NATL INST. OF STAND & TECH RIC

AlllOM TDl7bl Pl)BUCAT\ONS

NisrUnited States Department of Commerce

Technology Administration

National Institute of Standards and Technology

NIST Technical Note 1377

Radiated Emissions and Immunity of

Microstrip Transmission Lines:

Theory and Measurements

David A. Hill

Dennis G. Camell

Kenneth H. Cavcey

Galen H. Koepke

NIST Technical Note 1377

Radiated Emissions and Immunity of

Microstrip Transmission Lines:

Theory and Measurements

David A. Hill

Dennis G. Camell

Kenneth H. Cavcey

Galen H. Koepke

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

July 1995

^^t-'.e**^ °' Or

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U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, AratI Prabhakar, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1377, 36 pages (July 1994)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1994

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
CONTENTS

Page

1 . INTRODUCTION 1

2 . RADIATION FROM MICROSTRIP TRANSMISSION LINES 2 2 .

1

Radiation from Horizontal Currents 3 2 . 2 Radiation from Vertical Currents 5 2 . 3 Total Radiated Power 6

2 .

4

Low- Frequency Limit 7 2 . 5 Maximum Radiated Field 8

3 . REVERBERATION CHAMBER APPROACH 10

3 . 1 Radiated Emissions 10

3 . 2 Radiated Immunity 11

3 . 3 Measured Data 12

4 . CONCLUSIONS 15

REFERENCES 15

APPENDIX. RECIPROCITY DERIVATION OF FAR FIELDS 17

ill

RADIATED EMISSIONS AND IMMUNITY OF MICROSTRIP

TRANSMISSION LINES: THEORY AND MEASUREMENTS

David A. Hill, Dennis G. Camell, Kenneth H. Cavcey,

and Galen H. Koepke

Electromagnetic Fields Division

Nationai. Institute of Standards and Technology

Boulder. Colorado 80303

We analyze radiation from a microstrip transmission line and

calculate total radiated power by numerical integration.

Reverberation chamber methods for measuring radiated emissions

and immunity are reviewed and applied to three microstrip

configurations . Measurements from 200 to 2000 MHz are compared

with theory, and excellent agreement is obtained for two

configurations that minimize feed cable and finite ground plane

effects. Emissions measurements are found to be more accurate

than immunity measurements because the impedance mismatch of the

receiving antenna cancels when the ratio of the microstrip and

reference radiated power measurements is taken. The use of two

different receiving antenna locations for emissions measurements

illustrates good field uniformity within the chamber.

Key words: directivity; emissions measurement; immunity

measurement; microstrip; mode stirring; radiated power;

radiation efficiency; radiation pattern; reverberation chamber;

transmission line.

1 . INTRODUCTION

The increasing complexity of electronic systems has introduced an increased potential for electromagnetic interference (EMI) between

electronic systems [1]. The Federal Communications Commission (FCC) limits

on radiated emissions from digital equipment [2] cover the frequency range

from 30 to 1000 MHz, and increasing clock rates in digital circuits are likely to result in standards that cover even higher frequencies [3]. The

closely related issue of radiated immunity of electronic systems is also

receiving increased attention [4,5].

The FCC radiated emission limits specify the maximum radiated electric

field measured at a specified distance. In this report we propose an alternative (but related) approach based on the total radiated power over a
sphere enclosing the radiating system. The reverberation chamber [6] is a convenient facility for performing total radiated power measurements and

radiated immunity measurements [7] over a wide frequency range. It has the

advantage of not requiring any rotation of the unknown radiator, but it will

not give the directivity or the direction of maximum radiation. To

illustrate the theory and measurement procedure, we analyze the simple case

of radiation by a single microstrip transmission line [8] and integrate over

a far- field sphere to obtain the total radiated power. The reverberation

chamber measurement techniques are independent of the actual radiator (or

receiver), but we choose the example of a microstrip transmission line

because it is an open structure that does radiate and is commonly used in

dense circuits and interconnections [9].

The organization of this report is as follows. Section 2 covers the

theory of radiation by a microstrip transmission line. A closed- form

expression is obtained for the radiation pattern, and a numerical

integration is used to obtain the total radiated power. For low

frequencies, the numerical integration is performed analytically to obtain

an approximate expression for the total radiated power. The directivity of

the microstrip is also approximated from its electrical size and compared

with the actual directivity. This allows an estimation of the maximum

radiated field and hence provides a link to the FCC limit. Section 3 covers

the theory for radiated emissions and immunity measurements in a reverberation chamber and includes a comparison of theory and measurements.

Section 4 contains conclusions and recommendations for further study. The

appendix includes a reciprocity derivation of microstrip, far-field

radiation.

2. RADIATION FROM MICROSTRIP TRANSMISSION LINES

The currents on microstrips can be approximated by transmission line

theory [10] or determined by a numerical method, such as the method of

moments [11]. We choose transmission line theory because of its simplicity,

and we also use a line current approximation to compute the radiated field

[8]. Both methods typically assume an infinite ground plane and substrate
so that the radiated field can be written in terms of the known Sommerfeld

integrals or derived by reciprocity [12].

2.1 Radiation from Horizontal Currents

We consider the microstrip geometry shown in figure 1. The x-directed

microstrip has length L and width w and is centered at the origin. The

substrate has thickness h and permittivity e , and the ground plane is

located at z = -h. The region above the substrate has free -space

permittivity €„, and both regions have free -space permeability /i„ . It is

also convenient to define a relative substrate permittivity ^ e = e /e„. -' sr s The surface current is approximated by a line current I(x') located along

the X axis. The feed and termination currents are modeled by vertical

currents, I(-L/2) and -I(L/2), and their radiation will be considered in the

following section.

The field radiated by an infinitesimal horizontal dipole Idx' can be

written in terms of scalar and vector potentials [13] or an electric Hertz

vector [8]. Either method results in Sommerfeld integrals [14] which can be

evaluated numerically in the near field or by the saddle -point method in the

far field [14]. Since we require only far-field expressions, an alternative

method is to use reciprocity [12] as described in the appendix.

The 9 and 6 components, E., , and E,, ,, of the electric field radiated

Sha (phd

by a horizontal dipole can be written in the following forms (see appendix)

:

jw/x^Idx' e"^^

^ou^ = ; (R - l)cos^cos^ (1) ^hd 4a V r and

jw^ Idx' -jkR

where

"v

= '1 - 7^ tan(kvh)l/[l + ^J^ tan(kvh)), (3) sr sr
\- |l+i77cot(kvh)I/[l - J^coC(kvh)l, (4)

1/2 1/2 V = (e -1) , R = r - x'sin^cos(^, k = w(/i e ) , and (r,^,^) are the

standard spherical coordinates. These results are in agreement with the

far- field results in [8] obtained by saddle-point evaluation of Sonunerfeld

integrals. Surface waves are neglected.

To determine the electric field components, E., and E ,, , radiated by

the entire horizontal line current, we need to integrate eqs (1) and (2) for

x' = -L/2 to L/2 . We assume that the microstrip is fed at x = -L/2 and is

terminated in a matched load at x = L/2. Then the current is a pure

traveling wave which we write as

I(x') = Iq exp(-jk^x'), (5)

1/2 where k = ke and e is the relative effective dielectric constant for

e er er the microstrip mode [15]. Its value lies between e and 1. The only x'

dependence in eqs (1) and (2) arises from the exponentials. Since the

exponentials in eqs (1) and (2) are identical, we have only one integral to

evaluate. We can write that integral as an array factor A of the type that

appears in antenna array theory [16];

L/2

- j (k -ksin^cos(^)x' sin[(k -ksin^cos'^)L/2

]

e

^ ^ = dx' = —, ,, ^ . ^ .., „, . (6) i I - • ••

[(k--ksin^cos<ji)L/2]

L/2 ®

Using eqs (1), (2), and (6), we can write E , and E,, as

jcoMqIqLA -jkr

E^, = 7-^— (R - l)cos^cos^ (7) ^h 47r V r and
2 . 2 Radiation from Vertical Currents

A vertical electric dipole radiates only a B far- field component E

which can be written (see appendix)

:

jw/i Idz' jkv(z'+2h) -jkvz' -jkR

(R„ + l)sintf[^ tt:^ ] ^-—

.

(9)

tfvd Ajre v , ikv2h

sr 1 + e-^

From eq (5) , we see that the vertical feed current is I„exp(jk L/2) and

the vertical termination current is -I_exp(-jk L/2). The 6 component of the

electric field E is given by the integral of eq (9) over the locations of

the feed and termination currents:

z'=0

^^v I

~

J ^^^vd

z' = -h x'=-L/2

" '^^^ ). (10)

x'=L/2

The differential dz' is contained in E ,. By substituting eq (9) into eq avd

(10) and carrying out the z' integration, we obtain

E. = tan(kvh)(R + l)sin^ , (11)

tfv 47r€ kv ^ ' ^ V r sr

where

F = 2j sin[(k - ksin^cos^)L/2]

.

(12)
2.3 Total Radiated Power

The total radiated d component is the sum of the radiation by the

horizontal and vertical currents: E = E , + E . Only the horizontal

u uu. u'V current radiates a 4> component: E, = E,, . <p <pi\ We assume that the ground plane and substrate in figure 1 extend to

infinity so that the radiated field exists only in the upper half space

(^ < n/2) . We can write the total radiated power P as an integral over a

hemisphere at large r:

P_ - r' I I ^ sln« d»d^, (13)

^ /I i\ f i.

1/2 where r? = (/i_/e_) . Generally the double integration in eq (13) needs to

be performed numerically, but in the following section we perform the

integration analytically for the low- frequency limit.

For the usual case of weak radiation, most of the input power P. to the

microstrip is dissipated in the matched load. Thus we can approximate P. .by

Pj = |iol%. (14)

where Z_ is the characteristic impedance [15] of the microstrip. Thus we

can approximate the radiation efficiency Eff by

P

Eff = P^/P. = h^. (15)

l^ol ^0

If the detailed expression for P is substituted into eq (15) , the

2 denominator | I^ | cancels out.
2.4 Low- Frequency Limit

In the low- frequency limit, we assume that the quantities, kh , kL, and

k L, are small. Consequently we make the following small argument

approximations

Aj^ == 1, (16)

F^ = i (k - ksin^cos(^)L, (17)

i e

cot(kvh) ~ (kvh)'"*", (18)

and

tan (kvh) == kvh. (19)

If we use the approximations in eqs (16) through (19) and retain only

2 the leading term in k Lh , we can write the squares of the electric field

components as

2 2 1/2

2 ( V ^^^>

2 'er 'er 2 \E.\ ^ — — (cos^^ - 2-^^ sin^cos,^ + "T^ sin^) (20)

o , » . Z € Z (27rr) sr e sr and

2 2 2 (Igk Lhr?) ^

2 |E.| ~ ~ cos ^sin <f)

.

(21) ^ (2.r)2

If we substitute eqs (20) and (21) into eq (13) , we can perform the 9 and 4> integrations analytically with the following result:

2 2 2

I r,(k Lh)"

P^ - -^^

(1 + -F). (22)

r JTT z

e sr
If we had assumed a constant current I(x') = I„ on the microstrip in eq (5),

then we would have obtained the result in eq^ (22) with e =0:

er

2 2 2 I^r,(k^Lh)''

P

r

e =0 ^" er

(23)

The result in eq (23) agrees with the power radiated by a vertical half loop

of area Lh carrying a constant current I_ over a perfectly conducting plane

in free space [16]. The result for the traveling-wave current in eq (22) is

4 2 larger, but both eqs (22) and (23) have the expected k 'area dependence

which is characteristic of an electrically small loop.

2.5 Maximum Radiated Field

In order to relate total radiated power to the maximum radiated

electric field (as specified in the FCC regulations), we need to know the

directivity^ D of the radiator. The maximum radiated electric field E at

max

a distance r is

1 ^°^r 1/2

E = - (—^) / . (24) max r 47r

Normally the directivity of an inadvertant radiator, such as a microstrip

transmission line, is not known. However, if we exclude the unlikely

possibility of supergain, the maximum directivity of a radiator is

determined by its electrical size [17,18]. For a radiator that can be

enclosed within a sphere of radius a, the maximum directivity D is

3, ka < 1 D = { (25) m

(ka) + 2ka, ka > 1.
The case of ka < 1 corresponds to combined electric and magnetic dipole

radiation in the form of a Huygens source [18] , and the case of ka > 1 corresponds to general spherical multipole radiation.

For our microstrip geometry, we can set a = L/2 . However, we must also

account for the effect of the infinite ground plane. By image theory

considerations, we ca.< show that the maximum directivity of a source over a ground plane can be twice that of the same source in free space

.

Consequently, we can write the maximum directivity of the microstrip as

6, kL/2 < 1 D = { (26) m

2 2(kL/2) + kL, kL/2 > 1.

To illustrate the frequency dependence of the directivity, we consider

an example with the following parameters: h= 1.55 mm, L= 10 cm, w=4.8

mm, and e = 2.2. These parameters yield a characteristic impedance

Z„ = 50 Q and an effective permittivity e = 1.881 [15]. In figure 2 we

show the maximum directivity determined from eq (26) and the actual

directivity determined by numerically evaluating the radiated power in eq

(13). Both quantities increase with frequency, but the actual directivity

is always less than the maximum directivity. Consequently, eq (24) yields

an upper bound for the maximum radiated electric field.

The direction of maximum radiation is also of some interest. Our

program for directivity also computes the direction of maximum radiation.

For our simple microstrip geometry, the maximum radiation always occurs at zero or 180° azimuth. Figure 3 shows a plot of the zenith angle 6 of

maximum radiation for the same parameters as considered in figure 2. In

order to plot zenith angle continuously, we allow d to range from -90° to

90°. As the frequency increases, the array factor A in eq (8) forces the

radiation from the back direction toward 90° like an end- fire, traveling- wave antenna

.
3. REVERBERATION CHAMBER APPROACH

Although we computed the total radiated power by integrating the power

density over a large hemisphere as indicated by eq (13) , we prefer not to

require radiated field measurements over a surface enclosing the radiator.

The reverberation chamber [6] is a convenient facility for measuring total

radiated power without requiring motion of the receiving antenna or the

radiator. It is equally convenient for performing radiated immunity

measurements [ 7 ]

.

3.1 Radiated Emissions

Consider the reverberation chamber (cavity) geometry shown in figure 4.

In our measurements the radiator was a microstrip transmission line, but the

theory and measurement method apply to any radiator. A mode stirrer (tuner)

is used to obtain uniform power density throughout the chamber, and this

technique requires that the cavity be electrically large (so that the cavity

is multimoded)

.

A power conservation approach [19,20] shows that the power density S

in the chamber is

AQP

where A is the free -space wavelength, V is the cavity volume, Q is the

cavity quality factor, and P is the radiated power. Strictly speaking, S

in eq (27) and received power in equations to follow are ensemble averages

over all stirrer positions. The chamber Q is normally measured by a transmission loss method [6,17] or a pulse decay method [21], but it can calculated if all the loss mechanisms are known [20].

In a reverberation chamber, the effective area of a lossless,

2 impedance -matched receiving antenna is A /Stt [19,20]. The received power

P is then given by the product of the power density and the effective

area

:

10
2 3

P = s ^ = ^-^

P . (28) rec c 8n , / 2., r

lOTT V

If we measure the received power, eq (28) can be used to measure the

radiated power. If we also measure the input power P., then we can

determine the radiation efficiency from eqs (15) and (28):

le^r^VP

Eff = P /P. = —^^. (29)

r 1 2 A QP.

1

3.2 Radiated Immunity

Typically both a receiving (reference) antenna and a transmitting

antenna are used for radiated immunity measurements as shown in figure 4.

The power ^ received by-^ the reference antenna is P ^, and the power received

ref ^

by the microstrip is P . We define the shielding effectiveness (SE) in

decibels in terms of the received power ratio [7]:

SE = 10 1og^^,(P^^^/P^J, dB. (30)

With the above definition, SE is normally positive.

Another valid descriptor of the receiving properties of the microstrip

is the effective area A [71. The effective area can be determined from the

e received power as follows:

P P ,2 rm rm A /oi x e S P _ OTT c ref

From eqs (30) and (31), we can also write SE in terms of A

11
SE = 10 log^Q[A^/(87rA^)] , dB

(32)

By reciprocity [12], we can show that the effective area of the

microstrip is simply related to the radiation efficiency:

a2

A = Eff T-. (33)

e OTT

From eqs (31) and (33) , we can also write radiation efficiency as

Stt-A P

Eff = -^ = -f^. (34)

A ref

For the emissions measurement, we can interpret P as the power measured by

the reference antenna when the microstrip^ is transmittine^ and P j- as the

ref

power measured by the reference antenna when the other antenna is

transmitting. Equation (34) is actually preferred to eq (29) for measuring

the radiation efficiency because the chamber Q is needed for use of eq (29)

Equations (29) and (34) are mathematically equivalent when the usual

expression for chamber Q [19,20] is substituted into eq (29).

Equation (33) indicates that both the emissions and immunity properties

of the microstrip could in theory be obtained from either emissions or immunity measurements. However, considering the relatively large

uncertainty of reverberation chamber measurements [6], it is useful to do

both eiiiissions and immunity measurements and to use eq (33) as a consistency

(reciprocity) check.

3 . 3 Measured Data

In order to study feed and mounting effects, we constructed two

microstrip transmission line boards with the same microstrip parameters, but

different ground plane dimensions and feeds. The microstrip transmission

line parameters (h = 1.55 mm, L = 10 cm, w = 4.8 mm, and e = 1.1) axe the

12
same as those used for the pattern calculations in Section 2.5. The first

board is 10 cm by 10 cm with a coaxial feed line oriented as shown in figure

5. This board can be tested in the conventional raised position away from

all chamber walls, or it can be bonded to the chamber floor to achieve an extended ground plane and to short circuit currents on the exterior of the

coaxial feed. The second larger board (23 cm by 30 cm) has the microstrip

line centered with the coaxial feed line and the termination located below

the board as shown in figure 6. This board is a closer approximation to the

theoretical model because the ground plane is larger and the coaxial feed

line is fairly well shielded by the ground plane.

Figure 7 shows a comparison of theory and measurements for radiated

emissions from 200 to 2000 MHz. The lower frequency limit of 200 MHz is

determined by the need for a sufficient mode density in the NIST

reverberation chamber [6]. The chamber mode density increases with

frequency, but the approximate microstrip theory in Section 2 is not valid

above 2000 MHz. The results are given as the power ratio (in decibels) for

the reference antenna transmitting compared to the microstrip transmitting,

but the results could be converted to radiation efficiency by using eq (34).

One advantage of presenting the results as a power ratio is that the

emissions results can be directly compared with the immunity results for SE.

The theoretical curve is smooth, but the measured curves show modal

oscillations that are characteristic of reverberation chamber data [6].

Log-periodic dipole antennas were used for the transmitting and receiving

antennas below 1000 MHz, and broadband ridged horns were used above 1000

MHz. The power radiated by the reference antenna is the incident minus

reflected power. Thus impedance mismatch loss is taken into account for the

reference transmitting antenna, but not for the receiving antenna. The

measurement data were generated by averaging received power over 200 stirrer

positions at each frequency.

4

At low frequencies (small kL) , the k dependence of the radiated power

in eq (23) indicates that the slope in figure 7 should be approximately

-12 dB/octave, and this behavior holds below 500 MHz. The measurements for

the raised microstrip with the end feed (as shown in figure 5 show the

poorest agreement with theory, probably because of radiation from the shield

of the coaxial line and edge effects of the small ground plane. The

13
measurements for the bottom feed show the best agreement with theory, but

the measurements for the bonded microstrip also show good agreement with

theory. This indicates that test objects that operate with a ground plane

can be conveniently mounted on chamber walls even though conventional

practice has normally placed test objects at least a wavelength from chamber

walls. The plane wave spectrum theory for fields in reverberation chambers

[7,22] indicates that wall mounting is appropriate for ground plane test

objects, and wall mounting also provides a convenient means for short

circuiting the shields of feed cables.

Figure 8 shows the analogous comparison of measurements and theory for

the radiated immunity case. These results can be considered to represent SE

in decibels. These measurements can be performed more quickly than

emissions measurements because the power received by the reference antenna

and the microstrip are measured simultaneously. However, the measurements

are not as accurate because the impedance mismatch of the reference

receiving antenna is not taken into account. (In emissions measurements,

the impedance mismatch of the receiving antenna approximately cancels when

the ratio for the reference antenna and microstrip transmitting is taken.)

However, the agreement is still satisfactory for the bottom feed and bonded

microstrip cases.

As indicated by eq (28) , we assume that the effective area of the

2 reference antenna is A /87r. In figures 7 and 8, the crossover frequency of

1000 MHz actually includes two points, representing the log-periodic dipole

antenna and the ridged horn antenna. The small difference (approximately 1 dB) between these two points is a good confirmation that the specific

antenna and antenna pattern are unimportant in reverberation chamber

measurements . Another assumption in reverberation chamber measurements is

field uniformity throughout the chamber. For emissions measurements, we measured power received by each antenna when the microstrip was transmitting. The two antennas (log periodics or horns) were placed on opposite sides of the chamber and thus provide a field uniformity check.

Figures 9 through 11 show the emissions measurements for the two different

receiving antennas, and the measured difference is typically on the order of

1 dB except at the lower frequencies. These results are consistent with

earlier field uniformity measurements [6]

14
4. CONCLUSIONS

We have derived closed- form expressions for the fields radiated by a microstrip transmission line and have calculated total radiated power by

numerical integration. For low frequencies, we have obtained an approximate

expression for the total radiated power that is similar to the expression

for radiation by a small loop. We have also approximated the directivity of

the loop in order to provide a link to FCC limits on radiated electric

field. We have reviewed the theory for radiated emissions and immunity

measurements in reverberation chambers and shown how they are related. A

comparison of theory and measurements for both emissions and immunity from

200 to 2000 MHz shows excellent agreement for the two microstrip geometries

(bottom feed and bonded to chamber floor) that minimize feed cable effects.

The emissions measurements show somewhat better agreement with theory than

the immunity measurements because the impedance mismatch of the receiving

antenna cancels when the ratio with the reference measurement is taken.

Field uniformity was demonstrated to be good by measuring the microstripradiation in two different chamber locations

A number of extensions to this work would be worthwhile. Emissions and

immunity calculations could be performed for more complex circuit boards

with complex wire connections [23] and finite ground planes [24]. The

reverberation chamber measurement technique described here is valid for

general test objects and higher frequencies that will continue to become

more important. Time -domain analysis [4] and measurements should be pursued

because of the importance of digital systems. Some pulsed measurements have

been performed in reverberation chambers [25] and other cavities [26], but

further study of pulsed measurements in reverberation chambers is needed.

Also, frequency stirring [27,28] should be employed on immunity measurements

for increased speed and possibly increased accuracy.

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15
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[22] Dunn, J.M. Local, high-frequency analysis of the fields in a modestirred chamber, IEEE Trans. Electromag. Compat., 32: 53-58; 1990.

[23] Hill, R.L. ; Van Doren, T.; Hubing, T. ; Drewniak, J. Common mode

currents induced on wires attached to multilayer printed wire boards

with segmented ground planes. IEEE Int. Symposium Electromag. Compat.:

116-120; 1994.

[24] Daijavad, S.; Pence, W. ; Heeb, H.; Ponnapalli, S.; Ruehli, A.

Methodology for eveluating practical EMI design guidelines using EM

analysis programs. IEEE Int. Symposium Electromag. Compat.: 30-34;

1992.

[25] Crawford, M.L.; Ma, M.T. ; Ladbury, J.M.; Riddle, B.F. Measurement and

evaluation of a TEM/reverberating chamber. Natl. Inst. Stand. Technol.

Tech. Note 1342; 1990.

[26] Hill, D.A.; Crawford, M.L. ; Johnk, R.T. ; Ondrejka, A.R. ; Caraell, D.G.

Measurements of shielding effectiveness and cavity characteristics of

airplanes. Natl. Inst. Stand. Technol. NISTIR 5023; 1994.

[27] Loughry, T.A. Frequency stirring: an alternate approach to mechanical

mode-stirring for the conduct of electromagnetic susceptibility

testing. Phillips Laboratory, PL-TR- -91-1036; 1991.

[28] Hill, D.A. Electronic mode stirring for reverberation chambers. IEEE

Trans. Electromag. Compat., 36: 294-299; 294.

APPENDIX. RECIPROCITY DERIVATION OF FAR FIELDS

A reciprocity derivation [12] of the far fields can be obtained by

locating a test source at a large distance from the origin and calculating

the horizontal electric field at the air- substrate interface and the

vertical electric field within the substrate. The geometry is the same as in figure 1, except that the source is now a distant electric dipole source

17
that radiates a plane wave in the vicinity of the origin. The elevation

incidence angle is 6 , and for simplicity we set the azimuthal incidence

angle <^ equal to 0.

Consider first the horizontal polarization case. The total electric

field E _ in free space (z > 0) can be written

jkxsin^ jkzcos^ -jkzcos^. ^

yO " ^ \^ ^

' ^ ^

where

di is the dipole moment, and r is the dipole distance from the origin. The

magnetic field has both x and z components. The magnetic field x component

H _ in free space is

xO ^

E-.cos^

H n = xO ry ikxsin^ . ikzcos^ „ -ikzcos^. ,.ove-" (e-^ - R, e

-^ ) . (A3)

Within the substrate (-h < z < 0) , the electric field E can be

ys

written

„ „ jkxsin^ .. ikvz , „ -jkvz. .... E = E_ e-^ (Ae-^ + Be -^ ) , (A4)

ys

where A is the unknown coefficient of the downgoing wave and B is the

unknown coefficient of the upgoing wave. From Maxwell's curl equation, we

can write the x component of the magnetic field H in the substrate as

H = ^

^jkxsin^ ^^jkvz _ B^-jkvz 5 xs r)

18
The boundary conditions are that the tangential components of the

electric and magnetic fields are continuous at the air- substrate interface

and that the tangential electric field is zero at the ground plane:

(E ^

- E )| ^ = 0, (A6) yO ys I z=0

<»xO - »xs)|z=0 = 0' f'^')

and

El , = 0. (A8)

ys I z=-h

Simultaneous solution of eqs (A6) through (A8) yields expressions for A, B,

and R, . A and B are not needed to determine the electric field at the

interface, and the expression for R, is that given in eq (4).

The tangential electric field at the air- substrate interface is

V|z=0 = ^0<\ - 1> ^'"^^'"'- (*')

If we generalize eq (A9) to allow arbitrary azimuthal incidence angle, we obtain the reciprocal result for horizontal dipole radiation in eq (2)

.

We now consider vertical polarization where the source is a ^-oriented

electric dipole. The total magnetic field H _ in free space can be written

-E „ jkxsin^ , ikzcos^ _, -ikzcos^. /»ir>\ H _ = e-^ (e-^ + R e

-^ ) . (AlO) yO

The electric field has both x and z components. The electric field

component E „ in free space is

„ „ - jkxsin^ , jkzcos^ „ -jkzcos^. /aiin E - = E-Cos^ e-^ (e-^ - R e

-^ ). (All)

xO V

Within the substrate, the magnetic field H can be written

19
H = ^

3Jkxsin(? ^^jkvz

^ De-J^"), (A12)

ys ry

where C is the unknown coefficient of the downgoing wave and D is the

unknown coefficient of the upgoing wave. From Maxwell's curl equation, we can write the x component E and z component E of the electric field in

xs zs the substrate as

vE

E = -0 Jkxsin^ ^^jkvz _ De-J^^^) (A13) xs e sr and

17 jkxsin^ ._ jkvz , ^ -jkvz. /m/x E = e-^ (Ce-" + De -^ ) , (A14) zs 6 sr

The boundary conditions are again that the tangential components of the

electric and magnetic field are continuous at the air- substrate interface

and that the tangential electric field is at the ground plane:

(H _ - H )| ^ = 0, (A15) yO ys |z=0

<^0 - ^s>lz.O-0' <'^"'

and

El u = 0- (A17) xs z=-h

Simultaneous solution for eqs (A15) through (A17) yields the result for R

given in eq (3) and the following expressions for D and C:

D = (1 + R )/(l + eJ^^'^) (A18)

and

C = DeJ2^^^. (A19)

The tangential electric field at the air-substrate interface is

20
^x0|z=0 = V^ - \^^°'' ^'''""' (*20)

If we generalize eq (A20) to allow for arbitrary azimuthal incidence angle, we obtain the reciprocal result for horizontal dipole radiation in eq (1).

The vertical electric field E within the substrate is given by eq

(A14) where C and D are given by eqs (A18) and (A19) . This result is

independent of azimuthal incidence angle. So the reciprocal result for

vertical dipole radiation in eq (9) is obtained directly from eq (Al4)

.

Other electric and magnetic dipole source results could be obtained

easily by this reciprocity method, but they are not shown here because they

are not required for computing microstrip radiation.

21
/////////////////////////////////////////////>//////////////// z = -h

Figure 1. Geometry for a microstrip transmission line of length L.

22
m

>

13

12

1 1

10

9

O 8

7

6

o o o o o [)

Q-B-e-B-Q [)[^

5

0.0

J L I I I I I I II I L 0.5 1.0 1.5

FREQUENCY (GHz

2.0

Figure 2. Maximum (DM) and actual (D) directivity of microstrip line

Parameters: L=10cm, w=4.8mm, h=1.55mm, and e^^ = 2.2.

23
-90

0.0 0.5 1.0 1.5

FREQUENCY (GHz

2.0

Figure 3. Angle of maximum radiation of microstrip line. Parameters:

L= 10 cm, w=4.8 mm, h= 1.55 mm, and e sr 2.2

24
Reverberation Chamber

Transmit

Antenna

Stirrer

Microstrip

Reference

Antenna

Figure 4. Reverberation chamber configuration for emissions or immunity

measurements

25
Microstrip Coaxial Line

10 cm

Figure 5. Microstrip transmission line with end feed.

26
Coaxial Line

Figure 6. Microstrip transmission line with bottom feed.

27
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