lecture15

ECE529/ECE539 Section XV Antenna reflector

XV-1

Section XV. Antenna reflector

1. Introduction Virtually every antenna has a certain ground plane or a reflector. The primary goal of the ground plane is to shape the antenna pattern (increase gain in the desired directions) and minimize the backlobes or sidelobes or, which is the same, minimize radiation to or reception from unwanted directions. The ground plane also has an effect on the antenna impedance, but this effect is typically not very profound, except for metal or dielectric surfaces closely spaced to the antenna. The design of a proper ground plane may be a significant challenge. The most straightforward example is the ground plane of the path antenna – the PCB ground – or that of the VHF car monopole antenna – the car exterior. Similarly, for airborne antennas, the ground plane is the airplane fuselage. For other symmetric dipole-like antennas, the ground plane is typically a metal or wire conducting reflector; which should simultaneously serve as a neutral or common voltage reference for the antenna feeding circuit. In particular, for vertical wireless communication dipoles, the ground is represented by the Earth surface. In this section, we will review basic analytical models of solid metal ground planes and reflectors for dipole-like antennas including the edge effects and the simple diffraction mechanisms. There are three common analytical models that greatly help us to understand and analyze the effect of a ground plane or a reflector on the antenna, and to design an appropriate antenna ground plane. They include

- Wave reflection and Geometrical Optics (GO); - Diffraction, Geometrical Theory of Diffraction (GTD), and Uniform Theory of

Diffraction (UTD); - Physical Optics (PO).

1. Introduction 2. Ground plane for an electric dipole. The /4-rule 3. Method of images 4. Extensions of the image method-corner reflector 5. Finite ground plane – Geometrical Optics 6. Front-to-back ratio 7. Phase center of an antenna with the ground plane/reflector 8. Example – parabolic reflector 9. Edge diffraction model for the sheet ground plane

Problems


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and are commonly referred to as high-frequency methods. These models will be outlined in the present Section, as applied to some simple antenna examples. Even today, a large ground plane or a large reflector present a challenge for numerical modeling of antennas as the electrical size of the complete structure increases and often can no longer be handled with a full wave simulator, whether FEM, or FDTD, or MoM, with an accuracy necessary for low-noise applications. Hence, the value of the related analytical models greatly increases. Interestingly, the same analytical models find applications in a broader area of wireless communications that includes wireless channel estimation and path loss estimation for an indoor or outdoor environment. The ray tracing model, which is currently widely employed for channel estimation in terrains, uses these theories. 2. Ground plane for an electric dipole. The /4-rule Consider a radiating dipole above a PEC ground plane as shown in Fig. 1. We will use the geometrical optics approximation for the incident/reflected signals and will only consider the signal in the dipole E-plane – the xz-plane. The dipole radiates two signals – the forward wave that propagates in the positive direction of the z-axis and the backward wave that is incident upon the metal ground plane. The backward wave is reflected by the ground plane and is eventually added to the forward wave with a proper phase shift. The resulting radiated field is thus a combination of the direct forward wave and the reflected wave.

Fig. 1. Concept of a reflecting (metal) ground plane.


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Neglecting field divergence, one has a solution in the form of plane waves in time domain, with zero phase at dz

z

zxx

zxx

zxx

kk

dzktEztE

dzktEztE

dzktEztE

))(cos(),(

))(cos(),(

))(cos(),(

0ref

0inc

0for

(1)

The reflected wave has been selected in such a way that it satisfies the boundary condition on the metal surface at z=0, i.e.

)0,()0,(0 refinc ztEztEEE xxtxt (2)

Note that Eq. (1) is the exact solution to the reflection problem in plane wave geometry. Thus, the total forward radiated field becomes a combination of two signals

)2cos(cos),(),(),( 0reffortotal dkttEdztEdztEdztE zxxxx (3)

phase shifted by

dkz2 (4) The result is clearly obtained in the form

)2/cos()2/cos(2),( 0total tEdztE xx (5)

but Eq. (4) for the phase shift is generally more important to us. It says that the resulting phase shift includes two contributions:

i. the shift of or the E-field phase reversal due to the reflection from a PEC boundary;

ii. plus the shift /42 ddkz , which corresponds to time delay of a reflected signal over the travel distance of 2d.

If there were no phase reversal (e.g. a perfect magnetic boundary was present instead of a metal boundary), the dipole close to ground plane would radiate forward twice the field (and four times the power) compared to that in free space. When phase reversal is present, the total forward radiation is nearly zero when )(0 d . So is the input resistance of the antenna. This is why the dipole close to a metal surface is a very poor antenna radiator. The physical reason for it is the appearance of surface induced currents on the metal ground plane that are oppositely directed and radiate a field in anti-phase with the main current.


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The optimal separation distance – optimal for the maximum total radiated field – is achieved when in Eqs. (4) and (5) is zero or a multiple of 2 , i.e. when 4/d , or

4/3d , or etc. This separation distance is an important design parameter, not only for the horizontal dipole above a ground plane but also in many other cases. The phase reversal at the reflection from the metal boundary is responsible for a number of interesting effects. In particular, the right-handed circularly polarized (RHCP) signal will be reflected as the left -handed circularly polarized (LHCP) signal from a metal boundary, and vice versa. Here, we can immediately see a way to eliminate or reduce a multipath: if an antenna is intended for RHCP reception then it will only receive the original RHCP signal but will reject at least its first reflections that are LHCP. Finally we note that the above derivation (Eq. (5)) can indeed be done in the phasor form and the final result becomes

)exp()sin(2)2/exp()2/cos(2),( 00total djkEdkjjEdzt zxzxx E (6)

The term in square brackets is recognized as the array factor – see below in the following Sections. The term )exp( djk z is of little importance to us– it contributes to the absolute solution phase only. As another example of the general character of the “/4” rule, which may appear in many other situations, let us consider a feed of a standard horn that is typically given by a coaxially-driven monopole in the cavity – see Fig.2. The length of the monopole is mostly defined by the impedance matching criteria and may vary from horn to horn. However, the horn’s feed separation from the side wall is again close to 4/d , to enable the proper reflection.

Fig. 2. Feed placement for a horn cavity. 3. Method of images The next question is how does the ground plane work for other separation distances and elevation angles different from zenith? Unfortunately, the answer cannot be given in closed form for an arbitrary antenna and a finite ground plane. However, the exact answer can be given for an arbitrary antenna over an infinite ground plane using the so-called image method – see Fig. 3.


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The idea of the image method is simple and powerful: let’s remove the PEC ground plane but put another (image) dipole at the distance –d from the origin, i.e. symmetrically versus the ground plane position. It is then clear that the required boundary condition

0tE (7)

is satisfied everywhere on the plane of the ground due to field cancellation – see Fig. 1. This fact can proved for the tangential E-component by field superposition, not only for normal incidence direction, but also for any point in the xy-plane. Thus, the two dipoles will radiate in the upper hemisphere as one dipole above the ground plane since both configurations satisfy Maxwell’s equations and the same boundary conditions. The radiation to the lower hemisphere must be zero, but this is not the case for the image method. Therefore, the image method works only for the exterior problem (upper hemisphere that includes the dipole) and cannot formally provide the null in the interior (lower hemisphere).

Fig. 3a. Method of images for a horizontal dipole above a ground plane. The ground plane effect is replaced by that of the image dipole. How does current in the image dipole flow? Fig. 3b illustrates the answer to this question: remember that the radiated E-field is always directed parallel with the current. To satisfy the PEC boundary conditions we thus need two oppositely directed currents. When put close to one another, the opposite currents radiate two oppositely directed fields that also cancel each other – one might say that the antenna is “shorted out” and becomes itself a non-radiating transmission line.


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Fig. 3b. Current and field directions. The ground plane alters both the antenna impedance and the radiation pattern. The image method allows us to find the corrected dipole impedance and the resulting radiation pattern analytically. Let’s start with the impedance first. The treatment in what follows will be essentially that of Ref. [3]. Two dipoles: the original one and the image are mutually coupled through the impedance matrix (see Section II)

2

1

2221

1211

2

1

I

I

ZZ

ZZ

V

V (7)

where index 1 corresponds to the original dipole; index 2 – to the image. For reciprocal identical antennas, the mutual and self- impedances are identical, i.e.

22112112 , ZZZZ (8) The active or driving-point impedance of the original dipole (the impedance under presence of the image dipole) becomes from Eq. (7)

12112

11211

1

11 ZZ

I

IZZ

I

VZ d (9)

since currents 21 , II are equal in magnitude but are oppositely directed – see Fig. 3b. Once

12Z is known as in Section II, the dipole impedance above the ground plane simply

coincides with dZ1 from Eq. (9). One special case that should be evaluated is when the

separation d approaches zero. Then 1112 ZZ (two dipoles tend to coincide) and

01 dZ , which again means that the original antenna is “shorted out”.

Now, let’s proceed with the radiation pattern of a single horizontal infinitesimally small dipole [1] centered at origin and oriented along the x-axis. The pattern is conveniently


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presented in spherical coordinates. The phasor of the electric field has the form (Section IV)

220 sinsin1

4

)exp(

r

rjklkIjE (10)

If the dipole were of a finite length, with the sinusoidal current distribution, the pattern expression would become as in Section IV. We know that neither the pattern magnitude nor its polarization should change when we move the dipole (or any other antenna) by a certain finite distance, say d, from the origin – any linear translation cannot change the pattern magnitude or add new polarization components. What changes, however, is the phase since the signal from the dipole spaced closer to the observation point will arrive earlier, no matter how large the absolute distance to that point is. Thus, for d translation along the z-axis, one needs to replace r in the phase factor )exp( rjk in Eq. (10) by a new distance r. According to the law of cosines and Fig. 4

Fig. 4. Radiation geometry in spherical coordinates. The dipole offset from the origin is given by d.

cos21cos2;cos22

22222

r

d

r

drrddrrrddrr

(11)


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with d/r being a small parameter , 1 . Using Taylor series expansion and keeping only dominant terms, one has

)(cos;)(cos1 22 rOdrrOr

drr

(12)

which leads to (after substitution of r instead of r)

222,12,1 sinsin14

)cosexp()exp(

r

djkrjklkIjE (13)

where 2,1

E are the fields radiated by the original dipole and by the image dipole,

respectively; 02,1 II . The total field is given by their sum, i.e.

220total sinsin1

4

)exp()]cossin(2[),,(

r

rjklkIjkdjrE (14)

Once again we recognize the factor in square brackets as the array factor – one may want to compare this result to Eq. (6) at =0. One reason for emphasizing this name is that the linear pattern of two dipoles (or one dipole above the ground plane) is obtained as the single pattern multiplied by the array factor – see Eq.(10). Another reason is that the array factor does not really change from antenna to antenna: if we repeat the above derivation not for the infinitesimally small dipole but for a dipole of arbitrary length we will have exactly the same array factor in front of the corresponding single-dipole pattern. However, the array factor will change if the current directions in two dipoles were not the opposite (phase shift of ) but the same (phase shift of 0). This happens, for example, for a horizontal dipole above a PMC ground plane. Another (and more important) example is that of the vertical dipole above the ground plane, when the image current flows in the same direction as the current in the actual dipole [1]. Instead of subtracting two exponents in the case of the opposite current flow

)cossin(2)cosexp()cosexp( kdjdjkdjk (15a) one should add them together in order to obtain the array factor in the form

)coscos(2)cosexp()cosexp( kddjkdjk (15b) The antenna arrays considered in the following text extensively use the array factor for pattern synthesis. However, more generic (phased) arrays usually use a certain prescribed (not necessarily 0 or ) phase shift between the individual elements.


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4. Extensions of the image method-corner reflector 4.1 General The method of images can be quite helpful in other antenna problems like that of the corner reflector antenna shown in Fig. 5a to 5d. The dipole pattern is shaped by two finite ground planes, with corner (or flare) angle . The corner-reflector antenna is made up of two plane reflector panels and a dipole element. This arrangement prohibits radiation in the back and side directions and hence makes the antenna more directional. The antenna is useful in obtaining gains of up to 12 dB. Its major advantage is construction flexibility and simplicity. The method of images is shown schematically in Fig. 5a. It assumes three image dipoles (one for every plane plus one “balancing” image dipole). All four dipoles (the original one plus three images) form two polar dipole pairs that cancel the tangential E-field on both corner planes. Again, the field outside the corner angle is non-physical and should be ignored. The method of images is a reasonable assumption when the corner plates are rather long and the dipole is located far way from the corner edges. Again, this method allows us to find the dipole impedance and the resulting radiation pattern. These calculations are done in particular in [3] and in many other sources – see [6]; we will consider them below as an example.

Fig. 5a. The corner reflector with corner angle of 90 deg – top view – and the related method of images.

The method of images could be applied to other corner angles, for example 60 deg – see Fig. 5b ([6]). There are five image dipoles now, according to the anticipated symmetry.


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Fig. 5b. Corner reflector with corner angle of 60 deg – top view – and the related method of images. According to [6] , the method of images is applicable for corner angles equal to 180 deg/n, where n is any positive integer. This is a well-known fact in electrostatics. Corners of 180 degrees (flat sheet), 90, 60, 45 degrees, etc., can be treated by this method. The performance of corner reflectors of intermediate angles cannot be determined by this method but can be interpolated approximately from the others. Could the method of images be applied to a corner angle higher than 180 deg? Such an opportunity would be really inviting since the corner becomes an infinite metal wedge and one might be able to treat the wedge diffraction problem – one of the most complicated problem for any kind of ray tracing algorithm – by a simple mean. Unfortunately, the images need to be place into free space for such an approach; this circumstance violates the image method itself. 4.2 Corner reflector antenna Returning back to the corner reflector antenna, in general, its feed element is almost always a dipole or an array of collinear dipoles placed parallel to the vertex a distance s away, as shown in Fig. 5a (top view). To obtain a greater bandwidth the feed elements are thick cylindrical or biconical dipoles instead of thin wires. The aperture of the corner reflector (D) is usually made between one and two wavelengths 2 D [1]. The feed-to-vertex distance (s) is usually taken to be between a third and two-thirds of the wavelength 3/23/ s [1]. For each reflector, there is an optimum feed-to-vertex spacing. If the spacing becomes too small, the radiation resistance decreases and becomes comparable to the loss resistance of the system which results in an inefficient antenna. For very large spacing, the system produces undesirable multi-lobes, and it loses its directional characteristics [2]. The length of the sides of the 90 corner reflector is mostly taken to be twice the distance from the vertex to the feed sL 2 . The height (H) of the reflector is usually taken to be about 1.2


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to 1.5 times greater than the total length of the feed element, in order to reduce radiation towards the back region from the ends. Figs. 5c and 5d show typical UHF corner reflectors at 433 MHz, with variable flare angle. The antenna impedance bandwidth covers the band from 415MHz to at least 465 MHz. The fine impedance tuning could be made by a slight variation of the distance s, without affecting much the radiation pattern. The corner ground plane is not floating; it is always connected to the outer conductor of the coaxial split-tube balun.

Fig. 5c. Typical dimensions for a 433 MHz corner reflector.

Fig. 5d. Typical wide-radiation angle UHF corner reflector dipoles at 433 MHz. Antenna Lab, ECE Dept./WPI.


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5. Finite ground plane – Geometrical Optics A question of significant practical importance relates to understanding the impact of the finite ground plane. The geometry of a dipole antenna above the finite ground plane is shown schematically in Fig. 6. A first naive guess about the field distribution of the antenna is that of pure geometrical optics (GO)– see Fig. 6.

Fig. 6. Geometrical optics approximation for a dipole above a finite ground plane. In terms of geometrical optics, the total field is considered to be a combination of rays emanating from the dipole and then reflected from the metal surface according to the Snell’s law, that is [3]

0)(,0)( 1212 SSnSSn

(16a)

where 21 , SS

are the incident and reflected ray directions, respectively; n

is the unit outer

normal to the reflector surface. From Eq. (16a), one can express 1S

through 2S

and vice versa

nSnSSnSnSS

)(2,)(2 112221 (16b) Both expressions indeed coincide due to reciprocity. According to geometrical optics, the field everywhere within the reflection boundary (RB) in Fig. 6 is a combination of the incident (line-of-sight or LOS signal) and the reflected signal (RS). The field everywhere outside the reflection boundary but still inside the


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shadow boundary (SB) is the LOS signal from the dipole. The field below the shadow boundary is zero. The diffracted field in Fig. 6 is ignored. 6. Front-to-back ratio Is the field in the shadow zone really zero? If it is, then a small reflector might be completely sufficient for a dipole antenna. In practice, however, one prefers to use reflectors as large as possible, despite apparent size and weight constraints. This points us to the fact that a finite ground plane does not quite follow the laws of geometrical optics. In order to estimate the accuracy of geometrical optics, consider first a numerical example of a half-wave strip (o blade) dipole spaced a quarter wavelength apart from the square ground plane of a variable size. The geometry is shown in Fig. 7; the dipole width is /150.

Fig. 7. Geometry for a half-wave dipole with quarter wave separation. The xz-plane is the E-plane of the dipole; the yz-plane is the H-plane. The size, G, of the ground plane shown in this figure is approximately 1.7. We will further vary the ground plane size, G, as 0.5, 1.0, 1.5, and 2.0 and start looking at two radiation patterns: total gain in the E-plane of the antenna (the xz-plane) and total gain in the H-plane (the yz-plane). For the selected dipole geometry, total gain is close to elevation gain. The corresponding results (both rectangular and polar plots) are shown in Fig. 8 that follows. These are obtained with Ansoft HFSS software (using radiation boundary of a sufficiently large size and fine meshes with about 100,000 tetrahedra).


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Fig. 8. Radiation pattern of the dipole above a finite ground plane of variable size. Left – rectangular plot; right – the equivalent polar plot. One can see from Fig. 8 that there is in fact no shadow zone beneath the ground plane. Moreover, the radiation in the backward direction is quite significant. The backward radiation indeed decreases, when the ground plane size increases, but not monotonically. The ratio of power gain at zenith (in the direction of maximum radiation) to the gain in the opposite direction (at nadir) is called the front-to-back ratio. For the gain in dB, this ratio is just a difference between two gain values. For the present example, this ratio is given in Table 2.


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Table 2. Front-to-back ration for horizontal dipole as a function of ground plane size (square ground plane).

Ground plane size, G Front-to-back ratio, dB 0.5

~8 dB

1.0

~14 dB

1.5

~21 dB

2.0

~21 dB

One can see that performance obviously improving as the size of the ground plane increases. From the view point of performance/size ratio, the most beneficial is perhaps the ground plane on the size of about 1.5 – the incremental performance deteriorates after this point. Note that the full-wave numerical simulations for a large ground plane meet considerable difficulties. For example, in Ansoft HFSS, even the flat ground plane of 3-4 can hardly be simulated at present (accurate front-to-back ratio) on an ordinary PC, either with the radiation box or with the PML. For antenna modeling on large metal platforms, including aircrafts and ships, other methods are necessary that are considered below.

7. Phase center of an antenna with the ground plane/reflector The phase center of an antenna is the local center of curvature of the far-field phase front. In other words it is the center of a sphere that is tangent to the far-field phase front for a given direction (observation angle). It is clear that the phase center is in general frequency- and angle-dependent. The phase center is important in applications such as GPS navigation and indoor geolocation as long as the antenna has a significant physical dimension. If not, then the antenna phase center may be approximated by its geometrical center. The direction-of-arrival (DOA) or time-or-arrival (TOA) methods for source location will rather point to the antenna phase center but not to its physical center. For a simple dipole without the ground plane, the far-field is given by Eq. (10). Its inspection shows that the phase front is angle-independent and is also frequency-independent. The phase center coincides with the physical center of the antenna.

Note: Why is the ground plane important? Consider one example: a GPS antenna. You already know how weak the received signal could be. Now, imagine all the noise that is coming from Earth ground and surroundings. If the ground plane does not block it properly, this noise may entirely mask the useful signal.


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For a dipole (and any other antenna) with a reflector the situation changes. Consider Eq. (14) for the horizontal dipole pattern above the infinite ground plane. Even though there seems to be no explicit phase variation in the far field, remember that radius-vector r

is

now measured from the ground plane. Therefore, the phase center is now the center of the ground plane, for any frequency and for any dipole height. In other words, it is located exactly in the middle between the dipole and its image. When the ground plane is that of finite size or just small, the phase center is expected to be located somewhere between the ground plane and the dipole – see Fig. 9 – for the effect of the ground plane is less profound. It is also expected to have an angular dependence, especially at low elevation angles versus the ground plane.

Fig. 9. A schematic that illustrates the phase center of the dipole antenna above a finite ground plane.

Tranquilla and Best [7] have investigated the phase center of a monopole above a concentric wire-made ground plane. The ground plane was an array of eight quarter-wavelength radial ground plane rods. They found that the computed distance from the monopole base to the phase center varies between 0.12-0.15 except at zenith where the abrupt phase reversal in the field leads to a discontinuity in the plot of the phase center location. The angular position information indicates that the phase center remains near but not on the actual monopole axis (z>0) except for observation angles approaching the zenith or ground plane angles. This is contrary to DeJong’s formulation [8], which places the phase center along the negative monopole image axis. See also early Carter's work [9]. The phase center of horn antennas has been investigated in great detail - see [1] for a list of references - for it is critical in the proper design of a horn-fed reflector antenna.


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8. Example – parabolic reflector Despite all critics of the geometrical optics, it may be quite useful for certain problems; among them is the concept of the parabolic reflector schematically shown in Fig. 10. We require that any ray emanating from the center F of a reflecting surface will propagate as a plane wave in the positive x-direction. The opposite is also true: a plane wave incident into the negative direction of the x-axis should be focused at point F.

Fig. 10. A concept of the parabolic reflector. Based on the GO model, one can establish the form of the surface needed to satisfy those specifications. We will use the Snell’s law given by first expression in Eq. (16b). It is also more convenient to use indexes inc and ref instead of 1 and 2. With reference to Fig. 10, Eq. (16b) reads

nSnSS

)(2 refrefinc (16c)

where ynxnn

21 is the unit surface normal, yxSyxS

01,sincos refinc .

Symbols yx

, denote unit vectors in the x- and y-directions, respectively. Substitution into Eq. (16c) gives

2121 2sin,21cos nnn (16d)

The solution to the surface normal is straightforward:

2cos,

2sin 21

nn (16e)


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The surface equation can then be established based on the expression of the surface normal. An easier way, however, is to use the identity

fSQFS 2 (16f) which again states that the plane wave of constant phase is created by the reflector. With reference to Fig. 10, we arrive at the surface equation in the form

2/sin,cos,

2 f

SQSF (16g)

which is the familiar parabolic surface equation. 9. Edge diffraction model for the sheet ground plane Aside from a numerical simulation, is there an analytical model that can provide us with an explanation of the radiation in the shadow zone of the antenna ground plane? The answer is indeed yes and it relies upon a famous solution for the wedge diffraction obtained by Arnold Sommerfeld some 120 years ago (in 1896) – see, for example, [10]-[13]. The approach was based on the fact that wave functions (solutions) to a diffraction problem may not be unique in a physical space but are unique in Riemann space, which is a generalization of Riemann surfaces used in the theory of functions of complex variables. A certain contour integral on the complex plane was therefore formulated and then solved. Other methods have been considered for both metal edge and metal wedge that are based on Bessel functions expansions [11]-[15]1. The literature on modern diffraction methods related to antenna problems is extensive – see, for example, [3], [4],[11],[15]. Consider Fig. 6 again. When the signal from the dipole reaches the edge of the ground plane a diffraction occurs so that the edge will start to radiate in all directions, including the antenna backlobe. Hence, the shadow zone disappears. The diffraction phenomenon becomes significant in a number of antenna applications. In particular, the design of a large parabolic reflector antenna is impossible without taking into account the diffraction effects (see below in this Section). The scattering problem on a 2D metal wedge might appear to be similar to that of the corner reflector but the corner angle should now be greater than 180 deg – see section 1.4 above. Unfortunately, this circumstance makes it impossible to apply the image method. Instead, one has to develop a more complicated solution that extensively uses Bessel functions. Below, we will generally follow the Sommerfeld’s solution given in terms of

1 The original Sommerfeld’s derivation is complicated and it is skipped not only in the classic antenna books [1],[3],[4] but also in more advanced EM sources [11]. The derivation of the Sommerfeld’s result from a series of Bessel functions on the order of n+1/2i s given in [12],[15].


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Fresnel integrals. We will first consider the “knife edge” diffraction and then proceed to a more complicated wedge diffraction. 9.1. Incident field Consider the geometry shown in Fig. 11a. The incident field is a plane wave (not a dipole) that is polarized in z-direction – it is thus TM to z wave according to the generally accepted terminology2. We introduce cylindrical coordinates

krRzryrx ;,cos,cos (17) where k is the free-space wavenumber.

Fig. 11a. Metal edge excited by a plane wave (“knife edge” diffraction). In the phasor form, the incident plane wave field in Fig. 11a is given by

0,sin,cos),exp( incinc0inc kkkrkjEEz

(18)

or, when expanded results in Eq. (19),

])sinsincos[cosexp()exp( incinc00inc jkrExjkxjkEE yxz (19)

2 In this and in some other contexts, acronyms TM and TE are mostly used to specify the direction of the electric field and magnetic field in a plane traveling wave. Originally however, the separation into TM and TE modes comes from waveguide physics where it has a significant physical meaning.


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Eq. (19) yields the well-known representation of the plane wave in cylindrical coordinates

))cos(exp( inc0inc jREEz (20a)

Simultaneously, for the plane wave reflected back from the infinite metal ground plane,

))cos(exp( inc0ref jREEz (20b)

9.2.Scattered field and total field The total solution to the problem is a combination of the incident field and the scattered field. The total field must satisfy Maxwell’s equations (usually in the form of wave equations) and boundary conditions, i.e.

0at 0; totalt

scattinctotal EEEE

(21)

It is usually the matter of convenience what field is sought: the scattered field or the total field. Since the incident field is known a priori both approaches should yield the same result. In the present problem, we will proceed with the total field. The solution for the total field is assumed to have the same TM polarization form. In cylindrical coordinates, this can be represented by

0,,1

);,(,0

totaltotal

totaltotal

stotal

totaltotaltotaltotal

zzz

r

zzr

HR

EjkH

E

R

jkH

REEEE

(22)

9.3. Exact solution The form in which the solution is sought is critical for the subsequent analysis. The solution to the present problem will use the prerequisites of geometrical optics. A more simple solution to the problem is obtained when we replace the edge (a half of the infinite ground plane) by the infinite ground plane itself. One has

))cos(exp())cos(exp( inc0inc0scattinctotal jREjREEEE zzz (23)

In that case, the scattered field is just the reflected field and no special full-wave analysis of the scatterer is necessary in contrast to other examples. Now, for a semi-infinite ground plane (the edge), one may extend Eq. (23) by

),(),( incincscattinctotal REREEEE zzz (24)

where E is a solution to Maxwell’s equations with the augmented boundary conditions that becomes the plane wave far enough from the edge. Sommerfeld has found that


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djj

jRERE )exp())cos(exp(),( 2inc0inc (25)

where

2cos2 inc

R (26)

The integral

jdjdjF

2

1)exp(,)exp()(

0

22

(27)

is the original the Fresnel integral; it is now common to use Fresnel integrals C and S3 in the form

jFFjSC

jF

)()(,

22

22

1)( (28)

The asymptotic behavior, which is mostly needed for the analytical solution in the far field, is as follows

at 2

)exp(- )(

at 2

)exp()(

2

2

j

j

jF

j

jF

(29)

Note that for the TE plane wave incidence4 (E-field is perpendicular to the edge corner), the E-field in Eqs. (23)-(25) is replaced by the H-field, which is now parallel to the edge,

3 Integrals dxxuSu

0

2

2

1sin)( and dxxuC

u

0

2

2

1cos)( are implemented in Mathematica and

MATLAB (in Version 7.1.0 R14 of 2005 but not in the most recent versions of MATLAB).

4 Quite often in the literature, the TE incidence case is designated as the hard-surface case whereas the TM incidence case is designated as the soft-surface case, by analogy with acoustics (Neumann or Dirichlet boundary conditions, respectively).


XV-22

and the minus sign on the right-hand side of Eqs. (23),(24) is replaced by plus, that is [11],[12],[15]

),(),( incincscattinctotal RHRHHHH zzz (30a)

djj

jRHRH )exp())cos(exp(),( 2inc0inc (30b)

9..4. Far field The Fresnel integrals are highly oscillatory fields close to origin in Fig. (9), i.e. at 0 . However, they stabilize at large argument amplitudes. We are mostly interested in the radiation pattern that, except for the non-decaying incident and reflected plane waves

includes the decaying diffracted field. That diffracted field behaves as R/1 , which corresponds to a finite contribution into the far-field radiation pattern for a 2D geometry

since the radiation intensity behaves as 1~)/1( 2 RR . Thus, Eq. (24) may be in general rewritten as

diffrefinctotalzzzz EEEE (31)

where the separation is now made of the scattered field into the reflected one and the diffracted one. There are three solution regions marked in Fig. 11a. We’ll consider every of them separately: Region I, above the reflection boundary RB-Fig. 11b. In that region, both the incident wave and the reflected wave exist. Furthermore,

Rincinc at ,02

cos,02

cos

in both terms on the right hand side of Eq. (24). According to Eqs. (24)-(29),

)2/)cos((

1

)2/)cos((

1

22

1

)exp(),(

))cos(exp(

))cos(exp(

incinc

inc0diff

inc0ref

inc0inc

diffrefinctotal

kjD

r

jkrDEE

jREE

jREE

EEEE

z

z

z

zzzz

(32a)

where D (a linear complex field pattern) is called a diffraction coefficient.


XV-23

Fig. 11b. Region I for edge diffraction. Region II, between the reflection boundary RB and shadow boundary SB-Fig.11c. In that region,

Rincinc at ,02

cos,02

cos

and there should not be a reflected wave. The use of Eqs. (24)-(29) gives precisely the expected result:

r

jkrDEE

E

jREE

EEEE

z

z

z

zzzz

)exp(),(

0

))cos(exp(

inc0diff

ref

inc0inc

diffrefinctotal

(32b)


XV-24

Fig. 11c. Region II for edge diffraction. Region III, the shadow boundary SB-Fig. 11d. In that region,

Rincinc at ,02

cos,02

cos

Neither the incident wave nor the reflected wave is expected to exist. The use of Eqs. (24)-(29) again confirms this conclusion,

r

jkrDEE

E

E

EEEE

z

z

z

zzzz

)exp(),(

0

0

inc0diff

ref

inc

diffrefinctotal

(32c)


XV-25

Fig. 11d. Region III for edge diffraction. Note that the diffraction coefficient has the same form in all three cases. Again, for TE plane wave incidence on a edge, the diffraction coefficient has the same form as in Eq. (31) but relates to the H-field, with the minus sign in curled brackets replaced by a plus [11]. In the dimensionless form,

)2/)cos((

1

)2/)cos((

1)exp(

22

1

incinc0

diff

R

jR

jEEz (33a)

)2/)cos((

1

)2/)cos((

1)exp(

22

1

incinc0

diff

R

jR

jHH z (33b)

One can now generalize the above result to state that the diffracted ray (diffraction coefficient) from an edge is also given by Eq. (31) when an arbitrary incident field at the edge is given by 0E . This field can be that of a plane wave, a line source, a finite-length

dipole, or the field from a previous diffraction.


XV-26

References [1] C. A. Balanis, Antenna Theory. Analysis and Design, Wiley, New York, 2005 3rd

ed., pp. 883-884. [2] R.C. Johnson and H. Jasik, “Antenna Engineering Handbook”, third edition,

McGraw-Hill Book Company, 1993. [3] T. A. Milligan, Modern Antenna Design, Wiley-IEEE Press, New York, 2005. [4] W. Stutzman and G. A. Thiele, Antenna Theory and Design, Wiley, New York,

1998, 2nd edition. [5] L. Diaz and T. A. Milligan, Antenna Engineering Using Physical Optics, Artech

House, Boston, 1996. [6] J. D. Kraus, “The corner reflector antenna,” Proceedings of the I.R.E., Nov. 1940,

pp. 514-519. [7] J. M. Tranquilla and S. R. Best, "Phase center considerations for the monopole

antenna, IEEE Trans. Antennas Propagation, vol. AP-34, no 5, pp. 741-744, May, 1986.

[8] G. DeJong, “The phase centre of a monopole antenna,” Radio Sci., vol. 17, no. 2, pp.

349-355, 1982. [9] D. C. Carter, “Phase centres of microwave antennas,” IRE Trans. Antennas

Propagat., vol. AP-4, pp. 597-600, 1956. [10] A. Sommerfeld, Optics, Academic Press, New York, pp. 245-265. [11] A. Ishimary, Electromagnetic Wave Propagation, Radiation, and Scattering,

Prentice Hall, Upper Saddle River, NJ, 1991. [12] H. Bateman, The Mathematical Analysis of Electrical and Optical Wave Motion on

The Basis of Maxwell’s Equations, Diver Publications, Inc., 1955. [13] J. Van Bladel, Singular Electromagnetic Fields and Sources, IEEE Press,

Piscataway, NJ, 1991. [14] J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans.

Antennas Propagation, vol. 20, pp. 442-446, 1954. [15] C. A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, 1989.


XV-27

Review Questions

1. Please, formulate the /4 rule for the dipole with a metal ground plane in your own words.

2. A dipole operates at 1GHz. At which distance from the dipole should the ground plane be located?

3. What is the maximum gain for the dipole with a properly located ground plane?

4. Which ground plane size is most beneficial for the dipole from the viewpoint of performance/size ratio, in terms of operating wavelength ?


XV-28

Problems Problem 1. Equations of subsection 3 give an approximation for the dipole field above a metal ground plane, i.e. above the perfect electric conductor (PEC) in the ideal case. One could consider an impedance ground plane in Fig. 1 with a non-zero surface impedance (boundary condition on the reflecting surface) given by

0,)0(

)0(

Sy

xS Z

zH

zEZ

1. Derive the dipole field above the ground when

)0)0(( zHZ yS . This case

is known as perfect magnetic conductor (PMC).

2. Find separation values d (in terms of wavelength) necessary for optimal separation distance – optimal for the maximum total radiated field.

Problem 2. Repeat Problem 1 when

jZ S (an inductive surface

impedance) 5. Consider normal incidence only. Problem 3. The field of a infinitesimally small dipole above a ground plane is given by:

22

0

total

sinsin1

4

)exp(

)]cossin(2[),,(

r

rjklkIj

kdjrE

5 More precisely, units for surface impedance are not exactly Ohms but Ohms per square – see Section III.

Find analytical expression for total radiated power in terms of feed current for 4/d 6. Problem 4. The combination of two horizontal perpendicular dipoles shown in the figure that follows is known as a turnstile antenna. It is commonly used for creating dual independent polarization or circular polarization. In this problem we will assume infinitesimally small dipoles. By applying the method of images it is possible to evaluate parameters of a circularly-polarized antenna above a metal ground plane.

1. Obtain an analytical expression for the total radiated electric field of the turnstile above the ground plane assuming independent currents (phasors) in the dipole feeds 2,1I and dipoles of the same

length l. 2. Plot total normalized directivity

in the H-plane to scale when

021 III and 4/d , and

show on the same graph

6 This is a difficult question; integrate over azimuthal angle first and then introduce a new

integration variable cost .


XV-29

directivity of only dipole 1 ( 0, 201 III ).

Problem 5. Let’s denote dipole 1 in the figure to Problem 4 as dipole X and dipole 2 as dipole Y, according to the axes of Cartesian coordinates. The total

electric field, E

, of two dipoles in the H-plane of dipole X becomes

)(2

1

)(2

1

YX

YXYX

jEEL

jEERyExEE

where )(2

1, yjxLR

are the so

called right-handed circular polarization (RHCP) and left-handed circular polarization (LHCP) orthogonal unit vectors, respectively. These vector determine the RHCP and LHCP components of the total field – see, for example, [3]. From this equation, one obtains (RHCP) polarization isolation, or cross-polarization ratio C , in the

form L

RC E

E . Polarization isolation is

generally measured in dB and shows how is good is quality of circular polarization produced by an antenna. With all other data identical to these from Problem 3 and 4/d ,

1. Plot C (dB) in the H-plane of

dipole X to scale when

0201 , jIIII ;

2. Plot C (dB) in the H-plane of

dipole X to scale when

0201 , IIjII ;

3. Plot normalized RHCP gain7 for either case 1 or 2.

4. Polarization isolation of a GPS RHCP antenna should be 15 dB or better. Over which beamwidth is this value achieved?

5. Will your result change if the E-plane of dipole X is considered?

6. Why cannot we have a good polarization isolation over the entire hemisphere? Explain your answer. Could you suggest a possible solution?

Problem 6. Figure that follows shows a vertical electrical dipole above a ground plane. In this problem we will assume an infinitesimally small.

1. Obtain an analytical expression for the total radiated electric field of the vertical dipole above the ground plane assuming a short dipole of length l.

7 Gain in dBic - antenna gain, decibels referenced to a circularly polarized, theoretical isotropic radiator of the same total power. The realized gain of a circularly-polarized antenna is also often measured in dBic, which not quite correct since the realized gain additionally takes into account impedance mismatch losses.


XV-30

3. Plot normalized directivity in the E-plane to scale;

4. Repeat for directivity in the H-plane.

Problem 7. Show how to apply the image method to a dipole-fed corner reflector with corner angle of 45 deg. The feeding dipole is centered. Problem 8. Could we apply the image method when the corner angle is 270 deg? Why yes or why no? Problem 9. In this problem, we consider a vertical monopole over an infinite ground plane. The monopole length is not critical for the following analysis.

1. Explain how does the method of images works for the vertical monopole over an infinite ground plane? Present an equivalent dipole.

2. If the impedance of an equivalent dipole is Z, what is the monopole impedance?

Problem 10. Consider an idealized case of tunnel environment shown in the figure that follows. Both 2D walls are PEC boundaries. A more realistic tunnel environment is important from the geolocation point of view in mines.

Is it possible to apply the image method to the present problem? Do not rush to give a negative answer. Problem 11. Evaluate the following Fresnel integrals:

1.

0

2 )exp( djF

2. 10

0

2 )exp( djF

3.

10

2 )exp( djF

Problem. 12. What was unclear in the assigned reading. What needs to be extended?

lecture15

Documents

handed circularly

shadow boundary

pec ground

ground plane

ground plane

metal boundary

backward wave

edge diffraction