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`

`ANALYSIS AND DESIGN
`
`SECOND EDITION
`
`~. CONSTANTINE A.BALE1QMS§A-
`Ex 2017
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`“4
`
`

`

`4.5 Flnlte Laigfli Dipole
`
`153
`
`p = tk
`
`y = 1:112
`
`W some mathematical manipulations. (+60) takes the faint of
`
`(4-61b)
`
`(4—61c)
`
`(4-62a)
`
`In a similar manner, or by using the established relationship between the E, and
`H‘ in the far-field as given by (3-58b) or (4-27). the total H, component can be
`written as
`
`(46%)
`
`4.5.3 Power Density, Radiation Intensity, and Radiation
`Resistance
`
`For the dipole, the average Poynting vector can be written as
`
`w., - -;-Rc[E x m] = éRelfigng 5,113: = £12: [3,,on 9,53]
`no]! mg“ a) _ cos(‘7!) I
`
`_
`_ L 2 _
`wav - 51W” " i’ZTIIE'l — "8172’:
`sine
`
`and the radiation intensity as
`
`(4-63)
`
`"0'2
`U=12Wu=fl§7r1
`
`("’
`
`’)
`
`cos —cos 0 — cos -
`2
`2
`
`(”)2
`
`sin 8
`
`(4-64)
`
`The normalized (lo 0 (IR) elevation power pallems. as given by (4-64) for
`l '— A/ll. A/Z. 3AM. and A fll‘C shown plotted in Figure 4.6. The cuITent distribution of
`each is given by (4-56). The power patterns for an infinilcsliutal dipole I < A
`(II ~ sin2 9) is also included for comparison. As the length of the nntcnna increases,
`the beam hccomcs nan'owcr. Because of that. the dilectivity should also incncase with
`length. It is found that the 3—dB beumwidth of each is equal to
`
`9
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`I < A
`
`1 = M4
`
`3-db beamwidth = 90°
`
`3-dB beamwidlh = 87°
`
`
`
`
`1 = M2
`3-dB beamwidlh = 78"
`I = 3H4
`3-dB beamwidth - 64°
`
`
`3-dB beamwidth = 47.8" I = A
`
`(4-65)
`
`

`

` Elevation plane amplitude paltcrns for a thin dipole w
`
`As the length of the dipole increases beyond one wavelength (l > A), the number
`of lobes begin to increase. The normalized power pattem for a dipole with I = 1.25)
`is shown in Figure 4.7. In Figure 4.7(a) the three-dimensional pattern is illustrated
`using the soflware from [5], while in Figure 4.7(b) the two-dimensional (elevation
`pattern) is depicted. For the three—dimensional illustration 3 90° angular section of the
`
`attem has been omitted to illustrate the elevation lane directional attern variations.
`
`

`

`Figure 4.7 Tln'ec- and two-dimensional amplitude pallcrns for a
`
`thin dipole of] = 1.25A and sinusoidal cun‘enl distribution.
`
`I _-I -
`
`I
`
`I
`
`I
`B
`
`I
`
`R
`
`I
`
`[All
`
`[ENE
`
`

`

`
`
`(‘IIIIIIlI‘IIIIbIIIII III: 11qu IIII-IL1|I:-IIII
`
`II?lInu l'll\\‘Il'I‘.
`
`:HH‘VIIHIl
`
`Fract-usS.A.
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`Fractus S.A.
`Ex. 2017
`ZTE (USA), Inc. v. Fractus S.A.; IPR2018-01461
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`

`

`11.3.2 Conical Spiral
`
`LOG-PERIODIC ANTENNAS
`
`619
`
`(cid:16)
`
`df
`
`dθ
`
`The shape of a nonplanar spiral can be described by defining the derivative of f (θ )
`to be
`(θ ) = Aδ(β − θ )
`= f
`(11-19)
`in which β is allowed to take any value in the range 0 ≤ β ≤ π. For a given value of
`β, (11-19) in conjunction with (11-8) describes a spiral wrapped on a conical surface.
`The edges of one conical spiral surface are defined by
`r2 = r
`e(a sin θ0)φ = r
`ebφ
`r3 = r
`ea sin θ0φ = r
`ea sin θ0(φ−δ)
`
`(11-20a)
`
`(11-20b)
`
`(cid:16)2
`
`(cid:16)2
`
`(cid:16)2
`
`(cid:16)3
`
`where
`
`= r
`−(a sin θ0)δ
`(11-20c)
`and θ0 is half of the total included cone angle. Larger values of θ0 in 0 ≤ θ ≤ π/2 rep-
`resent less tightly wound spirals. These equations correspond to (11-15a)–(11-15c) for
`the planar surface. The second arm of a balanced system can be defined by shifting each
`◦
`of (11-20a)–(11-20c) by 180
`, as was done for the planar surface by (11-17)–(11-18a).
`A conical spiral metal strip antenna of elliptical polarization is shown in Figure 11.4.
`The conducting conical spiral surface can be constructed conveniently by forming,
`using printed-circuit techniques, the conical arms on the dielectric cone which is also
`used as a support. The feed cable can be bonded to the metal arms which are wrapped
`around the cone. Symmetry can be preserved by observing the same precautions, like
`the use of a dummy cable, as was done for the planar surface.
`A distinct difference between the planar and conical spirals is that the latter pro-
`vides unidirectional radiation (single lobe) toward the apex of the cone with the
`maximum along the axis. Circular polarization and relatively constant impedances
`are preserved over large bandwidths. Smoother patterns have been observed for uni-
`directional designs. Conical spirals can be used in conjunction with a ground plane,
`with a reduction in bandwidth when they are flush mounted on the plane.
`
`e
`
`(cid:16)2
`
`(cid:16)3
`
`r
`
`11.4 LOG-PERIODIC ANTENNAS
`
`Another type of an antenna configuration, which closely parallels the frequency inde-
`pendent concept, is the log-periodic structure introduced by DuHamel and Isbell [4].
`Because the entire shape of it cannot be solely specified by angles, it is not truly
`frequency independent.
`
`11.4.1 Planar and Wire Surfaces
`
`A planar log-periodic structure is shown in Figure 11.5(a). It consists of a metal strip
`whose edges are specified by the angle α/2. However, in order to specify the length
`from the origin to any point on the structure, a distance characteristic must be included.
`
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`

`620
`
`FREQUENCY INDEPENDENT ANTENNAS, ANTENNA MINIATURIZATION, AND FRACTAL ANTENNAS
`
`
`
`Figure 11.4 Coniml spiral metal strip antenna. (SOURCE Antennas. Antenna Masts and Mount—
`ing Adaptors, American Electronic Laboratories, Inc., Lansdale, Pa., Catalog 7.5M-7-79. Cour-
`tesy of American Electronic Laboratories, Inc., Montgomeryville, PA 18936 USA).
`
`
`
`Radiation
`pattern
`
`'
`
`=V
`
`.
`
`.
`
`‘
`
`.
`
`.
`
`.
`
`'
`__
`
`-
`
`7— _ _‘
`
`Pal tern goes;
`.
`to yer-u
`
`Z
`
`3
`I
`
`-
`
`Pn'. turn gun
`In [L m
`
` '
`
`.
`--.,~
`ll‘l Lug-periodic "1ch strip antenna FraciugJS‘A
`-
`Figure 11.5 Typical metal strip log—paiodic configuration and Etc
`-
`xT'fifi‘i‘?“
`
`.
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`

`LOG-PERIODIC ANTENNAS
`
`621
`
`In spherical coordinates (r, 0, 45) the shape of the structure can be written as
`
`0 = periodic function of [b ln(r)]
`
`( 11-21)
`
`An example of it would be
`
`0 = oosin [bIn (in
`
`'0
`
`(11-22)
`
`It is evident from (1 1-22) that the values of 0 are repeated whenever the logarithm
`of the radial frequency ln(w) = ln(27rf) differs by Zn/b. The performance of the
`system is then periodic as a function of the logarithm of the frequency; thus the name
`logarithmic-periodic or log-periodic.
`A typical log-periodic antenna configuration is shown in Figure ll.5(b). It consists
`of two coplanar arms of the Figure 11.5(a) geometry. The pattern is unidirectional
`toward the apex of the cone formed by the two arms, and it is linearly polarized.
`Although the patterns of this and other log-periodic structures are not completely
`frequency independent, the amplitude variations of certain designs are very slight.
`Thus practically they are frequency independent.
`Log-periodic wire antennas were introduced by DuHamel [4]. While investigat-
`ing the current distribution on log-periodic surface structures of the form shown in
`Figure 11.6(a), he discovered that the fields on the conductors attenuated very sharply
`with distance. This suggested that perhaps there was a strong current concentration at
`or near the edges of the conductors. Thus removing part of the inner surface to form a
`wire antenna as shown in Figure 11.6(b) should not seriously degrade the performance
`of the antenna. To verify this, a wire antenna, with geometrical shape identical to the
`pattern formed by the edges of the conducting surface, was built and it was investigated
`
`Feed
`
`ta] Planar
`
`(b) Wire
`
`Figure 11.6 Planar and wire logarithmically periodic ameEiIaC’tus S-A_
`Ex. 2017
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`

`

`622
`
`FREQUENCY INDE’ENDENT ANTENNAS, ANTENNA MINIATURIZA'HON, AND FRACTAL ANTENNAS
`
`Fred
`
`la) Planar
`
`(bl wire
`
`Figure 11.7 Planar and wire trapezoidal toothed log-periodic antennas.
`
`experimentally. As predicted, it was found that the performance of this antenna was
`almost identical to that of Figure 11.6(a); thus the discovery of a much simpler, lighter
`in weight, cheaper, and less wind resistant antenna. Nonplanar geometries in the form
`of a V, formed by bending one arm relative to the other, are also widely used.
`If the wires or the edges of the plates are linear (instead of curved), the geometries
`of Figure 11.6 reduce, respectively, to the trapezoidal tooth log-periodic structures of
`Figure 11.7. These simplifications result in more convenient fabrication geometries
`with basically no loss in operational performance. There are numerous other bizarre
`but practical configurations of log-periodic structures, including log-periodic arrays.
`If the geometries of Figure 11.6 use uniform periodic teeth, we define the geometric
`ratio of the log-periodic structure by
`
`
`
`and the width of the antenna slot by
`
`rn
`
`Rn+l
`
`x =
`
`(11-23)
`
`(11-24)
`
`The geometric ratio r of (1 1-23) defines the period of operation. For example, if two
`frequencies f, and f2 are one period apart, they are related to the geometric ratio r by
`
`(ll-25)
`
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`

`x'
`
`’
`
`LOG-PERIODIC ANTENNAS
`
`623
`
`‘7
`
`IX
`
`(3:11:1(113)
`
`°
`
`b4
`
`6.;
`
`14
`
`m Ant-mun
`
`Frequency (GHLI
`
`(‘1) Gau'u rlramucustiu
`
`Figure 11.8 Linearly polarized flush—mounted cavity-backed log—periodic slot antenna and typ-
`ical gain characteristics. (SOURCE: Antennas, Antenna Masts and Mounting Adaptors, American
`Electronic Laboratories, Inc., Lansdale, Pa., Catalog 7.5M—7-79. Courtesy of American Elec-
`tronic Laboratories, Inc., Montgomeryville, PA 18936 USA).
`
`Extensive studies on the performance of the antenna of Figure 11.6(b) as a function
`of 01,19, 1', and x, have been performed [9]. In general, these structures performed
`almost as well as the planar and conical structures. The only major difference is that
`the log-periodic configurations are linearly polarized instead of circular.
`A commercial lightweight, cavity-backed, linearly polarized, flush-mounted log-
`periodic slot antenna and its associated gain characteristics are shown in Figures 11.8(a)
`and (b). Typical electrical characteristics are: VSWR—Zzl; E-plane bearnwidth—
`70°; H-plane bearnwidth—70°. The maximum diameter of the cavity is about 2.4
`in. (6.1 cm), the depth is 1.75 in. (4.445 cm), and the weight is near 5 oz (0.14 kg).
`
`11.4.2 Dipole Array
`
`To the layman, the most recognized log-periodic antenna structure is the configuration
`introduced by [shell [5] which is shown in Figure 11.9(a). It consists of a sequence of
`side-by-side parallel linear dipoles forming a coplanar array. Although this antenna has
`slightly smaller directivities than the Yagi—Uda array (7-12 dB), they are achievable
`and maintained over much wider bandwidths. There are, however, major differences
`between them.
`
`While the geometrical dimensions of the Yagi—Uda array elements do not follow
`any set pattern, the lengths (ln’s), spacings (Rn’s), diameters (dn’s), and even gap
`spacings at dipole centers (sn’s) of the log-periodic array increase logarithmically as
`defined by the inverse of the geometric ratio r. That is,
`
`( 1 1-26)
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`FREQUENCY INDEPENDBJT ANTENNAS, ANTENNA MINIATURIZATION, AND FRACTAL ANTENNAS
`
`
`
`(b) Stranghl connection
`
`It) Crissvross connection
`
`Id) Coaxial connection
`
`Figure 11.9 Log-periodic dipole array and associated connections.
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`

`Another parameter that is usually associated with a log-periodic dipole array is the
`spacing factor σ defined by
`
`LOG-PERIODIC ANTENNAS
`
`625
`
`σ = Rn+1 − Rn
`2ln+1
`
`(11-26a)
`
`Straight lines through the dipole ends meet to form an angle 2α which is a characteristic
`of frequency independent structures.
`Because it is usually very difficult to obtain wires or tubing of many different diame-
`ters and to maintain tolerances of very small gap spacings, constant dimensions in these
`can be used. These relatively minor factors will not sufficiently degrade the overall
`performance.
`While only one element of the Yagi–Uda array is directly energized by the feed
`line, while the others operate in a parasitic mode, all the elements of the log-periodic
`array are connected. There are two basic methods, as shown in Figures 11.9(b) and
`11.9(c), which could be used to connect and feed the elements of a log-periodic dipole
`array. In both cases the antenna is fed at the small end of the structure.
`The currents in the elements of Figure 11.9(b) have the same phase relationship as
`the terminal phases. If in addition the elements are closely spaced, the phase progression
`of the currents is to the right. This produces an end-fire beam in the direction of the
`longer elements and interference effects to the pattern result.
`It was recognized that by mechanically crisscrossing or transposing the feed between
`◦
`adjacent elements, as shown in Figure 11.9(c), a 180
`phase is added to the terminal
`of each element. Since the phase between the adjacent closely spaced short elements is
`almost in opposition, very little energy is radiated by them and their interference effects
`are negligible. However, at the same time, the longer and larger spaced elements radiate.
`The mechanical phase reversal between these elements produces a phase progression
`so that the energy is beamed end fire in the direction of the shorter elements. The
`most active elements for this feed arrangement are those that are near resonant with a
`combined radiation pattern toward the vertex of the array.
`The feed arrangement of Figure 11.9(c) is convenient provided the input feed line is
`a balanced line like the two-conductor transmission line. Using a coaxial cable as a feed
`◦
`line, a practical method to achieve the 180
`phase reversal between adjacent elements
`is shown in Figure 11.9(d). This feed arrangement provides a built-in broadband balun
`resulting in a balanced overall system. The elements and the feeder line of this array are
`usually made of piping. The coaxial cable is brought to the feed through the hollow part
`of one of the feeder-line pipes. While the outside conductor of the coax is connected
`to that conductor at the feed, its inner conductor is extended and it is connected to the
`other pipe of the feeder line.
`If the geometrical pattern of the log-periodic array, as defined by (11-26), is to
`be maintained to achieve a truly log-periodic configuration, an infinite structure would
`result. However, to be useful as a practical broadband radiator, the structure is truncated
`at both ends. This limits the frequency of operation to a given bandwidth.
`The cutoff frequencies of the truncated structure can be determined by the elec-
`trical lengths of the longest and shortest elements of the structure. The lower cutoff
`frequency occurs approximately when the longest element is λ/2; however, the high
`cutoff frequency occurs when the shortest element is nearly λ/2 only when the active
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`626
`
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`
`region is very narrow. Usually it extends beyond that element. The active region of
`the log-periodic dipole array is near the elements whose lengths are nearly or slightly
`smaller than λ/2. The role of active elements is passed from the longer to the shorter
`elements as the frequency increases. Also the energy from the shorter active elements
`traveling toward the longer inactive elements decreases very rapidly so that a negligible
`amount is reflected from the truncated end. The movement of the active region of the
`antenna, and its associated phase center, is an undesirable characteristic in the design
`of feeds for reflector antennas (see Chapter 15). For this reason, log-periodic arrays
`are not widely used as feeds for reflectors.
`The decrease of energy toward the longer inactive elements is demonstrated in
`Figure 11.10(a). The curves represent typical computed and measured transmission-
`line voltages (amplitude and phase) on a log-periodic dipole array [10] as a function
`of distance from its apex. These are feeder-line voltages at the base of the elements
`of an array with τ = 0.95, σ = 0.0564, N = 13, and ln/dn = 177. The frequency
`of operation is such that element No. 10 is λ/2. The amplitude voltage is nearly
`constant from the first (the feed) to the eighth element while the corresponding phase is
`uniformly progressive. Very rapid decreases in amplitude and nonlinear phase variations
`are noted beyond the eighth element.
`The region of constant voltage along the structure is referred to as the transmission
`region, because it resembles that of a matched transmission line. Along the structure,
`◦
`phase change for every λ/4 free-space length of transmission line.
`there is about 150
`This indicates that the phase velocity of the wave traveling along the structure is
`υp = 0.6υ0, where υ0 is the free-space velocity. The smaller velocity results from the
`shunt capacitive loading of the line by the smaller elements. The loading is almost
`constant per unit length because there are larger spacings between the longer elements.
`The corresponding current distribution is shown in Figure 11.10(b). It is noted that
`the rapid decrease in voltage is associated with strong current excitation of elements
`7–10 followed by a rapid decline. The region of high current excitation is designated
`as the active region, and it encompasses 4 to 5 elements for this design. The voltage
`and current excitations of the longer elements (beyond the ninth) are relatively small,
`reassuring that the truncated larger end of the structure is not affecting the perfor-
`mance. The smaller elements, because of their length, are not excited effectively. As
`the frequency changes, the relative voltage and current patterns remain essentially the
`same, but they move toward the direction of the active region.
`There is a linear increase in current phase, especially in the active region, from the
`shorter to the longer elements. This phase shift progression is opposite in direction to
`that of an unloaded line. It suggests that on the log-periodic antenna structure there is
`a wave that travels toward the feed forming a unidirectional end-fire pattern toward
`the vertex.
`The radiated wave of a single log-periodic dipole array is linearly polarized, and
`it has horizontal polarization when the plane of the antenna is parallel to the ground.
`Bidirectional patterns and circular polarization can be obtained by phasing multiple log-
`periodic dipole arrays. For these, the overall effective phase center can be maintained
`at the feed.
`If the input impedance of a log-periodic antenna is plotted as a function of frequency,
`it will be repetitive. However, if it is plotted as a function of the logarithm of the
`frequency, it will be periodic (not necessarily sinusoidal) with each cycle being exactly
`identical to the preceding one. Hence the name log-periodic, because the variations
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`

`Figure 11.10 Measured and computed voltage and current distributions on a log-periodic dipole array of 13 elements with frequency such that l10 = λ/2.
`(SOURCE: R. L. Carrel, “Analysis and Design of the Log-Periodic Dipole Antenna,” Ph.D. Dissertation, Elec. Eng. Dept., University of Illinois, 1961,
`University Microfilms, Inc., Ann Arbor, Michigan).
`
`627
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`

`628
`
`FREQUENCY INDEPENDENT ANTENNAS, ANTENNA MINIATURIZATION, AND FRACTAL ANTENNAS
`
`Figure 11.11 Typical input impedance variation of a log-periodic antenna as a function of the
`logarithm of the frequency.
`
`(cid:11)
`
`are periodic with respect to the logarithm of the frequency. A typical variation of the
`impedance as a function of frequency is shown in Figure 11.11. Other parameters that
`undergo similar variations are the pattern, directivity, beamwidth, and side lobe level.
`The periodicity of the structure does not ensure broadband operation. However, if
`the variations of the impedance, pattern, directivity, and so forth within one cycle
`are made sufficiently small and acceptable for the corresponding bandwidth of the
`cycle, broadband characteristics are ensured within acceptable limits of variation. The
`total bandwidth is determined by the number of repetitive cycles for the given trun-
`cated structure.
`The relative frequency span of each cycle is determined by the geometric ratio
`∗
`as defined by (11-25) and (11-26).
`Taking the logarithm of both sides in (11-25)
`reduces to
` = ln(f2) − ln(f1) = ln
`(11-27)
`The variations that occur within a given cycle (f1 ≤ f ≤ f2 = f1/τ ) will repeat
`identically at other cycles of the bandwidth defined by f1/τ n−1 ≤ f ≤ f1/τ n, n =
`1, 2, 3, . . ..
`◦ ≤ α ≤ 45
`◦
`Typical designs of log-periodic dipole arrays have apex half angles of 10
`and 0.95 ≥ τ ≥ 0.7. There is a relation between the values of α and τ . As α increases,
`the corresponding τ values decrease, and vice versa. Larger values of α or smaller
`values of τ result in more compact designs which require smaller number of elements
`separated by larger distances. In contrast, smaller values of α or larger values of τ
`require a larger number of elements that are closer together. For this type of a design,
`there are more elements in the active region which are nearly λ/2. Therefore the varia-
`tions of the impedance and other characteristics as a function of frequency are smaller,
`because of the smoother transition between the elements, and the gains are larger.
`Experimental models of log-periodic dipole arrays have been built and measurements
`were made [6]. The input impedances (purely resistive) and corresponding directivities
`
`1 τ
`
`(cid:10)
`
`∗
`In some cases, the impedance (but not the pattern) may vary with a period which is one-half of (11-27).
`That is, = 1
`2 ln(1/τ ).
`
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`

`LOG-PERIODIC ANTENNAS
`
`629
`
`Input Resistances (Rin in ohms) and Directivities (dB above isotropic) for
`TABLE 11.1
`Log-Periodic Dipole Arrays
`τ = 0.81
`
`τ = 0.89
`
`τ = 0.95
`
`α
`
`Rin(ohms)
`
`D0(dB)
`
`Rin(ohms)
`
`D0(dB)
`
`Rin(ohms)
`
`D0(dB)
`
`10
`
`12.5
`
`15
`
`17.5
`
`20
`
`25
`
`30
`
`35
`
`45
`
`98
`
`—
`
`—
`
`—
`
`—
`
`—
`
`80
`
`—
`
`65
`
`—
`
`—
`
`7.2
`
`—
`
`—
`
`—
`
`—
`
`—
`
`5.2
`
`82
`
`77
`
`—
`
`76
`
`74
`
`63
`
`64
`
`56
`
`59
`
`9.8
`
`—
`
`—
`
`7.7
`
`—
`
`7.2
`
`—
`
`6.5
`
`6.2
`
`77.5
`
`10.7
`
`—
`
`—
`
`62
`
`—
`
`—
`
`54
`
`—
`
`—
`
`—
`
`—
`
`8.8
`
`—
`
`8.0
`
`—
`
`—
`
`—
`
`(SOURCE: D. E. Isbell, “Log Periodic Dipole Arrays,” IRE Trans. Antennas Propagat., Vol. AP-8,
`pp. 260–267, May 1960.  (1960) IEEE.)
`
`(above isotropic) for three different designs are listed in Table 11.1. Larger directivi-
`ties can be achieved by arraying multiple log-periodic dipole arrays. There are other
`configurations of log-periodic dipole array designs, including those with V instead of
`linear elements [11]. This array provides moderate bandwidths with good directivities
`at the higher frequencies, and it is widely used as a single TV antenna covering the
`entire frequency spectrum from the lowest VHF channel (54 MHz) to the highest UHF
`(806 MHz). Typical gain, VSWR, and E- and H -plane half-power beamwidths of com-
`mercial log-periodic dipole arrays are shown in Figures 11.12(a), (b), (c), respectively.
`The overall length of each of these antennas is about 105 in. (266.70 cm) while the
`largest element in each has an overall length of about 122 in. (309.88 cm). The weight
`of each antenna is about 31 lb ((cid:9)14 kg).
`
`11.4.3 Design of Dipole Array
`
`The ultimate goal of any antenna configuration is the design that meets certain specifi-
`cations. Probably the most introductory, complete, and practical design procedure for
`a log-periodic dipole array is that by Carrel [10]. To aid in the design, he has a set of
`curves and nomographs. The general configuration of a log-periodic array is described
`in terms of the design parameters τ , α, and σ related by
`1 − τ
`4σ
`
`(cid:20)
`
`(cid:21)
`
`α = tan
`−1
`
`(11-28)
`
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`

`

`630
`
`FREQUENCY INDEPENDENT ANTENNAS, ANTENNA MINIATURIZATION, AND FRACTAL ANTENNAS
`
`Figure 11.12 Typical gain, VSWR, and half-power beamwidth of commercial log-periodic
`dipole arrays. (SOURCE: Antennas, Antenna Masts and Mounting Adaptors, American Electronic
`Laboratories, Inc., Lansdale, Pa., Catalog 7.5M-7-79. Courtesy of American Electronic Labora-
`tories, Inc., Montgomeryville, PA 18936 USA).
`
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`
`

`

`0.22
`
`LOG-PERIODIC ANTENNAS
`
`631
`
`0.20
`
`0.18
`
`0.16
`
`0.14
`
`0.12
`
`0.10
`
`0.08
`
`0.06
`
`
`
`Relativespacinga
`
`0.04
`l .0
`
`0.96
`
`0.92
`
`0.88
`Scale factor I
`
`0.84
`
`0.80
`
`0.76
`
`Figure l 1.13 Computed contours of constant directivity versus a and r for log-periodic dipole
`arrays. (SOURCE: R. L. Carrel, “Analysis and Design of the Log—Periodic Dipole Antenna,” Ph.D.
`Dissertation, Elec. Eng. Dept., University of Illinois, 1961, University Microfilms, lnc., Ann
`Arbor Michigan). Note: The initial curves led to designs whose directivities are 1—2 dB too
`high. They have been reduced by an average of 1 dB (see P. C. Butson and G. T. Thompson, “A
`Note on the Calculation of the Gain of Log-Periodic Dipole Antennas,” IEEE Trans. Antennas
`Pmpagat., AP—24, pp. 105—106, January 1976).
`
`Once two of them are specified, the other can be found. Directivity (in dB) contour
`curves as a function of r for various values of or are shown in Figure 11.13.
`The original directivity contour curves in [10] are in error because the expression
`for the E-plane field pattern in [10] is in error. To correct the error, the leading sin(0)
`function in front of the summation sign of equation 47 in [10] should be in the denom-
`inator and not in the numerator [i.e., replace sin0 by 1/ sin(0)] [12]. The influence
`of this error in the contours of Figure 11.13 is variable and leads to 1—2 dB higher
`directivities. However it has been suggested that, as an average, the directivity of each
`original contour curve be reduced by about 1 dB. This has been implemented already,
`and the curves in Figure 11.13 are more accurate as they now appear.
`
`A. Design Equations
`In this section a number of equations will be introduced that can be used to design a
`log-periodic dipole array.
`While the bandwidth of the system determines the lengths of the shortest and longest
`elements of the structure, the width of the active region depends on the specific design.
`Carrel [10] has introduced a semiempin'cal equation to calculate the bandwidth of the
`active region Ba, related to a and r by
`
`Ba, = 1.1 + 7.70 — r)2 cota
`
`Fractuss-fi).
`
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`

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`
`In practice a slightly larger bandwidth (Bs ) is usually designed than that which is
`required (B). The two are related by
`Bs = BB ar = B[1.1 + 7.7(1 − τ )2 cot α]
`
`(11-30)
`
`where
`Bs = designed bandwidth
`B = desired bandwidth
`Bar = active region bandwidth
`The total length of the structure L, from the shortest (lmin) to the longest (lmax)
`element, is given by
`
`(cid:10)
`
`L = λmax
`4
`
`1 − 1
`
`Bs
`
`(cid:11)
`
`cot α
`
`(11-31)
`
`where
`λmax = 2lmax = υ
`
`fmin
`
`(11-31a)
`
`From the geometry of the system, the number of elements are determined by
`
`N = 1 + ln(Bs )
`ln(1/τ )
`
`(11-32)
`
`The center-to-center spacing s of the feeder-line conductors can be determined by
`specifying the required input impedance (assumed to be real), and the diameter of the
`dipole elements and the feeder-line conductors. To accomplish this, we first define an
`average characteristic impedance of the elements given by
`
`(cid:20)
`
`(cid:10)
`
`ln
`
`ln
`dn
`
`(cid:11)
`
`(cid:21)
`− 2.25
`
`Za = 120
`
`(11-33)
`
`where ln/dn is the length-to-diameter ratio of the nth element of the array. For an
`ideal log-periodic design, this ratio should be the same for all the elements of the
`array. Practically, however, the elements are usually divided into one, two, three, or
`more groups with all the elements in each group having the same diameter but not
`the same length. The number of groups is determined by the total number of elements
`of the array. Usually three groups (for the small, middle, and large elements) should
`be sufficient.
`The effective loading of the dipole elements on the input line is characterized by
`the graphs shown in Figure 11.14 where
`
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`

`

`LOG-PERIODIC ANTENNAS
`
`633
`
`σ
`
`Figure 11.14 Relative characteristic impedance of a feeder line as a function of relative char-
`acteristic impedance of dipole element. (SOURCE: R. L. Carrel, “Analysis and Design of the
`Log-Periodic Dipole Antenna,” Ph.D. Dissertation, Elec. Eng. Dept., University of Illinois, 1961,
`University Microfilms, Inc., Ann Arbor, Michigan).
`√
`τ = relative mean spacing
`(cid:16) = σ/
`Za = average characteristic impedance of the elements
`Rin = input impedance (real)
`Z0 = characteristic impedance of the feeder line
`The center-to-center spacing s between the two rods of the feeder line, each of
`(cid:11)
`(cid:10)
`identical diameter d, is determined by
`
`s = d cosh
`
`Z0
`120
`
`(11-34)
`
`B. Design Procedure
`A design procedure is outlined here, based on the equations introduced above and
`in the previous page, and assumes that the directivity (in dB), input impedance Rin
`(real), diameter of elements of feeder line (d), and the lower and upper frequencies
`(B = fmax/fmin) of the bandwidth are specified. It then proceeds as follows:
`
`1. Given D0 (dB), determine σ and τ from Figure 11.13.
`2. Determine α using (11-28).
`3. Determine Bar using (11-29) and Bs using (11-30).
`√
`4. Find L using (11-31) and N using (11-32).
`(cid:16) = σ/
`5. Determine Za using (11-33) and σ
`6. Determine Z0/Rin using Figure 11.14.
`7. Find s using (11-34).
`
`τ .
`
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`

`

`634
`
`FREOUBdCY INDEPENDENT ANTENNAS, ANTENNA MINIATURIZA'HON, AND FRACTAL ANTENNAS
`
`Example 1 1 .1
`Design a log-periodic dipole antenna, of the form shown in Figure 11.9(d), to cover all
`the VHF TV channels (starting with 54 MHz for channel 2 and ending with 216 MHz for
`channel 13. See Appendix IX.) The desired directivity is 8 dB and the input impedance
`is 50 ohms (ideal for a match to 50-ohm coaxial cable). The elements should be made of
`aluminum tubing with % in. (1.9 cm) outside diameter for the largest element and the feeder
`line and % in. (0.48 cm) for the smallest element. These diameters yield identical I/d ratios
`for the smallest and largest elements.
`Solution:
`
`1. From Figure 11.13, for D0 = 8 dB the optimum a is or =0.157 and the
`corresponding t is r = 0.865.
`2. Using ( 11-28)
`
`a = tan" i865 = 12 1
`4(0.157)
`
`. Using (1 1-29)
`
`Ba, = 1.1 + 7.7(1 — 0.865)2 cot(12. 13°) = 1.753
`
`and from (ll-30)
`
`216
`B; = BB“, = $0.753) = 4(1.753) = 7.01
`
`. Using (1 1-31a)
`
`A
`
`_ 1) _3x10“
`m—fnfin—54X106
`
`From (ll-31)
`
`1
`
`0
`
`= 5.556 In (18.227 ft) 5.556
`
`L = 7(1— m) cot(12.13 )_ 5.541 m (18.178 ft)
`
`and from (1 1-32)
`
`N = l +w = 14.43 (14 or 15 elements)
`ln(l/0.865)
`
`_ 0.157 =0.169
`0.865
`
`At the lowest frequency
`
`Am 18.227
`In.” _ T _ T _9.ll35 ft
`
`[m 9.1135(12)
`m _T _ 145.816
`
`ZTE (USA), Inc. v. Fractus S.A.; |PR2018—01461
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`
`Fractus S.A.
`
`Ex. 2017
`
`

`

`I
`
`I péing (.1.}'_.3.3.)._'._.,_
`
`Zn = 120[1II(145.816)—2.25]=327_33 ohms
`
`.
`
`.
`
`.
`
`.
`
`_
`
`,
`
`.
`
`.
`
`-
`
`_
`
`I
`.é'E‘°“"Fig'ufe'fi;1.4.
`
`.Za -_- 327.88 655—8": _
`.
`_ Rm '—'"'-ISO_-I—'—'
`-
`
`.
`
`I
`
`.
`
`.
`
`.
`
`-
`
`_
`
`20:1.2Rm=1.2(50)=600hms
`
`I...--""
`
`7. Using (1 1-34), assuming the feeder line conductoris madeof the same size
`tubing as the largest. element of the array: the center-to—center spacing of the
`feeder condu'ctors1s_ ,
`.-
`.
`.
`-
`-
`
`_ _‘_._v-“"','_.'--I '
`_
`.
`,
`.
`-
`
`'
`
`I
`
`I
`
`.
`
`s= 3cosh
`4
`
`60 _=0.846?0.85 in.
`120
`
`which allows for a 0.1-in. separation betweentheir 'COnHucting_surfac'e‘s'ff—-
`
`'
`
`'
`
`.- ._For such a high-gain antenna, this is obviously a good practical design. If a lower
`gain is specified and designed for, a smaller length will result.
`
`._.-
`
`A commercial log-periodic dipole antenna of 21 elements is shown in Figure _l'l_..15. --~ '
`The antenna is designed to operate in the 100—1,100 MHz with a_ gain of about 6 dBi,
`
`
`
`— --1«1giin4é '11.15 Commercial log—pedodic dipole antenna of 21 elements.
`Research Associates, Inc. Beltsville MD).
`
`elxa
`rac USA
`Ex.2017
`
`ZTE (USA) Inc.v Fractus-8.AL: IPR2018—01461
`
`-
`
`Pa