# Axial Mode Helix

A very useful mode of operation for the helical antenna is the axial or endfire mode. In this mode the radiation pattern has a single main beam along the axis of the helix (+z direction), that is, it is an endfire antenna. Experiments have shown that the axial mode occurs when the circumference of the he­lix is approximately one wavelength and when the helix has several turns. A primary component of current on the helix is a wave traveling outward from the feed along the wire at approximately the speed of light and the radiation is a beam off the end of the helix. Because the electric field vector ro­tates around in a circular fashion as does the current on the helix, we expect that the radiation field is circularly polarized along the helix axis. One very important feature of the axial mode helical antenna is its broadband character. As a rule of thumb, the approximate bandwidth for the axial mode is given as follows (2):

— <Ck<- 4 x 3

ground plane can be made from either solid metal or wire mesh.

Analysis of Radiation Pattern. The axial mode helix has a circumference of approximately one wavelength, so the cur­rent distribution would not be uniform, and we assume that there is an outgoing current wave, traveling along the helical conductor at phase velocity v = pc (p is the phase velocity relative to the speed of light c in free space). Then

I(£) = I0 e

where ( is the distance measured along the helix from the beginning of the turn closest to the ground plane, I0 the input current, fi = k/p the phase constant of the current wave. When the total length of one turn is approximately a wave­length, the current distribution in Eq. (12) has opposite phase (180° out-of-phase) on opposite sides of a turn, because they are separated by about a half-wavelength. Also the helical coil physically reverses current direction for opposite points. Thus the currents at opposite points of a turn are essentially in phase, giving rise to reinforcement in the far field along the helix axis. We can find the radiation pattern by using the principle of pattern multiplication because a helix with uni­form cross section can be considered as an array of N identical elements (or turns). We have a uniformly excited, equally spaced array with spacing S, so the total pattern is the prod­uct of the pattern for one turn (the element pattern) and the pattern for an array of N isotropic point sources (an array factor). When the helix is long (say, NS > A), the array factor is much sharper than the element pattern and hence deter­mines the shape of the total far-field pattern.

Array Factor. The array factor (AF) of a uniformly excited, equally spaced, linear array of N elements is given by

sin[(iV/2)’I/] N sin(^/2)

and

Ф = kS cos в + S

where в is an angle measured from the array axis (z axis) and S is the phase shift between adjacent elements. Explaining the phase relationships of the axial mode helix is difficult. Begin by finding the phase shift S required for endfire opera­tion, because we know that the radiation is endfire. For ordi­nary endfire, we find the conditions such that Ф is zero at в = 0°:

S = —kS — 2mn m is an integer

where the term (— 2mn) reflects the basic ambiguity of phase. Next we find conditions for increased directivity Hansen— Woodyard (H—W) (3) endfire, since this is an optimum form of endfire: (16)

The bandwidth ratio, the ratio of the upper and lower fre­quencies is 4/3 3/4 or 1.78, which is close to a 2: 1 band­

width. The helix is usually fed axially or peripherally with the inner conductor of the coaxial line connected to the helix and the outer conductor attached to the ground plane. The

S = —kS — 2mn — n /N = — 2n — (kS + n /N) m = 1

To be more accurate, ther term n/N should be replaced by 2.94/N (3). However, the choice of n/N hardly changes the
radiation field (4) and is convenient for expressing other quantities. We have tried the arbitrary choice (m = 1), be­cause this term —2n corresponds roughly to one turn of the circumference at midband (CA = 1). Later it will be clear that other choices are not possible solutions. Experiments show

(1,5) that the phase shift S obtained is close to that of Eq. (16) at midband (CA = 1). How does this happen? We have already pointed out that the term (— 2n) corresponds roughly to one turn of the circumference. In addition, the length around one turn of the helix is greater than a wavelength (LA > 1) at midband. More importantly, the velocity of travel is less than that of light (p « 0.9) at midband (5). These two additional contributions account for the minor terms —(kS + n/N). Thus the phase at midband is explained. However, the experimen­tal data for S tracks Eq. (16) fairly well over the entire band­width of the axial mode. How is this possible? We cannot alter our choice of m; one choice must work for the entire frequency range. And if p remained constant, Eq. (16) could not be satis­fied over the entire band. Fortunately, p does vary quite a bit (from 0.73 to 0.97) over the axial mode frequency range; the result is that H-W endfire described by Eq. (16) is tracked quite well over most of the band, falling off a little toward the high end. All in all, this is quite a remarkable story. The phase, so to speak, locks in to H-W endfire over the band­width of almost 2: 1. When first reported by Kraus, this was called an anomalous phase progression. It still continues to mystify succeeding generations.

To summarize, the phase progression along the helix wire is relatively simple; it corresponds roughly to that of the speed of light along the wire. The phase progression in z, which determines the phase difference S between turns, fol­lows the phase progression of the wire. Taking into account the phase ambiguity (2m n), we see that H-W endfire is ob­tained at midband. The relative phase velocity p then changes with frequency just enough to maintain the H-W endfire. An­other point worth noting is that we have not discussed back­fire (в = 180°) radiation. It does, in fact, occur along with the axial mode but is usually suppressed by the ground plane. It will return, to our advantage, with the multifilar helix.

Assuming, then, the validity of Eq. (16),

Equations (13) and (19) provide the complete normalized array pattern of the axial mode helical antenna. For the ele­ment pattern we will need an analysis of the radiation from a circular loop, which is covered in the next section.

Circular Loop Radiation. In this section we consider the ra­diation from a circular loop carrying a current Дф). The result is useful in understanding the operation of the helix in both the mono — and multifilar forms. In addition, it will yield an approximate element factor for a single turn of the helix. The loop of radius a is centered at the origin and lies in the xy plane. The current distribution Дф) may be represented in terms of a complex Fourier series representation:

 (20)

I(ф) = Ё Ine

n

Consider the typical term Inenф of the current distribution Дф). First we evaluate components A^, Aen of the far-field magnetic vector potential as follows. Directions ф, в are asso­ciated with the field point rather than the source point.

e-jkr r2n

Афп = — J ІпЄ>пф COS(ф — ф’ука^пвсо,(ф-ф -)аЛфі (21)

e-jkr r — 2ж

Авп = ————- Іп^пф [-яЫф’-ф^сояв^^^^^аф’

4n r J0

(22)

To evaluate Aф, we introduce the change of variables Ф ф’ — ф and change limits to obtain

Єпфe-jkrI a f2 / ej^ I e-jФ

Афп = 4jtr j Є7’1’1′

Next, we use the following integral expression for the Bessel function of the first kind Jm(x):

/>2П

/ ex cos 6 e>me de = 2n jmJm (x)

J0

 2W S 2N + 1′. = 77 І + -2лН (17)

 kS + 2n

 1 + W

 k

 L,

 (18)

 в Sx + (2N + 1)/2N

 ф n

 Aф is then evaluated directly to obtain є? пф (Ina)e-jkr in+1 Афп =— — [Jn+1(ka sin6») — Jn_1{ka sin0)]—————— (23)

 (24)
 (19)

Using p as obtained from Eq. (18) to calculate the array factor yields patterns in good agreement with measured patterns. The p value calculated from Eq. (18) also is in closer agreement with measured values of p (1). Therefore, it ap­pears that the Hansen-Woodyard increased directivity condi­tion is a good approximation for helices radiating in the axial mode. For a typical case where C = A, a = 14°, N = 10, we find from Eq. (18) that S = C tan a = 0.249A, L = 1.031A, and p = 0.79. Thus the traveling current wave has a phase velocity less than that of free space. Finally, substituting Eq.

(16) into Eq. (14) yields

Ф = kS(cos в — 1) — (2л +

Using similar methods, we obtain the following evaluation of Ae.

ejn,^(Ina)e-jkrjn+1(j cos 6)rT ABn=—— — [Jn+1(ka sm0) (25)

+ Jn-1 (ka sin 6)]

Ee n = ~iM^xAe n

(26)

The total fields may of course be obtained by adding contribu­tions of all Fourier modes.

Now let’s evaluate the far fields along the z axis (в = 0°, 180°) for each of the separate Fourier modes. We note that Jn(0) = 0 (n Ф 0) and J0(0) = 1. Evaluating the cases n = ±1,
we find that, along the z axis,

J — = ±J (27)

Ee

In other words, the modes n = ±1 representing traveling waves yield circular polarization along the z axis. Note that all other traveling-wave modes yield a null on axis. Of all the Fourier modes, only n = ± 1 radiate in the forward endfire or backfire directions. For the helix, we define forward or back­ward radiation as radiation away from or toward the feed point, respectively.

This result can also be seen by considering currents around the loop for various modes. For n = ±1, each current element is matched by its opposite across the loop that is in the same direction such as to add along the z axis and to rotate polarization as time progresses. All of the other modes cancel along the axis. For even modes, each element is can­celled by its opposite across the loop. For odd modes, a group of elements will cancel. For the general odd case n, any group of n elements each separated by 180°/n yields zero contribu­tion. For n = 5, for example, any group of five elements each separated by 36° yields zero contribution.

Any currents on the cylindrical surface may be resolved into ф — and z-directed currents. The z-directed currents do not radiate along the axis. Thus, for currents of any direction, only the n = ±1 Fourier modes can contribute to endfire or backfire. These results will be useful when considering multifilar helices.

The above discussion makes it easier to understand the operation of the helical antenna. At low frequencies the ze­roth mode (n = 0) is strongly excited, because there is little variation of phase around the cylinder on one turn. In addi­tion, the impedance of the higher modes is highly reactive. As frequency increases and CA approaches unity, we have one complete cycle around the cylinder on one turn, and we expect the e—ф mode to be excited for a right-hand helix. The phase velocity of the helix is lower than that associated with the speed of light and the impedance of the mode n = — 1 is rea­sonable, and so the axial mode begins at approximately CA = 0.75. Similarly, as frequency increases we expect the mode n = —2 to appear; this mode would produce beam pattern deterioration. The axial mode continues until about CA = 1.33.

Element Pattern of the Axial Mode Helix. For the element pattern of one turn of the helix, the current distribution is assumed to be

І(ф’) = I0 e-jet = I0 е—ваф’ (28)

where fi = k/p, а = D/2 and ф’ is the angle measured from the x axis. For accurate analysis of the element pattern, Eq.

(28) should be used to calculate the radiation integral. How­ever, when the helix with several turns operates in the axial mode (CA « 1), the array factor dominates the endfire beam pattern and the element pattern provides minor corrections. Thus it suffices to consider the radiation field of a planar loop with Cx = 1, instead of a three-dimensional one-turn helix. If we also assume that p « 1, then

І(ф’) ъ 1^е-ыф’ = I0е-(2ж/к)аф’ ъ I0e—jф’ (29)

Using the simple form of the current distribution in Eq. (29), we can easily calculate the radiation fields for the element pattern from Eqs. (23)-(26) for CA = 1 (n = —1):

Еф(в, ф) = C(r)[J0(sin9) + J2(sinв)]е-ф (30) Ев(в, ф) = C(r)[J0 (sin в) — J2 (sin 9)](j cos в)е—ф (31)

where C(r) gives the r dependence of the fields. Note that ka = 1 when CA = 1. If we plot the radiation patterns of |E0| and |Еф| using Eqs. (30) and (31) we obtain a figure-eight pat­tern for Ee with a null at 90°, and a nearly omnidirectional pattern for Еф (6). From the plot, it is interesting to note that the normalized E0 can be approximated by cosO. We also ob­serve that EO and Еф are 90° out of phase. In particular, when 0 = 0°, |E0| = |Eф|, thus the radiation field is circularly polar­ized in the endfire direction. As one departs from 0 = 0°, EO decreases more rapidly than does Eф, so the polarization be­comes elliptical. Finally, it should be noted that Kraus (1) has analyzed the element pattern, by using a single turn of a three-dimensional helix with uniform traveling wave current.

Beam Patterns. The complete total far-field pattern is given by the product of the array factor shown by Eq. (13) and the element pattern in Eq. (30) or Eq. (31). However, the array pattern is much sharper than the element patterns. Thus the total E0 and Eф patterns are nearly the same, in spite of the difference in the single-turn patterns. The main lobes of the E0 and Eф patterns are very similar to the array pattern. Therefore, for long helices (NS > A), a calculation of only the array factor is sufficient for an approximate pattern of any field component of the helix.

The measured patterns of a six-turn helix with a = 14° as a function of frequency are presented in Fig. 3. Patterns are shown over a range of circumferences from approximately 0.66A to 1.35A. The solid patterns are for the horizontally po­larized component ^ф) and the dashed for the vertically po­larized (E0). Both are adjusted to the same maximum. We ob­serve that the endfire beam patterns are preserved over the range of 0.73 < CA < 1.22, indicating that the axial mode helix is a broadband antenna.

Important Parameters. Four important parameters for prac­tical design of an axial mode helical antenna are beamwidth (BW), gain or directivity, input impedance, and axial ratio (AR). They are all functions of the number of turns, the turn spacing (or pitch angle), and the frequency. For a given num­ber of turns, the behavior of the BW, gain, impedance, and AR determines the useful bandwidth. The nominal center fre­quency of this bandwidth corresponds to a helix circumfer­ence of about 1A.

Beam Width. Based on a large number of measurements King and Wong (7) give the following quasiempirical formula for the beamwidths:

k

HPBW (half-power beam width) = B [degrees] (32)

c, vm;.

 Cx = 0.66 Cx = 0.73 Cx = 0.85 Cx = 0.97 Cx = 1.09 Cx = 1.22 Cx = 1.35 275 MHz 300 MHz 350 MHz 400 MHz 450 MHz 500 MHz 550 MHz Figure 3. Measured beam patterns of the monofilar axial mode helix. From Kraus (1). © 1988 by McGraw-Hill, Inc. Reprinted with permission of the McGraw-Hill Companies.

where KB varies from 61 to 70, for 3/4 < CA< 4/3, 12° < a < 15°, and 8.6 < N < 10. Note that as N increases the beam- width decreases. Figure 4(a) shows measured HPBW of a six — turn, 14° axial-mode helix as a function of the normalized cir­
cumference (CA). We observe that HPBW changes slowly over the range of approximately 0.7 < CA< 1.25.

Gain. The gain of the axial mode helix can be approxi­mately obtained (8) by  Circumference (Cx ) Figure 4. Measured performance of the monofilar axial mode helix. (a) Beamwidth. (b) Axial ratio. (c) VSWR. From Kraus (1). © 1988 by McGraw-Hill, Inc. Reprinted with permission of the McGraw-Hill Companies.

G = KGCfNSk (33)

where KG is the gain factor which depends on the design pa­rameters. King and Wong (7) report that KG varies from 4.2 to 7.7. Experiments show that the gain is peak when C is slightly larger than 1A.

Axial Ratio. We have shown from the approximate analysis described in a previous section that the radiation field is cir­cularly polarized in the mainbeam direction ( 0 = 0°), implying AR = 1. With a more accurate analysis including the effect of relative phase velocity for increased directivity, Kraus (1) obtains the axial ratio along the helix axis as follows:

2N + 1

AR = ~2ДГ — (в = °0) (34)

If N is large, the axial ratio approaches unity and the polar­ization is nearly circular. For example, for a six-turn helix, AR = 13/12 = 1.08 according to Eq. (34). This axial ratio is independent of frequency or circumference. In Fig. 4(b), the measured values of the axial ratio for the six-turn, 14° axial­mode helix are plotted as a function of the circumference (CA). We observe that AR is nearly 1 over the range of about 0.73 < CA < 1.4. The sense of circular polarization is deter­mined by the sense of the helix windings.

Input Impedance. The input impedance of the axial mode helical antenna is nearly purely resistive. The empirical for­mulas for the input resistance are given (1) by

Rm = 140C, (35)

within 20% for the case of axial feed, and

Rln = 150/VQ (36)

within 10% for the case of peripheral feed. Both relations are valid when 0.8 < CA < 1.2, 12° < a < 14° and N > 4. With a suitable matching section, Rin can be made any desired value from 50П to 150П. In the inset of Fig. 4(c), trends of input resistance R and reactance X are shown as a function of the relative frequency or circumference. Note that R is relatively constant and X is very small for 0.7 < CA < 1.5. Figure 4(c) also shows the voltage standing wave ratio (VSWR) measured on a 53П coaxial line. We observe that the VSWR nearly re­
mains constant (approximately 1), and equivalently the input impedance of the helix remains unchanged, over the range of about 0.7 < CA < 1.6.

 Ends open I -D—>
 Ends shorted I Feed region (phases 0°, 90°, 180°, 270°) Broadband Characteristics. Considering all the characteris­tics of beam pattern, input impedance, and polarization as a function of circumference, we find that the performance of the axial mode helix is satisfactory over the range of about 0.75 < CA < 1.25 within the restrictions given on a and N. Thus the bandwidth, defined by the ratio of upper and lower frequencies, is almost an octave. The broadband characteris­tics of the helix can be explained by the natural adjustment of the phase velocity. As the helix size CA, or equivalently the frequency, varies over rather wide range, the phase velocity adjusts itself automatically such that the fields from each turn add nearly in phase in the axial direction.

Variations and Applications of the Helical Antenna

 (a)

A slight taper on the end of the helix (9,10) reduces the axial ratio at the expense of a slight reduction in gain. Axial ratio is improved both on and off axis. A taper is also used at the input to improve impedance characteristics. A circular-cavity backing is sometimes used to reduce the back radiation and increase the forward gain. Dielectric-tube support has been used with the helix antenna. This lowers the frequency for the onset of axial-mode operation and has an effect on the terminal impedance. A solid dielectric core has also been used with the helix (the polyrod helix). A helix with an inner con­centric metal core has been used as a TV transmitter (11). The antenna utilizes higher order Fourier modes such as e±i2ф, е±^Ф which radiate sidefire rather than endfire. This is particularly useful with towers and masts whose circumfer­ence is much larger than a wavelength. An array of helices is stacked along the mast to produce the required beam pattern. The helical antenna has often been used as an element in various types of arrays. Large planar arrays of helices have been used in radio astronomy (12). An array of axial-mode helices has been used for global positioning system (GPS) sat­ellite transmitters (1). Helices are also used as feeds for para­bolic dishes. Applications of the helix are legion.

Updated: 24.03.2014 — 05:08