4. Instrumentation for generating a known electromagnetic field near a pyramidal horn to calibrate and evaluate an rf radiation monitor.

32
transmitting aperture and the calculated field point is adjustable (at NBS) up

to 7 meters above the waveguide flange. The on-axis field intensity can be

calculated in terms of the net power delivered to the transmitting OEG and the

measured gain of the OEG launcher determined by a technique similar to those

used for pyramidal horns [3 and 9 to 14].

2.4 Pyramidal horn antennas used in an anechoic chamber, 500 MHz to 10 GHz

The approach used at NBS to establish a standard field at frequencies above

500 MHz is to calculate the radiated field intensity in the near zone of standardgain horn antennas. These antennas consist of a series of rectangular pyramidal

horns. Figure 4 shows a sketch of the instrumentation used to calibrate the

amplitude-versus-frequency response of a radiation monitor or to evaluate the

various pattern responses.

It would be possible to calibrate an rf radiation monitor in far-zone,

plane-wave fields [13 and l&j ; however, the transmitter power levels required

2 to produce intense fields (up to 10 mW/cm or higher) would generally exceed

1 kW. An alternative approach is used at NBS in which lower power rf sources

are adequate. A simple equation for accurately calculating the field strength

close to a horn aperture will be given later. However, a brief discussion of near- vs. far-zone effects is presented first.

Plane-wave conditions exist in the far zone of a transmitting antenna. The

E and H field vectors are orthogonal to each other, and both are perpendicular

to the direction of propagation. In addition, the ratio of the magnitudes (E/H)

is a constant given by the free-space wave impedance (Z ); that is:

Z = (E/H) of free space ^ 377 ohms. (6) o

In tliis case E and H also have a definite relation to the average power density

given by:

E^ 2 S = ^ = Z H^ (7) o Z o

o

Equation (7) is not necessarily valid for conditions other than a single

plane wave. It is important to remember that the usual commercial radiation

monitor does not actually measure power density, even though the dial indication

is in units of mW/cm~. Yet, taking some liberties with technical exactness, it

is possible to convert a measured E or H field to an "equivalent" plane-wave

power density. However, this common practice is not necessarily valid.

33
OiH' w>jy Lo avoid llic lonfusioii ol pijwcr dfiisiLy iin i I .s would I)L' for the

American National Standards Institute (ANSI) to state the maximum permissible

field values in terms of the quantity actually measured; namely, E field in V/m

or H field in A/m. For isotropic probes having three mutually orthogonal

sensors, the total field magnitude is measured in terms of orthogonal field

components by the following equations:

/ 2 2 2 E^ ^ T = / E + E + E and H^ ^ ^ total / X y z total H+H+H 2 2 z (8) X y z

The total field as defined by equation (8) is also called the Hermitian

magnitude of E or H. It is closely related to the electric or magnetic field

energy density. The familiar power density or Poynting vector, however, is not

directly relatable to the energy density except for a simple field structure

such as a plane wave.

If electromagnetic fields always occurred in simple plane-wave configurations, the question of which physical quantity to measure and how to relate

this quantity to biological effects would be more easily answered. The regions

very close to radiating systems are most likely to approach hazardous levels.

Unfortunately, they are also characterized by complicated field structure

including reactive (stored) and propagating fields, multipath reflections,

standing and traveling waves, irregular phase surfaces, and unknown field

polarization.

In addition to the ambiguity of using power density units, a complication

arises when calculating field strength very close to a transmitting antenna,

known as near-zone gain reduction. Unlike a field traveling in a waveguide, the

electromagnetic field across a horn aperture has a somewhat spherical (rather

than planar) wavefront. The phase at the rim of the horn lags that at the

center, causing a nonequiphase front across the aperture, which reduces the

effective gain in the near-field (Fresnel) region. A further reduction occurs, even for an equiphase aperture, due to the difference in distance between the

various elements in the radiating aperture and the on-axis field point in question. Both of these "defects" reduce the power density to less than that pre--

dieted by the simple far-zone, inverse-square relation of a distant point source. In the far-zone (Fraunhofer) region of an antenna, plane-wave conditions will exist, and:

34
S = -^ (9)

4TTd

where:

2

S = On-axls power density of the radiated field, W/m , P = Power delivered to the transmitting antenna, W,

G = Gain of the antenna, and

d = Distance from the antenna aperture to the field point, m. A computer program was written at NBS for the effective gain of a pyramidal

horn at any distance. It is used so that high-level standard fields can be

produced with low-power transmitters. It involves an integration of the amplitude and phase contribution from all the aperture elements of the transmitting

antenna. An approximation, in terms of Fresnel integrals, , was compared with this

direct numerical integration and found to agree within ±0.1 dB. The Fresnel

integrals are useful for solving many problems in optics and fields near aperture antennas. Unfortunately, the integrals cannot be evaluated easily

without a computer. Graphical constructions and tables of Fresnel integrals are

sometimes employed, but both are tedious to use. Mathematical approximations

have also been given for the Fresnel integrals; some of these involve infinite

series expansions and others, a combination of trigonometric and algebraic

functions. However, recently a short paper was published by E. V. Jull [17]

which gives tables of values for near-zone gain reduction factors of a pyramidal

horn. His technique begins with an equation for the far-zone gain of a rec- tangular in-phase aperture. Then two separate factors, R^ and Rp,, are calculated for the gain reduction due to the H-plane and E-plane flare of the horn.

The author has generated simple polynomial expressions, similar to the

various algebraic equations which have been published for approximating the

Fresnel integrals, for determining these R^ and R„ factors with a calculator.

The pertinent horn dimensions used in the equations are shown in figure 5,

where the dimensions have been normalized to wavelengths by letting:

A = f. B=i, L„.^,L^3,andD.f . (10)

Here:

a, b, I and I - Horn dimensions, meters,

rl b

d = Distance in front of the horn aperture Co the field point, meters, and

A = Free-space wavelength, meters.

35
Figure 5. Sketch of a pyramidal horn. All dimensions shown are in wavelengths,

36
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ino

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37
After defining:

a = A and B = B (11)

the two gain reduction factors R and R„, in dB, are:

H L

R^ = (O.Ola) (1 + 10.19a + 0.51a^ - 0.0970"^)

(Due to H-plane flare of horn) (12)

Rg = (O.IB )(2.31 + 0.0533)

(Due to E-plane flare of horn) (13)

by:

Finally, the theoretical gain of the horn (near zone or far zone) is given

GAIN, dB = 10 log(AB) + 10.08 - R^ - R, (14)

The first two terms of equation (14) represent the gain of an in-phase

aperture. For a sectoral horn, either R^ or R would be zero. The calculated

values of R^ and R , using equation (14) for pyramidal horns, were found to

agree with reference [17] within 0.05 dB for R^ < 5 dB and Rp < 6 dB, assuming

D > L and D > L„. The calculated gain values given in NRL Report 4433,

n t, reference [18], for the 15-cm band horn were compared with equation (14) at

frequencies of 1.6, 2.0, 2.4, and 2.8 GHz, using Brauns near-zone correction

curves. The distances involved were 1.2 m, 1.8 m, 2.4 m, and <». The largest

discrepancy found for any of the 16 comparisons was less than 0.1 dB. The

average value of the differences between the NRL report and the calculations of

equation (14) was only 0.001 dB.

The above approach was also tested experimentally at two frequencies using

two different standard-gain horns covering the frequency range of 450 to 1100 MHz.

The horns were first calibrated at NBS by the well-known, three-antenna method

[10, 11, 12], and these gains were compared with that given by equation (14).

The distance used was about 5 meters. The gain difference between the experimental determination and the theoretical value was 0.02 dB at 470 MHz and

0.21 dB at 850 MHz.

38
Another near-zone gain calibration of the large pyramidal horn (1.225 x 0. m aperture) was performed at 750 MHz in order to compare experimental results w. those calculated from equation (14). Figure 6 gives a graph of the data obtaint

as a function of distance from the horn flange. As can be seen, the agreement

between the measured near-zone values and the theoretical equation was within

0.5 dB for all distances checked up to 7 meters above the horn flange.

For calibrating a radiation monitor, the on-axis field intensity of the

transmitting horn is calculated in terms of the net power delivered to the

pyramidal horn, using equation (9) . The proper gain value to use for any

distance is obtained from equation (14). The overall uncertainty in the 0.5 to

10 GHz frequency range is estimated to be within 0.5 dB3. S ummary and Conclusions

The method used at NBS for calibrating an rf radiation monitor involves

inserting the probe in a field of known magnitude. The optimum instrumentation

to use for generating this field depends on the frequency and required accuracy.

Up to 300 MHz a TEM cell is convenient since it produces a uniform electromagneti

field. Both the E and H field strengths can be calculated in terms of the plate

spacing and measured voltage (or power and cell characteristic impedance). This

known field can then be used to calibrate a radiation hazard meter or a small

transfer probe.

In the 300 to 1000 MHz range a series of rectangular waveguide transmission

lines is used for a calibrating field. The E and H magnitudes can be calculated

approximately in terms of the guide dimensions, power flow and frequency. However

the uncertainty of the calculated field is relatively large and it is necessary to

check the amplitude and uniformity by using a small transfer probe which is known

to be flat vs frequency across the 300 to 1000 MHz band. Another approach now bei

evaluated at an outdoor field site, for generating high-level fields in the 100 to

500 MHz range, makes use of large open-end waveguide launchers. This technique is

required only for larger antennas and probes which would cause perturbation of the

field inside a TEM cell or waveguide chamber.

Calibrating fields above 500 MHz are generated in a new anechoic chamber at

NBS, using a series of standard-gain pyramidal horns. Intense fields can be

produced very close to the antenna aperture, even with low power transmitters.

Simple algebraic equations are given for accurately calculating these near-zone

39
fields. The procedure involves a computation of the intensity produced by an in-phase aperture and then applying two near-zone correction factors. The

values of these gain reduction factors depend on frequency, horn dimensions

and the distance to the field point.

It is .concluded that radiation monitors (and other antennas) can easily

be calibrated over the entire frequency range of 10 kHz to 10 GHz with an uncertainty less than ±0.5 dB. It is possible to produce high level fields

(greater than 200 V/m) using only low-power rf sources (less than 50 W)

.

4. Refer enc es

[1] M. L. Crawford, "Generation of standard EM fields using TEM transmission

cells," IEEE Trans. Elec. Compat., Vol. EMC-16, pp. 189-195 (Nov. 1974).

[2] G. A. Skaggs, "High frequency exposure chamber for radiobiological

research," NRL Report 2218, Naval Research Laboratory (Feb. 1971).

[3] R. C. Baird, "Methods of calibrating microwave hazard meters," Proc . of

symposium on Biologic effects and health hazards of microwave radiation,

Warsaw, Poland, pp. 228-236 (Oct. 1973).

1 4] M. L. Crawford, "Generation of standard EM fields for calibration of

power density meters, 20 kHz to 1000 MHz," NBS Interagency Report,

NBSIR 75-804, 40 pages (Jan. 1975).

[5] R. R. Bowman, "Some recent developments in the characterization and

measurement of hazardous electromagnetic fields," Proc. of symposium on Biologic effects and health hazards of microwave radiation, Warsaw, Poland,

pp. 217-227 (Oct. 1973).

[6] P. S. Ruggera, "E- and H-field instrumentation and calibration below

500 MHz," Proc. of symposium on Biological effects of electromagnetic

waves, USNC/URSI meeting, Boulder, Colo., pp. 281-296 (Dec. 1976).

[7] E. Asian, "Simplify leakage probe calibration," Microwaves, pp. 52-54,

57 (Dec. 1975).

[8] D. Woods, "Standard intensity electromagnetic field installation for

calibration of radiation hazard monitors from 400 MHz to 40 GHz,"

Non-ionizing radiation. Vol. 1, pp. 9-17 (June 1969).

[9] R. R. Bowman, "Field strength above 1 GHz: measurement procedures for

standard antennas," Proc. IEEE, Vol. 55, pp. 981-990 (June 1967).

[10] W. H. Kummer and E. S. Gillespie, "Antenna measurements—1978," Proc.

IEEE, Vol. 66, pp. 483-507 (April 1978).

40
[11] "Microwave antenna measurements," Edited by J. S. Hollis, T. J. Lyon, and

L. Clayton, Jr., Scientific Atlanta, Inc., Chapter 8, "Measurement of gain,"

pp. 8-1 to 8-29 (July 1970).

[12] A. C. Newell, R. C. Baird and P. F. Wacker, "Accurate measurement of

antenna gain and polarization at reduced distances by an extrapolation

technique," IEEE Trans. Ant. & Prop., Vol. AP-21, pp. 418-431 (July 1973).

[13] "Electromagnetic horn antennas," Edited by A. W. Love, IEEE Press,

selected reprints prepared under sponsorship of the IEEE Ant. & Prop.

Society, 453 pages (1976).

[14] "Calibration technique replaces large antenna measurement ranges,"

Communications Designer's Digest, pp. 30-31 (Dec. 1969).

[15] H. I. Bassen and W. A. Herman, "Precise calibration of plane-wave

microwave power density using power equation techniques," IEEE Trans.

Microwave Theory and Techniques, Vol. MTT-25 (Aug. 1977).

[16] M. Swicord, H. Bassen, W. Herman, J. Duff and J. Bing, "Methods and

instrumentation for the evaluation and calibration of microwave survey

instruments," Proc. of symposium on Biological effects of electromagnetic

waves, USNC/URSI meeting, Boulder, Colo. pp. 297-309 (Dec. 1976).

[17] E. V. Jull, "Finite-range gain of sectoral and pyramidal horns,"

Electronic Letters, Vol. 6, pp. 680-681 (Oct. 15, 1970).

[18] W. T. Slayton, "Design and calibration of microwave antenna gain standards,"

NRL Report 4433, Naval Research Laboratory (Nov. 9, 1954).

41

Appendix B

Donaldson, E. E.; Free, W. R. ; Robertson, D. W. ; Woody, J. A. "Field

measurements made in an enclosure."

Reprinted with permission from Proc . IEEE 66(4) :464-472 ; 1978 April.

Copyright (c) 1978 by IEEE.

43

Field Measurements Made in an Enclosure

ERNEST E. DONALDSON, member, ieee, WILLIAM R. FREE, member, ieee.

DOUGLAS W. ROBERTSON, senior member, ieee, and JIMMY A. WOODY, member, ieee

Abimct-This paper presents a survey of radiated emission and sus- ceptibility measurement methods which can be employed to evahjate

the electromagnetic interference (EMI) characteristics of electronic systems. The paper reviews conventional anechoic-chamber and shielded- endosure-measurement methods, as well as alternate methods which

have been developed to circumvent measurement problems associated with radiated measurements in conventional shielded enclosures. Alternative methods presented include hooded antennas, mode perturbation, statistical sampling, transverse electromagnetic (TEM) cells, and

other techniques such as pa^aIle^plates,low-5-enclosu^es,and long-wire- antennas. A qualitative comparison of each technique in terms of advantages and limitations is also given.

I. Introduction

THE ACHIEVEMENT of electromagnetic compatibility

(EMC) in electronic systems is highly dependent upon the

availability of accurate and reliable measurement tech- niques for defining and controlling system electromagnetic

interference (EMI) characteristics. Over the last two decades,

considerable effort has been directed to the development of

better EMI measurement techniques. In particular, emphasis

has been placed on test methods for performing radiated emission and susceptibility measurements which provide a valid description of system EMC/EMI performance in actual operating environments.

There are three common test environments which are employed for radiatedemission and susceptibility measurements:

open-field, shielded-anechoic<hamber, and shield ed-enclosure. Each of these environments has advantages and disadvantages

for radiated measurements, both technical and economical

Open-field measurements constitute a fairly straightfoward

approach to evaluating system EMI performance, but require

a field measurement site, provide no isolation between the

test setup and external environment, and are subject to weather

conditions. The shielded anechoic chamber will simulate an

open-field environment while providing isolation between the measurement setup and external envirorunent. However,

anechoic-chamber costs can become prohibitive, particularly

as size requirements increase. A shielded enclosure will satisfy isolation requirements but can produce measurement problems

due to multiple reflections, resonances, distortion of radiated

fields, and changes in the emission and susceptibility characteristics of systems under test. However, primarily because of

economic considerations, the shielded enclosure is widely used

to perform radiated type measurements, and efforts to develop

improved EMI measurement methods have revolved about the shielded enclosure test environment

This paper briefly summarizes current measurement methods

which are employed in performing radiated type EM I measureManucript received October 17, 1977. The authors are with Engineering Experiment Station, Georgia In- stitute of Technology, Atlanta. GA 30332.

ments, and provides a qualitative comparison of the advantages

and limitations of each technique. These methods include: 1) anechoic chambers; 2) shielded enclosures;3)hooded antennas;

4) mode perturbation; 5) statistical samples; 6) TEM cell; and

7) other methods such as parallel-plates, long-wire-antennas, and

low-C-enclosures.

II. Measurement Methods

A. Anechoic Chamber

A shielded anechoic chamber [1], (2) represents the best approach for performing indoor radiated electromagnetic field measurements. This type chamber provides a nearly free-space

test volume isolated from the outside environment. A good

quality shielded enclosure should provide 100 dB or more of attenuation over the IjD-kHz to 20-GHz frequency range. Hence,

signals present in the outside environment will have little ef-

*"ect on measurements made in the chamber, and signals radiated

in the chamber during testing are contained within the chamber

and thus will not interfere with external operations. Also, a shielded anechoic chamber eliminates the need for FCC approval

to radiate, and can reduce the possibility of compromising

classified radiation characteristics. The anechoic material mounted on the inside walls, floor, and ceiling of the shielded enclosure absorbs the energy radiated inside the enclosure and

provides a nearly reflection-free volume for performing measurements. The disadvantages associated with shielded anechoic

chambers are the high cost of the chamber and the low-fre- quency performance limitations of currently available an- echoic materials. A rule-of-thumb estimate of shielded anechoic

chamber cost is SI 5 to S2S per cubic foot of interior usable space (not including the interior volume occupied by the an- echoic material) for typical laboratory size enclosures. The

cost per cubic foot tends to decrease as the chamber size increases, but the total cost is quite high for any size enclosure.

In general, the anechoic material must be at least one-quarter

wavelength thick for any appreciable absorption of radiated energy to take place. An exception is ferrite absorbing material; however, the cost of the material is so high (approximately

SI 00 per square foot) that it is not considered economically

feasible for anechoic chamber designs. Since wavelength is inversely proportional to frequency, the thickness of the an- echoic material must be increased as the low-frequency limit

is decreased. For example, to obtain any appreciable absorption at 100 MHz, the anechoic material would have to be at least 30 in thick, and at 50 .MHz the material would have to be 60 in thick. As the thickness of the anechoic material in- creases, there are three related factors which b'mit the low-fre- quency threshold of anechoic chambers: 1) the cost increases: 2) the size of the shielded enclosure must increase to retain the same test vohJme;and 3) it becomes moredifficult to mount

the material and prevent sagging of the pyramids. For these

(C) 1978 IEEE,

i^ril.

Reprinted with permission frcsm Proc. tfff 66(4): 464-472; 1978

45
DONALDSON et al.: FIELD MEASUREMENTS IN AN ENCLOSURE

reasons, most existing anechoic chambers have low frequency

limits of 200 MHz or higher. A few large chambers have 100- MHz low-frequency thresholds.

It is not possible to specify the absolute accuracy of measure- ments made in a shielded anechoic chamber (3J without specifying the test setup in some detail, since the measurement ac- curacy is dependent on several test parameters, including:

1) location of the test setup in the chamber; in particular, distance from the internal surface of the chamber;

2) directivity of transmitting and/or receiving antennas;

3) separation between source and receptor;

4) frequency; and

5) reflectivity of the anechoic material.

In addition, the accuracy is dependent upon the magnitude of

the energy being measured relative to the maximum energy

being radiated in the chamber. This can be illustrated by considering an antenna pattern measurement being made in an

anechoic chamber with a reflectivity level of 40 dB. On boresight, where measurements of maximum energy levels are being

performed, reflections will have little effect on measurement

results. Thus maximum errors on the order of 0.1 dB can

be expected. However, at points on the pattern which are below

the maximum, reflected energy can significantly influence measurement results. For example, at a level of - 30 dB relative to the beam maximum, a maximum error of 3.3 dB is possible.

It should be pointed out that a similar situation exists on out- door ranges due to reflections and the background interference environments which exist on the ranges.

• YIELDED E»aOS>;«t ni^LSFig. 1. Diagram of a typical measurement setup in a shielded enclosure showing multiple-signal paths.

SHIELDED

EhClOSuBI

Fig. 2. Diagram of a hooded-antenna measurement setup in a shielded

enclosure.

B. Shielded Enclosures

A large percentage of indoor radiated electromagnetic-field measurements are made in shielded enclosures without anechoic

material on the inside surfaces of the enclosure. The shielded enclosure provides a high degree of isolation from the electro- magnetic environment and local sources of interference. However, the reflections from the enclosure walls significantly affect any radiated measurements made in the enclosure. Measurement results are extremely sensitive to the size and

shape of the enclosure, the location of the test setup in the

enclosure, the spacing between the equipment under test and

the test antenna, and the presence and location of personnel

and test equipment in the enclosure. A typical measurement setup in a shielded enclosure is illus- trated in Fig. 1. The diagram also illustrates some of the multiple signal paths which exist with this measurement configuration. It is apparent from the figure that a great deal of radiated energy is reflected into the receiving antenna by the enclosure

walls. This energy would not be present if the measurement

setup was in the open-field or in an anechoic chamber. It is also apparent from the figure that the phase of a signal arriving at the antenna over any given reflection path relative to the phase

of the signal arriving at the antenna over the desired signal path

will be influenced by the frequency of the signal, the size of

the enclosure, the location of the measurement setup in the

enclosure, and the spacing between the radiating source and

the antenna.

If the majority of the reflected energy arriving at the receving antenna over all of the reflection paths is of the proper phase

to cause a cancellation with energy arriving at the antenna over the desired signal path, the measurement result will be less than would have been obtained in the open-field or in an anechoic

chamber. By the same token, if the majority of the reflected

energy is of the proper phase to add to the energy arriving at the antenna over the desired signal path, the measurement

result will be greater than would have been obtained in the open field or in an anechoic chamber. When the magnitude

of the reflected energy isequal to or greater than the magnitudeof the desired signal path energy, very large errors may be en- countered.

Results from experimental investigations [4]-[6] have shown

that errors as great as ±40 dB are possible in radiated measure- ments made in shielded enclosures at frequencies greater than

50 MHz. Results from these investigations have also shown

that a change of one-half inch in the spacing between the test antenna and the equipment under test could change measure- ment results approximately 15 dB. It is obvious that measure- ments made under these conditions are of little value and the

possibility of correlating these measurements with measure- ments made in the open-field is small

C. Hooded-Antenna Technique

A search for a technique which would reduce the multipath

reflections in shielded enclosures without the cost, fabrication,

and space disadvantages associated with an anechoic chamber

led to the development of the hooded antenna technique

[4) -[6] . This concept, illustrated in Fig. 2, involves shielding the test antenna in all but the desired signal direction by means

of a metal hood. The inside of the hood is lined with an an- echoic material to reduce reflections from the hood to the test antenna. The shielded enclosure wall behind the equipment

under test is also covered with absorbing material to prevent

reflections on or near the desired signal path. The concept

of the hooded antenna differs little from the concept of an anechoic chamber. The absorber-lined walls of the hood,

together with the absorber-lined enclosure wall opposite the

46
PROCEEDINGS OF THE IEEE. VOL. 66, NO. 4. APRIL 1978

aperture of the hood, look essentially the same to the test antenna as the six absorber-lined walls of an anechoic chamber.

The edge diffraction around the aperture of the hood may be minimized by extending shaped absorbing material beyond

the edge of the hood shield. Two hooded antennas covering the frequency range from

200 MHz to 12 GHz have been fabricated and evahiated. A

UHF hooded antenna designed to operate over the 200 to 1500-MHz frequency range consists of a balanced log-conical antenna and a cylindrical hood 24 in in diameter and 49 in long. The hood was fabricated with l/8-in sheet aluminum

and was lined with Eccosorb® NZ-1 absorbing material This

is a ferrite absorbing material, approximately 1 in thick, with

good reflectivity characteristics from 200 MHz to 15 GHz. A

microwave-hooded antenna designed to operate over the 1-to I2-GHz frequency range consists of a microwave log<onical antenna and a cylindrical hood 8 inches in diameter and 19.5

in long. This hood was also lined with NZ-1 absorbing material The use of these hooded antennas in a shielded enclo- sure with one wall covered with absorbing material reduced the

errors in "boresight" measurements from ±40 dB to ±3 dB. Subsequent investigations indicated that planar cavity-backed

spiral antennas with shorter hoods gave equivalent results (6) , [71. As discussed for the anechoic chamber, the accuracy of radiated measurements made in a shielded enclosure with

the hooded antenna technique is dependent on various test parameters and the magnitude of the energy being measured

relative to the maximum energy being radiated in the enclosure.

D. Mode Perturbation Because of convenience and economy, acoustical engineers

have historically used reverberation chambers as an alternative to anechoic chambers for the measurement of sound power

output I8)-llO]. Subjective experience with echoes led to early recognition that the reflective nature of the reverberation chambers precluded measurement of the radiated sound in- tensity at specific ranges or angular orientations. In order to provide a degree of field uniformity in the measurement area, Sabine (11) introduced the concept of placing a moving

reflector within the chamber. Since then other investigators

have developed techniques based on rotating field perturbators

and/or microphone-source movements. As a result, significant

improvements in energy density uniformity were obtained.

1) Strirred Mode: Recognition of a degree of similarity be- tween electromagnetic field measurements and the approach

of acoustical engineers to sound field measurements in rever- beration chambers has led to adoption of room perturbation

concepts for electromagnetic measurements in shielded enclosures. In one approach (12), ( 13] , a combination of mode

and spatial perturbation is obtained by rotating a probe antenna and a large reflecting surface within the enclosure. In effect, the perturbator "tunes" the enclosure through many resonant modes at the test frequency.

As the perturbator varies the room mode structure, the measured electric field will pass through a maximum value (^max) foi^ each sequentially excited mode. Since the losses remain relatively constant, fmax w'U exhibit considerable consistency from mode to mode and its value can be readily deter- mined. £"max is related to the power radiated iPnd) by the source by P,^^ =*(w)£"max* where Jfc(w) is a measured cal- ibration quantity for each specific room. Each enclosure can

b€ calibrated for the frequency range of interest by driving an

omnidirectional antenna with a CW source, setting the radiated47

power with a directional wattmeter, and recording fniax f*5' each calibration point

The technique has been found to be suitable for frequencies

slightly above the first resonant frequency of a room (approximately 35 MHz for a 3m X 6m X 6m enclosure) to over

1 GHz. The source or equipment under test (EUT) can be ar- bitrarily located at any point within the enclosure except in close proximity to the walls. The field perturbation technique provides for simple and

straightforward determination of the radiated power, but, for many cases of interest (specifications and standards) the field intensity at a specified range is required. A good approximation to the field intensity (Eq) for open-field conditions can

be obtained from P^d by assuming hemispheric radiation and

by use of the relation

_ >/607^

Eq =

r V/m (1)

where /'^ad is in watts and the distance in meters (R) is larger

than X/6 at the frequency of interest. The method has been tested for both simple and complicated

sources and the calculated field intensities based on the en- closure measurements have consistently been within ±6 dB of

open-field values from 45 MHz to 1 GHz.

A similar radiation measurement approach [14] -{171 is based on the use of two large rotating vanes, each rotating at uncorrelated speeds of approximately i.O to 2.0 r/s around

axes perpendicular to adjacent walls. When a radiating source

is introduced into the enclosure and the vanes activated, the

field, within the enclosure is perturbed in both time and space.

If the resultant field is sampled by means of a test antenna and

a detector with a low-pass filter for removing the vane rotational components, the measured field will exhibit considerable uni- formity. Field uniformity to within ±0.5 dB has been obtained

from 1 to 18 GHz for any antenna orientation and for any

location of the source or test antenna within the working area of the enclosure. Of course the electric field rapidly falls to zero near the metallic walls as the boundary conditions are

satisfied. Again, a room calibration coefficent curve, kicj) -

P^d/^avg. can be plotted for the frequency range of interest. Once the curve is obtained, a single measurement of the field

at any point in the room will suffice for determination of the

radiated power of other equipments at given frequencies.

The similarity of the two approaches is evident with the dif- ference in the E„^ of the first case and the fgvg above being

accommodated within each of the empirically established curves. Also, the reported results indicate that the 45-MHz to 1-GHz frequency range for the first case complements the 1-GHz to 18-GHz range of the second. The parameters which

define the frequency range limits have not been determined,

but they are undoubtedly related to the perturbator and room

dimensions in terms of test frequency wavelength.

A number of factors related to perturbed mode enclosures measurements should be noted. The technique is relatively insensitive to differences in the spectral content of the radia- tions from the reference source and the EUT because the

losses (skin current, etc.) in an individual enclosure are nearly constant for small frequency differences. However, the tech- nique requires an inherent assumption that the total power

radiated into, and dissipated within, the enclosure is equal

to that which would be radiated by the EUT under free space

or operational conditions. In a related sense, the use of * calibration curve for a specific enclosure requires the assurop-
DONALDSON tt si.: HELD MEASUREMENTS IN AN ENCLOSURE

tTRUCTURC UNDER TEST - z

. INTERNAL ANTENNA Y

^

REFERENCE ANTENNA I 1

MM

nr SMEiioco ENCIOCURE

-LONOWIRE TRANSMirrmc ANTENNA

DOUBLE STUB TUNER

DOUBLE

STUB TUNER

RF GENERATOR

PAOOLE WHEEL TUNEAS

RF RECEIVER

Fsg. 3. MIL-STI>-1377 type test setup.

tjon that the losses within the enclosure remain the same for each EUT configuration tested. Some quantitative data directed to bounding the above two assumptions has been reported

[18], but as yet the significance of the assumptions has not been determined. It is conceivable that an empirically deter- mined calibration factor could be obtained for individual en- closures which would permit comparison of EUT radiated power

in an enclosure to that in a free space or anechoic chamber en- vironment.

In spite of the assumptions and the limited quantitative data reported to date, the reverberation enclosure technique does

provide an attractive concept for a simple and repeatable shielded enclosure measurement technique for the determination of radiated power.

2) _Tuned Mode: A shielded-enclosure measurement tech- nique which involves mode tuning rather than mode stirring was developed by Jarva [19] and has been incorporated into MIL-STI>1377 (Navy) (20, 21] as a test method for measuring

the shielding effectiveness of cables, connectors, enclosures,

and filters over the frequency range of 1 to 10 GHz. This tech- nique involves placing the component or equipment to be tested inside a multimoded, tuned shielded enclosure in which paddle

wheel .tuners are used to redistribute the energy within the en- closure. A typical test setup is shown in Fig. 3. To determine shielding effectiveness, the equipment to be

tested is placed within the enclosure and the enclosure is en- ergizrd at specific frequencies of interest via the long-wire input antenna. The paddle wheel tuners are then adjusted to maximize the power coupled through the test specimen equipment

to the external receiver. Because the equipment under test is subjected to a composite field pattern, both in position and

polarity, many local maxima are possible as the tuners are moved. The largest local maxima is indicative of the intrinsic shielding effectiveness of the test specimen equipment. Comparison of this value with the maximum power coupled to the receiver via the unshielded reference antenna yields the

shielding effectiveness of the test specimen.

The MIL-STf>-1377 type test method has several advantages

as a shielding effectiveness test technique. Elaborate test equipment is not necessary, and testing can be performed with an

enclosure only slightly larger than the equipment under test. Test techniques are relatively simple, and experience to date indicates that measurements are repeatable within a few decibels. A basic limitation of the technique is that its use is re-

-12 -10 -• -« -4 -2 FIELD STRENGTH IdBWm^l

(*)

1.0 -

-8 -S -4 -2 +2 44 FIELD STRENGTH (dBm/m^)

(bl

Fig. 4. Sample statistical descriptions of case emission and case sus- ceptibility.

stricted to frequencies above a few hundred megahertz in order

to obtain a sufficient number of modes.

Efforts are currently underway to modify the MIL-STD- 1377

type test method to permit its use in performing radiated emissions and susceptibiLty type measurements |22].

E. Statistical Sampling

It is often necessary to predict, using measured case emission

and case susceptibility data, the mutual interference between

electromagnetic emitters and receptors. Current techniques

for collecting the necessary data involve emission or susceptibility measurements made at a fixed point with respect to the equipment case orientation, or at a point at which the maximum

case emission or susceptibility level is detected. Both of these techniques have definite disadvantages with regard to mutual

interference predictions. Measurements made at either selected

fixed points around an equipment case or at the point of maximum emission or susceptibility are not adequate for defining

emission or susceptibility levels at the infinitude of other case

orientations. To account for all possible emitter-case to receptor-case orientations which may exist in a typical equipment

lash-up would seemingly require that emission and susceptibility

levels be measured in a three-dimensional pattern centered around an equipment case. Unfortunately, this approach would

lead to an infinite number of measurements to completely

describe case emission or susceptibflity levels. Another approach which has been investigated involves a

statistical description of case emission and case susceptibility

(23] -(28]. The concept of this approach is illustrated in Fig.

4. Fig. 4(a) illustrates a probability distribution function

statistically describing the strength of a three-dimensional

radiated field at a given frequency which might surround a particular culprit case. The function shows the probability that the field strength at any point 35 in from the center of the

case would be less than any given level if the case were randomly

oriented in three space. Fig. 4(b) shows a corresponding prob48
PROCEEDINGS OF THE IEEE. VOL. 66. NO. 4. APRIL 1978

t

>- JO 55

is

• ANECHOIC CHAMBER

• LABORATORY SITE

. SHIELDED E^LOSURE

HOOOED

-16 -12 -S -4 RELATIVE POWER LEVEL (dbl

Fig. S. Average cumulative dbtributions for a simulated source in three envbonments.

CENTER

PLATE

E: CENTER PLATE IS SUPPORTED WITH

OlELECTniC RODS AND SPACERS Rg. 6. Geometry of TEM cell.

ability distribution function statistically describing the susceptibility of a particular victim case in a radiated field of the same

frequency. Here, the function yields the probability that the

victim case will fail in a field the strength of which exceeds any

given level if the case is randomly oriented in three space. The probability of mutual interference can be predicted from

the data of Fig. 4. For example, in Fig. 4(b), the probability

of a failure by a case in a field corresponding to a level of -4

dBm/m^ is 0.3. If the culprit is at a range of 35 in, the probability of a field corresponding to a level of at least -4 dBm/m'

at the victim case is 0.6. Therefore, the joint probability of a failure is given by the product, 0.3 X 0.6 = 0.18. In the above example, a 35-in separation between source and

receptor cases was assumed for purposes of illustration. The

probability of mutual interference at a separation of other than

35 in can be calculated by appropriately modifying the field strength 1evek of Fig. 4(a) using the distance/inverse-squafe

law for radiated fields. Also, where the sample calculation as- sumed only one interference frequency, the probability of mutual interference can be determined for any number of in- terferences frequencies.

A power distribution measurement technique for statistically

describing case emissions in the manner indicated in Fig. 4(a) has been evaluated (23 J. An experiment was conducted in which measurements were made on simulated culprit sources

to determine probability distribution functions for the source

emissions. This experiment consisted of statistically describing the three-dimensional radiated fields about each of the sources at different frequencies under three different environmental conditions: an anechoic chamber, a typical work loca- tion in a laboratory, and a shielded enclosure. Shielded-enclosure measurements were made using the hooded antenna

technique previously described.

Fig. 5 shows the probability distribution functions established from measurements at one frequency for one simulated source

in the anechoic chamber, laboratory, and shielded enclosure. The distribution functions were expressed in terms of relative power levels for purposes of convenience in illustrating the

characteristics of the functions; conversion to absolute levels of field strength can be accomplished through the use of ap- propriate calibration factors. The data of Fig. 5 and data re- corded on other simulated sources and at other frequencies show

that the power distribution measurement technique can be used

to obtain highly repeatable radiated emission measurements in a laboratory and/or shielded enclosure that routinely lie within

±1.5 dB of corresponding values measured in a free-space or anechoic chamber environment.

Used in conjunction with appropriate statistical descriptions

of case susceptibility, case emission data accumulated with the power distribution measurement technique could be used to predict directly the probability of electromagnetic compatibility (or incompatibility) in actual field equipment lash-ups.

Although not yet reduced to practice, the concept of statistical based measurements of case emission and case susceptibility

represents a potentially significant method for defining mutual

interference between emitters and receptors.

F. TEM Cell The transverse electromagnetic (TEM) cell is an expanded

coaxial transmission line operated in the TEM mode. As shown

in Fig. 6, it consists of a rectangular metallic box enclosing a flat-platecentef condCictor. The TEM cell provides shielded enclosure isolation but does not introduce the reflection problems associated with the conventional shielded enclosure. The

TEM mode field characteristics between the center plate and

either the top or bottom wall of the cell are used to establish a relatively uniform TEM field for susceptibility testing or to couple the EUT's emissions to the cell's measurement ports. The geometry of the TEM cell is designed based on the follow- ing considerations (29] -(33)

:

1) maximum usable test area; 2) maximum upper frequency limit; 3) maximum uniformity of the field characteristics; and

4) maximum cell mismatch or VSWR (50-12 system).

These design considerations are functions of the cell's dimensions above and below the center plate (6/2, W, and L as shown

in Fig. 6) and, therefore, are related to each other. The maximum dimensions of the EUT must be less than 1/3 of 6/2, W, and L to keep the field perturbations in the cell within ac- ceptable limits. Also, to prevent multimoding effects the upper

frequency limit must be less than the lowest order TE mode

(TEjo) cutoff frequency (fc io\ where

/cio ~ 3X 10* 2W

(2)

Thus as the cell's dimensions are increased to accommodate

larger EUT's, the upper frequency limit of the cell decreases

proportionally. Tests have shown that the optimum geometry

49
DONALDSON et «/.: HELD MEASUREMENTS IN AN ENCLOSURE

MOMITOR OR

DETECTOR

y -'^>N

.

GENERATOR

DUAL

DIRECTIONAl. COUfLER ./ LOAD \ TEMCELL / \

i 1 INCIDENT POWER

METER

REFLECTED K>WER

METER

Hf. 1. Susceptibility test connguration using TEM cell from 1 MHz

to fCIOfor maximum test area and maximum test frequency is one

in which b equals W. With this geometry, the f-field uni- formity in the usable test area is within ±2 dB. Modifying

the cell geometry by reducing b to 0.6 W, i.e., reducing the usable test area, improves the f-field uniformity such that the variations in the relative f-field are less than ±1 dB. A typical TEM cell test setup for susceptibility testing above

1 MH"z is shown in Fig. 7. The EUT is placed midway between

the outer wall and the center plate of the cell. The £-field in the cell is given by the following equation:

Ec =

bl2

(3)

where

Eg magnitude of the F-field in the cell, P„ net power flowing through the cell, Rg real part of the cell's characteristics impedance, and

fc/2 distance between the cell's outer wall and center plate.

For frequencies below 1 MHz, the dual directional coupler

and power meters in Fig. 7 are replaced with a voltage monitor

tee and an RF voltmeter. The f-field in the cell is then given

by the following equation:

" b/2

(4)

where V^ is the measured voltage at the input port of the cell An error analysis of this susceptibility measurement technique

indicates that the uncertainty of the value of E^. is less than ±2 dB depending on the EUT's perturbation of the field in the

ceU.Emission measurements have been made as shown in Fig. 8, based on the assumption of reciprocity [29], [331. The emissions of the EUT are coupled to the TEM mode of the cell and

propagated to the cell's ports. The RF energy at the measure- ment port of the cell was used to determine the relative (as a function of frequency) emissions of the EUT based on the

receiver characteristics and the impedance and coupling properties of the cell. When an EUT is placed in a TEM cell, its radiation resistance

is changed relative to its free-space radiation resistance. Thus

to determine the absolute magnitude of the EUT's emissions

relative to free space, a correction factor must be applied.

Theoretical and experimental investigations (31), (321 have been conducted to determine this correction factor for elec- trically small dipoles in a TEM celL This correction factor can be applied to emissions from equipments that can be modeled

as electrically small (dimensions much less than a wavelength)

dipoles with a resulting estimated uncertainty in the measured

DETECTOR

RECEIVER OR , / :^yi] \ toatm \ TEMCELL /

LOAD

Fig. 8. Emission test connguration using TEM cell.

f-field of less than 5 dB. Correction factors for larger EUT's

have not been determined and the absolute magnitude of such

equipment's emissions relative to free space cannot be deter- mined from TEM cell measurements. However, work in im- proving the TEM cell is presently being conducted (33 J

.

The TEM cell offers several advantages in measuring the susceptibility and emission characteristics of small equipments and

devices. It is portable, simple to build, and can be used from

dc to fcio to provide fields from 10 ^V/m to 500 V/m. The

cell's construction cost is lower than conventional anechoic

chambers and shielded enclosures, and it is a relatively accurate (1-2 dB) susceptibility test chamber. Because of its TEM mode

of operation, the cell has a linear phase response from dc to near fdQ and can thus be used for swept frequency measurements. The major limitations of the TEM cell are: 1) the inverse proportionality between its size and the upper frequency limit; 2) the restriction on EUT size imposed by cell dimensions; and

3) the inaccuracy of the emission measurements on devices that

are not electrically small.

G. Other Methods

Various methods, other than those previously discussed,

which are used for making susceptibility and/or emission measurements include long-wire-antenna, parallel-plate, and

low-^-enclosure methods.

1) Long-Wire Antenna: The long-wire-antenna method in- volves a wire suspended on insulators belween two opposite

walls of a conventional shielded enclosure as shown in Fig. 9. This method is generally used for making susceptibility measure- ments at frequencies below 30 MHz. The wire is installed along

the longest dimension of the enclosure and at a distance from

the ceiling between one-fourth and one-thiid the interior height of the enclosure (34]. In a manner similar to the TEM cell, the enclosure operates in the TEM mode with the wall as the

outer conductor. Therefore, the termination resistances are chosen to match the characteristic impedances of the concentric

feed-hne and the antenna, respectively (34|, (35]. The EUT

is located on a ground plane in the center of the enclosure and

directly under the center of the long-wire antenna. The field intensity (£"</) at the EUT location is given in microvolts per meter by the following equations:

and

where

El

£</ = — E,

\__ 2.36 X 10^ /l

2d, - d 2d: ^)

(5)

(6)

voltage in /iV to the input of the concentric

line, Kj attenuation factor, Z; characteristic impedance of the line, and

d, rfj , and rfj distances in meters as shown in Fig. 9.

50
PROCEEDINGS OF THE IEEE, VOL. 66, NO. 4. APRIL 1978

-SHICLOEO ENCLOSURE

-CONCENTRIC

LINE l<1t AWG

KVIRE IN V TUBE)

Fig. 9. Long-wire antenna test chamber.

•3

•

3U

XV

a.B

ou

c«

3 •

« C i-3

1c

« w

C S ^^

41 w

U.

>s

>

C

V

S

•

•

eo

o c

3 • o e

41 « • 3 .25

•*-( at w s

co

o

41M

VJ 41ec a u

X 4» O "

ki 1-1 o. u

u

ou

• .-1 u ^

« 1-t a.jt o to

4> >s U U

u x: C *J o -^ o > Anechoic Chamber H H L H H M^'' M L M H H M M H

Shielded Enclosure M L<1) L H H h<" M L Ha) H H U M L

Hooded Antenna M H M H H m'^ M L M H H L M H

Stirred Mode M M H H H f^6) L M L H H L L H

Tuned Mode ^^' M M M H H H L L L H H L L H

Statistical Sapple L H H U H

m(6) H H H hW H L H H

TEM Cell L 11(3) L L M H L M M H L L L H

Long-wire Antenna H h(3) L L H H L H M H H H M H

Parallel Plate L h(3) L L M H L M H L L L L H

Low-Q Enclosure H M L H H

«(6 H L M H H M M H

Key:

B High M Moderate

L Low

Notes:

1. Invalid measurement method.

2. Susceptibility measurements only.

3. Shielding effectiveness measurements only.

4. High in anecholc chamber and shielded enclosure, but low in laboratory.

5. For emission measurements, the ratio of field intensity to sampled power.

For susceptibility measurements, the ratio of field intensity to Inmit power.

6. Function of antenna gain.

Fig. to. Qualitative comparison measurement of methods.

The long-wire antenna test chamber can accommodate

relatively large EUT's. It provides isolation from the ambient

electromagnetic environment and can develop relatively high

field intensities. Also, the long-wire antenna is simple and can

be installed in a shielded enclosure in a short period of time. The major limitation of the long-wire antenna test configura- tion is the upper frequency Umit of approximately 30 MHz.

2) Parallel Plate: Parallel-plate structures are widely used for making susceptibility measurements up to approximately 30

MHz [34] . A common structure consists of two parallel plates 24-in wide and up to 1 8-in separation [36). The length of the

plates is normally 10 ft although there is no maximum limit on

the length other than practical considerations. The ends of the

parallel-plate structure are terminated with matched loads so that it operates in the TEM mode. A generator is connected to one end to drive the line and a receiver or RF voltmeter is con- nected to the other end to measure the voltage between the

plates (V). The magnitude of the field intensity between the

plates (f ) is given by £"= Vlh, where /i is the separation distance between the plates. The EUT is placed between the parallel

plates and its operation is monitored for malfunction or de- gradation of performance. This structure is capable of providing fields up to several hundred volts per meter with the only

practical limitation being the power rating of the terminating

resistors. The parallel-plate technique has several disadvantages. First, the size of the EUT is limited by plate separation. Equipment

dimensions which approach the plate separation distance will substantially perturb the TEM field. When large perturbations

exist in the field, accurate calculations of the field intensities can no longer be obtained from the equation E = V/h (this ef- fect is the same as when a large EUT is placed in a TEM cell). Another disadvantage is that the parallel-plate structure is not

truly a shielded enclosure. Thus, the EUT is exposed to the ambient electromagnetic environment and the structure itself radiates electromagnetic fields. 3} Low-Q Enclosures: Low-^ (lossy wall) enclosures are a modified form of the conventional shielded enclosure in which

nonmetallic lossy walls of low refiectivity are used to reduce

reOections and attenuate wall coupling. Two investigations of

51
DONALDSON et al.: FIELD MEASUREMENTS IN AN ENCLOSURE

the feasibility of low-Q enclosures have been conducted. The

first investigation [37] consisted of coating the inside walls of

a conventional shielded enclosure with a lossy material This

material reduced the Q of the enclosure and, thereby, reduced

the ±40-dB measurement errors in conventional enclosures by more than 20 dB over the 20- to 200-MHz frequency range. The

lossy material was a mixture of graphite and spackling com- pound with a resulting conductivity of 1.0 mho/m. This lossy wall approach, even though promising, has been evaluated only

in scaled models of full-sized enclosures. The second investigation [29) of \ow-Q enclosures evaluated

the use of an underground room or tunnel as a shielded enclosure. The tunnel walls were soHd granite with a relative permittivity of 6. It was found that if the tunnel was suffi- ciently deep, adequate isolation from the ambient electro- magnetic environment could be obtained. Also, measurements

indicate that the ±40 dB measurement errors in conventional

shielded enclosures can be reduced to less than ±5 dB in the low-Q tunnel over the 20 to 100 MHz frequency range. The

major disadvantage of performing measurements in low-Q tunnels is their limited accessibility.

III. Conclusions

Each of the measurement methods presented has merits as well as limitations in terms of its application to a particular measurement requirement. Fig. 10 provides a qualitative comparison of the different methods as a function of various measurement parameters. The ordinate tabulates the measurement methods presented, whereas the abscissa lists the parameters

selected to permit measurement method comparisons.

As an example of the manner in which Fig. 10 was derived, consider radiated emission or susceptibility measurements

performed in an anechoic chamber. Relative to other methods,

anechoic chamber measurements are high cost because of the

'required shielded enclosure, anechoic material, and measure- ment instrumentation. The accuracy, frequency range, EUT

size range, and field intensity levels which can be achieved are high, limited primarily by such factors as instrumentation and

anechoic material characteristics and chamber size. Further- more, shielded anechoic chamber measurements provide high

isolation between the measurement environment and the external environment and have a well established theoretical

basis. The operator skill, test time, and calibration require- ments for anechoic-chamber measurements are moderate, as are proximity effects (effects of equipment, personnel, etc.) on measurement accuracy. Also, the measurement sensitivity

is moderate, i.e., equipment emissions can be measured at moderately low levels and the field intensity necessary to measure equipment susceptibility can be attained with sources

of moderate power. Anechoic chamber measurements are

relatively simple and straightforward (low complexity and

generally yield a direct measurement result with little or no

data reduction required.

It is to be noted that Fig. 10 is intended only to provide a broad qualitative comparison of the various measurement methods based on selected measurement parameters. The

selection and use of a specific method for a particular applica- tion may depend upon measurement parameters not listed in the figure or other factors. For example, the availability of a particular method (facilities, instrumentation, etc.) may be

the overriding factor in the selection of a method for a new

measurement task. Also, measurement methods may be dic- tated by^contractual requirements.

16

17

18

19

20

21

22

23

24

25

26

References

S, Galahan, "Understanding microwave absorbing materials and

anechoic chambers," Microwaves, Pt. I, Dec. 1969; Pt. II, Jan. 1970;Pt. Ill, Apr. 1970;Pt. IV, May 1970. W. H. Emerson, "Electromagnetic wave absorbers and anechoic chambers through the years," IEEE Trans. Anetennas Propagat.,

vol. AP-21, pp. 484-489, July 1973.

J. Appel-Hansen, "Reflectivity level of radio anechoic chambers," IEEE Trans. Antennas Propagat, voL AP-21, pp. 490-498, July 1973. W. R. Free, "Radiated EMI measurements in shielded enclosures,"

in IEEE Electromagn. Compat. Symp. Rec., pp. 43-53, 1967. W. R. Free et al., "Electromagnetic interference measurement

methods-shielded enclosure," Tech. Rep. ECOM-02381-F, Contract DA28-034, AMC-02381 (E), Dec. 1967. W. R. Free and C. W. Stuckey, "Electromagnetic interference methodology, communication equipment," Tech. Rep. ECOM0189-F, Contract DAAB07-68-C-0189, Oct. 1969.

J. C. Toler and J. A. Woody, "An analysis of errors in radiated EMC measurements," ii\ Proc. of the 2nd Symp. Technical Exhibition on Electromagnetic Compatibility (Montreux, Switzerland)

pp. 199-204, June 28-30, 1977.

R. Waterhouse, "Noise measurement of reverberant rooms,"

J. Acoust Soc. Amer., vol. 50, no. 4, pp. 931-934, Oct. 1973.

J. Tichy and P. K. Baade, "Effects of rotating diffusers and sampling Techniques on sound-pressure averageing in reverberation rooms," J. Acoust. Soc. Amer., vol. 56, no. 137-142, July 1976.

C. E. Ebbing and G. C. Maling, Jr., "Reverberation room qualiflca- tions for determination of sound power of sources of discrete fre- quency sound," / Acoust. Soc. Amer., vol. 54, no. 4 pp. 935-941,

Oct. 1973. W. B. Sabine, Collected Papers on Acoustics. New York: Dover

Publications, 1964. H. A. Mendez, "A new approach to electromagnetic field strength measurements in shielded enclosures," in WESCON Tech. Papers, Session 19, 1968.

, "Meaningful EMC measurements in shielded enclosures," presented at 1969 IEEE Int. Symp. Electromagnetic Compatibility,

Asbury Park, NJ, June 17-19, 1969.

P. Corona et al., "Use of a reverberating enclosure for measure- ments of radiated power in the microwave range," IEEE Trans. Electromagn. Compat, vol. EMC- 18, pp. 54-59, May 1976.

P. Corona and Latmiral, "Evaluation and use of the reverberating

chamber," presented at 1st National Meeting of Applied Electro- magnetism, National Committee for Engineering Sciences and

Architecture of the National Research Council, Institute Universatario Navale, Naples, Italy, 1976.

E. Paolini and L. Picciolo, "Measuring technique for assessing in- terference radiations produced from industrial scientific and

medical (ISM) apparatus at microwaves," in IEEE Int. Electro- magnetic Compatibility Symp. Record, pp. 279-287, June 1973.

H. Anderson, Litton Microwave Products, Minneapolis, MN, per- sonal communication.

P. Corona et al., "Performance of a reverberating enclosure for power measurements in the microwave range," in Proc. 2nd Symp.

and Technical Exhibition on Electromagnetic Compatibility

(Montreux, Switzerland), pp. 419-423, June 28-30, 1977. W. Jarva, "Shielding tests for cables and small enclosures in the 1 to 10 GHz range," IEEE Trans Electromagn. Compat, vol. EMC12, Feb. 1970. MIL-STD-1377 (Navy), "Effectiveness of cable, connector, and weapon enclosure shielding and filters in precluding hazards of electromagnetic radiation to ordnance; measurement of," Aug.

20, 1971.

J. D. Lee, "MIL-STD-1377 Versus MIL-STD-28S Microwave

Shielding Effectiveness measurements," in IEEE Int. Symp. Rec. on Electromagnetic Compatibility, Oct. 7-9, 1975.

J. R. Cummings, "Translational electromagnetic environment chamber, A new method for measuring radiated susceptibility and emissions," in IEEE Electromagnetic Compatibility Symp.

Rec, pp. 4AIel-4Ale5, 1975.

C. W. Stuckey and J. C. Toler, "Statistical determination of electromagnetic compatibility," IEEE Trans Electromagn. Compat, vol. EMC-9, pp. 27-34, Sept. 1967.

C. W. Stuckey et al., "Statistical Description of Near Zone Spurious

Emissions," Georgia Institute of Technology, Atlanta, Georgia, Technical Report AFAL-TR-67-37, Contract AF 33 (61 5)-3329, February 1967.

J. C. Toler and C. W. Stuckey, "The power distribution measure- ment technique— A method for measuring radiated fields from

equipment cases," presented at the 20th meeting of the SAE

Electromagnetic Compatibility Committee, Wright-Patterson Air Force Base, OH, May 1967.

R. C. Johnson, "Mutual gain of radar search antennas," in Proc. 9th Tri-Service Conf. Electromagnetic Compatibility, 1963.

52
PROCEEDINGS OF THE IEEE, VOL. 66, NO. 4, APRIL 1978

1 37 1 R. D. Wetherinfton ei al., "Analysis of some near zone micro- wave antenna patterns recorded by the Naval Ordnance Test Sta- tion," Georgia Institute of Technology. Atlanta, GA, Tech. Rep., Contract NOrd-1 6 189, 1958. |33|

{28| M. W. Long, "Wide angle radiation measurements," in Proa 6th

Conf. on Radio Interference Reduction and Electronic CompatibiUty. 1960. |34|

(39] M. L. Crawford, "Techniques for measurement of electromagnetic

radiation and susceptibility of electronic equipment," in Proc 1st (3S|

Int. Symp. and Technical Exhibition on EMC (Montreux, Switzerland), pp. 38-44. I97S. (30| , "Generation of standard EM fields using TEM transmission |36|

ceDs." IEEE Trans. Electromagn. Compat, voL EMC- 16. pp. 189-

195. Nov. 1974. (31 1 J. C. Tippett and D. C. Chang, "Radiation characteristics of elec- |37|

trically small devices in a TEM transmission cell." IEEE Trans. Electromagn. Compat., voL EMC-18, pp. 134-140. Nov. 1976. (32 1 M. L. Crawford, "Experimental evaluation of the radiation characteristics of dipole sources enclosed in a TEM transmission cell."

in Conf. Precision Electromagnetic Measurements Digest, pp. 57-

59. 1976. —, "Improved techniques and instrumentation for EMC measure- ments." in IEEE Int. Electromagnetic Compatibility Symp. Rec,

pp. 369-371. 1976. MIL-STD-462. Notice 3. "Electromagnetic interference characteristics, measurement of." Feb. 9, 1971.

D. R. J. White. "Electromagnetic interference and compatibility,

vol. 4. EMI test instrumentation and systems,' Don White Consultants, Inc.. Germantown. MD. o. 4.7. para. 4.2. 1971.

B. E. Roseberry and R. B. Schulz. "A parallel-strip line for testing RF susceptibility." IEEE Trans. Electromagn. Compat., yo\.EMC7. pp. 142-150, June 1965. W. R. Free et al., "Electromagnetic field probes," Georgia In- stitute of Technology, Atlanta, GA, Research and Development

Tech. Rep. ECOM0306-F. Contract DAABO7-71-C-0306, Mar.

1973.

53
<
Appendix C

Crawford, M. L. "Generation of standard EM fields using TEM transmission

cells.

Reprinted from IEEE Trans. Electromagn. Compat. EMC- 16(4) : 189- 195 ; 1974

November

.

55

Generation of Standard EM Fields Using TEM Transmission Cells

MYRON L. CRAWFORD

Abstract—A new technique developed at the National Bureau

of Standards (NBS) for establishing Standard, uniform, electro- magnetic (EM) fields in a shielded environment is described. The

technique employs transverse electromagnetic (TEM) transmission

cells that operate as 50 Q impedance-matched systems. A uniform

TEM field is established inside a cell at any frequency of interest below that for which higher order modes begin to propagate. Standard field strength levels from 10 nV/m to 500 V/m can be established with imcertainties of less than 1.0 dB to 2.0 dB inside the NBS cells for frequencies from dc to 500 MHz. The cells are especially useful for calibrating EM radiation hazard meters, for emission and susceptibility testing of small to medium sized equipment, and

for special low level calibration of very sensitive field strength meters.

I. INTRODUCTION

THE proliferation of electronic/electromechanical sys- tems in our environment is causing a rapid rise in the

level and number of potential interfering signals. C!ompatible operation of such systems in the presence of electro- magnetic (EM) interference is a fimction of their EM

susceptibility, the abUity to accurately measure and

control it, and/or to effectively shield against it. A number of techniques exist for establishing knowii,

uniform levels of electromagnetic energy for susceptibility

testing over limited frequency ranges and for limited

applications Cl}~[33- For example, high level fields can

be generated quite accurately above a few hundred megahertz using standard gain horns and below a few megahertz

using parallel plate lines. Both these techniques are

•widely used, but suffer a major disadvantage; they radiate

electromagnetic energy into the surrounding space which

may interfere with the measurements, be hazardous to the operator, or interfere with other experiments being

conducted within transmission range. The technique described in this paper contains the

EM field inside the transmission cell. It is extremely

broad band in frequency, being limited only by the waveguide multimode frequency associated with the cell size. (Donstruction costs of the cells are minimal and the use of

expensive anechoic chambers or shielded enclosiu-es are unnecessary. The cells can be used to establish known

field strength levels from 10 nV/m to 5(X) V/m with

uncertainties less than 1.0 dB to 2.0 dB (depending on

frequency) . The cells are especially useful for calibrating

EM radiation hazard monitors, for equipment emissions

and susceptibility testing, and for special low-level'sensor

calibrations.

II. DESCRIPTION, DESIGN, AND EVALUATION

OF THE TEM CELLS

The design of the cells was patterned somewhat after a large cell constructed for the USAF School of Aerospace

Medicine at Brooks AFB £4"}. The NBS cells were designed

as shielded chambers for EM susceptibility and emissions

testing of test specimens, and for calibration of EM

field probes. A cross sectional view of a typical cell is shown in Fig. 1. The cell consists of a section of rectangular

coaxial transmission line tapered at each end to adapt

to standard coaxial connectors. The line and tapered

transitions have a nominal characteristic impedance of 50 ohms along their length to insure minimum voltage

standing wave ratios ( VSWR) . The EM field is developed

inside the cell when RF energy is coupled to the line from

a transmitter connected at the cell input port. A matched

50 ohm termination is connected to the output port. The

fields inside the cell are monitored using special electric and magnetic field probes designed and constructed by

NBS [5]. A differential power measuring system, designed

by NBS [6J for measiiring energy absorbed from the cell by test subjects, can also be used to determine the field strength if the cell's impedance is known.

Experience with the Brooks AFB cell showed that at

frequencies for which only the principal wave (TEM

mode) propagates through the cell, a reasonably uniform

electric field could be generated. The main limitation of

their cell was its large size, restricting its usefulness to

frequencies below 50 MHz. A series of smaller cell^ were

developed at NBS to increase this range up to 500 ^IHz.

The major design considerations were:

1) maximize usable test cross sectional area;

2) maximize upper useful frequency limit

;

3) minimize cell impedance mismatch or voltage

standing wave; and

4) maximize vmiformity of EM field pattern characteristic of the cell. The cells were designed using experimental modeling

and the approximate equation for the characteristic

impedance of shielded strip line £4]

94.15

zo^

Min w

.6(1 - t/b)

+

C 1 0.0885«J

(ohms) (1)

Manuscript received April 16, 1974. The author is with the National Bureau of Standards, Buulder.

Colo. 803a2. Reprinted from IEEE Trans. Electroroagn.

November.

where €, is the relative dielectric constant of the mediimn

between the conductors, C/' is the fringing capacitance in

C^onpat. EMC-16(4): 189-195; 1974

57
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, NOVEMBKR 1974

-0.92L

TOP VIEW

. .157 Al. sheet

.127 Al. aheat Dlslsetrle Rod

(- Accea*

Door

-Olalactrio Roda -

SIPB Ylfili

Fig. 1. Design for rectangular TEM transmission cell.

b

w

i

t

t

Fig. 2. Shielded strip line.

picofarads per centimeter, and w, b, and t are shown in Fig. 2. The problem of modifying the shielded strip line into a "rectangular coaxial" line was primarily one of deter- mining experimentally the value of C/. This was done

using a time domain reflectometer (TDR) to evaluate

small scale models of cells with the cross sectional geometry

shown in Figs. 3 and 4. C/ was found to be approximately

equal to 0.087 pF/cm, respectively. Dimensions for b were determined from the design criterion that as much

as 1/3 * of the volume between the septum and outer

plates can be occupied by the equipment under test (EUT) to meet design considerations 1, 3, and 4. Once

b was calculated, an experimental estimate of C/ deter- mined, and an available metal thickness t selected, w

can be calculated from (1), assuming a nominal 52 ohms

for the line characteristic impedance. (Fifty-two ohms

The one third factor is considered a maximum. The impedance

loading effect from inserting the EUT should not exceed a few ohms

if a reasonable VSWR and EM field pertuxbation is to be maintained.

Fig. 3. Cross sectional view of optimum geometry of rectangular

transmission line for maximum test area and frequency.

w

w —

b

it.

Fig. 4. Cross sectional view of rectangular transmisison line with improved E field uniformity.

58
CRAWFORD: STANDARD EM FIELDS TABLE 1 TEM Cell Dimensions

Square Cell (Fig. 3) Rectangular Cell (Fig. 4)

Cutoff/

Multimode

Frequency

(MHz)

Plate Separation

b (cm) w

(cm)

t (cm)

C'f (pF/cm)

Plate Separation

b(cm) w

(cm)

t (cm) (pF/cm)

100 150 123.83 .157 .087 90 108.15 .1.57 .053 300 50 41.28 .157 .087 30 36.05 .157 .053 500 30 24.77 .157 .087 18 21.83 .157 .053

1

1 1 1 1 1 1 MAIN LINE SECTION1

1

52 -

1''\^/'N ^„^___/^

S \

t 51 — 1 \ —

Ul 1

\ ^ 1

\ TAPER ^ 1

\ TRANSITION 7

SO ^ TAPER / \

-?

v TRANSITION / N. /

49

1 1 1 1 1 1 1 -1 1

Shflrp Rtsoftonct Ptah VSWR I 92

LENGTH (FEET FROM MALE TYPE N PORT)

Fig. 5. Time domain reflectometer trace of distributed impedance

of empty cell. (Form factor Fig. 3.)

FREQUENCY (MHz)

Fig. 6. Port input VSWJJ of empty cell. (Form factor Fig. 3.)

was chosen to allow for some impedance loading effect when inserting the EUT inside the cell.) Table 1 gives

the dimensions for constructing the two cell forms with

specified upper frequency limits. The TDR was then used

to make refinements by trimming w until the proper

characteristic impedance was obtained.

The cross section of Fig. 3 is used at high frequencies

where maximimi cell size is limited and requires some

compromise in electric field uniformity. The cross section

of Fig. 4 achieves greater field imiformity at the cost of

vertical test space restriction. Fig. 5 gives a typical TDR

trace of the distributed impedance along the length of a

cell of Fig. 3 cross section, and Fig. 6 shows the VSWR

as seen at the cell's input and output ports.

MAPPING THE FIELDS INSIDE THE CELLS

Measurements were made using a calibrated short

dipole to probe the electric field inside the empty cells. The variations in relative field strength versus position were determined in the longitudinal, transverse, and

vertical directions within the cells. The electric field E

is essentially vertically polarized in the region near the

center of the cells and p-adually becomes horizontally

polarized as one moves in the horizontal direction toward

the gap at the side. Both vertical and horizontal components of E were measured at each point to determine the

total electric field, E= (E,^ + Eh'V'^ where E, and

Eb are in phase. The electric field distributions for each

form factor are shown in Figs. 7 and 8. The electric field in the test regions shown is primarily vertically polarized or E, » Eh. The relative field distribution is independent

of the magnitude of the test field used and the frequency

so long as the frequency is less than the first order TE

mode (TEio) cutoff frequency given by the following

equation:

ifc)io = c/2W. (2)

The equation fgr determining the cutoff frequency for any higher orderjmode in general is given by

\ Je) m,n 2bW (3)

For (2) and (3), c is the velocity of propagation of

light :^ 3.0 X 10* m/s, b and W are as shown on Figs. 3

and S and m and n are integers related to the half sine variations of the field in the vertical and transverse

directions. If higher order modes are allowed to propagate,

the field configuration, which is the vector sum of each

contributing mode, no longer has the simple pattern

shown in Figs. 7 and 8. Thus higher order modes would

greatly complicate interpretation of the measured results of the cell. Variations in the electric field strength for the empty

cells were less than 2 dB for the cross section of Fig. 3

and less than 1 dB for the cross section of Fig. 4 over the

area typically occupied by the EUT. Inserting the EUT

shorts out part of the electric field due to the metal in the case and increases the field strength proportional to the percentage cross section occupied. Fig. 9 shows an

example of the results of inserting a solid metal case inside a ceU of the form factor sho\\Ti in Fig. 3. This case occupied

1/3 the vertical separation between the upper waU and

59
IEEE TRANSACTIONS ON ELECTROMAGNETIC COBiWATIBIUTT. NOVEMBER 1974

0.4 W 0.6 W 0.6 W

Dlstanc* Fr<M Prob« Sld« of C»ll

0.7 IT

Fig 7. Relative electric field distribution inside cell. Cross sectional cut through upper half at center of cell. (Form factor Fig. 3.)

T

0.4 W 0.6 W

Di«t*nc« From Frob* Sid* of Cvll

0.8

Fig. 8. Relative electric field distribution inside cell. Cross sectional cut through upper half at center of cell. (Form factor Fig. 4.)

the septum or center conductor and increased the field strength 3 dB and 6 dB, respectively, in the regions

directly above and below the case. This increase in field strength for a constant test input power must be taken

into account when determining the absolute test field, and occiu-s if significant cross section within the cell is occupied by the EUT.

III. STANDARDIZATION OF THE FIELD INSIDE

THE CELLS

The absolute electric field strength E, at the cell's center between the upper wall and the center conductor

is determined using the equation

E. = {PnRcY'yd (4)

where P„ is the net power flowing through the cell. Re

is the real part of the cell's complex characteristic impedance, and d is the separation distance between the

cell's upper wall and its center plate or septum.

A brief discussion of the sources of error is given below.

The total fractional error £^E, in determining the absolute

field strength £', inside the cell is given as A£,~|i(6H + tp) +ed + £|f I (5)

where ip = AP„/Pn, «« = ARc/Rc, <d = Aj/d, and (g is the error due to the nonuniformity of E, determined

experimentally by mapping the field distribution in the

test region of the cell. Equation (5) was derived by substituting P„' = I P„ + APn I, Re' = \Rc + ARc\, and

60
i'hawford: standard em fields

I.. T

5

i/a b (1/2

1/8 b (1/2

3/6 b (1/6

»/4 b (1/8

1/8 b (1/6 L)

0.4 W O.t 0.« 1

Dlituice FroH Probe Sid* of G«ll

Fig. 9. Relative electric field distribution iniside cell with metal case mounted in center of upper half space. Cross sectional cuts

at center and off end of case.

d' = \d + Ad\ into (4) and adding eg (obtained from E

field mapping). Higher order terms contributing to small

errors were then dropped in the derivation to arrive at (5).

The error €p, in determining P„, is due to uncertainties

in coupler calibration, absolute measurement of RF

power on the side ai'm of the coupler, and impedance

mismatch between the cell, coupler, RF source, and cell termination. If a precision caUbrated coupler and power

meter are used and the cell and its termination are impedance matched (VSWR < 1.05), tp should be less than

±5%.

The error €« in determining Re is a function of the

measurement accuracy of the TDR and the impedance

loading of the EUT inside the cell. If the EUT occupies a small portion ( < 1/5) of the cross section of the cell, €« will be small (<3%), and is typically neglected in the calculation of Er. For larger EUT's (occupying up

to 1/3 the cross section of the cell) the impedance loading

effect must be determined with the TDR and used to correct Re when using (4) to calculate E^. «« for these

cases can be much larger but typically would be less than

10% if the EUT is centered inside the cell. Exceeding the

1/3 load factor is not reciommended.

Determining ts is more difficult. Introducing the

EUT inside the cell perturbs the electric field distribution as described in the section on field mapping. This loading

factor (increase in J5^) is determined using the small

calibrated probes referred to earlier. If the size of the

EUT is less than the l/5th factor and if care is taken to properly orient or eliminate interconnecting leads to the

EUT, (g can be reduced to less than 6% for cells with

the form factor of Fig. 3. Larger EUT's would necessitate

measurement of the field distribution around the EUT

and a resulting higher estimate of es.

TABLE 2 Summary of Measurement Errors

Percent Uncertainty

Source of Error Form Factor

Fig. 4 Form FactorFig. 3

a)

b)

c)

d)

e)

Absolute measurement of incident RF power on the

side arm of coupler ± 3.0 ± 3.0

Coupler calibration ± 2.0 ± 2.0

tp, total error in deter- mination of RF power

passing through cell ± 5.0 ± 5.0

Real part of cell complex

impedance tR Cell plate separation u

do 3.0 ± 3.0 ± 1.0 ± 1.0 Nonuniformity of electric

field inside cell tg ± 6.0 ±20.0

maximum field strength

error ±11.0

<± l.OdB

±25.0

<± 2.0dB

AEr I i(0.03 + 0.a5) + 0.01 + 0.06 I X 100

(0.20)

The sources of errors for the two form factors are summarized in Table 2.

IV. APPLICATIONS

The block diagrams for making EM susceptibility

measurements using the cell are shown in Fig. 10. The

EUT is mormted inside the cell in the desired orientation

with the interconnecting leads/cables and power cord

extended through a side or end wall of the cell, as required.

The orientation, size, and type (shielded or unshielded)

of interconnecting leads and power cord of the EUT can have a large effect upon the equipment's susceptibility. Tests can be performed v\ith the leads oriented for minimum or maximum field couphng while making the

61
IKKE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBIUTY, NOVKMBKR 1974

HIGH POWER

GEN

ISOLATOR

(Optional) LOW -PASSFILTER SIGNAL SAMPLER

FREQ COUNTER

PWR MTR

Pine

COAXIAL SWITCH

PWR MTR TEST

MONITOR

DUAL

DIRECTIONAL COUPLER TEM CELL.

H-PWR 90A

LOAD

H-PWR

50 n

LOAD

(a)

HIOH POWER

GENERATOR

LOW- PASS

FILTER

SnSNAL SAMPLER

VOLTAGE MONITOR

TEE

FREO COUNTER RF

VOLTMETER

E » • A

TEST MONITOR

/|EUT|.>^I H-PWR

SO a \TEM CELly/^ LOAD

EUT • Equipment Undar Tnt

(Volt* / Mat«r)

EUT • EguipmaM Undir Tm>

(b)

Fig. 10. Block diagrams of system for susceptibility testing of equipment.

susceptibility measurements, thus providing data for evaluating the lead contributions to the overall EUT

susceptibility. Minimum lead interaction with the test

field is achieved by orienting them in a direction per- pendicular to the E field and by properly shielding them.

Maximum interaction is achieved by allowing the unshielded leads to sag inside the cell and/or by aligning

with the polarization of the test field (leads through top

of cell). The susceptibility level for the EUT is deter- mined at frequencies from 1 MHz to the cell multimode

frequency by means of the block diagram of Fig. 10(a).

The generator output is progressively in' reased untU

the EUT's normal operation is noticeably elected. The

test field can be further increased until the EUT fails if failure tests are required.

The cells can also be used to calibrate various field detection devices such as hazard probes, field sensors, or small field intensity meters. In this type of operation

the device being calibrated is placed inside the cell in a imiform field region. The standard test field is deter- mined by the procedure outlined in Section III, and the

field intensity indicated by the instrument is compared

with the known value of the standard test field. Measurements below 1 MHz are made using a voltage

monitor tee and RF voltmeter as shown in Fig. 10(b),

because directional couplers are not available at these

lower frequencies. Directional couplers and power meters

are used above 1 MHz because of the ability to monitor

impedance variations in the system when installing

EUT's inside the cell and orienting them for various

tests. Swept measurements can be made for a fixed orientation

of the EUT and its cables using the block diagrams of

Fig. 10(a) or (b) assiuning the circuit components remain

matched over the desired frequency range and have

acceptable frequency bandwidth characteristics. When

using the system of Fig. 10(b), the input voltage to the

cell is monitored directly and (4) becomes.

E, = VJd (6)

where V^ is the measured input voltage to the cell and

is equivalent to Vc = (PnRc)^'^- The error analysis then

involves evaluating the uncertainty in measuring Vc as compared to P„ and Re, and at the frequencies indicated

for these tests, the errors would be about equivalent.

Pulsed RF susceptibility measurements can also be

made by replacing the signal source and monitor/detector

with an appropriate pulse generator, sampler, and detector

(oscilloscope). The exposure level standardization -would

be similar to that outlined in Section III, but would

involve determining the calibration measurement accuracy

of the detector, the sampling (coupling) accuracy, and

the time domain response of the cell. Additional work

needs to be done in this area but it is believed that measure- ments using pulses with frequency components less than

500 MHz could be made.

The measurement systems shown in Fig. 10(a) and (b)

have been used to calibrate or determine the EM sus62
chawford: standard em fields

ceptibility of a number of test items. Examples are calibra- tion and evaluation of radiation hazard probes and meters

(both NBS and commercial types) ; calibration of smaU

E field sensors; calibration of small field strength meters and sensitive receivers; susceptibility of fire alarms (smoke

detectors) for malfunction and false alarm indication;

equipment cable susceptibility (shielded and unshielded)

and evaluation of RF conductance/interference on high

resistance dc transmission Unes. Intercomparison of the standardized field strength data

obtained using the cells with data taken using caUbrated

probes (caUbrated with parallel plate lines and in a imiform

field over a large ground screen) indicate excellent agree- ment, i.e., well within the uncertainties attributed to the

diflFerent techniques. Table 3 presents a sample of these

results at 15 MHz.

V. SUMMARY AND CONCLUSIONS

The objective of the work described in this paper was

to develop an alternate technique for making susceptibility

measurements of electronic equipment at frequencies up

to 500 MHz. Tasks included designing and constructing

the cells, optimizing the field distribution and usable

test area inside the cells, evaluating the characteristics

of the ceU and measurement system, and performing

susceptibihty measurements on typical electronic equip- ment. The technique offers a vmique way of determining not only EUT susceptibility to CW RF fields but also EUT susceptibility to pulsed RF fields. The cells can

also be used to determine relative (as a function of fre- quency) levels of radiated emissions from electronic

equipment by using the cell's inverse coupling characteristic with 50 ohm impedance-matched RF detectors

connected to the input and/or output ports [7]. However, more work is needed to determine if these radiation

measurements can be interpreted quantitatively.

The real advantage of using TEM transmission cells for making sxisceptibility or emissions measurements is the elimination of background interference without the

introduction of measurement problems associated with

Bhielded or anechoic enclosures. Furthermore, no EM

fields are generated external to the cell and the cells produce uniform and readily determined fields. Shielded

enclosures on the other hand reflect the emitted energy

from their waUs in such a compUcated manner that

prediction of the enhancement or interference of the

desired signal is extremely diflicult. Measurements using

the cell are simple to make and require a minimum of

detection equipment, e.g., no additional antennas are

required.

The main handicap of the cell is its size limitation due

to multimoding at higher frequencies. The largest cell at NBS can accomodate equipment 8 in by 19 in by 25 in at frequencies up to about 150 MHz. Smaller cells useful at frequencies up to 500 MHz have much less usable

test space. Fig. 11 shows a photograph of the large NBS

ceU.

TABLE 3 Intercomparison on NBS Cells with Parallel Plate Line

AT 16 MHz Using 10 cm Dipole Probe

Probe Output

Voltage

NBS

0.6 m by 1.0 m

Cell E.(V/m)

NBS

1.2 m by 1.2 m

Cell E,(V/m)

Parallel Plate Line E,{V/m)

0.1 21.3 21.5 27.0

0.3 34.5 34.8 32.0

0.3 47.2 47.6 47.0 0.5 71.0 71.5 71.5

1.0 127.0 127.5 131

Fig. 11. Photogrt^h of NBS rectangular TEM transmission cell.

VI. RECOMMENDATIONS FOR FUTURE WORK

Additional work is needed to solve the problems of

size limitation of the cells, to evaluate the loading effects of large equipment inside the cells, to determine possible

mode suppression and alteration techniques, and to derive

the mathematics of cell modal coupling. The error analysis

included in this report is rough and needs further re- finement.

Only limited susceptibility measurements have been

made using the cells. Additional measurements should be

made with other pieces of equipment to establish the degree

of generality of the method.

REFERENCES

[1] V. P. Musil, "Generating High-Intensity Electromagnetic Fields

for Radiated-Susceptibility Test," IEEE EMC Symp. Rec., Seattle Wash., pp. 185-194, July 1968.

[2] B. E. Roseberry and R. B. Schulz, "A Parallel-Strip Line For

Testing RF Susceptibility," IEEE Trans. Electromagn. Compat.,

vol. EMC-7, pp. 142-150, June 1965.

(3] R. R. Bowman and W. E. Jessen, "Calibration Techniques for RAMCOR Densiometer Antennas," NBS Rep., unpublished, Dec. 1970.

[4] G. A. Skaggs, "High Frequency Exposure Chamber for Radiobiological Research, "NLR Memo. Rep. 2218, Feb. 1971.

[5] F. M. Greene, "Design and Calibration of E and H Field Probes

for HF Band Application," Proc. Dep. Defense Eleclromagnelic

Res. Workshop Washington, D.C., pp. 50-76, Jan. 1971.

[6] M. L. Crawford, C. A. Hoer, and E. L. Komarek, "RF Differential Power Measurement System for the Brooks AFB Electromagnetic Radiation Hazards Experiment," NBS Rep. unpublished, Apr.

1971.

[7] M. L. Crawford, "Measurement of Electromagnetic Radiation from Electronic Equipment using TEM Transmission Cells," NBSIR 73-306, Feb. 1973.

63

Appendix D

Tippet, J. C. ; Chang, D. C. "Radiation characteristics of dipole sources located inside a rectangular, coaxial transmission line."

Nat. Bur. Stand. (U.S.) NBSIR 75-829; 1976 January.

65

NBSIR 75-829

RADIATION CHARACTERISTICS OF DIPOLE SOURCES LOCATED

INSIDE A RECTANGULAR, COAXIAL TRANSMISSION LINE

John C. Tippet

David C. Chang

University of Colorado

Sponsored by:

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

January 1976

Prepared for: Electronics Systems Division (AFSC)

Hanscom Air Force Base, Mass. 01731

'. •"•t.u ^o« */

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

67

CONTENTS

Page

FOREWORD ,...., iv

1. INTRODUCTION 1 2. PARAMETERS CHARACTERIZINn A TEM CELL 2

3. APPROXIMATE EXPRESSIONS FOR THE CAPACITANCE AND 5 CHARACTERISTIC IMPEDANCE

4. RADIATION RESISTANCE OF ELECTRIC AND MAGNETIC DIPOLES 9 IN A TEM CELL

5. SUMMARY AND CONCLUDING REMARKS 13

6. REFERENCES 16 APPENDIX A: Calculation of the Capacitance of a TEM Cell 17 APPENDIX B: Calculation of the Electric i^leld Distribution 21 Inside a TEM Cell APPENDIX C: Numerical Evaluation of Comolete Elliotlc Integrals and 29 Jacohian Elliptic Functions

69

FOREWORD

This report describes theoretical analyses of Transverse Electromagnetic (TFX)

Transmission cells developed at the National Bureau of Standards. The effort is part of a program sponsored by the Electronic Systems Division, Hanscom Air Force Base, under Contract

number Y75-917 with the National Bureau of Standards (NBS) . The purpose of this effort is to evaluate the use of TEM cells for measuring total RF power radiated by a small electronic

device.

The theoretical analyses were developed by staff from the University of Colorado under contract with NBS. Myron L. Crawford of the Electromagnetics Division was the technical

monitor for NBS and Charles E. Wright of the Electronics Systems Division was the technical

monitor for the Air Force. The period of performance covered by this report extends from

May 1975 to October 1975.

Results of these analyses Include exnressions for the capacitance and characteristic

Impedance of a cell, the electric field distribution inside a cell, and the radiation

resistance of electric and magnetic dipoles in a cell. These results are prerequisite to determining the radiation characteristics of a device under test relative to its operational

environment or to its operation in a free space environment.

71

RADIATION CHARACTERISTICS OF DIPOLE SOURCES LOCATED INSIDE

A RECTANGULAR, COAXIAL TRANSMISSION LINE

When making EMC measurements inside a shielded enclosure, the

radiation characteristics of the device being tested changes. In this report the change in radiation resistance of dipole sources located inside a National Bureau of Standards TEM transmission cell

is determined. In many cases a practical device can be modeled by

dipole sources. In these cases, the analysis allows one to predict

the device's radiation characteristics in other environments, e.g.,

free space.

Key words: EMC measurements; radiation resistance; rectangular coax;

shielded strip line; TEM cell.

1. INTRODUCTION

There are currently many researchers interested in developing a method

of measuring the radiated EM emissions and EM susceptibility of electronic

equipment [1]. At the National Bureau of Standards work is progressing on the design and experimental evaluation of a TEM transmission cell for this

purpose. A typical NBS cell consists of a section of rectangular coaxial

transmission line tapered at each end to connect to standard SO il coaxial

line as shown in figure 1.

Figure 1. Design for rectangular TEM transmission cell

73
To minimize reflections, the cell is designed to have a nomlhal characteristic

impedance of 50 £1. The description, design, and evaluation of these cells is

described in reference [2].

In order to interpret the measurements made in this cell, a knowledge of

the propagating TEM mode is required. This report contains a theory for

calculating some basic properties of the TEM mode. These include: 1) capacitance; 2) characteristic impedance; and 3) electric field distribution inside

the cell.

Once we possess a basic knowledge of the TEM mode, we can study the radiation characteristics of some simple sources located inside the cell. Since

these characteristics will not be the same as if the source were located in

free space, we need to determine the effect the TEM cell has on the radiation

resistance of the source. This will allow measurements made in the TEM cell

to be correctly interpreted for sources in other environments such as free

space.

The power radiated by and the radiation resistance of both electric and

magnetic dipoles located in the TEM cell are derived and compared to their

values in free space. We indicate how these expressions can be used to determine

both the magnitudes and directions of the respective dipoles through measurements

made in the TEM cell

2. PARAMETERS CHARACTERIZING A TEM CELL

The TEM cell is basically a rectangular, coaxial transmission line as shown in figure 2

Y

*- X

Figure 2. Cross-section of a TEM cell.

In order to understand how an electromagnetic wave can be guided by this struc

tare, a brief review of the results from standard transmission line theory

is presented.

74
Any multi-conductor system, of which the TEM cell is one, can propagate

at least one TEM mode. This mode has many unique properties, not the least

of which is that it has no lower cut-off frequency. That is, the TEM mode

can propagate through the guide at frequencies all the way down to DC. Another

characteristic property of the TEM mode (as its name implies) is that the

electric and magnetic field components of this mode lie totally in the transverse

plane (i.e., E - H 0) . In the transverse plane, the electric field satisfiev

Laplace's equation: V^E" = 0, which in the case of the TEM cell reduces to:

(^~2 •• •KTrr)^(.x,y) = 0. This means that the transverse field distribution can

dX dy • be obtained from the solution of a related static problem. The magnetic field

is easily obtained from the electric field as

i-± (2.1)

"o

where

r- -

1

a^ X E""

t ' ^(x,y)<Jwt+YZ.

» H" = ff(x,y)«Jwt + YZ.

»

^n v^.

IJ is the magnetic permeability;

e is the dielectric permittivity;

Y is the propagation constant;

and ? is a unit vector in the z direction.

z The total fields are then given by:

E = Re[E + E"] (2.21

and H = RefH* + H'] (2.3)

Many times, it is desirable to characterize TEM waves in terms of the voltage

and current on the line instead of the field quantities, E and H. The voltage

and current are given by the following equations:

V(z) = V%"^^ mm+ V'e^^ (2.4) * -'

where

and

^C^) = '>''" * V'' t2.5)

V; = - / F- . ai (2.6)

p

i; = M-.aT (2.7)

75
p is any path connecting the two conductors in a constant cross-sectional

plane, and J. is a closed path ^ encircling" the inner conductor. Vmmand I are related by a constant which is called the characteristic impedance, Z of the

line, and is given by:

Z„ - ± X? - ± ^

(2.8) m I z

- %

Thus, eq. (2.5) may be written:

V_ . V, 1(2) . jn e"^' - -^ e^^ (2.9)

o

'^ + If one measures the amplitudes of the forward and backward voltage waves, V~

,

then, with a knowledge of the characteristic impedance, eqs . (2.4) and (2.9)

determine the voltage and current anywhere on the line.

The characteristic impedance, Z , can also be expressed in terms of the

distributed capacitance per unit length of the transmission line, C as follows;

Z„ - -^— (2.10)

^^o

where v = —3^3 is the phase velocity.

o o Thus, a knowledge of the distributed capacitance, C , completely determines the

characteristic impedance.

The capacitance of the TEM cell can be obtained analytically using the

method of conformal transformation. The details of the analysis are contained

in Appendix A where it is shown that the capacitance per unit length, C , is

given by:

_o » 2 Hi^ (2.11)

Eq K(X')

where K(X) and K(X') are complete elliptic integrals of the first kind of

modulus X and X' respectively [3]. X' is termed the complementary modulus and

is related to X by:

The modulus, X', is given by:

[1 - XM** (2.12)

1,, sn_C

'en C

)*

(2.13)

where ^ = mg and sn E and en C are Jacobian elliptic functions of modulus k [4],

As shown in figure 2, g

= a-w is the width of the gap between the center septum

76
and the side wall, k' is the complementary modulus to k defined analogously

to eq. (2.12), and m is given by:

m - i^^ (2.14)

k, and thus, k' can be determined from the requirement that:

^^<^ -^

(2.15)

K(k') b

where K(k) and K(k') are complete elliptic integrals of the first kind of modulus

k and k' respectively. We note that in reference [5] the value of k^ is tabulated for a given ratio, K(k')/K(k).

Using eq. (2.10), the characteristic impedance is then found to be:

^-i^^i^ (2.16)

n^ ^ K(X)

where n is the intrinsic impedance of free space. Equations (2.11) and (2.16) are exact but are not easy to use. There is an efficient algorithm [6], however, for evaluating numerically the complete elliptic integrals appearing in

eqs . (2.11) and (2.16); it is contained in Appendix C. In the next section,

approximate formulas are derived for the capacitance and characteristic

impedance

.

3. APPROXIMATE EXPRESSIONS FOR THE CAPACITANCE AND

CHARACTERISTIC IMPEDANCE

In view of the complete elliptic integrals appearing in the formulas in

the last section, it is desirable to investigate the possibility of obtaining

formulas which do not involve special functions, but nevertheless, yield

numerical results sufficiently accurate to be of practical engineering use. We hope to obtain a formula which expresses the capacitance in terms of the

dimensions, a, b, and w of the TEM cell, and which can thus be used to design

TEM cells with a given characteristic impedance.

Since eqs. (2.11), (2.15), and (2.16) all involve ratios of complete

elliptic integrals, the following approximation is particularly useful (7).

J[AL.= 1 inf2l-l^ (6^4) (3.1)

(6') TT ^ 1 - /^J ^

Using cq. (3.1) we can write approximate expressions for eqs. (2.15) and (2.11)

respectively as follows:

Co

lA.llMl-LJ^ (k^4) (3.2)

2 in[2 1_L_J^1 (x^4) f2.3)

77
Equations (3.2) and (3.3) may be written alternatively as follows:

= ^ ln[2(l + /Tr)^(l*k)] - - ln(l-k*)

b ir IT

p

-2- = i ln[2(l + /X)Hl*X)] - ^ ln(l-XM

(3.4)

(3.5)

Subtracting eq . (3.4) from eq. (3.5) we obtain:

ii=iln

and substituting from eq . (2.13)

C.

5 4 i . 2 ^^

In eq. (3.2), the restriction that k^ > y is equivalent to requiring

h 1 ^ < 1, since when k^ = y, K(k') = K(k). If we make the somewhat more stringent

K requirement that — < 1, which is equivalent to k^ > .97, then eq. (3.7) may be

cl further simplified by noting that for k = 1 , en C

= sech C*. sn C = tanh C, and

5 = 5f. Thus,

7hJ — In sinh

IT •> 2bJ

AC

(3.8)

where

e^ TT ^h + /X^ ^1 + xJ^

(3.9)

An alternate form of eq . (3.8) may be obtained by using the following identity:

.(^g/2b)

with the result

sinh IS. 2b [1 * coth(5|)]

(3.10)

i^^ iln 1 + coth

^

2b

AC_

e_

(3.11)

In this form, it is easy to identify the first term in eq. (3.11) as the plate

capacitance between the center septum and the horizontal walls, and the second

term as the fringing capacitance between the edges of the septum and the side

a walls. For large gaps, the fringing term approaches - In 2, as expected [8].

78
It is interesting to note that the term in the square brackets in eq. (3.11)

is the identical formula given by T.-S. Chen [8] and originally derived by

S.B. Cohn [9]. Cohn's formula was derived assuming that the width of the center

septum, 2w, was very large compared to the plate separation, 2b. This is

equivalent to assuming that the two edges of the septum do not interact. AC,

then in eq. (3.11) can be interpreted as a correction term needed to account

for the interaction between the two edges. From eq. (3.9) it can be seen that

AC will be negligibly small if X is near one (or X' is near zero) since k is near one. From eq. (2.13) X'^ is given approximately by:

X'^ = k'2 sinh-'C^) (3.12)

It can be seen from eq . (3.12) that for small gaps, X' is always much less

than one. By using the approximate expression for the modulus, k given by

G.M, Anderson [10], it can be shown that for large gaps eq. (3.12) further

reduces to: w

-2lT = e " (3.13) I 2 - - E"

From eq. (3.13) it can easily be, verified that X' will be negligibly small,

and hence AC may be neglected if:

F^r (3.14)

From eq. (3.3) we have the restriction that X* > y or equivalently X'^ < y. From eq. (3.13) it can be seen that X'* < i- if:

^

> ^ In 2 = 0.1 (3.15)

1 w 1 So for the range: Tn ^ K

"^ T' ^^ ^^ "°^ negligible and must be calculated using

eqs. (3.9), (3.13), and k = 1.

Using eq. (2.10), the characteristic impedance is found to be:

Thus we have obtained approximate expressions for the capacitance and

characteristic impedance of the TEM cell. These are given in eqs. (3.8), (3.9),

and (3.16). Due to the limitations of the approximations, these formulas are

w 1 valid for a/b >_ 1 and r i t^jt In 2. The approximate formula for the capacitance

given in eq. (3.8) is plotted in figure 3 with a dashed line for AC = 0. The

exact formula using eqs. (2.11), (2.13), and (2.15) is plotted using a solid

line. The two curves agree almost identically except where w/b < y. This

discrepancy can be attributed, however, to the AC term which was neglected.

79
1.0 10.0

a/b

Figure 3, Capacitance per unit length of a TEM cell.

80
4. RADIATION RESISTANCE OF ELECTRIC AND MAGNETIC DIPOLES

IN A TEM CELL

The power carried by the TEM mode of a transmission line that is excited

by an elementary electric dipole is given by [11]:

V

= (4.1)

where

and

A is the excitation factor;

V is a unit voltage;

Z is the characteristic impedance.

From Green's theorem, A is given by:

where

and

-5- / 7 • E^'^ dv'

2V^ T (4.2)

J is the source current density;

E^ ' is the electric field of the negative (-z) propagating wave;

T is the volume enclosing all sources.

Since our source is an electric dipole, the volume integral in eq, (4.2) reduces

to

/ J . E^-) dv' = E fx .y„,zj I l.^ cos 9 (4.3)

where (x000 ,y ,z ) is the source point;

is the angle between the dipole and the electric field at the source point

i. ^r is the effective dipole length;

1 is the magnitude of the dipole current;

and E is the magnitude of the electric field at the source point

Using eqs . (4.2) and (4.3), eq. (4.1) can be written as

P = Z

e

E„I cos

"""^

'eff 2V

(4.4)

If, instead of an electric dipole, our source were a magnetic dipole,

then following a similar analysis, the radiated power is given by

2 2ttA X_

E„I cos If) m

2V

(4.5)

where A is the wavelength;

A=Trr^ is the area of the loop representing the magnetic dipole;

I is the equivalent magnetic current;

and (|) is the angle between the loop normal and the magnetic field at the source point.

81
In order to determine the change in the radiation characteristics of the

source, we will calculate the ratio of the radiation resistance inside the cell

to that in free space. The free space radiation resistances of electric and

magnetic dipoles are given in reference [12] as:

2 I"

'^ nj4^] "^.o — (4.6)

"n-J^ole)' (4.7)

From eqs. (4.4) and (4.S), the radiation resistances inside the guide are given

by:

e j2 ^ o

E„ cos 9 o ^eff 2V

(4.8)

Km

''. Ih

2uA ^o ^°'

2V

(4.9)

If we define the following, normalized parameters;

(4.10)

E - V/b

(4.11)

then the ratios, Qemand Q , of the radiation resistances are respectively: '^

E cos e

K'^

4.12)

Km 3-^z; E cos ^ ^o^

C4.13)

where k = 2tt/X . Thus, in order to determine the "correction factors," Q^ and

o e Q , which determine the change in the radiation characteristics of dipole sources

inside a TF.M cell as compared to free space, we need to know the squared magnitude

of the electric field E , at the dipole location. An analytical expression for

the electric field squared is shown in Appendix B as:

^o

Vm'

K(a')

dn^Cm'z)

sn*(m'w) - sn*(m'z)

(4.14)

82
where

a' - [1 - a*]'';

a - sn(m'w,k);

b KCk')*

(4.15)

(4.16)

m'

(4.17)

(4.18)

K(k) .» a and z = X + iy;

where x and y indicate the location of the dipole as measured from the center

of the TEM cell in the transverse plane. Thus, E in eqs . (4.12) and (4.13)

is given by:

1/2 bm'

K(a')

dn^m'z)

sn^(m'w) - sn^(m'z)

(4.19)

We further note that eqs. (4.12) and (4.13) are very similar. In fact, if we were to average cos 6 and cos <^ they would be identical. That is.

<Q > ^e <Q > ^m %n\h)

where

r- K E'

4

(4,20)

(4.21)

So that the same correction factor can be applied whether the dipole is electric

or magnetic. It is interesting to note that this is the same result obtained by

D.M. Kerns in his analysis of dipoles located in an ordinary coaxial line [13].

His result for the correction factor also contains the same (1/frequency)

dependence, but the constant Q which contains the geometrical constants characteristic of the particular transmission line (through Z ), is different. Equation (4.20) is plotted in figure 4 and shows how the radiation resistance is

12 3

Figure 4. Correction factor as a function of frequency.

83
reduced as the frequency is increased. Values of Q are included in tables 1-2

for dipoles located at various places inside a TEM cell. Each table corresponds

to a different TEM cell geometry.

Table 1

. the ratio of the radiation resistance inside the TEM cell to

^o

that in free space for k b = 1. Dimensions of the TEM cell:

a = 25 cm, b = 25 cm, w = 20.64 cm,

y(cra)

y=b—25 0.220 0.205 0.158 0.091 0.027 0.000

20 0.236 0.222 0.181 0.119 0.057 0.031

15 0.248 0.277 0.255 0.217 0.172 O.ISO

10 0.357 0.365 0.390 0.435 0.482 0.497

5 0.432 0.460 0.567 0.866 1.507 1.696

0.465 0.503 0.665 1.281 14.322 4.216

/

Center of

TEM cell

10 15 20. x=w

25

t x=a

x(cm)

Table 2

Q , the ratio of the radiation resistance inside the TEM cell to

^o

that in free space •^ for k^b = 1. Dimensions of the TEM cell:

25 cm, b = 15 cm, w = 18.025 cm.

yCcm)

y=b—IS5 0. 303 291 0. 247 0.159 0.051 0.000

2 0. 307 297 259 0.177 0.071 0.020

9 318 314 294 0.238 0.140 0.087

6 332 336 350 0.367 0.291 0.217

3 344 .356 412 0.622 0.642 0.403

349 .364 .441 0.901 1.262 0.512

Center of

TEM cell

10 15

x^w

20 25 xfcm)

f x=a

84
5. SUMMARY AND CONCLUDING REMARKS

We have calculated in this report the exact capacitance and characteristic

impedance of a TEM cell, as given by eqs . (2.11) and (2.16). In addition,

approximate formulas given by eqs. (3.8), (3.9), and (3.16) were obtained

which agree very closely with the exact results. Finally, the electric field

distribution of the TEM mode, given by eq. (4.14), was obtained analytically

and was used to derive expressions for the radiation resistance of simple dipoles

located inside a TEM cell as given in eqs. (4.8) and (4.9). These expressions,

when compared to their free-space counterparts, allows one to evaluate the

change in the radiation characteristics of those sources which can be modeled

by dipoles when located inside the TEM cell as compared to free-space.

How can these results be used to interpret measurements made on a practical

source radiating inside a TEM cell, such as a piece of electronic equipment?

When can a practical source be modeled by a collection of elementary dipoles?

If the source can be modeled by a collection of dipoles, how can we determine

the magnitudes and directions as well as the type (electric or magnetic) of the

dipoles which represent the source? These are all questions which should be

asked, and which, as yet, have not been completely answered. We will, however,

indicate some of the problems and some ideas that might be used to solve them.

When we model the source by a dipole, we are assuming that the source is

"electrically small"; that is, its dimensions are small compared to the operating

wavelength. The useful upper frequency limit of the TEM cells used at NBS is

typically 100 MHz. At this frequency the wavelength is 3 meters, so that the

dimensions of most electronic equipment will be only a fraction of a wavelength

and can therefore be modeled by dipoles. of both electric and magnetic type.

This means, of course, that we are not interested in determining where leaks occur from a piece of equipment since we have assumed a "point" source. Assume, now, that we have a collection of electric and magnetic dipoles

all operating in phase at the same frequency, which is under our control. By

superposition, all of the electric dipoles can be combined into one equivalent

electric dipole. Similarly, all of the magnetic dipoles can be combined into

an equivalent magnetic dipole. We would like to be, able to determine the magnitudes and directions of these equivalent dipoles. From eqs. (4.4) and (4.5) the

total power radiated is given by:

2V (^e£f I cos 8)^ + 27rAI cos ((i m ^

2 1 (5.1)

85
At low frequencies, eq. (5.1) is dominated by the first term, so that P « P . At high frequencies, the second term dominates, and P « P . Therefore, by

judiciously choosing the frequency, we can make the source appear as if it were either an electric or magnetic dipole. Whether or not we have chosen

a frequency such that the source appears as a single type of dipole and not a "hybrid" dipole should be verified by measuring the power as a function of

frequency over a narrow frequency band. The electric dipole will exhibit a characteristic that is independent of frequency, whereas the characteristic

of the magnetic dipole will increase as the square of the frequency. If we are not able to adjust the frequency so that the source appears as only one type of

dipole, then we will have to take measurements at two selected frequencies and

fit the data to eq. (5.1); so that even in this case, the separation of the

contributions from the electric and magnetic dipoles is not a problem.

Now assume that we are able to obtain a single dipole source by adjusting

the frequency. By measuring the power, we can determine the magnitude of the

dipole moment times the cosine of some angle; (fc_r£ I) cos 6 for the electric

dipole, and (AI ) cos 4) for the magnetic dipole. In order to separate the

angular dependence from the dipole magnitude we can perform the following

experiment

(1) Rotate the source until the radiated power is zero, thus aligning the

electric or magnetic dipole perpendicularly to the respective field.

(2) Rotate the source about the longitudinal axis of the TEM cell (z-axis).

If the power radiated does not remain zero, return the source to its

original position.

(3) Rotate the source slightly about the x axis for magnetic dipoles or the

y axis for electric dipoles.

(4) Repeat steps (2) and (3) until the radiated power remains zero upon

rotation about the z-axis.

(5) The dipole is then aligned along the z-axis.

Thus, we can rotate the electric or magnetic dipole 90** around the x or y axis

respectively, and obtain cos 6 = 1 or cos 41 1 . With the dipole so oriented,

we can measure the magnitude of the dipole moment by measuring the power

radiated.

The procedure just described is more complicated if we have both an electric

and a magnetic dipole. In this case, we may not be able to obtain a null in

the power by rotating the source. By taking measurements as a function of frequency and rotation angle of the source, however, we can separate the contributions from both the electric and magnetic dipoles, as well as their respective

orientations. Thus, having found the equivalent dipoles that represent the

source, we are able to predict the radiation characteristics of the source in

free space. It should also be noted that in this report we have addressed

ourselves to the problem of radiated emissions testing of electrically small

devices; by reciprocity, however, the results are applicable also to the problem

of susceptibility testing.

86
The usefulness of the technique just described hinges on the validity of

the following assumptions that were made initially. (1) The equivalent dipoles

that represent the source have dipole moments that are constants, independent

of frequency. (2) The directions of the dipoles are not a function of frequency. (3) The electric and magnetic dipoles operate in phase at the same frequency, and (4) the operating frequency is under our control. These

assumptions will now be examined.

In order to understand why it seems reasonable to model a practical source

by an equivalent electric and magnetic dipole operating in phase at the same frequency, we will borrow some of the results from the theory of excitation of

waveguides by small apertures [14] . We will consider a practical source to

consist of an electrically small conducting box housing low-frequency AC circuits. The box can have numerous apertures through which electromagnetic energy

may radiate; however, we restrict these apertures to be small compared to the

size of the conducting box. The coupling through a small aperture can be modeled

by replacing the aperture by an equivalent electric and magnetic dipole. The

magnitude of the electric dipole is proportional to the normal electric field

that would be present at the aperture assuming the aperture is replaced by a perfect conductor. Similarly, the magnitude of the magnetic dipole is proportional to the tangential magnetic field that would exist at the aperture.

It seems reasonable to assume that at the apertures no normal electric fields

would exist, since this would imply that a net accumulation of charge would

exist inside the box. However, there could exist tangential magnetic fields as

a result of any current- loops in the circuitry. Thus, we will model the apertures by magnetic dipoles only. These dipoles will induce currents and charges

on the outside surface of the conducting box, which can, in turn, be modeled by

an equivalent electric dipole and a magnetic dipole. These dipoles will obviously

not have magnitudes that are independent of frequency; however, at each frequency,

the relationship between the magnitude of the electric and magnetic dipole will

remain fixed, if we assume that the box contains only one source which induces

both dipoles. Since the box is electrically small and contains only one source,

the equivalent dipoles will also operate in phase. By the same argument, their

directions will not be a function of frequency. Therefore, we can separate

the contribution to the total power radiated from each dipole as described

previously by measuring the power radiated at two selected frequencies. Since

in some cases we do not have the operating frequency under our control, this

will not always be possible unless our source radiates at more than one frequency, or we know a priori from physical grounds that the source can be modeled

by only one type of dipole.

87
6. REFERENCES

1] Adams, J.W., Electromagnetic Interference Measurement Program at the

National Bureau of Standards, USNC/URSI, Commissions I-VIII, Annual

Meeting, p. 33 (Oct. 20-23, 1975).

2] Crawford, M. L., Generation of standard EM field using TEM transmission

cells, IEEE Trans, on Electromagnetic Compatibility, EMC- 16 , No. 4 (Nov. 1974).

3] Bowman, F. , Introduction to Elliptic Functions with Applications, Chapter

II (Dover Publ. Inc., New York, 1961).

4] Ibid. , Chapter I.

5] Abramowitz and Stegun, Eds., Handbook of Mathematical Functions, Sth Ed.,

p. 612 (Dover Publ. Inc., New York, 1968).

6] Ibid. , Sec. 17.6, p. 598.

7] Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis, 4th Ed.,

p. 486 (Cambridge Univ. Press, 1969).

8] Chen, T.-S., Determination of the capacitance, inductance, and characteristic impedance of rectangular lines, IRE Trans, on MTT, MTT-8 , No. 5,

510-519 (Sept. 1960).

9] Cohn, S.B., Shielded couplecJ-transmission line, IRE Trans, on MTT, MTT

, No. 5, 29-38 (Oct. 1955).

Anderson, G. M. , The calculation of the capacitance of coaxial cylinders

of Tectangular cross-section, AIEE Trans., 69^, pt. II, 728-731 (1950).

Collin, R.E., Field Theory of Guided Waves, pp. 200-204 (McGraw-Hill, New

York, 1960).

Ramo , Whinnery and Van Duzer, Fields and Waves in Communication Electronics,

pp. 645, 657 (Wiley, New York, 1967).

Kerns, D.M., unpublished work (Aug. 1965).

Collin, R.E., op cit. , pp. 285-302.

Walker, M. , The Schwarz-Christof fel Transformation and Its Applications -- A Simple Exposition (Dover Publ. Inc., New York, 1964).

Bowman, F., op cit

, p. 58.

Bowman, I"., 0£ cit

, p. 57.

Churchill, R.V. , Complex Variables and Applications, 2nd Ed., p. 209

(McGraw-hill, New York, 1960).

Abramowitz and Stegun, 0£ cit

, Sec. 17.6, p. 598, Sec. 16.4, p. 571.

Abramowitz and Stegun, 0£ cit

, Sec. 16.21, p. 575.

88
APPENDIX A

Calculation of the Capacitance of a TEM Cell

A cross- section view of the TEM cell is shown in figure 5 with an x-y

coordinate system superimposed.

Figure 5. Cross-section of a TEM cell.

The center septum of width 2w is located symmetrically inside the cell of height

2b and width 2a and is assumed to have negligible thickness. In addition, the

septum is located a distance g from each vertical side wall and is embedded in

a homogeneous dielectric of permittivity, e . For convenience, some key points

in the cell have been labeled A through F. The reason for choosing an unsymmetrically located coordinate system is to facilitate obtaining the approximate

formula for the capacitance given in section 3. To determine the capacitance, the method of conformal transformation will

be used, whereby the structure in figure 5 is transformed into a simpler con- figuration whose capacitance is already known. Since it is well known that

capacitance is invariant under a conformal transformation, the formula obtained

will also be applicable to the TEM cell.

Since we have symmetry with respect to the septum, we will calculate the

capacitance between the upper plate, A-F-E-D and the center septum B-C. The

total capacitance is then twice this figure, since we have effectively two

capacitors in parallel. The region A-D-E-F may be mapped into the upper-half

of a complex t-plane via the Schwarz -Christof fel transformation [15] which, due

to symmetry, can be expressed in terms of Jacobian elliptic functions. The

transformation is given by [16]:

or alternatively by;

mz / dt

[4t(l-t)(l-k2t)]

t = sn* (mz ,k)

89

(A.l)

(A. 2)
where sn is a Jacobian elliptic function of modulus, k;

mil

b

(A. 3)

and

z - X + iy. (A. 4)

Here K(k) and K(k') are complete elliptic integrals of the first kind of moduli

k and k' respectively and:

k' - [1 - kM** (A. 5)

The modulus k can be determined from the requirement that:

K(k) . 2a

K(k') b

(A. 6)

Under the transformation given by eq. (A. 2), the region, A-D-E-F, in the z- plane is mapped into the upper half of the t-plane as shown in figure 6.

°o ^0 1 1/k^

H h A B C D

Figure 6. Complex t-plane.

Using eq. (A. 2) and elliptic function identities, a and P can be calculated

as

:

a =• sn^mg = sn^5

Sq " sn^m(2a-g) - cnH/dn^

(A. 7)

(A. 8)

where

5 = mg (A. 9)

.nd en and dn are also Jacobian elliptic functions all of which have modulus, k,

For convenience, we now make an intermediate transformation from the t-plane to

a complex u-plane defined by:

u-^a

t

- %'

' ^-

°oJ

(A. 10)

The u-plane is shown in figure 7.

90
1 H h

B C

1/X' —HFigure 7. Complex u-plane.

X in figure 7 can be found using eq. (A. 10) and substituting t 1. Thus:

'l - a

8 - a ^^o o-*

(A. 11)

Substituting for a and g from eq. (A. 7) and eq . (A. 8), we obtain:

^2 ^ cnH - snH dnH

cnHll-sn^]

Using elliptic function identities, eq. (A. 12) reduces to

.2 fsnir 1 - k en E, and defining a complementary modulus, X' as:

X' =

We have from eqs . (A. 13) and (A. 14):

X' = [1 - X^**

(A. 12)

(A. 13)

(A. 14)

X- - k'(|^]' (A. 15)

In the final transformation, we map the upper half of the u-plane into a rectangular region in a complex X'Pl^ne. The transformation is given by:

u = sn2(x,X) (A. 16)

and the X'Plane is shown in figure 8.

B K(X)

Figure 8. Complex X'Pls^®-

91
From figure 8, it is evident that the capacitance is just given by the ordinary

parallel plate capacitor formula, that is:

C - ^

CA.17)

where A is the cross-sectional area;

d is the plate spacing;

and e is the permittivity of free space.

Substituting for A and d in terms of K(A) and K(X') we obtain: — = ^^^ (A. 18)

Eq K(X')

where C is now interpreted as the capacitance per unit length. Therefore, the

total capacitance of the TEM cell per unit length is just twice that given by

eq. (A. 18).

_2_ = 2 ^^^^ (A. 19) e« K(X')

92
APPENDIX B Calculation of the Electric Field Distribution

Inside a TEM Cell

In order to evaluate eqs . (4.12) and (4.13), we need to calculate the

magnitude of the electric field, E , inside the guide. This is most easily

done using a different coordinate system as well as a Schwarz -Christof fel transformation similar to that used in Appendix A. A cross-section view of the TEM

cell is shown in figure 9 with a symmetrical x-y coordinate system superimposed,

Y

- X

Figure 9. Cross-section of a TEM cell.

The region A-D-E-F can be mapped into the upper-half of a complex t -plane

via the following transformation [17]

where

t = sn(m'z,k)

- MXL = K(k')

a b

(B.l)

(B.2)

Nute that this transformation is not the same as the one used in Appendix A to

solve for the capacitance, since a different coordinate system was used. The

symbols used in this appendix should not be confused with those in Appendix A. The complex t -plane is shown in figure 10.

-1 —^- -a

Figure 10. Complex t -plane,

93
From eq. (B.l), a is given by:

a - sn(m'w,k) (B.3)

For convenience, we now make an intermediate transformation from the t -plane

to a complex u -plane defined by:

% " ^o/«

The u -plane is shown in figure 11

(B.4)

-1 -4-

1 a H h

Figure 11. Complex u plane.

Finally, we map the upper-half of the u -plane into a complex X-'Plane

defined by:

"o

" ^"(XQ.ct)

The Xfj'Plane is shown in figure 12.

(B.5)

^oi

1 A

1 D

, 1

K(a')

-• 2K a) •- C or

Figure 12. Complex Xo'Plane.

In order to calculate the electric field, we must find the complex potential, F,

which is given by [18]:

F = ^CXq) * HUq) (B.6)

where 4i(Xq) is the potential function;

and

ij/(x ) is the stream function;

F satisfies the Cauchy-Riemann equations.

In addition the potential <KXq) must satisfy the following boundary conditions

and

(Xq) = on BC

(Xq) = V on AD.

94
It is easily verified that the following solution satisfies all of the above

requirements

and

MxJ

I^CXq)

Vx •

K(a')

Vx

or KCa')

(B.7)

(B.8)

where

Y - Y + lY . ''o ''or ^0 1 The electric field, E, is defined by;

E = -V* = -(1^ * * i ly *) 3x

^ ^ i Irr li-l = ^

F*

3x ^) ar (B.9)

The real part of E gives the x-component of the electric field, and the imaginary

part of E gives the y-component of the electric field, t— may be calculated

as follows:

dF ^ d^

^^ ^

dz dXo du^ dt^ dz

(B.IO)

dF Using eq. (B.l) andeq. (B . 4) .through eq. (B.8), g— may be evaluated as

dF ^ -iVm' dnCm'zj)

dz K(a')[P^(z)]

(B.ll)

where

Pq(z) = [snMm'w) - snMm'z)]

Thus, the magnitude-squared of the electric field, E^ is given by

^0

Vm' ^K(a')

5.1 pccted from the edge condition.

dnMm'z)

sn^ (m'w) - sn^ (m' z)

2

(B.12)

(B.13)

It is easy to see from eq. (6.13) that for z = ±w, E^ goes to infinity as ex95
Cciuation (B.ll) was evaluated numerically for some typical TEM cell

;:ometries and used to calculate the x and y components of the electric field,

;is well as the magnitude and polarization angle of the electric field defined by;

6 » arctan !i (B.14)

Tlitvse results are included in tables 3 through 10 where all field quantities

li.ive been normalized to V/b. In addition, the relative electric field distri

iMition is plotted in figure 13. The top graph in that figure contains experi'

icntal data measured by M.L. Crawford [2] for comparison.

K w l>

(1) 'a3

+J •HC

CJi (0

•a

(U •H

UH

U

+JV

o

0.8 -

0.6

-1.0 -.8 -.4

x/a

.2 .6 .8 1.0

1.0 -

0.8 -

O.f)

-1.0 -.8 -.6 -.4

x/a

1.0

Figure 13. Relative electric field distribution inside a TEM

cell. Parameter indicates distance from the

center septtun (y/b) .

96
Table 3 x-component of the electric field in a TEM cell of dimensions

a = 25 cm, b = 25 cm, w = 20.64 cm normalized to V/b

.

y(cm)

y=b-*25 000 000 0.000 0.000 0.000 0.000

20 000 060 0.129 0.208 0.278 0.307

15 000 108 0.245 0.422 0.600 0.680

10 000 127 0.311 0.620 1.029 1.237

5 .000 090 0.248 0.647 1.684 2.285

000 000 0.000 0.000 0.000 3.603

Center/o TEM cell

f

5 10 15 20, x=w x=a

x(cm)

Table 4 y-component of the electric field in a TEM cell of dimensions

a = 25 cm, b = 25 cm, w = 20.64 cm normalized to V/b.

yCcm)

y=b^25 824 793 0.698 .530 .289 000

20 a5 3 825 0.736 .568 .315 000

15 935 917 0.852 .699 .410 000

10 1 049 1 052 1.051 .977 652 000

5 1 153 1 186 1.298 1 499 1 343 000

1. 196 1 245 1.431 1 986 6 640 000

Center of

TEM cell

10 15 20. x=w

25

f x=a

x(cm)

97
Table 5 Magnitude of the electric field in a TEM cell of dimensions:

a = 25 cm, b = 25 cm, w = 20.64 cm normalized to V/b.

y=by(cm)

-*25

20

15

10

5

Center of

TEM cell

0.824 793 0.698 0.530 0.289 0.000

0.853 8^2 7 0.747 0.605 0.420 0.307

0.935 924 0.886 0.817 0.727 0.680

1.049 1 060 1.096 1.157 1.218 1.237

1.153 1 189 1.321 1.633 2.154 2.285

1.196 1 245 1.431 1.986 6.640 3.603

10 15 x=w

25

1 x=a

x(cm)

Table 6

Polarization angle of the electric field in degrees in a TEM

cell of dimensions: a = 25 cm, b = 25 cm, w = 20.64 cm.

y(cm)

y=b-*2 5 90 .00 90 .00 90 .00 90 .00 90 .00 -• 20 90 .00 85 86 80 .05 69 .89 48 .54 00 00

15 90 00 83 27 73 97 58 89 34 35 00 00

lo' 90 .00 83 14 73 50 57 60 32 36 00 00

5 90 .00 85 64 79 .20 66 67 38 56 00 00

90 00 90 00 90 00 90 .00 90 00 00 00

Center of

TEM cell

10 15 20 25

f f x=w x=a

x(cm)

98
Table 7 x-component of the electric field in a TEM cell of dimensions:

a = 25 cm, b = 15 cm, w = 18.025 cm normalized to V/b.

y(cm)

y=b-*15 000 000 000 .000 000 000

12 000 024 067 .143 220 249

9 000 040 121 .284 462 517

6 000 043 141 .410 763 .817

3 000 028 101 .440 1 247 1 112

000 000 .000 .000 1 969 1 .254

10 15

Center of

TEM cell

f x=w

20 25

f x=a

x(cm)

Table 8 y-component of the electric field in a TEM cell of dimensions

a = 25 cm, b = 15 cm, w = 18.025 cm normalized to V/b.

y(cm)

y=b—15 966 946 .872 .698 .394 000

12 972 956 890 .724 .411 000

9 989 981 944 807 464 000

6 1 010 1 015 1 028 979 557 000

3 1 028 1 046 1 120 1 311 645 000

1 035 1 058 1 164 1 664 000 000

Center of

TEM cell

10 15

1 x=w

20 25

4 x = w

x(cm)

99
Table 9 Magnitude of the electric field in a TEM cell of dimensions:

a = 25 cm, b " 15 cm, w =» 18.025 cm normalized to V/b.

y(cm)

=b—15 0.966 0.946 0.872 0.698 0.394 0.000

12 0.972 0.956 0.892 0.738 0.466 0.249

9 0.989 0.982 0.951 0.856 0.655 0.517

6 1.010 1.016 1.038 1.062 0.945 0.817

3 1.028 1.046 1.125 1.383 1.404 1.112

1.035 1.058 1.164 1.664 1.969 1.254

10 15

Center of

TEM cell

t

. x=w

20 x=ax(cm)

Table 10

Polarization angle of the electric field in degrees in a TEM

cell of dimensions: a = 25 cm, b = 15 cm, w = 18.025 cm.

y(cm

y=b-^l-5

12

9

6

3

Center of

TEM cell

90 00 90 00 90 .00 90 .00 90 .00 _

90 00 88 58 85 .68 78 80 61 76 00 00

90 00 87 66 82 70 70 .61 45 12 00 00

90 00 87 60 82 20 67 26 36 15 00 00

90 00 88 48 84 82 71 43 27 36 00 00

90 00 90 00 90 00 90 00 90 00 00 00

10 15

1 x=w

20 x=a

x(cm) i

100
APPENDIX C Numerical Evaluation of Complete Elliptic Integrals and

Jacobian Elliptic Functions

The method used to evaluate elliptic functions and elliptic integrals is

given in reference [19], a summary of which follows. Assuming that the modulus

of the particular function desired is k, we begin by defining the following:

a^ - 1 b^ - k- c^ = k (C.l)

Then we set up an Arithmetic-Geometric Mean (AGM) table defined recursively

as follows:

^1 = ifV^) h = (%»'o)'* ^1 I^^-^o)

32 = jCa^^b^) b2 = (a^b^)'* c^ = yCa^-b^)

^N = I^^N-l^-l^ ^N = ^^N-lN-l^'' '^N " I^^N-r^N-l)

Stopping at n = N when Cjg = to the degree of accuracy required.

The complete elliptic integral K(k) is then given by:

K(k) = j^ (C.2) ^a^

The Jacobian elliptic functions, sn(^), cn(C), and dn(C) can be determined

by calculating 4ijg in degrees where

*N " 2 ^N ^ "¥— (C-5)

and then computing recursively, 4>jg.n, 4'm.2» *'*

"^i* "^o ^^ using:

c sin(24.^.1-(^ ) • — sin ^^ (C.4)

%

The Jacobian elliptic functions are then given by:

sn(5,k) - sin g (C.5)

cn(5,k) - cos o (C.6)

cos (|) dn(C,k) - 2_ (C.7)

COS(^j^-(*q)

101
In eqs . (C.5) through (C.7), the argument, 5» was assumed to be real. If the

argument is complex then the following formulas from reference [20] can be

used.

s*d, + ic'd'S, 'C,

c^ + k^'s^'s^

C'C, - is'd'S, 'd,

cn(C,k) = i

2 2 2 ^^'^^

c, + k 's 'S,

2 d«c, 'd, - ik 'S'C'S,

dn(^,k) = S 3 2 2 (CIO)

cj + k 'S 'S^

where C

" C^ + iC^^

and s = sn(^^,k) s^ = sn(C^,k')

c = cn(5^,k) c^ - cnCC^.k')

d = dn(Cj.,k) dj = dn(q,k')

Two Fortran subroutines, AGM and SNCNDN, were used to perform the calculations

described in this appendix. When AGM(K) is called, the AGM table described

above is defined. SNCNDN (X, SN, CN, DN) can then be called with real argument X to return the functions desired. A copy of these subroutines follows.

SUBROUTINE AGM(K)

COMMON A«20),3(20)ie(20)*L

REAL K A(l)=l.

B(1)=S0RT(1.-K«K) . C(1)=K

DO 10 1=2,20

A(I) = .5*(A(I-n*B(I-l))

B(l)=SCRT(fl(l-l)«8<I-l))

C(I)=.5»{A(I-l)-B(I-l))

1F(C(I) .LT.l.E-6) GOTO 30

10 CONTINUE

PRJinT 20

20 FORMAT (IX, *AGM FAILF.D TO CONVERGE IN 2o I7ERaTI0In'S«)

30 L=l

RETURN

Elv-p SUBROUTINE SNCNDN(X.SN,CN,DN)

COMMON A(20) »B(20) iC(20) iL

DIMENSION P(20)

P(L)=A<L)*X«»2.»»(L-1)

I=L

10 P (I-l

s.5»{ASlN( (C(I ) /A (1))»SIN(P (I) ))?{!)>

1 = 1-1

IF(I.NE.O) GOTO lo

SN=S1N(P(1))

Cn=C0S(P(1) ) 0n=Cn/C0S(P<2)-P(i))

RETURN

END

102
NBS.1UA (REV. 7-73)

U.S. OEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. WUHLKAI ION OK KKI^OKT NO. NBSIR 75-829

2. Ciov 't Aciission

No.

3. Kii- ipiiiii 's Ai L i-,..ioii \,

4. in LI". AND SUBTITLI-: RADIATION CHARACTERISTICS OF DIPOLE SOURCES

LOCATED INSIDE A RECTANGULAR, COAXIAL TRANSMISSION LINE

5. Puhlioiion D.iii- January 1976

6. Ptrformiii^ Or^.uu/.ii loii < 276.00

7. Ai;illOH(S)

John C. Tippet and David C. Chang

8. I'crforniiiif; t)r^aii. Kipmi Ni NBSIR 75-829

9. PLKIORMINC; ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Pro|iit/T.isk, Work Unii No 2768281

n . ( onu .11 I (ir.inl N'o.

12. Spunsoring Organization Name and Complete Address (Street, City, Siete, ZIP)

Same as Item 9.

13. Type of Ripori & PiriDil Covered

14. Sponsorinj; Aginc y < o>l<

IS. SUPPLEMENTARY NOTES

16. AUSTRAC.T (A 2 00- word or less tactual summary of most signidcanl information, ff document includes a signidcarjl

bibliography or literature survey, mention it here.) When making EMC measurements inside a shielded enclosure, the radiation

characteristics of the device being tested changes. In this report the change

in radiatioa resistance of dipole sources located inside a National Bureau of

Standards TEM transmission cell is determined. In many cases a practical device

can be modeled by dipole sources. In these cases, the analysis allows one to

predict the device's radiation characteristics in other environments, e.g., free

space.

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) EMC measurements ; radiation resistance; rectangular coax;

shielded strip line; TEM cell.

18. AVAILABILITY £] Unlimited

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USCOMM UU 103

Appendix E

Kanda, M. "Electromagnetic field distortion due to a conducting rectangular

cylinder in a transverse electromagnetic cell."

Reprinted from IEEE Trans. Electromagn. Compat. EMC-24(3) : 294-301 ; 1982

Augus t

.

105

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-24, NO. 3, AUGUST 1982 Electromagnetic-Field Distortion Due to a Conducting

Rectangular Cylinder in a Transverse

Electromagnetic Cell

MOTOHISA KANDA, member, ieee

Abstract—This paper deals with the distortion efTects in a TEM cell resulting from loading by the object under test. To insure that higher order modes are ruled out as contributing factors, a brief discussion of the cutoff frequencies of these modes is presented. The paper then proceeds to a theoretical and experimental analysis of the loading

effects. In the theoretical analysis for the loading effects, i.e., the electromagnetic-field distortion caused by an object under test in a TEM cell, the frequency-domain integral equation for the magnetic

field, or equivalently, the current density on the surface of a perfectly conducting cylinder in a parallel-plate waveguide, is solved by the method of moments to predict the degree of magnetic-field distortion. The experimental investigations are performed by mounting a number of electrically small half loops on the surface of the con- ducting cylinder in a TEM cell. The loading effects in terms of magnetic-field distortion are analyzed as the ratio of one of the object dimensions (height) to the separation distance between the inner conductor and the ground plane of the TEM cell. Also, the response of an electrically small loop to both the magnetic and electric components of the electromagnetic field is used to measure the phase

relation between the magnetic and electric fields, which in turn can be used to determine the degree of degradation of the TEM mode due to the presence of the conducting cylinder. These theoretical and

experimental results are compared with the available quasielectrostatic results. Key Words—Electromagnetic compatibility (EMC), Green's

function, integral equation, linear equation, method of moments,

parallel-plate waveguide, quasi-electrostatic, TEM cell.

I. INTRODUCTION

THE STUDY of electromagnetic compatibility (EMC) (in- cluding the electric and biological effects due to electro- magnetic (EM) radiation) and EM calibration require accurate measurement techniques for defining EM interference (EMI)

characteristics. Thus, fully enclosed, rectangular, transverse electromagnetic (TEM) transmission lines with thin inner conductors are often used for generating standard known test

fields. In all cases, it is desirable that only the dominant TEM

mode should propagate. Thus the limitations of these struc- tures are discussed in this paper in terms of the higher order modes and the loading effects. First, presented below is a brief discussion of the modal considerations -teased on existing

theory. Results are presented which upper bound the fre- quency to which the TEM cell should be limited in order to suppress the higher order modes. Then, the loading effects,

i.e., the electromagnetic-field distortions caused by an object

Manuscript received April 10, 1981; revised February 10, 1982. The author is with the Electromagnetic Fields Division, National Bureau of Standards, Boulder, CO 80303. (303) 497-5320.

t'

1 t, L-2H 2b

t -2a—"

I I I L

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4,5 5.0

a/b Fig. 1. Curve of equal TEqi and TEjo cutoff frequencies for a trans- verse electromagnetic cell.

under test in a TEM cell, are treated in detail giving analysis

and experimental support.

II. MODE SUPPRESSION

The higher order mode cutoff frequencies of the rectangular TEM cell have been well studied by many workers ( 1 ] - [4]

In a rectangular hollow waveguide, the dominant mode is always the TEiq mode as long as the width exceeds the

height. However, the same conclusion does not hold for the

rectangular line with an inner conductor, even if its thickness is infinitesimaUy small. In fact, it is found [1], [2], [4] that, depending on the width of the inner conductor and the size of

the TEM cell, the cutoff frequency of the TEqi mode can be

much lower than that of the TE, q mode as illustrated in Fig. 1

While the cutoff frequency of the TEjo mode is simply cal- culated from

/.=

2\/m€ — + —

a h

n=0

lay/ile

(1)

the cutoff frequencies of the TEqi mode are much more in- volved and have been calculated by many workers [l]-[4] as shown in Fig. 2. Let us consider, for example, a typical SO-fl TEM cell having a width of a = 1 m, and a height of b =

0.6 m, and a width of inner conductor of j = 0.72 m. While

the cutoff frequency of the TEjo mode is 150 MHz, the cut- off frequency of the TEoi mode is approximately 135 MHz,

according to Fig. 2. It can also be shown that the cutoff fre- quencies of all TM modes of a TEM cell are always higher than

Reprinted

August.

U.S. Government work not protected by U.S. copyright

frcm IEEE Trans. Electromagn. Ciarpat. E1K:-24(3)

107

294-301; 1982
KANDA: ELECTROMAGNETIC-FIELD DISTORTION

0.9 1.0

Fig. 2. Normalized TEqi cutoff wavelength for a transverse electro- magnetic cell.

those of their hollow-waveguide counterparts. Thus, the dominant cutoff frequency (i.e., the lowest) is either the TEqi

mode or the TEjo mode. It is interesting to note that the cut- off frequency of the TEqi mode decreases as the gap between

the inner conductor and the wall of the TEM cell becomes nar- nower. This phenomenon has also been observed in a ridge

waveguide [5] and is associated with the infinite-gap capacitance.

III. LOADING EFFECTS

In EMC measurements, an object under test is placed

inside a TEM cell. The field from the TEM-mode incident upon the scattering object is identical to that of a plane wave

in free space. However, the scattered field produced by the

object in the TEM cell is different from the scattered field pro- duced by the object in free space because of multiple reflec- tions from the TEM-cell walls, or equivalently, because of the mutual coupling between the object and the TEM cell. Placing a test object in a TEM cell is equivalent to introducing a capacitive discontinuity. For low frequencies, where the

transverse dimensions of the object are negligible compared to the wavelength, the discontinuity due to the object is a pure

capacitive reactance, and may be regarded as the fringing

capacitance of the corresponding electrostatic problem. Under

this assumption, the ratio of the electric-field strength near

the metal object in the TEM cell to the unperturbed electric-field strength at the same location in a pure TEM mode

has been calculated by Meyer [6] and is shown in Fig. 3.

In the theoretical analysis of the loading effects, i.e., the

electromagnetic-field distortion caused by an object under test

in a TEM cell, the frequency-domain integral equation for the magnetic field, or equivalently, the current density on the

surface of a perfectly conducting cylinder in a parallel- plate waveguide, is solved by the method of moments to predict the degree of magnetic-field distortion. For the purpose of

mathematical tractability, a parallel-plate waveguide is used to

30

2.5

o

1 2.0

S 1.^

1.0 0.0 0.2 0.4 0.6 8

h/b

1.0

Fig. 3. Electromagnetic-field distortion due to a metallic cube in a transverse electromagnetic cell (electrostatic approximation).

'SdScm):^

b (30 cm)

li^

Fig. 4. Coordinate system of a rectangular cylinder in a parallel-plate waveguide.

model a TEM-cell structure. The results given in this paper are

for the magnetic-field strength on the surface of the cylinder.

Other related quantities, such as the electric-field strength

and the Poynting vector, can be readily derived from the sur- face current by use of Maxwell's equations.

A. Theory

The problem of detenmining the electromagnetic field scat- tered by a perfectly-conducting rectangular cylinder has been

studied by many workers [7] -[9]. The coordinate system

used to analyze the current density on the surface of the rec- tangular cylinder in a parallel-plate waveguide is shown in Fig.

4. The source of the TEM wave is a delta-function voltage source of the form

M^yV^iuy^iz-z^) (2)

where Vq is the magnitude of an equivalent voltage source

located at the source location Zq. Since there is no variation in the>'-direction and the voltage source has only a>'-component, we will have a scalar wave equation in Hy, i.e.,

l^Hy+k'^Hy^'ji^eMy- (3)

where V^ = (d^/9x^) + (9^/9z^) and k is the free-space wave- number. The boundary condition is that the tangential E field be zero on ail of any reflecting conducting surfaces, i.e..

= (4)

where n = x and z. The solution of (3) may be found through the Green's

108
Function technique

V^G(r, r) + k^G{r, r) = -5(x -x')b(z - z').

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-24, NO. 3, AUGUST 1982

at X = and x = b, one finds

(5)

Multiplying (3) by G(r, r), (5) by Hy(r'), subtracting the two results, and integrating over the free-space volume v

f

[G(ry)V^Hy(r')-Hy(r')S/^G(r,r')] dv'

= /cje j G(r,r')My(r')du'

+ JHy(r')5(x-x')8(z-z')dv: (6)

u

Evaluating the second integration of the right-hand side of (6) and using Green's theorem on the left-hand side, we get

Gix,z;x',z') = — re/*<"*^-) -t-e'*' 2kb

z-z h

rnrx rnrx cos — cos b b

jg/T„(z + r) ^.gyT„!z-z|j

where

r„=v^ =

(12)

(13)

/[

,JHy{r') „

.„9G(r,r')

^^^^'^-tr'"^''' 3.'

ds'

I

= /we / G(r, r')My{r') dv + Hy(r). (7)

Once the Green's function is obtained and the source is specified, (9) can be solved by the method of moments. The tech- nique used is discussed briefly below. A detailed discussion on

this subject is given by Harrington [10]

The unknown Hy is expanded in terms of a set of known

basis functions /)• with unknown coefficients a,, i.e.,

On the perfectly conducting surface s of the rectangular cylinder, we set the boundary condition as Hy=X °^ifi- (14)

1=1

^Hy{r-) = 0. (8)

Substituting (14) into (9) we obtain

, , bG{x,z\x',z')

Equation (7) then reduces to

Hy(r) = -/we j G(r, r')My(r') dv

V

, 9G(r,r')

Soi

i=\

ff(x,z) + n dn'

ds'

=fr'"'(x,z). (15)

-/ fiy(r)

dn'

ds'

Equation (15) is a single equation with n unknowns a,. To

create at least n linear equations, a set of testing functions w.- (9) is introduced, and the inner products (integral over the sur- face) of both sides of (15) for each Wj are set equal. The result where s indicates the surface of the rectangular cylinder. The "s a set of linear equations of the f6rm

first term on the right-hand side of (9) corresponds to the in- cident field, and the second term to the scattered field. The

known incident magnetic field is given by

//'"'=(x,z) = -/we / f

Vo8(z'-Zo)

where the matrix element Zij is given by

•G{x,z,x',z)dx' dz

(10)

Zii= [ fiir)+ //.<r')^%^ ds'

and the column matrix element Vj is given by

Vj = j fr''\r)wi(r) ds.

(16)

wj(r)ds (17)

(18) Equation (9) is an integral equation which can be solved for Hy(r) once Green's function G(r, r) is obtained.

The Green's function is the solution of (5). By impos- The unknown coefficients Oy are found by solving the matrix

ing the proper boundary condition for the parallel-plate (16). The expression for //_^ is then given by (14).

waveguide, i.e., B. Theoretical and Experimental Results _ _ Q /"l n

'^ number of electrically small half-loops whose diameters

bx are 1 .5 cm are mounted transversely across a perfectly con109
KAN DA: ELECTROMAGNETIC-FIELD DISTORTION

II

c8

13

12

I

vo ^ 09

S 0.7

1 MHz

H^yc

Ph«M ol inctdcni fidd

IIIBcml

C cm

B

h(5 S cm) his 5 cmf

X(kh= 1.15 X 10'^) Z(M= 3.72 x 10'^) X(kh= 1.15 x itf^)

Fig. 5. Magnetic-field distortion along path ABCD due to a conducting

cylinder (width / = 18 cm, height h = 5.5 cm) in a transverse electro- magnetic cell at 1 MHz.

ducting rectangular cylinder. The cylinder is placed in a TEM

cell which acts approximately as a parallel-plate waveguide. A

vector voltmeter is used to measure both the magnitude and

phase of the magnetic-field strength and therefore the current

density on the surface of the cylinder. In the theoretical analysis, the frequency-domain integral equation for the magnetic

field on the surface of the cylinder in a parallel-plate waveguide given in Section IIIv4 is solved by the method of moments.

Other related quantities, such as the electric-field strength and

the Poynting vector, can be readily derived from the surface

current by use of Maxwell's equations.

In this part of the paper, the loading effects due to the conducting rectangular cylinder in a TEM cell are indicated in terms of the magnetic-field distortion, which is defined as the

ratio of the magnetic-field strength on the surface of the cyl- inder in a TEM cell to that at the same position in an empty

TEM cell. The separation distance of the parallel-plate waveguide, i.e., the distance between the center conductor and the ground plane in the TEM cell, is chosen to be 30 cm. Three

different conducting rectangular cylinders are used, all of

which have the same widths of 18 cm, but have different heights of 5.5 cm, 15 cm, and 18 cm. The magnetic-field dis- tortion and the corresponding phase are shown in Figs. 5-10

for frequencies from 1 MHz and 100 MHz. The phase refer- ence is taken to be at the center of the top of the cylinder

(x = h and z = //2). Figs. 1 1 and 12 show the magnetic-field dis- tortion at the center of the top of the cylinder as the ratio of

its height to the separation distance of the TEM cell. For a comparison, the electric-field distortion due to a metal cube

reported by Meyer [6] is also shown in these figures. In general, it is found that the magnetic-field distortion due to a per- fectly-conducting rectangular cylinder is quite small, and much

less than the electric-field distortion reported by Meyer [6]

.

In order to confirm the result by Meyer [6] , an electrically small dipole is mounted on the center of the top of the perfectly-conducting rectangular cylinder. The electric -field distortion (defined as the ratio of electric field at the surface

X (kh=

. 1.16 X_

H)-'') Z{W= 3.72 x »'') X(kh= CD1.16 X)-!) Fig. 6. Magnetic-field distortion along path ABCD due to a conducting

cylinder (width / = 18 cm, height h = 5.5 cm) in a transverse electro- magnetic cell at 100 MHz.

of the cylinder to that at the same position in an empty TEM

cell) is shown in Fig. 1 3 as a function of the ratio of the cylindrical height to the separation distance of the parallel-plate

waveguide. Fig. 13 indicates that the electric-field distortion is much larger than that predicted by Meyer [6] . The discrepancy

may be due to the fact that the Meyer calculation is based on

the quasi-electrostatic approximation.

When the conducting rectangular cylinder is placed in the TEM cell, the' scattered field produced by the cylinder can be

far different from the original incident TEM mode. In order to determine the degree of degradation of the TEM mode due to the presence of the cylinder, the response of an electrically small loop to both the magnetic and electric components of an

110
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. EMC-24. NO. 3. AUGUST 1982

rf3 X(kh= 6.28 X 10") " Z{Ui=7.4B x KJ ) X(kh= 6.28 X W^l °

Fig. 7. Magnetic-field distortion along path ABCD due to a conducting

cylinder (width / = 18 cm, height A = 15 cm) in a transverse electro- magnetic cell at 2 MHz.

100 MHz

Kit on) X

cMltcml D - MISew)

X(kh= 3.V« x 10"^ B Z(k/= 3.72 x 10^ C x(kM 3.H x K)"')

Fig. 8. Magnetic-field distortion along path ABCD due to a conducting

cylinder (width / = 18 cm, height h = 15 cm) in a transverse electro- magnetic cell at 100 MHz.

X(kh = 1.06 X

10"') 2(k(= 7.46 x W") X(kh = 1.06 x 10 )

Fig. 9. Magnetic-field distortion along path ABCD due to a conducting

cylinder (width / = 18 cm, height A = 25 cm) in a transverse electro- magnetic cell at 2 MHz. Ill
KANDA; ELECTROMAGNETIC-FIELD DISTORTION

100 MHz

X(kh 3 5.23 X 10") Z(K( = 3.72 X 10' X{kh = 5.23 X

Fig. 10. Magnetic-field distortion along path ABCD due to a conducting

cylinder (width / = 18 cm, height A = 25 cm) in a transverse electro- magnetic cell at 100 MHz.

2 MHz

2.0

1.9

1.S

1.7

i.a

1.5

1.4

1.3

1.2

1.1

1.0

—-• Thaory A EipwiiMnl _• —— Elactrie FnM Enhwic*. I mint rtporud by ( Q. Mryw I b(30cml

0.2 0.4 O.t h/b

Fig. 11. Magnetic-field distortion at the center of a top of the cylinder

versus the ratio of its height to the separation distance ot the parallel- plate waveguide at 2 MHz.

h/b

Fig. 1 2. Magnetic-field distortion at the center of the top of the cylin- der versus the ratio of its height of the separation distance of the

parallel-plate waveguide at 100 MHz.

112
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-24, NO. 3, AUGUST 1982

O so

• --• lOMHt lEKpOTinMnll A —A 2MHz !E»p*riin«ntl

by G. M«v«f

Fig. 13. Electric-field distortion at the center of the top of the cylinder

versus the ratio of its height to the separation distance of the parallel- plate waveguide.

r SQUARE LOOP \

-CONDUCTING

CYLINDER

HYBRID

Fig. 14. Simultaneous electric- and magnetic-field measurements using

a doubly loaded electricaUy small loop.

electromagnetic field is examined. Consider the loop loaded at each of the diametrically opposite points as shown in Fig. 14. One can show that the sum of the two load currents /j; is given by [11]

and their difference /^ is given as

/^ = KgEx

(19)

(20)

where Kf, and Kg are, respectively, the loop scsnsitivity constants for magnetic and electric fields. These formulas clearly

show that the use of the sum current gives a measure of the magnetic field, whereas that of its difference gives a measure

of the electric field. While the magnetic and electric fields are normally in phase

for the pure TEM mode, the presence of the perfectly conducting rectangular cylinder in the TEM cell will cause a dis- tortion, and thus a phase degradation of the TEM mode

around the cylinder. The phase degradation of the TEM mode

thus obtained is shown in Fig. 15 as a function of the ratio of

Empty TEM cHI TEM cdl with r«ctan^l»f cylind«f 1h "> 5.5 cm hvtqht) TEM c*ll with rectangular cyhr>d«f (h - 15 cm haight) TEM c*4l with ractangular cv(tr>d«r (h - 25 cm h«i^t)

J ' I I I MM ± I I I I MM

1 2 3456789 10 20 304050 70 100

Friquancy (MHz) Fig. 15. Phase degradation of the TEM mode versus the ratio of the

cylinder height to the separation distance between the center con- ductor and the ground plane in the TEM cell.

the cylinder height to the separation distance of the TEM cell.

It is very interesting to note that the phase degradation at the center of the cylinder increases as the frequency approaches

the cutoff frequency of the TEqi mode, and also as the height

of the cylinder becomes large compared to the separation of

the parallel-plate waveguide. The phase degradation observed

at the low frequencies around 2 MHz is not well understood,

but is probably due to the experimental problems caused by

the vector voltmeter and the hybrid junction used in the ex- periments.

IV. CONCLUSIONS

This paper discussed the loading effects of the TEM cell, i.e., the electromagnetic-field distortion caused by a metal object

placed in a TEM cell. In the theoretical analysis the frequencydomain integral equation for the magnetic-field strength on the

surface of a perfectly conducting cylinder in a parallel-plate

waveguide was solved by the method of moments to obtain the degree of magnetic-field distortion. The results given in the paper are for the magnetic-field strength on the surface of the

cylinder. Other related quantities, such as the electric-field strength and the Poynting vector can be readily derived from

the surface current by use of Maxwell's equations.

The experimental investigations were performed by mounting a number of electrically small half-loops on the surface of a conducting cylinder placed in a TEM ceil. The loading effects

in terms of magnetic-field distortion were expressed as a function of the ratio of the object height to t^ie separation distance between the inner conductor and the ground plane of the TEM

cell. Also, the response of an electrically small loop to both the magnetic and electric components of an electromagnetic field was used to measure the phase relation between the magnetic

and electric fields, which in turn was used to assess the degree

of degradation of the TEM mode due to the presence of the

perfectly conducting cylinder. These theoretical and experimental results were compared with the available quasi-electrostatic results. REFERENCES

(l| W. Brackelmann, D. Landmann. and W. Schlosser. Die Crenz- frequenzen von hoheren Eigenwellen m Sireifenleilungen AEU. vol. 21, pp. 112-120. Mar. 1967.

113
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. EMC-24, NO. 3, AUGUST 1982

(2) L. Gruner, "High order modes in rectangular coaxial wave- (7|

guides," IEEE Trans. Microwave Theory Tech., vol. MTT-15, pp. 483--485, Aug. 1967.

(3) R. Mittra, and T. Itoh, "A new technique for the analysis of the [8] dispersion characteristics of microstrip lines," IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 47-56, Jan. 1971.

(4) J. C. Tippet, "Modal Characteristics of Rectangular Coaxial 19) Transmission Line," Ph.D. thesis. Univ. Colorado, Boulder, CO,

June 1978.

(5) S. Ramos and J. R. Whinnery, fi>/d5 and Waves in Modern Radio. [ 10] 2nd ed. New York: Wiley, 1953.

[6] G. Meyer. "Application of a broadband measuring line in field (II] immunity testing, in 2nd Symp. Tech. Exhibition Electromagn.

Compat.. Montreux. Switzerland. June 1977. pp. 241-46.

K. Mei and J. G. Van Blade, "Low-frequency scattering by

rectangular cylinders," IEEE Trans. Anitnnas Propagat.. vol. APII, pp. 52-56, Jan. 1963. K. K. Mei and J. G. Van Bladel, "Scattering by perfectly con- ducting rectangular cylinders," IEEE Trans. Antennas Propagat..

vol. AP-II, pp. 185-192, Mar. 1963. M. G. Andreasen, "Comments on scattering by conducting rec- tangular cylinders," IEEE Trans. Antennas Propagat.. vol. AP-12.

pp. 235-236, Mar. 1964. R. F. Harrington, Field Computation by Moment Methods. New

York: Macmillan, 1968. H. Whiteside and R. W. P. King, "The loop antenna as a probe," IEEE Trans. Antennas Propagat.. vol. AP-19, pp. 291-297. May

1964.

114
Appendix F

Spiegel, R. J.; Joines, W. T.; Blackman, C. F. ; Wood, A. W. "A method for

calculating electric and magnetic fields in TEM cells at ELF."

Reprinted from IEEE Trans. Electromagn. Compat. EMC-29(4) : 265-272 ; 1987

November.

115

A Method for Calculating Electric and Magnetic

Fields in TEM Cells at ELF

R. J. SPIEGEL, MEMBER, IEEE, W. T. JOINES, MEMBER, IEEE. C. F. BLACKMAN, AND A. W. WOOD

Abstract—A method is presented whereby the electric and magnetic

field distributions within rectangular-strip transmission lines (TEM cells) can be calculated. Quasi-static approximations are employed, thereby

restricting the validity of the results to operational frequencies well below

the cell cutoff frequency. The method is illustrated by calculating the

fields within an existing structure used in biological experimentation.

Where possible, calculations are compared with measured data. Key Words—TEM cells, rectangular-strip lines, Crawford cells, quasistatic solution, electric field, magnetic field. Index Code—A3d.

I. Introduction

RECTANGULAR-STRIP transmission lines, commonly

termed TEM or Crawford cells, have found increasing use for exposing biological objects to electromagnetic (EM)

radiation [l]-[4]. Their most popular configuration consists of

a center conductor (septum) surrounded by a rectangularshaped shield; these structures are commercially available in a wide variety of sizes. Other "open" versions consist of a conductor-over-ground plane, or a conductor sandwiched

between two ground planes. In general, it is desirable to operate these cells using exposure frequencies well below the

cell cutoff frequency [5], [6], i.e., the frequency at which

higher order modes begin to propagate. When operated below

cutoff and terminated in its characteristic impedance, a

propagating TEM field can be generated within the cell structure. This TEM field approximates a plane wave, and,

consequently, reasonably well-defined and uniform fields exist over a central portion of the cell. Because of the inherent simplicity of the structure relative to design and construction, several research laboratories have

built customized cells suitable, to their requirements. However, very few researchers have either modeled or measured

the electric and magnetic field distributions within their exposure volume. Rather, it has generally been assumed that the electric (E) field strength at the center of the cell, midway

between plates, is determined by dividing the voltage applied

to the center conductor by the conductor-to-shield separation

distance. It is further assumed that the magnetic (H) field component is equal to the E field intensity divided by 377 fi. Of course, the rationale for the E field calculation is based on

Manuscript received March 24. 1986; revised May 18. 1987.

R. J. Spiegel. W. T. Joines, and C. F. Blaclunan are with the Health Effects Research Laboratory, Environmenial Protection Agency, Research Triangle

Park, NC 27711. Tel. (919) 541-7542.

A. W. Wood is with the Department of Physics. Swinburne Institute of

Technology. Hawthorn, Victoria, Australia. IEEE Lx>g Number 8717789.

the premise of a parallel-plate capacitor configuration. The H

field determination relies on plane-wave propagation in free space, where, according to theory, the E and H fields are

related by the intrinsic impedance of free space, i.e., 377 fi. For many applications, these rather rough calculations will

suffice. However, it is certainly desirable to have much more

information relative to the E and H field distributions within

the cell. This can be achieved by measurement, but the process

is lengthy and can be difficult or impossible in small cells. Additionally, especially at low frequencies (<100 kHz),

accurate field measurement is difficult due to probe lead

interference. This research presents a relatively simple method whereby

the E and H fields can be calculated inside the cell. Quasistatic methods are employed, which consequently restrict the

calculations to frequencies below the cell cutoff frequency.

The E field distribution in a TEM cell exposure system is determined by numerical means, under the assumption that essentially no axial variation of the fields occurs down the length of the cell (at least in the volume of the cell utilized for object placement). An integral equation is formulated in terms of the line charge density over the transverse contour of the

cell. This integral equation is solved by moment-method

procedures for the line-charge density, and the E field distribution is subsequently calculated. The H field distribution is determined from the vector potential where the integral

is evaluated analytically from knowledge of the current

distribution on the center conductor and outer shield.

n. Electric and Magnetic Field Distributions

A general formulation of the EM field distribution inside a TEM cell is beyond the scope of this analysis. Rather, for the reason that most TEM cells are operated well below the cell upper frequency limit (cutoff frequency), quasi-static approximations are used to calculate the internal field (E and H)

distribution. Basically, the quasi-static approximation is based on the premise that the dimensions of the cell are much smaller

than the wavelength of the field produced by the cell current

distribution. Further, it is also tacitly assumed that the distance

from the source (current distribution) to field points (location

where object is placed) is very small (relative to the wavelength). Under these conditions, the scalar and vector potentials can be approximated by [7]

</>(70»- \p{r)G(T,7')dV (1)

U.S. Government work not protected by U.S. copyright

Reprinted frcm tftftr Trans. Electroroagn. Conpat. EMC-29(4): 265-272;

November.

117

1987
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. EMC-29, NO. 4. NOVEMBER 1987

and

(2)

respectively, where

J current distribution,

p charge distribution,

G Green's function, r field point, r' source point,

€o = 8.85 X 10-'2F/m, and

fio = 47r X 10-' H/m.

With these provisions imposed, the resulting oscillating E and

H fields are identical to static fields, except for the timedependence factor of exp (yw/), with oj being the angular

frequency. Thus, the E and H fields are given by

and

S=-v<i>

H=—VxA.

Mo

(3)

(4)

With tapered transitions at each end, the rectangular-shaped

cell can be adapted to standard coaxial connectors. These

tapers, along with a matched terminal impedance, help in providing a good electrical match between source and cell, thereby ensuring minimal voltage standing-wave ratios. Consequently, there is very little axial variation of the field along

the cell (in the direction of EM propagation), and the E and H

distributions essentially vary only over the transverse plane of

the cell. A two-dimensional analysis is thereby warranted,

with the Green's function expressed as [7]

G{r, r') = (l/2T) In il/^/(x-x')^ + {y-y'y) (5)

where the coordinate (x, y) denotes the field point and (x ' , >»

'

)

locates the source point in the transverse plane.

A. Electric Field Distribution

First consider the determination of the E field distribution. A general solution for the potential produced by a surface- charge distribution p is then a superposition of due to all elements of charge pdx'dy' , or

<i>(x,y) = —\^\^p{x\y')\n

{\/^{x-x'Y + {y-y')^) dx' dy' (6)

where the integration is over the cross section of the TEM cell, including the septum. The boundary conditions are

^(x, y) = over the cell walls

<i>(x, y)=V over the septum.

Because the charge density is everywhere except over the

cell walls and septum, as defined by contours C\ and Ci in

Ixi. Vl)

C^m.^m'

- • (xj- va' Ci 1x3' V2>--

IXn.Yn)

-iCn

•SEPTUM

OUTER WALL -x

-^C„

Fig. I. TEM cell segmentation used in the computer program.

Fig. 1, the surface integral of (6) becomes a line integral.

Substituting the boundary conditions into (6) results in the following integral equation:

F=-^ ( p{x\ y') In (\/^{x-x'y^{y-y'y) dc'

(7)

where V = when the integration is performed over contour

Cz, C = C\ + C2, and it is now understood that p is a line- charge density.

Equation (7) can be solved for p by using well-known

moment-method procedures [8]. The numerical procedure

consists of dividing the contour C into A/ small segments AC„,

as illustrated in Fig. 1; p is assumed to be constant over each

segment. This allows the integral (7) to be approximated by a

set of linear, algebraic equations from which the p's associated

with each segment can be determined by inverting the matrix

equation. Details of the moment-method solution can be found

in Appendix A.

In order to determine the accuracy of this formulation,

calculations were performed for a TEM cell designed according to National Bureau of Standards (NBS) specifications [9]. The insert in Fig. 2 contains the cross-sectional dimensions of

the cell; longitudinal dimensions are immaterial for this study

since it is assumed that no axial variation of the fields occurs down the length of the cell. Calculations for the E field distribution were made by dividing the contour of the cell, including the septuni, into 145 segments, each 1 cm in length.

Fig. 2 is a graph of the calculated results for the magnitude of

the total E field normalized with respect to the electric .field at the central position (x = 15 cm, >> = 7.5 cm and x = 15 cm, y = 22.5 cm in the lower and upper half of the cell, respectively). Measured values [9] are also included in the graph. As shown, the calculated values agree quite well with

the measured values. To further illustrate the technique. Fig. 3 contains plots of

the magnitude of both the vertical (Ey) and horizontal (E^)

components of the E field strength for a TEM cell used in calcium-ion efflux experiments [l]-[4]. This cell is fairly small (refer to Fig. 3 for dimensions) and was custom-made

118
SPIEGEL el al.: CALCULATING ELECTRIC AND MAGNETIC HELDS IN TEM CELLS

Y NBS CELL (difnentioni in cm)

I 12 o

z

H

V)5 10

u.u 0.9

ccu

uj 0.8

ui> 0.7 <

i

06

0.5

0.4

03

0.2

0.1 I30 1

Y = 3.75 cm ^ 26.25 cm

J

2 4 6 8 10 12 14 16 18 20 22 24 26 DISTANCE FROM SIDE WALL Icml X DIRECTION

Fig. 2. Comparison of calculated results with measured values in an TEM cell.

NfBS

[1]. The E field is plotted horizontally across the cell for constant vertical positions. The y positions (y = 1.53, 2.0,

and 2.47 cm or^ = 6.47, 6.0, and 5.53 cm) represent scans through the volume occupied by the biological samples. No E

field measurements were made in this cell because its small

size precluded the use of available probes. However, based on

the excellent agreement obtained with the NBS cell, little useful information would have been obtained by this exercise. Note that the field strength is essentially in the vertical direction near the center of the cell, and the horizontal component gradually becomes dominant toward the side walls. Also, the field distribution is relatively flat near the center of

the cell. The field strength was calculated for a voltage of

4.16-V peak applied to the cell septum.

B. Magnetic Field Distribution

The H field is easier to determine than the E field. This is a

result of the fact that the current distribution can be determined

from knowledge of the charge distribution. However, as will be demonstrated below, this approach leads to inconsistencies when the frequency approaches the static case. Therefore, at lower frequencies another method is presented.

With the cell terminated in its characteristic impedance, a

traveling wave with a propagation constant /3 exists in the cell. The equation of conservation of charge for a surface current J

is given by

v-7=-j<j}p (8)

where p is the surface-charge density previously discussed.

Neglecting transverse current flow, one can show that the flow

of surface current down the cell is related to the surface charge

via

y^ = (w//3)p. (9)

The substitution of (9) into (2) shows that the vector potential

is identical to the scalar potential, except that the magnitude is different by the factor (eo/xow)/|8- This difference means that from (3) and (4), the H field is related to the E field according

to H=(Sxa,)i€oOi/0) (10)

where d^ is a unit vector in the axial (z) direction. Thus, the E

field distributions of Figs. 2 and 3 also represent the H field distributions when the proper scaling factor and vector component conversions are applied through (10).

It should be pointed out that there is a small transverse

current flow in the cell as a result of the nonuniform charge

distribution (e.g., the edge of the septum produces a very large

charge buildup). From Maxwell's equations, this current flow

119
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-29, NO. 4, NOVEMBER 1987

CALCIUM EFFLUX CELL

(difTwnuont in cm)

1 23456789 10 11 DISTANCE FROM SIDE WALL (cm) X DIRECTION

Fig. 3. Electric field distribution for both the horizontal (E,) and vertical (E,) components.

would produce axial fields. However, in the central portion of

the cell, measurements [10], [11] have shown these components to be minimal, and they can therefore be neglected for

this discussion. To determine the scaling factor requires knowledge of /3. From transmission-line theory, a well-known expression for

the complex propagation constant is

y=j(j(,LCy^^[\-{RG)/(LCo:^)-j(G/(Cu}) + R/(Lu))V'^

(11)

where

L = (jj^h)/iV = inductance/unit length,

C = {eoW)/h = capacitance/unit length,

G = {aoW)/h = shunt conductance/unit length,

R = 2/{acWt) = resistance/unit length,

h height between the septum and shield,

t thickness of septum,

W equivalent width of septum (including correction for fringing field), oq conductivity of free space « 1 x 10"'^ S/m, and

Oc conductivity of metal (aluminum) = 3.72 x 10^ S/m.

For higher frequencies (below cutoff) and low-loss cells (small R and G), the smallness of the imaginary term allows

the propagation constant to be given by

/3«o;(IC)'^2 = w(Moeo)'''. (12)

Since G/(cjC) = ao/(ojeo) » 1.8 x 10" V/, at lower

frequencies (but above/ = 0) G/(a)C) < 1 and (11) may be

expressed as

y=jo>{LCy'^[\-(JRy(.Lo})Y'^. (13)

Upon separation of the real and imaginary parts of (13), it is possible to show

j8 = aj(LC)'/2[o.5(H-Vl+/?V(La))2)]"2. (14)

Notice that (14) becomes proportional to oj"^ as the frequency

approaches the static case (w = 0), and consequently equals zero when co equals zero. This is also consistent with the exact value 7 = {RG^^ = real number in (11) at w = 0, or/3 = 0. The foregoing analysis, however, cannot be utilized to determine the H field for arbitrarily small frequencies because

(10) approaches zero as oj goes to zero. This, of course, makes

no sense because a cell terminated in its characteristic impedance, say 50 12, has a current flow given by

/= V/50. (15)

Thus, another technique must be used for frequencies below a

120
SPIEGEL el al.: CALCULATING ELECTRIC AND MAGNETIC FIELDS IN TEM CELLS

yet undetermined frequency and the static case. The approach

taken here is described below.

It is possible to determine [7] the surface current distribution on a thin metallic plate for a current injected at one point

contact and removed at another. With the electrodes separated

a given distance, the current pattern takes the approximate

shape of the septum of a TEM cell. Because the plate can be

cut along a line of current, the current distribution on the TEM

cell septum likewise takes the same form. For electrode

separation distances that correspond to the length of a TEM

cell, it is easy to show that the current distribution in the

central portion of the septum is essentially uniform. Likewise,

the nature of the boundary conditions and the geometry of the outer shield of the cell suggest that its current distribution is also uniform.

Therefore, the current distribution on Cz (shield) can be

represented as

7=0, -5(/-/')

i^2

and on Ci (septum) by

7= -a,- 8(1-1')

^i

(16)

(17)

where

Li

8{ltotal length of Cj,

total length of C2, and

/') Dirac delta function evaluated on either Ci or CiWriting (2) in component form yields

H.=— \ ur)G{7,r)dc'

by

"y=-T- \ Mr')G{r,r)dc'

(18)

(19)

where 7, is given by either (16) or (17), depending on whether

the integral is evaluated over C| or C2. The Green's fiinction is again expressed according to (5). Appendix B presents more

detail on the evaluation of the integrals associated with (18)

and (19). Fig. 4 contains the results for the previously described TEM

cell, where the current flow in the cell is determined from (15)

with V = 4. 16. As before, the magnitude of the field is plotted

for horizontal scans across the cell for constant vertical

positions. The field pattern is as expected: the H^ component

is relatively constant in the center of the cell and falls off

rapidly near the edge of the septum, while Hy is zero in the center of the cell and becomes larger near the side walls.

III. Discussion and Conclusions

A technique for numerically computing the E and H field distributions inside TEM cells has been described. The method

is based on quasi-static approximations, and is consequently

valid for cells that are operated well below their cutoff

frequency value. The E field determination relies on momentmethod procedures, whereas analytical techniques can be used

for the H field calculation. Both methods are implementable on small personal computers.

It does not appear that a technique for calculating H field distributions within TEM cells has previously appeared in the

literature. Therefore, the data presented herein are novel. Because of a similar paucity of measured data, the numerical

results could not be compared with accepted measurements. A further interesting result of this study is the observation

that the standard method for determination of the magnetic

field via E/H=Zo (20)

where

Zo = wave impedance = (Ato/co) '^^ = 377 Q

is valid only for higher frequencies in the cell. This fact is demonstrated by the data contained in Table I for the TEM cell previously utilized for the determination of the E and H field distributions. The column headed by the words "Transmission-Line Theory" represents the wave impedance according

to E/H= Zo/[0.5(l + Vl-i-/?V(Z-w)2)] 1/2 (21)

which is easily determined from a consideration of (14) and

the wave equation for transmission lines. Note that E/H does

not equal Zq until the frequency becomes approximately 1 kHz. The other column contains the value for E/H in the center of the cell as determined via quasi-static theory. It only

contains one value because the theory is based on static current and charge distributions under the assumption that the results are valid for nonzero frequencies when the corresponding

wavelengths are large compared to the cell dimensions. Note

also that the quasi-static results are equal to the transmissionline data around 100 Hz. This indicates that, for frequencies

above this point, the wave impedance predicted by (21) is most

likely the most valid result. Finally, recent experimental

results [11] tend to support these results. For the cells tested, E/H was on the order of 300 Q for frequencies less than 200

Hz. Above 200 Hz, E/H tended to rise, and 377 Q was

achieved for frequencies in excess of 1 or 2 kHz.

Appendix A

Evaluation of Matrix Elements

The solution to (7) in the text was accomplished by using

well-known moment-method procedures [8] in which pulse

ftinctions were used for basis functions and delta functions were used for testing. The basic procedure consists of

partitioning the contour of Fig. 1 into N small segments and

assuming that p is constant over each segment. The ;ith segment will be denoted by AC„ and the position of the midpoint of each segment, by ix„, y„). By requiring that (7) be

satisfied at each (x„, y„), it is possible to show that the integral

equation can be approximated by a set of linear, algebraic

equations that can be cast in matrix form as

[yn.] = [G„n][Pn] (Al)

121
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-29, NO. 4, NOVEMBER 1987

TEM CELL (NOT TO SCALE)

TScnn

-10 cmt

4ein _L

-14 cm0.5

|H„|

Fig. 4.

4 5 6 7 8 9 10 11 DISTANCE FROM SIOEWALL (cm) x DIRECTION

Magnetic field distribution for both the horizontal (//,) and vertical (Hy) components.

14

TABLE I Calculated Wave impedances for a TEM Cell

FREQUENCY [HzJ WAVE IMPEDANCE [Q]

Transmission-Line Theory Quasi-static Theory

325

1 50

10 160 100 340

1000 377

,

equation. Usually, however, it is numerically more efficient not to calculate the inverse of the [Gmn] matrix, but to solve

the system of equations by employing a Gauss-Jor<Jon

elimination method in which the matrix is triangularized and

the unknowns are determined by systematic elimination. When m = n, this approach implies that the field point

(Xm, ym) lies on the integration contour. Consequently, the

integral of (A2) is singular because there are a finite number

of points where Xm = x' and )>„ = y' simultaneously. In a

physical sense, the diagonal elements G„„ represent the

potential at the center of each AC„ due to a unit charge over its length. Centering the coordinate system on the segment AC„

allows (A2) to be written as where

2X€o •'AC'n V" ,-y'y/

dc'. (A2)

"" 2-Kto "-^ l^-^c^2 \cj

^-Jix^-x'y + iym-y'Y'

The unknowns [p„] can be determined by inverting the matrix

J- „(!).c] (A3,

122
SPIEGEL el al.: CALCULATING ELECTRIC AND MAGNETIC FIELDS IN TEM CELLS

where the singularity at c = has been excluded from the Now (18) is evaluated over a vertical surface, i.e., integration by surrounding it with an infinitesimally small

interval, and after the integration is performed, by collapsing ,, _l_ f* f" £. , _ „\ _^ Q/y f>\ j^./ j„/ /dc\

the interval to 0. Performing the indicated integration yields * L Jo Jo dy

AC„ ( AC„\ where x' = a defines the vertical surface. Performing the Gnn = -— ( 1 -In —— 1 . (A4) indicated integration over the x' coordinate yields

Off-diagonal elements are treated by assuming the charge //^= f In [l/V(x-fl)^ + (^->'')^] dy' . (B6) on AC„ is a point charge and by using ^tpL Jo dy

/ ^ ^ From the functional form of the Green's function, It is possible GAC„

27reo In ( ] (y^5) to show that dy = -dy' . This knowledge allows (B6) to be

V ^(Xm -x„y + (ym -ynY ' integrated easily, with the result given by

If more accuracy is desired, then (A2) can be numerically or ,, I , . n t^—

;

rr^ , n -^ ;, ,^^^

,

. „ . . Hx = \n[^ix-ay + {y-bY/s(x-ay+y^. (B7)

analytically integrated. 2tL -' v-^ f v / j' i

\ j Once the potential has been determined, the electric field distribution can be calculated by expressing the electric field The Hy component can be determined in exactly the same

as the gradient of the potential and then integrating the result, fashion as H^. However, the derivative is performed on the The resultant expressions for E^ and Ey are Green's ftmction in the x direction (see (19)). Consequently,

the result for the Hy component from a vertical surface is

\ N ix-x ^ given by (B4), while the horizontal surface contribution is Ex = -— yi Pn^C„ :;

(A6) obtained via (B7). In other words, the contributions from both

types of surfaces are reversed from the previous case. 2x60 „t', " " ix-x„y-^{y-y„Y

From this information, it is possible to determine the

2"*^ resultant H^ and Hy components inside the TEM cell by

. ^ sui)erposition. The result is

Note that, in (A6) and (A7), the numerical integration associated with the integrals was achieved by merely adding +tan~' {.x — x 'l/v-tan"' (x/v)l I the contributions from each of the elements of charge p„AC„. lirL^

Appendix B

Evaluation of Magnetic Field Integrals an (y/ix-x^y + iy-y^y/^ix-Xiy+y^)

I The evaluation of the integrals associated widi (18) and (19) + In (V(x^ + ( y-yxV/^x^+y^)] +

^^^

in the text is straightforward, albeit tedious. First, consider

(18) evaluated over a horizontal surface; the integral becomes . [tan" ' (x-X2)/(y~y2) - tan~ ' {x-Xi)/iy —yi)]

I ro rb d (B8)

ffx=j\ \ 8(y'-b) — G(r,r')dy' dx' (Bl)

L Jo Jo dy _j

Hy = ^nn {^{x-x,y + (y-y,y/^x^ + (y-y,y)

where y' = b defines the horizontal surface and L is the 2tL2

length of the surface. Performing the indicated differential operation on the Green's function yields +ln (^-J{x - Xiy \- y^ / ^x^ +y"^)] 27rL3 dG-—= -{^y-yy[{^x-x'y + {y-y'y\lT:. (B2) • [tan-' (jF-7,)/Ar-tan-' {y/x) dy

The substitution of (B2) into (Bl) and subsequent integration +tan"' (y-y )/(x-x )-tan-' y/{x-x )]-(-—

produces lirL^ H,= -I(y-b)/(2TrL) [' dx' /[(x-x'y + {y-by]. (B3)

Jo • In (y/(x-x,y + iy-y2)y^ix-X2y + (y-y2y).

It is easy to show that (B3) gives (B9)

H^ = I/2rL [tan-' {x-ay{y-b)-lan-^ x/(y-b)\. (B4) The parameters x,, X2, Xi, >»,, and y2 are defined in Fig. 1,

123
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-29, NO. 4, NOVEMBER 1987

while Z.2 and L^ define the lengths of the horizontal and

vertical sections of the shield, respectively. As before, Li is the transverse length of the septum.

References

[1] C. F. Blackman el a/.. "Induction of calcium-ion efflux from brain

tissue by radio-frequency radiation: Effects of modulation frequency and field strength." Radio Sci., vol. 14, no. 6S, pp. 93-98, 1979.

[2] C. F. Blackman et al., "Induction of calcium-ion efflux from brain

tissue by radio-frequency radiation: Effects of sample number and

modulation frequency on the power-density window," Bioelectro- magnelics, vol. 1, pp. 35-43, 1980.

[3] C. F. Blackman, S. G. Benane, W. T. Joines, M. A. HoUis. and D. E. House, "Calcium-ion efflux from brain tissue: Power-density vs

internal field-intensity dependencies at 50-MHz RF radiation," Bioelectromagnetics, vol. I, pp. 277-283, 1980.

[4] C. F. Blackman, W. T. Joines, and J. A. Elder, "Calcium-ion efflux in brain tissue by radiofrequency radiation," in Biological Effects of

Nonionizing Radiation, K. H. Illinger, Ed. Washington, DC:

Amer. Chem. Soc., 1981, pp. 299-314.

[5] D. A. Hill, "Bandwidth limitations of TEM cells due to resonances,"

J. Microwave Power, vol. 18, pp. 182-195, 1983.

[6] C. M. Weil, W. T. Joines, and J. B. Kinn, "Frequency range of largescale TEM mode rectangular strip lines," Microwave J., vol. 24, pp. 93-100, 1981.

(7] J. Van Bladel, Electromagnetic Fields. New York: McGraw-Hill,

1964, pp. 126 and 274.

(8) R. F. Harrington, Field Computation by Moment Methods. New

York: Macmillan. 1968.

[9] M. L. Crawford, "Generation of standard EM fields using TEM

transmission cells," IEEE Trans. Electromagn. Compat.,\o\. EMC16, pp. 189-195, 1974.

[10] D. A. Hill, "Human whole-body radiofrequency absorption studies using a TEM-cell exposure system," IEEE Trans. Microwave Theory

Tech., vol. MTT-30, pp. 1847-1854, 1982.

[II] A. W. Wood, W. T. Joines, and C. F. Blackman, "The characteristics of transverse electric and magnetic field (TEM) cells at extremely low

frequencies," Bioelectromagnetics, to be published.

(

124
Appendix G

Gruner, L. "Higher order modes in rectangular coaxial waveguides."

Reprinted with permission from IEEE Trans. Microwave Theory Tech.

MTT-15(8): 483-485; 1967 August.

Copyright (c) 1967 by IEEE.

125

Higher Order Modes in Rectangular

Coaxial Waveguides

A precise deierminaiion of the characteristic impedance of rectangular coaxial waveguides has been recently undertaken by Cruzan and Garver [1 i and it is well known

that these structures have many applications

in the design of shielded stnplines, varactor mounts, etc. While their operation is, in general, confined to the TEM mode, there are

instances uhen higher-order modes must be taken into account; the effect of the latter on

stnplines has been studied by Oliner [2], but there appears to be no record of a similar

investigation applicable to rectangular coaxial waveguides.

In what follows, a symmetrical structure

will be considered, i.e.. the centers of the inner and outer conductors will be assumed to coincide. Furthermore, reference will be made to TE,-. and TM,,. modes to conform

with standard notation for single-ndge waveguides [31 as well as rectangular uaveguides.

Inspection of Fig. 1 shows that when

the subscripts m or n or both are even, the planes of symmetry MN or KL or both are

electric walls and the field pattern may be deduced from that of the corresponding single- ndge wavesuide. On the other hand, the

TEj._n.:.,.;andT.Mi„4.,.)-.t:(''=0, 1, 2, • • ,

Manuscript received Januar\ H, 1967; revised April 4, 1967.

(c) 1967 IEEE. Reprinted with permission frcan IEEE Trans. Micrcfwave Theory

Tech. MIT-15(8): 483-485; 1967 August.

127
IF.LE TRANS VCTIONS ON MICROWWE THtORV \NI) TECHNIQI flS. M Gl ST 1967 m=0, 1, 2. • ) mode paitcms cannot be

studied in this manner. Furthermore, while

the TE,3 modes in ridge waveguides have

been very adequately studied [}], [6\, published information concerning other modes is either incomplete or not available. The solution of the problem may be

readily accomplished (without recourse to

finite difference methods) using a procedure due to Butcher [4] as well as Collins and

Daly 15). Thus, in order to determine the cutoff frequencies of TM:,+ i.;„,, modes, it is neces- sary to solve the wave equation for the cross

section TLBNPF (Fig. 1) subject to the boundary conditions which require the longi- tudinal component of the electric field intensity £, to vanish along PFT and LBN while

the normal derivative dE,/d/i is equal to zero along PN and TL. Furthermore, it is advantageous [5) to divide the cross section into two rectangular areas and to match the fields along the boundary FF'. Using subscripts ] and 2 to refer to the cross sections TLBF'

and PFF'N, respectively, we find that

£'i = 21 01' sinh pu ( -)

rry b

•Sin —— < y < —

2

r = ], 3, 5, • • , odd (1)

iij = 2- '^i" cosh p2m — sjn m ad < y <d d 9^

7n = 1, 2, 3, • • (2) where

P;- -A'a» +

p„- = - fc'a' + k =

X.

(3)

and X, represents the cutoff wavelength. Continuity of E, along the boundary implies that

E,i = E,: when x = s/2, st^o; when the equation is multiplied by sin mrv b and both sides integrated from tc 6'2 we find that, noting

the orthogonality properties of trigonometric

functions.

«„.^inhp,„(^-i-)

4d '

s -r l^<P:-^h^.„ cosh p2„— (4) m 2a where

1 f^ . n>ru n A^„ = — I 5U) sin —

a J d

^y

dy. (5)

Similarly, continuity of Hy along the boundar> FF' requires that dE,x,dx be equated to BE,', dx at .r = 5 2; when the re- suiting equation is multiplied by sin nmy d

and both sides integrated from to J, we find that H <?iP\rK„, cosh pi, (- ) \'2a 2 /

= — 6:-nP;m sinh p>„ — (6)

0.2 0.4 0.6 0.8 S/O. I

Fig. 1. Typicol mode characiensiics of rectangular coaxial »a\ecuides (fc/o=0.8. 2j1 b =0.6).

Eliminating <?:„ from (4) and (6) we have

2^ 2^ <Plf<lnf(«) s-l.] r-1.1

"°-^'^"(t-^)='^ (7)

w here

a^,,c,) Wp,

V

K^..K„n coth Pirn —

I SI

— Jn- tanh p.. (

1 \2 2a/

and 0,,= 1 if n = r whereas n,., =0 otherwise. Equation (7) has a solution only if the deter- minant vanishes and the cutoff frequencies

ma_\ be deduced b) setting

det[a,.,„,l = 0. (9) Let us designate d,.-^ an expression obtained from (8) by letting the summation extend over

ail even n and r (commencing with r = n = 2);

similarly, let a^ruD refer to (8) subject to coth

p:„(s,'2n) being replaced by tanh p:„{s/2a), etc. Then we find that the cutoff frequencies of various modes can be deduced by solving the following equations

TM:,, ,.;.,: deiK.,,,,) = (10) TM:,.:„.,: det[a„,„„] = (11)

T.M:,.:-.: del[a„.„„] = 0.' (12) The cutoff frequencies of TE modes may

be obtained in an analogous manner if the magnetic field intensity H, in regions 1 and 2

is represented as suitable Fourier series and

then the conunuity of H, and Ey is satisfied along the boundary FF'. introducing a nota- tion

b.,:^, = S d

l-p.^.

V

P'.->Cr^C^„ coth pi„ —

— 0,- Tanh p;, 1 \2 2a/

(13)

128
CORRESPONDENCE

vkhcre

( -nr = — I CDS—— CO:, —— Jl/ U-J)

li J a ^^ = l if m=0, X = 2 if r = 0, and ^-. = A,= 1 oiherv^ise and finally letting the subscnpts o,

e, c, and / have the same significance as applied to both a„, and b^ (but for the fact that even summation includes r = «=0), we find that the cutoff frequencies of various modes

may be deduced by solving the following

equations

TE:„.,.:„^,: det[b„,,..>] = (lo) TE:„+,.:„: det[fe„,(.c)l = 06)

TEj„.;„.„: dct[6„,;„„] = (17) TEj„.:„:dct[6„„.<)] = 0. (ISj

Evidently (10) to (12) and (16) to (18) may be used to study modes in single-ridge waveguides; (16) and (18) are, with some changes

in notation, due to Collins and Daly [5].

It can be shown that the cutoff frequencies of all modes reduce to those of the rectangular w aveguide as the dimensions of the inner con- ductor tend to zero. Thus, for example, for TM:,wfi.:m modes we note that when 2J/b= 1, then for all s (including s=0) we have p\, =p~.m, Km, = K„n = \ subject to ;i = r = m. Hence (10) reduces to

[^cothp„^ + tanhp„(--^)]

X const. = (.10) which has a solution

the presence of a root \, a= 1.249 while for

the rectangular waveguide corresponding to j=0, 2^6 = 1. wc find that X< a =1.249 as

well.When t/ = and s -a, the coaxial structure

is transformed into two waveguides and the equations cannot be expected to hold in general.However, a study of the field pattern suggests that when s approaches a, in the limit the ratio X^/a of a coaxial TE,i mode tends

to that of a TEio mode in a rectangular waveguide, viz., K/a = 2; similarly, K/a of the

coaxial TEu mode tends to that of a TEu

mode in a rectangular waveguide of reduced

height (when A/a =0.8, 2d/b=0.6, replacing

b by 36 we find that X^/a =0.466) etc. Numerical calculations confirm these conjectures. The convergence with respect to m in both

(8) and ( 1 3) is quite rapid and both expressions

vary asymptotically as 0(m~'). Furthermore,

the cutoff frequencies are primarily deter- mined by a single diagonal term of det[an,] or det(A„,] and a 3X3 determinant is likely to be adequate for most purposes.

It is of interest to note that the above pro- cedure entails no approximations other than

those inherent in the assumption that the walls

are lossless. Normalized cutoff wavelength

ratios K/a obtained by Pyle [6] for the TEio mode (A /a = 0.45) were compared with those

obtained by the evaluation of 2X2 and 3X3

determinants of (16) and truncating the summation with respect to m after 8 terms. Typical results are shown below.

(31 S. Hopfer, "The design of hdgcd »«vtguide»." IRE Trans. %ficrowart Theory and Ttchniquts, vol. MTT-3. pp. ;0-29, October 1955.

(4) P. N. Butcher, "A theoretical Jtudy of propjgition

jlong tape ladder lines." Proc. lEE (Lx)ndon). pt. B. vol. 104. pp. 169-176. March 1957.

|5] J. H. CollinJ and P. Daly. "Orthogonal mode

theory of lingle-ridge waveguides," J. Electronics and Control, vol. 17. pp. 121-129. August 1964.

16] J. R. Pyle, "The cutoff wavelength of the TEio mode

in ridged rectangular waveguide of any aspect ratio." IEEE Trans. Micropore Theory and Technqiuts,

vol. .MTT-U. pp. 175-183, Apnl 1966.

coth

Plr = (20;

and hence ^=-/(^)*-(Ty

n = 0, 1,2, w = 1, (21) •

Equations (20) and (21) hold for TEc^.+i.:., modes as well subject to m =0, 1, 2, • • •

. Similarly, when TM2„.2„+i and TE:„.t-.-i modes are considered, we find that the plane

5 = represents an electric wall and hence, for

all values of 2d/b, the determinant vanishes when

tanh

Pi-

(22)

and _ / rlny rlni -f- 1\'

With reference to the curves when 5 = 0, for the TEci mode X</a=1.6 and for the TMmode x,/a = 0.848. Furthermore, TM:„.:„ and TE;,.:,^ modes

have two planes of symmetry 5 = and 2J b=\

and hence (22) applies in conjunction with

//_'n\' /'2m \-

^' = -M'(7) -(t)- "''

Specifically, when 5 = 0, for the TE:o mode

X<.'o = l and for the TM- mode X< 'a = 0.625. Finally, TE;„+),:,„^i andTM:,.,i.;.,-i modes

have no planes of symmetry. When 5 = and

2J b=\ determinants (10) and (15) diverge;

however, in practice when Id b-\, b'a = O.S and 5 = 0.03, numerical computations reveal

d/b s/a 2X2

Determinant 3X3Determinant Reference (61

0.2 0.2

0.2 0.9

0.8 0.2

0.« 0.9

3.769 2.892

2.121

2.051

3.920 3.985 2.916 2.961 2.140 2.163 2.051- 2.«57

Some \c/a characteristics of a rectangular coaxial waveguide (such that A/a =0.8, 2d I'b = 0.6) are presented in Fig. 1. An examination of these curves shows

that as the frequency is increased and the propagation of higher-order modes becomes

possible, the TEio (TEoO mode appears fol- lowed by the TEu and TE20 modes (the TMn

mode precedes the TE^o mode for some combinations of 2d/b and 5/a subject to the same

aspect ratio A/a = 0.8); it also shows under what conditions two modes have the same

cutoff frequency and hence velocity of propagation. AciCNOWLEDGME>rT

All computations were carried out on the Monash University CDC3200 computer.

L. Gruner

Dept. of Elec. Engrg. Monash University

Clayton, Victoria Australia

References

(11 O. R. Cruzan and R. V. Carver, "Characteristic impedance of rectangular coaxial transmission

lines." IEEE Trans. Microvace Theory and Tech- mques, vol. MTT-12, flp. 488-495, September 1964.

12] \. \. Oliner, "Theoretical developments in symmetrical strip transmission line," in Proc. of the Symp. on Modern Advances in .Kfcrotace Techn'ques, vol. 4. Brooklyn. N. Y.. Polyiechruc Press of

the Polytechruc Institute of Brooklyn, 1954.

129

Appendix H

Tippet, J. C; Chang, D. C. ; Crawford, M. L, "An analytical and

experimental determination of the cutoff frequencies of higher order TE

modes in a TEM cell,"

Nat. Bur. Stand. (U.S.) NBSIR 76-841; 1976 June.

131

NBSIR 76-841

AN ANALYTICAL AND EXPERIMENTAL DETERMINATION

OF THE CUTOFF FREQUENCIES

OF HIGHER-ORDER TE MODES IN A TEM CELL

John C. Tippet

David C. Chang

University of Colorado

and

Myron L. Crawford

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

June 1976

Prepared for:

Electronics Systems Division (AFSC)

Hanscom Air Force Base, Mass. 01731

M

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

133

CONTENTS

Page

FOREWORD • iv

1. INTRODUCTION 1

2. FORMULATION OF THE INTEGRAL EQUATION 2

3. SMALL GAP APPROXIMATION FOR THE GREEN'S FUNCTIONS 6

4. SOLUTION OF THE APPROXIMATE INTEGRAL EQUATION 9

4.1 Cutoff Frequencies of the Higher-Order Modes 9

4.2 Field Distribution of the Higher Order Modes 12

5. EXPERIMENTAL DETERMINATION OF THE RESONANT FREQUENCIES OF TEM CELLS 13

6. RESULTS AND CONCLUSIONS 14

7. REFERENCES 17

APPENDIX. COMPUTER PROGRAM FOR FINDING THE ROOTS OF AN EQUATION

USING THE BISECTION METHOD 18

LIST OF FIGURES AND TABLES

Figure 1. Cross-section of a TEM. cell 19

Figure 2. Nomograph for determining the cutoff wavelength of the first

higher-order perturbed mode in a TEM cell given the ratios g/a

and b/a 20

Figure 3. Functional diagram of an automatic network analyzer (ANA) system- 21

Figure 4. Typical data obtained from an automatic network analyzer 22

Figure 5. Comparison of Mittra and Itoh's exact solution and our approximate solution for the cutoff wavelength of the first higherorder perturbed mode in a TEM cell 23

Table 1. Calculated and measured resonant frequencies of the first higherorder perturbed mode in nine different NBS TEM cells 24

135

FOREWORD

This report describes the theoretical and experimental determination of

the cutoff frequency of the first higher-order mode for several TEM cells of

differing geometry developed at the National Bureau of Standards (NBS) . In

addition, the field distribution of the first higher-order mode is found

explicitly

The effort is part of a program sponsored by the Electronics Systems

Division, Hanscom Air Force Base, under contract number Y75-917 with NBS.

The purpose of this effort is to evaluate the use of TEM cells for determining

the radiated emission characteristics of small electronic devices.

The theoretical analyses were developed by staff from the University of

Colorado under contract with NBSk The experimental work was performed at

NBS. Myron L. Crawford of the Electromagnetics Division was the technical

monitor for NBS and Charles E. Wright of the Electronics Systems Division was

the technical monitor for the Air Force. The period of performance covered by

this report extends from November 1975 to April 1976.

Results of this effort will be used to develop higher-order mode suppression

or rejection techniques to increase the useful frequency range of a TEM cell.

137
\

.^:^^n

M

i^-' ':*•*,..
AN ANALYTICAL AND EXPERIMENTAL DETERMINATION OF THE CUTOFF FREQUENCIES

OF HIGHER-ORDER TE MODES IN A TEM CELL

John C. Tippet, David C. Chang and Myron L. Crawford

ABSTRACT

In addition to the TEM mode, higher order TE and TM modes can propagate in a TEM cell. Since only the TEM mode is desired, the

higher-order modes restrict the useful frequency range of the TEM

cell. In this paper, the cutoff frequency of the first higherorder mode is obtained both analytically and experimentally for

several TEM cells of differing geometry. The difference between

the experimental and theoretical results is shown to be only a few

percent. In addition the field distribution of the first higherorder mode is found explicitly.

Key words: Shielded strip line; TEM cell.

1. INTRODUCTION

In our previous report on the TEM cell, an effort was made to investigate

the properties of the dominant TEM mode [1] . In this paper, the first few

nigher-order TE modes in the TEM-cell will be studied. They are of interest

since they determine the upper frequency limit below which only the TEM mode

can propagate. Thus, knowledge of these modes may enable us to increase the

useful frequency range of a TEM-cell, for example, by placing selectively

absorbing material in the cell at locations Where the effect is least on the

propagation of the TEM mode.

As is obvious from the sketch shown in figure 1, the TEM cell can be

viewed as a special case of a shielded strip line for which the strip is

embedded in a homogeneous dielectric. Solutions leading to the cutoff frequencies of the higher-order modes in shielded strip line structures usually

involve a numerical search for the roots of an infinite determinant obtained

from a mode matching at the interface between the partial regions separated

by the center septum in the cell. A problem arises, however, since the

infinite determinant must be truncated in order to obtain numerical results.

As Mittra has shown [2], the partitioning of the set of equations (i.e. the

number of modes used in each of the partial regions) must be chosen in the

correct ratio in order that the solution of the truncated set of equations

converges to the correct value. In fact, an incorrect choice of the partitioning ratio leads to a violation of the edge condition [3].

In order to bypass the difficulties associated with relative convergence,

an alternative approach was sought in this report. An integral-equation formulation for the electric field in the gap was found, and an approximate, but

analytical solution of the integral equation was obtained using the assumption

of a small gap between the center septum and side walls. (All of the TEM cells

currently in use at NBS indeed have small enough gaps so that the analysis

139
applied.) This allows us to represent the cutoff frequency of the first higher

order mode as a perturbation to the TE,q mode in a rectangular waveguide. The

method developed in this paper is applicable to TEM cells for which the center

septum is displaced vertically from its center position, and may also apply to

TEM cells for which the regions above and below the center septum have different dielectric constants (i.e. a shielded strip line). In addition to the

cutoff frequencies, the electric field distribution of the higher-order modes

can be found explicitly.

In order to check our solution, we compared our results to those obtained

by other authors [4,5,6]. In particular, Mittra and Itoh [4] used a singularintegral equation formulation. Their results reduce exactly to ours when a small

gap approximation is used in their equations. However, the results obtained by

Brackelmann, et al. [5] and Bezlyudova [6] were not useful for comparison since

their result for a zero-thickness septum reduces to the cutoff frequency of the

unperturbed TE,q mode in a TEM cell, independent of the width of the center

septum. If we regard the upper and lower regions of the TEM cell as two rec- tangular waveguides coupled through a small aperture, then the use of the so- called coupled mode theory predicts that two system modes (symmetric and antisymmetric) can exist, both of which have approximately the same propagation

constant. The symmetric mode is unaffected by the edge condition at the ends of

the septum, while the anti-symmetric mode is more strongly affected by the

width of the septum [7]. It is clear that the results of Brackelmann et al. [5]

and Bezlyudova [6] were intended for the symmetric mode. However, because the

excitation of the TEM cell is basically of a TEM nature which yields an. oppositely-directed electric field in the upper and lower regions of the cell,

it is more relevant to obtain results corresponding to the anti-symmetric case. The resonant frequency of the first higher-order perturbed mode was also

obtained experimentally in this paper. Using the well-known relationship between

the resonant frequency and the cutoff frequency, excellent agreement was obtained

between the theoretically and experimentally measured resonant frequencies.

2. FORMULATION OF THE INTEGRAL EQUATION

A cross-section of a TEM cell with an off-center septum is shown in

figure 1. The strip, of width 2w, is located a distance g from each vertical

sidewall and is assumed to have negligible thickness. In addition it is

embedded in a homogeneous dielectric of permittivity, c and permeability, y . The outer wall of the cell, of width 2a and height 2b = b, + b-/ and the strip

are assumed to be perfect conductors. The region above the strip is designaterd

region 1 while that below the strip is designated region 2.

140 (
We will attempt in this paper to obtain only the cutoff frequencies of the

TE-type modes since the TE,- mode is usually the lowest order mode next to the

dominant TEM-mode in the cell. For the TE modes it is well known that the

fields in region 1 can be found in terms of an auxiliary scalar function,

f , which satisfies the following relations.

(V^ + kM*^'''^ =0 in S^, region 1 (1)

and

3 t^-"-^ = on the metal walls, i.e., (2) n B-C and A-F-E-D

where

K^ = k^ - Y^; (3)

and we have assumed that the fields propagate as e

"^ - Y

_ V?^ is the transverse

Laplacian, which, for our geometry is given by 8^/9x^ + 9^/9y^, and 9 represents the outward normal derivative. w is the angular frequency, X is the wavelength, k is the wave number and y is the propagation constant which is equal

°

(1) to zero at cutoff. In addition to (2), we will also require that 9_'i' be non-

(1) zero in the gap regions, that is, A-B and C-D. If 9 4* were zero there also,

then we could write the solution to (1) immediately as just an unperturbed

rectangular waveguide mode unaffected by the presence of the center septum.

In

follows

terms of I* , the field components in region 1 can be written as

H = K^y^^^I (5) z z and

(1)

E. = iwy^V. (y^-'Mx a, (6)

w O u Z where a is a unit vector in the z-direction and V represents the transverse

gradient (a^ \- ^ ^^ ^ .

In order to determine ^^ ' , we will write it as a superposition of a complete set of basis functions ii/ ^

' as follows mn

In S, *^^^ = I A^J;S^JNx,y). (7)

1 _'• mn mn

m, n

A„^' are the unknown amplitudes, and the basis functions, ^J^J , must satisfy

mn ^ mn

the following

(^t ^^rii^'^^iii^ = ° (9^

141
and

3 ^^^^ = (9)

on H, , the boundary of S, , i.e. A-B-C-D-E-F. It may appear that (7) is logically

(1) (1) inconsistent since the basis functions, \i) satisfy (9) while their sum, * ' (1) "^

satisfies 3 4" ^ in the gap regions. As shown by Friedman (8], however,

(1)

"i* IS discontinuous at y = and thus the limit y = and the sum in (7)

cannot be interchanged. In other words, the requirement of 3 4* ^ at the gap

should be understood as applied to the plane y = where = e and e - 0.

The solution of (8) and (9) is obviously

h

with K given by mn ^ '

C<-y> = U;

mn

miT , , , niT cos \j- (x+a) cos e

(S' ^23-" * IB" '•bj^

(10)

(11)

and the normalization constant appearing in (10) is included for convenience.

Using the orthogonality properties of the basis functions, the amplitudes,

A , can be written as follows mn

^mn = Vn /

'1' ^^'(x' ,y ' ) H-^Mx' ,y ) ds' (12)

where

1

A. = \ (13)

1 otherwise

i =

1

In order to find an alternative expression for the surface integral appearing

in (12), we multiply (1) by i)^^^ and (8) by f^^^. Subtracting the resulting

equations and integrating over S,, we obtain the following

(K^lV - K') (14) / "i^^^iV ds = / [ij;ii^^4'^^^ - l-^^^V^ii^^ih ds.

mn ' '^mn t mn t t mn

^1 ^1

Using the two-dimensional form of the divergence theorem and Green's second

identity [9], the right-hand-side of (14) can be written as

R.H.S. = (i [^'^^h ^^^^ - *^^^3 4/iih di . (15)

'q mn n n mn

*1

The second term in (15) is zero by virtue of (9) . Thus the unJcnown amplitudes

given in (12) reduce to

A^^^ = /iW^" <f i|-^^^3„4'<^^ d£. (16) mn 1

142
Because of (2), however, the line integral in (16) only contributes in the

gap. Thus,

^ ^mn n

^1

-w a

/ + / -a w

"^lii^ (x,0)3yy^^Nx,0) dx

since

n 9y at y = 0.

(17)

(18)

In order to combine the two integrations appearing in (17), we will consider

two cases: (1) the odd modes for which 4*^ Mx,y) = -4' ^

' (-x,y) ; and (2) the even modes for which t (x,y) = t (-x,y) . The following analysis will apply

for the odd modes for which (17) can be written as

(1). u.(l) (1) (1) (1) 6 ii^-^'d "V^^' d£ = -/ 8 ^^^Mx,0) l^^^' {x,0) - ij;^-^' (-x,0)] dx. ^. ^mn n j y ^ ' ' '^nm ^mn r ;j >ja. x, w

(19)

The term in the square brackets in (19) is zero if m is even. Thus (16) gives

f ^^

(m=0,2,4,...) mn

(K^ - K^-"-^ ) w

mn

(1) (1)

/ d^^^^' (x,0)^^' (x,0) dx (m=l,3,5,...)

(20)

Equation (20) can now be inserted into (7) which then determines the scalar

function, * in region 1 in terms of the unknown cutoff wave number, K, and

an integration of the unknown electric field in the gap- (8 "v] (x,0)a: e (x,0))

y X Thus

4-^^^ (x,y) = 2 I P.^n (K- k"^) ^) ^2pll,n^--y^ / V'''^-'''^4lil,n^-'''^ ,n IK ^2p+l,n' ^

where p = 0, 1,2, ...

dx'

(21)

A similar relation can be written in region 2 by replacing all subscripts and

superscripts 2 by 1 and by replacing y by -y. In the gap, we require con- tinuity of H and E which is equivalent to requiring the following

V^-'-^ (x,0) = H*^^^ (x,0) (w < x < a) (22)

and

3y^^^^ (x,0) = 3^4-^^^ (x,0) (w < X < a) . (23)

When (22) and (23) are imposed on (21) and its equivalent expression in region

2, we obtain the following integral equation.

/ 3 4'(x',0)G (x,x')dx' =0 (w < x < a) w ^

(24)

143
wnere

G„(x,x') = I I. j=l p,n (K' rrr-T7*2p+i^^'°)'*'2p+i(^''°J (25)

^2p+l,n^

and we have dropped the superscript on 3 f(x',0) since it is continuous across

tne gap as well as the subscript on the basis functions since they are evaluated at y = . The Green's function given in (25) is singular at x = x*.

Thus, as shown in [10, pp. 197-198] the integral equation (24) is to be

interpreted in the principal value sense. A similar expression can be obtained for the even modes. The resulting

integral equation is identical to (24) if we replace G by G which is given as

Gg(x,x') = I I - ® j=l p,n (K^- K j)2

) i|;-„(x,0)ij;,^(x',0). '2p 2p

2p,n

If the septum is located in the center of the TEM cell (i.e. b.

(26)

b2 = b)

then the summation on "j" in (25) and (26) can be dropped and replaced by the

factor 2.

3. SMALL GAP APPROXIMATION FOR THE GREEN'S FUNCTIONS

Tne integral equation given in (24) could be solved numerically at this

point using, for example, the moment method. In this method, the unknown function, 3 4'(x',0), can be approximated by an M-term Fourier series. Using the

orthogonality properties of the basis functions, an M x M matrix equation

ensues, and the requirement that the determinant vanish then determines the

cutoff frequencies of the higher-order modes. However, since a strictly

numerical technique does not furnish much insight as to how the cutoff frequencies change as the gap varies and may also lead to relative convergence

difficulties [11], we will solve (24) by approximating the Green's function

using a small gap assumption, i.e. Kg << 1. The solution to the approximate

integral equation can then be obtained analytically.

In order to find an approximation for G , we note that the summation on

"n" in (25) can be written as follows

n=0 [K^ - K

(j) 2 2p+l,n

"jl

2 1

00

I n=l '"= -4lil>\ TT ^'^'^2p+l

(27)

wnere

'2p+l

- (2p+l)M^) (281

144
The summation on the right-hand-side of (27) is given as [12]

1 1 TT nil (n^ - ai^lh ~ 2ai^)? 2 ITT ^°^ ^" "2pli)

2p+l 2p+l

(29)

so that (27) reduces to

b^

1 I D

n=0 [K^ - KJ^|,^2j 2Tr a^^,,

Thus, G given in (25) can be written as

2 b^ cot(7Ta^|^)

jpCOt(7T a^^l^)

G^(x,x') = I fl ^£^^ (x,0)* (x',0).

]=1 2TT p ^2^^!

(30)

(31)

Tne summation on "p" in (31) cannot be done in closed- form; however, we notice

tnat for large "p"

2a

(2p+l)

(32)

so tnat if we add and subtract the right-hand-side of (32) to G we obtain the

following

G^(x,x') = G^^Nx,x') + G^^Nx,x') (33)

wnere

5<^Nx,x') = I -^ -2^

I -^^^ ^^^

°

j = l 2tt ^ hJ p (2p+l)

0)

(34)

and

^(2)

2 b^

(x,x') =1^1

j=l 2tt p

cot (Tia^^l . ) ^""2p-H^ +

[2al 1 ai^!, 'bJ (2p+l 2p+l (2p+l)J 2p+l (x,0).J;2p^^(x',0) . (35:

-2 „ ^(2) The term in the square brackets in (35) now decays rapidly as p . Hence, G

approacnes a constant whenever | x-x '

| ->

. On the other hand, the terms involved in G^ decay only as p~ and thus will have a logarithmic singularity

as I x-x 'I

-> 0. Therefore under the assumption of a small gap the dominant

contribution to G will come from G . o o Substituting for the basis functions from (10) , and making the following

change of variables

t = a - X (36)

and

f = a - x' (37)

145
G can be written as

o

(1) (t,f) = -± I p=0 (2p+l)

which can be summed as [13]

cos

(2p+ l)TT(t + t')

2a

+ cos

(2p-H)Tr(t-t')

2a

(38)

G^^^ (t,t') = Ijln tan

^

|t+t'

4a

+ In tan I —

'4a

t-f (39)

Tne largest value of | t+t '| is 2g so that if we assume that g/a << 1 we can

replace the tangents in (39) by their arguments and obtain

1

'

G^-"-^ (t,t') = - In It^-f^l O TT

TT4a (?« !)• (40)

(2) In order to obtain an approximate form for G we notice that if we restrict attention to only the first few higher-order modes for which K is not

too large, then the sum in (35) converges very rapidly, so rapidly in fact that

tne arguments of the cosines appearing in the basis functions in (35) remain

(2

)

close to zero. Thus, G is essentially independent of x and x' for gK - and we can write

j=l ^ a p=0

cot(.a <J|^)

2p+l

2a

bj^ (2p+l)J

(41)

(2gK << 1) [42)

Since we have assumed a small gap, it seems reasonable to represent the modes

,in the TEM cell as perturbations of rectangular waveguide modes; that is, we let

qr qr

with

(q=l,3,5...) (r=0,l,2, ) :s=i,2; (43)

;(S) << K

(s)

qr

(44)

and K

(s)

qr

cannot be so large that (42) is violated. Then the main contribution

to the summation in (41) occurs when a

(J) = n or when K K(J) There may 2p+l 2p+l,n

be more than one term for which this occurs since it is possible to have two or more rectangular waveguide modes with the same cutoff frequency. We will take»

for example/ q = 1 and r = which corresponds to a perturbation of the TE,

q mode in a rectangular waveguide which is the first higher-order mode that we would expect to appear, K

(s) does not depend on "s" in this case since r = 0, q^

(2

)

and only one term in (41) is dominant. For this case, G can be written as

<=i^'> TL10

2

I

j = l

cot(b .6^q)

+

2 a6 10

V p=l (2p+l)

1 - coth

irb

2a

(2p+l) (45)

146
Tne summation on "p" in (45) is very small compared to the dominant p = term, so we will neglect it, subject to a consistency check when 6,q is determined.

In any case the summation is rapidly convergent and can be evaluated numerically

if needed. Thus,

(G<^'

TE,

4 ^ cot b^6.^Q -H cot b^&^Q

a6.

(46)

'10 ""10

In the general case for non-degenerate odd modes, we obtain in a similar way

qr

2

I cot(b.^^^^)

-(s:

'qr

TtqJ

P?f

I p=0 (2p+l)

(q-1)

1 - coth

Tib,

2a

{2p+l)

(47)

where

'qr ^bj

^ (s) ^ qr

(48)

The preceding analysis has dealt with the approximation of G . A similar

analysis also applies for G , and we therefore only quote the results.

.(1) (t,t') = I ln[(^)' |t^-t'^|) (49)

and in the general case for non-degenerate even modes we obtain

2 ^^e >.^,^(s)

qr

2

'I cot(b.5^^M

3 qr

c (s)

qr

irqj TT p=l p

p?^q/2

Tib .p coth —

2 cot ^ 2a '

TT q

(50)

It is interesting to note that all of the dependence on position (through the

variables t and t') is contained in G while all of the dependence on the as

(s) °'® (2) yet undetermined K (through 6

^ ' ) is contained in G ' '.

^ qr o,

4. SOLUTION OF THE APPROXIMATE INTEGRAL EQUATION

4 . 1 Cutoff Frequencies of the Higher-Order Modes

he approximate form for G into (24) we obtain

/ f(f) ji ln[(Jr)' |t^-t'^|] + G^^^l df = :5i)

where

f(t') = 3yy(a-f,0) (52;

147
If we make the following substitutions

Tit u = 4j; (53)

u' = e^^- (54)

-o = ?f'- (")

A^ = / g(t') dt; (56) o

h(u') a f(i^); (57)

then (51) reduces to

u

(2)

II VU I ±11 I u - u I uu — — -^-— \3 o But the solution of (58) is known to be [14]

_C

'o

J ° h(u') In |u'- u"| du' = - J^ ^o ^o' ^^^^

h(u') = ^ ,- (59)

[u^ - u'^]^

where C is an arbitrary constant which is related to A defined in (56)

Inserting (59) into (58) we obtain

/"° ^ r In |u^-u'^| du- = ^

G^^U (60) o [u^ - u'^]"* 4a ° °

o which can be written alternatively as follows

/"° ^ r In |u-u'i du' = ^

G^2), . (gi)

-u^ [u^ - u'^]^ 4a ° °

If we make the following substitutions

u = u^ cos 9 (62) o and

u' = u^ cos (p (63) o then (61) reduces to

C / ln[uQ |cos e - cos 4>|]d$ = Jj- Gq^^A^. (64) o However, since [10, p. 171]

In I cos 6 - cos

1 = -ln2 -25"- cos me cos mij (65) t* iTl m=l

148
(64) reduces to

(2 J 4a o o

(66;

A in (66) can be found in terms of C by changing variables and substituting

o

(59) into (56) with the result that

Thus, (66) and (67) give

(2)

A = 2aC, o

7T = 0.

(67)

(68)

(2 ) Substituting for G and u from (46) and (55) respectively we obtain the fol- ^ o o lowing characteristic equation for the perturbed TE,- mode.

a6

j[cot b^6^Q + cot t>26^Q:

TT Trg 2] (69)

Provided that the remaining sum in (45) is indeed small under the small gap

assumption, this is then the equation for determining &-,n- In particular, when

the gap ratio (g/a) is much smaller than one, the perturbation solution can be

obtained simply from the small argument expansion of the cotangent to yield

TTb/a

10

^1^2 1"(^)

(70)

and from (43) the cutoff wavenumber is then given by

K - TTb/a

2a ^1^2 ^"(^^

(71)

It is of interest to point out that the form of K as given by (71) is essentially the same as the one obtained in [15] which treats a similar problem in

which coupling occurs through only one aperture.

As a trial solution for any numerical or graphical search of the root of

(69) , we note that since the value of the cotangent can vary from «= to when

b,6,Q increases from to tt/2 (assuming b, > h^) , the expected solution therefore lies in the range where 6,» varies from to T:/2b, which is the midpoint

between the unperturbed TE, „ mode and the unperturbed T^^jn ^°*^®" ^°^ 9/^

ratios from .01 to .3 and a/b ratios from .5 to 1, a nomographic solution of

(69) was obtained which allows one to easily determine the cut-off wavelength

graphically. This is included in figure 2. The integral equation with G instead of G is identical if we replace "a"

by a/4. Thus, for example, for the perturbed TE-q mode we obtain

|[cot b^62o + b^a^Q] = ^[in(^) - 2] (72)

from which the perturbed cutoff wavenumber can be obtained as above.

149
4.2 Field Distribution of the Higher Order Modes

The field distribution of the higher-order modes can be obtained by substituting the aperture field found in (59) into (21), but first we can evaluate

advantageously the unknown constant, C, in terms of the "voltage" drop across the

gap. Substituting for f(t') from (52) into (56) and realizing from (6) that

E^(x',0)

3 ^(x' ,0) = -^T (73)

we obtain for A^.

o

= KIT /^ Ex(^''°^^^' = iV

(74) o a-g o where V is the potential difference across the gap. Thus, C in (67) is given as

iV

C =

2ei(ii\i, (75)

and the aperture field from (59) can then be written as

(76)

Transforming the integration in (21) from x' to u' via (37) and (54) and

inserting h(u') from (76) we obtain

(1) (1)

/ 3 y^-'Mx',0)i^]^;|. (x',0)dx' = - ±

W ^ ^ IT

2 ab.iV •» u

^a)UQ^

o cos [ (2p+l)2u']du' .^^j

/u" - u'^

o but the integration in (77) is given as [16]

u.

r o cos[(2p->-l)2u']du' ^

7T J r2(2p+l)u ] /u^ - u'^

o

= 1 J^[(2p+1) ^] (78)

where J is a Bessel function of order zero. Thus by analogy with (21) the

fields in regions 1 and 2 can be written as A^(-1)P J [(2p+l)^]sin[(2p+l)5^]cos(g^)

T^=*Nx,y) = -i-

^ab.

3

iV

^coM

n' ~' -o'""*" ~'2a

' - za- b

I ^r 5

^ (j = 1,2)

P/n (k' - K^l'., 2p+l,n;)

(7gj

The summation on "n" in (79) can be written as follows

cos ^n <=- E^

n=0 3f (K^ - k;-"', ')

2p+l,n'

[^1

2

TT - I ^^ ' -^ f"^ - »gi?'

(80)

150
wnere a

(J)

2p+l

is given as [12]

was defined in (28). The summation on the right-hand-side of (80)

cos

n-ny b. , COS

n=l (n^ - ag|2^) 2a^^^^ 2a (J)

ira (j)

2p+l

sin(TTa(^)^^)

(8i;

so that (79) reduces to

-2b *^^^ (x,y) =

Tia

iV

- (-l)Pj [(2p+l)^]sin[(2p.l)g] ^°^ (&-^l)^«2p;i

•^^^o p=0

2a'

'2p+l sin(.ag|;7

(j) From (28) and (43) we have that a^

attention to the perturbed TE, _ mode we have

-^-j^— for r = so that restricting

'i'^^^ (x,y)

iV cos

J sin

afi^Q^ ^^V^' sin 6, „b . 10 :

TTX 2a

(82)

2b iV

COM I P=l

(-l)Pj^[(2p+l)g]sin[(2p+l)g] ^°^[^F^l)^^2p+l

2a

2p+l

J_

sin(T,aj^|^)

(83:

Since the potential difference, V, is still arbitrary, we can normalize V in such

,TTX, a way that the amplitude of sin (p—) is unity with the result that

:j) (x,y) = sin

, b . 6, n sin 6, f^b . Q TTX 3 10 10 ] ^ 2a

(84;

COS [6j^Q (y+b.) ]

where

(-l)Pj [2p+l)^ sin^ (2p+l)

p=l

2a

TTX 2a cos

2p+l

/b . ' 2p+l

sin(T,a^^|^)

(j = 1,2). (85)

In this form the first term in (84) can be recognized as the dominant TE,q mode

of a rectangular waveguide and the second term as a small perturbation. As

g/a - 0, 6 10 0, and the perturbation term in (84) vanishes

5. EXPERIMENTAL DETERMINATION OF THE RESONANT FREQUENCIES OF TEM CELLS

Measurements were made using an automatic network analyzer to determine

the network scattering (S) , parameters for a number of TEM cells as a function

of frequency. These S-parameters comprise a total characterization of the cell

as a two terminal network from which the cell's complex impedance, insertion

loss, and resonant frequencies can be determined.

151
The operational details of the system used to make these measurements has

been published [17] and are not repeated here. However, the test system's

functional diagram is shovm in figure 3. The system has three main sections,

namely, source, measurement, and computer. The signal generator provides

the RF power over the frequency range of interest (0.1 GHz and above) required

to pre-calibrate the system and test the unknown device or the TEM cells. The

frequency and level of the RF source is stabilized, level controlled, and preselected by command to the computer. The RF is applied to the device under

test via the S-parameter test set. Then depending upon how S, and S_ are set,

the parameters S,,, S-p* ^,2' °^ S-i are measured.

The "complex ratio detector" measures the amplitude ratio and phase difference between the reference and test channels. This information is digitized

and routed to the computer via the instrument interface.

The computer then takes the S-parameter data and stores it either as calibration data, if measuring standards, or raw, corrected data if measuring the TEM

cell. This data is then processed for correction and routed to the displays

and to other computer peripherals.

Summarizing, the measurement sequence essentially is broken into two

steps. The first is the preparation of the system through pre-calibration to

make the desired measurement. The second is to measure, digitize, and store

the S-parameter data in the computer and then convert, correct, and output the

data in the desired form. Examples of data obtained for a typical TEM cell are given in figure 4. Resonant frequencies are indicated by the high VSWR.

Note how narrow the bandwidth is for each resonance. This is indicative of the

small RF losses within the cell's conductive walls and center plate and is an indication of the cell Q factor.

6. RESULTS AND CONCLUSIONS

In this section we will discuss the result obtained for the resonant frequency of the first higher-order, perturbed mode in a TEM cell with a symmetrically placed center septum via three different techniques; 1) the experimental measurement as described in the last section, 2) theoretical calculations

based upon (69), and 3) a numerical result based upon an exact integral equation

formulation as obtained from equation (58) in the work of Mittra and Itoh [4].

In both of the last two methods, the cutoff frequency, f , is actually calculated, and this must then be converted to the resonant frequency, f , via

the following well known result

^res = K ' ^y'^"" (86)

where

c = l/ATT^ ; (87)

and d is the resonant length,

152
As shown in the figure included with table 1, the two ends of the TEM cell

are tapered, so that the resonant length is not very well defined. As a first

approximation/ however, we will take the average overall length as the resonant

length.

In order to find the zero of (69), a root-finder based upon the bisection

method was used. This is contained in the Appendix. This Scune root-finder

was also used to find the zeroes of Mittra and Itoh's equation (58), which for

a symmetrically-placed center septum and a homogeneous dielectric filling

reduces to the following for the TE modes

Y, (0)

'^l

= (88)

where

and

1 TTW

Y. (0) S' (0) + 2d, (0)E, -^— = -4i i i . (90)

k^ kj^[S(0) - 2Ig)

The symbols in the above equations are defined in Mittra and Itoh's paper and

in general must be evaluated numerically. However, for a small gap, it can be

shown that the solution of (88) is given approximately by

S(0) - 21 = (91)

or just the denominator of (90) equal to zero. Note that K. - in (88) as w -> a. Furthermore (91) can be shown to reduce to our result given in (69)

for g/a -^ 0.

Shown in figure 5 are the calculated cutoff wavelengths computed via

Mittra and Itoh's eq. (58) and our result, eq . (69) for two different TEM cell

geometries. As expected, the two curves become indistinguishable for very

small gaps. The w/a ratios of the TEM cells currently being used at NBS are

all greater than 0.72. Thus figure 5 shows that only a few percent error

occurs even for those cells with the largest gaps. It is also interesting to

note that Mittra and Itoh's result predicts a cutoff wavelength given by that

of the TE, , mode in the full cell for septum widths up to nearly half of the

width of the cell.

153
Shown in table 1 are the calculated and experimentally measured resonant

frequencies of the first higher-order, perturbed mode in nine different TEM

cells used at NBS. The column labeled "Form Factor Group" refers to those

cells which are scaled versions of one another. As can be seen, the largest

percentage error between our approximate solution and the measured value is

6.4% for w/a = .72 or g/a = 0.28. For smaller gaps (w/a = .83) the percentage

difference is typically 1%.

Thus we are able to predict the useful frequency range of a TEM cell within

a few percent. Using the same technique, additional work needs to be done to

extend our solution to TEM cells which have different dielectrics above and

below the center septum, i.e., a shielded strip line. In addition, if we re- formulate the problem in terms of an integral equation for the current on the

septum, then we should be able to obtain an equivalent form for small center

septums, i.e., w/a << 1. The formulation of the problem for TM modes has yet

to be accomplished. When this is completed, we should be able to look at the

TEM mode solution (a degenerate TM mode) and obtain a formula for the characteristic impedance of a TEM cell with an off-center septum. Thus, the solution

of the conventional strip-line problem can be obtained by the same technique

described in this report.

154
7 . REFERENCES

[I] Tippet, J.C. and D.C. Chang, "Radiation characteristics of dipole sources located inside a rectangular , coaxial transmission line," NBSIR 75-829

(Jan. 1976)

[2] Mittra, R, "Relative convergence of the solution of a doubly infinite set

of equations," J. Res. Nat. Bur. Stand. (U.S.), 67D, (Radio Propagation)

No. 2, 245-254 (March-April 1963).

[3] Jones, D.S., The Theory of Electromagnetism, pp. 566-569 (Pergamon Press,

New York, N.Y. , 1964)

[4] Mittra, R. and T. Itoh, "A new technique for the analysis of the dispersion characteristics of microstrip lines," IEEE Trans, on MTT, Vol. MTT-19,

No. 1, 47-56 (Jan. 1971) .

[5] Brackelmann, W. , D. Landmann, and W. Schlosser, "Die grenzf requenzen von hoheren eigenwellen in streifenleitungen, " Arch. Elek. Ubertragungstech.

,

Vol. 21, No. 3, 112-120 (March 1967).

[6] Bezlyudova, M.M. , "Cutoff wavelength of waveguide-type oscillations in

transmission lines having internal and external conductors of rectangular

cross section," Radio Engineering and Electronic Physics, Vol. 8, No. 11

1727-1733 (Nov. 1963).

[7] Kopp, E.H., "Coupling between dissimilar waveguides," IEEE Trans, on MTT,

Vol. MTT-16, No. 1, 6-11 (Jan. 1968).

[8] Friedman, B. , Principles and Techniques of Applied Mathematics, p. 271

(Wiley, New York, N.Y., 1956).

[9] Jackson, J.D., Classical Electrodynamics, p. 15 (Wiley, New York, N.Y.,

1962) . [10] Lewin, L. , Theory of Waveguides (Newnes-Butterworths, London, 1975).

[II] Mitta, R. , T. Itoh, and T.-S. Li, "Analytical and numerical studies of

the relative convergence phenomenon arising in the solution of an integral

equation by the moment method," IEEE Trans, on MTT, Vol. MTT-20, No. 2,

96-104 (Feb. 1972)

[12] Collin, R.E., Field Theory of Guided Waves, p. 581 (McGraw-Hill, New York,

N.Y., 1960).

[13] Jolley, L.B.W., Summation of Series, pp. 96, 97 (Dover Publ. Inc., New

York, N.Y. , 1961)

[14] Chang, D.C, "Equivalent-circuit representation and characteristics of a radiating cylinder driven through a circumferential slot," IEEE Trans, on Antennas and Propagation, Vol. AP-21, No. 6, 792-796 (Nov. 1973).

[15] Voytovich, N.N. , B.Z. Katsenelenbaum, and A.N. Sivov, "The excitation of

a two-dimensional metal cavity with a small opening (slotted cylinder),"

Radio Engineering and Electronic Physics, Vol. 19, No. 12, 8-17 (Dec. 1974)

[16] Gradshteyn, I.S. and I.M. Ryzhik, Table of Integrals Series and Products,

p. 419 (Academic Press, New York, N.Y., 1965).

[17] Rytting, D.K. and S.N. Sanders, "A system for automatic network analysis,"

Hewlett-Packard Journal, Vol. 21, No. 6, 2-10 (Feb. 1970).

155
APPENDIX

SUBROUTINE «OOT(A,e.rtX.JMtl.E.En

c This suMHouTiNt usFS th£ bisection metmoo to solve fcp one ojo

c onor or f(X) = o on the intewval (A.b). Tnt functics p^bStJ

C Thkou'Gh F xubT PF DEcLAWEO EXTERNAL In ALL C-iLLING PS0'.><4mS. E IS

C INFEHVAL CF uNCEPTAImTY DESISEO FOB ThE ROOT i and muST riE S-iLLt-'

C Than the STAMTINr, INTFBVALt « » b-A. THfc NU'^oEU OF HISLCTICNS IS

C UETEMmINEU d' f-HAX » LN(W/E)/LN(?) . £FTE*> HISECTINC. T-E F„-jct:un

C VALUE IS COMPAHEn TO El. If AR?(F(X0)1 > El TmEN Trt. f VCRCwT INL

C PMlNTS! nISC'iNTlNUlTy AT X « XO, A ></iNOOM S£4»Ch OCCjPI'-^ -Max

C Tints IS USfO TO LOOif FOH A CHANGE OF SIGN IF SIGN(F(i)) = C SION(F(B)).

RE*L LN2 DIMENSION Y(J) I

C QUESTION; OOES Fja) , o.

YI«F(A)

IF(n,N£.0.) GOTO 10 Xc* GOTO SO

C QUESTION: DOES F (8) « 0.

in Y?«F(B)

1F(V2.nE.O.) GOTO iO

X = B GOTO BO

C UUESTION: Af^E The SIfSNS OF F(A) AND F(8) iUFFERt

?0 Il=Slr,N(l..Yl)

l2=SlGMtl. tYZ) W«B-A

IFfri.NE.i;?) Guro 60

C SEARCH FOP 4 CHANGE IN SIGN.

DO 30 J«ltJMAX

X = fl»n4,NF to. 1 »»i I3 = SIGIJ(1..F(X) ) IF (IJ.NF.ln I'OTO 50

30 CONTINUE

PRINT 40

4n FORMAT (1X»N0 CHANCE o'' SIGN FOUND'/)

HE IuRn

50 8 = X

C DETERMINE NUHWER OF BISECTIONS

60 LN2r0.693M7l81

NMAX = AL0{i(W/E)/LN2*l.

Y(2»ni=4

Yti-Il)iH

C BEGIN BISECTION

DO 70 Nal«HwAA

X=(Ytll YlS) '''a. Y3«F (X) IF (Y3.EQ.0.) GOTo 80

13 = MGN(1.,YJ)

•»0 Y(2-I3)=X

80 IFIAUS(F(A) ) .LE.Fl ) GOTO 85

C CONVEHOENCE TO A DISCONTINUITY

PHluT 82.

flj koRhat(U»diScontinutTy at X = •EIZ.'./I RETURN

C CONVERGENCE TO A ROOT

«S PRI^T "O.X

QO FORMAT (ix«OfiE 000 ROOT AT X = •E12.'"' RETURN

ENO

156
t t

b. B|„ 2w.

®

>- g

I ©

2a

Figure 1. Cross-section of a TEM cell.

157
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1^ o

00 inCO o ir> o

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= -3 O O Z B

158
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1 oevKC UNOCM TUT

V 1 \f

IL

L Jh , Rt<*nnc* -KmI

SIGNAL CCNUUTOR a—

CHz

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S««r«mct«r T«t M Ma

Output

Canpulw Diipiir

1

1 S«toct

INSTRUMEffT INTERFACE COMPUTED

^ OomputtrStctiM COMPUTER

PCRIPHERAU —OU«f

Figure 3. Functional diagram of an automatic network analyzer

(ANA) system,'

159
toc

u

o

3u

01c

•T3(Uc

•a

(0y

•Ha

H

3

UASA

160
s:

3 OO

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to OS=3

in t— > q:

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—I a:

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3.0

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, ^ , , i_ MITTRA i ITOH'S SOLUTION

APPROXIMATE SOLUTION FOR SMALL GAP

(g/a <<1)

TE^^ MODE OF THE FULL GUIDE

4.0

3.0 -

2.0

1.0 -

a = b g < » g

1

-*— 2w —> 2b

* <:a *

UNPERTURBED

•TE^Q MODE

5, Comparison of Mlttra and Itoh's exact solution and our approximate solution for

the cutoff wavelength of the first higher-order perturbed mode In a TEM cell.

161
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162
MP.S-IUA IHEV. 7.73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR RKPORT NO. NBSIR 76-841

2. Go»'t Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

An Analytical and Experimental Determination of the

Cutoff Frequencies of Higher-Order TE Modes in a TEM Cell

S. Publication Date July 1976

6. Performing Organization Code

276,06

7. AUTHOR(S)

John C. Tippet, David C. Chang, and Myron L. Crawford

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2766281

11. Contract/Grant No.

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

Electronics Systems Division (AFSC)

Hanscom Air Force Base, Mass* 01731

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary ol most sigpilicant mlormation. If document includes a significant

bibliography or literature survey, mention it here.) In addition to the TEM mode, higher order TE and TM modes can

propagate in a TEM cell. Since only the TEM mode is desired, the

higher-order modes restrict the useful frequency range of the TEM

cell. In this paper, the cutoff frequency of the first higherorder mode is obtained both analytically and experimentally for

several TEM cells of differing geometry. The difference between

the experimental and theoretical results is shown to be only a few

percent. In addition the field distribution of the first higherorder mode is found explicitly.

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)

Shielded strip line; TEM cell.

18. AVAILABILITY X; Unlimited

I For Official Distribution. Do Not Release to NTIS

I Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. Cn

!XX Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

163

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

28

22. Price

$4.00

USCOMM-DC 29042-P74

ISCOMM UL

Appendix I

Tippet, J. C; Chang, D. C. "Higher order modes in rectangular coaxial line

with infinitely thin conductor."

Nat. Bur. Stand. (U.S.) NBSIR 78-873; 1978 March.

165

NBSIR 78-873

HIGHER ORDER MODES IN RECTANGULAR COAXIAL LINE

WITH INFINITELY THIN INNER CONDUCTOR

John C. Tippet

David C. Chang

Department of Electrical Engineering

University of Colorado

Sponsored by:

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80303

March 1978

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

167

CONTENTS

Pa^e

ABSTRACT 1

I . INTRODUCTION 1

II. FORMULATION 3

III. EXTRACTION OF THE SINGULAR PARTS OF THE KERNEL FUNCTIONS 5 IV. SOLUTION OF THE SINGULAR INTEGRAL EQUATIONS 8 V. DERIVATION OF THE SECULAR EQUATIONS 11 VI. CONCLUSIONS • — ~ 15

VII . ACKNOWLEDGMENTS 17 VIII . REFERENCES 18

APPENDIX 1. EXPRESSION FOR C/, IN TERMS OF P., ~ 21

ij ij.n

APPENDIX 2. EVALUATION OF THE CANONICAL INTEGRALS 27 APPENDIX 3. DERIVATION OF THE INFINITE MATRIX EQUATIONS— 30

169

HIGHER ORDER MODES IN RECTANGULAR COAXIAL LINE WITH INFINITELY THIN INNER CONDUCTOR

JOHN C. TIPPET, Student Member, IEEE, and

DAVID C. CHANG, Senior Member, IEEE

ABSTRACT

The singular integral equation approach is used to derive

the secular equations for both TE and TM waves in a rectangular

coaxial line with zero thickness inner conductor. Approximations

for the secular equations are found that reduce to simple expressions in terms of well-known special functions (elliptic integrals)

When the strip width is exceedingly small or nearly equal to the

width of the outer conductor, closed form expressions for the cutoff frequencies can be found by replacing the elliptic Integrals

by their asymptotic forms for modulus either near zero or one.

Key words: Higher order modes; rectangular coaxial line; stripllnes.

I. INTRODUCTION

In many applications Involving rectangular coaxial waveguiding structures

an understanding of the propagation characteristics of higher order modes is

just as important as that of the fundamental TEM mode. One such application

is that of the so-called "TEM cell" which is used in some EMI measurement

systems [1]. This device consists of a section of rectangular coaxial line

which is used as a transducer for coupling EM energy from a device under

test into the TEM mode of the transmission line. The useful frequency range

of the TEM c-ell is limited by the cut-off frequencies of the higher order

modes. Thus, in order to design a cell for use over a specified frequency

range, these cut-off frequencies must be known.

This work was supported by the U.S. Department of Commerce, National Bureau

of Standards, under Contract No. NOAACST-8393.

The authors are with the Electromagnetics Laboratory, Department of Electrical

Engineering, University of Colorado, Boulder, CO 80309.

171
The device analyzed in this paper consists of a zero thickness metal

plate located inside a rectangular waveguide. The inner conductor may be

offset vertically from its central position but is located symmetrically

about the y-axis. Both the inner and outer conductor are perfectly conducting

and the medium separating the two conductors is a homogeneous dielectric. A

cross-sectional view of the rectangular line is shown in Fig. 1.

Although the transmission line properties of the TEM mode in these

structures have been extensively studied, comparitlvely little work has

been done in analyzing the higher order mode structure. One can find in the

literature, however, this problem [2-4) as well as related problems, such

as shielded strip lines [5-7], various coupling configurations [8-9], rectangular lines with thick Inner conductors [10-13], and ridge waveguides

[14-16] analyzed using a variety of techniques. These include functional

equation techniques, finite difference techniques, mode matching techniques,

integral equation techniques, and methods based on transmission line theory.

The purpose of this paper is to obtain a relatively simple, closed form

expression for the characteristic equation for both TE and TM modes which is

valid for arbitrary strip widths. Thus, very little computer programming is

necessary to calculate the cut-off frequencies. In addition, approximate

solutions for either small or large strip widths can be obtained without

resorting to any numerical analysis. This Is achieved by using the

singular Integral equation technique similar to that used in waveguide

diaphragm problems [17]. This method has the advantage of handling the

edge condition exactly and eliminates the problems encountered in any

numerical solution associated with the discontinuities of the fields near

the sharp edges of the inner conductor. More specifically, the problem

is formulated using an integral equation—Green's function type of approach.

172
The singular part of the kernel of the resulting integral equation is

extracted (as is done, e.g., in [18-20]) and the nonsingular part of the

kernel that remains is expanding in terms of Chebyshev polynomials as

suggested by [19].

II. FORMULATION

The cut-off frequencies of the higher-order modes of a rectangular

coaxial line are just the eigenvalues K of the reduced wave equation

(72 + k2)

z =

(1)

where

V2 =. 32/3x2 + 32/3y2

and the unknowns satisfy the following boundary conditions on the metal walls

3 H = n z

and

In the first case, H represents the z-component of the magnetic field of a

TE mode; and in the second case, E represents the z-component of the electric

field of a TM mode.

In order to solve this eigenvalue problem, we will convert (1) into an

integral equation. This is accomplished by expanding „ Into a complete set

of basis functions appropriate to each of the subregions above and below the

strip (j>l,2, respectively), aa

(J - 1,2)

Mj^" [a^J^oI

z '^mn m,n

mn

e(j) OB<J> /J)

z mn ^mn

173
where the A ^•'''s and B

^^ '

' s are as yet undeterminded coefficients,

mn mn

For our geometry, u- and <li mn

are just the TE and TM basis functions

of a rectangular waveguide of height b , i.e.,

mn ^^\x,y) = f^^cos miT , , . 2l ^""^^^ cos

rnryy

and

(j) mn

(x.y) = ab

jj

sin miT 2a

(x+a) sin mry

jJ

These basis functions also satisfy (1) but with known eigenvalues

K

(j) mn

They do not, except in the rather trivial case of an unperturbed rectangular

Th

*

2 waveguide mode, satisfy the same continuity conditions as the unknowns

in the gap regions. We can still expand in terms of these basis functions,

however, since 3 H or E need not be uniformly convergent at y = 0. In this

case the limit y

->• cannot be taken inside the appropriate expansion. See

[21] for a discussion of this phenomenon.

Using the orthogonality properties of the basis functions to solve for

the unknown expansion coefficients, applying Green's theorem, and matching

boundary conditions in the gap regions, results in the following integral

equations for the TE and TM modes, respectively.

fa E^(x',0)G'^^(x,x')dx' = (2)

and

r^

3 ,E (x',0)GTM(x,x')dx' = . (3)

174
where P denotes that the integral is to be Interpreted in the principal value

TE TW sense, E (x,y) « 9 H (x,y), and the kernels G and G which contain the

X y z unknown eigenvalues K,_ and KL,^ are given by

,TE (x.x') •

I y A A

. ,

' m n j=l m,n

/^^(x,0).i/f^\x'.0) mn mn

[^I

_ K^J>2i mn

(A)

and

G (x.x') =

I -r-

(j)

ni,n K

(J) mo

O^^^'y^0^^(^''°>

f'S^ - ^m^'^^l y =

(5)

where

"iI ^

i >

The symmetry about the y-axls allows the integrations appearing in (2)

and (3) to be written only over the one gap from x = w to a and permits

a decomposition of the TE and TM polarizations into even and odd modes.

This is manifest in the summations on "m" in (4) and (5). For TE modes

with odd symmetry, i.e., tt (x,y) = -H (-x,y), (4) is summed over odd

values, while for TM modes with odd symmetry, i.e., E (x,y) = -E (-x,y),

(5) is summed over even values. The reverse is true for modes with even

symmetry.

III. EXTRACTION OF THE SINGULAR PARTS OF THE KERNEL FUNCTIONS

The summations on "n" in (4) and (5) can both be summed exactly using

the residue series technique [22]. The derivative in (5) can then be

taken and when evaluated at y = one finds

175
aad

where

fr. 2 b."^ ^m cot "Y /,. ,.^

C"(x.,') -

I J- I ^

*J>(x.O)V<»(x',0, (6,

j-1 m yJ

2 r^J^cot nr^J>

• mo

(J) =^1 PK ' - K ^J^'l'' ^m Ti ^*^TE mo ^ (8)

and

m IT l^TM "^mo ^

Although the sunmiations on "m" in (6) and (7) cannot be done in

closed form, we can extract the singular part in each sum by replacing

the coefficients in front of the basis functions by their asymptotic

form for large "ra", i.e.,

cot TTY '^

Co 1 -, m 2a I 1

(9)

m

-*

and

r^J^cot .r^J) .

'" m m 2a I

The kernels will then consist of two parts, a singular part G which

can be summed in closed form (see, e.g., [22]) and a correction series

G = G-G. Recognizing that the summations on "m" are either over even or

odd indices, four cases need to be considered. These are not independent,

however, since it is easily verified that

176
GTM(x,x') - 3^g'''^(x,x') (10)

Thus, only the kernels for the TE polarization are needed, i.e.,

:te 1 Godd = i^"f^^"iii^^^'i^«"yt-^'ii (11)

and

G^^ = - in [4 sin ^|t+t'|Bin ^It-t'l]

even n 2a' 2a I (12)

where the following change of variables was introduced

t = a-x (13)

and

t' = a-x'. (14)

The singular kernels for the TM polarization can then be found using (10),

Although the correction series G cannot be summed in closed form,

they are rapidly convergent and are given as

and

G'^^(t,t') - y A cos

' m

m

mrrt [2a J cosmTTt'

I 2a J (15)

GTM(t,t') = y B sin m

m

mut

[2a J cos

'mTTt'' 2a

(16)

wnere

A = y -i

m '. air

Am cot Try (j)

M)

2a mo

.bj.

(17)

"> jii ""^j

r^J^cotTrr^^^

2a (18)

177
<iiiJ i is tlio Kronekcr delta defined as mo

;i m =

6 =

\

IV. SOLUTION OF THE SINGULAR INTEGRAL EQUATIONS

Having extracted the singular parts of both kernels we can move the

nonsingular correction series to the right-hand sides of both integral

equations and treat these terms as forcing terras. Since the singular

parts of the kernels of (2) and (3) are related according to (10), it is

convenient to differentiate (2) with respect to x. Both equations will

then be of the same canonical form. In terms of the new variables defined

in (13) and (14), (2) and (3) can be rewritten as follows

P U(t')GTM(t,t')dt' = U(t')3 G'^^(t,t')dt' (19)

and

where

and

f8 ~ r8 P V(t')GTM(t,t')dt' = - V(t')G"'(t,t')dt' (20)

JQ Jo

U(t') = E^(x',0)

V(t' ) H 3^,E^(x',0)

Equations (19) and (20) are both singular integral equations. The

standard form of the singular integral equation, however, has integration

limits -1 and +1 respectively. These limits can be achieved through the

use of the Schwinger transformation [17]

178
and

cos

cos

TtC N

a+3u

a+6v

(21)

(22)

where

and

a = i [cos(^) + 1]

3 1 jeos(J£} -1]

Defining

T [F(u)] -='h{ ^Mdu

and transforming to the new variables u and v defined in (21) and (22) we

find that (19) and (20) can be rewritten in standard form as

T^[F^j(u)] = H^j(v) (i.j = 1,2) (23)

where F^^(u) =u(f)cosfe^/sin[!Lf

F^^M = U(t')/sln[^

F2^(u) » V(t')cos[g^ /sin m

F22(u) - V(t*)/sln ^t'"*

H,,(v)

H,2(v)

- f%(t')3j.G'^^(t,t')dt'/2sln[g

U(t')3|.G"'(t,t')dt'/sln

and

H2i(v)

r8V(t')GTM(t,t')dt'/2sin UaJ

H22(v)- -

j V(t')GTM(t,t')dt'/sin[^

179
In (23) the subscript "i" refers to the type of mode, TE or TM respectively,

and the subscript "j" refers to the type of symmetry, odd or even respectively.

As shown in [17] (23) can be inverted exactly for the unknown F. (v) as

F, .(v) = —^

(C.. - T [/T^H (u)]} (24)

where the C, 's are constants to be determined.

In order to make use of (24) we need to express H in terms of u.

The most convenient expansion to use is one in terms of Chebyshev polynomials

of the second kind U • In terms of these polynomials H^ . can be written as n

'^ ' ij

»ij("> = ^, ^j.n\("> (25)

-* n=0 -*

where the P., 's are expansion coefficients to be determined, ij.n

[23]

To express F as a function of v we can use the following identity

T^[/l^ U^(u)] = -Vi(v) (26)

where T is the Chebyshev polynomial of the first kind.

Thus, F is given as

F..(v) ^{C., + [ P.. nT .,(v)} . (27)

180
V. DERIVATION OF THE SECULAR EQUATIONS

In order to complete the solution given in (27) we need to evaluate

the unknown constants C and P., . We begin by first expressing C, . in

ij ij»" ij terms of the P.. 's. For the TM polarization this is accomplished by ij,n

requiring

I 3^,E^(x',0)dx' - E(a,0) - E(w,0) - 0. (28)

'w

For the TE polarization, we substitute the solution given in (27) back

into the undifferentiated form of the integral equation, i.e., eq. (2). As

shown in Appendix 1, one then finds for C. . the following

'^r-^o Jo '«.^i '^i.° ""

where

11, m

"

11, m

"

4A ' ^2n+l ^mn

m n=m

- -^ y A

4A /• *

m n=m

12, m 12, m 2A ^ 2nTnn m n=m

^l,m 21,

and

R^- » (m > 0) . ^^,m

I, J, I,2» 3"d I-, are three canonical integrals, the recursion formulae

for which, are derived in Appendix II. For modes with odd symmetry, the

integrals I and I ^ can be expressed in terms of complete elliptic integrals

of modulus /a for the TE polarization and /^ for the TM polarization,

while for modes with even symmetry, I,^ contains no special functions.

181
p and q are the coefficients of the following expansions

5" PTM„ T (") ^2a

and

fmrt' cos

nt' ' 2a_

CTTt' 1

'

f. "^mn 'm'

cos(2n+l)(||-) n

in=0

We are now in a position to substitute our solution for F, . (expressed

solely' in terms of the unknown P.. 's), back into the definition of the ij.n "

H . ' s defined in (23)- Upon matching coefficients of U (v) we then obtain

an infinite set of equations for the unknown P 's. Setting the

ij ,n determinant of the coefficients to zero we obtain the secular equation

from which K can be determined. The details of this procedure are outlined

in Appendix III. Four infinite sets of equations are derived of which the

following for the odd TM polarization is representative.

a3

CO n

^22, m

- 2" ^ J^ ^2n+2 ^mn ^k+l.n+1 ^22, k (30) n=m k=0

where s is the coefficient in the following expansion

sin[(n+l)—] n ~- = y s U (v) .

. (-"t'l '-„ mn m

sin —

m=0 ^ a-^

Equation (30) is the simplest of the four equations given in Appendix III

since C^„ = for this case only. Fortunately, however, these equations can be

greatly simplified. In most cases, only a few of the A 's or B 's, given respectively in (15) and (16), are significant, and thus the infinite matrix equations

given in Appendix III reduce to equations of very small order. The number

182
of Amm's or B 's chat we must keepr depends r upon r the number of hieher a order

(j) m

modes that we allow to propagate. We can see from (8) or (9) that once

"m" becomes large enough so that K > K or K,^, respectively, then y

or r

-^ becomes imaginary. The cotangents then in (15) or (16) can be

replaced by hyperbolic cotangents and thus the succeeding A 's or B 's

become negligibly small. Thus, we can truncate our infinite matrix

equation to a finite one at the sacrifice of being able to calculate all

of the higher order modes. We will only find those modes for which

K,^ or K,^ < K^ where we have truncated the matrix to one of the finite

order M.

In many applications (e.g., [24]) one is only interested in the first

few higher order modes. This method is especially applicable for finding

these modes since we can usually truncate the matrices to matrices of order

one. In this case, (30) reduces to

1 = ^

B2 (31)

since s-^ = 1 and q^ =6. Substituting for B- from (16) we then find the

following secular equation for the odd TM modes

2 ar

^^>

I -ITT- ^ot.A^^ - 1 - h

2b ----9 - 12 (32)

J=l J

where r^"' was defined in (8). Similarly, for the even TM modes we find

I !Il_ „„r/J> - j^^^ (»od

^ ) (33)

183
for the odd TE modes,

2 b.cot.y^ (l-B)K-2E , ^ r-,

I -^ (jy- = 2E+6K ^"°'^ "^ > (34)

j = l 4aYj^ -^

and for the even TE modes,

2 b cotTTY^^^

I -^ TTT— = -^n(-6) (35)

j=l 2ay^J^

In (34) and (35) for the TE modes we have also made an additional

approximation in neglecting all of the A 's except A and A respectively

in expressing C^ in terras of the P. 's through (29).

The roots of the secular equations given in (32) through (34) were

found using a simple bisection method. These data were then plotted in

Figs. 2-4 which are graphs of the cutoff wavelengths of the first higherorder modes as a function of strip width. In these figures we have made

the simplifying assumption that the strip is symmetrically located, i.e.,

b^ = b- = b, and the mode designation refers to the limiting value to which

the cut-off frequency reduces as the strip width goes to zero (3 -> -1).

We have not plotted a similar curve from the data of the solution of

(35). This equation shows an anomalous behavior for small gap. In this

case Y must tend to zero in order to balance the logarithmic divergence

which occurs when 6

-* 0. Since K ^ "^ y we thus have a mode with no low- TE o frequency cut-off. This is not the TEM mode, however, since it only exists

in the limit of zero gap. Brackelmann, et al [11] have also observed this

phenomenon in their analysis of a rectangular line with finite thickness

inner conductor.

184
VI. CONCLUSIONS

In addition to the higher-order modes whose cut-off wavelengths are

plotted in Figs. 2-4, there are also some higher-order modes which are unperturbed by the presence of the inner conductor. If b^ = b^ b, these ihodes

are just the TE „ and TM » modes of the full guide of cross-section •^ m,2n m,2n ^

2a X 2b. Alternatively, we can view this system as two rectangular subwaveguides of cross-section 2a x b which are coupled through the gap. The

sub-waveguide modes can then be combined to form a system mode of the entire

structure which is either symmetric with respect to the inner conductor

(i.e., E (x,y) = E (x,-y) or antisymmetric. As found also in [27] only the

symmetric combination is unperturbed by the presence of the inner conductor.

If the inner conductor is offset then there exists fewer -unperturbed modes.

Since the perturbed modes reduce to rectangular waveguide modes in the

limit of either zero strip width (6 -> -1) or zero gap (B * 0) we can obtain

approximate solutions for the cut-off wavelengths by replacing the cotangents

and elliptic integrals in (32) to (34) by their asymptotic forms, for beta

either near or -1. One can then find that the cut-off frequencies are

perturbed as follows. For small strip widths

2 . . f8a 'TM11 ' ''" l^.

TM,21

and

TE

11

while for small gaps one finds

185
'TM11

6^TM21

2

4

6„_. « l/£n —

where

2

2 2 K - Kunperturbed

f6

The significance of the logarithmic perturbation of the cut-off wavelength

of the TE^- mode for small gap and the TM mode for small strip width is

apparent in Figs. 3 and 4. For any finite gap width or strip width, respectively, the perturbation is quite significant. Thus, in contrast to the other

cases for which the perturbation is algebraic, the cut-off frequencies are

not accurately predicted by their unpertu^*"°d values. This same type of

logarithmic or algebraic behavior was found by [8] in an analysis of a related

problem of two rectangular waveguides coupled through a longitudinal slot.

In reference [11] are plotted similar curves for the case of a thick

center conductor. The cui-ves show the same qualitative behavior as a function of strip width. In particular, the cut-off wavelength of the TE^

mode for a small gap tends to be that of a "coaxial TE^„ mode" in each

sub-v;aveguide

Although we have only presented the numerical results for a one-term

approximation to the secular equations, higher-order solutions can be

easily found, since the canonical integrals needed in the computation of

the matrix elements are given recursively. For example, if we keep two

of the B ' s in the expression for the secular equation for the odd TM

mode, (30) reduces to

186
1 -

2 ^2 1 - ^ B^l = . (36)

The first term In (36) gives the same roots as we found earlier in our

one-term approximation as given in (31). In addition, however, we have an

extra term which gives rise to additional higher-order modes. One can see

how continuing this process would ultimately recover all of the higher-order

modes.

VII. ACKNOWLEDGMENTS

The authors would like to thank Dr. Leonard Lewin and Dr. Paul Wacker

for their critical comments and helpful suggestions and Mr. Chuck Miller

and Mr. Myron Crawford for their constant support and encouragement.

187
VIII. REFERENCES

[1] Crawford, M. L. , Generation of standard EM fields using TEM transmission

cells, IEEE Trans. Electromagnetic Compatibility, vol. EMC-16, pp. 189-

195 (Nov. 1974).

[2] Belousova, L. I., and Sal'nikova, L. P., Computation of the cutoff

frequencies of higher order modes in shielded symmetric stripline,

Radiotekhnika (Khar'kov), No. 17, pp. 72-76 (1971).

[3] Shestopalov, V. P., The Method of Riemann—Hilbert Problems in the

Theory of Diffraction and Propagation of Electromagnetic Waves ,

Izdat. Khar'kov. University (1971).

[4] Belousova, L. I., and Sal'nikova, L. P., Higher wave modes in a shielded

nonsymmetrical strip line. Radio Eng. and Elect. Physics, Vol. 16,

No. 7, pp. 1199-1202 (July 1971).

[5] Lerer, A. M. , and Mikhalevskii, V. S., Using paired integral equations in

certain problems involving the diffraction of waveguide modes by bodies

of finite size, Radiophys. Quantum Electron., vol. 18, pp. 859-865

(1975).

[6] Mittra, R. , and Itoh, T. , A new technique for the analysis of the dispersion characteristics of microstrip lines, IEEE Trans. Microwave Theory

Tech., vol. MTT-19, No. 1, pp. 47-56 (Jan. 1971).

[7] Davies, J. B., and Corr, D. G., Computer analysis of the fundamental and

higher-order modes in single and coupled microstrip. Electronics Letters,

Vol. 6, No. 25, pp. 806-808 (Dec. 1970).

[8] Garb, Kh. L. , and Fridberg, I. Sh. , Dispersional properties of a Rectangular

waveguide, loaded in half -space by a narrow slot, in a wall of finite

thickness, Radio Eng. and Elect. Physics. Vol. 19, No. 1 (1974).

188
[9] Voytovich, N. N. , Katsenelenbaum, B. Z. , and Sivov, A. N., The excitation

of a two-dimensional metal cavity with a small opening (slotted

cylinder), Radio Eng. and Elect. Physics, Vol. 19, No. 12, pp. 8-17

(Dec. 1974).

[10] Bezlyudova, M. M. , Cut-off wavelength of waveguide-type oscillations in

transmission lines having internal and external conductors of rectangular

cross section. Radio Eng. and ELect. Physics, Vol. 8, No, 11, pp. 1727-

1733 (Nov. 1963).

[11] W. BrMckelmann, D. Landmann and W. Schlosser, Die Grenzfrequenzen von

HSheren Eigenwellen in Streifenleitungen, Arch Elek. Ubertragungstech.

,

Vol. 21, No. 3, pp. 112-120 (Mar. 1967).

[12] Gruner, L. , Higher order modes in rectangular coaxial waveguides, IEEE

Trans. MTT, Vol. MTT-15, No. 8, pp. 483-485 (Aug. 1967).

[13] Brackelmann, W. , Wellentypen auf der Streifenleitung mit rechteckigem

Schirm, Arch. Elek. Ubertragungstech., Vol. 21, No. 12, pp. 641-647

(1967).

[14] Pyle, J. R. , The cutoff wavelength of the TE^« mode in ridged rectangular

waveguide of any aspect ratio, IRE Trans. Microwave Theory Tech., Vol.

MTT-14, No. 4, pp. 175-183 (April 1966).

[15] Collins, J. H. , and Daly, P., Orthogonal mode theory of single ridge

waveguide, J. Electronics and Control, vol. 17, pp. 121-129 (Aug. 1964).

[16] Cohn, S. B., Properties of ridge wave guide, Proc. IRE, pp. 783-788

(Aug. 1947).

[17] Lewin, L. , Theory of Waveguides , London: Newnes-Butterworths (1975).

[18] Chang, D. C. , Equivalent-circuit representation and characteristics of

a radiating cylinder driven through a circumferential slot, IEEE Trans.

Ant. Prop. Vol. AP-21, No. 6, pp. 792-796 (Nov. 1973).

189
[19] Gladwell, G. M. L. , and Coen, S., A Chebyshev approximation method for

raicrostrip problems, IEEE Trans. MTT, vol. MTT-23, No. 11, pp. 865-870,

(Nov. 1975).

[20] Chang, W. , and Otto, D. V., Current distributions on electrically thick

tubular antennas of limited length. Int. J. Electronics, Vol. 39,

No. 4, pp. 393-413 (Apr. 1975).

[21] Friedman, B. , Principles and Techniques of Applied Mathematics , New York:

Wiley, p. 271 (1956).

[22] Collin, R. E. , Field Theory of Guided Waves , New York: McGraw-Hill (1960).

[23] Tricomi, F.G.. Integral Equations , New York: Interscience (1957).

[24] Tippet, J. C, Chang, D. C, and Crawford, M. L. , An analytical and experimental determination of the cutoff frequencies of higher-order TE modes

in a TEM cell. National Bureau of Standards Internal Report NBSIR 76-841

(June 1976).

[25] Gradshteyn, I. S., and Ryzhik, I. M. , Table of Integrals Series and Products ,

New York: Academic Press (1965).

[26] Byrd, P. F., and Friedman, M. D. , Handbook of Elliptic Integrals for

Engineers and Physicists , Berlin: Springer-Verlag (1954).

[27] Kopp, E. H. , Coupling between dissimilar waveguides, IEEE Trans, on MTT,

Vol. MTT-16, No. 1, pp. 6-11 (Jan. 1968).

190
APPENDIX 1. F.XPRESSION FOR C_ IN TERMS OF P ij ij.n

We begin with the odd TM Polarization and transfonn (28) into an

equation in terms of u as follows

(a fg r 1

3 ,E (x',0)dx' = V(t')dt' = F,-(u)du = (1.1) Jw '^ •'O •'-1

If we now substitute for F^-C") from (27), we obtain

du

22 i-l /l^

-- I P n=o

22,

ri V,(u)du

-1 / 1-u^

(1.2)

The right-hand side of (1.2) vanishes because of the orthogonality property

of the Chebyshev polynomials, i.e., [25]

1 T^(u) T„(u)du m n IT -1 /1-u^

2A mn m

(1.3)

where

A = m

2 Tn=0

1 m>0

Thus,

S2 = ° (1.4)

For the even TM Polarization, we transform (1.1) into an equation in terms of

a new variable x defined as

X

= cos [^] (1.5)

to obtain

191
ra j-g 3 ,E (x',0)dx' = 2] F (u)sin

V, ^ ^ '0

f^t'l 2a

''^' - ^ I ^21

^ ^2x2-l-a

V5"

dx-O.

If we now substitute for F^, from (27) we obtain

fl C2^dx

J^ /(l-X'^)(x^-a) n=0

21,

dx

'^ /(i-x')(x'-a)

since

^T^ = - I /(i-x^)(x^-ot)

Defining I„ as

21,

^

2x^-i-« .dx. /(l-X^)(X^-a)

we find for C„ the following

00 '^^'~h^0 n=0

''^l.n ^21,n+l (1.6)

Now for the odd TE polarization we transform (2) into an equation in

terms of t and t', and without loss of generality, evaluate the resulting

equation at t = to obtain

I U(t')G''^^(0,t')dt' = -I U(t')G^^(0,t')dt'

Jo 'o (1.7)

The left-hand side (L.H.S.) of (1.7) can be transformed into an equation in

terms of the variable x defined in (1.5) as follows

G(0,t') = i m tan2

^ = i £nf^ 4a n |l+x

192
Thus,

L.H.S. =

. ,. ^ii^"^^"[iSj sin

Tit' dt' = Aa (-i..! 11

2x^-l-a

e tn 1-y

U+xJ

dX ^"

If we now substitute for F^ from (27) we obtain

L.H.S. = 2a

1 In 1+X

ji-xj

dX

^ J^ /(l-X^)(X^-a)

Cn + y. P,-, „ T

11 ^l^Q 11, n n+ll^ B

[ 2x^-1-0

(1.8)

Defining I,, as ^ 11,

11, n TT J n^

/a.

1 ^ 2x^-l-a

£n 1±X

li-xj

.ir

v'(l-X^)(x''-a)

we see that (1.8) reduces to

L.H.S. = -^

[C,, I,, o + y P,, I,, ^,] , IT 11 11,0 '-^ 11, n ll,n+l n=0

(1.9)

Now the right-hand side (R.H.S.) of (1.7) can be written in terms of u as

follows

r'o

sin( J

«>

, R.H.S. = -I F^^(u) -—f.- y A^^^^ cos[(2n+l)||-]df = c°si^^J n=0

cos[(2n+l)-^]

n=0 ^ -1 cos (27-J

a6

IT

^2a

1

If we now make use of the following expansion in terms of Chebyshev polynomials

of the first kind T

193
TTt' cos[(2n+l)3—) n

= y P T (u)

m«0

^2a

<=°s ill")

(1.10)

and substitute for F^ , (u) from (27) we find the following

R.H.s. =M ^ y A n=0 in=0 2n+l '^mn l:

1 T (u)du CO

/1-u^ k.=0 ' (1.11)

The integrations in (1.11) can be evaluated by using the orthogonality

relation given in (1.3) and thus (1.11) reduces to

(1-6 ) n R.H.S. = a6 I A„ ^, [P_C ' n=0

2n+l ^ on 11 ^ -^—

J-, ^l.k-l Pkn^

k=l

(1.12)

where 6 is the Kroneker delta. Finally (1.9) and (1.12) when equated and

solved for C gives

^. ^11, n ^ll,n+l "

4 ^^ ^2n+l^-'-"'^no^. S ^ll,k-l ^kn

n=0 n=0 k=l

'11 ^11,0 2 If. ^2n+l Pon

n=0

(1.13)

For the even TE polarization we again start with (1.7), but we transform

the left-hand side of that equation into an equation in terms of u as follows

G "(O.t') = i £n[(-26)(l+u)]

Thus,

ag f 1 L.H.S. = - ^ J F^2(">^"f(-2^^(1+^>J'^"

194
If we now substitute for F,2^"^ from (27) we obtain

L.H.S. = aB f ^ £n[(-2e)(l-Hi)]du

-1 /l-u'^

f'l2-^ ^, ^2.n Vl(">5- (1-1^)

n=0

Defining I^» as

r 1 T (u)£n(l+u)

I,„ =- du + 6 Jln(-2B)

we see that fl.l4) reduces to

L.H.S.

--^"

7T ^^2 ^12,0 "^ ^„ ^12, n ^12.n+l^ ' n=0 ' • (1.15)

Now the right-hand side of (1.7) can be written in terms of u as follows

R.H.S. = - F^2(") sin mi A- cos

2n

mrt'

dt'

TT ^_ ^2n J , 12 n=0 •' -1

F^„(u) cos

niTt' du.

If we now make use of the following expansion

cos

niTt' I q T (u) ^» ^mn m

m=0

and substitute for F^2(u) from (27) we find the following

o « n

r 1 '^ (u)du »

R.H.S. . i^

I l^2n%.L ^^=T f'l2 * J, ^2.k \^&^^ ' ^^'"^

n»0 m=0 '-1 /1-u^ k"0 '

195
Again, the integrations in (1.16) can be evaluated using the orthogonality

relation given in (1.3) to obtain

R.H.S. = a6 I A. [q_ C, , +

(1-6 ) n no -

n"=0 2n ^on 12 l^ ^2.k-l ^kn^ (1.17)

Finally (1.15) and (1.17) when equated and solved for C^ - gives

'12

oo °** n

^, ^2.nh2,n+l +f ^, ^2n^lAo>

J- ^2.k-l ^kn

n=0 n=0 k=l *

. (1.18)

1,0 n + ^ y A- q 12,0 . ' 2n ^on n=0

Equations (1.4), (1.6), (1.13.) and (1.18) can all be expressed in one

equation as

where

and

00

'S.l.m " -^11,10 " 4A~ i- ^2n+l ^rni m n=m

I, „ = 1,~ + -x-T— ) A„ q

12, m 12, m 2A ^ 2n mn

m n=in

21, m 21,

R„ ^

= (m > 0)

196
APPENDIX 2. EVALUATION OF THE CANONICAL INTEGRALS

The canonical integrals appearing in (29) are defined by

^11, n V

J 'n[ B /a

dn

1+

1-X

M.

^a-X^)(x^-a)

1 T (u)iln(l+u)

.,, - , " du + 6 )ln(-26) 12, n TT [^ y^I^p: no

and

l,n J n ^ e

^ /a

.d2L Ai-x'^Xx^-oi)

The evaluation of I^ „ can be found in [19] and is given by

^12, n

^

in(-e)

(.l)n+l

n :=

n >

The remaining two integrals can both be evaluated by recognizing that [25]

T (u)-T (u) = 2(l-u")U (u) n n+z n

where

-2 /l^ = -I /(l-x^)(x^-a)

and

2x2-l-cx

197
Thus, if we Cake I^, - I^, . ^ we can convert the integral into omt for il,n il,n+^

which the square root is now in the numerator. This integral can then be

integrated by parts to obtain the following recursion relation

rl+ct^ 2{~-] (2n+2) I. - ^. - (2n+l) I +j

^il,n+2 (2n+3) ^^ ^'^'' ^ "^^

where

^11, n 1^ ] n^ 6

J [TiPj

/a

X dX

and

^21,n = °

J^, is zero because the integrand of I--, does not contain a logarithmic

function as does I-,-,. The evaluation of J^ ^ is easily accomplished by

substituting

X^ = cos^e + asin^e

to give

11, n B

The starting values for the recurrence relations given in (2.1) are

given as [26]

hl.O = ^ ^^^

^11.1 "

e |2[1+K(^)-E(^)] - (1+a) U/^)

^21.0 = ^('^>

198
L

and

21,1 6 2E(/^) - (1-hi) K (/=?)

where K and E are complete elliptic integrals of the first and second kind,

respectively.

199
APPENDIX 3. DERIVATION OF THE INFINITE MATRIX EQUATIONS

The procedure for deriving the infinite matrix equations for the four

cases, even and odd TE and TM, is basically the same. Therefore, we will

derive only the odd TE case and quote the results for the remaining cases.

We begin with the definition at H^ ^ defined in (23), i.e..

H,^(v) =

r8 U(t')3^G^^(t,t') dt' /2sin(g) (3.1)

"TE From the definition of G given in (15) we can derive the following result

a^G

TE

2sin(l^) 4l I (2n+l)A

n=0

2n+l ^°" ^2a i^)

sin[(2n+l)-^] cos [(2n+l) ^]

M^] /-TTt'l (3.2)

If we use the following expansion in (3.2),

sin [(2n+l)^]

la

I r U (v)

^-, mn m

m=0

insert (3.2) into (3.1), and use the expression for H given in (25) we

obtain the following

\ ^l,n ^n(-> = h I, ^, (2"^^)^

n=0 ' n=0 m=0

fg

2n+l mn U(t')cos(^}

cos[(2n+l) ^]

cos

TTt' 2a

dt'U (v) m

We can now equate coefficients of U (v) to obtain

11, m Aa I (2n+l)A. ,, r„„

f V(t') cos (^}

n=m

2n+l mn j 2a

.TTt" cos[(2n+l)-^]l

C08{^]

(3.3)

(3. A)

200
If we now transform the integration in (3.4) from t' to u via (21) and use the

expansion given in (1.10), we obtain the following

11, m A

I I (2n+l)A. ^- r^^ p

n=ra k=0 2n+l mn ^kn

ri

-1

Fj^j^(u) Tj^(u) du . (3.5)

Next, we insert the expression for F (u) given in (27) and note that the

integration in (3.5) can be performed using the orthogonality relation

given in (1.3). Equation (3.5) then reduces to the following

11, m 8

„

oo n (2n+l) A„ ^T r p. P., , , _ n_g r

Y 2n-H mn ^kn ll,k-l

- a L L A n=m k=0

where we have defined P^

^

^ = C^

The final step involves substituting for C^ ^ from (29) with the result that

Pn = ^ I (2n+l) A„ ^. r 11, m 8 ^ 9n4-1 n n=m

2n+l mn

n-1 2P,

U.

k=0

^'^•" 1^'^ ^1,0 k=0 ^^'^^^ ^^'^

(3.6)

Equation (3.6) is thus the infinite matrix equation desired. The following

equations for the other three cases can be derived In an analogous manner.

Pio = I (i^+l) A- ^o s 12, m 2

^ 9114.9 „ n=m

2n+2 mn

n

I q k=0

2q o,n+l

I

R. k+l.n+l 12, k R^2

q k=0 ^2,k+l 12,

aB

n-1 2p

^21, m

" T ^ ^2n+l "^mn P"'^no^ J^ ^k+l.n ^21, k

" rITT J^ ^21,k+l ^21,

201
and

'22, m "2 ^ J-n ^2n+2 \n \+l,n+l ^22,

n=m k:*© '

where we have made use of the following expansion

.ln(("^^>^J n

I ^.. U (V)

sin [—J m=0

202
X

>- -*- 2

CM

-0>

CM

CSJ

CSJ

(0 •HX

«

o

o

Id

(d iH3

60C

u

0)

o

<u CO

(0

CAO

l-i u

•H

203
b^/bg = 1

2.0

1.8

1.6

1.4

1.2

1.0-

.8-

-^

-J 1 11 1 1 1 1 4-^

^a

7^^

1 ^

_ -^ ^^.^^ -

-

- ^vs^a=.25 -

-

1 1 1 1 1 1 1 1 1 .1 .2 .3 .4 .5 .6 w/a

.7 .8 .9 1

Fig. 2. Cutoff wavelength vs. strip width of the TM-, mode.

204
b^/bg = 1

Fig. 3. Cutoff wavelength vs. strip width of the TM^^ mode.

205
b^/b^ = 1

Fig. 4. Cutoff wavelength vs. strip width of the TE^ ^ mode.

206
N8S-1I4A (HEV. 7. 731

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. I'l lU.K A HON OH Ki;PORT NO.

NBSIR 78-873

2. ('ov't Accc-s.sion No.

3. Recipient's Accession No.

4. TITI.K AN!) SURm I.H

Higher Order Modes in Rectangular Coaxial Line with

Infinitely Thin Inner Conductor

5. Publication Date March 1978

6. Performing Organization Code

276.00

7. AUTHOR(S) 8. Performing Organ. Report No. John C. Tippet and David C. Chang

9. PFKFORMING ORGANIZATION NAMK AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2763280

11. Contract /Grant No. NOAACST-8393

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

Same as #9.

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 2 00- word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The singular integral equation approach is used to derive the secular

equations for both TE and TM waves in a rectangular coaxial line with zero thickness inner conductor. Approximations for the secular equations are

found that reduce to simple expressions in terms of well-known special

functions (elliptic integrals) . When the strip width is exceedingly small

or nearly equal to the width of the outer conductor, closed form expressions

for the cut-off frequencies can be found by replacing the elliptic integrals

by their asymptotic forms for modulus either near zero or one.

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

Higher order modes; rectangular coaxial line; striplines.

18. AVAILABILITY [X; Unlimited

I ! For Official Distribution. Do Not Release to NTIS

I I Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. C13

[x] Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151 207

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

42

22. Price

$4.50

USCOMM.DC 29042-P74

Appendix J

Weil, C; Gruner, L. "High-order mode cutoff in rectangular striplines."

Reprinted with permission from IEEE Trans. Microwave Theory Tech.

MTT-32(6):638-641; 1984 June.

Copyright (c) 1984 by IEEE.

209

Short Papers.

High-Order Mode Cutoff In Rectangular StripHnes

CLAUDE M. WEIL, member, ieee, and LUCL^N GRUNER,

MEMBER, IEEE

Abstract —The higher order mode cutoff characteristics of rectangular

stripline structures, «ith thin center conductors, are discussed. Data are

given, using an alternative method of presentation, on the normalized

cutoff of the first eleven higher order modes. Discussions are included on

the physical reasons why cutoff is altered for some modes, relative to that

in rectangular waveguides, but not for others.

I. Introduction

Large-scale rectangular strip-transmission lines containing a propagating transverse electromagnetic (TEM) field are now

uidely used for such purposes as electromagnetic susceptibility

and emissions testing, calibration of field probes and survey

Manuscript received April 4, 1983; re\ised Januar>' 27, 1984.

C. M. Weil is with the Boeing Militan' Airplane Companv, Mail Slop 40-35.

P.O. Box 3707, Seattle, WA 98124.

L. Gruner is with the Department of Electrical Engineering. Monash University, Oavton. Victoria. Australia 3168.

meters, and studies on the biological effects of radiofrequency

(RF) radiation exposure. These structures are characterized by an

air dielectric and a thin center conductor (septum) surrounded by

a rectangularly shaped shield. This provides for an optimally

sized test space within the line in which equipment, field probes, or experimental animals,- etc., are exposed to a well-defined and

reasonably uniform field. Crawford [1] has discussed the properties of such lines and has described a family of TEM "cells" constructed at the National Bureau of Standards. These devices

are commercially available and have been termed "Crawford

Cells" or "TEM Transmission Cells" by their manufacturers. The usable frequency range of these de\'ices is of ob\'ious

importance to those invoh'ed in their use. Whereas it had been

thought that these structures could not be used above the cutoff frequency where the first higher order mode is predicted to occur

[2], it has recently been shown by Hill [3] that such is not

necessarily the case. In his important study. Hill has shown that

significant perturbation of the internal fields within the structure

exists primarily at certain discrete frequencies where resonances

of the higher order mode fields occur. Such resonances will occur when the equivalent electrical length of the strip- transmission line

001 8-9480/84/0600-063 8S0L00 ^1984 IEEE

(c) 1984 IEEE. Reprinted with permission from

Ttech. MrT-32(6): 638-641; 1984 June.

'TTT. Trans. Microwave Iheory

211
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL. MTT-32, NO. 6, JUNE 1984

l(mn) is

gimn)

i.e.

equivalent to multiples of a half-guide-wavelength

/2, for the particular higher order mode being considered.

/, P^«,mn)/2; ;?=1.2,3,--- . (1)

The subscripts m,n denote the higher order mode. Substituting

(1) into the well-known relationship for wavelength in waveguides

J_ J_^ 1_ X' (2)

'r-'(mn) where X <.,;„„) represents the cutoff wavelength value, gives an

expression by which the various resonant frequencies /«(„„^)

be predicted

can

flR(mnp) Jc{mn) l2/,„„,/

(3)

where /,c(mn) ; 'c' is the velocity of light.

Note that the eqiuvalent electrical length /,„„, given in (3) generally exceeds the actual physical length of the transmission

line due to the presence of fringing fields at the line terminations. The magnitude of this difference varies with the mode being

excited as well as the cross-sectional dimensions of the line and

depends on whether the termination is abrupt (i.e., the line has a

box-like shape with square ends) or gradual (tapered ends). Hill

[3] was able to derive empirical values of l„„ for two different tapered cells, based on measured values of the resonant frequency

fRimnpr Attempts at predicting the fringing field correction are

presently being undertaken in order to confinn the accuracy of

HiU's empirical estimates. To what extent these structures are usable when higher order mode resonances are present and whether or not they are usable

at frequencies between such resonances depends very much on

the particular application for which the transmission line is being used and the manner in which it is loaded (i.e., the composition

and size of the object placed in the line). Some modes have been

shown to interact strongly with any sizable load within the line while others interact little, owing to the differing field patterns of

these modes. For some applications such as, for example, field- probe calibrations, it is possible to correct for the presence of the higher order mode fields by alternately positioning the probe on

both sides of the center plate and averaging the two response

cun'es versus frequency.

Acciuate prediction of the various resonant frequencies in rectangular stripline structures using (3) requires a knowledge of

the cutoff frequency /,„,„, for a number of the first higher order modes that can propagate in such structures. The purpose of this short paper is to review some of the existing data on this subject, as well as to present some additional data, as yet unpublished, in a form that is readily usable by those working wih TEM-mode

cells.

II. Cutoff Data

The higher order mode problem in rectangular coaxial struc- tures (i.e., those with a center conductor of appreciable thickness) has been independently studied by Brackelmann ei al. [4] and

Gruner [5]. Baier [6] published additional data on cutoff in rectangular coaxial lines of varying dimensional parameters. More

recently, Gruner [7] pubhshed data on the TEoi mode cutoff in rectangular lines with thin center conductors (i/b < 0.1 where '/'

is the conductor thickness and '6' is the vertical side-wall dimension: see Fig. 1). Details of the numerical techniques employed

are pronded in both of Gruner's papers and will not be further

Rg. I. Cross section of rectangular stripline structure.

1 1 1 1 1 1 1 1 "41 M —

- -^

"•ii

«J -

u - y -

u - A

i^ -

1 " /

?

1^

"Ji "^

^^^"^v^ >.",,

1J - "^^

IJ _ —

• •«•» .^Tfj, u - * - b !*•• >^

u -

1 1 1

1 1

1

-

1 a 1 U 1,s u u u •J U u

Fig. 2. Normalized cutoff frequency versus the parameter w/a for five of the

altered modes (case a/b - 2.0. t/b - 0).

elaborated on here. Similar data for the TEji, TM,i. and TM;i

modes were published by Tippet and Qiang [9] in a NBS report

that has not been widely disseminated.

It has been shown [7], (9] that, for rectangular structures having

a zero- thickness center conductor, the cutoff frequency for all modes with n-odd subscripts will be altered relative to that of its rectangular waveguide counterpart where no center conductor is present; i.e.

TE„.2„., (m,n-0,l,2--)

TMm.2,-1 (m,n-l,2.3-)-

Cutoff for all of the remaining modes having n-even subscripts

TE„.:, (m,n-0.1,2--.m#«#0)

TM„.3„ (m,ii-l,2.3- •)

remain unchanged relative to that of their waveguide counterpart.

The physical reasons for this effect are discussed in the next

section. For the unaltered modes, the normalized cutoff frequency,

relative to that of the dominant TE,o mode, can be conveniently

212
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQCES. VOL. MTT-32. NO. 6. JUNE 19\

una RATIO laM

Fig. 3. Noraialized cutoff frequency versus a/b for the lowest higher order

tnodes.

expressed as follows:

Jclmn) la

Uc(10) ' c( m »i

)

'\lm^-hn-{a/bf (4)

where 'a' is the horizontal width dimension; see Fig. 1. Of the modes where cutoff values are altered, the first four are of

particular significance, namely the TEqi, TEh, TE21, and TMu

modes. Fig. 2 shows the normalized frequency cutoff

(/f(mn)//c(io)) versus the dimensional ratio w/a, where 'w' is the horizontal width of the center plate for five of the altered modes.

These curves are shown for an aspect ratio value of a/ib = 2.0; changing the aspect ratio significantly alters the normalized cutoff

values. The curves of Fig. 2 represent the traditional maimer in which

cutoff data have been presented. This method of presentation is not optima] because, in practice, most rectanguliu" striplines possess a fixed characteristic impedance, Zq (usually 50 fl) but

use differing aspect ratio values that cover the range 0.5 < a/b <

2. It is felt that the data can be better interpreted if the normalized cutoff data are presented as a function of the aspect ratio over the range < a/b < 2 for various chosen values of Zq. This has been done in Fig. 3 which shows normalized cutoff data for the four altered modes; seven of the unaltered modes are also included for comparison purposes. The dotted Unes represent the Umiting case for a rectangular waveguide with no center strip present. Additional higher order modes such as, for example, the TE03, TEo4,- • •,TEi3,TEi4,- • •,TE;,,TE24,- • •, etc., are only

significant for structures with a/b < 0.5 and have been omitted

from Fig. 3 in order to avoid excessive detail. The curves of Fig. 3 were generated using the characteristic impedance data shown in Fig. 4. i.e., the required value of w/a could be determined for given values of Zg and a/b. The data of Fig. 4 were obtained

M M M IJ

Fig. 4. Characteristic impedance. Zq versus w/a for various a/b values

(case i/b - 0).

using the technique described by Weil and includes a correction

for the edge-interaction effect [10]. Note that the data given in

Figs. 3 and 4 of Tippet and Chang [9] for the TEn and TMji

cutoff agree well with the data shown in Fig. 3.

III. Discussion

The data presented in Fig. 3 show well how the presence of the center conductor in the rectangular structure alters the cutoff of

certain modes relative to that in rectangular waveguide. It can be

clearly seen that, for the three altered TE-modes (TEqi, TEu,

and TE21)' ^^ cutoff frequencies are reduced, with the reduction

increasing markedly as the characteristic impedance value is lowered. It is evident that, for a 50-ft line, the first higher order mode is always the TEqi for a/b ^1.94. Such effects are physically explainable as follows: for all of the TE-modes haxing even numbered values of the >-axis subscript

'n\ there exists an electrical wall along the x-axis, parallel to the center conductor (see Fig. 1). This means that for the n-evcn case, the £-field level is zero along the x-axis and no capacitive

coupling can exist between the center-strip edges and the vertical side walls. Consequently, the presence of the center strip does not

alter cutoff for the n-even modes. For the TE-modes haNing

n-odd subscripts, however, the reverse situation exists. A magnetic wall, where maximum £-field exists, is present along the

X-axis. Capacitive coupUng now exists between the center-strip

edges and the vertical side walls that causes the structure to appear electrically larger in the vertical fr-dimension than its actucil physical dimension. Consequently, cutoff frequencies are lowered for these modes relative to their rectangular waveguide

counterparts, as seen in Fig. 3. The electrical enlargement of rectangular structures in the fc-dimension is a useful concept for explaining the changes in the cutoff characteristics of the altered TE-modes. If b' represents

the enlarged dimension (i.e., the vertical height of the equivalent

rectangular waveguide without center strip), then, for the TEqi

mode, the normalized cutoff frequency is given by '^ "

(5)

'f(Ol)

a^ b'

By inverting the TEqi cutoff data in Fig. 3, a plot of the normalized cutoff wavelength against b/a is obtainable, i.e., X,,oi)/2fl = fe'/a against b/a.

213
1EL£ TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL. .MTT-32. NO. 6. JUNE 1984

For lines with very low characteristic impedance ( Zg - 0), the center plate occupies an increasingly larger proportion of the rectangular width, a of the structure. For this case, the center

strip coupling becomes large so that b''»b and a/b'-*Q.

Hence, from (4), it is evident that, for this case, 2 a

7X^,01, - for

all a/b. Similarly, for the cases of the TE,i and TE21 modes, it can be seen that 2a/Xf„,)-»l and 2a/\^^2\)-*l for all a/b.

Both Gniner's and Baier's results for the rectangular coaxial line

[5], [6] confirm the above. Referring back to Fig. 3 again, it is apparent that, for the only

altered TM-mode shown (TMn), the cutoff frequency is in- creased. Whereas when no center conductor is present (waveguide

case), the TMn TMode will always propagate before the TE21

mode, this situation generally becomes reversed when the center

strip is present. For the case of a 50-0 line, it is apparent that the TEii cutoff is below that of the TMn mode for all a/b > 0.9. Note that the presence of a relatively narrow center strip ( w/a <

0.2) causes a marked increase in the TMn cutoff, but that this increase does not exceed that corresponding to the TM12 cutoff. In fact, for lines with Zq < - 70 Q, the TMu. cutoff is essentially the same as that for the TMjj mode. In this case, when the center conductor occupies an appreciable fraction of the width (0.6 a or more), it apparently acts as an electrical wall, causing the TMjj

mode field structure to break up into a TM12 structure that contains an //-field null along the .x-axis. These results are confirmed by Gniner's data [5] which show the curves for the TMjj and TM12 cutoff, as well as those for the TM^i and TM;,

modes merging for values of vv/a > 0.6.

ACKNOWLEDG.MENT

The first author wishes to recognize tlie significant contributions to this work of Dr. W. T. Joines, Duke University, Durham,

NC

References

[1] M. L. Crawford, "Generation of standard EM fields using TEM trans- mission cells." IEEE Trans. Electromagn. Compat.. vol. E.MC-16. no. 4, pp. 189-195. Nov. 1974.

I2J C. M. Weil, W. T. Joines. and J. B. Kinn, "Frequeno' range of large- scale TEM mode rectangular strip bnes." Microwave J., vol. 24. pp. 93-100. Nov. 1981.

(31 D. A. Hill. "Bandwidth limitations of TE.M cells due to resonances." J. Microwave Power, vol. 18. no. 2. pp. 181-195. June 1983.

[4] W. Brackelmann. D. Landmann. and W. Schlosser, "Die Grenzfrequenzen von Hohren Eignewellen in streifenleitungen," Arktv der Elekinschen

Uhertragung. vol. 21. pp. 112-120. Mar. 1967.

(51 L. Gruner. "Higher-order modes in rectangular coaxial waveguides." IEEE Trans. Microyiove Theory Tech.. vol. MTT-15. pp. 483-485. .Aug. 1967.

[6] W. Baier. "Wellentvpen in Leitungen aus Leitem rechieckigen Querschnitts (Modes in waveguides consisting of conductors of rectangular cross

section)." .irkiv der Elekinschen L'beriragung, vol. 22, no. 4. pp. 179-185.

1968. Portions of the above are also reproduced in; Microwave Engineers Handbook.. T. S. Saad, Ed. Dedham, .Mass: Anech House, 1971. vol. 1, pp. 145-146.

(7J L. Gruner. "Estimating rectangular coax cutoff." Microwave J., vol. 22. pp. 88-92. Apr. 1979.

18] J. C. Tippet. D. C. Chang, and -M. L. Crawford. "An analytical and

experimental determination of the cut-off frequencies of higher-order TE

modes in a TE.M cell," National Bureau of Standards Report NBSIR

76-841. June 1976. Available from NTIS. PB 256319.

(9) J. C. Tippet and D. C. Chang. "Higher order modes in rectangular

coaxial line with infiniielv thin inner conductor." National Bureau of Standards Report NBSIR '78-873, Mar, 1978.

[10] C. M. Weil. "The characteristic impedance of rectangular transmission

lines with thin center conductor and air dielectric." IEEE Trans. .Micro- wave Theory and Tech.. vol. .MTT-26, pp. 238-242. Apr. 1978.

0018-9480/84/0600-0641$01.00 £1984 IEEE

214
Appendix K

Hill, D. A. "Bandwidth limitations of TEM cells due to resonances."

Reprinted with permission from J. Microwave Power. 18(2) : 181-195 ; 1983

Copyright (c) 1983 by the International Microwave Power Institute.

215

Bandwidth Limitations of TEM Cells

due to Resonances

Douglas A. HilP ABSTRACT

The first eight to ten TE„,„p resonances have been identified in one large (6. 1 x 7.3 x 13.0 m) and one small (1.0 x 0.6 x 2.0 m) TEM ceil. The resonant frequencies fit a new

equivalent coaxial box model with the effective length of the box depending on the mode.

Weakly propagating TE^n modes were also detected at frequencies above their respective

first-resonance frequencies. A biological body or metal box at the center of the test zone

interacts strongly with the TE|i,p and TEiip resonances, but not with the TE)ip resonances. These interactions cause bandwidth limitations in the use of TEM cells for bioeffects dosimetry studies, EMC testing, and probe calibration work.

I. INTRODUCTION

A TEM cell is a rectangular-coaxial transmission line sharply tapered at each end to form a closed

cell. These cells are currently used for radiofrequency radiation dosimetry [l— 5] and bioeffects [2,

6, 7] studies, for the calibration of field-strength meters [8- 10], and for EMC testing of electronic equipment [11-13].

The center section of a TEM cell is a uniform-cross-section transmission line that will propagate

a single TEM mode at all frequencies and a set of TE„,n and TM„,n higher-order modes at frequencies

above their respective cut-off frequencies. Field patterns for the higher-order modes have been

reported by Baier [14, 15] and the cut-off frequencies calculated by Baier [14, 15], Gruner [16— 18], Crawford [8], and Tippet et al. [19, 20] using different methods. Weil et al. [6, 21] have reported

possible evidence of a strongly propagating TEn mode in two different TEM cells and have noted that

this is consistent with Gruner's finding [17] that the TEm mode is usually the first to propagate in a TEM cell. The TEM mode propagates through the tapered ends of the cell without significant alteration. Each higher-order mode, however, is always reflected at some point within the taper where it becomes

too small to propagate the mode. The propagating energy in the higher-order mode undergoes multiple

reflections, end to end, within the cell until it is dissipated. At certain frequencies a resonance

condition is satisfied, in which the cell's effective length for the mode is 'p' half guide wavelengths

long ip = 1, 2, 3, . . .). At these resonant frequencies, /Rminpi. a TEnmp resonant field pattern exists. Thus the TEn,n mode in a given TEM cell has one cut-off frequency, /„„,„,. associated with it and an

infinite set of resonant frequencies, /Rmmpi. with p - I, 2, 3 The same is true for the TM„,n

higher-order modes, although these only occur at higher frequencies. While many of these resonances have been observed several times [1 1 — 13, 19, 22] and one has been calculated, neither of the first two resonances, TEqh and TE|oi, have been properly identified or calculated to date. This paper will identify, by resonant frequencies and field patterns, the first eight to ten TEn,np resonances in two different TEM cells. Propagating higher-order modes will then be discussed. We

will also report the interactions between the cell resonances and two test objects: a human body in a

large (6. 1 x 7.3 x 13.0 m) TEM cell used for dosimetry studies [5]; and a rectangular metal box in a small (1.0 x 0.6 x 2.0 m) TEM cell used for EMC studies (Narda Microwave model 8802).

The goal of this study is to determine the extent to which TEM cells may be used above the cut-off frequency of the first higher-order mode,/,„ii,. More specifically. /,„iii for the large cell is 15.2 MHz

and it is desired to extend the usable frequency range as high as possible, to the ISM frequency of

27.12 MHz and perhaps even to 50 MHz in an attempt to measure the predicted peak absorption in a grounded person at about 47 MHz [23, 24].

Manuscript received AugusI 2.^. l^Hl: in revised lorm Januar\ y\. 1483 Work perlnnned jl ihe Division ol Biological Sciences. National Research Council ol Canada. Ottawa. Issued as DREG Report No ; Xfi4 'Radiati(<n Biology Section. Delence Research Establishment Ottawa. Ottawa. Canada. KI.A 0Z4.

Copyriithi I' IVS.I h\ l\tPI OiM»ii. Caiuhlu Journal of Micnmaie Power. I»t2l. I9HJ

Ccpyright (c) by IMPI, Ottawa, Canada. Reprinted with permission frcatn J.

Microwave Fewer. 18(2): 181-195; 1983.

217

^
JOURNAL OF MICROWAVE POWER. 18(2). ]W}

0-i20 FREQUENCY (MHz)

Figure I Detection of the TEon resonance at 20.70 MHz by swept-frequency measurements, (a) Electric field (£b) at position Bl . (Field components and positions are defined in Fig. 3 and the text), (b) Incident (I), transmitted (T), and reflected (R) power. Ordinate scales are linear in (a) electric

field, or (b) power, but otherwise arbitrary.

II. RESONANCE ANALYSIS OF A LARGE TEM CELL

A. Detection of Resonances

The cell resonances were detected by swept-frequency measurements of electric field strength

within the cell, and of reflected and transmitted power at the cell ends. As Crawford reported [8].

filtering of second to fourth harmonics to levels at least 40 to 50 dB below the fundamental was found

to be essential. Field-strength measurements were made using a dipole and loop, each 10 cm in size,

of Greene's type [25] which use high-resistance leads to avoid RF pick-up problems. Electric-field measurements were found to be a reliable way of detecting all cell resonances, whereas reflected power

(or input-port VSWR) and transmitted power (or insertion loss) were not. This is clearly illustrated in Figs. 1 and 2 where the detection of the first two cell resonances is shown. The strong resonance at 30.0 MHz, later shown to be the TEioi resonance, is not detected in the reflected and transmitted power

measurements. If the strength of a resonance is defined by the peak-to-peak change in power or in the

largest resonant E-field component, it was found that the first eight TE^np resonances could be better detected using E-field measurements than using power measurements; the signal-to-noise ratio is 20

to 30 dB larger. (The resonance at 20.7 MHz is 10 dB stronger for a different E-field component than shown in Fig. 1(a).) The four weakest of the first eight resonances were undetectable using power

measurements.

B. Identification of Resonances

The first seven cell resonances were positively identified using two kinds of evidence, taken

together: the measured resonances (and antiresonances) of the main component of the TEM-mode E

field at selected sites in the transverse plane; and the comparison of measured and calculated resonant

frequencies.

In order to understand the expected E-field pattern of the resonances the mode patterns for the higher-order modes are first compared to the TEM-mode field pattern in Fig. 3. That figure shows the more general case of a finite-thickness septum, while the septum is actually infinitesimally thick in

218
HILL: BANDWIDTH LIMITATIONS OF TEM CLLLS

("10)

(b)

Oi

30 31 FREQUENCY (MHz)

Figure 2 Detection of the TEi,,, resonance at 30.01 MHz by swept-frequency measurements, (a) Electric field (£h) at position Bi. (b) Incident (1). transmitted (T). and refiected iR) power.

b CL CL

i '1 ^ ' r . E« B«

i 0» A*

1 F. C«

CL G CL

(a) KEY TO POINTS

a (vertical)

/ (transverse)

I (longitudinal)

(b) DIRECTIONS

(C) TEM MODE (e) TE,„MODE

(d) TEo, MODE (t) TE,, MODE

Figure 3 Cross-section of a rectangular-coaxial transmission line with a septum of finite thickness. Points and dimensions are defined in (a), and directions in (b). E-fieid patterns are sketched for the fundamental mode in (c). and accurately drawn (from [14]) for the first three higher-order modes

in (d)-(f).

the two cells investigated. A number of reference points in the transverse plane are defined in Fig. 3(a): points A to F are all located midway between septum and wail; points A and D are also midway

between the other two walls. Figure 3(b) also defines the directions ci. h. and ( in the transmission

line. The size of a cell will be denoted by its dimensions in the a. b. and ( directions, in that order. The ( dimension is the cell length, tip to tip. The use o\' the directions a. b. and ( . rather than directions

in space, will facilitate intercomparison of results between TEM cells oriented differently in space (compare, for example. [5], [7], and [8]).

219

\
JOURNAL Oh MICROWAVE POWER. ISO. I'>S.^

Tabic I Idcntitieation of resonances in the large (6.1 x 7.3 x 13.0 m) TEM cell from E-ficId and

circumferential-current characteristics. Positions AO to XI are defined in the text

' t-Minp resonance Form of £., resonance n (\R /I \ Circun" fcrcntial currents Detected E-n

Largest h ^^ ^V- on floor

:ro: N

inu wail in VSWR eld component

or trans. (^) (MHz) m. n. P AO Al Bl CI El WO Wl GO Gl XO XI power? e,

20.70 1 1 O R A R N N O N N yes E. 16 31.51 1 2 O R A R O N N no E.. 3.1 42.06 1 3 o o A R A N N N N no E. 1.3 30.01 1 1 R R R R A N N N N no E, 4.0

40.98 1 2 A A A R N N yes E. 4.6 36.08 1 1 1 R R R R R N N yes E, 11 47.2 1 1 2 O R R R R o O N yes E, 27

44.77 2 1* O O R A A N N N N N N no £. 1.8

•This idenlitlcalion not certain (see text).

The total E-field pattern due to the presence of some RF energy in a propagating higher-order

mode can be qualitatively understood by adding (or subtracting) the pattern of the higher-order mode

to (or from) the TEM-mode pattern. Thus, for example, when the TEdi pattern is added to the TEM

pattern, the relationships for £(, at the six points A to F become; E^iB) = £'h(E) > £b(A) =

Ei,(D) > £'b(C) = £b(F). Similarly, the addition of the TE|u pattern to the TEM pattern results in the relationships £b(A) > £,(B) = £,(C) > £h(D) > £b(E) = E^{F).

Returning now to the resonances in a transmission line of finite length, the fields at the points

A to F will depend not only on the mode but also on the longitudinal position in the cell. For example,

when two half guide wavelengths of a TEnm mode equals the cell length, the TEn,n: standing wave

pattern is created, which has an E-field null at the mid-point of the cell, and is not detectable there.

In assessing E-field patterns, it was found useful to employ positions located either meters (AO,

BO. . . XO) or I m (Al, 81, ... XI). approximately O.I \. towards the load from the cell center.

It was also found very useful to measure the wall currents associated with the resonances. They

were determined by measuring the H fields parallel to the metal walls and using the well known

boundary condition that the tangential H field is equal to the surface current density in the perpendicular direction. (Surface charges and current patterns for the case of higher-order modes propagating

in circular coaxial transmission lines have been well illustrated by King et al. [26].) For example, for the TEoii resonance, the maximum charge on the wall (maximum concentration of E-field lines) is at position GO and there is no surface charge at WO. The current maximum, however, occurs at WO a

quarter of a cycle later, and a current null occurs at GO. Similarly, the TEmi resonance has a current

null at WO and a maximum at GO. while the TEm resonance has a current null at both WO and GO.

For the p = 2 resonances, similar results to the above occur at the positions Wl and Gl. and the currents at WO and GO are all zero regardless of the mode.

The measured E fields and wall currents at the resonant frequencies are characterized in Table

I. The frequency evidence for this identification of the resonances is given in Table 2. All the observations in Table I for the TE„ip. TEmp, and TEnp resonances can be shown to agree with the

theoretical expectations from the kind of arguments illustrated above. The results for the TEo:i resonance likely fit the theoretical expectations as well, but this is not certain because a drawing of

the TE,i: mode pattern has not been published.

The only theoretical ambiguity in identifying resonances is that the pattern of E-field resonances

(R) and antiresonances (A) may be inverted when the pha.se of the higher-order mode with respect to the TEM mode is inverted. For example, the pattern of R.A.R for the positions 81. CI, and El for the TE„|| and TE,,:: resonances is inverted into A.R.A for the TE„|, resonance. Apart from the above theoretical ambiguity, the E-field and current characteristics listed in Table

I are universal characteristics for all TEM cells of any size or shape. The currents at XI are all non-zero, as expected theoretically. The same is true for the four positions marked CL in Fig. 3(a).

In a future paper, we will report on an attempt to suppress the resonances by cutting the cell longitudinally at the four cutting lines, CL. in order to eliminate the circumferential currents that

220
HILL: BANDWIDTH LIMITATIONS OF TEM CELLS

Tabic 2 Identification of resonances in the large (6. 1 x 7,3 x 1 3.0 m) TEM cell from the resonant frequencies.

/«„„„p, ^ 53 MHz

Higher-order mode Resonance

Type A„„„, (MHz)

X,„„ (

"^ \

Type

TE^np or TM,„„p

/R,n„„p, (MHz)

TE^„ or TM„„:

Waveguide

formula

Series expansion

Calculated using

\2lJ

m n [7] [16,27] (±0.02) (MHz) m n p u; L^„ Measured

TE TE

1 20.5* 15.2 0.74 13.7 1 I

1 2

1 3

llA

35.4

50.3

20.5 31.3 43.820.70

31.51 42.06

1 24.6 24.6 0.45 16.6 1

1

1 229.3 40.429.7 41.330.01 40.98

1 1 32.0* 31.3 0.35 17.9 1

1

1 1

1 2

35.2 44.835.9 47.436.08

47.2

2 41.0 41.0 0.34 18.0 2 1 44.0 44.8 44.77

2 1 53.3* 45.4 -^ — 2 1 1 48.1 — —

1 2 47.8 47.8 — 1 2 1 50.4 — —

2 49.2 49.2 — — 2 1 51.7 — —

TM TUX

1 1 32.0* 47.5 I.OO 11.6 1 1 1 50.1 48.9 48.9

1 2 47.8 47.8 — — 1 2 1 50.4 — —

*Figures known wrong bul shown for comparison purposes.

+i.„ = t, + 0.5 /-K. (^) = 16.0 MHz.

^Resonance not delected. +See note added in proof.

support the resonances. The effect of the cuts on the resonant currents will be monitored at position

XL The last column of Table 1 gives the relative strength of each resonance, it is defined as the ratio of the largest peak-to-peak change in any of the three components of E to the magnitude of £b near

the resonance: all measurements are taken at position Bl since all resonances are detectable there.

In Table 2, the resonant frequencies have been theoretically predicted with the data arranged by

modes in increasing order of/..imn,. The latter is calculated from the cell dimensions in the transverse

plane. While Crawford [8] has suggested the use of the waveguide formula to calculate /cmm for all TE„„ and TMmn modes, Gruner's analysis [16, 17] shows that the waveguide formula is only valid for n even. Gruner calculated/,., mm for n odd using a series expansion method. His results [27] for the large

TEM cell are compared to the waveguide-formula results in Table 2. Some of the correct values of

/t(mni may be obtained from other publications. For example, the value of 15.2 MHz for/ion is also obtained by interpolating in Fig. 2 of Weil et al. [21, 28] for the large cell (70 fi [5], and

ajb - 0.833). Furthermore, the one-term approximate solution of Tippet and Chang [20] gives values

of/,(mni for the TEn and TMn modes which are in agreement, within 5%, with Gruner's results. Finally, it is noteworthy that/,.,31, < /.coi for the TE modes since this ordering of the modes is reversed from the predictions of the waveguide formula. The resonant frequencies, /Rmmpi. are ideally calculated from the values of/imm and the cell's length and taper dimensions using a resonance mode chart. While such a chart has not been published

for TEM cells, a good approximate calculation can be done by modeling the cell as a transmission line with flat transverse ends instead of tapered ends — the 'equivalent coaxial box' model. The condition

for the resonances of the TEn,„ mode is that the effective length of the box for that particular mode,

Lmn. equals an integral number of half guide wavelengths for the mode:

Z-mn = p(X„„,n,/2); p = 1,2.3,... (1)

Z.,nn will be taken to be an adjustable parameter which is the same for all resonances of a given mode.

This is based on the reasonable assumption that the septum and wall current patterns in the tapered

ends of the cell, which determine the effective cell length, are similar for all resonances of the same

221
JOURNAL OF MICROWAVE POWER, 18(2). 1983

Tabic 3 Analysis of resonances in the small (1.0 x 0.6 x 2.0 m) TEM ccll.*/R,„,np, < 360 MHz

Higher -order mode Resonance

Detected Type / '

\ Type

y Rdnnp) (MHz) in VSWR Strength

TETM„ /c.,„n, \2lJ ' ^mnp or trans. m n (MHz) x,„„ (MHz) m n P calc. meas. power?

1 129 0.8! 83 1 1 155±3 156.3 no 0.4 ±3 ±0.04 ±1 1 2 210±4 212.9 no 2.0

1 3 280 ±6 282.1 no 0.4

I 4 356 ±8 351.3 no 0.3

1 150 0.49 101 1 181±4 184.3 yes 4.3 ±1 ±0.04 ±2 2 252 ±5 255.7 no 1.9 3 338±7 329.6 no 1.2

1 1 260 0.52 98 1 1 278±9 272.7 yes 4.8 ±8 ±0.11 ±2 1 2 326±10 331.8 yes 9.2

2 —

— — 2 1 —

318.0 no 0.5

•Narda M crowave Model 8802: a = 100.3. b == 60.0. w = 72. \.L = 100.3 /.,. = 101.2 (cm): Z„ = 50 0.

'''/cKUi and/g,,,,,, not known; resonance identiHed from field characterisitcs.

mode. Thus, one parameter, L„,„, is adjusted to get good agreement between measured and calculated frequencies for between two and four resonances, depending on the mode. £„,„ is related to the cell dimensions by the equation

^mn ^c ' •^mn^E (2)

where L^ is the length of the uniform-cross-section center part of the cell (6.10 m), L^ is the length

(along the center line) of the two tapered ends of the cell (6.86 m), and X„„ is the fraction of the two

ends included in the value of L^„.

By using the standard waveguide formula for Xgimm [29], (\"' = \J

conversions from wavelength to frequency (/Rmmpi = <^/X>/c(i (eq. 1) becomes

/R(mnp) /td [ilJ ' (p > 0; /M,/i > 0)

+ \>.|;;„,) and the usual = f/^tinin)) the resonance condition

(3)

C. M. Weil has pointed out (private communication) that \g in the tapered regions of the cell is actually larger than given by the waveguide formula applied to the center section of the cell. Hence,

\g as used in the resonance condition (eq. (1)) and the waveguide formula is an average of the values applying to the center and tapered sections of the cell. Using eq. (3) with (c/lL^n) as an adjustable parameter, good agreement between measured and

calculated frequencies for the first eight TEn,n resonances was obtained. /«,„„?! was also calculated for

all resonances, for comparison purposes, using the average cell length of Lo = L^ + 0.5 L^ =

9.38 m. The agreement between measured frequencies and those calculated using the average cell length can be seen to be significantly worse than for the calculation using an adjustable cell length.

The values oi X^n corresponding to the best-fit values of LTM are also given in Table 2. The first TM resonance was uniquely identified by the presence of a resonant longitudinal E field, as well as by the measured frequency.

III. RESONANCE ANALYSIS OF A SMALL TEM CELL

A. Resonance Analysis

The resonances in a smaller (1.0 x 0.6 x 2.0 m) TEM cell were also investigated using the same methods as for the larger cell. The results are presented in Table 3. The cut-off frequencies were

calculated from the waveguide formula foryi^^iioi and from Figs. 1 and 2 of Weil et al. [21, 28] for/^^m,, and/^,01), respectively. The uncertainties shown for/,in,„| result primarily from interpolation in the

figures of Weil et al. The uncertainty shown for Xn,„ takes into account both the uncertainty in /,.,„,„,

222
HILL: BANDWIDTH LIMITATIONS OF TEM CELLS

Table 4 Comparison of Xm„ determined for the first three TE„,„ modes in two TEM cells

Cell outer dimensions Cell parameters x,„„ axbx e (m)

ab w

a ZotQ) TE,„ TE,„ TE,,

6.1x7.3x13.0 0.83 0.67 70 0.74 ±0.02

0.45 ±0.02

0.35 ±0.02

1.0x0.6x2.0 1.67 0.72 50 0.81 ±0.04

0,49

±0.04

0.52 ±0.11

Average 1.25 0.70 60 0.77 0.47 0.43 ±2 (difference) ±0.42 ±0.03 ±10 ±0.04 ±0.02 ±0.08

and the estimated uncertainty from adjusting (c/2 L^„) for the best fit to the measured frequencies. The

agreement between measured and calculated frequencies is excellent, the largest difference being 3%

for the TE|03 resonance. As with the large cell, all resonances except for the TEo:i were positively

identified in terms of the universal E field characteristics of Table 1 . In the small cell the largest resonant E-field component for the TE resonances was always found to be E^,, so that the strength of

the resonance is simply defined as ^EjE^, for the purposes of Table 3.

B. Comparison With Large Cell The results for the resonance field strengths in the two cells were compared. In both cases it was

found that resonances with a relative E-field strength (^EJE^,) of less than 4 were consistently not observable by power measurements, while stronger resonances were. Two of the strongest resonances

in both cells were found to be the TEn: and TE,ii resonances, in that order. This is likely explained

by the similarity of the TEn mode to the TEM mode since a transfer of power between two similar modes is more likely than between two dissimilar modes. The group at NBS [12, 13, 19] also found

the first large resonance to be the TEm resonance (which they called the first TE|o resonance using

a different notation — see section VI. B).

Generally, the resonances in the large cell are more strongly excited. This is particularly true of

the TEoM resonance. The stronger excitation of the resonances could be due to a number of factors but the most likely one is the presence of the two wooden platforms in the large cell (see Fig. 1 of

[5]), which would scatter radiation into the higher-order modes.

The effective lengths of the two cells, expressed in terms of the fraction of the tapered ends

included for each mode, are compared in Table 4. Considering the large differences in cell parameters,

the agreement in the values of X^n is considered good.

IV. PROPAGATING HIGHER-ORDER MODES

A search was made for the presence of propagating higher-order modes in the empty cell at all non-resonant frequencies. The modes were detected by comparing (within ±2%) the £h fields at appropriate sites in the transverse plane. In searching for the presence of the TEoi mode, £h at position

B was compared to E^ at position C. A difference of 8% in the two fields would be recorded as a 4%

contribution of the TEoi mode to the total E field since that mode adds to the TEM field at position

B and subtracts from it at position C (or vice versa). Similarly, the presence of the TE|o mode was

detected by comparing E^, at positions E and C. The presence of the TEn mode can only be detected by comparing £h at a given point above and below each resonant frequency; it does not produce

asymmetries in the region of points A to F. Using these methods, it was established that none of the tlrst three higher-order modes (TEt,,, TE|o, and TE,,) is present between its own cut-off frequency and its own first-resonance frequency in either of the two cells tested. For example, the TE^i mode is undetectable in the large cell between

/cod (15.2 MHz) and/R,oi 1,(20.7 MHz) but makes a 4% contribution to the E-field pattern above /r,,,,,,. Similariy, the TE|o mode is undetectable between /.noi and/R,,,,,, but makes a 5% contribution to the

field pattern above /«, ion- The field contributions of each of the first three higher-order modes in each

TEM cell are given in Table 5. A range is shown when the results depend on frequency. The p = 2

or 3 resonances were found to cause negligible changes in the contribution of each propagating

higher-order mode.

223
JOURNAL OF MICROWAVE POWER. IXCi. 1483

Table 5 E-field contributions of the propagating higher-order modes above their respective first- resonance frequencies. Upper left figures apply to the large (6. 1 x 7.3 x 13.0 m) TEM cell and

lower right figures to the small (1 .0 x 0.6 x 2.0 m) TEM cell

Higher-order mode (TE^n) TE,„ TE,„ TEm

/r.TM., (MHz)

20.7

156

30.0

184

36.1 273

Contribution to E^ from

propagating TE„,n mode (%)

4 <3

5 lo- ts 8-10

10-15

It may be that each TEn,n mode does not propagate at frequeticies between/,in,„, and/Ri^n,, because

the cell length is less than half a guide wavelength in that range and a length of at least half a guide

wavelength is required for the wave to develop. For example, in the large cell./,,ioi = 24.6 MHz and

/r(ioi) = 30.0 MHz; at an intermediate frequency of 27.12 MHz, \g,:oi/2 is 13.1 m which is much

larger than the effective cell length for that mode, 9.0 m. Weil et al. [22] investigated E-field patterns in a SO-H TEM cell of cross-sectional dimensions

33.5 X 30.5 cm. At 425 MHz they found large E fields near the edge of the septum, which they

interpreted as evidence for the propagation of the TEoi mode. Using the parameters:

L^ — Le = 1.52 m (private communication); Xoi = 0.77 (from Table 4); and/-,oi, = 275 MHz; the

first six TEoip resonant frequencies were calculated. Their measurement frequency was found to lie between/Riois, (392 MHz) and/R,oi6| (434 MHz). Thus, the field pattern they observed could have been

either the propagating TEqi mode or one of the higher {p — 5 or 6) resonances of that mode, or a combination of both. Despite the presence of resonances and propagating higher-order modes, 50-fl TEM cells can still be used to reliably calibrate field probes at most frequencies between/„oi, and/RHini- This is done by

averaging calibration data taken at points AO and DO. Neither the TEji mode nor its resonances are detectable at these points and the effect of the propagating TE|o mode is eliminated by averaging the two results. The TEiop resonances, however, must be avoided. Thus, for example, the small cell is usable for probe calibration work up to 273 MHz (/rhh,) except for frequencies within ±5 MHz of

the TEiop resonances at 184 and 256 MHz. This is consistent with the manufacturer's stated frequency

limit of 250 MHz except for frequencies of 184 ± 5 MHz, near the TEioi resonance.

V. BANDWIDTH LIMITATIONS FOR BIOEFFECTS DOSIMETRY STUDIES

A. Resonance Interactions in the Large TEM Cell The interactions between resonances and RF-absorbing biological bodies can be qualitatively studied by measuring E^, as a function of frequency near each resonance for the empty and loaded cell. The results for the TEioi resonance are shown in Fig. 4. It can be seen that the presence of the human

body in the cell inverts, shifts, and broadens the resonance. The effect is more pronounced for the grounded case since the interaction is stronger and the subject absorbs more RF power. The results for the TEni resonance at 36 MHz are very similar to those for the TEio: resonance at 30 MHz.

The interaction of the body with the TEon resonance at 20.7 MHz. however, is quite different. The traces of Fig. 5 show clearly that the body does not interact significantly with the resonance, but does cause a slight frequency-independent reduction in E^. This is thought to be nothing more than

a slight shadowing effect of the body on the probe. There is negligible interaction because the E^, fields of the TEy, mode and its resonances are zero at position A. Consequently, the body does not interact with, or absorb energy from the resonant fields. The use of multiple bodies at positions including B,

C, E, and F could result in interactions with the TE„ip resonances. The same pattern of interaction between the body and the resonances also holds for the higher

resonances of the same modes; a body in an E orientation at position AO or Al interacts with all TEmp

and TEiip resonances but with none of the TEoip resonances.

Plots such as those of Fig. 4 give semiquantitative evidence that for frequencies near the resonances the effective exposure fields are altered significantly from tho.se measured in the empty

cell. That this is so may be seen in the dosimetry measurements of Table 6. Compared to the expected

approximately frequency-squared dependence [30], the absorption rates for the saline-filled cylinder

224
HILL: BANDWIDTH LIMITATIONS OF TEM CLLLS

(a) FREE SPACE

(b) GROUNDED

LOADED\

28 29 30 FREQUENCY (MHz)

Figure 4 The interaction of a human subject with the TEi,,i E-field resonance. £h is recorded for the empty ceil and again when loaded with a subject in an E orientation under (a) free-space, or (b) grounded, conditions. For (a) the subject is centered on position AO (defined in the text) and the probe at Al. For (b), the subject's bare feet are touching the wall and the probe is in the same

position, relative to the subject, as for (a).

FREE SPACE

20 21 FREQUENCY (MHz)

Figure 5 The lack of interaction between a human subject and the TE,im E-field resonance. The

probe records Et, at position CI. The "loaded' trace is with a subject in an E orientation at position CO.

and human subject F are anomalou.sly high at the two frequencies (27. 12 and 34.30 MHz) just below

the two main interacting resonances (30.0 (TEioi) and 36.0 (TEm) MHz), and anomalously low at the two frequencies (30.5 and 38.5 MHz) Just above the resonances. This result is exactly as suggested

by the traces of Fig. 4. Unfortunately, we do not have a way of quantitatively evaluating the altered effective exposure fields. Because of this, the large TEM cell cannot be used, in its present form, for bioeffects dosimetry work above about 25 MHz, or 837f of/R,iiih- the first interacting resonance.

Tests were also done for a second kind of possible dosimetry error resulting from the propagation

of higher-order modes. Radiation scattered by the body from the TEM mode into one of the higherorder modes would never leave the cell and would eventually be absorbed by the cell. This would make

a false addition to the detected absorption of the human subject. This possibility was tested by

measuring the apparent absorption of a test object found to interact with the resonances like a human

subject, but presumed to be nonabsorbing: a full-sized commercial manikin covered with a layer of

thick aluminum foil. It can be seen in Table 6 that the measured "absorption" of the metal manikin

is detectable at some frequencies above 25 MHz. but that it is within acceptable limits (< lO'^ of the absorption of the human subject or saline model).

225
JOURNAL OF MICROWAVE POWER. 18(2). 1983

Table 6 Normalized specific absorption rates. NSAR, for three test objects exposed to RF fields. The objects are in an E orientation at position AU in the large (6.1 x 7.3 x 13.0 m) TEM ceil

Frequency

(MHz)

NSAR (mW/kg per mW/cm") Mean ± SE(N)

Subject F Saline cylinder+ Metal manikin

18.50 22.3±3.1(8) 28.3±2.4(8) 3.3±3.0(10)

1.0±3.1(10)

23.25 38. 6± 1.3(6) 47.9±4.6(6) -1.9±1.5(9)

27.12 63.1 ±3.8(6) 106.4±4.7(6) 6.5± 1.9(10)**

8.6±4.9(9)

30.50 32.3±2.1(6) 13.8±4.4(11) 0.4± 1.9(9) . 34.30 106.4± 1.5(6) 129.4± 10.0(6) 4.9±2.4(9)*

38.50 75.2±2.9(6) 88.9±3.0(6) 3.5±I.1(9)*

Height (cm) 175.3 183.0 183.0 Mass (kg) 84.0 62.0 (70.0)

^Saline at 23°C. t' = 80. ct = 0.43 S/m.

Signincance levels using a *p < 0.05

one-tailed Student's t lest **p < 0.01

In conclusion, two different bioeffects dosimetry limitations due to resonances and higher-order

modes have been investigated and only one has been found to be important — the interaction between some of the resonances and the human subject.

B . Discussion of Other Bioeffects Dosimetry Work in TEM Cells While several authors are using TEM cells for RF bioeffects studies, only Allen et al. [1], Joines

et al. [2] and Marshall et al. [3] have used their measurement systems to determine the ratio of

absorption rate to exposure power density for the animals or preparation used. The possible application

of resonance interaction concepts to their work will be briefly considered.

Allen et al. [I] measured RF absorption in rhesus monkeys and monkey phantoms from 10 to 50 MHz in a 50-0, 2.82 x 1.45 x 9.15 m TEM cell. Using their cell dimensions [1, 31] and

assuming L^ = L^, the first two cut-off and resonant frequencies were calculated to be

/cioi) =/ciiO) - 53 MHz./r,oii) = 56 MHz, and/R,iou = 58 MHz. Their highest operating frequency,

50 MHz, appears to have been sufficiently below /rho,, to minimize the interaction effect. This is confirmed by the lack of anomalous absorption results at 50 MHz in their measured curves. Joines et al. [2] operated at 147 MHz, which is so far below the calculated value of/„oi, for their

cell, 930 MHz, that resonance interactions would not have been a problem in their experiments.

Marshall et al. [3] measured the absorption of mice and prolate-spheroidal models of mice from

200 to 500 MHz, in a 50-il, 0.50 x 0.30 x 1.00 m TEM cell. Since their TEM cell is a half-sized

scale model of the cell analysed in Table 3, their resonant frequencies are all double those given

in Table 3. in particular, /rioui = 313 MHz, /r.iou = 369 MHz. /r,oi2i = 426 MHz and /r,io2i =

51 1 MHz. Their E-polarization absorption results for the models (Fig. 6 of [3]) show an anomalously

high absorption rate at 500 MHz, consistent with the field anomalies they detected, and likely due to

the TE,o: resonance. Other anomalies are also apparent in the 300- to 400-MHz range and are likely

due to interactions of the six mice models with the TEoi,, TEioi. and TEoi: resonances. In their experiments, an interaction with the TE,j|p resonances is possible since four of the six mice in their cell are offset from position AO in both the a and t directions.

VI. BANDWIDTH LIMITATIONS FOR EMC STUDIES

A. Resonance Interactions in the Small TEM Cell The small TEM cell is rated by the manufacturer for use in EMC studies up to 250 MHz provided

the equipment under test (EUT) is no larger than 10 x 20 x 30 cm. The interaction of a metal box

of that size with the TE resonances was studied using the methods described above for the large cell. Several of the interactions are shown in Figs. 6 and 7. In Fig. 6 the interaction with the TEioi resonance

is seen to be strong, while the TEoh resonance is barely detected, with or without the metal box

226
HILL: BANDWIDTH LIMITATIONS OF TEM CELLS

160 170 FREQUENCY (MHz)

Figure 6 The interaction of a small rectangular metal box with the TE,,,, resonance in the small

(1.0 X 0.6 X 2.0 m) TEM cell. The box is centered on position AO and oriented with its dimensions of 30 X 20 X 10 cm in the (, a. and b directions, respectively. The probe is at position A2

(0.2 \. towards the load from position AO). Vg is the E-probe voltage (in arbitrary units) and is approximately proportional to f^ at the low power levels used.

TE,

Ve

0-i

TE,v

LOADED

EMPTY

250 260 270 FREQUENCY (MHz)

Figure 7 The interactions of a small rectangular metal box with two of the higher resonances in the small (1.0 x 0.6 x 2.0 m) TEM cell. Details regarding box. probe, and ordinate scale are the same as for Figure 6.

present, because both box and probe are centered on position A in the transverse plane. In Fig. 7 the metal box is shown to interact strongly with both the TEio: and TEm resonances. On the scale used

to show the resonances in Fig. 7, E^ for the TEM mode is barely measurable. It can be seen in both

Figs. 6 and 7 that the presence of the EUT sometimes strengthens a resonance. It is also noteworthy

that some resonances are shifted up in frequency when interacting with an EUT at position AO. This was found for the TE102 resonance (Fig. 7) and the TEn: resonance (not shown).

In general the interactions of the metal box or human subject with the resonances are similar. Both

test objects are conductors, with the metal box being a good conductor (tan 8 > 1) and the human

subject being a poor conductor (tan 8 = 4 at 20 MHz [30]). The values of tan 8 account for the small

relative changes in Q of the resonances interacting with the metal box and the larger changes in Q of

the resonances interacting with the human subject.

The small TEM cell is usable for EMC studies up to 250 MHz with the exception of a range near 180 MHz. The range would appear, from Fig. 6, to be 20 MHz, but Crawford's experience with actual EMC testing would correspond to 40 MHz for this resonance (private communication). It would also be usable, with reduced accuracy, at inter-resonance frequencies above 250 MHz.

227
JOURNAL OF MICROWAVE POWER. IXC). I9«3

Tabic 7 Analysis of resonances in the NBS 3.0 x 3.0 x 6.0 m TEM cell,* /k,„.„p, < 100 MHz

Higher-order Mode Resonance

(MHz)

-

/r. n„p, (MHz)

TE„„

Detected by:

or TM„„ m (MHz) a:,„„ Type m n P

calc. from (3) VSWR or

fh"*" insertion loss

TE,„ 0.58 29.0 0.77 28.2 1 I 40 -^

1 2 63 — —

1 3 89 — —

TE... i.OO 50.0 0.47 34.0 1 1 60 60 —

1 .0 2 84 — —

TE„ 1.27 63.5 0.43 35.0 1 1

1 1

127395 74 —

TMm — — — — 1 1 11 — 97 97

*[I0]. o = b = /,, = /,,: = 3 00 m; Z„ = 50 O: (2o/Xj Irom [26]; ,V„,„ from Table 4. i'Mea.sured al position AO. ^Resonance not detected in original study flO]. In recent measurements of £k at positions AO and Al. Crawlbrd has detected resonances in this

cell (without absorber) at the following frequencies: 38. 50. 61. 65. 74. and 85 MHz. The frequency was stepped digitally in 2-MHz increments

(private communication).

+See note added in proof.

B. Discussion of the NBS TEM Cells The group at NBS, in a series of papers [8, 11 — 13. 19-21]. has been developing the use of TEM

cells in EMC work. The application of our findings to their work will be discussed below.

Unfortunately, they use a different nomenclature regarding the identification and labeling of the

first cell resonance. They take the first cell resonance to be either the first resonance [1 1], or the first large resonance [19], detected in input VSWR. In either case the one they are referring to is called

the first resonance of the "TEio mode" [12, 13] or of the "perturbation to the TEio mode in a rectangular

waveguide" [19]. The mode in question is described in [19] as being the antisymmetric type of

perturbed TE|o mode, in which the electric field is oppositely directed on each side of the septum. In our notation of Fig. 3(a), this is the TEu mode. That they are referring to the TE,, mode is further confirmed by using their formula [12] to calculate/,, for the large TEM cell: the calculated value of

30.4 MHz is much closer to /ill, = 31.3 MHz than to /no, = 24.6 MHz. Thus, what they refer to as

the first resonance is actually the TEm resonance. Our results are consistent with Crawford's in the sense that we both found the TEm resonance to be the first resonance which is consistently strongly

excited. I should also add that the lower resonances were often observed in their data (e.g., Figs.

17-20 of [13]) but never identified or discussed. The NBS 3 x 3 x 6 m, 50-fl, TEM cell has been re-analysed in Table 7. The cell dimensions,

from [1 1], are given in the table and w is calculated to be 2.48 m since w = 0.825 b for any 50-fl cell of square cross section [8, Table 1]. Assuming that the resonance at 97 MHz is the first TM resonance, as postulated, and that the TEmnp resonances are weakly excited, all of the observations in Table 7 are consistent with our

resonance analysis. With only weak excitation, the TE^np resonances would not have been observable

in the VSWR and insertion loss measurements (Figs. 6 and 7. respectively, of [1 1]). Furthermore, with

a field probe at the center of the test zone (position AO), only the TEim and TEm resonances at 60

and 74 MHz were observable while all the TE„ip and p = 2 resonances were not, as expected from

the 'AO' column of Table I . Thus the fact that five of the first seven (TE) resonances were not detected by Crawford et al.'s [II] measurements of £'b(A0) does not mean they were not present — on the contrary, a field probe at position Bl would likely have revealed them all. Based on our observations on the small TEM cell, the four TE^p and TEup resonances at 60. 75,

84, and 95 MHz would all interact with an EUT in the NBS cell. Furthermore, the interacting resonances could be stronger in the EUT-loaded cell than in the empty cell, as was found for the TEiop

resonances in the small cell. Additionally, any cables between the EUT and points outside the cell

228
HILL: BANDWIDTH LIMITATIONS OF TEM CELLS

Tabic 8 Dependence of the unusable bandwidth near the first interacting resonance (TEf„i ) on the TEM-cell application*

Type of

application

Discussed

in section

/riiiii) (MHz)

Unusable bandwidth

(MHz) '7f of /«,,.,.,

Bioeffects dosimetry

EMC/EMI

Probe calibration

V.A

Vl.A

IV

30

184

184

10 20-40

10

33 11-22

6

*The probe or tesi object is in Tree space al position AO. the center of the lest /one.

might interact with the resonances, possibly even with the TEmp resonances, depending on the cable

positions. For the above reasons, further studies of possible EUT-resonance interactions in the NBS

cells are warranted. Crawford and Workman [13] noted the same thing when they said that measured

fields "obviously will be influenced by inserting the EUT, especially at frequencies above the cell's resonances and muitimode, and will require further evaluation". The absorber loading of the NBS cells does reduce the strength of the E^, resonances and this may

prove, after further investigation, to be an effective damper on the interactions of the EUT with the lower-frequency resonances. SUMMARY

The first eight to ten TEmnp resonances have been identified in one large (6. 1 x 7.3 x 13.0 m)

and one small ( 1 .0 x 0.6 x 2.0 m) TEM cell. Many of the weaker resonances were undetectable by means of cell VSWR or insertion-loss measurements. Furthermore, many of them (e.g., the TEoip and

TEm„2 resonances) were not detected with a probe measuring £b at the center of the test zone (position

AO). They were all detectable, however, by means of swept-frequency measurements of Ef, and

circumferential wall currents at selected sites within the cell. They were identified by the field characteristics of the resonances and by matching measured and calculated resonant frequencies. The

resonant frequencies fit the equivalent coaxial box model where the effective length of the box, !„„,

depends on the mode (nu n), but not on the number (p) of half guide wavelengths in the resonance. The effective cell length for the TE^np resonances was found to include between 0.3 and 0.8 of the length of the cell's two tapered ends. Each of the first three higher-order modes (TEoi. TE|o, TEn) was found to propagate at fre- quencies above its first resonance, but not between its cut-off frequency and first resonance. The

propagating higher-order modes alter the TEM-mode E field in the test volume of the cell by up to ±15%.

A biological or metallic test object at the center of the test zone interacts with all TE^p and TEnp.

resonances but not with the TEoip resonances. As a result of the interaction the resonances shift in frequency and broaden, with the effect being stronger for a grounded test object. The frequency shift

is downwards for most resonances but upwards for the TEio: and TE|,: resonances. The test object may interact strongly with a resonance too small to be detected in VSWR

measurements, such as the TEioi resonance in both cells. There can also be a large interaction with

a resonance not detected by a probe at the center of the test zone, such as the TEio: resonances in both

cells. For these reasons, it is important to find all resonances within the frequency range u.sed. This

is done with swept-frequency measurements of E^ at position Bl. At frequencies near interacting resonances, the power absorbed by a biological object can be

measured but the effective exposure field cannot be determined. For this reason the large cell is usable

for bioeffects dosimetry work only to about 25 MHz. which is 657c larger than/, (15.2 MHz) and

2l7o larger than/R,oii, (20.7 MHz) but \19c below /r,,,,!, (30.0 MHz), the first interacting resonance. A TEM cell is usable for field-probe calibration work up to/R,iii,. except near/R.ioi,. The effects of the TE„ip resonances and the propagating TE„i and TE,o modes are all eliminated by averaging the two calibration curves taken at positions AO and DO on opposite sides of the septum. In particular, the small cell is usable up to 250 MHz, except for frequencies within ±5 MHz of the TEuh resonance at 184 MHz.

229
JOURNAL OF MICROWAVE POWER. 18(2). 1983

The relative size of the unusable bandwidth near an interacting resonance depends on the

particular application of the TEM cell. This effect is illustrated for the first interacting resonance, the TEioi resonance, in Table 8. As expected, the relative unusable bandwidth is seen to be smallest for the small metal object (probe undergoing calibration), intermediate for the larger metal object (EUT

undergoing EMC testing) and biggest for the larger lossy object (human subject).

To design a TEM cell for maximum usable bandwidth, the frequency of the first interacting resonance, TEioi, should be as high as possible. This is accomplished primarily by making cell dimension a as small as possible which causes /^.,,0) to be as large as possible. To a lesser extent, keeping L^ and L^ (and hence L|o) as small as possible also helps.

A method of partially suppressing the TE resonances will be published soon [32]. ACKNOWLEDGMENTS

The technical assistance of J. A. Walsh throughout this work is greatly appreciated. The author

would also like to thank L. Gruner for his calculation of the cut-off frequencies of the large TEM cell, C. M. Weil and M. L. Crawford for useful suggestions and information, and J.-L. Schwartz for the loan of the small TEM cell. Note added in proof. E. D. Mantiply has suggested that p = is possible for the TM resonances. Based on the discussion of resonances in cylinders and coaxial lines in Moreno [29], I agree. Thus,

the first TM resonance is TM,io and not TMm as given in Tables 2 and 7. REFERENCES

1 S. J. Allen. W. D. Hun. J. H. Krupp. J. A. Ratliff, C. H. Dumey, and C. C. Johnson. Measurement of

radiofrequency power absorption in monkeys, monkey phantoms and human phantoms exposed to 10-50 MHz fields. Biological Effects of Electromagnetic Waves (Selected Papers of the USNC/URSI

Annual Meeting, Boulder. CO), Vol. 11. Oct. 20-23. 1975. pp. 83-95. (HEW Publication (FDA) 77-801 1, U.S. Govt. Printing Office, Washington. OC 20402).

2 W. T. Joines, C. F. Blackman, and M. A. HoUis. Broadening of the RF power-density window for calcium-ion efflux from brain tissue. IEEE Trans. Biomed. Eng. BME-28. 568-573, Aug. 1981. 3 S. V. Marshall, R. F. Brown, C. W. Hughes, and P. V. Marshall. An environmentally controlled exposure system for irradiation of mice at frequencies below 500 MHz. Proc. 1981 IEEE Int. Symp. on Electromag.

Compat., 1981. pp. 99-104.

4 D. A. Hill. Human radiowave absorption from 7 to 40 MHz. Proceedings of a Workshop on the Protection of Personnel Against Radiofrequency Electromagnetic Radiation, J. C. Mitchell. Ed.. USAF School of Aerospace Medicine. Aeromedical Review. 3-81, Sept. 1981. pp. 170-175.

5 D. A. Hill. Human whole-body radiofrequency absorption studies using a TEM-cell exposure system. IEEE

Trans. Microwave Theory Tech., MTT-30. 1847-1854. November 1982. 6 C. F. Blackman. J. A. Elder. C. M. Weil. S. G. Benane. D. C. Eichinger. and D. E. House. Induction of

calcium-ion efflux from brain tissue by radiofrequency radiation: effects of modulation frequency and field strength. Radio Science, 14(6S), 93-98. Nov. -Dec. 1979. 7 R. J. Smialowicz, J. S. Ali, E. Berman. S. J. Bursian. J. B. Kinn. C. G. Liddle, L. W. Reiter. and C. M.

Weil. Chronic exposure of rats to 100-MHz (CW) radiofrequency radiation: assessment of biological effects. Radiation Research, 86, 488-505. June 1981. 8 M. L. Crawford. Generation of standard EM fields using TEM transmission cells. IEEE Trans. Electromagn.

Compat. EMC-16, 189-195, Nov. 1974. 9 P. S. Ruggera. E- and H-field instrumentation and calibration below 5(X) MHz. Biological Effects of

Electromagnetic Waves (Selected Papers of the USNC/URSI Annual Meeting, Boulder, CO) Vol. 11, Oct. 20-23, 1975, pp. 281-296. (HEW Publication (FDA) 77-801 1 , U.S. Govt. Printing Office. Washington.

DC 20402.)

10 E. B. Larsen. Techniques for producing standard £M fields from 10 kHz to 10 GHz for evaluating radiation monitors. Proceedings of the 1978 Symposium on Electromagnetic Fields in Biological Systems. Ottawa. Canada. June 28-30. 1978. IMPI (1979). pp. 96-1 12.

11 M. L. Crawford. J. L. Workman, and C. L. Thomas. Generation of EM susceptibility test fields using a large absorber- loaded TEM cell. IEEE Trans. Instrum. Meas. IM-26. 225-230. Sept. 1977.

12 M. L. Crawford. J. L. Workman, and C. L. Thomas. Expanding the bandwidth of TEM cells for EMC

measurements. IEEE Trans. Electromag. Compat. EMC-20. 368-375, Aug. 1978.

13 M. L. Crawford and J. L. Workman. Using a TEM cell for EMC measurements of electronic equipment.

National Bureau of Standards. Technical Note #1013. April 1979.

14 W. Baier. Wellentypen in Leitungen aus leitem rechteckigen Querschnitts. (Modes in waveguides consisting of conductors of rectangular cross section), Arkiv der Elektrischen Ubertragung. 22. 179- 185. 1968. 230
HILL: BANDWIDTH LIMITATIONS OF TEM CELLS

15 T. S. Saad (Ed.), R. C. Hansen and G. J. Whcclcr (Co-editors). Microwave Engineers' Handb()ok. Dcdham.

Mass.: Artcch House Inc.. 1971, Vol. 1. pp. 145-147.

16 L. Gruner. Higher order modes in rectangular coaxial waveguides. IEEE Trans. Microwave Theory Tcchn. MTT-15. 483-485, Aug. 1967.

17 L. Gruner. Estimating rectangular coax cutoff. Microwave Journal, 22, 88-92, April 1979.

18 L. Gruner. Characteristics of crossed rectangular coaxial structures. IEEE Trans. Microwave Theory Techn. MTT-28. 622-627, June 1980.

19 J. C. Tippet, D. C. Chang, and M. L. Crawford. An analytical and experimental determination of the cutoff

frequencies of higher-order TE modes in a TEM cell. National Bureauof Standards Report #NBS1R 76-841,

June 1976. 20 J. C. Tippet and D. C. Chang. Higher order modes in rectangular coaxial line with infinitely thin inner conductor. National Bureau of Standards report #NBSIR 78-873. March 1978.

21 CM. Weil, W. T. Joines. and J. B. Kinn. Frequency range of large-scale TEM mode rectangular strip lines. Microwave Journal. 24. 93- 100, Nov. 1981. 22 M. L. Crawford, and J. L. Workman. Asymmetric versus symmetric TEM cells for EMC measurements. IEEE 1978 International Symposium on EMC, Atlanta, GA, June 20-22. 1978. (78-CH-I304-5 EMC)

23 M. F. Iskander. C. H. Dumey, H. Massoudi, and C. C. Johnson. Approximate calculation of SAR for planewave irradiation of man model near a ground plane. Proceedings of the 1978 Symposium on Electro- magnetic Fields in Biological Systems, Ottawa, Canada. June 28-30. 1978. IMPI (1979). pp. 304-323.

24 M. J. Hagmann and O. P. Gandhi. Numerical calculation of electromagnetic energy deposition in models

of man with grounding and reflector effects. Radio Science. 14(6(S)). 23 — 29, Nov. -Dec. 1979. 25 F. M. Greene. Development of electric and magnetic near-field probes. National Bureau of Standards

(Boulder. Colo.) Technical Note No. 658. Jan. 1975. 26 R. W. P. King. H. R. Mimno. and A. H. Wing. Transmission Lines. Antennas and Waveguides. New York:

Dover. 1st. edn.. 1965. Ch. III. pp. 254-257.

27 L. Gruner. Private communication, calculations of/„,„„, for the large TEM cell using the computer program

of [16.17]. Cell dimensions are (meters): a = 6.\0, b = 7.32, w = 4.06. / = 0. 28 CM. Weil. Private communication, full-sized copies of Figs. I -3 of [21] were made available for accurate

calculations. 29 T. Moreno. Microwave Transmission Design Data, New York: McGraw-Hill. 1st. edn.. 1948, Ch. 7. p. 123, equation 7-44. 30 C C Johnson. C H. Dumey. P. W. Barber, S. J. Allen, and J. C Mitchell. Radiofrequency Radiation Dosimetry Handbook. Rep. SAM-TR-76-35, USAF School of Aerospace Medicine/RZP, Brooks Air Force

Base. TX 78235. Sept. 1976.

31 J. C. Mitchell. A Radiofrequency Radiation Exposure Apparatus. USAF School of Aerospace Medicine.

Report SAM-TR-70-43, July 1970. 32 D. A. Hill and J. A. Walsh. Resonance suppression in a TEM cell. J. Microwave Power, in press.

231

Appendix L

Wilson, P. F. ; Ma, M. T. "Simple approximate expressions for higher order

mode cutoff and resonant frequencies in TEM cells."

Reprinted from IEEE Trans. Electromagn. Compat. EMC-28(3) : 125-130 ; 1986

August.

233

Simple Approximate Expressions for Higher Order

Mode Cutoff and Resonant Frequencies in TEM

Cells

PERRY F. WILSON, member, ieee, and MARK T. MA, fellow, ieee

Abstract—Simple approximate expressions for determining the cutoff

frequencies of the first few higher order modes and the associated resonant frequencies in transverse electromagnetic (TEM) cells are presented. Both symmetric (seven TE and two TM modes) and asymmetric (three TE modes) cells are discussed. Key Words—Higher order modes, resonant frequencies, TEM cells. Index Code—A3 d/e.

I. Introduction

THE transverse electromagnetic (TEM) cell, as shown in Fig. 1 (configured for a susceptibility measurement), is essentially a large section of rectangular coaxial transmission

line (RCTL) used to establish an isolated known test environ- ment. High field levels can be achieved with minimal power

requirements and without interfering with nearby equipment.

A basic TEM-cell limitation is the appearance of resonances which tend to destroy the desired TEM-mode field distribution. Recent papers by Weil and Gruner [1] and Hill [2] have

addressed this resonance problem. Weil and Gruner present

numerical solutions for the normalized cutoff frequency of the

initial higher order modes as a function of the inner conducting

width. These results may be used to predict the appearance of

higher order TEM-cell resonances if the effective length of the

cell is known. However, determining the resonant length of a

cell is nontrivial since the tapered sections affect each higher

order mode differently. Therefore the effective length needs to be determined mode-by-mode and cannot be readily specified

universally, or theoretically. Hill discusses TEM-cell reso- nances in some detail based on experimental data from various

cells. He specifies both how to detect and how to identify the various resonances, and compares measured results to predicted values based on Weil and Gruner's numerical calculations. Each paper stresses that, because the TEM cell is a highQ cavity, the higher order resonances appear at sharply

defined frequencies. Thus there may well exist windows

between resonances where TEM-cell usage is still quite valid. Knowing where to expect these resonances and the associated windows should be of aid in many TEM-cell applications.

This paper outlines some approximate analytical expressions for determining the cutoff frequencies of the first few

higher order modes. The expressions are readily evaluated on

Manuscript received March 25, 1985; revised January 21. 1986. The authors are with the Electromagnetic Fields Division, National Bureau

of Standards, Boulder, CO 80303. (303) 497-3842. IEEE Log Number 8608934.

a hand calculator. As such, they should supplement the more

accurate results of Weil and Gruner if a numerical solution is not available, or if interpolation from existing figures is undesirable. Many of the results presented in this paper are derived from various National Bureau of Standards Technical Notes based on the work of Tippet et al. [3], and Tippet and

Chang [4], as well as of the present authors [5], [6]. Because

many TEM-cell users may be unfamiliar with these scattered

results (Weil and Gruner make a comment to this effect), this paper serves to collect them in a convenient reference. Some

additional formulas have also been generated as needed.

Throughout, we will assume that the gaps between the inner

conductor and the outer side walls are small, both with respect

to the relevant TEM-cell dimensions and with respect to the operating wavelength. In practice, this small-gap requirement

is usually met. A wide inner conductor is desirable to increase

the usable test section as well as to achieve a 50-Q impedance,

thus matching the cell to standard cables and equipment. The

basic analytical approach leading to the cutoff-frequency

expressions given in this paper is to formulate an integral

equation for the unknown gap fields. The small-gap assumption can then be used to simplify the integral equation kernel

[3], [5], [6], which leads to approximate modal equations, or,

if the integral equation is solved exactly [4], to simplify the

resulting infinite matrix equations. The details of these

derivations may be found in the references, and highlights of

the TE-mode analysis are given in the Appendix. The modal

equations listed here will be used to analyze various cells that have appeared in the literature. Good agreement is found.

Both symmetric and asymmetric TEM cells will be discussed.

II. Cutoff Frequencies of Higher Order

Modes—Symmetric Cell

The cross section of a general asymmetric TEM cell is shown in Fig. 2. The relevant dimensions (a, b\, bi, b, w, g)

conform to [3]-[6], and therefore differ slightly from those used in [1] and [2]. For a symmetric cell, b\ = bi = b. The

modal equations for determining the cutoff frequencies of the

first nine higher order modes in a symmetric TEM cell are

listed in Table I. The small-gap approximation used to generate the gap-dependent formulas involves neglecting terms like {irg/2a)- and {keg)' when compared to unity, where

kc is the wavenumber at cutoff for the mode of interest. As is well known [1], the TE„_2n and TM„,2n modes are unaffected

U.S. Government work not protected by U.S. copyright

Reprinted from IEEE Trans. Electroraagn. Ccsitpat. EiyiC-28(3): 125-130; 1986

August.

235
CQAUAL

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. EMC-28. NO. 3. AUGUST 1986

- S£PTUM (MNED CONOJCTORl

OUTDtSMElD

OeiCTIK

SffTUM SUMWTS

ACCESS OOW

ruT/ouTPin/fow^ ijads IREO TWU GONNiaORS)

COAXIAL CONNEC

IBOTXEMOSi BUnCfT—' '^—DtlfCTWC EOUrMtWT mam test sumwT

Fig. 1. An NBS TEM cell. TABLE I SYMMETRIC TEM-CELL SMALL-GAP MODAL EQUATIONS

Hode

[ref]

Modal Equation (f^ -^ |

(MHz) )

«01

[3]

w b Za ->

« tan X - - H [in (—) R ] 2 a ,9 TE^^

R„ . r 1 (coth £^ -1) cos^H^)

TElO «-7(l)

[3.5]

X [y2 (^)' ]>/2

If t »nif _--.,"., / ^ rinl...\,7 rnr? {^2\ t p l" y tany ( ) Lin( )2 COS' ( ) R J cos2 (-^) '' 11

1 (2p*l).b 2 2p*l

«Tr • 2 r, T—r (COth-^—! -1) cos TE p-1 {-f^-9) 2p*l 2a 2a

«02 X • »

TEi2.TNi2 X w Cl (fj)']'/'

TEZO x..|

TEji

[3]

cot X , ^-''r' '«'

. 2 . 2. .,^ , „ X

' y , 'b' ^'"^g' "' '« ' ' "tEjj"'

R . t, - (eoth ^ -1) eoi» (SlS)

TE p-2 p • 1

[5] • : r .. (2P*l).b ,, , ,(2p*l).9,», • — t [coth — -1] J,[— ^J TH *9 p«l 2a > 2a

2b

2w

2o —*

Fig. 2. The cross section of an asymmetric TEM cell. For a symmetric cell, b, = bi = b.

meaningftil. For each mode, the cutoff frequency fdm.n) is related to the dimensionless modal equation variable x via

/«..«=^ {1} (HZ)

= l^^(MHr) (1)

where c * 3 x 10* m/s and MKS units are used throughout.

Alternately, the normalized cutoff wavelength is

V(m,/i) It / b

a X \a

(2)

Once the cutoff frequency of a particular mode has been

determined, the associated resonance frequencies fR(m,n,p) follow via [2]

by the presence of the inner conductor for a symmetric cell and, therefore, straightforward rectangular waveguide {la x

2b) cutoff-frequency results apply. Expressions for these simple modes (TE|o, TE02, TE12, TE:o. TM12) are included in Table I for completeness and to make the modal ordering more

(3)

where L„„ is the effective length of the cell for that particular

mode. Hill expresses L„„ as the length of the cell central 236
WILSON AND MA: MODE CUTOFF AND RF.SONANT FREQUENCIES IN TEM CELLS

section Lc plus a taper-dependent term, that is.

(4) where Le is the length of the two taper sections, and X„„ is an

empirically determined fraction, different for each ceil as well

as for each mode.

The modal equations in Table I may be applied to a symmetric TEM cell of dimensions o = 3.05 m, 6 = 3.66 m,

and g = 1.02 m and compared to results given in Hill's paper

[2] based on numerical calculations by Gruner, as well as the exact waveguide formulas. The results are given in Table II. The perturbed-mode cutoff frequencies agree quite well except

for the TE21 (45.4 versus 52.9 MHz) and the TM,, (47.5 versus 52.2 MHz) modes. However, the TE^i-mode cutoff frequency should be greater than that of the TE20 mode (49.2

MHz), thus Hill's entry is likely in error. Based on Weil and

Gruner's results [I], the TE2i-mode cutoff frequency should

be approximately 50.4 MHz.

The TMII -modal equation result does not agree as well. The

integral equation for the TM case is somewhat different from

that for the TE case. The basic small-gap approximation is weaker and agreement is not expected to be quite as good [5]. A second example is a symmetric NBS cell (a = 6 = 1.5 m; g = 0.26 m; Lc = Le = 3.0 m) also analyzed in Hill's paper. Weak resonances were detected at 38, 50, and 65 MHz,

while relatively strong resonances appeared at 61, 74, 85, and

97 MHz. From these data. Hill generates the results listed in Table HI. We have added a column showing fc{m.n) values based on Table I, which agree well with Hill's figures. Hill

essentially leaves the TMn entries empty so we have an added

/c(i,i) value of 1 15 MHz interpolated from a figure found in [4] and completed the line based on this value. The fR(m,n,p)

column shows both Hill's numbers and those found via (3) with Table I fc(m,n) values inserted. The X„„ values used in Table HI are averages generated by Hill (see Table IV, [2]) based on an analysis of two different cells. Table HI may now

be used to identify the measured resonances. The weak

resonances at 38 and 65 MHz are the TEoii and TE012

resonances, respectively. Resonances at 61, 74, 85, and 97

MHz can be recognized as TEioi, TEm, TE102, and TE112

resonances, respectively. The expected TE013 resonance at 89

MHz may have been masked by the large TE112 resonance or

the probe placement may have been such that this resonance was not detected. The apparent weak resonance at 50 MHz

was probably a voltage standing-wave ratio (VSWR) problem

due to mismatches.

The weak/strong resonance effect illustrates an important

point. The TEoi mode in a small-gap TEM cell has strong gapfringing fields while not exhibiting significant fields in the

central test area. Therefore, although the TEon is the first possible resonance, it is usually not excited unless a large test object or some other perturbation is present. The TEoi-mode

gap confinement also accounts for its large taper factor (A'oi) relative to those associated with the other TE modes, such as

A'lo for the TE|o or X\ 1 for the TEi 1 . The TEoi-mode gap fields penetrate beyond the cutoff point more effectively than modes

with more uniformly distributed fields. For the above TEM cell, depending on the usage, we may

be able to extend its usable frequency range beyond the

TABLE II CUTOFF FREQUENCIES IN A LARGE (a = 3.05, b = 3.66. AND

g = 1.02 m) SYMMETRIC TEM CELL

Exact agreement for the TEio, TE02, TE12, TE20, and TM,2 mcxles follows from the use of rectangular waveguide formulas.

Mode ^(m.n) ^^^)

Hill [2] Table 1

TEoi 15.2 15.5

"10 24.6 24.6

"11 31.3 31.9

TE02 41.0 41.0

"12 47.8 47.8

TE20 49.2 49.2

TE21 45.4 52.9

THji 47.5 52.2

TH12 47.8 47.8

nominal limit of the TEou resonance (40.6 MHz). For

example. Hill [2] suggests that the unusable bandwidth about

the TEioi resonance ranges from 33 percent of /^doi) (60.5

MHz) for bioeffects dosimetry to 6 percent for probe

calibrations. The earlier TEon resonance (40.6 MHz) interacts negligibly with centrally placed test objects. Thus this cell could be used up to 50 MHz for bioeffects dosimetry and up to 53 MHz for probe calibrations. Additional windows above the TE:oi resonance may exist, especially for small test objects.

in. Cutoff Frequencies of Higher Order

Modes—Asymmetric Cell

Most TEM cells are symmetric. However, some thought

has been given to the use of asymmetric cells for EMI/EMC

measurements [7]. A vertical offset of the inner conductor, as shown in Fig. 2, would create two chambers of unequal size. These might allow the testing of both large equipment and

small probes without increasing the overall size of the cell and,

consequently, reducing its bandwidth. However, one sacrifices field uniformity and strength in the larger chamber, and

the asymmetric cell will have a more complicated modal

structure. The modal equations listed in Table I are actually special cases of the asymmetric formulas derived in [3]-[6]. We will

list in Table IV only the first three asymmetric TEM-cell

modal equations, all TE modes. Crawford and Workman [7] have published some data on resonances in an asymmetric cell based on VSWR measurements. However, a number of problems present themselves when

237
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-28, NO. 3, AUGUST 1986

TABLE UI ANALYSIS OF A LARGE (a = b = 1.5. g = 0.26, L, = Le = 3.0 m) NBS

TEM CELL

Nod*

'c(b nl (HH.)

>U

c Rtioninc*

Typt

n p

'R(a.n d) (MHi) 2Un

(NHl)

Hill [2] Tiblt 1 Hill [2] Tlblt 1

TEoi 29.0 29.2 0.77 28.2

1 1 40.0 40.6

1 2 63.0 63.

1 3 89.0 89.5

^W SO.O SO.O 0.47 34.0

1 1 60.0 60.6

1 2 84.0 84.4

"u ii.i 64.3 0.43 3$.0

1 1 1 73.0 73.2

1 1 2 9S.0 96.0 TMU "S-OCO 11&.9 1.00 11.6 1 t n5.0[4] 116.9

TABLE rV ASYMMETRIC TEM-CELL SMALL-GAP MODAL EQUATIONS

Hod*

[rtf]

Modll Equ.tlon: f^,,

,,, ^ | HHi

TEoi

i Ccot ^ X cot ^ X] . i (|) [u, (^) R^g^]

[3] \/?pil^--V' -0X.i^.2,c.sMi!f5)

TElo X . f (|)

X [y (|j) ]

'

TEll coi2 (^) ^

[cot ^ y cot ^ ] •7(|'C"<v|>-2"»Mg)*RTEj

[3.5] \: I

^^"^•^^'^ * -''^''^ '' '»*''''^'

equations are simple to use and should aid TEM-cell users in predicting and detecting cell resonances, thereby extending

usable bandwidth of a cell.

Appendix

The derivation of the gap-perturbed modal equations listed in Tables I and FV follows essentially the same steps in each

case. First, the unknown fields in the entire RCTL cross

section are expanded in terms of a set of orthogonal basis functions [3]. These basis functions correspond to the rectan- gular-waveguide modes in either the upper (2a x ^i) or lower

(la X bz) RCTL chambers (see Fig. 2). Field continuity across the gap yields integral equations of the form [3]

AnCit, n dt'=0 (5)

trying to identify resonances. Hill [2] points out that VSWR

measurements do not necessarily detect all resonances. Another difficulty is setting the proper taper fraction X„„. In order to analyze properly their cell [7], a more careful resonance-detection measurement procedure, as in [2], would

be desirable.

IV. Conclusion

This paper has reviewed a number of modal equations

governing higher order modes and resonances in TEM cells. Both symmetric and asymmetric cells are discussed. These

where / and /

' are gap variables, (/, t')e [0, g], fit ' ) is an unknown field component in the gap, and C(/, /

' ) is a known

symmetric kernel. In each case, the kernel becomes singular as

|/ - ^'

I -^ 0. This dominant singular behavior G<'>(/, /')

may be extracted leaving a finite secondary term C^>(/, /'),

i.e.,

G(r, r) = G<'>(r, /')+G<2)(/, r'). (6)

Assuming a small gap allows one to approximate both kernel

contributions. The resulting simplified integral equation may

then be solved directly.

As examples, consider the gap-perturbed TE modes. As

derived by Tfppet et al. [3], the TE-kemel components G^'^/,

/

' ) and G<2K/. /' ) are given by

G<'>(/, /') = - In tan -^ \t + t'\ tan — \t-t'\

238
WILSON AND MA: MODE CUTOFF AND RESONANT FREQUENCIES IN TEM CELLS

for the TE2m+i, zn+i (odd, gap-perturbed) modes; and be retained. Thus the cosine terms (9) may no longer be near

unity over the range (/, t ' )e [0, g\. An alternate approximation

is to replace the cosines by their value at gap edge. The

justification is that, for TE modes, /(/ ' ) in (5) is proportional

to the horizontal component of the electric field which

diverges at the gap edge. Thus we will weight the approximate

kernel toward this point. This again reduces C^^K'. t') to a

constant, which aids in solving the integral equation (5), but

for the TE2m.2n (even, gap-perturbed) modes. All dimensions also retains more information about the gap size. Thus we

(a, bu t>2, g) are defined in Fig. 2, and

t, /') = - In 4 sin -^ \t+t'\ sin — \t-t'\

K \_ 2a la ]

.. fcot -Koi^ 2a "I p-Kt cos pvt'

'la'

<'=n-©1

1/2 (9)

lO = k^-y^

where k = 2ir/X, and 7 is the propagation constant in the RCTL.

Now, assume that the gap is small. In the singular terms this allows us to replace the tangent and sine functions by their small argument approximations, i.e.,

GC)(/. n«^ln |^(^£y|/2_f'2|j (TE2^^,2„^,)

G<')(r. /')«^hi [^(^ly |/2_^'2|j (TE2^^^,) (10)

for the odd and even cases, respectively. Since \t ± t'\ is bounded by Ig, terms like (ir^/2<jr)V3 and {-Kg/lay/6 are neglected when compared to unity in approximating the tangent and sine terms. Next, consider the finite kernel terms G<^H^ ''). noting

that for large p, i.e., (pv/a)^ > K^,

cot

Ttt, «a)la

pbj

(11)

implies that the summand gj/^ (/, /') decays rapidly.

Thus only a few terms should contribute significantly. If, in addition, (picg/la) < 1 for these initial terms, then the cosines may be replaced with unity and G^^^/, /') « G<^\ a

constant, in each case

G(2) = i 2^1; ^^'.,(0, 0) (TE2«^,.2^^,)

^" = -27[^^-|;^S(0.0)](TE^...,).

These are the proper limits for G|J','. as g -» and, in fact, those used in [3]. These approximations tend to yield good

results for the TEoi and TEn modes. However, as additional modes are considered, Kincreases and larger/? indices need to

replace g^>(0, 0) in (12) with g^Kg, g), resulting in

G^^' = - t-i. s'iM, g) (TE2„+i.2.+ i)

1).

(13)

Clearly, these reduce to (7) as {g/a) -* 0. The approximate integral equation now becomes

[' f{t'){G^^Kt, /') + G<2'} dt'^Q (14)

Jo

where G<'^(/, ?' ) is a simple logarithmic kernel and G'^^ is a

constant. This type of equation may be readily solved [3] with

the result that

G(2) =

2 [80] =- In — (TE2;„+i,2fl+i)

\la^

G0) = — (TE2ot2/,+ i). - In I —

IT (15)

Together, (13) and (15) generate the TE modal equations. For example, begin with the TE2/„.2„+i modal equation

which is

1 2 6y Ttt cot bjK - f cot Tta^ al

~r^,~^l IbjK ^f^X c.'C VJ

cos' a J TT

la

(16)

If the gap is very small, then the right-hand side of (16) will be

large. One possible solution is to let bjK be small; thus the

first term in parentheses dominates. Note that K small implies

that 7^ « k^, the TEM-mode propagation constant. Thus this mode will not be cut off in the limit as ig/a) -* 0, and may be

identified as the TEoi mode. Noting that K^ < {prlaY for p

> 1 in a^, we arrive at TEoi asynunetric-cell modal equation

(12) cot bxK+ cot b

laK

-21 COS' pifbx P'^bz

coth hcoth a a v) (17)

239
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. EMC-28, NO. 3. AUGUST 1986

This reduces to the TEoi form given in Table IV if we let x =

bK, and to the symmetric-cell equation given in Table I if we

let bi = b2 = b. The next solution is to let the cot a^/aW term dominate for p = 1. This means K^ « (ir/a)^; thus 7^ « Ar^ - (ir/a)2,

which is the TE20 propagation constant. Thus this solution will yield the TE21 modal equation. Letting K^ < (pr/aY forp ^

2 in OL^, we arrive at

cot b\K->rco\. b

laK H

b\ cot Ttoz bi cot ira2 aira (I) ara(2)

cos'

TTg 2 —= - In m•cos^

TTg

2.-:/^

pvbi pvb,

coth hcoth ^-2) (18)

set p = 1 as the dominant term and let y = [x^ -

{Tcb/la)^] "^. The TM modal equations are based on a similar

analysis as outlined in [5] and [6].

Letting X = bKan^y = [x^ - (tZj/a)^"^ reduces this to the

result listed in Table IV, and so forth. The TEn modal

equation follows from the TE2m+\,2n+i set of equations if we

[11

[2]

[31

[4]

[5]

[6]

[71

References

C. M. Weil and L. Gniner, "Higher order mode cutoff in rectangular

striplines," IEEE Trans. Microwave Theory Tech., vol MTT-32

pp. 638-641, June 1984. D. A. Hill, "Bandwidth limitations of TEM cells due to resonances,"

J. Microwave Power, vol. 18, pp. 181-195, June 1983.

J. C. Tippet, D. C. Chang, and M. L. Crawford, "An analytical and

experimental determination of the cutoff frequencies of higher-order

TE modes in a TEM cell," Nat. Bur. Stand., Boulder CO Rep

NBSIR76-841, June 1976.

J. C. Tippet and D. C. Chang, "Higher order modes in rectangular

coaxial line with infuiitely thin inner conductor," Nat. Bur. Stand., Boulder, CO, Rep. NBSIR 78-873, Mar. 1978.

P. F. Wilson, D. C. Chang, and M. T. Ma, "Excitation of a TEM cell by a vertical electrical hertzian dipole," Nat. Bur. Stand., Boulder CO, Tech. Note 1073, Mar. 1981.

P. F. Wilson, "On slot-coupled waveguide and transmission line structures," Ph.D. dissertation, Dept. Elec. Eng., Univ. Colorado,

Boulder, CO, 1983. M. L. Crawford and J. L. Workman, "Asynunetric versus symmetric

TEM cells for EMI measurements," in IEEE 1978 Int. Symp. EMCSymp. Rec. (Atlanta, GA), June 1978, pp. 204-208.

*U.S. GOVERNMENT PRINTING OmCE:l 9 8 8 -576-835/ B 5 2 3 G

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1. PUBLICATION OR

REPORT NO.

NBS/TN-1319

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

August 1988

4. TITLE AND SUBTITLE

Generation of Standard Electromagnetic Fields in a TEM Cell

5. AUTHORiS)

Motohisa Kanda and R. David Orr

^. PERFORMING ORGANIZATION (/f joint or other than N8S, see instructions;

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11. ABSTRACT (A 200-word or /ess factual summary of most si gnificant informatior). If document includes a significant

bibliography or literature survey, mention it here)

This paper documents the facilities and procedures used by the

National Bureau of Standards to calibrate radio frequency electric

field probes using a transverse electromagnetic (TEM) cell. The

advantages, limitations, and physical characteristics of TEM cells

are presented. Impedance, field uniformity, and mode structure,

critical aspects of a cell as a standard field enclosure, are discussed. The paper concludes with sections on setup and measure- ment procedures for users, uncertainty in the standard field, and

statistical control of the calibration system. Copies of key

references are included to provide ready access to the details of

topics summarized in the text.

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

antenna; antenna calibration; calibration; electromagnetic field; radio frequency;

standard field; TEM cell; transverse electromagnetic cell.

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