### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Determination of the Large Period of Slot Group
- 3. Suppression of the Higher-Order Harmonics
- 4. Diagrams for Determination of the Slot Sizes and Shapes
- 5. Conclusions
- Acknowledgments
- References

[1] Based on the analysis of spatial harmonics of periodically slotted structures, the formulas for determining the slot period of the leaky coaxial cables according to the desired frequency band are given. Methods of suppressing the higher-order harmonics for extending the frequency bandwidth are discussed. Diagrams for determining slot sizes and shapes according to the desired coupling loss are given.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Determination of the Large Period of Slot Group
- 3. Suppression of the Higher-Order Harmonics
- 4. Diagrams for Determination of the Slot Sizes and Shapes
- 5. Conclusions
- Acknowledgments
- References

[2] The earlier theoretical works of the radiating cables are mainly concerned about the propagation properties, radiation characteristics, and the interferences with the environments of the relatively simple structures, such as braided cables [*Wait and Hill*, 1975; *Seidel and Wait*, 1978], helically wound cables [*Wait*, 1976; *Hill and Wait*, 1980a], and axially slotted coaxial cables [*Hassan*, 1989; *Delogne and Laloux*, 1980]. These studies provide better physical understanding of the radiating cables, but not better design practices. This is because those simple surface-wave-based radiating cables are subject to larger longitudinal attenuation and higher coupling loss than the periodically slotted leaky cables, which are designed based on the leaky wave theory. However, the design procedure of leaky-wave-based cables is much more complicated than surface-wave-based cables. To our knowledge, there is no publication in open literature which has systematically discussed the general design theory of the leaky cables, though the characteristics and application problems of the periodically slotted leaky cables have been studied by many researchers [*Hill and Wait*, 1980b; *Richmond et al.*, 1981; *Delogne and Deryck*, 1980; *Gale and Beal*, 1980; *Liénard and Degauque*, 1999], and several new leaky wave cables have already been designed by distributing the slot angles in a triangle function [*Kim et al.*, 1998]. In our previous work [*Wang and Mei*, 2001a], a new method was proposed, in which the field distribution in the cable slots was calculated by FDTD method, and the far field was obtained by integrating the aperture field of the slots together with dyadic Green's function. By this method, the coupling loss of different kinds of leaky cables with periodic slots can be determined accurately. As an application, a new kind of directional leaky coaxial cable was proposed and analyzed by this method [*Wang and Mei*, 2001b]. This paper is intended to provide accurate general formulas and diagrams for designing a leaky cable for a given frequency band and a desired coupling loss.

### 3. Suppression of the Higher-Order Harmonics

- Top of page
- Abstract
- 1. Introduction
- 2. Determination of the Large Period of Slot Group
- 3. Suppression of the Higher-Order Harmonics
- 4. Diagrams for Determination of the Slot Sizes and Shapes
- 5. Conclusions
- Acknowledgments
- References

[4] For a desired frequency band which satisfies (*f*_{0} + Δ*f*) : (*f*_{0} − Δ*f*) > 2: 1, the higher-order harmonics from -2 to *M* (less than -2) should be removed simultaneously. As we know, the field around a periodically slotted cable can be expressed by [*Collin*, 1992]

This means there are infinite numbers of spatial harmonics, but only those with minus orders may produce leaky wave. In order to remove the higher-order radiation harmonics, additional arrays of slots should be cut with the same period of *P*, as shown in Figure 2. The new arrays essentially have the same field distributions as the original one, but with different amplitudes and shifts of the source location (the field point is not changing), and can be written as

The complete field of the cable can be written as

where

In (7), *a*_{1}, …, *a*_{N} are associated with the field amplitudes of different slot arrays due to different sizes or shapes of slots, while exp[*j*2*m*π(*s*_{1} + … + *s*_{N})/*P*] comes from the shifts of the source location.

[5] The task is now focused on arranging the slots so that *F*_{m} = 0 for certain *m*. There are basically two ways to do so; one is to let *a*_{1} = … = *a*_{N} = 1, and adjust *s*_{n} (*n* = 1, …, *N*), the other is to fix *s*_{n} and adjust the amplitude *a*_{n}.

[6] In the first way, *s*_{n} should not be equal to one another, otherwise the characteristics of the cable will be reduced to that of a cable with slot period *s*_{n}. We may arrange the slots so that *F*_{m} can be written as

Let *P*_{1} = *P*/4, *P*_{2} = *P*/6, …, then *F*_{−2} = *F*_{−3} = … = 0, and the −2nd, -3rd, … higher-order harmonics are removed. For the case of inclined slots as shown in Figures 3a and 3b, *E*_{ϕ} components of the aperture field are opposite in geometry (regardless of the phase difference due to propagation), so the slots can be arranged to let *F*_{m} have the following form for *E*_{ϕ} component.

This indicates that the configuration of inclined slot cable itself has suppressed the even higher-order harmonics for *E*_{ϕ}, and only the odd-order harmonics need to be suppressed. Let *P*_{1} = *P*/6, *P*_{2} = *P*/10, …, then the −3rd, -5th, … higher-order harmonics are removed. Figure 3a illustrates the configurations designed by (8) and (9) to have the frequency bands of *f*_{h} : *f*_{l} = 3:1 (for vertical slot cables) and *f*_{h} : *f*_{l} = 5:1 (for inclined slot cables) with -2nd, and −2nd, -3rd higher-order harmonics suppressed. The frequency bands of the cables shown in Figure 3b have been extended to 4:1 and 7:1 respectively. This method of suppressing the higher-order harmonics is simple and easy to realize, but we can see that for removing one higher-order harmonic, the number of slots should be doubled. So this method cannot be used to design the cables with extremely wide frequency bands because there would be many slots crowded together and this brings difficulty in fabrication.

[7] In the second way, we let *s*_{1} = … = *s*_{N} = *P*/*N*, and adjust *a*_{n}. In this case, (7) can be written as

where *a*_{0} = 1. This can be seen as a summation of values equally sampled from *a*(*z*) exp(*j*2*m*π*z*/*P*) within one period. If *a*(*z*) is also a periodic function of *z* with period of *P*, then . This is equivalent to saying that the summation of the equally sampled values vanishes. So in order to make *F*_{m} = 0, the coefficients of (10) should be distributed as a periodic function. Here we use the cosine function. Equation (10) can be rewritten as

It evident that by this formula, the high-order harmonics with order number *m* that satisfies −*N* ≤ *m* < −1 can be removed simultaneously, *N* (even number) is the total number of slots within one period (except the slots with zero inclined angle or zero size). This method can only be used for those structures that can distribute the slots by triangle functions, the geometry can be reversed within a period. This can only be the cables with inclined slots designed for azimuthally polarized radiation, similar to that shown in Figure 3. From (7) we can see that it is the radiating field amplitude, not the slot size itself, that should be distributed as a periodic function, so it is difficult to realize the distribution accurately. However, because the radiating field comes from the integral of the field excited in the slot aperture, which is proportional to the slot size and shape, so an alternative approximate way is to distribute the sizes or shapes of the slots in a periodic function. Two basic configurations may achieve this goal. One is to fix the slot length and distribute the slot angle in a sine or cosine distribution as done by *Kim et al.* [1998], and the other is to fix the slot angle and adjust the slot length, as shown in Figure 3c. But these two configurations are more difficult to fabricate and will cause high-level surface wave when the slot number increases.

[8] The methods described above are done toward the goal of fully removing the higher-order harmonics. These methods are limited by the slot number. For an extremely wide frequency band, it is possible to optimize the slot distances or the slot lengths to control the higher-order radiation harmonics to an average level, rather than removing them entirely.

### 4. Diagrams for Determination of the Slot Sizes and Shapes

- Top of page
- Abstract
- 1. Introduction
- 2. Determination of the Large Period of Slot Group
- 3. Suppression of the Higher-Order Harmonics
- 4. Diagrams for Determination of the Slot Sizes and Shapes
- 5. Conclusions
- Acknowledgments
- References

[9] The configuration and coordinate of the leaky cables used in this paper are shown in Figure 3d. *a* and *b* are the radii of the cable's inner and outer conductors, ε_{r} is the relative permittivity of the dielectric in the cable. *l*_{0} is the half-length of the slot, *w* is the slot width, θ is the inclined angle of the slot (between the axes of the slot and the cable). The coupling loss is defined by the ratio of the power received by a standard half-wavelength dipole antenna located 1.5 m or 2 m in the front of the cable (*P*_{r}) and the power propagating in the cable (*P*_{t}), usually expressed by *A*_{c} = −10log(*P*_{r}/*P*_{t}). The key problem is to find the radiation field around the cable accurately. Numerous attempts have been made in this direction, without apparent success until recently when numerical methods became available. But the relationship between the radiation field and the slot structure still cannot be given in an obvious form even in terms of approximate formulas. Therefore, the inverse problem, determining the slot's structure from given coupling loss, is a complicated one. Here, we only give the curves for a 1 5/8″ cable whose parameters are *a* = 8 mm, *b* = 20.65 mm, and ε_{r} = 1.26, and the cable's jacket is about 2.5 mm in thickness. The characteristic impedance of this cable is *Z*_{0} = 50 Ω. The curves for other cables can be similarly obtained. The coupling losses are calculated by using the above definition, but *P*_{r} is evaluated by multiplying the Poynting vector with the effective area of the half-wavelength dipole. The radiating field used to calculate the Poyting vector was computed by the method proposed by *Wang and Mei* [2001a]. For vertical slot cables, the coupling loss is calculated from the dominant component *E*_{z}, and, for the inclined slot cables, is calculated from *E*_{ϕ}. This corresponds to the dipole polarizations used in the realistic cases.

[10] The coupling losses as functions of slot size, inclined angle and arrangement for three typical kinds of cables shown in Figure 3 are given in Figures 4–7. From these figures, we can see that below the lower limit of the frequency band, the coupling losses are very large; this is due to the fact that the surface wave is dominating. When frequency increases, -1st leaky wave radiates, so the coupling losses suddenly go down. Above the upper limit of the frequency band, higher-order harmonics begin to radiate, so the coupling losses decrease faster than that in monoharmonic radiation band, but larger fluctuation occurs. Figures 4–7 also show that the coupling losses are decreasing with the frequency, especially in the case of inclined slot cables. This is because the field stimulated in the slot aperture becomes larger when frequency increases, as can be seen from Figure 8, which illustrates the amplitudes of the distributions of the electric field in the slot aperture. From Figure 8 we can see that the field will be very large when the frequency reaches the resonant points at which the slot length is about half the wavelength. Since the radiation at the resonant points is very strong, the coupling losses at these points are very low, as the depressed parts in Figures 6 and 7 show. By the way, because more slots per unit length on cable will transfer more energy into environments, so the cables with smaller slot period will have lower coupling losses, as can be seen from Figures 5–7.

[11] The frequency bands of the cables shown in Figures 4a and 4b have been extended to 4:1 and 5:1 respectively. The frequency bands of the curves bounded by *P* = 0.71 m and 0.36 m in Figure 4a coincide with (200 MHz, 800 MHz) and (400 MHz, 1600 MHz) respectively, and the frequency bands of curves bounded by *P* = 0.71 m and 0.48 m in Figure 4b coincide with (200 MHz, 1000 MHz) and (300 MHz, 1500 MHz) respectively. Figure 4 also shows the curves with different slot lengths and widths.

[12] Figures 5–7 give the curves of inclined slot cables with slot lengths of 60 mm, 90 mm, and 120 mm respectively. The frequency bands in parts a, b, and c of the figures have been extended to 5:1, 7:1 and 15:1 respectively. The frequency bands of the curves bounded by *P* = 1.41 m, 0.71 m and 0.47 m in Figures 5a, 6a, and 7a correspond to (100 MHz, 500 MHz), (200 MHz, 1000 MHz), and (300 MHz, 1500 MHz) respectively. The frequency bands of the curves bounded by *P* = 1.41 m, 0.8 m, 0.56 m in Figures 5b, 6b, and 7b correspond to (100 MHz, 700 MHz), (180 MHz, 1260 MHz), and (250 MHz, 1760 MHz) respectively. The frequency bands of the curves bounded by *P* = 1.41 m and *P* = 1.1 m in Figures 5c, 6c, and 7c correspond to (100 MHz, 1500 MHz) and (1250 MHz, 1750 MHz) respectively.

[13] When designing, the first step is to determine the large slots period from the desired frequency band using (1)–(3), then select one type of the cable shown in Figure 3, and calculate the small periods of slots using (1), (8), (9), and (11). The second step is to determine the size and shape of the slots. If the desired frequency band and coupling loss are within those shown in Figures 4–7, then the slot size and shape can be obtained directly from the figures. If the desired frequency band and coupling loss are not any of those shown in the figures, one may use interpolation to determine the slot size and shapes. In addition, at the resonant points, the voltage standing wave ratio will be very large, and the properties of the leaky cable are distorted. So the resonant frequencies should also be evaluated and are usually pointed out by the manufacturers in their product guidebook. The method of determining the resonant points is presented by *Wang and Mei* [2001a].