LCTUR 18: Horn Antennas (Rectangular horn antennas. Circular apertures.) 1 Rectangular Horn Antennas Horn antennas are popular in the microwave bands (above 1 GHz). Horns provide high gain, low VSWR (with waveguide feeds), relatively wide bandwidth, and they are not difficult to make. There are three basic types of rectangular horns. The horns can be also flared exponentially. This provides better impedance match in a broader frequency band. Such horns are more difficult to make, which means higher cost. The rectangular horns are ideally suited for rectangular waveguide feeds. The horn acts as a gradual transition from a waveguide mode to a free-space mode of the M wave. When the feed is a cylindrical waveguide, the antenna is usually a conical horn. Nikolova 16 1
Why is it necessary to consider the horns separately instead of applying the theory of waveguide aperture antennas directly? It is because the so-called phase error occurs due to the difference between the lengths from the center of the feed to the center of the horn aperture and the horn edge. This makes the uniformphase aperture results invalid for the horn apertures. 1.1 The H-plane sectoral horn The geometry and the respective parameters shown in the figure below are used in the subsequent analysis. l H R x a α H R A z R H H-plane (x-z) cut of an H-plane sectoral horn α l H H Nikolova 16 A = R +, (18.1) A = arctan lh RH = ( A a) A R, (18.) 1 4. (18.3) The two required dimensions for the construction of the horn are A and R H.
The tangential field arriving at the input of the horn is composed of the transverse field components of the waveguide dominant mode T 1 : where Z g β = η λ 1 a λ g = β 1 a π y( x) = cos x e a H ( x) = ( x)/ Z x y g jβ z g is the wave impedance of the T 1 waveguide mode; is the propagation constant of the T 1 mode. (18.4) Here, β = ω µε = π / λ, and λ is the free-space wavelength. The field that is illuminating the aperture of the horn is essentially a spatially expanded version of the waveguide field. Note that the wave impedance of the flared waveguide (the horn) gradually approaches the intrinsic impedance of open space η, as A (the H-plane width) increases. The complication in the analysis arises from the fact that the waves arriving at the horn aperture are not in phase due to the different path lengths from the horn apex. The aperture phase variation is given by e j β ( R R ). (18.5) Since the aperture is not flared in the y-direction, the phase is uniform along y. We first approximate the path of the wave in the horn: x 1 x R= R + x = R 1+ R 1 R + R. (18.6) The last approximation holds if x R, or A/ R. Then, we can assume that 1 x R R. (18.7) R Using (18.7), the field at the aperture is approximated as Nikolova 16 3
π a ( x) cos x e y A β j x R. (18.8) The field at the aperture plane outside the aperture is assumed equal to zero. The field expression (18.8) is substituted in the integral for I y (see Lecture 17): I = x y e dx dy, (18.9) (, ) (, ) jβ( x sinθ cosϕ y sinθsin ϕ) y θϕ + a S A y + A/ β + b/ π j x I x e R e jβx sinθ cosϕdx e jβ y sinθsinϕdy y ( θϕ, ) = cos.(18.1) A A/ b/ I ( θϕ, ) The second integral has been already encountered. The first integral is cumbersome and the final result only is given below: where (.5β b sinθ sinϕ) ( b ϕ ) 1 π R sin I y ( θϕ, ) = I( θ, ϕ) b, (18.11) β.5β sinθ sin and R π j β sinθ cosϕ+ β A R π j β sinθ cosϕ β A [ ( ) ( ) ( 1 ) ( 1 )] I( θϕ, ) = e C s js s C s + js s [ ( ) ( ) ( 1 ) ( 1 )] + e C t js t C t + js t s1 = 1 βa πr Rβ u πβ R A ; s = 1 βa πr + Rβ u πβ R A ; t1 = 1 βa πr Rβ u+ πβ R A ; t = 1 βa πr + Rβ u+ πβ R A ; (18.1) Nikolova 16 4
u = sinθcosϕ. Cx ( ) and Sx ( ) are Fresnel integrals, which are defined as x π ( ) cos = τ τ ; ( ) = ( ), Cx d C x Cx x π ( ) sin = τ τ ; ( ) = ( ). Sx d S x Sx (18.13) We note that more accurate evaluation of I y ( θϕ, ) can be obtained if the approximation in (18.6) is not made, and is substituted in (18.9) as or ( ) π π a ( x) = cos x e e cos x e y = A A ay + jβ R β jβ R + x R j R + x The far field can be calculated from I ( θϕ, ) as (see Lecture 17): e jβ r (1 cos )sin θ = jβ + θ ϕ Iy ( θϕ, ), 4π r e jβ r j (1 cos )cos I ϕ = β + θ ϕ y ( θϕ, ), 4π r y ( ) ( b ) πr e jβ r 1+ cosθ sin.5βb sinθ sinϕ = jβ b β 4πr.5β sinθ sinϕ ( θˆ + φˆ ) I( θϕ, ) sinϕ cos ϕ. The amplitude pattern of the H-plane sectoral horn is obtained as ( βb θ ϕ) ( βb θ ϕ). (18.14) (18.15) (18.16) 1+ cosθ sin.5 sin sin ( θϕ, ) = I( θϕ, ). (18.17).5 sin sin Principal-plane patterns Nikolova 16 5
-plane ( ϕ = 9 ): F ( βb θ ϕ) ( βb θ ϕ) 1+ cosθ sin.5 sin sin ( θ ) =.5 sin sin (18.18) The second factor in (18.18) is exactly the pattern of a uniform line source of length b along the y-axis. H-plane ( ϕ = ): 1+ cosθ FH( θ) = fh( θ) = (18.19) 1+ cos θ I( θϕ, = ) = I( θ =, ϕ = ) The H-plane pattern in terms of the I( θϕ, ) integral is an approximation, which is a consequence of the phase approximation made in (18.7). Accurate value for fh ( θ ) is found by integrating numerically the field as given in (18.14), i.e., + A/ π x fh ( θ ) cos e e dx A jβ R + x jβx sinθ. (18.) A/ Nikolova 16 6
- AND H-PLAN PATTRN OF H-PLAN SCTORAL HORN Fig. 13-1, Balanis, p. 674 Nikolova 16 7
The directivity of the H-plane sectoral horn is calculated by the general directivity expression for apertures (for derivation, see Lecture 17): D 4π = λ S a S A ads A. (18.1) ds The integral in the denominator is proportional to the total radiated power, + b/ + A/ π ds x dx dy ηπ rad = a = cos = A S b/ A/ A Ab. (18.) In the solution of the integral in the numerator of (18.1), the field is substituted with its phase approximated as in (18.8). The final result is where ε t = 8 ; π D H b 3 A 4π = εh ph = εε t λπ λ λ H ph {[ ( ] [ ] } 1 ) ( ) ( 1 ) ( ) π ε H ph = C p C p + S p S p ; 64t 1 1 p1 = t 1 +, p = t 1+ 8t 8t ; 1 A 1 t =. 8 λ R / λ ( Ab), (18.3) The factor ε t explicitly shows the aperture efficiency associated with the aperture cosine taper. The factor ε H ph is the aperture efficiency associated with the aperture phase distribution. A family of universal directivity curves is given below. From these curves, it is obvious that for a given axial length R and at a given wavelength, there is an optimal aperture width A corresponding to the maximum directivity. Nikolova 16 8
R = 1λ [Stutzman&Thiele, Antenna Theory and Design] It can be shown that the optimal directivity is obtained if the relation between A and R is or A= 3λR, (18.4) Nikolova 16 9 A R = 3. (18.5) λ λ
1. The -plane sectoral horn l R y b α R B z R -plane (y-z) cut of an -plane sectoral horn The geometry of the -plane sectoral horn in the -plane (y-z plane) is analogous to that of the H-plane sectoral horn in the H-plane. The analysis is following the same steps as in the previous section. The field at the aperture is approximated by [compare with (18.8)] Here, the approximations and π a = cos x e y a β j y R y 1 y R= R + y = R 1+ R 1 R + R Nikolova 16 1. (18.6) are made, which are analogous to (18.6) and (18.7). (18.7) 1 y R R (18.8) R
The radiation field is obtained as 4 j r βr βb a π R e β j sinθsinϕ ˆ ˆ ( sin cos ) = jβ e θ ϕ + φ ϕ π β 4πr βa cos sinθcosϕ (1 + cos θ ) [ C( r) js( r) C( r1) + js( r1) ]. βa 1 sinθcosϕ The arguments of the Fresnel integrals used in (18.9) are r β B βb = R sinθsin ϕ, 1 π R r β B βb = + R sinθsin ϕ. π R (18.9) (18.3) Principal-plane patterns The normalized H-plane pattern is found by substituting ϕ = in (18.9): βa cos sinθ 1 cosθ + H ( θ ) =. (18.31) βa 1 sinθ The second factor in this expression is the pattern of a uniform-phase cosineamplitude tapered line source. The normalized -plane pattern is found by substituting ϕ = 9 in (18.9) : [ Cr ( ) Cr ( 1) ] + [ Sr ( ) Sr ( 1) ] (1+ cos θ) (1+ cos θ) ( θ) = f ( θ) = 4 C( r θ= ) + S ( rθ= ) Here, the arguments of the Fresnel integrals are calculated for ϕ = 9 :. (18.3) Nikolova 16 11
and r β B βb = R sin θ, 1 π R r β B βb = + R sin θ, π R (18.33) B β rθ = = r( θ = ) =. (18.34) π R Similar to the H-plane sectoral horn, the principal -plane pattern can be accurately calculated if no approximation of the phase distribution is made. Then, the function f ( θ ) has to be calculated by numerical integration of (compare with (18.)) B/ jβ R ( ) + y f j sin y θ e e β θ dy. (18.35) B/ Nikolova 16 1
- AND H-PLAN PATTRN OF -PLAN SCTORAL HORN Fig. 13.4, Balanis, p. 66 Nikolova 16 13
Directivity The directivity of the -plane sectoral horn is found in a manner analogous to the H-plane sectoral horn: where ε t 8 =, ε π ph D = a 3 B 4π ε ph εε t phab λπ λ = λ, (18.36) + = =. λr C( q) S( q) B, q q A family of universal directivity curves λd / a vs. B/ λ with R being a parameter is given below. R = 1λ [Stutzman&Thiele, Antenna Theory and Design] Nikolova 16 14
The optimal relation between the flared height B and the horn apex length R that produces the maximum possible directivity is B= λr. (18.37) 1.3 The pyramidal horn The pyramidal horn is probably the most popular antenna in the microwave frequency ranges (from 1 GHz up to 18 GHz). The feeding waveguide is flared in both directions, the -plane and the H-plane. All results are combinations of the -plane sectoral horn and the H-plane sectoral horn analyses. The field distribution at the aperture is approximated as π a cos x e y A β x y j + R H R. (18.38) The -plane principal pattern of the pyramidal horn is the same as the -plane principal pattern of the -plane sectoral horn. The same holds for the H-plane patterns of the pyramidal horn and the H-plane sectoral horn. The directivity of the pyramidal horn can be found by introducing the phase efficiency factors of both planes and the taper efficiency factor of the H-plane: where ε t 8 = ; π D 4π = εε ε ( AB), (18.39) λ H P t ph ph {[ ( ] [ ] } 1 ) ( ) ( 1 ) ( ) π ε H ph = C p C p + S p S p ; 64t 1 1 p1 = t 1 +, p = t 1 +, 8t 8t ε ph C( q) + S( q) B, q q λr = =. Nikolova 16 15 1 A 1 t = ; 8 λ R / λ The gain of a horn is usually very close to its directivity because the radiation H
efficiency is very good (low losses). The directivity as calculated with (18.39) is very close to measurements. The above expression is a physical optics approximation, and it does not take into account only multiple diffractions, and the diffraction at the edges of the horn arising from reflections from the horn interior. These phenomena, which are unaccounted for, lead to only very minor fluctuations of the measured results about the prediction of (18.39). That is why horns are often used as gain standards in antenna measurements. The optimal directivity of an -plane horn is achieved at q = 1 [see also (18.37)], ε ph =.8. The optimal directivity of an H-plane horn is achieved at t = 3/8 [see also (18.4)], ε H ph =.79. Thus, the optimal horn has a phase aperture efficiency of ε P = ε Hε =.63. (18.4) ph ph ph The total aperture efficiency includes the taper factor, too: εp = εεhε =.81.63 =.51. (18.41) ph t ph ph Therefore, the best achievable directivity for a rectangular waveguide horn is about half that of a uniform rectangular aperture. We reiterate that best accuracy is achieved if ε ph H and ε ph are calculated numerically without using the second-order phase approximations in (18.7) and (18.8). Optimum horn design Usually, the optimum (from the point of view of maximum gain) design of a horn is desired because it results in the shortest axial length. The whole design can be actually reduced to the solution of a single fourth-order equation. For a horn to be realizable, the following must be true: R = RH = RP. (18.4) The figures below summarize the notations used in describing the horn s geometry. Nikolova 16 16
b α R y B a α H H R x A z R R H It can be shown that RH A/ A = = R A/ a/ A a, (18.43) H R B/ B = = R B/ b/ B b. (18.44) The optimum-gain condition in the -plane (18.37) is substituted in (18.44) to produce B bb λr =. (18.45) There is only one physically meaningful solution to (18.45): ( 8λ ) 1 B= b+ b + R. (18.46) Similarly, the maximum-gain condition for the H-plane of (18.4) together with (18.43) yields R H A a A ( A a) = A A 3λ =. (18.47) 3λ Since R = RH must be fulfilled, (18.47) is substituted in (18.46), which gives Nikolova 16 17
1 8 AA ( a) B= b b + +. (18.48) 3 Substituting in the expression for the horn s gain, 4π G = ε ap AB, (18.49) λ gives the relation between A, the gain G, and the aperture efficiency ε ap : 4π 1 8 Aa ( a) G = ε ap A b+ b +, (18.5) λ 3 + 3bGλ 3G λ =. (18.51) 4 A4 aa3 A 8πε 3 ap π ε ap quation (18.51) is the optimum pyramidal horn design equation. The optimumgain value of ε ap =.51 is usually used, which makes the equation a fourth-order polynomial equation in A. Its roots can be found analytically (which is not particularly easy) and numerically. In a numerical solution, the first guess is usually set at A() =.45λ G. Once A is found, B can be computed from (18.48) and R = RH is computed from (18.47). Sometimes, an optimal horn is desired for a known axial length R. In this case, there is no need for nonlinear-equation solution. The design procedure follows the steps: (a) find A from (18.4), (b) find B from (18.37), and (c) calculate the gain G using (18.49) where ε ap =.51. Horn antennas operate well over a bandwidth of 5%. However, gain performance is optimal only at a given frequency. To understand better the frequency dependence of the directivity and the aperture efficiency, the plot of these curves for an X-band (8. GHz to 1.4 GHz) horn fed by WR9 waveguide is given below ( a =.9 in. =.86 cm and b =.4 in. = 1.16 cm). Nikolova 16 18
[Stutzman&Thiele, Antenna Theory and Design] The gain increases with frequency, which is typical for aperture antennas. However, the curve shows saturation at higher frequencies. This is due to the decrease of the aperture efficiency, which is a result of an increased phase difference in the field distribution at the aperture. Nikolova 16 19
The pattern of a large pyramidal horn ( 1.55 f = GHz, feed is waveguide WR9): Nikolova 16
Comparison of the -plane patterns of a waveguide open end, small pyramidal horn and large pyramidal horn: Note the multiple side lobes and the significant back lobe. They are due to diffraction at the horn edges, which are perpendicular to the field. To reduce edge diffraction, enhancements are proposed for horn antennas such as corrugated horns aperture-matched horns The corrugated horns achieve tapering of the field in the vertical direction, thus, reducing the side-lobes and the diffraction from the top and bottom edges. The overall main beam becomes smooth and nearly rotationally symmetrical (esp. for A B). This is important when the horn is used as a feed to a reflector antenna. Nikolova 16 1
Nikolova 16
Comparison of the H-plane patterns of a waveguide open end, small pyramidal horn and large pyramidal horn: Nikolova 16 3
Circular apertures.1 A uniform circular aperture The uniform circular aperture is approximated by a circular opening in a ground plane illuminated by a uniform plane wave normally incident from behind. x z a y The field distribution is described as a = x ˆ, ρ a. (18.5) The radiation integral is ˆ I j x e β ds S = a rr. (18.53) The integration point is at r = xˆρ cosϕ + y ˆρ sinϕ. (18.54) In (18.54), cylindrical coordinates are used, therefore, rˆ r = ρ sin θ(cosϕcosϕ + sinϕsin ϕ ) = ρ sinθcos( ϕ ϕ ). (18.55) Hence, (18.53) becomes a π a I j sin cos( ) x = e βρ θ ϕ ϕ dϕ ρ dρ = π ρ J( βρ sin θ) dρ. (18.56) Here, J is the Bessel function of the first kind of order zero. Applying the identity Nikolova 16 4
to (18.56) leads to xj ( x ) dx = xj 1( x ) (18.57) a I x = π J1( βasin θ) βsinθ. (18.58) Note that in this case the equivalent magnetic current formulation of the equivalence principle is used [see Lecture 17]. The far field is obtained as ( ˆ cos ˆ cos sin ) e jβ r = θ ϕ φ θ ϕ jβ I x = π r j r 1( sin ) ( ˆ e β J βa θ = θcosϕ φˆ cosθsin ϕ) jβπa. πr βasinθ Principal-plane patterns -plane ( ϕ = ): θ (18.59) J1( βasin θ) ( θ ) = βasinθ (18.6) H-plane ( ϕ = 9 ): ϕ J1( βasin θ) ( θ) = cosθ βasinθ (18.61) The 3-D amplitude pattern: J 1( βasin θ) ( θϕ, ) = 1 sin θsin ϕ βa sin θ f ( θ ) (18.6) The larger the aperture, the less significant the cosθ factor is in (18.61) because the main beam in the θ = direction is very narrow and in this small solid angle cosθ 1. Thus, the 3-D pattern of a large circular aperture features a fairly symmetrical beam. Nikolova 16 5
xample plot of the principal-plane patterns for a = 3λ : 1.9 -plane H-plane.8.7.6.5.4.3..1-15 -1-5 5 1 15 6*pi*sin(theta) The half-power angle for the f ( θ ) factor is obtained at βasinθ = 1.6. So, the HPBW for large apertures (a λ ) is given by 1.6 1.6 λ HPBW = θ1/ arcsin 58.4 βa =, deg. (18.63) βa a For example, if the diameter of the aperture is a = 1λ, then HPBW = 5.84. The side-lobe level of any uniform circular aperture is.133 (-17.5 db). Any uniform aperture has unity taper aperture efficiency, and its directivity can be found directly in terms of its physical area, 4π 4π Du A π a λ λ = p =. (18.64) Nikolova 16 6
. Tapered circular apertures Many practical circular aperture antennas can be approximated as radially symmetric apertures with field amplitude distribution, which is tapered from the center toward the aperture edge. Then, the radiation integral (18.56) has a more general form: a I = π ( ρ ) ρ J ( βρ sin θ) dρ. (18.65) x In (18.65), we still assume that the field has axial symmetry, i.e., it does not depend on ϕ. Often used approximation is the parabolic taper of order n: a ρ ( ρ ) = 1 a Nikolova 16 7 n (18.66) where is a constant. This is substituted in (18.65) to calculate the respective component of the radiation integral: n a ρ Ix ( θ) = π 1 ρ J( βρ sin θ) dρ a. (18.67) The following relation is used to solve (18.67): 1 n n! (1 x) nxj( bx) dx = J 1 n 1( b) bn+ + a b= βa θ. Then, I x ( ) π a I x ( θ) = f( θ, n) In our case, x= ρ / and sin where. (18.68) n + 1 n+ 1( n+ 1)! Jn+ 1( βasin θ) n+ 1 ( βasinθ) θ reduces to, (18.69) f( θ, n) = (18.7) is the normalized pattern (neglecting the angular factors such as cosϕ and cosθsinϕ). The aperture taper efficiency is calculated to be
εt = C 1 C C + n + 1 C(1 C) (1 C) + + n+ 1 n+ 1. (18.71) Here, C denotes the pedestal height. The pedestal height is the edge field illumination relative to the illumination at the center. The properties of several common tapers are given in the tables below. The parabolic taper ( n = 1) provides lower side lobes in comparison with the uniform distribution ( n = ) but it has a broader main beam. There is always a trade-off between low side-lobe levels and high directivity (small HPBW). More or less optimal solution is provided by the parabolic-on-pedestal aperture distribution. Moreover, this distribution approximates very closely the real case of circular reflector antennas, where the feed antenna pattern is intercepted by the reflector only out to the reflector rim. [Stutzman&Thiele] Nikolova 16 8
[Stutzman&Thiele] Nikolova 16 9