REFERENCE

NBS

PUBLICATIONS

A111DB 247TES

NBSIR 86-3047 WKSUWlf Hill, David A/oJ o,0

h22w47925

OUT-OF-BAm jse of antenna

ARRAYS

David A. Hill

Michael H. Francis

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

June 1986

-QC

100

- U 5 6

86-3047

1986

NBSIR 86-3047

KBS RESEARCH

INFORMATION CENTER

OUT-OF-BAND RESPONSE OF ANTENNA

ARRAYS

David A. Hill

Michael H. Francis

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

June 1986

Supported by

Defense Nuclear Agency (DNA)

Alexandria, Virginia 22310

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

Abstract 1

1. INTRODUCTION 1

2. ANTENNA ARRAY THEORY 1 2.1 Effective Aperture 2 2.2 Directivity Theory

2.3 Directivity Calculations

2.4 Impedance Mismatch

3. NEAR-FIELD MEASUREMENTS

3.1 ULSA Measurements

3.2 AWACS Measurements 1 3.3 Sampling Considerations 10

4. CONCLUSIONS 12

5. REFERENCES 14

APPENDIX 16

iii

O

VO

VO

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O'.

CO

OUT-OF-BAND RESPONSE OF ANTENNA ARRAYS

David A. Hill and Michael H. Francis

Electromagnetic Fields Division

National Bureau of Standards

Boulder, CO 80303

The response of antenna arrays to out-of-band frequencies has

been analyzed using the effective aperture approach. An average

value of effective aperture can be obtained by averaging the

incidence angle and the polarization of the incidence field.

Far-field patterns have also been calculated by treating the

array element excitations as random variables. The randomness

in the element excitations causes a decrease in directivity and

an increase in sidelobe level. Out-of-band measurements of

reflection coefficient and near-field response have been made on two large slotted-waveguide arrays for frequencies from 2 to 1

8

GHz. Both arrays are narrow band, and this is easily explained

by the large impedance mismatch at out-of-band frequencies.

Key words: antenna array; directivity; impedance mismatch;

near-field measurements; out-of-band response; polarization;

slotted waveguide.

1 . INTRODUCTION

The response of antennas to out-of-band frequencies [ 1 — 3 ] plays an important role in interference and jamming problems. The initial National

Bureau of Standards (NBS) work in this area included an analysis of

reflector antennas [4] and a comparison with previous measurements at outof-band frequencies [5]. In this report we consider antenna arrays.

The organization of this report is as follows. Section 2 contains an analysis of antenna arrays for out-of-band frequencies. Effective aperture,

directivity, and impedance mismatch are treated in detail. Section 3 contains near-field measurements at out-of-band frequencies on two large

arrays of slotted waveguides. Some considerations on sampling in frequency

and space are also included. Section 4 summarizes the results of this study

and includes recommendations for future work.

2. ANTENNA ARRAY THEORY

In this section we analyze the receiving characteristics of antenna

arrays at out-of-band frequencies. Sections 2.1 and 2.4 on effectiveaperture and impedance mismatch are actually applicable to arbitrary antenna

types, but sections 2.2 and 2.3 are specific to antenna arrays.

1
2.1 Effective Aperture

The effective aperture A of an antenna is defined as the ratio of the

received power to the incident power density. If we follow Tai's definition

and notation, then A can be written [6]

A PQ

X D(e,4>)

Mtt

( 1 )

where p is the polarization mismatch factor, q is the impedance mismatch

factor, X is the free space wavelength, and D is the directivity of the

antenna in the direction of the incident field (0,<J>).

The result in (1) is quite general and can be used to derive several

useful results. If the antenna is polarization matched to the incident

field, then p

= 1. If the load impedance is conjugate matched to antenna,

then q

= 1 . If the antenna is isotropic, then D = 1 for all 0 and <j>. In

this case, (1) reduces to the well known value,

A -i - < 2 >

for an isotropic antenna [7].

For out-of-band frequencies, the antenna will generally not be

impedance matched, and this subject will be discussed further in section

2.3. Also, the antenna polarization will not generally be matched to the

incident field. If we assume that the incident field is randomly polarized,

then the average value of p is [6]

<p> = 1/2 ,

where < > denotes average value. In this case, (1) reduces to

<A>

D( 0 , 4>) 8tt

(3)

(4)

This is probably the most useful form for out-of-band calculations because

it allows us to include the frequency dependence of q and the angular

dependence of D. However, if we choose to average D over the incidence

angles 0 and <j>, then we obtain

<D> = 1 . (5)

This yields the following simple expression for <A>

<A> Q

8tt

( 6 )

2
For cor.piicated antennas, it is usually not possible to compute q at out-ofband frequencies, and q must be measured [8]. For cases where the antenna

feed is multimoded, each mode can be treated as a separate antenna. An

open-ended waveguide example is shown in the Appendix.

2.2 Directivity Theory

In this section we analyze the far-field pattern of antenna arrays at

out-of-band frequencies. Although we are interested primarily in the

receiving case, we assume that the antenna array is reciprocal and that the

receiving and transmitting patterns are the same. Thus we car. treat the

transmitting case, which is more convenient for analysis. For large arrays

operating at out-of-band frequencies, the array element excitations have

large uncertainties and are best represented as random variables 13 1 • Thus

we require a statistical analysis, and some of the required statistical

antenna theory has already been developed Z 91 . Much of the earlier work on statistical antenna theory dealt with the effect of random errors ir. array

construction and feed networks [10-13]. In the out-of-band case the

randomness in the element excitations is actually due to our lack of

knowledge of the out-of-band -characteristics of the components in the feed

system and the array elements.

As a starting point, we follow the analysis of Gilbert and Morgan

[121. The power pattern $(u) of an antenna array of identical elements can

be written

§ (u) = s“[u) |f (u)

|

2

,

2 - where s (u) is the element power pattern, f(u) is the array factor, and u is

a unit vector representing a direction in space. For an array of N

elements, the array factor is

N

f (u) = I A exp(-ikR *u) , 15' n * n

n=1

where k is the wavenumber (=2ir/A), A^ is the excitation coefficient of the

nth element, R is the oosition vector of the nth element, and the time

n dependence is exp(-iojt).

>Te now assume that at out-of-band frequencies the exc::a: .

• coefficients have some random scatter about their mean or expectei values,

and we can write them as [12]

r. + a

r.

( 9 )

3
where a is the expected value of A and the a 's are independent random

n n n

K complex variables with zero mean. Gilbert and Morgan [12] assumed that the

position vectors R^ were also random in order to model construction errors,

but we will not consider this case. The expected or mean value of the power

pattern <$(u)> can be written

<<D(u)> = s

2 (u) <|f (u) | 2

> (10)

p _ _ p ^ p

= s ( u ) [|f

0 (u)r + E <|a

n r>] , n=1

where f^ is the mean value of the array pattern

N

f A (u) = E a exp(-ikR *u) . (11)

0 „ n n n=1

2 - In (10) we have assumed that the element power pattern s (u) is not random.

The expected value of the power pattern in (10) depends only on the expected

value of la I , and we do not need to know the detailed statistics of a 1 n

1 n unless we wish to know the statistics of $. If we assume that the real and

imaginary parts of a

n are Gaussian with zero mean, then 0 has a modified

Rayleigh distribution [10].

Following Gilbert and Morgan [12], we assume that the random and

deterministic parts of A^ are related by

where e is independent of n. If we obtain

<$(u)> = s

2

(u) [

|

f

Q

(u)

|

2

substitute (12) into (10), then we

2 N2

+ e 2

I |aj 2

] . n=1

(13)

In order to do any out-of-band calculations, we need to know how c depends on frequency. For a well designed array we know that e is small at

the design frequency f

Q

. Also, we know from limited measurements and from

the out-of-band characteristics of feed system components [3] that e increases as the frequency departs from the design frequency. Consequently
we postulate that e is proportional to the difference between the actual

frequency f and the design frequency

e (14)

where c is a constant on the order of unity. The frequency dependence of e in (14) can be modified for cases where a better knowledge of e is

available. Also, it is possible to treat the case where the element pattern

is random [9], but we will not consider this case.

So far we have considered only the relative power pattern <f>(u). If we wish to compute directivity, we need to normalize $(u) to the total radiated

power. For the deterministic case (e = 0), the directivity D^(u) can be

written

D

0

(u) = Vu)/[ %

1 VU) df2] ’ (15)

__ 2 — — 2

where $

Q

(u) = s (u) |f Q (u)|

The integration in (15) is over all solid angles, and dft is the differential

of solid angle. For the random case (e * 0), a convenient definition of

directive gain D(u) is [12]

D (u ) = <$(u)>/[-7-^- / <$(u)> dfi] . 4 TT

( 16 )

Strictly speaking, (16) is not correct because it assumes that the expected

value of the ratio is equal to the ratio of the expected values [14].

However, (16) is a useful expression because it is generally accurate and is

fairly easy to compute [12].

If we substitute (13) and (15) into (16) and perform some algebra, we obtain

2 ,- N

D_ (u ) + [s (u)/I ]e l a

0 r

D(u) = n=1

(17)

1 + (I /I Je‘

s $ n=1

The integrals I g and I are given by

5
I ( 18 ) = -jT~~ / s

2 (u) dP and I. = / $.(u) dft * 0 . s 4 tt $ 4tt 0

In general the evaluation of (17) is rather tedious because of the summation

over n and the two integrations over solid angle Q. However, there are two

fairly simple limits of interest. At the design frequency when e = 0 , we have

D ( u) e=0

D q(u) (19)

This is simply the gain for the deterministic case as given by (15). When

the frequency is well above band and e approaches °°, we have

D(u) = s

2 (u)/I (20)

This is the directive gain of a single element where the array pattern has

become isotropic.

2.3 Directivity Calculations

The expression for D(u) in (17) is the primary result of the previous

section, and specific array results can be calculated if the element

weightings, the element pattern, and the appropriate value for c are known.

To illustrate the effects that occur at out-of-band frequencies, we consider

the example of a rectangular planar array of magnetic dipole elements as shown in figure 1 . The number of elements elements N is N N , the x spacing is d , x y

and the y spacing is d We choose the planar array geometry because it is so common and the magnetic dipole element because it is a simple model for a small slot as in a slotted waveguide array [15].

We first consider the element pattern. It is easy to show that the

power pattern of a magnetic dipole on a ground plane is

2 2 2 cos cp + cos 0 sin <j> , 0 < ir/2 s (u) = { (21 ) 0 , 0 > tt/2

The elevation angle 0 and the azimuthal angle <f> are defined in figure 1 . By

substituting (21) into (18), we can calculate I s

! 2tt tt/22 2

I = -7T— / / (cos <() + cos 0 sin <j>) sin0 d0 d<}> = 1/3 . (22)

s 4lT 0 0

From (21) and (22), we see that the directivity of a magnetic dipole on a ground plane is 3 (or 4.77 dB), which is twice that of a magnetic or electric dipole in free space.

6
For simplicity we consider a uniform broadside array (a = 1 for all

n). Then the required sum in (17) is

N

E n=1

N N N . x y

(23)

The mean value of the array pattern is

1

N -1 N -1 x y

fo ( u) = n £_ 0 n

E_

0 exp(-ikn

x

d

x sin6cos<j))exp(-ikn d sin6sin<j)) , (24)

n

x

- n

y

" X X Y Y

where u = x sin0cos<() + y sin0sin<)> + z cos0 and x, y, and z are unit vectors.

The summations in (24) can be performed analytically, and the square of the

array pattern can be written

, ,-,.2 „2 r Sln ( N yV2)

,2 r Sln ( NxV2)

l2 f

0 u I N N sin(Y /2) ] [ N sin(Y /2)

] (25)

where Y = kd sin0sind> and Y = kd sin0cosd>. The evaluation of 1^ as

y y

Y x x $

2 - defined in (18) is generally tedious, but if we approximate s (u) by unity

and sin0 by 0 in the vicinity of the main beam where 0 is small, then 1^ can

be evaluated analytically as shown by Kraus [7, p.121]

I

<i> NX

4nd d

Y x

( 26 )

We now have expressions for all of the quantities required for evaluation of

the directivity in (17).

We consider first a square array example with the following

parameters: Nxy’xyO = N 10, d = d A_/ 2, and f = 2f_, where and f n are

0 0 0

the design wavelength and frequency. In figure 2 we show the directivity in

the plane <f> = 0 for various values of c. Since the element pattern is

2 isotropic (s = 1 ) in that plane, we are seeing the array pattern. The

pattern for c = 0 is as expected for a uniform array with no randomness in

the element weightings. There are nulls at N

x Yx /2 equal to integeral

multiples of tt, and the first sidelobe is 1 3 dB below the main beam. As c is increased, the directivity decreases, and the nulls are filled in. This

behavior is consistent with some of the results of Shifrin [9]. As c approaches infinity, the directivity approaches that of a single element as shown by (20). The curve for c = 5.0 is starting to show that behavior, and

the directive gain for large N^Y^/2 is approximately that of a sing]-

7
element (4.77 dB) . The difficult problem of choosing the appropriate value

of c for a specific array would be best answered experimentally. In the

absence of experimental results, a value of c near unity appears to be a reasonable choice.

For our second example, we consider a large rectangular array with the

following parameters: N = 120, N = 20, d = d A_/2, and c = 1. These

x y x y 0

parameters are close to those of the ultra low sidelobe array (ULSA) [16]

which was measured on the NBS near-field antenna range as described in the

following section. The design wavelength is approximately 10 cm, and the

array dimensions are approximately 1 m by 6 m. The directivity in the plane

<(> = 0 is shown in figure 3. The curve at the design frequency (f = f ) is

again the curve for a uniform array with nulls at N^T^/2 equal to integer

multiples of it. The above-band curve (f = 4f^) shows a reduced gain,

filled-in nulls, and a nearly constant level at large values of N^Y^/2. The

constant level is the element directivity (4.77 dB) in the plane <j> = 0. The

array has no grating lobes at the design frequency, but there can be grating

Yy/2 equal integeral

multiples of tt as indicated by (25). However, the level of the grating

lobes will be reduced just as the level of the main beam is reduced for f = 4f in figure 3. In figure 4 we show the directivity of the same array as

lobes at above-band frequencies where either Y^/2 or

a function of frequency. If we neglect the randomness of the array (c = 0),

2 then the directivity increases as f . However, when the randomness is

included (c = 1), the directivity decreases with frequency and approaches

the directivity of a single element.

Both of our examples in this section have considered a broadside array

with uniform element weighting. In practical applications, the elements

could have a linear phase shift for beam steering or some other weighting,

such as D ol ph-Chebyshe v for low sidelobes [17]. In either case, the

qualitative effect of randomness of element weighting at out-of-band

frequencies will be the same as in the uniform array examples. The

directivity will decrease, the pattern nulls will fill in, and the distant

sidelobe level will rise. A close examination of (17) shows that these

effects occur regardless of the particular form of the weighting

coefficients a . n

2.4 Impedance Mismatch

Tai [6] gives the following expression for the impedance mismatch

factor q

q

4RR

Li

(R + R

l

) 2

+ (X + X

L

) 2 (27)

8
where Z = R + iX = antenna impedance and = R^ + iX^ = load impedance.

This expression is in agreement with that of Kraus [ 7 , ch. 3] for the case

of a lossless antenna. When the antenna has both loss resistance R as well

as radiation resistance R , it is most convenient to combine impedance

mismatch and resistive loss in the following expression for q,

q

HR Rt r L

(R + R + R. ) 2 + (X + X. ) 2

1 Li Li

( 28 )

If both the antenna loss resistance R^ and the load reactance X T are

zero, then (28) can be written

HRR

q

* | 2

(R + R ) + X

Li

where r is the input reflection coefficient of the antenna when R is the

real characteristic impedance of the transmission line or waveguide feed.

Since the magnitude of r can be measured, q is easily determined from (29).

There is no need to measure the input resistance and reactance of the

antenna. The main restrictions on (29) are that the antenna resistive

losses must be negligible and the transmission line or waveguide feed must

be single moded at the antenna terminals. Some measurements of reflection

coefficient are shown in the following section.

3. NEAR-FIELD MEASUREMENTS

In this section we present the results of near-field measurements on two large slotted waveguide arrays, ULSA and AWACS. In all cases, we transmitted with the large array and received with a small probe. Reference

[16] contains a discussion of the design of ULSA and AWACS and photographs

of both arrays.

3.1 ULSA Measurements

In June 1985, the NBS acquired a new receiver in order to do swept

frequency measurements with the near-field range. At that time a slotted

waveguide array, ULSA [16], was on the near-field range for a regular nearfield measurement. In order to test the receiver, the opportunity was taken

to perform some swept frequency measurements at selected points in the near

field. These measurements were taken over the frequency range of 2-10 GHz.

Some sample results are shown in figures 5 and 6. Figure 5 data warn

acquired with the probe at the center of the ULSA while figure 6 data w<v

acquired at a position 25 cm below the center of the ULSA. Both sets ‘ data were acquired using the same S-band open-ended waveguide probe that was

9
used in the regular near-field measurement. In both cases the probe was

located 65 cm from the plane of the array.

The character of both figures 5 and 6 is similar in that they both

have a low frequency cutoff at about 2.5 GHz. This corresponds to the

cutoff for the slotted waveguide of the ULSA. The response at higher

frequencies is seen to decay to a level which is about 20 dB below the

maximum in-band response.

3.2 AWACS Measurements

Swept frequency measurements for the AWACS were performed over the

frequency range 2-18 GHz using a wideband horn probe with good response over this entire frequency range. The swept frequency measurements determined

the amplitude and phase of the output at the receiver relative to the input

at the transmitter as a function of frequency for each point of measurement

in the near field. Measurements were made at the center of the AWACS array,

30 cm above and below the center, 61 cm below the center, 89 cm to one side

of the center, and 178 cm to one side of the center. The probe was always

located 35 cm from the plane of the array. Typical swept frequency results

are shown in figure 7 (for the array center) and figure 8 (for 89 cm to one side of center). It can be seen from these results that the AWACS like the

ULSA is narrow band. The low frequency cutoff corresponds very nearly to

the cutoff of the slotted waveguide. The low response at higher frequencies

is due to the large input reflection coefficient (see figure 9) which is

nearly 1.0 or 0 dB.

If a Fourier transform is done on the swept frequency response, the

impulse response of the antenna can be obtained. These results are found in

figures 10 and 11. Time equal to zero corresponds to the signal at the

input. The peak response is at 50 ns and correlates well with the time

required for the signal to travel the length of the antenna and then to the

probe. A secondary peak is seen at about 80 ns and is believed to be due to

part of the signal reflecting off the antenna termination.

If we examine the swept frequency results we see there are two small

peaks in the antenna response just above band at 3*83 GHz and 3-94 GHz.

They are about 10 dB below the antenna’s peak response. A one dimensional

near-field measurement was made at 3*94 GHz and compared to the in-band

results. The 3.94 GHz results are found in figure 12 (near field) and

figure 13 (far field). The near field at 3.94 GHz is much more skewed than

the in-band result. The far-field main beam at 3.94 GHz is much closer to

being steered on-axis than is the in-band far-field main beam. The location

of the main beam in both cases is in good agreement with the theory of

slotted waveguide arrays [15]. The sidelobes at 3.94 GHz are slightly

higher than the in-band sidelobes, and this is in agreement with (17).

3.3 Sampling Considerations

The measurements on ULSA and AWACS in the previous sections provide a great deal of information on the out-of-band response of these arrays, but
we need to examine sampling requirements for complete and efficient

characterization. In out-of-band problems, we do not require precise

measurements of far-field patterns and frequency response, but we would like

to be sure that we do not miss any major features, such as peaks in the

frequency response or major lobes in the far-field pattern. Since large

arrays at above-band frequencies have a large electrical size, we would also

like to be efficient in taking the measurements.

We first consider frequency sampling. This is most easily done by

examining the frequency dependence of the input reflection coefficient r. This quantity is related to the impedance mismatch as shown by (29). It is

generally difficult to derive an expression for r for large phased arrays at

out-of-band frequencies, but we can expect that r can be represented

approximately in the form,

r = Z B exp( i2kL ) . (30) n n

n

Here represents a normalized field scattered from some nth point on the

antenna, and represents the path length from the nth point back to the

feed point. For example, the nth point could be a slot in a slotted

waveguide array, and would be the distance from that slot to the feed

point. In such a case the energy could travel either along the exterior of

the antenna or back down the waveguide. The form of B^ is unknown, but we expect that B^ will be slowly varying with frequency compared to the

exponential factor in (30). The most rapid frequency variation in (30) will

come from the exponential with the longest path length L. If we replace k

by 2nf/v, where v is the free space velocity of light, then the minimum

sampling increment in frequency Af is given by

MirAfL/v = it or Af = v/(ML) .

For example, if we consider the ULSA where L is approximately the length of

the waveguide sections (= 6 m) , then Af = 12.5 MHz. This value is in

agreement with the rapid variations for both the ULSA and AWACS in figures

5-9. Although the expression in (31) was derived for the input reflection

coefficient, a similar expression can be derived for sampling the near field

where 2L is replaced by a path difference that is less than L. Thus the

input reflection coefficient controls the sampling requirement, and it

depends primarily on the size of the array. Details of the array, such as the element type or spacing, affect B^ in (30), but not Af in ( 3 1 )

A related sampling issue is the spatial sampling requirement for a fixed frequency. The planar near-field sampling requirement for calculation

of the far-field pattern is well known [18], and the planar sample spacing?

are

(32

1

1

Ax = Ay S A/2 .
If the requirement is that peaks in the near-field response are not missed,

then the spacings can be somewhat greater than A/2 as shown by geometrical

considerations. Note that the distance between the peaks and valleys of the

small ripples in figure 12 is just slightly greater than A/2. In any case,

the near-field response of large arrays at above-band frequencies requires

an enormous number of samples.

The most efficient out-of-band measurement is probably input

reflection coefficient because it yields the mismatch factor and the average

effective aperture as given by (6). This will not yield any pattern

information, but approximate principal plane patterns could be obtained from

near-field centerline measurements as shown in figures 12 and 13. For such

measurements it would be logical to choose frequencies where the input

reflection coefficient is small.

4. CONCLUSIONS

The response of antenna arrays has been studied theoretically and

experimentally. The effective aperture A is the most useful receiving

characteristic in interference calculations, and in general A depends on incidence angle, polarization, and the load impedance. It is possible to

average over incidence angle and polarization to obtain an average value as

in (6). However, it is difficult to calculate the impedance mismatch factor

at out-of-band frequencies, and this factor (or the reflection coefficient)

must generally be measured. A typical experimental result for the

reflection coefficient of the AWACS is shown in figure 9.

When pattern information is needed, then the directivity must be

computed or measured. Accurate values of the element excitations are not

generally known at out-of-band frequencies, and in section 2.2 the element

excitations are treated as random variables. An expression for the expected

value of the directive gain has been derived, and some far-field patterns

are computed in section 2.3. The effects of the randomness in the element

excitations are to decrease the directivity, to fill in the pattern nulls,

and to raise the sidelobe levels.

Near-field measurements on two large slotted waveguide arrays, ULSA

and AWACS, are shown in section 3. It is not possible to do a detailed

comparison with the theory, but the qualitative trends are consistent with

the theory. The narrow band nature of both arrays is evident in the

measurements, and this property is most easily determined by doing a sweptfrequency measurement of the antenna reflection coefficient. Near-field

measurements for far-field patterns are very time consuming for large arrays

at out-of-band frequencies, but approximate principal plane patterns can be

obtained from centerline measurements. Some examples are shown in figures

12-15.

A number of extensions to this work would be useful. Other antenna

arrays with different types of elements could be studied experimentally at

out-of-band frequencies. The slotted waveguides which were studied here had

sharp cutoffs below band because the waveguides were cut off. It would be

1 2
interesting to do measurements on another type of array that did not use a waveguide feed and did not have a low frequency cutoff. Input reflection

measurements should have high priority, and some near-field measurements and

far-field calculations could be done. The pulse response of general

antennas could be studied theoretically and experimentally. Some near-field

pulse responses were synthesized for the AWACS as shown in figures 11 and

12. It would be useful to study the feasibility of computing the far-field

pulse response of antennas from near-field pulse data (real or synthesized).

13
5. REFERENCES

[1] Lind, W.R. Out of band performance of antennas. Moore School of

Electrical Engineering Report No. 65-18; 1965.

[2] Levinson, D.S. Transmission-line measurement techniques. Rome Air

Development Center Technical Report No. RADC-TR-65-1 87 ; 1 965.

[3] Cown, B.J.; Cain, F.L. ; Duffy, E.G. Statistical prediction model for

EMC analysis of out-of-band phased array antennas. IEEE Trans.

Electromag. Compat., EMC-18: 1 63-170; 1976.

[4] Hill, D.A. Out-of-band response of reflector antennas. Nat. Bur.

Stand. (U.S.) Interagency Report 85-3021; 1985.

[5] Cown, B.J.; Ryan, C.E. ; Weaver, E.E. Antenna pattern measurements to

characterize the out-of-band behavior of reflector antennas. Georgia

Institute of Technology, Engineering Experiment Station, Report ECACTR- 83 -OO 3 ; 1 983.

[6] Tai , C.T. On the definition of the effective aperture of antennas.

IEEE Trans. Ant. Propagat. AP-9: 224-225; 1961.

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

[8] Engheta, N. ; Lee, K.S.H. ; Yang, F.C.; Madle, P. Archimedean spiral

antenna. Dikewood Memo Series on HPM Generic Coupling; 1985.

[9] Shifrin, Y.S. Statistical antenna theory. Boulder: Golem Press; 1971.

[10] Ruze, J. The effect of aperture errors on the antenna radiation

pattern. Suppl . Nuovo Cimento, 9: 364—3 80 ; 1 952.

[11] Bailin, L.L.; Ehrlich, M.J. Factors affecting the performance of

linear arrays. Proc. IRE, 41: 235-241; 1953.

[12] Gilbert, E.N. ; Morgan, S.P. Optimum design of directive antenna arrays

subject to random variations. Bell Sys. Tech. J. , 34: 637-663; 1955.

[13] Elliott, R.S. Mechanical and electrical tolerances for two-dimensional

scanning antenna arrays. IRE Trans. Ant. Propagat., AP-6: 114-120;

1 958.

[14] Ott, R.H.; Dybdal , R.B. Statistics for the directivity of an antenna

array. Aerospace Corporation, Report ATR-84 ( 8502) -1 ; 1984.

[15] Compton, R.T. ; Collin, R.E. Slot Antennas. in Antenna Theory (ed.

Collin, R.E.; Zucker, F.J.). New York: McGraw-Hill; 1969.

[16] Schrank, H.E. Low sidelobe phased array antennas. IEEE Antennas and

Propagation Society Newsletter, 25(2): 5-9; 1983.
[17] Dolph, C.L. A current distribution for broadside arrays which

optimizes the relationship between beam width and side-lobe level.

Proc. IRE, 34: 335-348; 1946.

[18] Kerns, D.M. Plane-wave scattering-matrix theory of antennas and

antenna-antenna interactions. Nat. Bur. Stand. (U.S.) Monograph 162;

1 981 .

[19] Brownell, F.H. Extended asymptotic eigenvalue distributions for

bounded domains in n-space. J. of Math, and Mech., 6: 119-166; 1957.
APPENDIX MULTIMODE ANTENNA EXAMPLE

For multimode antennas, (1) can be generalized to

2 M

A = jr- l P q D , 4tt . m m m

m=1

( A1 )

where M is the number of modes and subscript m denotes the properties of the

mth mode. For the open-ended waveguide example, we assume that each mode is

matched (q =1). If we assume that the incident field is randomly

m polarized and we average over all incidence angles, then (A1 ) becomes

<A> ( A2

)

For an open-ended waveguide, M is the number of propagating modes. In

general, M depends on the shape of the waveguide cross section and the

frequency in a rather complicated manner. However, if the cross-sectional

dimensions of the waveguide are electrically large, then M has a fairly

simple asymptotic value [19],

M

2ttS

(A3)

where S is the cross-sectional area of the waveguide as shown in figure 16.

It is easy to check (A3) for the case of a rectangular waveguide, but it is

more difficult for other shapes, such as circular. If we substitute (A3)

into (A2), then <A> reduces to

<A> = S/M . (AM)

The result in (AM) can also be derived directly by geometrical optics.

If we ignore edge diffraction, then the effective aperture is independent of

polarization and is approximately the projected waveguide area [M],

S cos9 , 0 S tt/2

A « { (A5)

0 , tt/2 < 0 < it

where 0 is the incidence angle shown in figure 16. If we average (A5) over

all incidence angles, then we obtain

2ir tt/2

<A> = t— / / S cos0 sin0 d0 d<}> = S/M , (A6)

Mtt - ~

16
>

17

Figure

1.

Geometry

for

an N

y

by

N

y

array

of

horizontal

magnetic

dipoles

on

a

ground

plane.
Figure 2

Nx yx 12

Directive gain D for a square array of magnetic dipole elements

for various values of c.

18
Q

Nx/x/2

Figure 3. Directive gain for a large rectangular array of magnetic d::o:*-

elements at the design frequency f and an above-band fr<-qu»-r: v

)i ^ ^

19
60

50

QQ

s

40

4—

> 30

H—

o

0

5 20

10

0

0

Maximum (c=0)

\

\\

\ Statistical (c=1)

dx — dy — X0I2

Nx = 120

N

y = 20 Limit

1 2 4 6 8 10

Figure 4. Directivity of a large rectangular array of magnetic dipole

elements as a function of frequency.

20
Amplitude

in

dB

Figure

Frequency in GHz

5. ULSA swept frequency response for probe at array center.

21
Frequency in GHz

Figure 6. ULSA swept frequency response for probe 25 cm below array center.

22
37.00

33.00

77.00

Figure 7. AWACS output-to-input ratio as a function of frequency for pr :

• at array center.

23
Output-to-input

ratio

in

dB

28.00

•33.00

-38.00

-43.00

48.00 I

-63.00

• 68.00

-63.00

- 68.00

-73 00

78.00

1.60 3.20 4.90 6.60 8.30 10.00 11.70 13.40 iS.10 16.80 18.50

Frequency in GHz

Figure 8. AWACS output-to-input ratio as a function of frequency for probe

located 89 cm to the side of array center

24
Reflection

coefficient

amplitude

in

dB

Frequency in GHz

Figure 9. Amplitude of AWACS input reflection coefficient as a function cf

frequency.

25
Relative

power

in

dB

-OOOO

-.0000

- 10. 0000

15.0000

- 20.0000

-25.0000

30.0000

35.0000

-40.00C

-45.0000

50.0000

.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 8C.CH) 90.00 100.00

Time in ns

Figure 10. AWACS time domain power spectrum for probe at array center.

26
Relative

power

in

dB

.0000

• 5 . 0000

10.0000 -

- 15.0000

• 20.0000

-25.0000

30.0000

• 35.0000

40.0000

•^5.oooo

- 50 . 0000

.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 9: .oo roo OO

Time in ns

Figure 1 1

.

AWACS time domain power spectrum for probe located 89 cm to the

side of array center.

27
Amplitude

In

dB

Figure 12. AWACS one-dimensional near-field x-scan at 3.94 GHz.

28
Amplitude

in

dB

Azimuth in degrees

Figure 13. AWACS one-dimensional fai—field x-scan at 3 - 9

J • .

29
Amplitude

in

dB

Figure 14. AWACS one-dimensional near-field x-scan at in-band frequency.

30
Amplitude

In

dB

I

•

i

Figure 15. AWACS one-dimensional far-field x-scan at in-band

31
Figure 16. Open-ended waveguide of cross-sectional area S. The incident

elevation angle is 0.

32
HBS-1I4A REV. 2 -80) U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

SHEET (See instructions)

REPORT NO.

NBSIR 86-3047 June 1986

4 . TITLE AND SUBTITLE

Out-of-Band Response of Antenna Arrays

5. AUTHOR(S)

David A. Hill and Michael H. Francis

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No.

national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Defense Nuclear Agency

Alexandria, Virginia 22310

10. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

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 response of antenna arrays to out-of-band frequencies has

been analyzed using the effective aperture approach. An average

value of effective aperture can be obtained by averaging the

incidence angle and the polarization of the incidence field.

Far-field patterns have also been calculated by treating the

array element excitations as random variables. The randomness

in the element excitations causes a decrease in directivity and

an increase in sidelobe level. Out-of-band measurements of

reflection coefficient and near-field response have been made on two large slotted-waveguide arrays for frequencies from 2 to 1

8

GHz. Both arrays are narrow band, and this is easily explained

by the large impedance mismatch at out-of-band frequencies.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by : a.ntenna array; directivity; impedance mismatch; near-field measurements;

uut-of-band response; polarization; slotted waveguide.

13. AVAILABILITY

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