The magnetic surface current density is defined in terms of the electric field at an aperture as follows: 2E n (6.1)

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1 Chapter 6. Aperture antennas Antennas where radiation occurs from an open aperture are called aperture antennas. xamples include slot antennas, open-ended waveguides, rectangular and circular horn antennas, and reflector antennas. Because aperture antennas are typically large in terms of electrical wavelength, they are used mostly at microwave frequencies. Very high gains can be obtained with reflector antennas, while other types of aperture antennas have low to moderate gains. Slot and waveguide elements are useful in arrays. In this chapter we will discuss the operation and design of several popular types of aperture antennas. We begin with formulas to find the farzone fields radiated by aperture antennas. Far-zone fields from a radiating aperture: In Chapter 3 we showed how the far-zone fields from an antenna element could be found from the electric current distribution on the antenna. This method is applicable to elements such as dipoles and related elements where the radiation is due to a well-defined electric current that can be approximated easily. For aperture antennas it is more convenient to view the fields as being produced by a fictitious magnetic current that exists over the radiating aperture. In most cases, the magnetic current can be well-approximated in terms of the electric field of the aperture. A rigorous derivation of these results requires the use of the electromagnetic equivalence theorem and image theory, a procedure which the interested reader can pursue in the references listed in Chapter. Here we will only give some general results that will be useful for finding the radiated fields from slots, waveguides, and horn antennas. The magnetic surface current density is defined in terms of the electric field at an aperture as follows: M = s n (6.1) where n is the unit vector normal to the surface of the aperture. Then, similar to the derivations in Chapter 3 for electric currents, the following expressions can be derived for the far-fields radiated by magnetic current components: θ φ ( x φ y φ) jk jkr jkg = e M M e dx dy dz 4πr sin cos (6.a) v ( x θ φ y θ φ z θ) jk jkr jkg = e M + M M e dx dy dz 4πr cos cos cos sin sin (6.b) v where g = x sinθcosφ+ y sinθsinφ+ z cosθ. The use of these results will be illustrated in the next section.

2 6.1 Slot antennas Slot antennas consist of an opening in a metallic ground plane. Fields in a slot antenna can be excited with a probe feed, such as a coaxial line across the slot, or from an open waveguide. A slot antenna fed with a probe or cable will radiate equally on both sides of the ground plane, whereas a slot or aperture fed with a waveguide will radiate only on the open side. Here we consider a canonical slot antenna element, consisting of a rectangular aperture in an infinite ground plane. The geometry is shown in the figure below. z y b a x Figure 6.1 Geometry of a rectangular slot antenna. Far-fields of a rectangular slot antenna: We assume a uniform electric field in the aperture: = y (6.3) As in the case of the short dipole with uniform electric current, this distribution of electric field is not completely realistic because it does not satisfy the proper boundary conditions at the edges of the slot. Nevertheless, it is a close approximation to many problems that occur in practice. The equivalent magnetic current in the slot is found from (6.1): M = s n = y z = x (6.4) Substituting this into the integrals of (6.) leads to the following required definite integral for both field components:

3 where a/ b/ jk ( x sinθcosφ+ y sinθsinφ) I = e dx dy x = a/ y = b/ ka kb sin sinθcosφ sin sinθsinφ = ab ka kb sinθcosφ sinθsinφ sin X siny = ab X Y ka X = sinθcosφ kb Y = sinθsinφ (6.5) The total far-fields can be expressed as, θ φ = = jkab πr jkab πr e e jkr jkr X Y sinφ sin sin X Y (6.6a) sin X siny cosθcosφ X Y (6.6b) These results further simplify for the principal plane patterns: -plane (φ=9 ): θ jkab = πr φ = e jkr kb sin sinθ kb sinθ (6.7a) (6.7b) H-plane (φ= ): θ = φ jkab = πr e jkr ka sin sinθ cosθ ka sinθ (6.8a) (6.8b) Notice that the field expressions for both planes are of the form of sinx/x, and that the -plane pattern is a function of b, the slot dimension in the y-direction, while the H-plane pattern is a function of a, the slot dimension in the x-direction. The beamwidth in the principal planes is inversely proportional to these

4 dimensions. Thus, it can be shown that the beamwidth between first nulls of the -plane pattern is approximately equal to, BWFN = 115 (λ/b) (6.9) The patterns of the rectangular slot with uniform field are similar to the patterns of a uniform linear array, in that the beamwidths will be equal for equal aperture sizes, and the sidelobes will be -13 db. xample 6.1 Plot the far-zone principal plane patterns for a rectangular slot antenna having dimensions a = λ and b = 5λ. Solution: We use PCAAD (the line source routine) to evaluate the patterns for this slot. The -plane and H-plane patterns are shown below (only the upper hemisphere patterns are shown): Observe that the sidelobe level is 13 db for both patterns. The beamwidth between first nulls for the -plane pattern (on the left) is 3, in close agreement with (6.9), which gives 3. Directivity of the slot antenna: It is also possible to derive a useful expression for the directivity of the slot element. From the definition of directivity, U D = 4π max P rad (6.1)

5 The maximum power density can be evaluated from (1.5) and (6.7a): U max * r kab ab = = = η 8π η λ η The total radiated power could be found by integrating the far-zone fields, but this does not lead to a closedform result. Instead, we can integrate the power density in the aperture, since all radiated power must flow out of the aperture. If we assume that the aperture field can be approximated as a TM plane wave, then we can express the magnetic field in the aperture using the impedance relation for plane wave fields, and approximate the radiated power as follows: a/ b/ 1 * ab Prad = H zdx dy = dx dy = η η x = a/ y = b/ Using these results in (6.1) gives the directivity as, a/ b/ x = a/ y = b/ 4πab 4πA D = = λ λ (6.11) Although this result was derived for the special case of a rectangular aperture antenna, it actually is very general, and can be used to estimate the directivity of any type of aperture antenna or planar array. Its accuracy depends only on the requirement that the aperture be electrically large (a wavelength or more on a side), and that the field distribution in the aperture be uniform. Also, this result assumes that radiation occurs only from one side of the aperture. 6. Open-ended waveguide Another type of aperture antenna is the open-ended waveguide. This antenna can be considered as a slot element fed with a waveguide. The aperture distribution can be approximated by the field distribution of the waveguide, which is typically the dominant mode of the waveguide. Both rectangular and circular waveguide can be used, for either rectangular or circular apertures, respectively. The open-ended waveguide may be mounted against a ground plane (the case most similar to a slot antenna in a ground plane), or may be used without a ground plane. The ground plane shields the lower half-space from radiation, but in either case the open-ended waveguide radiates a main beam only on one side of the aperture. Another difference from the slot antenna discussed in the previous section is that the aperture field distribution is not uniform over the slot area, varying in amplitude in the x-direction according to the waveguide modal field. The open-ended waveguide thus has a more realistic field distribution than does the slot element with the uniform field assumption.

6 Far-zone fields of the open-ended rectangular waveguide: Here we consider the case of a rectangular waveguide opening into a ground plane, with a propagating T 1 mode. The geometry is shown in Figure 6.. y b z Figure 6. Cross-sectional geometry of a rectangular waveguide opening into a ground plane. The aperture field can then be approximated in terms of the T 1 mode of the rectangular waveguide: Then the equivalent magnetic current can be found from (6.1) as, x = (6.1a) πx y cos a for x a/, y < b/ (6.1b) M s = x x cos π (6.13) a where a in (6.1) and (6.13) is the wide dimension of the guide. Applying (6.13) to the expressions of (6.) gives the far-zone fields of the open-ended rectangular waveguide as, -plane (φ =9 ): θ = jkab πr e jkr kb sin sinθ kb π sinθ (6.14a) φ = (6.14b)

7 H-plane (φ = ): θ = (6.15a) φ = jkab π r e jkr ka cos sinθ cosθ π ka sinθ (6.15b) Note that the -plane pattern is the same as the -plane pattern for the slot element, as given in (6.7). This is because the field distribution is constant along this dimension (y) for both of these antennas. The H-plane pattern is different, however, because the slot element assumed a uniform distribution along the x-axis, while the waveguide has a cosine variation. This makes the H-plane beamwidth of the waveguide antenna larger than that of the slot antenna, for the same a dimension. See Problem 6.3 for the beamwidth between first nulls of this pattern. Another difference in the H-plane pattern for the open-ended rectangular waveguide when compared with the uniform slot antenna is that its sidelobes are reduced. While the sidelobe level of the uniform amplitude slot is 13 db, the tapered amplitude distribution of the waveguide results in a sidelobe level of 3 db. This is the tradeoff with the increased beamwidth. Directivity of the open-ended rectangular waveguide: Using the same method as in Section 6.1, we can compute the directivity of the open-ended rectangular waveguide antenna. From (6.14a) the maximum radiation intensity can be found as, U max * r k a b ab = = = 4 η π η λ η π and the total radiated power can be computed by integrating the Poynting vector over the rectangular aperture: a/ b/ 1 * πx ab Prad = H zdx dy = cos dx dy = η a 4η x = a/ y = b/ Then the directivity can be computed as, a/ b/ x = a/ y = b/ 3ab 8 4πA 4πA D = =.81 πλ π λ λ (6.16) Note that this directivity differs from that of the slot element by a factor of.81. This is because the tapered amplitude distribution of the H-plane of the rectangular waveguide aperture results in a wider beamwidth than for the slot antenna. This is true of aperture antennas and arrays in general: the smallest beamwidth, or highest gain, is achieved for a uniform amplitude distribution. The form of the directivity in (6.16) is expressed as the product of the directivity for a uniform aperture and the aperture efficiency:

8 D =η ap 4πA (6.17) λ where η ap is the aperture efficiency, with η ap 1. An aperture antenna with a uniform field distribution has an aperture efficiency of unity, while apertures with tapered amplitudes have aperture efficiencies less than unity. PCAAD xercise: Use PCAAD to find the directivity of a rectangular waveguide element, and compare with (6.17). From the main menu, select Arrays, then Uniform Linear Array. nter the frequency as 1 GHz, 1 element in the array, and a spacing of 1. cm. Then click the lement Type Select button and choose Rectangular Waveguide as the element type. nter dimensions of 1. cm and.9 cm, then click OK to return to the analysis window. Click the Compute button, and read the directivity as 4.1 db. quation (6.16) gives a value of 4. db. Comparison of rectangular aperture characteristics: The table below summarizes the pattern characteristics of rectangular apertures having uniform and T 1 aperture distributions. The expressions for the beamwidths and directivity are approximate, giving best results for apertures which are larger than a few wavelengths in size. Table 6.1 Summary of Rectangular Aperture Pattern Characteristics Parameter HPBW (-plane) HPBW (H-plane) Sidelobe level (-plane) Sidelobe level (H-plane) Directivity Uniform T 1 Distribution Distribution λ ( ) b 51 λ b ( 51 ) λ ( ) a 51 λ a ( 69 ) -13 db -13 db -13 db -3 db 4πab λ 4π ab λ (.81) xample 6. Plot the principal plane patterns for a rectangular X-band waveguide, with a =.86 cm and b = 1.16 cm, operating at 1 GHz. Compare the half-power beamwidths with the approximate expressions given in Table 6.1 Solution: PCAAD was used to compute and plot the -plane and H-plane patterns, which are shown below:

9 The beamwidths obtained from Table 6.1 are not in very good agreement with these values because the aperture is relatively small, electrically. Also note that these patterns do not have any sidelobes, again because of the small size of the aperture. Problem 6.4 deals with an electrically larger aperture. 6.3 Horn antennas Horn antennas are one of the most popular types of antennas at microwave frequencies. Horns have moderate gain, and are rugged and inexpensive. Horn antennas can be fed from either rectangular or circular waveguide, with the latter often used for dual linear or circular polarization. The four basic types of horn antennas are the -plane sectoral horn, the H-plane sectoral horn, the pyramidal horn, and the conical horn. These antennas are shown in Figure 6.3 below. The -plane sectoral horn is so named because its flare occurs in the plane of the electric field of the waveguide, while the H-plane sectoral horn has a flare in the H-plane of the waveguide. The non-flared dimensions of sectoral horns is generally the same as the corresponding dimension of the waveguide. The pyramidal and conical horns have flares in both dimensions of the feeding waveguide.

10 -plane sectoral horn H-plane sectoral horn Figure 6.3a. and H-plane sectoral horn antennas. Figure 6.3b. Pyramidal and conical horn antennas. The basic principle of operation of each of these variations is the same, in that the horn forms a smooth transition from the waveguide field to the larger radiating aperture of the antenna. Both the impedance bandwidth and the pattern bandwidth are very good, typically with good performance over the full operating bandwidth of the waveguide. Like the slot and open-ended waveguide elements, we can find the radiated fields of a horn antenna by integrating the equivalent magnetic current of the aperture. A critical difference from these cases, however, is that the electric field in the aperture of a horn antenna does not have a uniform phase. This is because the path length along the sides of the horn is generally longer than the path length through the center of the horn. This effect, known as phase error, is shown graphically in Figure 6.4 below:

11 y y' L ψ R y' δ(y') z ψ e b Figure 6.4 Geometry of a horn antenna for the calculation of aperture phase. Phase error of a horn antenna: As shown in Figure 6.4, the waveguide field expands from the mouth of the waveguide to the horn aperture to form a circular phase front. Thus the aperture field near the sides of the horn lags in phase relative to the center of the aperture. Using the notation of Figure 6.4, we can quantify this phase error in terms of the geometrical parameters of the horn. L is the slant length of the horn; R is the length from the apex of the horn to the aperture plane at z = R; ψ e is the angle from the z-axis to the side wall of the horn; and δ is the deviation from the circular phase front to the aperture plane. Then we have that, R = Lcosψ e (6.18) and, ( R+ ) = R + y δ (6.19) The phase error at any point in the aperture plane of the horn is proportional to δ(y'), the distance between the circular phase front and the end of the horn aperture. This can be found from (6.19) as, y y δ ( y ) = R+ R + y = R+ R 1+ R+ R L R 1 y = R (6.)

12 where the first two terms of the Taylor series expansion for the square root function have been used. This approximation is generally valid for horns that are longer than their half-width. The maximum phase error occurs at the wall of the horn, for which y' = b/: δ max = b 8R (6.1) The electrical phase error (in radians) is then, φ max πδ max πb = = λ 4λR (6.) Then the aperture field can be written as (assuming an -plane sectoral horn), where δ is given by (6.). y x a e jk = cos π δ (6.3) Note from (6.) that the phase error increases as the opening of the horn, b, increases, for a fixed axial horn length, R. Similarly, the phase error decreases as the horn length increases, for a fixed horn opening dimension. Design of horn antennas: As with the case of the aperture antennas we have previously treated, the far-fields from a horn antenna can be found by integrating the magnetic surface current density. The integrals are a bit more difficult in this case, because of the quadratic phase error term, so we will not present the results here. Instead, the far-field expressions can be evaluated using PCAAD, which implements these results for several types of horn antennas. In the example below we illustrate some of the basic points related to horn antenna design. In particular, we will see that increasing phase error leads to a pattern with a broader main beam and filled-in nulls, for a given aperture size. xample 6.3: Consider an -plane sectoral horn antenna with R = 6λ, and a =.5λ (the H-plane dimension of the waveguide). Find the maximum phase error, the directivity, and the -plane patterns for aperture sizes of b = 1λ, λ, 3.5λ, 5λ, and 8λ. Solution: We use PCAAD to compute the patterns, directivity, and phase errors for these three cases. The phase errors and directivity are given below (with results for additional values of b/λ):

13 b/λ Max. Phase rror Directivity db db db db db Observe that as the aperture size increases, for a fixed horn length of R = 6λ, the maximum phase error increases monotonically. The directivity, however, initially increases with aperture size, but reaches a maximum, and then decreases. This is because the increasing phase error quickly becomes large enough to counteract the increase in directivity caused by the increasing aperture area. The -plane patterns for b = λ, 3.5λ, and 8λ are shown below: 6.4 Reflector antennas Reflector antennas are probably the most commonly used antennas for high gain (>3 db) requirements, such as satellite communications, microwave radio links, and radar systems. Reflectors can be formed using flat plates, and these are sometimes used on microwave radio relay towers. Corner reflectors, consisting of two plates at right angles, are sometimes used with Yagi-Uda antennas for improved front-to-back ratio. But high gain reflector antennas typically use parabolic metal reflectors to provide a large electrical aperture with a constant phase distribution.

14 Geometry of the parabolic reflector: The cross-sectional geometry of a parabolic reflector antenna is shown below in Figure 6.5. The antenna consists of the parabolic metal reflector, and a feed antenna that illuminates the reflector. y P Q D r' z θ' θο o focal point r o f zo Figure 6.5 Cross-sectional geometry of the parabolic reflector antenna. The diameter of the parabolic reflector is D, and the focal length of the parabola is f. The subtended angle between the center line of the parabola and the edge is θ. The parabolic surface is characterized by the fact that the path length from the focal point () to a point P on the parabola, to a point Q on the focal plane (z = ) is a constant. We can express this mathematically as follows: Since OP = r and PQ = r cosθ, we have that OP + PQ = constant = f (6.4) Solving for r' gives, ( θ ) r 1+ cos = f f θ r = = 1 cos Applying this result to the edge of the reflector, where r = r and θ = θ, gives f sec (6.5) + θ

15 Multiplying both sides by sinθ gives, θ r f sec =. rsinθ = f sec θ sinθ = f sec θ sin θ cos θ Since r sin θ = D/, the following useful relation between the subtended angle of the dish and the f /D ratio is obtained: D f = cot θ (6.6) 4 Feed illumination of the reflector: The aperture plane of the reflector antenna is located at the z = plane in Figure 6.5. As derived above, the path length of any ray from a source located at the focal point at z = y = will be a constant over this plane. The aperture plane could also be assumed to be located at the z = z plane, as this plane is parallel to the z = plane. The reflector aperture is excited by placing a feed antenna at the focal point, to illuminate the reflector and provide fields over the aperture plane. The feed antenna should have a pattern selected so that the maximum amount of radiation is distributed uniformly over the reflector area. The feed antenna pattern is sometimes called the primary pattern, and the radiation pattern due to the reflector called the secondary pattern. While the phase of the aperture plane field distribution is constant, the amplitude is generally tapered toward the edges of the reflector. This is because the fields from the feed antenna must spread out over a larger area as the angle increases from the boresight of the feed antenna (along the z-axis) toward the edge of the reflector. This has the effect of reducing the aperture efficiency of the reflector, and is measured by the taper efficiency of the reflector system. In addition, a realistic feed antenna will have a pattern that extends beyond the subtended angle, θ, of the reflector, and so this power is not captured by the dish. This effect also leads to a drop of efficiency, and is called spillover efficiency. Thus the total gain of the reflector can be expressed as the maximum directivity based on the area of the dish, reduced by the taper and spillover efficiencies: D G = π λ η spillover η taper (6.7) where D is the diameter of the reflector dish. Other effects, such as surface roughness, blockage by struts or supports, and cross polarization, can also reduce the actual gain of a reflector antenna. Reflector antenna design: Design of a parabolic reflector antenna requires determination of the size of the dish, and the feed antenna beamwidth. The size of the dish is set by the specified gain and the aperture efficiency, as given in (6.7). The aperture efficiency is a function of the f /D ratio for the dish, as well as the beamwidths of the feed antenna. For a given feed pattern, there is an optimum value for the dish size that will maximize the aperture efficiency. This is because small dishes have low spillover efficiencies, while large dishes have low taper efficiencies.

16 The graph shown below in Figure 6.6 illustrates how the aperture efficiency (the product of spillover efficiency and taper efficiency) varies with reflector size and feed pattern. This data assumes an idealized feed antenna n with an azimuthally symmetric (no ϕ variation) pattern, and a power pattern that varies as F( θ) = cos θ. The spillover and taper efficiencies for this type of feed pattern can be evaluated analytically, as discussed in several of the references listed in Chapter. The graph gives the optimum values of reflector size (subtended angle) for various feed patterns. Note that the optimum reflector size is smaller for feed patterns having smaller beamwidths (larger n). Also observe that the maximum aperture efficiency is approximately 8% for all cases Aperture fficiency n= n=4 n= Reflector Half Angle (θo) Figure 6.6 Aperture efficiency (taper and spillover) versus reflector size for three feed (power) patterns of the form F ( θ) = cos n θ. xample 6.4: A parabolic reflector antenna has a diameter of.5 m, an f /D ratio of.5, and operates at 1 GHz. Find the n subtended angle, θ, for the reflector. What is the value of n, for a feed power pattern of the form F( θ) = cos θ, that will maximize the aperture efficiency of the reflector. What is the resulting gain? Solution: From (6.6), the subtended angle of the reflector is, f cot D θ 4 = =, θ = 53.1

17 From Figure 6.6, or using the PCAAD software, the optimum value of n for a 53 reflector half-angle is n = 4. This results in an aperture efficiency of about 8%. The resulting gain is, from (6.7), π D G = ηaperture = = 35. db λ A more rigorous analysis of the reflector antenna requires numerical integration of the aperture field of the reflector, as generated by the feed antenna. PCAAD performs such an analysis for prime focus reflector antennas, and thus can generate secondary patterns and directivity when the feed patterns are specified. PCAAD xercise: Use PCAAD to compute the patterns of a reflector antenna having a dipole feed element. From the main menu, select Apertures, then Parabolic Reflector (pattern analysis). nter a frequency of 3 GHz, an f/d ratio of.5, and a dish diameter of 1 cm. Use the supplied dipole feed patterns files, para.dat and parah.dat, and select a rectangular pattern plot with a step size of.5 and an azimuth angle of 45. Click the Compute button, and read the directivity as 47. db. Click the Plot Patterns button, and read a cross-pol level of about 6 db.

18 Review Questions for Chapter 6: Q6.1 A rectangular slot antenna has an -plane dimension of 4λ. What is the beamwidth between first nulls for the -plane pattern? (a) 5 (b) 1 (c) (d) 9 Q6. A square aperture antenna has a uniform field distribution, and is λ long on each side. What is the directivity of the antenna (assuming one-sided radiation)? (a) 3 db (b) 6 db (c) 17 db (d) db Q6.3 A square aperture antenna has a tapered amplitude distribution. If the aperture is λ long on a side, and its aperture efficiency is 5%, what is the directivity of the antenna? (a) 6 db (b) 1 db (c) 14 db (d) 15 db Q6.4 What would be the characteristics of an ideal reflector feed antenna that would lead to 1% aperture efficiency? (a) feed pattern would be omnidirectional (b) feed pattern would be cosθ (c) feed pattern would be a delta function (d) feed pattern would be uniform out to θ, and zero thereafter

19 Problems for Chapter 6: P6.1 Derive the result in (6.9) for the beamwidth between first nulls for the -plane pattern of a rectangular aperture antenna with a uniform electric field. P6. For the H-plane pattern of an open-ended rectangular waveguide given in (6.15b), show that the pattern ka π is defined at θ =. Also show that a null does not occur when sinθ =, which is where the first null occurs in the H-plane of the slot antenna pattern given by (6.8b). P6.3 Derive an expression for the beamwidth between first nulls for the H-plane pattern of an open-ended rectangular waveguide. P6.4 Plot the H-plane pattern for a T 1 -type aperture distribution with a = 3λ. Compare the actual halfpower beamwidth from the pattern with the result obtained from Table 6.1. P6.5 A reflector antenna has a diameter of 1 m, with f /D=.55, and f = 3 GHz. Find n for the optimum feed n pattern of the form F( θ) = cos θ, and the resulting antenna gain.