8/11/2019 ch0407

1/17

8/11/2019 ch0407

2/17

ECE 354 Lecture Notes, Chapter 4

33

The patterns of a short dipole pointing along the x-axis are plotted in Figures 4.25 (a-c).This pattern is identical to the dipole pattern of Figures 4.23 (a-c), except for aninterchange of the axis. The patterns of an N =5 endfire array with main beam along + z are once more plotted in Figures 4.25 (d-f). The products of the two patterns are plottedin Figures 4.25 (g-i). Note that, in contrast to the dipole array studied in the previousexample, the present does radiate along the z -axis. This is because the dipole pattern inthis case is maximum along the z -axis, and not zero as in the previous example.However, again in contrast to the array studied in the previous example, the pattern of thecurrent array is no more axisymmetric with respect to the z -axis.

________________________________________________________________________

Example 4.5: A non equall y spaced lin ear arr ay.

Consider the array depicted if Figure 4.26(a). The array consists of 4 nonequally

spaced but equally excited isotropically radiating elements. The center elements arespaced one wavelength apart, the outer elements are spaced only half a wavelength apart.As shown in Figure 4.26(b) and (c), this array can be considered as an equally excited,two-element array with an interelement spacing of 1.5 wavelengths, where the element of the array itself consists of a two element equally spaced array with an interelementspacing of half a wavelength. All antenna elements are fed in phase, hence k 0 0=

applies to both arrays. To construct the array factor associated with this antenna,construct the array factors associated with its constituents, and multiply out. The various

patterns associated with the array factor of the array of Figure 4.26(c) are shown in Figure4.27 (a-c). The patterns of the array factor of the "master" element of Figure 4.26 (b) areshown in Figure 4.27 (d-f). The array factor pattern of the antenna of Figure 4.26 (a) isobtained by multiplying the previous patterns and is shown in Figure 4.27 (g-i).

________________________________________________________________________

Example 4.6: the binomial arr ay.

A binomial array is an equally spaced array which is constructed to have nosidelobes. The array is obtained starting from a 2 element uniformly excited array withinterelement spacing d = /2 (Figure 4.28 (a)), the field patterns of which are shown in

Figure 4.29 (a-c). Higher order binomial arrays are constructed as illustrated in Figure4.28 (b-c): in general, an nth order binomial array is obtained by superimposing two (n-1)th order binomial arrays such that they overlap in n-1 elements. Hence, the nth order

binomial array can be considered to be a two element array: each element is an two (n-1)th order binomial array, and these two arrays are spaced d = /2 apart. All arrays are fedin phase, hence k 0 0= for both arrays. The fact that the two arrays overlap does not poseany difficulty when applying array theory. Hence, the pattern of nth order binomial array

8/11/2019 ch0407

3/17

ECE 354 Lecture Notes, Chapter 4

34

can be constructed by multiplying the pattern of the ( n-1)th order array with the pattern of the two element array of Figure 4.28 (a); for example, the pattern of the three elementarray is obtained by squaring the pattern of Figure 4.28 (a), and is shown in Figure 4.29(d-f). Patterns of fourth order binomial arrays are shown in Figures 4.29 (g-i).

_______________________________________________________________________

Example 4.7: A 3 x 2 planar arr ay.

Consider the two-dimensional array shown in Figure 4.30 (a), consisting of a three bytwo array of isotropic elements, all excited in phase. The interelement spacing along xand y measures d = /2. This antenna can be regarded as a two element LCPESA withinterelement spacing d = /2 and phase progression k y0 0= (the y subscript is added toindicate that this phase shift applies to the y-directed array), where each element itself is athree element LCPESA with d = /2 and k x0 0= , as is illustrated in Figures 4.30 (b-c).

The pattern of this array can be obtained by plotting the patterns of these constituents, and by multiplying out. The patterns of the LCPESA of Figure 4.30 (b) are shown in Figures4.31 (a-c). The patterns of the LCPESA of Figure 4.30 (c) are shown in Figures 4.31 (d-f). The patterns of the three by two array of Figure 4.30 (a) is obtained by multiplying the

patterns of the arrays of Figures 4.30 (b-c) and are shown in Figures 4.31 (g-i). The three-dimensional radiation pattern of this antenna is shown in Figure 4.32.

In a similar vein, Figure 4.33 shows the construction of the pattern of a five by fiveLCPESA planar array residing in the x-y plane. Again, it is assumed that d = /2 and thatall elements are fed in phase. Figure 4.34 illustrates the construction of a five by fiveLCPESA with interelement spacing d = /2 residing in the x-y plane, but assumes a phase

progression k x0 4= / , no phase progression is assumed along y, hence k y0 0= . Inother words, a phase progression from column to column is assumed. One observes thatthe scanning action of the beam in the y-z plane, the plane perpendicular to the axis alongwhich the phase progression is applied. In both Figures 4.33 and 4.34, the pattern of theLCPESA residing along x are plotted in insets (a-c), the patterns of the LCPESA residingalong y are plotted in insets (d-f), and the planar array patterns are plotted in insets (g-i).The three-dimensional pattern of the antennas considered in Figures 4.33 and 4.34 areshown in Figures 4.35 and 4.36.

8/11/2019 ch0407

4/17

8/11/2019 ch0407

5/17

ECE 354 Lecture Notes, Chapter 4

36

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.23: Patterns of the enfire array (Type I) of short colinear dipoles discussed in Example 4.3. (a-c)

Dipole patterns, (d-f) array factors, (g-i) antenna patterns.

8/11/2019 ch0407

6/17

ECE 354 Lecture Notes, Chapter 4

37

x

y

z

d

Figure 4.24: A LCPESA of 6 perpendicular short dipoles

8/11/2019 ch0407

7/17

ECE 354 Lecture Notes, Chapter 4

38

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.25: Patterns of the enfire array (Type I) of short perpendicular dipoles discussed in Example 4.4.

(a-c) Dipole patterns, (d-f) array factors, (g-i) antenna patterns.

8/11/2019 ch0407

8/17

ECE 354 Lecture Notes, Chapter 4

39

x

z

y

1

1

1

1

/2

/2

x

z

y

1

1

3/2

x

z

y

1

1

/2=

(a) (b)

(c)

Figure 4.26 (a) nonequally spaced array under consideration in Example 4.5, (b) equally spaced arrayequivalent to the array shown in (a), provided that the elements of the array are as shown in (c).

8/11/2019 ch0407

9/17

ECE 354 Lecture Notes, Chapter 4

40

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.27: Patterns of the arrays considered in Example 4.5 (a-c) patterns of the array shown in Figure

4.26 (c), (d-f) patterns for the array shown in Figures 4.26 (b), and (g-i) patterns of the composite array

shown in Figure 4.26 (a).

8/11/2019 ch0407

10/17

ECE 354 Lecture Notes, Chapter 4

41

z

/21

1=/2

/2

+/21

1

11

(a) (b)

/2/21

2

1/2

(c)

1

2

1

1

2

1

+ =

1

3

3

1

Figure 4.28. Construction of binomial arrays. (a) 2 element array, (b) 3 element binomial array obtained bysuperimposing two 2 element arrays, (c) 4 element array obtained by superimposing two three element

arrays .

8/11/2019 ch0407

11/17

ECE 354 Lecture Notes, Chapter 4

42

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.29: Patterns of binomial arrays (a-c) N =2, (d-f) N =3, (g-I) N =4. Note that as N increases, no

sidelobes are introduced, but the main beam becomes narrower and narrower.

8/11/2019 ch0407

12/17

ECE 354 Lecture Notes, Chapter 4

43

x

y

z

d

d d

(a)

d

x

y

z

d d

x

y

z

d

(b) (c)

Figure 4.30. (a) A 3 x 2 array and its constituents (b) a 3 element array along the x-axis, and (b) a two

element array along the y-axis, the elements of which are shown in (b).

8/11/2019 ch0407

13/17

ECE 354 Lecture Notes, Chapter 4

44

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.31: Patterns of the 3 x 2 array discussed in Example 5 (a-c) patterns of the LCPESA of Figure 4.30

(b), (b) patterns of the LCPESA of Figure 4.30 (c), and (c) patterns of the 3 x 2 array of Figure 4.30 (a).

8/11/2019 ch0407

14/17

ECE 354 Lecture Notes, Chapter 4

45

.

Figure 4.32: 3D Pattern of the 3 x 2 array discussed in Example 4.7 and shown in Figure 4.30 (a).

8/11/2019 ch0407

15/17

ECE 354 Lecture Notes, Chapter 4

46

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.33: Patterns of a 5 x 5 LCPESA planar array discussed in Example 4.7.

8/11/2019 ch0407

16/17

ECE 354 Lecture Notes, Chapter 4

47

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.34 : Patterns of a 5 x 5 LCPESA planar array discussed in Example 4.7 with phase progression

along x.

8/11/2019 ch0407

17/17

ECE 354 Lecture Notes, Chapter 4

48

Figure 4.35: 3D Pattern of a 5 x 5 LCPESA planar array discussed in Example 4.7 without phase progression.

Figure 4.36: 3D Pattern of a 5 x 5 LCPESA planar array discussed in Example 4.7 with phase progression.