IMBSIR 75-829

RADIATION CHARACTERISTICS OF OIPOLE 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

NBSIR 75-829

RADIATION CHARACTERISTICS OF OIPOLE 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 1975

Prepared for:

Electronics Systems Division lAFSC)

Hanscom Air Force Base, Mass. 01731

"1 ij.^ ' '

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

James A. Baker, 111, Under Secretary

Dr. Betsy Ancker-Johnson , Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

CONTENTS

Page

FORETTORD iv

1. INTRODUCTION 1

2. PARAMETERS CHARACTERIZINC, A TEM CELL 2

3. APPROXIMATE EXPRESSIONS FOR THE CAPACITANCE AND 5 CHARACTERISTIC IMPEDANCE

h. RADIATION RESISTANCE OF ELECTRIC AND MAGNETIC DIPOLES 9 IN A TEM CELL

5. STIM?<ARY AND CONCLUDING REMARKS 13

6. REFERENCES 16 APPENDIX A: Calculation of the Capacitance of a TEM Cell 17 APPENDIX B: Calculation of the Electric T'ield Distribution 21 Inside a TEM Cell APPENDIX C: Numerical Evaluation of Comnlete Elliptic Integrals and 29 Jacobian Elliptic Functions

iii
FOREWORD

This report describes theoretical analyses of Transverse Electromagnetic (TEM)

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 expressions for the capacitance and characteristic

impedance of a cell, the electric field distribution Inside a cell,' and the radiation

resistance of electric and magnetic dlpoles 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.

iv
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 50 Q. coaxial

line as shown in figure 1.

Figure 1. Design for rectangular TEM transmission cell.
To minimize reflections, the cell is designed to have a nominal characteristic

impedance of 50 Q. 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

F

E

2 b A B — 2w — C D ^

2a

Figure 2. Cross-section of a TEM cell.

In order to understand how an electromagnetic wave can be guided by this structure, a brief review of the results from standard transmission line theory

is presented.

-2-
Any mult i- 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^ = = 0) . In the transverse plane, the electric field satisfies

Laplace's equation: V^E = 0, which in the case of the TEM cell reduces to:

g

2 3

2 _

(g^2 + g:^)E(x,y) = 0. This means that the transverse field distribution can be obtained from the solution of a related static problem. The magnetic field

is easily obtained from the electric field as

a X t±

H- = ± ^-

(2.1)

o where

is the magnetic permeability;

is the dielectric permittivity;

Y is the propagation constant;

a is a

z The total fields are then given by:

and a is a unit vector in the z direction,

z

E = Re[E + E ] (2.2)

and iT = Re[H'^ + 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%"^^ + V"e^^ (2.4) m m ^ '

I(z) ^ ^ = 1%"^^ + lle^^ (2.5) m m ^

'

where

and

V* = - / E- . ai (2.6)

P

i; = ^ H"* . aj (2.7)

-3-
p is any path connecting the two conductors in a constant cross-sectional

plane, and i is a closed path encircling the inner conductor. and 1^^ are related by a constant which is called the characteristic impedance, o£ the

line, and is given by:

= ± ^ = ± — ^- (2.8)

a X Ei

• I %

Thus, eq . (2.5) may be written:

v: . v_

= JH e-^" - ^

e^" (2.9)

. Z Z - 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, as follows:

Z = J_ (2.10)

where v = is the phase velocity.

o o

Thus, a knowledge of the distributed capacitance, , 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:

t C_o = 2 ^LLA)_ (2.11)

K(A')

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:

X' = [1 - X^]'^ . (2.12)

The modulus, X', is given by:

X' = k'f^2-^]' (2.13)

'•cn C

where 5 = mg and sn ^ and cn E, 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

-4-
and the side wall. k' is the complementary modulus to k defined analogously

to eq. (2.12), and m is given by:

m = (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^iAli (2.16)

K(A)

where ri^ 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].

K(6 ' ) IT 1-/5- i&'>j) (3.1)

Using eq. (3.1) we can write approximate expressions for eqs. (2.15) and (2.11)

respectively as follows

:

. 2a^l^f2l_Wkl ^^.13 ^32)

b ^ ^ 1 - /F^

^

C

0 ^ 2 ,„f, 1 + /X ] - In 2 i^^>j) (3.3)

O

- /X

-5-
Equations (3.2) and (3.3) may be written alternatively as follows:

^ ^ I ln[2(l + /F)2(l-Hk)] - - Ind-k^) (3.4)

- - ln[2(l + /A)2(l + X)] - - ln(l-X2) (3.5)

Subtracting eq . (3.4) from eq. (3.5) we obtain:

4a

b

2 1In

TT

and substituting from eq . (2.13)

C

= 4

2 in|^

sn 5

+ - In

TT

1 + /X )

1 +

1 + X

1 + k

^ 2 , I! 1 + /X 1 M 1 + X

+ — In

1 + /F-' 'I + k

(3.6)

(3.7)

In eq . (3.2), the restriction that k^ > y is equivalent to requiring

b 1 ^ < 1, since when k^ = y, K(k') = K(k) . If we make the somewhat more stringent

b requirement that — < 1, which is equivalent to k^ > .97, then eq . (3.7) may be

3. further simplified by noting that for k - 1 , cn ^ - sech E,; sn E, - tanh 5, and

5 IF-

'Thus.

—2 1In s inh

^

2b

AC

(3.8)

where

AC A m

1 + /ri^r^ ^ (3.9) 1 + /X-' '-1 + X- u An alternate form of eq. (3.8) may be obtained by using the following identity:

.(TTg/2b)

with the result

s inh TTg 2b [1 + coth(5|)]

(3.10)

w ^ 2 , — + — In 1 + coth M

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

Q walls. For large gaps, the fringing term approaches — In 2, as expected [8].

-f-
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 A' is near zero) since k is near one. From eq. (2.13) X'^ is given approximately by:

A'^ - k'^ sinh'*(5|) (3.12)

It can be seen from eq . (3.12) that for small gaps, A' 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:

- 2lT r- A'2 . e

^ (3.13)

From eq . (3.13) it can easily be verified that A' will be negligibly small,

and hence AC may be neglected if: w 1

From eq. (3.3) we have the restriction that A^ > ^ or equivalently A'^ ^ J' From eq. (3.13) it can be seen that A'^ < y if:

^ > 1^ m 2 ^ 0.1 (3.15)

So for the range: TO F

^ 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:

o 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 valid for a/b >_ 1 and ^ _> j:^ 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 < ^. This

discrepancy can be attributed, however, to the AC term which was neglected.

-7-
32.5 -

30.0

27.5

25.0

22. 5

20.0

17.5

15.0

12.5

10.0

7.5

5.0

2.5

0

1 .0

--APPROXIMATE —EXACT

-2w^l

2a

t2b

1

a/w=l .25

^a/w = 2

'a/w=3.33

a/w=l

0

10.

a/b

Figure 3. Capacitance per unit length of a TEM cell.
4. RADIATION RESISTANCE OF ELECTRIC AND MAGNETIC DIPOLES

IN A TEM CELL

The power carried by the TEM mode o£ a transmission line that is excited

by an elementary electric dipole is given by [11]:

|A"'|2

where A is the excitation factor;

V is a unit voltage;

and is the characteristic impedance.

From Green's theorem, A* is given by:

Z _ _c ^ A = - — I J . e'^'-' dv' (4.2)

where J is the source current density;

E is the electric field of the negative (-z) propagating wave;

and 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^(x^,y^,z^) I l^^^ cos 6 (4.3)

T where (x^,y^,z^) is the source point;

6 is the angle between the dipole and the electric field at the source point ; ^eff effective dipole length;

I 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 (4.4)

e o

E I cos

I _2

eff 2V

If, instead of an electric dipole, our source were a magnetic dipole,

then following a similar analysis, the radiated power is given by

2 P = Z (4.5) m o

T . EI cos

2TrA o m

A 2V

o where is the wavelength;

A=7rr^ is the area of the loop representing the magnetic dipole;

1^ is the equivalent magnetic current;

and (() is the angle between the loop normal and the magnetic field at the source point. q_
In order to determine the change in the radiation characteristics o£ the

source, we will calculate the ratio o£ 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:

R

2tt eff)

°l ^o J (4.6)

n TT f27Tr] o (4.7)

From eqs. (4,4) and (4.5), the radiation resistances inside the guide are given

by:

R' = ^ e j2

E cos 0 o 'eff 2V

(4.8)

R' = -lli = Z m

-J.2 o m

T E cos * 2ttA o^

X 2V

(4.9)

If we define the following, normalized parameters:

Z_

Z' = o (4.10)

o V/b

(4.11)

then the ratios, and Q^, of the radiation resistances are respectively:

(4.12)

(4.13)

where k 2Tr/A^. Thus, in order to determine the "correction factors,' Q and ^e

Q^, which determine the change in the radiation characteristics of dipole sources

inside a TEM 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:

Vm'

K(a')

dn' (m'z)

sn^ (m'w) - sn^ (m' z)

(4.14)

-10-

I
where

a' = [1 - aM'^; (4.15]

a = snCm'w,k) ; (4.16)

(4.17)

(4.18)

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

a ^ K(k)

b K(k')

m- = ^

E = bm'

K(a')

dn^ (m' z)

sn^ (m 'v:) - 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 4) they would be identical. That is,

<Q > = <Q > ^e Qo/^V^ (4.20)

where

3tt ^o ^-^0^

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 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

k b 0

Figure 4. Correction factor as a function of frequency.

-11-
reduced as the frequency is increased

for dipoles located at various places inside a TEM cell

to a different TEM cell geometry.

Values of Q are included in tables 1-2 ^0

Each table corresponds

Table 1

Q , the ratio of the radiation resistance inside the TEM cell to

^o'

la a = 25 cm, b

y (cm)

that in free space for k^b

2 5 cm, w =

= 1. Dimensions of the TEM cell

20.64 cm.

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 0.150

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 0 .465 0 . 503 0 .665 1.281 14 . 322 4 . 216

/ 0 5 10 15 2 0. 25

Center of

TEM cell x=w x=a

X (cm)

Table 2

^o' the ratio of the radiation resistance inside the TEM cell to

that in free space for k^b = 1 Dimensions of the TEM cell

a = 2 5 cm, b = 15 cm , w = 18.025 cm

.

y (cm)

y=b—15 0 . 303 0 . 291 0 . 247 0. 159 0 .051 0 . 000

12 0 . 307 0 .297 0 .259 0.177 0 . 071 0 .020

9 0 . 318 0.314 0 .294 0 .238 0 . 140 0 .087

6 0 . 332 0 . 336 0.350 0.367 0 . 291 0 . 217

3 0 . 344 0 .356 0 .412 0 .622 0 .642 0 . 403

0 0 . 349 0.364 0 .441 0 .901 1.262 0 .512

0 5 10 15 20 25

Center of

TEM cell

\ x=w

+ x=a

X (cm)

-12-
5 . SU>P-L\RY .\ND CONCLUDING REMARKS

We have calculated in this report the exact capacitance and characteristic

impedance o£ a TEM cell, as given by eqs . (2.11) and (2.16]. In addition,

approxiir_ate 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,

w^hen 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 w^hen 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 5 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 t)'pe

.

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:

P = P + P =

e m

2V

(^g££ I COS 9)'

2-rrAI cos © m

Xo

(5.1)

-13-
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; i^Qff I) 6 for the electric

dipole, and (AI^^) cos <}) 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 5 we can rotate the electric or magnetic dipole 90° around the x or y axis

respectively, and obtain cos G = 1 or cos <i> = 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. -14-
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. C3) 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.

-15-
• • 6. REFERENCES

[I] 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 CNov. 1974).

[3] Bowman, P., 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, 5th 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, MTT8

, No. 5,

510-519 (Sept. 1960).

[9] Cohn, S.B., Shielded coupled"- transmission line, IRE Trans, on MTT, MTT -

3

, No. 5, 29-38 (Oct, 1955).

[10] Anderson, G. M. , The calculation of the capacitance of coaxial cylinders

of rectangular cross-section, AIEE Trans., 69^, pt. II, 728- 731 (1950).

[II] Collin, R.E., Field Theory of Guided Waves, pp. 200-204 (McGraw-Hill, New

York, 1960).

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

pp. 645, 657 (Wiley, New York, 1967).

[13] Kerns, D.M., unpublished work (Aug. 1965).

[14] Collin, R.E., op cit

.

, pp. 285-302.

[15] Walker, M. , The Schwarz -Christof fel Transformation and Its Applications A Simple Exposition (Dover Publ. Inc., New York, 1964).

[16] Bowman, F. , 0£ cit

.

, p. 58.

[17] Bowman, F., op^ cit. , p. 57.

[18] Churchill, R.V., Complex Variables and Applications, 2nd Ed., p. 209

(McGraw-hill , New York, 1960).

[19] Abramowitz and Stegun, 0£ cit

.

, Sec. 17.6, p. 598, Sec. 16.4, p. 571.

[20] Abramowitz and Stegun, op^ cit

.

, Sec. 16.21, p. 575.

-16-
APPENDIX A

Calculation of the Capacitance of a TEM Cell

A cross- sect ion view of the TEM cell is shown in figure 5 with an x-y

coordinate system superimposed.

2b

A B — 2wC D

-.^ 2a ^

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 f el transformation [IS] which, due

to symmetry, can be expressed in terms of Jacobian elliptic functions. The

transformation is given by [16]

:

mz

dt

o [4t(l-t3 (l-k^t)]^

(A.l)

or alternatively by:

t = sn^ (mz , k) CA.2)
where sn is a Jacobian elliptic function of modulus, k;

. (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 - k^]'^ (A. 5)

The modulus k can be determined from the requirement that:

Kfk) 2a

^

^ = — (A. 6) K(k') b

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.

0 % ^0 1 H \- \ h

A B CD

Figure 6. Complex t-plane.

Using eq. (A. 2) and elliptic function identities, and 3^ can be calculated

as :

= sn^mg = sn^C (A. 7)

= sn^m(2a-g) = cn^/dn^? (A. 8)

where

^ = mg (A. 9)

and cn 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:

o o

(A. 10)

The u-plane is shown in figure 7

-18-
1/X' —I—

Figure 7. Complex u-plane.

X in figure 7 can be found using eq. (A. 10) and substituting t = 1. Thus

'l - a

(A. 11)

Substituting for and from eq . (A. 7) and eq . (A. 8), we obtain:

cn C

^2 _ cn^g - sn^g dn^g

cnn[l-snH]

Using elliptic function identities, eq. (A. 12) reduces to

X2 = 1 . 1,.2 fsiL^

and defining a complementary modulus, A' as:

X' = [1 - X^]'^

We have from eqs . (A. 13) and (A. 14):

, f sn g cn g

(A. 12)

(A. 13)

(A. 14)

X' (A. 15) = k

In the final transformation, we map the upper half of the u-plane into a rectangular region in a complex x-pla.ne . The transformation is given by:

u - sn'(x,X) (A. 16)

and the X"Pl3.ne is shown in figure 8.

D

t K(X'

a-

—-K(A) ^

Figure 8. Complex X'Pl^ne

-19-
From figure 8, it is evident that the capacitance is just given by the ordinary

parallel plate capacitor formula, that is:

C = • (A. 17)

where A is the cross-sectional area;

o

' d is the plate spacing;

and is the permittivity of free space.

Substituting for A^ and d in terms of K(A) and KCX') we obtain:

C ^ K(X)

K(X')

(A. 18)

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).

_o_ ^ 2 (A. 19)

K(A')

1

-20-
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 easilydone using a different coordinate system as well as a Schwarz -Christof f el trans

formation 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

2b

2w

2a

9

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]

;

t^ = sn(m'z,k) (B.l)

where

m' K(k) ^ K(k')

a b (B.2)

Note 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.

Figure 10. Complex t -plane.

-21-
From eq. (B.l), a is given by:

a = sn(in'w,k) (B.3)

For convenience, we now make an intermediate transformation from the t -plane

0

to a complex -plane defined by:

The u^-plane is shown in figure 11

(B.4)

Figure 11. Complex u^ plane.

Finally, we map the upper-half of the u -plane into a complex x^'Pl^rie

defined by

u = sn(x ,ot) 0 ^^o

The Xg'Plarie is shown in figure 12.

(B.5)

'01

i A D

\

K(a')

^ 2K(a) ^

C or

Figure 12. Complex XQ'Plane.

In order to calculate the electric field, we must find the complex potential, F,

which is given by [18]:

F = ^HXq) + I^CXq) (B.6)

where ^jCXq) is the potential function;

and

iP(Xq) is the stream function;

F satisfies the Cauchy-Riemann equations

In addition the potential ^iCXq) must satisfy the following boundary conditions:

(('(Xq) = 0 on EC

and (^(Xq) = V on AD,

-22-
It is easily verified that the following solution satisfies all of the above

requirements

:

and

vx01

K(a')

Vx or K(a'

)

(B.7)

(B.8)

where

X. Y + lY • ^or ^01

The electric field, E, is defined by

E = -V(})

= -(

dx

9_

9x 3? p* (B.9)

The real part of E gives the x-component of the electric field, and the imaginary

dp part of E gives the y-component of the electric field. may be calculated

as follows :

dY„ du„ dt„ dF dr ^o 0 0 dz dx„ du„ dt„ dz

'^o 0 o

(B.IO)

dF Using eq. (B.l) and eq . (B.4) through eq. (B.8), may be evaluated as

dF

where

^ -iVm' dn(m'z)

dz K(a') [P^(z)] ^

P^(z3 = [sn^Cm'w) - sn^Cm'z)]

(B.ll)

Thus, the magnitude-squared of the electric field, E^ is given by:

E^

o Vm'

S. 1 pected from the edge condition.

dn' (m'z)

sn (m ' w) - sn (m'z)

2

(B.12)

(B.13)

It is easy to see from eq. (B.13) that for z = ±w, E^ goes to infinity as ex-

-23-
Equation (B.ll) was evaluated numerically for some typical TEM cell

geometries and used to calculate the x and y components o£ the electric field,

as well as the magnitude and polarization angle of the electric field defined by;

= arctan y (B.14)

These results are included in tables 3 through 10 where all field quantities

have been normalized to V/b. In addition, the relative electric field distribution is plotted in figure 13. The top graph in that figure contains experimental data measured by M.L. Crawford [2] for comparison.

oK

+JH

cs

TiiH

Q) •H

U

•H

i-l -Uu

0)

0.1 >

•H

-P

03

.—IQJ

1.6

1.4 -

1.2 -

1.0 -

0.8-

0 . 6-1.0 -. -.4 0

x/a

.2

1.6

1.4 _

1.2 -

1.0 -

0.8 -

0 . 6-1.0 -.8

.6 .8 1.0

1 . 0 x/a

Figure 13. Relative electric field distribution inside a TEM

cell. Parameter indicates distance from the

center septum (y/b)

.

-24-
Table 3 x-component o£ 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)

-*2 5 20

15

10

5

0

0 . 000

0.000

0.000

0.000

0.000

0.000

0 Center of

TEM cell

0.000

0 .060

0 .108

0.127

0.090

0 . 000

0.000

0.129

0.245

0 .311

0 .248

0.000

0 .000

0.208

0 .422

0.620

0-. 647

0 .000

0 . 000

0. 278

0 . 600

1 .029

1. 684

0 . 000

10 15 20,

X=1v

0.000

0.307

0.680

1.237

2.285

3.603

2 5 xfcm)

f x=a

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

•

v=byCcm)

*2 5 20

15

10

5

0

0.824

0.853

0. 935

1. 049

1.153

1.196

0 Center of

TEM cell

0.793

0.825

0. 917

1. 052

1.186

1 . 245

0.698

0. 736

0. 852

1. 051

1.298

1 .451

0 .530

0.568

0.699

0 .977

1 .499

1 .986

0.289

0.315

0.410

0 .652

1.343

6 . 640

10 15 20, x=w

0 .000

0.000

0.000

0,000

0.000

0 .000

25 xfcm) x=a

-25-
Table 5 Magnitude o£ 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-*2 5 0 824 0 793 0.698 0.530 0 .289 0 .000

20 0 853 0

8^2 7 0.747 0.605 0.420 0 .307

15 0 935 0 924 0.886 0.817 0.727 0.680

10 1 049 1 060 1. 096 1.157 1. 218 1.237

5 1 153 1 189 1. 321 1 . 633 2.154 2. 285

0 1 196 1 245 1.431 1 . 986 6.640 3.603

Center/of

0 5 10 15 x=w

25 x=a TEM cell

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)

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

10 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

0 90 00 90 00 90 00 90 00 90 00 00 00

10 15

Center of

TEM cell

20x=w25

t x=a

X (cm)

-26-
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)

b—15 0 000 0 000 0.000 0 .000 0.000 0.000

12 0 000 0 024 0.067 0.143 0.220 0.249

9 0 000 0 040 0 .121 0.284 0.462 0. 517

6 0 000 0 043 0.141 0.410 0.763 0.817

3 0 000 0 028 0.101 0.440 1.247 1.112

0 0 000 0 COO 0.000 0.000 1 . 969 1.254

/ 0 5 10 15 t 20 25

Center of

TEM cell x=w

t 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^l5

12

9

6

3

0

Center of

TEM cell

0 966 0 946 0.872 0.698 0 .394 0 . 000

0 972 0 956 0.890 0 .724 0.411 0.000

0 989 0 981 0.944 0.807 0.464 0 .000

1 010 1 015 1.028 0.979 0 .557 0.000

1 028 1 046 1 .120 1 . 311 0 . 645 0.000

1 035 1 058 1 .164 1. 664 0.000 0.000

0 5 10 15

1 25 x=w x=w

X (cm)

-27-
. • 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)

y=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

0 1 035 1 058 1.164 1.664 1 . 969 1 .254

/ 0 5 10 15

^ 20 25

t x=a Center of

TEM cell

X =w

X (cm)

Table 10

Polarization angle of the electric field in degrees in a TEM

cell of dimensions 25 cm, b = 15 cm, w = 18.025 cm,

y (cm)

b-^15 90 00 90 00 90 .00 90 .00 90 00

12 90 00 88 58 85 68 78 80 61 76 00 00

9 90 00 87 66 82 70 70 .61 45 12 00 00

6 90 00 87 60 82 20 67 26 36 15 00 00

3 90 00 88 48 84 82 71 43 27 36 00 00

0 90 00 90 00 90 00 90 00 90 00 00 00

0 5 10 15 20 25 x(cm)

-28-
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.I)

Then we set up an Arithmetic -Geometric Mean (AGM) table defined recursively

as follows:

a, = ^(a +b ) b, = (a b c, = ha -b ) 1 2^00-^ 1 ^00^ 1 z^oo-*

^2 i(a^+b^) b2 = (a^b^)*-^ C2 = jCa^-b^)

1 ^ 1 " 2'^^N-l^^N-l^ " '^^N-l^N-l^' " 2 '^^N- 1 "^N- 1^

Stopping at n = N when Cj^ = 0 to the degree of accuracy required.

The complete elliptic Integral K(k) is then given by:

K(k) = (C.2)

The Jacobian elliptic functions, sn(5), cn(5), and dn(?) can be determined

by calculating in degrees where

^ 180° ,^

*N

= 2 5 (C.3)

and then computing recursively, <ti|vj_2^> ^^-2' ***

'''l' '^o using:

c sin (2(t)^_^-(()^) = — sin (C.4)

an The Jacobian elliptic functions are then given by:

sn(C,k) = sin (j)^ (C.5)

cn(?,k) = cos (C.6)

cos (j) dn(?,k) = ^

(C.7) cos ((t'j^-fi'o)

-29-
In eqs . (C.5) through (C.7), the argument, was assumed to be real. If the

argument is complex then the following formulas from reference [20] can be

used.

where K

--=5 + iC- ^r ^1

and s

•= sn(?^,k) ^1

c

== cn(?^,k) ^1

d == dn(5^,k) ^1

s 'd, + ic 'd* s, •c-|

sn(C,k) =

i

2

2 2 ^^'^^

c^ + k • s • s^

c • c, - is 'd • s 'd,

cn(5,k) =

i

2

2 2 ^^-^^

c^ + k s • s-j^

2 d "c, •d-| - ik • s • c • s, dnCC.k] = ^2 2

2 2 ^ (C.IO)

c, + k "s 's.

sn(5i,k')

cn(C. ,k')

dn(5. ,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)

COMnON A 520) ,8 (20) ,C (20) »L REAL K A(l)=l.

B ( 1 ) =SORT ( 1 »-K«K) . . C(l)=K

DO 10 1=2,20

A(I)=.5*(A(I-1)-^B(I-1))

B(I)=SQRT(A(I-1)08(I-1) ) C(I)=.5*(A(I-1)-B{I-1)

)

IF(C(I) ,LT<.l.E-6) GOTO 30

10 CONTINUE

PRIInT 20

20 fcriiat (ix,*AGM faile:d to converge in 2o rrCRATlONS^}

30 L = I RETURN

£f^Q / SUBROUTINE SNCNDN (XtSN.CNfDN)

COMMON A (20) ,3 (20) € (20) «L DIMENSION P(20)

P (L) =A (L) *X**2.*« (L-1

)

I=L • 10 P (I-l ) =,5* (ASIN ( (C { I) /A (I )

) «SIN (P (I) ) ) +P (I) ) 1 = 1-1

IF(I.NE.O) GOTO lo

SN=SIN{P(1))

Cm=C0S{P(1))

DN=CN/C0S (P (2) -P ( 1 )

)

RETURN

END

-30-
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 75-829

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

RADIATION CHARACTERISTICS OF D I POLE SOURCES

LOCATED INSIDE A RECTANGULAR, COAXIAL TRANSMISSION LINE

5. Publication Date January 1976

6. Performing Organization f-odc 276 . 00

7. AUTHOR(S)

John C. Tippet and David C. Chang

8. Performing Organ. Report No. NBSIR 75-829

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2768281

11. Contract/Grant No.

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

Same as Item 9.

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. -ABSTRACT (A 200-word or less factual summary of most si^ilicant ir\ formation. If document includes a significant

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 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.

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.

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