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

Theory of Near-Field Phased

Arrays for Electromagnetic

Susceptibility Testing
NATIONAL BUREAU OF STANDARDS

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Theory of Near-Field Phased

Arrays for Electromagnetic

Susceptibility Testing

David A. Hill

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued February 1984
National Bureau of Standards Technical Note 1 072

Natl. Bur. Stand. (U.S.), Tech Note 1072, 109 pages (Feb. 1984)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1984

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

Page

1. Introduction 1 2. Review of previous work........ 3

3. Basic concepts , 4

3.1 Equivalence principl e 4

3.2 Relationship to scattering problem 6

3.3 Sampl ing and scanning considerations 10

4. Near fields of smooth current distributions 12

4.1 Two-dimensional geometry 13

4.2 Three-dimensional geometry 18

5. Near fields of arrays .. 22

5.1 Electric dipole array 23

5.2 Huygens' source array 26

6. Near-field array synthesis 28

6.1 A new uniqueness theorem 29

6.2 Least squares analysis 32

6.3 Line source example 36

7. Conclusions and recommendations 42

8. Acknowledgments. 44

9. References 44

Appendix A Asymptotic evaluation of two-dimensional fields 48

Appendix B Asymptotic evaluation of three-dimensional fields 53

Appendix C Fields of a Hertzian dipole 61

Appendix D Fields of a Huygens' source.. ,63

Appendix E A uniqueness example 65

Appendix F Line source fields 68

Theory of Near-Field Phased Arrays

for Electromagnetic Susceptibility Testing

David A. Hill

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

The feasibility of using a near-field array for electromagnetic

susceptibility testing is studied. The basic objective is to

control the element weightings such that a plane wave is generated

within the test volume. The basic theory is developed for

arbitrary array geometries, and numerical results are obtained for

finite planar arrays. A general near-field array synthesis

technique is developed, and the technique minimizes the mean

square error in the test volume while constraining the array

excitations. The constraint prevents large excitations and is

useful in minimizing the fields outside the test volume. The

basic idea looks promising, but some practical considerations,

such as bandwidth and angular scanning limitations, require

further theoretical and experimental investigation.

Key words: array synthesis; dipole; electromagnetic susceptibility; Huygens' source; line source; near field; phased array.

1. Introduction

The ideal incident field for electromagnetic susceptibility testing is one which is a uniform plane wave throughout the test volume. The field outside the test volume is arbitrary in principle because it does not affect the

response of the equipment under test (EUT). However, it is desirable to keep

the field outside the test volume small in order to avoid wasted power and

reflections from surrounding objects. Thus, a far-field antenna range is not

generally acceptable for susceptibility testing because only a small portion

of the total power is incident on the test volume. An additional disadvantage

with far-field ranges is that the far-field distance is often very large. It

is desirable that the test facility dimensions not be too much larger than the

dimensions of the test volume so that the facility could be located indoors.

Present methods of producing a plane wave can be divided into "low frequency" and "high frequency" methods. The low frequency methods utilize a section of a transmission line which supports a transverse electromagnetic
(TEM) wave. Both parallel plate lines and TEM cells have been used to produce

fields which are planar over a portion of their cross section. The transmission line dimensions are normally somewhat less than a wavelength so that

higher order modes cannot propagate. Consequently, transmission line methods

are most effective for testing EUTs which are electrically small. High frequency methods generally focus a spherical wave into a plane wave with a microwave lens [1] or a paraboloidal reflector. Both methods are designed on the basis of ray tracing and are thus limited to high frequencies. The paraboloidal reflector, which has been termed "compact range" [2], has been used

quite successfully for antenna pattern measurements and with less success for

radar cross-section measurements. There exists a frequency gap (about 50 MHz

to 1 GHz) between the low frequency and high frequency methods where another

method of producing a plane wave is needed.

The present methods generate a fixed plane wave and have no capability of

scanning the angle of arrival. Thus, the EUT must be rotated in order to

obtain angular information. The possibility of a compact range with a spherical reflector and a movable feed to allow angular scanning has been discussed

[3], but it has not been determined whether a spherical reflector will produce

a plane wave of acceptable quality.

The near-field phased array method, which we analyze in this report, has

the potential of electronically scanning in both elevation and azimuth. If

the EUT is located on a turntable [3], then an electronic scan in elevation is

sufficient. Another potential advantage of a near-field array over a compact

range is that arrays should be able to work down to lower frequencie*; since

they are not limited to the geometrical optics range. The frequency range of

interest runs from about 30 MHz up to microwaves.

The organization of this report is as follows. Section 2 contains a review of previous work on near-field arrays. Section 3 introduces some of

the basic concepts in near-field phased arrays and points out the relationships to near-field scanning and electromagnetic scattering problems. Section

4 presents analytical and numerical results for the near fields of continuous

current distributions. The abrupt truncation of the current at the edge of

the source region is shown to produce an undesirable edge diffraction field
within the test volume. Section 5 presents analytical and numerical results

for the near fields of discrete arrays, and the effect of element spacing is

studied. Section 6 presents a fairly general array synthesis procedure for

minimizing the difference between the actual field and the desired plane wave

field within the test volume. A constraint condition also provides a means of

keeping the field strength small outside the test volume. Section 7 summarizes the results of this study and makes recommendations for further work.

2. Review of previous work

The idea of generating a plane wave in the near field of a phased array

was discussed as early as 1968 by Martsafey [4] who analyzed an infinite

planar array. He showed that if the element spacing is less than a half

wavelength, the field of the array consists of a single propagating plane wave

and an infinite sum of decaying evanescent waves. Other geometries are more

complicated, but spherical [5,6] and cylindrical [7] arrays have also been

analyzed.

Ludwig and Larsen [6] pointed out that synthesizing a plane wave in a test volume is the reciprocal case to predicting the far-field pattern of an antenna which is transmitting from the test volume. Thus, the array element

weightings in the plane wave synthesis case are the same as the probe weightings in near-field scanning. They confirmed this relationship for a spherical

geometry with electric dipole elements by computing the field inside a sphere

where the element weightings were given by the spherical near-field scanning

theory. The plane wave quality was good when a full sphere was used, but

deteriorated when the array covered only a portion of the sphere. Actually,

when the array is truncated, the spherical near-field scanning theory no longer gives the best weighting values. Ludwig and Larsen [6] have suggested

that the determination of the optimum weights for truncated spheres is an interesting area for future work.

Bennett and Schoessow [5] determined the weighting function for a truncated spherical array by using an approximate diffraction integral. They

attempted to reduce the error field in the test zone by multiple iterations

and were partially successful. Many detailed calculations are contained in
the Ph.D. thesis by Schoessow [8]. A Fourier analysis of the field in the

test zone showed that much of the error field originated from the edges of the

array

.

The only experimental results have been obtained by Lynggaard [9] who

built and tested a five-element array of horns. The array was essentially

planar, but the theory for a large sphere [6] was used to determine the

element weightings. The test volume was a small sphere at a fairly large

distance from the array. Directional scanning of the plane wave was not

attempted.

3. Basic concepts

3.1. Equivalence principle

The electric and magnetic fields, E^j and H^, of a time harmonic,

exp(jwt), plane wave can be written in the following form:

E = E^ exp(-j k • r) ,

where

H = H exp{-j k • r) , (1) —p —

o

— —

E^. k = 0, H^-kx V^

k = k/jkl, |k| = 2tiA ,

T\ is the free space impedance, \ is the free space wavelength, _r is the

position vector, and E^ is a constant. All field and source quantities are assumed to vary as exp(jwt), and the time dependence will be suppressed

throughout this report. For a propagating plane wave, J^ is real in (1) and

the electric and magnetic fields have constant magnitudes. The plane wave

fields in (1) are solutions to the source-free Maxwell's equations in a homogenous, infinite region.

The equivalence principle can be used to show that the plane wave fields

in (1) can also be generated by sources in a finite region. Consider a volume
V enclosed by a surface S as shown in figure 1 Inside S, we specify planar

fields. En and ]Lj, as given by (1). Outside S, the exterior fields, E^ and H^. are arbitrary outgoing fields which satisfy Maxwell's equations By outgoing

fields, we mean that they satisfy the radiation condition at infinity- From

the equivalence principle [10], these fields can be generated by the following

electric and magnetic surface currents, J_ and M^, on S:

J = n X (H^ - H^) and M = (E^ - E^) X n , (2) — —P —c — —P —

G

where n is the inward unit normal to S ^ and M^ have units of amperes/m and

volts/m, respectively Because ^ and H^ in (2) are arbitrary, the source currents J^ and M^ are nonunique, and there are many current distributions which

will generate a perfect plane wave inside S. Nonuniqueness is typical of

inverse problems where the fields in a region are specified and the sources

are unknown. The practical problem involves finding a solution of (2) where

the currents J_ and M^ can be approximated well by a phased array of realistic

antenna elements. As an aside, the plane wave fields, E^ and hL^ in (2) could

actually be any desired fields which satisfy the source-free Maxwell's equations. So far only plane wave fields are of interest for electromagnetic

susceptibility testing, and we consider only plane wave synthesis throughout

this report.

The most desirable case of (2) is obtained when the exterior fields, J^

and H^, are zero. Then the surface currents, J^ and M^, are simply:

A. A. J^ = n X H and M^ ^- E x n (3) — -p — -^

Since the exterior fields of the current distribution in (3) are zero, it

makes no difference what the surrounding medium is as shown in figure 2. Consequently, surrounding objects would not contribute undesirable reflections

because they would not be illuminated by the sources J^ and M^.

Although the source distributions in (2) and (3) are highly idealized,

they show that there is no basic limitation to generating a perfect plane wave

in the near field of sources- The practical limitations arise from the use of
an array of realistic antennas in attempting to approximate the smooth electric and magnetic current distributions of (2) and (3).

3.2. Relationship to scattering problem

One disadvantage of the sources given by (2) is that both electric and

magnetic surface currents are required. It would be useful to have solutions

requiring only electric currents or magnetic currents rather than a combination of both. An electric current distribution might be approximated by an array of dipole antennas, and a magnetic current distribution might be approximated by an array of loops (magnetic dipoles). The desired solutions can be

found by examining the relevant scattering problems.

Consider electric and magnetic fields, E^ and H^^-, incident on a closed,

perfect electric conductor as shown in figure 3a. Only an electric surface

current J^ will be set up on the surface S, and that current distribution will

radiate scattered electric and magnetic fields, ^ and H^. The total field

everywhere is the sum of the incident and scattered fields. Inside S, the

total field must be zero in accordance with Waterman's extended boundary con- dition [11]:

E. +£5 = and H ^- + Hg = . (4)

If we choose the incident field to be the negative of the desired plane wave Uj = - Ej^, H^ = - ]L) , then from (4) the scattered fields inside S are exactly the desired plane wave:

E^ = E and H = H . (5) —s ^ —s -^

If we now remove the incident field and let the scattered current J_g radiate

in free space, the result will be the desired plane wave inside S and scattered fields, l^ and H^, outside S as shown in figure 3b. Thus, the problem

of finding an electric surface current distribution which generates a plane wave inside S is equivalent to solving the problem of plane wave scattering by

the same shape. Analytical solutions are available for plane wave scattering
by simple shapes [12] such as spheres and cylinders. Numerical methods are required for general shapes, and integral equation solutions [13] are typically utilized.

To obtain a solution to (2) with only magnetic surface currents, it is

possible to pursue the same method for a perfect magnetic conductor. An

alternative is to apply duality [10] directly to the electric current configuration in figure 3b by replacing the fields and sources as follows:

^ —p' -s —s

'

H ^ - E , HI ^ - E , (6)

-p --p -s -s

and J^ > M^ . -^ —

Thus, magnetic currents N[g on S will produce a plane wave inside S, but the

polarization will be orthogonal to that of the plane wave produced by J^ in

figure 3b. If the scattering solution pictured in 3a is known for both polarizations of the incident field, then the dual magnetic current solution can be

obtained for both polarizations from (5).

A more general solution, where the electric and magnetic currents have a fixed ratio over the entire surface S, can be obtained when the scatterer is

characterized by a surface impedance boundary condition [14] as shown in fig- ure 4a. If L|. and H^^. are the total (incident plus scattered) electric and

magnetic fields just outside S, they are required to satisfy

nx(nx£.)=Znxh[. , (7)

where Z is the surface impedance. Equivalent electric and magnetic currents J^ and M^g can be defined

A A J^ = - n X H. and M^ = n x E. . (8) —s —t —s —
From (7), we can relate J^ and M^:

J. = - 4 n X M, or M, = Z n x J^ . (9) —S L —

——

From (9), we can see that J_5 and N[g are orthogonal and the ratio of their

magnitudes is determined by Z. The case, Z = 0, represents a perfect electric

conductor as in figure 3a, and ^[5 = 0. The case, Z = °°, represents a perfect

magnetic conductor, and J_g = 0. As shown in figure 4b, if we let J_5 and M^

radiate in free space, they generate a plane wave field inside S and scattered

fields outside S. If we let Z equal the free space impedance n, then the

electric and magnetic currents in (9) radiate as a distribution of Huygen's

sources [15]. A Huygen's source is essentially a small portion of a plane wave and can be considered an approximate model for a small aperture antenna,

such as a horn or an open-ended waveguide. It has also been used to model the

probe antenna in the theory of near-field scanning [16].

In general, the scattering problem for impedance surfaces [17] is some- what more difficult than for perfectly conducting surfaces. However, the

planar case is useful for illustrating the significance of the value of Z.

Consider a plane wave incident on an impedance plane (z = 0) as in figure

5a. The incident fields, E •, and H^^^, and reflected fields, Eyg and H^^^, are given by:

E . = - E e^'^^ , H . = - ^e^"^ , yi ' XI T) '

E = - R E e"""^^ , H = ^e"""^^ , (10)

ys ' xs Ti '

R = |-^ , and A = Z/ti .

The equivalent currents are determined from (8) and (10):

Jys = - V 1-0- ^"^ "xs = - ^ Jys • '"'

If we let the currents in (11) radiate in free space, the desired plane wave

fields, E figure 5b;

and H^^ , are radiated in the negative z half space as shown in
^yp-h'''" and H^^ = ^ e^^^ . (12)

In the positive z half space, the reflected fields as given by (10) are radiated. The ratio of the reflected field to the desired plane wave at z = is

given by

E11

Eyp

1 + A • z =

^ - ^

(13)

For A equal to zero or infinity, the reflected field has the same magnitude as the desired plane wave and equal powers are radiated on each side of the z = plane. In other words, an electric current sheet by itself or a magnetic cur- rent sheet by itself radiates equally on both sides. However, a sheet of

Huygen's sources (A = 1) produces no reflected field, and all of the power is

radiated into the test zone (z < 0). Of course, this result holds only for

the infinite planar geometry of figure 5.

The solutions for the source currents in figures 3b and 4b are generally

fairly difficult to obtain because the related scattering problems in figures

3a and 3b are difficult to solve. One special case of practical interest is

when the perfect electric conductor in figure 3a is large compared to a wavelength. Then the physical optics approximation [18] yields the following

simple result of J.:

2 n X H , illuminated area

J = { "^ . (14)

, shadowed area

As shown in figure 6, the near field of J_g can be divided into three

regions. The desired plane wave exists iTi the center, a transition zone exists along the shadow boundary, and a very small field exists outside the

transition zone. The details of the transition zone will be covered in sec- tion 4, but the width d is on the order of /\ z. In the geometrical optics

limit (\ * 0), the width of the transition zone shrinks to zero, and a sharp
shadow boundary exists For nonzero \, the field at large z eventually

evolves into a spherical wave with inverse distance dependence.

3.3- Sampling and scanni ng co nsiderations

Up to this point we have considered only continuous current distributions. If such current distributions are to be approximated by a phased array

of discrete elements then sampling criteria must be considered in order to

determine the number of elements required In order to obtain some rough

estimates, we consider the simple geometry shown in figure 7. The test zone

has width D and is located at a distance R from the array. The array is of

width W, and the elements have an interelement spacing s. We assume that W is

large compared to a wavelength and that the smoothed current distribution can

be approximated by (14). Since n is constant for a planar surface, J_^ is

constant in magnitude, and we are dealing with a uniform array.

If the plane wave is to be scanned in angle, then s must be less than x/2

to avoid grating lobes [4]. In order that the truncated plane wave include

all of the test zone, W must be somewhat greater than D as seen in figure 7.

The number of elements in one dimension N is thus given by

N = W/s > 2 DA . (15)

If we assume the same parameters in the orthogonal planar direction, then the

total number of elements in the array N^ is given by

N^ > 4(DA)2 (16)

For comparison, the total number of elements required for a spherical array

for a test zone of diameter D can be computed as the product of the number of

elevation samples Nq times the number of azimuthal samples N^ [6]. For large

DA, this product is given by

N„ N^ - (uD/\) {2tiDA) = 2ii^(DA)^ • (17)

(p

10
It is not surprising that the number of elements required for a spherical

array in (17) is somewhat greater- than the number of elements required for a planar array in (16). The spherical array allows the plane wave to be scanned

over all angles in elevation and azimuth while the planar array allows scan- ning over somewhat less than a hemisphere. A possible means of increasing the

scanning range is to use several planar arrays with different orientations.

This idea has been discussed for the reciprocal case of near-field antenna

measurements [19], but apparently has not been implemented.

If the plane wave is not to be scanned in angle, then the planar array in

figure 7 is a broadside array, and the elements have uniform phase. In this

case, the interelement spacing s can be increased to a full wavelength before

grating lobes appear. The total number of elements required can then be

reduced from 4(DA) in (16) to (DA) . In practice this number can be

reduced even further by allowing grating lobes outside the test region. In

this case, s can be increased to [18]:

s = \/l + 4R^/(D + W)^ < \/l + (R/D)^ . (18)

p Then N is reduced to:

2

Some of the numerical results of Schoessow [8] show grating lobes outside the

test zone where he has used interelement element spacings greater than a wavelength.

The truncated plane wave produced by the planar array can be scanned in

angle by introducing a linear phase shift to the elements. However, if the

truncated plane wave is scanned too far, it will miss the test zone. By

purely geometric considerations, the maximum scan angle 8^^^^^ is given by [18]:

^.ax - tan-'[Vr] ' '^O'

11
This result is in agreement with the reciprocal problem of near-field antenna

measurements where the far-field antenna pattern cannot be determined for

angles larger than 9^^^ [20]. The result in (20) ignores diffraction effects,

and a term of the order A/R should be subtracted if diffraction effects are

important. If W is increased to infinity in (20), then 6 = nil and the ^ max

plane wave can be scanned over a hemisphere In order to cover a full sphere,

the array must surround the test volume.

It has been noted that an alternative to actually building a phased array

is to take a single element and step it through all of the element positions

[21]. In this way the plane wave is never actually generated, but the

response of the EUT is the sum of the separate responses properly weighted to

yield a plane wave response. This method has the advantage of not requiring

the potentially large number of elements as indicated by (16) and (17). Also,

the requirement for accurate control of the amplitude and phase of a large

number of elements is eliminated. The difficulty with this approach is that

the sum of the responses is equal to the total plane wave response only if the

EUT is linear, and this is not always true for strong fields [3]. The other

point in favor of actually constructing the array is that the EUT response is

obtained directly without further processing. Another interesting possiblity

of the constructed array is that it could yield the far-field radiated emissions pattern of an arbitrary source EUT if the array is operated in the

receiving mode.

4. Hear fields of smooth current distributions

Realistic near-field arrays of finite extent suffer from truncation

effects at the edges of the array as indicated in figure 6. The simplest

model for studying truncation effects is a smooth physical optics current

distribution as given by (14) which is abruptly truncated at the edges of the

array. For simplicity, we consider only planar current distribution, but the

asymptotic method of analysis could also be used for curved surfaces. In

section 4.1 we consider a simple two-dimensional geometry, and in section 4.2 we consider a more realistic three-dimensional geometry.

12
4.1. Two-dimensional geometry

We consider a semi-infinite surface current shown in figure 8. The electric current distribution J is given by (14):

where

2 z X H^

J.{x') =

{ -^

H = H e

-i) -0

- JKi • P

X = X

z = (21)

x' >

k. = k k. k. = X sin e. + z cos e.

£ = X X + z z ,

and 0^- is the incidence angle defined in figure 8. The electric and magnetic

fields are most easily derived via the magnetic vector potential A [10]:

A =

J J(x') G(|p - p'l) dx' , (22)

where

G(|P -£'|) =43-h|,^^ (k|p - P'|) ,

£' = X X' ,

(2) and H is the zero order Hankel function of the second kind.

The integral in (22) is difficult to evaluate in general, but for sufficiently large values of kz it can be evaluated by asymptotic methods [22].

The total vector potential A then can be written as the sum of two terms:

A = A3 + A^ . (23)

Ag is the plane wave term which arrives via the direct ray from the stationary

phase point as indicated in figure 8. It is derived in Appendix A, and the

result is:

13
-jk{x sin e. + z cos e.)

^^^•"^

1, e. - e^ > u{e. - e ) = {

' ^ 0, G. - 0g <

2 2

sin 9 = x/p and p = /x + z .

Because of the unit step function U in (24), A^ exists only in the illuminated

region to the left of the dashed line in figure 8. A^ can also be thought of

as the geometrical optics field which drops abruptly to zero to the right of

the dashed line (shadow boundary) in figure 8. The edge diffraction term Ap

is also derived in Appendix A, and the result is

-j(kp-J) 2

A ~ z X H ^ F (v) e^" sgn(e^ - e.) , (25) "^ "° jk cos 9 /? " ® ^ e

^^^^^ + 1, 9 - 9. > sgn(9 - 9 ) = { ^ ^ ^ ^ - 1, 9 - 9. < e 1

The Fresnel integral F and the argument v are defined in Appendix A. Both Ap

and A^ are discontinuous at the shadow boundary (9^ = 9^.), but the sum of the

two terms is continuous, as it should be.

At points which are not near the shadow boundary, the asymptotic form of

the Fresnel integral [22] can be used in (25), and Ag simplifies to

e'^'^P sgn(9^ - 9.)

ift = z X H —

^ ^

. (26) ~® ~° jk /27tjkp (sin 9 - sin 9.)

s 1

14
In the illuminated region (Gg < 9^), the edge diffraction term A^ will

interfere constructively and destructively with the desired plane wave term A^

and will cause ripples in amplitude and phase. Such ripples are very evident

in the results of Bennett and Schoessow [5] and a Fourier analysis of the

total field by Schoessow [8] has determined that the error field arrives

primarily from the edges of the array. From (24) and (26), the ratio of

undesirable edge diffraction term to the plane wave term is found to be

^

cos e.

^

. (27)

/2ukp I sin 9^ - sin 9^.

|

The (kp)" '^ dependence in (27) is characteristic of edge diffracted fields

[22]. The expression in (27) can be used to determine the width of the transition zone from small field strength to full plane field strength as indicated in figure 6. If we assume that 9^ and 9^. are small and we require |Ag/Ag|

be small outside the transition zone, then from (27) it follows that

/2i]cp |9^ - 9^. I > 1 . (28)

The width d of the transition zone is approximately the arc distance

2p|9g - 9J. From (28), d must satisfy

d = 2p|9 - 9.| > 2/X^ . (29)

Thus, the width d of the transition zone is somewhat greater than /Xp", and

this result is in agreement with that of Turchin and Tseytlin [18].

The actual field components, H_ and E_, are obtained by taking the

following curl operations on A:

H = V X A and

E = J^ V X V X A ,

(30)

15
where e is the permittivity of free space. The details of the curl operations

are carried out in Appendix A. In figures 9-14, we show some numerical

results for the electric field as a function of x. The main purpose of these

results is to illustrate the behavior of the field in the transition zone. The Fresnel integral, which is required for the evaluation of the edge

diffracted fields, was evaluated by power series for small argument and

asymptotic series for large argument [23].

In figures 9-12, the electric field is polarized in the y direction, and

Hq is given by

H = H. (x cos e. - z sin e.) . (31)

In figure 9, the magnitude of the electric field is shown at two distances

from the planar surface current. The distance z = 30 X was chosen to match

the example in the paper by Bennett and Schoessow [5]. In both cases, the

field decays smoothly in the shadow region (x > 0) and exhibits damped oscillations in the illuminated zone (x < 0). As predicted by (29), the transition

zone is wider for larger z. Phase results for the same cases are shown in

figure 10. Results for magnitude and phase as a function of incidence angle

Gj are shown in figures 11 and 12. Again, the geometrical optics results are

shown as dashed lines. The transition zones are seen to widen for oblique

angles.

In figures 13 and 14, the magnetic field is polarized in the y direction,

and IHq is given by

Thus, the electric field is polarized in the xz-plane. Figure 13 shows magnitude results for normal incidence (6^- = 0). The existence of a small E^

component is due to diffraction. Figure 14 shows results for various values

of 9^-, and a larger value of [E^l exists for oblique angles.

In figures 9, 10, and 13 and in (27), it is clear that the diffracted

field decays rather slowly and causes undesirable ripple in the field in the

16
illuminated region. The reason for the slow decay is that the edge diffracted

field decays only as (kp)"-*-'^ as seen in (27). The magnitude of the edge

diffracted field can be decreased by tapering the current at the edge of the

array (x' = 0). To see this, we consider first the untapered result. The

integral form of the edge diffracted field Ag is

A^ = - / f(x') eJ'^9^^ ' dx' , (33)

where f_ and g are defined in (A2). For the case with no stationary points

near zero or in the interval to <^y (33) can be integrated by parts to obtain

[22]:

Jkg(O)

A^ ~ 1(0) f^g^ . (34)

If we replace the abrupt transition in the current at x' = with a linear

taper as in figure 15, then x' near zero jF_ in (33) is replaced by a tapered

function f_^:

l^W) = f(x') (-x7L) , (35)

where L is the approximate length of the taper as shown in figure 15. The

tapered current is slightly smoothed at x' = L to avoid a discontinuity in the

derivative. Since jfj.(0) is zero, the leading term in the asymptotic evaluation of (33) now is [22]:

- fKO) eJ^9(0)

where

"^^

(jk)'^ [g'(0)]^

f|(0) = - f(0)/L.

Equation (36) can be written in the following form:

^t ~ jkLg ' (O)

^ ' (37)

17
where g' (0) = sin 9^ - sin 9^-. Thus, the diffracted field can be reduced by

tapering if kL is large. This effect will be shown quantitatively in section

5.

4.2. Three- dimensional geometry

In this section we treat the more realistic case of physical optics cur- rents on a finite surface. The asymptotic methods which we will use to

evaluate the field are applicable to curved surfaces, but for simplicity we

will treat only planar surfaces. The theory has much in common with some of

the earlier work on diffraction by apertures [24,25].

Consider first a general planar area as shown in figure 16 where the

surface currents are given by the physical optics expression in (14). For

high frequencies, the electric field (or the magnetic field) can be expanded

in an asymptotic series of the following form [25]:

i - is ^ Ee " ^ • '38)

where the three ray paths are shown in figure 16. For points not near the

shadow boundary, the three terms have the following dependence on wave number

k [25]:

E a k° , Fa k""^^^ , and E, a k""^ . (39) —

s

"^e —

c

E_5 is the stationary phase contribution which yields the desired plane wave

field. E^ is an edge diffraction term which emanates from a stationary phase

point on the edge of the surface and, in some cases, there could be multiple

stationary points on the edge. E^ is a corner diffraction term which emanates

from a point where the tangent to the surface is discontinuous. For large k,

E_^ is usually negligible compared to E^, and we will not include E_^ in the

field calculations in this section. The main objective is to determine how

much degradation of the desired field E^ is caused by the leading undesirable

term E^.

18
We begin by setting up the general integral for the vector potential A(r^)

in terms of the surface current J(r'):

^ j(7') e-J'^ll-Il'l

We assume that the surface S is located in the xy plane and that J_ is the

physical optics current given by (14):

2 z X H (r' ), r' on S

where

H (r') = H e ^ and k. = k. k . -p — -0 -11

We can then write _A in a form convenient for asymptotic evaluation:

A(r) = // f e^'^^ dS , (42)

S

where

z X H

f = -^

2Tc|r - r"

I

and

Integrals of the type in (42) can be evaluated as the sum of two terms [22]:

A

~ A + A . (43) — —s —

e

A_s is the stationary phase term which yields the desired plane wave field

within the illuminated region.

19
In order to evaluate the edge diffraction term _Ag , we write (42) in the

following form:

L a{u«) .. A =

/ /

'^ f e-^'^y du, du^ , (44)

where u-^, U2, L, and a are defined in figure 17. The edge diffraction contribution Ag is essentially the negative of the integral in (44) for u^ > a:

*e'

- / / f e^'^9 du. du,

a(u2) ^

"^ [45)

The Uj^ integration in (45) can be cast into a Fresnel integral form as in (25)

for the one-dimensional integral. The U2 integration is done by stationary

phase, and the general result is [22]:

A,~±

.,...2

feJ^Se'J^ /ttt^

^"l

F^(v) .

"2^2

,- 1/2

J T sgn(a, „ ) "2"2

Ui = «(u2e)

"2 = "2e

(46)

Vi^'^"2e^' "2eJ <0

where

V = /.

"l"l

20
In (46), subscripts u-^ or U2 imply differentiation, and all quantities are evaluated at U2 = Ug2 which is the stationary point determined from g^^

=0.

If there is more than one stationary edge point, then (46) becomes a summation

of the contributions from each point.

We now consider the specific example of a rectangular surface as shown in

figure 18. For simplicity, the plane wave is scanned only in the xz-plane,

and the propagation direction k. is given by

k^. = z cos e^. + X sin 9^. , (47)

where 9- is the incidence angle as shown in figure 18. Since the perimeter is

made up of four straight segments, there can be up to four stationary edge

points. The asymptotic evaluation of the fields is tedious, and the details

are given in Appendix B.

Results for both Ey polarization and Hy polarization are derived in

Appendix B, but in the following numerical examples we consider only Ey

polarization. The calculations have shown that polarization effects are fairly minor for cases where a, b, and z are large compared to X. For our

numerical results, we choose a/X = b/X = 30. These dimensions correspond

rather closely to those in the numerical results of Bennett and Schoessow

[5]. In figure 19, we show the magnitude of the electric field along a center

cut (yA = 15) at two distances from the current sheet. The geometrical

optics approximation is shown for comparison. The field quality is seen to

deteriorate at the larger distance, and this effect is predicted by the

angular dependence of the edge diffraction terms. The magnitude of the ripple

at zA = 30 is somewhat larger than that computed by Bennett and Schoessnow

[5] for a synthesized array, but their analysis involved only a one-dimensional array. Our two-dimensional current sheet is more physically realistic

because it includes diffraction from edges in both x and y directions. It is

21
the constructive and destructive interference of the four edge-diffracted rays

with the plane wave that produces the complicated interference patterns in

figures 19-21.

Figure 20 shows the effect of scanning the plane wave. The quality of

the plane wave is seen to decrease as the scan angle 9^ is increased, and the

field is no longer symmetrical about the center of the plane wave zone. Figure 21 shows the magnitude of the electric field on a cut off the center

line (y/X = 5). The ripple in Ey is slightly greater than that along the

center line in figure 19. A small E^ component is produced by diffraction

from edges 1 and 3.

Other calculations have produced similar results, and the main problem is

the large ripple caused by edge diffraction. As indicated in the previous

section, edge diffraction is strong because of the abrupt truncation of the

current at the edges. This edge diffraction could be reduced by tapering the

current near the edges as demonstrated in (37), but taper was not investigated

for the geometry in figure 18. It is investigated in the following section on

arrays.

5. Near fields of arrays

The numerical results for smooth current distributions in section 4 show

the effects of truncating the radiating surface. In this section, we analyze

discrete arrays in order to study the effect of element spacing. This effect

is analogous to sampling in the reciprocal problem of near-field scanning

[20]. Since we are not yet prepared to consider the details of the array

elements, we choose fairly simple array elements for the analysis. Hertzian

electric di poles are the array elements in section 5.1, and Huygens' sources

[15] are the array elements in section 5.2. In each case, the effect of

tapering the amplitude of the elements at the edge of the array is studied,

and the amplitude of the ripple field in the test volume is substantially

reduced.

22
5.1. Electric dipole array

In this section we analyze a planar array of electric dipole elements.

For simplicity, we consider Hertzian dipoles, but there is little difference

in the far-field pattern of Hertzian dipoles and finite dipoles of length up

to a half wavelength [26]. It is assumed that the test zone is at least a wavelength from the array so that only the far-field terms of a given element

are required. However, the test zone will normally be located in the near

field of the entire array.

The geometry of the rectangular array is shown in figure 22. The dipole

elements are located at x^-, y^ which are given by

where

X = Ax'(i - 1/2) , i = 1, 2, ... , N^

yj = Ay'(j - 1/2) , j = 1, 2, ... , N^

Ax' = a/Nj^ , Ay' = b/Ny

(48)

and the total number of elements is Nx^y. The fields of a single element (E^j,-

Hj^a) of arbitrary orientation in the xy-plane are derived in Appendix C. The

total fields of the array (E^, H) are given by double summations:

^x \

i=l j=l ^J

(49)

i=l j=l ^^

where E^^ is given by (C4) and ][^- • is given by (C3)

23
For simplicity, we consider the magnetic field of the plane wave to be

polarized in the x direction and propagating in the positive z direction:

Hp = X Hq e"-^"*^^ . (50)

From (14), the smoothed surface current density is

J^ = 2 2 X Hp = ; 2 H^ . (51)

The dipole moment p^-j is the surface current density times the area of the

patch as indicated in figure 22:

p.. = Ax' Ay' J = y 2 Ax' Ay' H^ . (52) —1

J

s o

The fields of uniform arrays have been calculated from (49), (52), (C2),

and (C3), and some results are shown in figure 23. The parameters, a = b =

30 X, were chosen to match the examples in section 4.2 and the earlier work of

Bennett and Schoessow [5]. Only half of the center cut for y/X < 15 is shown

since the fields are symmetrical about yA = 15. For an element spacing. Ax'

= Ay' = 1.5 X, large oscillations exist, and the quality of the field is

clearly not adequate. This is to be expected because the element spacing

requirement discussed in section 3.3 has been exceeded. The results are much

better for a spacing of one wavelength where most of the ripple is now caused

by edge diffraction which cannot be reduced simply by reducing the element

spacing.

The analytical result in (37) indicates that the edge-diffracted field

can be reduced by tapering the amplitude of the elements near the edge of the

array.

To explore the effectiveness of tapering, we utilize a linear taper

function tj^ defined by;

24
d-1 '

^' ^ T

t^(i, T, N) =

j 1 , T < i < N + 1 - T (53)

N + 1 - i ,• ^ M ^ 1 T ~TV1— ' 1 > N + 1 - T

where H is the total number of elements in a row and T is the number of

tapered elements on each end of the row. For example, for T = 1, tj^ = 1/2 for

i = 1 or N, and t^ = I for all other elements. For T = 0, t^ equals unity for

all elements. The linear taper function can be used in both the x' and y'

directions so that all edges of the array have the same taper. Thus, for the

tapered array, we replace p^-^ of (52) by:

... =y 2AX' Ay' H^ tjjd, T^. N^) yj, T^, N^) . (54)

The use of two taper parameters, T^ and Ty, in (54) allows for the possibility

of different tapers in the x' and y' directions, but in all our calculations

we assume T^ = Ty = T.

In figure 24, we examine the effect of the taper parameter T when the

interelement spacing is a half wavelength. The untapered result (T = 0) is

similar to the result in figure 23 when the interelement spacing is \ except

that the ripple has been slightly reduced. When T is increased to 2 and 6 in

figure 24, the ripple is seen to decrease markedly, and the field is quite

smooth near the center of the array. The main disadvantage of the taper is

that the width of the test zone is narrowed as the number of tapered elements

is increased. Thus, the best choice of T depends on the quality of the plane wave and the size of the test zone required.

The number of elements required for the array in figure 24 is 3600 (N^ Ny

= 60 X 60), and this number is so large because the electrical size of the

test zone is so large. It is only at microwave frequencies where the test

zone must be electrically large for EUT's of typical dimensions (on the order

25
of meters). Consider the case in figure 25 where the array dimensions have

been decreased by a factor of 10 (a = b = 3 \) . In this case, the number of

array elements is reduced to 36 (Nj^ Ny = 5 x 6) , but the field shows an amplitude taper across the test zone. (The flat geometrical optics field is

shown for comparison.) In this case, the linear taper does not really help

the problem. It is likely that some other type of nonuniform excitation would

provide an improvement, but it is not obvious how to find the best element

excitations. A general numerical method is probably required, and a promising

method is described in section 6. The preliminary conclusion is that the

linear edge taper is useful only for reducing the edge-diffracted field of

electrically large arrays.

5.2. Huygens' source array

In this section we analyze a planar array of Huygen's sources. A

Huygen's source [15] is essentially a small portion of a plane wave, and it is an approximate representation of a small aperture antenna [16] such as a horn

or an open-ended waveguide. It has a broad pattern in the forward direction

and a null in the backward direction. For this reason, it is a better aperture antenna model than single electric or magnetic di poles which have equal

radiation in the front and back directions. The directivity of a Huygens' source is 3 (or 4.77 dB) which is twice that of a a Hertzian dipole, but is

less than that of an open-ended rectangular waveguide [27].

Figure 26 shows the crossed electric and magnetic di poles which constitute a Huygen's source. For simplicity, we assume that the plane wave is

propagating in the positive z direction and the electric field is polarized in

the X direction:

P

" P -jkz ^ = ^ ^ox ^ ' (55)

H = ;

^

e-J"^^ . -P -^ T)

26
om the equivalence principle, the electric and magnetic dipole moments, ^^-j

and jn^-j, are [15]:

£.. = -iYAx' Ay' , (56)

^n = - ^ ^ox ^^' ^y' •

The far fields of a Huygens' source have been given in spherical coordinates

[15], but in Appendix D we derive the Cartesian components.

The individual elements are again located as indicated in (48) and in

figure 22, The fields are double summations:

^x ^

i=l j=l ^J

(57)

H = I I H . , i=l j=l ^"^

where E_jj and H^j are given by (D6). In figure 27, the normalized electric

field strength on a center cut is shown for two different interelement spacings. As with the dipole array, a spacing of 1.5 X is too large, and the

field has unacceptably large ripples.

In order to examine the effect of edge tapering, we again apply a linear

taper to the edge elements. Thus, for the tapered array, we replace £jj and

nijj of (56) by

-^ij = - ^ T^"' ^^' V^' ^x' \^ VJ' Ty' V ' (58) m.. —^E^^Ax' Ay' t,(i, T^, U^) t,(j, T^, V '

27
where Tj^ is given by (53). For the example in figure 28, T^^ = Ty = T. The

decrease in field ripple is comparable to that for the tapered dipole array in

figure 24. The case for T = represents a uniform array and, in all cases,

the interelement spacing is \/2. This spacing provides a slight decrease in

ripple for the uniform array as compared to the case for a spacing of X in

figure 27.

In summary, the use of edge tapering for an array of Huygens' sources decreases the edge-diffracted field as it did for the array of dipole

sources. Thus, a large array with tapering provides a high quality field over

a large area, but requires a large number of elements. The improvement of the

field quality in going from a uniform to a tapered array might have implications for the reciprocal problem of near-field antenna measurements [18] when

only a finite scan area [25] is used. Rather than abruptly truncating the

scan, some sort of tapering process at the edge of the scan area [28] might be

helpful

6. Hear- field array synthesis

The results for tapered arrays in section 5 indicate that large improvements over uniform arrays are possible. The linear edge tapering which was examined in section 5 was based on reducing the edge-diffracted field according to the analysis in section 4.1. The question naturally arises as to

whether other types of tapering or completely different source distributions

might yield better results. The goal of this section is to develop a general

method of near-field array synthesis which will yield an optimum source distribution.

The subject of antenna array synthesis is well developed for conventional

far-field applications [29,30], but near-field applications raise new difficulties. First, the convenient Fourier transform relationship which exists

between the source distribution and the far-field pattern is not available for near fields, particularly for arbitrary geometries. Second, far-field synthesis involves approximating the field over a surface of infinity whereas our near-field application requires a specified near field within a test volume.

28
In section 6.1, a uniqueness theorem is derived which allows us to

specify the desired fields on the surface surrounding the test volume. In

section 6.2, this theorem is used to derive a synthesis procedure based on surface fields rather than volume fields. The general numerical synthesis

method is based on earlier work by Mautz and Harrington [31] and does not

require the far- field Fourier transform relationship. The method minimizes

the mean square error in the fields and imposes a constraint on the source

norm. The source norm constraint is useful in keeping the fields small outside the test volume. Section 6.3 applies the method- to a planar array of

line sources with good results.

6.1. A new uniqueness theorem

Uniqueness theorems tell what surface field information is required to

uniquely determine the fields within the enclosed volume. The classical

uniqueness theorem for time-harmonic fields [10] states that the tangential

electric (or magnetic) field uniquely determines the fields in the enclosed

volume. Unfortunately, this theorem breaks down for lossless media, such as free space. A more general uniqueness theorem from Muller [32] states that

the tangential electric and magnetic fields are sufficient to uniquely specify

the fields in the enclosed volume. The proof of the theorem requires continuously differentiable constitutive parameters, but does not require loss.

Thus, it is applicable for our free space geometry.

In near- field synthesis problems, it is not as convenient to specify both

the tangential electric and magnetic fields as it would be to specify a single

quantity. A single quantity which has been found to uniquely determine the

fields in the enclosed volume is the tangential vector F which is defined as:

F = E^^^ - T) n X H , (59)

where E_^an is the tangential electric field, r\ is the free-space impedance,

and n is the inward unit normal as shown in figure 29. A uniqueness theorem

for F can be stated as follows. "The fields in a source-free region are

29
specified by F^ over the boundary." This theorem can be stated in the following equivalent form. "If F^ = on the surface of a source-free region, then

E^ = H^ = within the enclosed region." The meaning of the second statement is

that if there is no change in £_, then there can be no change in £ and H_ throughout the enclosed region. These are the difference fields which

Harrington [10] utilized, and if they are zero, then only one unique field

solution exists.

The proof of the second form of the theorem goes as follows. First,

scalar multiplication of the equation, f^ = 0, by its complex conjugate,

£* = 0, yields

(itan - 11 n X H) • (itan - n n x H*) = . (60)

If the scalar multiplication in (60) is carried out term by term, then

l^tanl^ + Ti^ (n X H) • (n x H*) - ti[e^^^ • (n x H*) + E*^^ • (n x H*)] = .

(61)

Using some vector indentities, (61) can be written:

lE^^j2 + n^ |H^^j2 ^ 2 r, R^n • (E x H*)] = , (62)

where Rg indicates the real part. We now integrate (62) over the closed sur- face S:

f^ d^tan'^ ^"^^

l"tanl^5 dS + 2 n R^ fcfr n • (E x H*) dS = . (63)

The second integral in (63) is the real part of the Poynting vector integrated

over a closed surface. Since the interior region is source free, this integral must be greater than or equal to zero [10]:

30
2ti R ^ n • (E X H^*) dS > (64)

^ S

All quantities in the first integral are either positive or zero Thus, the

only way for (63) to be satisfied is for both E^^^ and jij.^^ to be zero:

itan = Mtan = °" S. (65)

To this point, we have shown that if F^ = 0, then Lj.^^^ = Jitan

"^ on

S. Muller [32] has shown that (65) implies that E. = H

= inside S. Thus

the proof is complete.

Although we initially assumed that r] in (59) was the impedance of free

space, the proof of the uniqueness theorem requires only that r\ be some positive quantity. For the special case of a lossless medium, the second

integral in (63) is identically zero:

R (^ n . (E X H*) dS = . (66)

^ S

In this case, all that is required is that r\^ be positive Consequently, t) could be any positive or negative real quantity (but not zero). This freedom

to choose Ti positive or negative real corresponds to the positive or negative

real a in the combined field integral equation of Mautz and Harrington [33].

For our near-field synthesis procedure which follows^ we take ti to be the

free-space impedance in order to equalize the importance of the electric and

magnetic fields In order to illustrate the utility of the function JF^, an example is shown in Appendix E where ^^an ^^^ iitan^ ^^ ^^^ sufficient to

determine the interior fields, but F^ is sufficient.

An interesting feature of the function F_ is that it might have significance for measurements of the fields in a region. One way to determine the

fields in a source-free region, as in figure 29, would be to probe the fields

directly throughout the volume. For an accurate field characterization, this

31
would require on the order of (L/X)^ measurements of the three components of a vector field {E_ or H_) . L is some linear dimension of the region. In

prinicple, the same field information is available from measurements of F^ over

the surface S. This would require only on the order of (L/\)^ measurements of

the two components of F^. Each component of F^ involves an orthogonal component

of E_ and H_, and a special loop antenna probe [34] could obtain these two

components simultaneously.

6.2. Least squares analysis

The general configuation for the synthesis problem is shown in figure

30. The sources produce fields within the test volume, but we choose to work

with the tangential fields on the surface S. A symbolic relationship between

the sources s and the field F on the surface S can be written:

T s = F, (67)

where T is a linear operator. We assume that T is known which means that we know how to compute the fields given the sources (the forward problem). We

follow the method of Mautz and Harrington [31], but the details and some of

the notation are different because we are working with near fields and arbitrary geometries.

The problem is made discrete from the start and cast into matrix form:

[T] [s] = [F] . (68)

[s] is an N X 1 source column matrix, and the scalar elements s,- are typically

input currents or voltages to antennas:

rS^n

[S] Nxl(69)

32
[F] is an M X 1 column matrix, and the elements are vectors which are orthogonal to n:

[F] Mxl

rF^n

f2

'fw

(70)

where

F_. = u^ F^. + U2 F2^.

JK A u, and Up are tangential vectors as shown in figure 30

trie and magnetic fields F^^- is given by

In terms of the elecii - (Etan - ^ " ^ H)

;th

r = r.

r^j is the position vector of the i^" point as shown in figure 30.,

M X N rectangular matrix of vectors.

(71)

[T] is an

[T] MxN

l-Ill I12 •

-

• IlN

I21

•

-Imi . . Imn-*

(72)

The element J^j is the vector Fj produced by source Sj = 1 when all other

source elements are zero

In the synthesis problem, the right side of (68) is set equal to the

desired column matrix;

[T] [s] ^ [P] , (73)

33
where

Pi = 'ip tan - ^ " X Hp'

j: = £,-

and E_Q and FL are the plane wave fields given by (1). The method is actually

more general, and [P] could be a column matrix of any desired field values.

Since [P] is an M x 1 column matrix of two-component vectors and [s] is an

N X 1 column matrix of scalars, (73) actually represents 2M equations in N

unknowns. Normally 2M is greater than N, and (73) can only be solved approximately. The least squares solution to (73) is [31]:

[s] = [CT]* [T]]"^ [T]* [P] , (74)

where the tilde denotes transpose. Matrix multiplication in (74) and in the

following equations has the usual meaning. Whenever the elements multiplied

together are vectors, the dot product is understood. The solution for [s] in

(74) minimizes the quantity:

M

CFl - [P].2 = I \F^ - P^f , (75)

m=l

where [F] is given by (68). In some cases, (74) provides a useful solution;

but in other cases, the norm of [s] is large. Such solutions usually produce

large fields outside the test volume where small fields are desirable.

In cases where the source norm is large, a more useful solution can be

obtained by minimizing

£ = ll[F] - [P]il^ (76)

with a constraint on the source norm

il[s]ii^ < C , where ,, (77)

iiEsiii^ = I is,r • n=l "

34
C is a positive constant to be chosen. The simultaneous solution to (76) and

(77) has been derived by Mautz and Harrington [31], and we give only their

final results.

The solution requires the eigenvalues \^ and the eigenvectors M^ which

sati sfy

[T]* [T] M. = \. [<D]. . (78)

The matrix [T]* [T] is Hermitian, and efficient computer codes [35] are available for computing the eigenvalues and eigenvectors. The constrained solution

for [s] is [31]:

N C.

1=1 1

where (79)

C. = Mt [T]* [P] .

The Lagrange multiplier a satisfies

N ICJ^

C -

I

'- ^

= . (80)

i=l (\. + a)"^

(80) is a monotically decreasing function in a which is real and positive, and

the equation is easily solved numerically for a. If a is set equal to zero,

then (77) gives the unconstrained solution for [s] which is the eigenvector

expansion of (74). It has been shown [31] that as C is decreased, both a and

the error e^ increase. However, it is often the case that large decreases in

the source norm can be achieved with only small increases in the error e^,.

35
6.3 Line s ource ex ample

In order to demonstrate the effectiveness of the synthesis technique, we consider the two-dimensional problem shown in figure 31. An array of ydirected electric line sources is located in the z =. plane, and a rectangular test volume is centered about z = Zq on the z axis. All quantities are independent of y The two-dimensional geometry was chosen for simplicity, but

the line source array problem is not entirely academic. For example, Bennett

and Muntanga [7] have used a slotted waveguide to approximate a line source in

plane wave synthesis, and their measurements show that the fields are nearly

uniform in the y direction.

We consider only the broadside case where the plane wave is propagating

in the positive z direction. Thus, the desired plane wave fields, E^ and HL^, are given by

E = y E and % = >< "px

where

Sy = ^0 ^ jkz

(81)

and %x = - 'Eo/'i' ^"'''

For the geometry in figure 31, the elements of the matrices [T], [P], and [F]

are y-component vectors rather than two-component vectors. Thus we have a scalar problem, and the matrix [P] is.

Pp

[P] =

Pi

(82)

M

36
where

A A P. = [E - Ti y • (n X x) H ]

1 py

-^ px

X = X.

By using a vector identity and (81), we can rewrite P-,-:

A A P. = E {1 + n • z) e ^ 1 (83)

The value of n • z depends on which side of the test volume the point is

A A A A located. On side 1, n • z = 1, on side 3, n • z = -1, and on sides 2 and

A A 4, n • z =

The array consists of 2N line sources with equal spacings d as shown in

figure 31- Since the desired field is symmetrical in x, the array is symmetrical. The j'' element is located in the z = plane at x'- given by

x^ = ± d(j - |-) , j = 1, . M . (84)

(The integer j should not be confused with imaginary j = /-T ) The source matrix [s] consists of the unknown line currents:

,-I,^

[s] = (85)

37
The [T] matrix i an M x N matrix of scalars:

[T] =

^11 ^12

hi

Ml

Tin-

'mn.

where (86)

'u [^yj Ti y (n X Hj)]

= X.

= 1

Eyj and H^ are the electric and magnetic fields produced by unit line sources

at xj = + d(j - 1/2)

Appendix F

The specific forms of Eyj, ^Lj ' ^"^ ^ij ^^^ given in

A computer program was written to compute the least squares solution for

the source matrix [s] with constraint by (79) and without constraint by

(74). The Lagrange multiplier a required in (79) was determined from (80) by

Newton's method [36]. The solution of (74) was obtained from the QR decomposition [37] of [T]. As a check on the two methods, the solution of (79) with

a = was compared with the solution of (74) and, in all cases, the [s] matrix

agreed to at least 5 significant figures

The solution of (80) for a requires a constraint parameter C as defined

by (77). A gbod value for C can be obtained from the physical optics approximation for [s]. From (14) and (81), the smoothed current density J^ is given

by

38
J = - y 2 E^/Ti . (87)

For an interelement spacing of d, this results in an [s] matrix of uniform

current elements I,,:

[s] =

I ^

u

"v

, I, = - 2 d E^, (88)

If we require that the norm of the synthesized solution be no greater than

that of the uniform solution in (88), then C is given by:

C = N |IJ (89)

Other values of C might be more appropriate in some cases, and the computer

program allows any value of C However, all of the numerical results in this

section utilize the value given by (89).

The synthesized solution for the currents was found to be insensitive to

the number of surface points M (equals the number of equations) so long as the

points are spaced less than \/2. The points are equally spaced in u over the

range from to A + B, and the spacing Au is given by

Au = (A + B)/M (90)

The locations of the points are

u^- = Au (i - 1/2) , i = 1, 2, . . . , M (91)

39
For cases where the fields are not symmetrical (for example, a scanned plane

wave), the points must be distributed over the entire perimeter.

The numerical results in figures 32 - 40 apply to a square test volume

which is four wavelengths on a side (A = B = 4 \). In figures 32 - 34, the

spacing d = \/2, the number of elements 2 N = 12, and Zq = 4 \ The

unconstrained currents in figure 32 are quite large and variable, whereas the

constrained currents are much closer to those of a uniform array. The value

of the uniqueness function Fy as given by (71) is shown over half the

perimeter in figure 33. The curves are discontinuous at u = A/2 and u = B + A/2 because n is discontinuous at those points. For comparison, we define a normalized squared error e^;

M

2

ll[F] - [P]ll^ illJ!^ ' ^n ll[P]ll M

I

^^

I |Pil'

i=l ^

(92)

where Fy^ is the actual value of Fy at the i''" point and P.,- is the desired

plane wave value as given by (83). For the uniform array, e^ = 6.24 x 10"^,

for the constrained array, e^ = 4.35 x 10"^; and for the unconstrained array,

£^ = 5.52 X 10"^. The unconstrained array looks the most attractive from the error standpoint, but the difficulty with the unconstrained array is shown in

figure 34 where the electric field is actually larger outside the test volume

than inside. This feature is undesirable both because of wasted power and

because of the possibility of reflections from objects outside the test

volume. The constrained array has a fairly uniform field in the test volume,

and the external field falls off at about the same rate as the field of a uniform array- The results for other values of z and for the magnetic field

are similar to those in figure 34.

In figures 35 and 36, the interelement spacing has been increased to

3 ^/4, and the number of elements 2N has been decreased from 12 to 8 such that

the width of the array is about the same For this case, the unconstrained

array satisfies (77) and the constraint has no effect. Thus, only one solu40
tion for the current is shown in figure 35. For the uniform array, the

normalized error Sp = 8.87 x 10"^; and for the synthesized array, e^ = 1.57 x 10"^. As expected, the errors have increased as the number of elements has

been decreased from 12 to 8. The center cut of the electric field is shown in

figure 36, and the field of the synthesized array decreases even faster than

that of the uniform array outside the test volume. This is because the source

norm is smaller for the synthesized array

In figures 37 and 38, the interelement spacing d has been increased to \,

and the number of elements 2N has been decreased to 6. In this case, the

unconstrained solution again satisfies (77), and the constraint has no effect. The solution for the current is shown in figure 37. For the uniform

array, the normalized error e^ = 0.278; and for the synthesized array, e^ = 0.110. Thus, the synthesis procedure reduces the error, but both errors are fairly large. The center cut of the electric field is shown in figure 38, and

the quality of the field is poor. Larger values of d have been found to yield

even larger errors in the fields.

Part of the reason that the field quality is poor for d = X in figure 38

is that the test volume is located close to the array (z^ = 4 X). In figures

39 and 40, the test volume has been moved farther from the array (Zq = 8 X).

In this case, the constraint has a small effect as indicated by the two cur- rent solutions in figure 39. For the uniform array, the normalized error e^ = 8.81 X 10'^; for the constrained array, e^ = 9.22 x 10""^; and for the

unconstrained array, e^ = 5.85 x 10"^. Thus the errors have been decreased

significantly by moving the test volume farther from the array. The center

cut of the electric field is shown in figure 40.

In figures 41 - 43, we consider a smaller test volume located closer to

the array (A=B=Zq=2X). The solutions for the currents with and without

a constraint are shown in figure 41. For the uniform array, the normalized

1 9

error e^ = 1.00 x 10"-^; for the constrained array, e^ = 1.57 x 10"'^, and for

the unconstrained array, e^ = 9.10 x 10"^. The corresponding uniqueness

function Fy is shown over half the perimeter in figure 42. The electric field

41
on a center cut is shown in figure 43, and, as in previous cases, the

constrained synthesis field has a more rapid decay than the unconstrained

synthesis field outside the test volume.

Other calculations have produced similar results The uniform array

always has the largest error, and the unconstrained synthesized array has the

smallest error. The constrained synthesized array is a good compromise

because it has a small error and a rapid field decay outside the test

volume. The computer code for the array synthesis is very fast, and it is

easy to do further parameter studies.

One area which deserves further study is the relationship between the

normalized error in the surface fields defined by (92) and the error in the

fields throughout the test volume. By uniqueness, the volume error vanishes

when the surface error in F^ vanishes. Also, the numerical results indicate

that the normalized error in the fields throughout the test volume are small

when the errors in £ on the surface are small. However, it would be useful to

have a more definite relationship since our synthesis technique only minimizes

the error in F_.

7. Conclusions and recomnendations

In the wide frequency range of interest (approximately 30 MHz to several

GHz) for electromagnetic susceptibility testing, there is a gap between the

"low frequency" and "high frequency" methods of producing a plane wave test

field. For example, the TEM cell is useful below about 50 MHz, and the com- pact range is useful above about 1 GHz From the previous work reviewed in

section 2 and the theoretical results of sections 3 - 6, it is clear that a near-field phased array is a promising means of producing a plane wave in the

frequency range of interest. The reverberation chamber [38] can be used over

a similar frequency range, but it has not been discussed in this report

because it does not produce a plane wave.

As shown in section 2, there is no fundamental limitation on the quality

of the plane wave which can be generated by near-field sources. Many

different source distributions can produce the same plane wave field in a

42
given test volume, but they will produce different fields outside the test

volume. Thus, there is no unique solution to the array synthesis problem in

designing a near-field array, and many array geometries should be con- sidered. A number of practical questions involving bandwidth [39], array

scanning capability, and array elements and geometry must be addressed before

near-field arrays can be made useful for susceptibility testing.

The truncation effects at the edges of near-field arrays are analogous to

the errors due to finite scan area in near-field scanning [20]. As shown in

sections 4 and 5, the main effect is a ripple which results from interference

between the edge diffracted fields and the desired plane wave. As shown in

section 5, the ripple can be reduced by tapering the amplitude of the elements

at the edges of the array. The main practical difficulty at high frequencies

is that a large number of elements is required to produce a plane wave over an electrically large test volume.

In order to avoid trial -and-error procedures in determining the element

weightings, a very general near-field synthesis method based on a new uniqueness theorem was developed in section 6. The method minimizes the mean squared error between the desired field and the actual field over the test

volume, and it provides a direct numerical solution for the amplitude and

phase of the element weightings. In addition, a constraint can be placed on the source norm in order to prevent large fields outside the test volume.

This is important in minimizing wasted power and reflections from nearby

objects. The computer code is \/ery efficient, and parameter studies are easily performed.

In order to better understand the capabilities and limitations of near- field phased arrays, the following theoretical studies are recommeded. The

general near-field synthesis technique should be applied to realistic threedimensional geometries involving realistic array elements (such as di poles,

loops, or aperture antennas). The case where the test volume is located in

the near field of the individual elements should be included in order to

understand low-frequency effects where the test volume is less than a wavelength from the array. Bandwidth limitations involving both the array

geometry and array elements should be studied. Special wideband antennas

43
might be considered A related study could examine the feasibiity of transmitting narrow pulses for time-domain testing. The directional scanning

limitations imposed by the array geometry and the array elements should be

studied. Nonuniform element spacings [30] should be considered for reducing

the number of array elementsIt is premature to design and build a large near-field array at this

point, but valuable experience could be gained by building and testing one or

two small arrays Also, some of the potential difficulties, such as mutual

coupling between array elements and reflections from surrounding objects,

could be studied experimentally just as well as theoretically Planar arrays

appear to be the simplest to start with. The large frequency range of interest opens up numerous possibilities for array elements. Dipoles or loops

could be used at VHF, and aperture antennas such as horns or open-ended waveguides, could be used at microwave frequencies. Special wideband antennas

should also be considered. The element weightings would be determined from

the array synthesis theory An important part of the design would be the means of controlling the amplitude and phase of the element weightings. If

possible, both amplitude and phase of the electric and magnetic fields should

be measured throughout the test volume and compared with theoretical

predictions. Some measurements should be made as a function of frequency and

scan angle in order to evaluate bandwidth and scan limitations.

8. Acknowledgments

The author is grateful to Dr. M. T. Ma for technical discussions during

the course of this work. Many thanks also go to C. K. S. Miller and H. E.

Taggart for their continuous interest and support.

9

.

References

[1] Olver, A. D.; Saleeb, A. A Lens-type compact antenna range.

Electronics letters, 15: 409-410; 1979.

[2] Johnson, R. C ; Ecker, H. A., Moore, R. A. Compact range techniques

and measurements. IEEE Trans. Ant. Propagat. , AP-17: 568-575; 1969.

44
[3] Excell, P. S.; Gunes, Z. F. The compact range principle applied to

electromagnetic susceptibility testing. lERE Conf. Proc= Electromagnetic Compatibility: 317-322; 1982.

[4] Martsafey, V. V. Measurements of electrodynamic antenna parameters by

the method of synthesized apertures. Radio Engr. Electron. Rhys.,

13: 1869-1873, 1968.

[5] Bennett, J. C; Schoessow, E. P. Antenna near-field/far-field transformation using a plane-wave-synthesis technique. lEE Proc, 125:

179-184; 1978.

[6] Ludwig, A. C; Larsen, F. H. Spherical near-field measurements from a compact-range viewpoint. Technical University of Denmark Report R213;

1979.

[7] Bennett, J. C; Muntanga, N. E. Antenna near-field/far-field transformation with reduced data aquisition and processing requirements.

lEE Proc, 129(H): 229-231; 1982.

[8] Schoessow, E. P. Plane wave synthesis: a new approach to the problems

of antenna near-field/far-field transformation. University of

Sheffield, Ph. D. thesis; 1980.

[9] Lynggaard, S. K. Plane wave synthesis for antenna measurements.

Technical University of Denmark (in Danish), M. S. thesis; 1982.

[10] Harrington, R. F. Time-harmonic electromagnetic fields. New York:

McGraw-Hill; 1961.

[11] Waterman, P. C. Matrix formulation of electromagnetic scattering.

Proc. IEEE, 53: 805-812; 1965.

[12] Bowman, J. J.; Senior, T. B. A ; Uslenghi P. L. E. , ed. Electromagnetic and acoustic scattering by simple shapes. Amsterdam: NorthHolland Publishing Co. ; 1969.

[13] Mittra, R., ed. Computer techniques for electromagnetics. Oxford:

Pergamon Press; 1973.

[14] Senior, T. B. A Approximate boundary conditions. IEEE Trans. Ant.

Propagat., AP-29: 826-829; 1981.

[15] Jordan, E. C; Balmain, K. G. Electromagnetic waves and radiating

systems. Englewood Cliffs: Prentice Hall; 1968.

[16] Wood, P. J. The prediction of antenna characteristics from spherical

near-field measurements, part 1, theory. Marconi Review, XL: 42-68;

1977.

[17] Andreason, M. G. Scattering from cylinders with arbitrary surface

impedance. Proc. IEEE, 53: 812-817; 1965.

45
[18] Turchin, V. I.; Tseytlin, N. M. Antenna testing based on near-field

measurements- Radio Engr. Electron. Phys., 24: 1-26; 1979.

[19] Appel -Hansen, J. Antenna measurements using near-field scanning techniques. Technical University of Denmark Report NB 129, 1980.

[20] Yaghjian, A. D. Upper-bound errors in far-field antenna parameters

determined from planar near-field measurements, part 1: analysis.

Nat. Bur. Stand. (U.S ) Tech. Note 667; 1975.

[21] Appel -Hansen, J. Near-field scanning and intermediate distance techniques for antenna measurements. Technical University of Denmark

Report NB 134; 1981.

[22] James, G. L. Geometrical theory of diffraction for electromagnetic

waves. Stevenage: Peter Peregrinus; 1976.

[23] Abramowitz, M.; Stegun, I. A., ed. Handbook of mathematical

functions. Nat. Bur. Stand. (U.S.) Handb.; 1964.

[24] Keller, J. B. Diffraction by an aperture. J. Appl . Phys. 28: 426-

444; 1957.

[25] Keller, J. B.; Lewis, R. M.; Seckler, B. D. Diffraction by an aperture, II. J. Appl. Phys. 28: 570-579; 1957.

[26] Kraus, J. D. Antennas. New York: McGraw-Hill; 1950.

[27] Yaghjian, A. D. Approximate formulas for the far fields and gain of

open-ended rectangular waveguide. Nat. Bur. Stand. (U.S.) NBSIR 83-

1689; 1983.

[28] Crawford, M. L. Calibration of broadbeam antennas using planar near- field measurements. Conference on Precision Electromagnetic Measurements: 53-56; 1976,

[29] Collin, R. E.; Zucker, F. J., ed. Antenna theory part 1. New York:

McGraw-Hill; 1969. Ch. 7.

[30] Ma, M- T. Theory and application of antenna arrays. New York; John

Wiley & Sons, 1974.

[31] Mautz, J. R.; Harrington, R. F. Computational methods for antenna

pattern synthesis. IEEE Trans. Ant. Propagat. AP-23: 507-512; 1975.

[32] Muller C. Foundations of the mathematical theory of electromagnetic

waves. Berlin: Springer-Verlag; 1969.

[33] Mautz, J. R.; Harrington, R. F. H-field, E-field, and combined-field

solutions for conducting bodies of revolution. AEU, 32: 157-164;

1978.

46
[34] Kanda, M. An electric and magnetic field sensor for simultaneous

electromagnetic near-field- measurements--theory. Nat. Bur. Stand.

(U.S.) Tech. Note 1062; 1983.

[35] Boisvert, R.; Howe, S.; Kahauer, D.; Knapp-Cordes, J.; Knapp-Cordes,

M. Guide to available mathematical software. Center for Applied

Mathematics, Nat. Bur. Stand. (U.S); 1981.

[36] Hamming, R. W. Numerical methods for scientists and engineers. New

York: McGraw-Hill; 1973.

[37] Dongarra, J. J.; Mohler, C. B.; Bunch, J. R.; Stewart, G. W UNPACK

user's guide. Philadelphia: SIAM, 1979.

[38] Crawford, M. L. Evaluation of a reverberation chamber for performing

EM radiated fields susceptibility measurements. Nat. Bur. Stand.

(U.S.) NBSIR 81-1638; 1981.

[39] Schoessow, E. P.; Bennett, J. C. The plane wave synthesis technique

for antenna near-field/far-field transformation: further development. IEEE Conf. Pub. 195: 269-273; 1981.

47
Appendix A

Asymptotic evaluation of two-dimensional fields

The first step in the evaluation of the integral in (22) is to replace

the Hankel function by its asymptotic form for large argument [10]:

Substituting (21) and (Al) into (22), the integral can be cast in the

following form:

where

A = /° f(x') eJ'^a^^'^ dx' ,

2 X H f(x') = =^ , (A2)

/2iijk[(x - x')^ + z-]^~^

and

.;2 . .2 g(x') = - [x' sin e. + /(x - x')^ + z^ ]

.

In order to obtain the stationary phase contribution to (A2), we need the

first and second derivatives of g(x') with respect to x'

g'{x') = - sin e^. + -—

2 2 /(x - x')^ + z^

and (A3)

2

g"(x' ) = jlJ- —_ + _ ^^-^^-^

The stationary phase point Xg occurs for g'(Xg) = 0. This equation is easily

solved to yield

48
Xg = X - z tan 9^' (A4)

Since x^ must be negative, a stationary phase point exists only for z tan 9^- >

X. This is the region to the left of the dashed line in figure 8. At the

stationary point (x' = x^), f^, g, and g" have the following values:

g(x') = - (x sin 9. + z cos 9.) ,

•^ s 1 1

g"(x') = - cos^ 9./Z , and {A5)

cos 9.

f(x;) = z X H^ /

2-uTkT-

•

The stationary phase evaluation of integrals of the form of (A2) yields the

following general form [22]:

As

'" ^iTTg^TT-^^'s^ exp{j[kg(x;) + J

sgnCg"(x;)]]} , (A6)

where sgn is the sign of the argument Since we are interested only in positive z and values of |9^-| less than ti/2, g" in (A5) is negative. Substituting

(A5) into (A6), we obtain the final result for A^:

- jk(x sin 9.+ z cos 9.)

4 - "^ X H^ ^

jF^STTT U (z tan e. - X) . (A7)

The general form for the end point (or edge diffraction) term is given by

[22]:

-.. 2 J^9iO) J 2 ^ ,. Tjv

A^ ~1(0) e^^y^^' ^kTg^'foTT^t ^"^ ' sgn(g'(0)) ,

49
g"(0)

I

, where (AS)

The Fresnel integral F+ is defined as [22]:

CO F^{v) = / exp(+ jt^) dt. (A9)

V

By substituting (A2) and (A3) into (AS), we obtain the final value for A^:

-j(kp - J) 2

A_,~ixH^^^^^^,— F_{v) e^"^ sgn(e -9^.), (AlO)

where

_ /W2

cos

V = .::'q I sin 9g - sin 9.

2 2

sin 9 = x/p , and p = /x + z .

The electric and magnetic fields are given by the curl operations indicated in (30). In order to evaluate the curl, we use the following vector

identity [10]:

V X (£4)) = V(}) X a^ + (t)V X a^ . (All)

Using (A7) and (All), the curl of A^ is found to yield the expected plane wave expression Hrj in the illuminated region:

H^ = V X A —s —

s

U(e. - 9 ) -jk(x sin 9 + z cos 9 ). ^ (1 ^ u \ ~ __J e_ ^

r 1 1 J X (z X H ) , .^2)

jk cos 9. ^ Le -0 KRi^)

~ H U(9. - 9 ) , -p 1 e

'

50
where \\j^ is given by (21). The stationary phase value of the electric field

Is given by an additional curl operation:

E = J^ V X H^ —S jwe —

~Te_r U(e. - e^) -jk(x sin e. + z cos e.)

' '111

^ Vie J X H^

U(e. - e ) -jk(x sin e. !• z cos e.) , ~ ^-, ^

[- J k e

^ J k • X H (A13)

jcje - 1—0

~ - T, k. X Hp u(e, - e^) .

A where k. is given by (21). Strictly speaking, both (A12) and (A13) should

contain an extra term from the differentiation of the unit step U(9. - 9 ),

but this term is omitted here because it is later canceled by a term in Ag.

The curl of the edge diffraction term is somewhat more involved:

H = V x A —

—

F (V) eJ'^^ "'''*'

sgn (9 -e.) . ,. - 3i^^^-^^

-^- -_!.-. V X [z X H^ e-J^n (A14)

2

.|^ j(v + -rt/4) ~ [H cos e - z (H • p)] e~^^^ F (v) ^ sgn (9 - 9.) , -<i Q -0 - cos 9 /? e 1 e where

A A P

= X sin 9^ + z cos 9^

e e

51
In {A14) and in all other equations in Appendices A and B, only the leading

term in powers of k is retained. The electric field edge diffraction term is

obtained from the curl of H^:

E« = T^-

1 ^ X H„ ~ ti[H^ X AAA p cos e^ - y sin 9^ (H^ • p)] -e jwe -e "--o e "^ e -o -*

3-Jkp ^j(v2 +^/4) p(^j ^g^ (Q^ _ Q_j

cos 9 /ix e

The total electric and magnetic fields can now be written:

{A15)

£=E5 + E^ and H =

Jig + H^ • ^A16)

52
Appendix B

Asymptotic evaluation of three-dimensional fields

For the rectangular surface shown in figure 18, a stationary phase point

exists inside the tube:

< y < b and

(Bl)

z tan 9^ < X < a + z tan 9^- .

The stationary phase contribution from this point is the desired plane wave

field. Outside the tube, no stationary phase points exist, and only edgediffracted fields are present.

Each of the four edges in figure 18 generates an edge-diffracted field of

the form given by (46). For the rectangular geometry of figure 18, the specific form for f_ and g as given by (42) are:

z x H - ^ —^27iR^ ^"^ 9

" " ^^' ^^'" ®i

"^ ^^ ' where

' (82)

R = |r - r'l = /(x - x')^ + (y - y')^ + z^ .

For each of the four edges, u-|^, U2, and a as defined in figure 17 are different functions of x', y', a, and b. For each edge we will now define and

derive all of the required quantities in the edge diffraction term (46).

For edge number 1, we have

Uj^ = - y' , U2 = x' , and a = 0. (83)

The required derivatives of g are

53
The stationary edge point determined from g^^ = is:

2 2

Up = X - /y + z tan e. , where (B5)

< Uo^ < a . 2e

If the stationary point U2g lies outside the interval from to a, then the

contribution from edge 1 is zero. Substituting (B2) - (B5) into (46), the

contribution from edge 1 is:

2

2

2 "^^1 ^''^^l sgn (y) /y + z e F (v^ ) e A . = - z X H

^

, —el —0 —y 2

2" k cos 0. /JTi(y sin e. + z

where (86)

k(^ + z^)^'^cos e.

^ 2(y'^ sin"^ 9. + z^)

and

2 2

g^= - X sin e. - /y + z cos e. .

For edge number 2, we have

U]^ = x' , U2 = y' , and a = a (87)

54
The required derivatives of g are

X - X ^. Q „ _ y - y \ =-R— - ^^'"^- ' % R

» 2

.>2 , 2

X(j_^.xL)_..±j11

g

= zA1>l^j^)1±jl1

The stationary point determined from g^ = is:

U2g = y , where < U2g < b . (B9)

If the stationary point U2g lies outside the interval from to b, then the

contribution from edge 2 is zero. Substituting (B2) and (B7) - (89) into

(46), the contribution from edge 2 is:

sgn (g ) a7~-\)' . z' Z

, , ^^^^2

A^ =,XH^ ^----^I e 2-2

KZ /Jit

where

3„ = -trJLl-i - sin e , 1^ /(x - a)"^

9 + z"^9'

(BIO)

V2 = k[(x-aV^zW / 2

^ 2 z'^ ^"1

and

2 2

g2 = - a sin 9 . - /(x - a) + z .

For edge number 3, we have

Ui = y* , U2 = - x' , and a = b. (Bll)

55
The required derivatives of g are

- y - y' „ - .,•« Q ^ X - x' g u^ =^R-' \ - ^^"^- "—

g, ,

= ^(A--^^--^'! , and (B12)

,.^2 . .2

'2"2 R-^

g

= ^i-(yjLX)-±ja

The stationary point determined from g^j = is:

2 2

"2e

^ "^(y ' ^^ '"2 ^^" 6^- - X , where {B13)

- a < u^^ < . 2e

If the stationary point m^q lies outside the interval from - a to 0, then the

contribution from edge 3 is zero. Substituting (82) and (BU) - (BIS) into

(46), the contribution from edge Sis:

, 2 "~~2~~T J ^3

sgn (y - b) /{y -b) + z e F (v-) jkg^

k cos e. /J7[ [(y - b) sin 9- + z ]

where

k cos e. [(y - b)^ + Z-]

V = / — L,.^ X— - p- |y - b| , {B14)

2 2 [{y - b)2 sin^ e. +z2] '

and

2 2" g^ = - X sin 9^. - /{y - b) + z cos 9^. .

56
For edge number 4, we have

u^ = -X ', U2 = - y and a = (B15)

The required derivatives of g are \ = sine. +--^, g,^ -^~Y-

^^

= ^(y,.-.y;)^^z^1

, and (B16)

.n2 . 2 =

- [(x - x')^ + z^l

'U2U2

The stationary point determined from g„ = is

2

Uop = y , where < Uo. < b 2e (B17)

If the stationary point U2g lies outside the interval from to b, then the

contribution from edge 4 is zero. Substituting (82) and (815) - (817) into

(47), the contribution from edge 4 is:

sgn (g ) /x^ + / . 2 A . = z X H — e

^ F (v.) eJkg/

where (818)

g = SI n e . —

"1 ' ,J—z /x + y

and

,. , 2 ^ 2v372

V = ^k(x + y )

'

^ z/2

94 = - /x^ + z^

sin e /x2 . /

57
The remaining task is to derive the electric and magnetic fields from the

four edge-diffracted vector potential terms, A^j^, Ag2' Ae3' ^^^ tte^' ^^^^ °^

the terms can be written in the following general form:

jkgM

A = z X H h(r) e , (619) —

e

— —

where h and g are different for each edge. Since we are dealing with high

frequency fields, we include only the leading term in k for Hj^:

jkg(r)^ ^ jkg(r)

(B20)

H^=vxAg=Vx[zxH^ h(r) e ] = - (z x H^) x V [h(r) e ]

jkg(r) . ~ - jkh(r) e (z x H^) x V g(r) .

For the electric field JEg , we again retain only the leading term in k:

E^ = -A- V X H —e jwe —

e

jkg(r) . = - Ti^ V X [h(r) e (z x H^) x V g(r)] (621)

jkg(r)

~

Ti^ jk h(r) e [(z x H^) x V g] x V g .

As a check on (B20) and (B21), it can be shown that ^ and H^ are consistent

to the first order in k:

V X E = - jw^i H . (B22) —e e

E gand H_q also satisfy the expected orthogonality relationship for high

frequency fields:

E^ • H^ = . (B23)

58
In addition, the phase term g for each edge, satisfies the eiconal equation

[22] as expected for high frequency fields:

Vg • Vg = 1 . (824)

Equations (822) and (823) are satisfied for any phase function g« To show

that (824) is satisfied, we need to take the gradient of g for each edge

term. This requires no extra work because the gradients are required in (820)

and (821) anyway. For the four edges the gradients are given by:

cos e Vg, = - X sin 9. (y y + z z) ,

VQo = - ^ ^^ ' ^^ "" ^

^ , (B25)

7 7

/(x - a)"^ + zcos e. Vg^ = - X sin 0^. [y (y - b) + z z)] , /(y - b)^ + z^

/x2 . z2

8oth E and Hy polarizations are of interest. For Ey polarization, H^ is

given by

H^ = Hq (- x cos e^. + z sin e^.) . (826)

A In this case, z x H is given by

z X H^ = ~ Hq cos e^. y . (827)

Thus, the surface current is y-directed. For Hy polarization, H^ is given by

Hq =y Hq . (828)

and

59
In this case, z x H is given by

z X H = - H^ X —0 (B29)

Thus the surface current is x-directed.

60
Appendix C

Fields of a Hertzian dipole

The fields of a Hertzian dipole in spherical coordinates are well known

[10,26], but here we derive the Cartesian components of the fields. Consider

an electric dipole of moment p_^-j located at (xj, y\) in the x-y plane as indicated in figure 22. The magnetic vector potential k^^ is given by

-ij 4txR ^ij '

where (CI)

R = /(x - x;.)^ + (y - y^^ + z^ .

The magnetic field H^^-- is derived from the curl of (CI):

If p^jj is written in terms of x and y components, then (C2) can be written:

_ Jke-J'k^

"^-^ 4tiR'

H... = ^-j- (x z p..y - y z p.j^ + z [(y - yl)p.j^ - (x - xl) p..^]} ,

(C3)

where

p..=Xp.. +VD..

Only the leading terms in k have been retained in (C3).

The electric field £^-^ is determined from the curl of !H^-^-:

-jkR

iij = j^-^ ^ ^io

~ ^ ^,T- ^^ PijxC^' ^ ^y - ^i^'] ^ Pijy(^ - -V ^y - yV^

+ y {p^ (X - x;.) (y - yl) - p..y [(X - x;.)2 + z^]}

61
z[Pij, ( X- X') .p..y z(y-yl)] (C4)

Again, only the leading terms in k have been retained

62
11 X D

Fields of a Huygens' source

The fields of a Huygens' source are the sum of the electric dipole fields

and the magnetic dipole fields. The electric dipole fields can be obtained by

making the following substitutions in {C3) and (C4):

p. . = - E Ax' Ay'/ri ^ijx ox

"^ and (Dl)

p.. = .

This yields the following fields of the electric dipole source:

jk E^^Ax' Ay' .n_^ _2

-'^^ ^ ' M

3 {x [z^ + (y - y\.r] + y ix'.. - x) (y - y\.)

- z z (x - xl.)} , (D2)

jk Eq^ ax' Ay' e"'^*'^'^ . H,-,-^ =

o [y z - z (y - y'. .)] -

The electric and magnetic fields of the magnetic dipole moment can be

obtained from {C3) and {C4) by duality [10] and the following substitution:

p. . = - E Ax' Ay'/t] ^ijy ox -^ '

' and (D3)

p.. = .

By using duality on (D3), (C3), and (C4), we obtain the fields of the

appropriate magnetic dipole moment as defined in (55):

jk E^^ Ax' Ay' e'"^"^^ E.. = ^- 2" [x z - z (x - x'..)] , ijm ^^^i 13

63
jk E„^ Ax' Ay' . - o o H.. = 5-^^-5 (x (x'.. - x) (y - yl.) + y {z^ + (x - x'..)'^]

- z z (y - yjj)} . (D4)

The total fields are the sum of the electric and magnetic dipole fields:

lij = iije -^ iijm ^"d Hij = iije "^ iij^ . (D5)

Substituting (D2) and (D4) into (D5), we obtain the final forms:

^^ ^ox

^^' ^^' ^ r: r_2 . ,.. ... .2

4TtR^

+ z (xl . - x) (z + R)l

and (D6)

• Aw' «-JkR

H jk E Ax Ay e . . 2 ? -, ij = —^^""T"^ '"^ ^""ij

" ""^ (y - y^j) + y [^ + (x - x^)^ +

z r]

4titiR'

+ z (y^i - y) (z + R)}

Note that E^^-j and H^^-,- ae zero in the back direction (x = x^-j, y = y^-j, and z <

0). As in Appendix C, only the leading terms in k are included.

64
Appendix E

A uniqueness example

To illustrate the utility of the function F^ defined by (59), we consider

a circular cylindrical example. Let the surface S in figure 29 be a circular

cylinder of radius a, and assume that external sourc.es produce only an E^

electric field which is independent of z. Thus, the problem is two-dimensional, and the magnetic field components, H and Hx, are obtained from

Maxwell's curl equation:

1 5 E H = ^

(|) jcon 6p

- 1 ^ ^Z

H = T-

(El)

p jwup 5(t)

Standard cylindrical coordinates (p, 4), z) are used.

Inside the source-free cylindrical region, E^ satisfies the scalar

Helmholtz equation and can be written:

^z = ^ ^^n^^P^ ^^"^

' ^^2)

n=-oo

,th

coefficients. The vector F as defined by (59) is

where J^ is the n*-" order Bessel function [23] and a^ are the unknown

where (E3)

F = (E + Ti H^)l

z z a (J) |p = a

For now, we take x]^ to be some arbitrary real scalar, rather than the

intrinsic impedance of the medium r). Substituting (El) and (E2) into (E3), we obtain

65
z a 9

J' (ka) Ti . ^ {E4)

where Jp is the derivative of Jp with respect to the argument. To determine

a^, we multiply (E4) by e""^"^*^ and integrate from to 2 tx:

2u

-jn(})

Tl.

Both Jp and Jp have an infinite number of zeros on the real axis, but

never at the same argument [23]. Thus, the coefficient of a^ in (E5), which

is complex, never vanishes and a^ can be written:

^n

=

p = a e-J"* d^

E6)

2 u [J^ (ka) - J -^j; (ka)]

Thus a^ and the interior fields (E^, H , H.) are uniquely determined by F^ on

the boundary, p = a. If -n^ is very small, then a^ is primarily determined by

the electric field E^- In the limit, ti^ = 0, a^ is given by:

2n

z p = a e-J'"* d^

2ti J^ (ka)

(E7)

and ap is determined by E^- Unfortunately, this representation fails when

Jp(ka) = 0. In such cases, ap cannot be determined from E^, and the fields

are not uniquely determined.

If Tig is wery large, then ap is primarily determined by the magnetic

field H^. In the limit, ti^ = <=, dp is given by:

2%

il!o_%p = a ^n 2ti jyTkaT

n

e-J"* d^

(E8)

This representation fails when Jp (ka) = 0. In such cases, ap cannot be

determined from H., and the fields are not uniquely determined.

66
For the case where t)^ = n, a^ is given by

2n

-jnc})

a = _2 . (E9)

" Zn [J^ (ka) - j J^ (ka)]

In this case, a^ is determined approximately equally by the electric and

magnetic fields. Another advantage of the choice rig = ri is that the

denominator is numerically well conditioned. Whenever J^^ = 0, J^ is a maximum

and vice versa. So the choice of ti^ = ti appears to be a good one.

67
Appendix F

Line source fields

A y-directed line source located in the z = plane at x = x', produces

the following y component of electric field [10]:

Ey—V^H<2> (kp) . where (Fl)

// ^2 ^ 2

P

= /(x - X ) + z

and I is the line current. The Eyj term required in (86) is the sum of two

terms with I set equal to unity:

where

and

P."" = /(x - x;)^ + z^ {F2)

p." = /(x + x;)2 + z^ .

The magnetic field Hj is obtained from the curl of the electric field:

where -, c

1 o E . H . = J:

^

(F3) "xj jto^ 5 z ^^^' ^"^

, a E .

ZJ jwp. 5 x

Substituting (F2) into (F3), we obtain;

1 r 2 ..(2) ,, +, . z ..(2)

"xJ=4l[-V"l '<'^"--H}^'(^Pj-)].

"i "i

68
(F4)

zj ^-^-^Hf)ap/,.-^^H<^>Upp].

where Hp^ is the first-order Hankel function of the second kind. As with the

zero-order Hankel function in (Al), when the argument is large, the asymptotic

expansion can be used [10]:

h;"' '^ (x) ~

j / |ie"^^

1XX

The expression for T^,- is obtained by substituting {F3) into (84)

A A T^-[E . -,y. nx(xH . ZH )] xj ZJ X = X.

z = z,

Using vector identities, (F6) can be written

A A

X = X.

z = z.

(F5)

(F6)

(F7)

The values of the dot products in (F7) depend on which side of the test volume

the point is located:

A A side 1 : n • z

=

1 , n • X = ,

side 2

A A

: n • z

=

A A

,

A A n • X =

A A

- 1 ,

side 3 n • z

= - 1 , n • X= ,

side 4

A A

: n • z

= ,

A A n • X

= 1 •

{F8)

69
Ee, H

Figure 1. Electric and magnetic surface currents, J_ and M, on a surface S,

Ee= He=0

Arbitrory Medium

Figure 2. Electric and magnetic surface currents, J and M , which generate

a plane wave inside S and a zero field outside S.

70
Perfect Electric

Conductor

Ipi "bp

Test Volume

(b)

Is»Hs

Figure 3. Electric surface current J^ flowing on S. (a) Scattering problem

for plane wave incidence (b) Source problem for plane wave synthesis.

71
Surface

Impedance, Z -Atfr

-ip.-H P' ^p

Test Volume

\ Es.Hj

(iO

Figure 4. (a) Equivalent currents, J_g and M_g, excited on an impedance

surface. (b) Source currervts, J_g and M_g, for plane wave synthesis.

72
H XI y

^ \\ \ /

ys

H xs

(a)

Surfoce Impedance

Plane (z = 0)

H

ys

xs

(b)

-^ Test Volume

Mxs

'ys

'xp

yp

Figure 5. (a) Plane wave incident on a surface impedance plane, (b) Radiation by equivalent electric and magnetic sheet currents, J^^ and

M"xs*

73
Shadow

Boundary

Small Field

Plane Wave As

Transition Zone

Figure 6. Near fields of the physical optics current, J.^ = 2 n x H

Array

D H

Test

Zone

W

Figure 7. Planar array of width W with i nterel ement spacing s. zone is of width D and is located at a distance R. The test

74
Field

Point

Diffrocted Roy

Direct Ray

' X

Stationary Phose Point

Figure 8. Physical optics current J_(x' ) on a semi-infinite plane. The total

field can be described by a direct ray plus a diffracted ray.

75
1.2

yO

z/X = 5

Figure 9. Magnitude of the normalized electric field of a semi -infinite

current sheet (Ey polarization).

76
Figure 10. Phase of the normalized electric field of a semi -infinite current

sheet (E polarization).

77
-t-> C ^-'

•t- c

4-

O

C •!-

•r- +J 1 fO

•rN

ir

•4- >» O LiJ o .,- I— CD

•I- to 4- O) , , cnc

03

p oc

-o

-o

u

0) c N •-- E O o t->

<+- +^

-o

•I- O) c s- cn s- fO 3

U111 uj

78
X

ii~

Z3U

14- O

c ••-

•r- !->

I fO

•I- N

CL

"+- on X ^

"^^

N o

fOis

14- O

O itO) +J

</) (U

O. CO

(0^3/ Ag) 3S0(4d

79
1.2

TJ3

o

TJ

O

E

Geometrical

Optics

Figure 13. Magnitude of the normalized electric field of a semi -infinite

current sheet (Hy polarization).

80
C

x/X

Figure 14. Magnitude of the normalized electric field of a semi-infinite

current sheet for various incidence angles 9^- (H polarization).

81
Abrupt

Truncation

Tapered

Current

-L

Figure 15. Comparison of truncated current and tapered current.

Field Point

Figure 16. Stationary phase i^) , edge diffracted (E^), and corner diffracted

iE^) field contributions from surface currents J^.

82
Perimeter L

Figure 17. Coordinate system for calculation of edge diffraction.

Field

Point

(o,b, 0)

Figure 18. Rectangular current sheet of dimensions a by b. Each of the four

edges (1-4) contributes to the edge diffracted field.

83
14

z/X = 30

z/X = 10

x/X

Figure 19. Magnitude of the normalized electric field radiated by a square

current sheet. The geometrical optics approximation (g.o.) is

shown for comparison.

84
1.4

±

10 20 30

x/X

\^y

40 50 60

Figure 20. Magnitude of the normalized electric field radiated by a square

current sheet for two scan angles, 9- = 30° and 60°.

85
14

1.2

1.0

"co

E

y/X =5

z/X =10

o/X = b/X =30

^: = 0*>

10 20

x/X

Figure 21. Magnitude of the normalized electric field radiated by a square

current sheet for a cut off the center line, y/X = 5.

86
Field

Point

(XtVtZ )

Figure 22. Geometry for a rectangular array of equally-spaced electric dipole

elements or Huygens' sources.

87
!.4

Figure 23. Normalized electric field strength of a uniform square array of

electric dipoles. An interelement spacing of 1.5 X violates the

sampling criteria of section 3.3.

88
12

z= 20X

x= I5\

Q=b=30X

Spacing = X/2

y/X

15

Figure 24. Normalized electric field strength of a tapered square array of

electric di poles. The curve for T = represents the uniform

array.

89
Spocing = X/2

y/X

Figure 25. Normalized electric field strength of a small tapered array of

electric dipoles. The geometrical optics field (g.o.) is shown

for comparison.

90
Field Point

(x,y,z )

("ij'. yij'.O)

Figure 26. Electric and magnetic dipole moments, p^-j^ and m^-j^, of a Huygens'

source.

2.0

y/X

Figure 27. Normalized electric field strength of a uniform square array of

Huygens' sources. An interelement spacing of 1.5 X violates the

sampling criteria of section 3.3.

91
12

2=20X

X = I5X

I0|- = b = 30X

Figure 28. Normalized electric field strength of a tapered square array of

Huygens' sources. The curve for T = represents the uniform

array.

Source- Free Region

External

Sources

Figure 29 Source-free region enclosed by S. F_ is a tangential vector which

uniquely determines the interior fields

92
F= u^F^ ^ U2F2

Origin

Sources

Figure 30. Geometry for near-field synthesis,

the surface S. The vector F is tangential to

(Xj, Zj)

Test

Volume

«»z,

u •^-

In I2 I1 -• • • •-

©

®

-• • • o •- H

N

Figure 31. An array of y-directed electric line current sources illuminating

a rectanqular test volume.

93
I80--

90»-

o

Q.

-90«

-180'

No Constraint

Constraint

d= X/2

4X

2 3 4 5

Element No.

Figure 32. Magnitude and phase of the synthesized currents for an array of 12

elements with and without a constraint.

94
2.5

2.0

1.5

•0

0.5

Uniform Array

Synthesis

Constraint

Plane Wave

A= B= zo d= X/2

N= 6

= 4X

2 4 6 8 u/X

Figure 33. Normalized magnitude of the uniqueness function for a uniform

array, a synthesized array, and a constrained array. The perfect

plane wave result is shown for comparison.

95
2.0

Uniform Array

Synthesis

Constraint

Plane Wave

0

x/X

Figure 34. Normalized magnitude of the electric field for a uniform array, a synthesized array, and a constrained array The perfect plane

wave is shown for comparison.

96
10' H 0»

M

-10*

0)

(0o -20*

•30*-

1.5-

1.0-

0.5-

Element No.

Figure 35. Magnitude and phase of the synthesized currents for an array of 8

elements. In this case, the constraint has no effect.

97
1.4

0.4

Uniform Array

Synthesis

Plane Wove

N = 4

Zo= z = 4X

A = B = 4X

d = 3X/4

Figure 36. Normalized magnitude of the electric field for a uniform array and

a synthesized array. The perfect plane wave is shown for comparison.

98
"h 10 M

t2 0*

Si 5 -I0»

-20«

• _

1.5-

>d)-

0.5

-

(d= X

A=B = Zo = 4X

-

2 Element No.

Figure 37. Magnitude and phase of the synthesized currents for an array of 6

elements The constraint has no effect.

99
•0

Uniform Array

Synthesis

Plane Wave

x/X

Figur-; 38. Normalized magnitude of the electric field for a uniform array and

a synthesized array. The perfect plane wave is shown for comparison.

100
20'

C -20" ^ -40°

60"

1.5

1.0

0.5

A= B=4\

Zo= 8X

d = X

No Constraint

Constraint

12 5 Element No.

Figure 39. Magnitude and phase of the synthesized currents for an array of 6 elements with and without a constraint.

101
•0

x/X

Figure 40. Normalized magnitude of the electric field for a uniform array, a synthesized array, and a constrained array. The perfect plane

wave is shown for comparison.

102
20»

0« - -20"

(A

-60*

1.5

0.5

A = B=Zo= 2X

d =X/2

Constraint

No Constroint

I 2 3

Element No.

Figure 41. Magnitude and phase of the synthesized currents for an array of 6

elements v/ith and without a constraint.

103
3.0

2.5

2.0

1.5

0.5

Uniform Array

Synthesis

Constraint

Plane Wave

A =B =2\

d = X/2

2 u/X

Figure 42. Normalized magnitude of the uniqueness function for a uniform

array, a synthesized array, and a constrained array. The perfect

plane wave is shown for comparison.

104
1.4

.2-

EO

1.0

0.8

0.6-

0.4-

0.2

\

1

1 1 1

\'v Synthesis _ \ Constraint

arr.!k^.:.:::>

Plane Wave

KJ » * * ••••y ••••••••• •• X

~Ar~ —

\

N. \^

^^•

\ \

\ \

\ \ \ -^

\ "^

- N = 3

d = X/2

Zq = z = A= B = 2X \

1

1 1 1 0.5 1.5 2.0 2.5 x/X

Figure 43. Normalized magnitude of the electric field for a uniform array, a synthesized array, and a constrained array. The perfect plane

pave is shown for comparison.

105
NBS-n4A (REV. 2-80) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1072

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

February 1984

4. TITLE AND SUBTITLE

Theory of Near-Field Phased Arrays for Electromagnetic Susceptibility Testing

5. AUTHOR(S)

David A. Hill

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

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, Stote. ZIP)

10. SUPPLEMENTARY NOTES

\2^ Document describes a computer program; SF-185, FlPS Software Summary, is attached.

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

bibliography or literature survey, mention it here)

The feasibility of using a near-field array for electromagnetic susceptibility

testing is studied. The basic objective is to control the element weightings

such that a plane wave is generated within the test volume. The basic theory

is developed for arbitrary array geometries, and numerical results are obtained

for finite planar arrays. A general near-field array synthesis technique is

developed, and the technique minimizes the mean square error in the test volume

while constraining the array excitations. The constraint prevents large currents

and is useful in minimizing the fields outside the test volume. The basic idea

looks promising, but some practical considerations, such 'as bandwidth and angular scanning limitations, require further theoretical and experimental investigation.

12. KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

array synthesis; dipole; electromagnetic susceptibility; Huygens' source;

line source; near field; phased array.

13. AVAILABILITY

KXl Unlimited

(331 For Official Distribution. Do Not Release to NTIS

^X] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

(331 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

109

15. Price

-'^U.S GOVERNMENT PRIMTINQ OFFICE:18e4—778-311 / 0197 USCOMM-DC 6043-P80


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