3

BASICS OF ANTENNA THEORY

The antenna is a device that radiates or receives electromagnetic waves. In general, antennas can be classified as either single-element radiators or multiple (usually identical) radiating elements. Dipoles, horns, printed patch radiators, and reflectors are considered single-element radiators. Linear, circular, and area arrays consist of multiple (usually identical) radiating elements. Antennas play a very important role in radar-related applications. In this chapter, antenna theory is reviewed, starting with the Maxwell equation in Section 3.1. The infinitesimal dipole is described in Section 3.2, and the half-wavelength dipole is covered in Section 3.3. The 1D linear phase array is briefly reviewed in Section 3.4, and the 2D linear array is discussed in Section 3.5. The key antenna-related parameters are covered in Section 3.6. Some commonly used antennas, including the microstrip antenna, are reviewed in Section 3.7.

3.1 MAXWELL AND WAVE EQUATIONS

The classical Maxwell equations serve as the fundamental basis for all electromagnetic phenomena and are shown below:

Since for any vector , the equation 0 implies the existence of , such that ; that is

or

The quantity is curlless and, mathematically speaking, for any scalar φ. Thus can be represented as the gradient of a scalar potential φ, namely, . Therefore, the two Maxwell equations

imply the existence of the magnetic and electric potentials and , such that

The potentials and φ are not uniquely defined and may be changed by adding constants to them. This freedom in selecting the potentials allows us to impose some convenient constraints between them. In discussing radiation problems, it is customary to impose the Lorenz condition:

The remaining two Maxwell equations become, with and c² = 1/με,

Substituting Eqs. (3.2) and (3.3) into Eq. (3.4), together with the equation and , one obtains the Maxwell equations as the equivalent wave equation forms, in terms of electric and magnetic potentials φ and :

It is assumed that sinusoidal time dependence exists for all quantities ω, ρ, , and :

φ(r, t) = φ(r) e^jωt, ρ(r, t) = ρ(r)e^jωt, ....

With the sinusoidal time dependence characteristics, Eq. (3.5a) can be further simplified as

where k = ω/c. The Maxwell equations will now be explored further.

3.1.1 Harmonic Time Dependence

Consider the following equation, where δ⁽³⁾() is the delta function and serves as the excitation source:

The solution G() of Eq. (3.6) can be considered as an impulse response of the system to have the form

Equation (3.6) states that δ⁽³⁾() can be considered as the excitation source, while G(), from Eq. (3.7), is the impulse response of Eq. (3.6). With ρ/ε and μ as the excitation sources, from linear system theory, the solutions of φ and A in Eq. (3.5b) can be obtained as the convolution of excitation function with the impulse response function:

FIGURE 3.1 Potentials generated by current/charge distribution.

Figure 3.1 shows the geometry of magnetic and electric potentials and (, located at distance and caused by the excitation source ρ located at distance ′.

With R = | – ′|, then

Following replacement of /t with jω, the Lorenz condition takes the form

The electric and magnetic fields of Eq. (3.2) then become

From Eqs. (3.9) and (3.10a), with ω²με = ω²/c² = k², one obtains the and fields as

In the following sections, we will investigate the electric and magnetic fields due to various current sourcing dipoles. We will start from the point source, followed by the line source, and then the planar source. Fields as a function of distance will also be discussed.

3.2 RADIATION FROM AN INFINITESIMAL CURRENT DIPOLE

The infinitesimal dipole is a dipole whose length dl is much smaller than the wavelength λ of the excited wave, namely, dl λ. An infinitesimal dipole is equivalent to a current element Idl with I as the current. The infinitesimal dipole, with dl 0, serves as a point radiator or excitation source of the electric and magnetic fields. A point radiator is also called an isotrope. Physically, it does not exist. Yet, theoretically it serves as a reference for many types of antennas whose performance is expressed in terms of a basic radiator. It is used as a building block to solve many problems related to complicated antenna design. It radiates energy equally in all directions, and the radiation pattern in any plane is a circle.

Figure 3.2 is a graphical representation of radiation from an infinitesimal dipole (point radiator), where the point radiator is located at the origin of the coordinate.

FIGURE 3.2 Radiation from a point radiator.

3.2.1 Magnetic Vector Potential Due to a Small but Finite Current Element

The magnetic vector potential due to a current source and observed at point P, with distance R from the dipole, can be obtained from Eq. (3.8) as

For a linear current source, , it follows that

Since current I is a constant along Δl, and the dipole is very small, the integration shown above can be approximated to its integrand. In addition, the distance R r will be replaced by r in both the exponential term and in the denominator. Therefore, with

which can be represented in spherical coordinates as

where θ is the angle between () and the z axis.

3.2.2 Field Vectors Due to Small but Finite Current Radiation

From Eqs. (3.10b) and (3.11b), the fields , can then be obtained as

Therefore

Since

it follows that

where is the impedance of free space.

The radiation field vectors can then be tabulated as

3.2.3 Far-Field Region

The far-field region is defined as a region with kr 1. By neglecting the second order term 1/r² and keeping only the first order term 1/r, one can obtain the far field of the infinitesimal dipole from Eqs. (3.12a) and (3.12b) as

From these electric and magnetic fields, one can see that no radial component exists at the far field and that the E field is perpendicular to the H field.

As an example, consider a case where a time-varying standing wave with the current distribution along the dipole's length is the sinusoidal one:

The finite-length dipole can be considered as formed by an infinite number of infinitesimal dipoles with length dz′. Each infinitesimal dipole produces the elementary far field and can be described as follows, from Eq. (3.13):

Here, I_e(z′) is the current value of the current element at dz′. Using the far-field approximations, as shown in Fig. 3.3, then

R r, for the amplitude factor.

R r−z′ cos θ, for the phase factor.

FIGURE 3.3 Far-field approximation of z-oriented dipole.

Therefore

and

For current distribution as defined in Eq. (3.14), the integration factor expressed above becomes

By utilizing the following two integrals,

one obtains the far field of the finite-length dipole as follows

3.2.4 Summary of Radiation Fields

On the basis of the preceding discussion, we can divide the antenna radiation field into three regions, which are functions of wavelength λ and dimension of antenna D. The definition of these three regions are listed below.

3.3 RADIATION FROM A HALF-WAVELENGTH DIPOLE

For current distributed along a dipole with length l = λ/2, the electric and magnetic fields can be obtained from Eq. (3.15) as follows:

The received power density, or far-field power density, is defined as

where η is the impedance of the free space, or η = |E/H|. For a half-λ dipole, this is

The normalized power pattern is the ratio of received power density to the maximum power density:

The radiation pattern can be either a normalized field |(θ, )| or a normalized (θ, φ)|²/η. The radiation pattern can be plotted as either a 2D or 3D pattern. Figure 3.4 displays the radiation pattern of a half-wavelength dipole in both 3D and 2D views. Figure 3.4a is a 3D cross-sectional view. The left side of Fig. 3.4b shows the azimuth pattern, looking down from the z axis into the 3D pattern. The right side of Fig. 3.4b shows the elevation pattern, looking from the x axis into the 3D pattern.

FIGURE 3.4 Radiation pattern of a half-wavelength dipole.

3.4 RADIATION FROM A LINEAR ARRAY

A single antenna with a given radiation pattern may be grouped and arranged to yield a different radiation pattern. There are several design variables that can be modified to achieve the desired antenna radiation pattern. These variables consist of the array shape (linear, planar, and circular), element spacing, element-exciting amplitude and phase, and element radiation pattern.

Linear array is one of the patterns that utilize identical radiating elements arranged along a straight line. Figure 3.5 is an example of a 10-element linear array, with the infinitesimal dipole serving as the radiating element. Each dipole is equally spaced with distance d along the x axis. Assuming that the current amplitudes and phases are all equal for the array elements, the far field at observation point P(r, θ, 0) due to the infinitesimal dipole located at the origin is, from Eq. (3.13), expressed as follows:

FIGURE 3.5 A 10-element linear array.

Notice that E_θ0 = E₀ is a fictitious field with no radiating element at the origin and serves as a reference field. Notice also that θ is the observation angle between the receiver (point P) to origin vector and the z axis. The far field due to the individual element n of the 10-element array is

where the parameter r_n can be approximated as r_n r for the magnitude term. For the phase term, r_n can be replaced with r_n r −(n − 5.5) d sin θ. In terms of reference signal E₀, the immediately preceding equation becomes

The array's far field can then be obtained as

Let

ψ = kd sinθ;

then

In general, if the number of radiating elements in the array is N, then the total field observed at far distance can be derived as

Here F_θ(θ) = E₀ = jηk(IΔl) sinθ · (e^−jkr /4πr) is the element factor, and

is the array factor.

The array factor F_A(ψ) has its maximum value equal to N, which occurs at the peak of the mainlobe. Therefore, the normalized array factor is

Figure 3.6 displays the magnitude of normalized array factor F_A(ψ) in terms of degrees for N = 10.

As can be seen in Fig. 3.6, there are one mainlobe and eight sidelobes for the angle ψ ranging from 0° to 360°. The nulls of the array function can be computed by determining the zeros of the numerator term where the denominator is not simultaneously zero:

sin(N ψ/2) = 0 or N ψ/ 2 = ± nπ.

Therefore

FIGURE 3.6 Normalized linear antenna array factor for N = 10.

and

The peaks of the array factor can be found by determining the zeros of the numerator term where the denominator is simultaneously zero:

Therefore

The phase of the linear array elements, based on the infinitesimal dipole, may be chosen such that the mainlobe of the array pattern lies along the array axis (x axis in Fig. 3.5) or normal to the array axis (z axis in Fig. 3.5). The former is known as the end-fire array, while the latter is called the broadside array. For finite-length dipoles with the same configuration as shown in Fig. 3.5, each dipole is equally spaced along the x axis but is oriented along the y axis direction in the end-fire array for the mainlobe along the x axis, and along the z axis direction in the broadside array for mainlobe along the y axis. The dipoles in the end-fire array are usually closer together than they are in the broadside array, and the currents are usually 180° out of phase with each other. The dipoles in the broadside array usually are driven by currents with the same phase.

FIGURE 3.7 Normalized linear antenna array factor for N = 10, d = λ/2.

From Eq. (3.20), one can observe that by varying the angle θ, the received far field will change. Figure 3.7 displays the plot of normalized F_A(θ) versus θ for a broadside array with radiating elements separated by d = λ/2 and N = 10. As can be seen, the total-field array factor has the maximum value at θ = 0 and π, which is the angle when the receiver is perpendicular to the antenna array.

3.4.1 Power Radiation Pattern from a Linear Array

The normalized power radiation pattern of a linear array is defined as the square of the normalized array factor. According to Eq. (3.20a), the power radiation pattern,

Since (N πd/λ) sin θ (πd/λ) sin θ for large N, with Nd = L (the length of the antenna) and sin θ θ for small angle θ, the radiation pattern becomes

Let [(L sin θ)/λ] = x; then

The half-power beamwidth θ_3dB is computed by setting G_α(θ = θ_3dB)= 0.5:

This equation can be solved graphically, resulting in

or

If θ_3dB is small, then sin θ_3dB θ_3dB. Therefore, the one-way half-power beamwidth can be computed as follows:

Equation (3.22) serves as a very important parameter used in radar detection and radar image processing.

For a N-element linear array with d = λ/2, Nd = L implies λ/L = 2/N, and

With small-angle approximation, one obtains

If N is sufficiently large, the antenna can be considered as a uniformly illuminated aperture.

The peak of the first sidelobe of G_a(θ), from Eq. (3.21c), can be computed by finding the first sidelobe peak of sinc(x). Since sinc(x) equals zeros at x = 1, 2, 3, ..., the peak of the first sidelobe of G_a(θ) appears at x = 1.5; therefore

or

3.5 POWER RADIATION PATTERN FROM A 2D RECTANGULAR ARRAY

The 2D power radiation pattern may be approximated as the product of the patterns of the two planes that contain the principal axes of the 1D antenna, namely

where L and W are the lengths of the two arrays in the θ and φ directions, respectively.

The half-power beamwidth of G(θ, φ) in the θ and φ directions are, from Eq. (3.22), θ_3dB λ/L and φ_3dB λ/W, respectively.

3.6 FUNDAMENTALS OF ANTENNA PARAMETERS

The antenna parameters describe the antenna performance with respect to space distribution of the radiated energy. Key antenna parameters are radiation beamwidth, directivity, gain, antenna efficiency, antenna impedance, and radiation efficiency. A brief description of these parameters now follows.

3.6.1 Radiation Beamwidth

The radiation beamwidth can be defined in two ways, namely, the half-power beamwidth and the first-null beamwidth. The half-power beamwidth (HPBW) is the angle between two vectors where the electric field intensity equals 0.707 of the maximum mainbeam field intensity. The first-null beamwidth (FNBW) is the angular difference between the first null on each side of the mainbeam. In general, FNBW is about twice the size of the HPBW. Radiation beamwidth can be defined in terms of either power radiation pattern or field pattern. The field pattern (the absolute value of array factor) will be used for illustration.

A graphical representation of the radiation beamwidth, together with the relative power level against the angle θ, is shown in both rectangular and polar formats in Figs. 3.8 and 3.9, respectively. A linear array with 6 dipoles, located at half wavelength from each other, is used here for illustration. Figure 3.8 displays the field pattern in rectangular format. Here the mainlobe with normalized field = 1 is located at angle θ = 0°. A backlobe with normalized field = 1 is located at angle θ = 180°. In between, there are 4 sidelobes symmetrically located at angles θ = 0^° to θ = 180°, and θ = 180° to θ = 360°, respectively. The HPBW and FNBW are also shown around the mainlobe at θ = 0°.

FIGURE 3.8 Field pattern in rectangular format for N = 6.

FIGURE 3.9 Field pattern in polar format for N = 6.

Figure 3.9 shows the same field pattern of Fig. 3.8 with relative field level plotted against the angle θ in polar format. The outside circle, shown as a solid line, corresponds to the field level of 1. The inner circles, shown as dashed lines, correspond to various levels of the field.

3.6.2 Solid Angle, Power Density, and Radiation Intensity

Given a sphere of radius r (see Fig. 3.10), the solid angle Ω is defined to be a spherical surface S_Ω divided by r²:

The definition of one steradian (1 sr) is the solid angle with its vertex at the center of a sphere with radius r, which is subtended by a spherical surface area equal to that of a square with each side of length r. There are 4π steradians in a sphere.

The infinitesimal area ds on a surface of a sphere, with radius r, in spherical coordinates is

ds = r² sin θ dθ dφ.

FIGURE 3.10 Graphical representation of a solid angle.

Therefore

Letting P_T be the radiator transmitting power, the power density P_d is defined as

The radiation intensity U is defined as the power per solid angle:

For an ideal isotropic point radiator, the power densities P_d at all points on the surface of the sphere of radius r are equal, and can be expressed as

The radiation intensity on any surface surrounding the sphere is given by

The relationship between power density and radiation intensity is then

3.6.3 Directivity and Gain

The directivity D of an antenna is defined from the radiation pattern as

or

The antenna gain is defined as the ratio of the maximum radiation intensity U_max to the maximum radiation intensity of a reference antenna U_ref, which has the same power input to the measured antenna:

Quite often, the terms directivity and the gain are used interchangeably. The difference is that the directivity ignores antenna losses, which are quite small for most classes of antennas.

The radiation intensity averaged over all directions is equal to the total power radiated by the antenna divided by 4πr². The radiation intensity averaged in a given direction is equal to r²θ_3dBφ3dB. The antenna gain is then equal to the area of the isotropic sphere divided by the sector (cross section) area:

where φ_3dB and θ_3dB are the azimuth and elevation beamwidths in radians, respectively.

3.6.4 Antenna Impedance

An antenna is seen from the generator as a load with impedance Z_A represented as

Z_A = (R_L + R_rad) + jX _A,

were R_L represents the ohmic resistance where energy is lost (or transformed into heat) in the antenna structure and R_rad represents the radiation resistance where energy radiates into space. X_A is frequency dependent and represents energy stored in electric and magnetic fields.

The maximum power is delivered to or from the antenna when the impedance of an antenna is matched to the impedance of an equivalent generator or load. Impedance-matching devices, such as a balum or matching transformer, are used to serve this purpose.

3.6.5 Antenna Efficiency

The antenna power radiated into air is represented as

The antenna power loss is represented as

The antenna efficiency η_a is a measure of the power dissipated in the ohmic losses of the antenna. It is the ratio of the total power radiated from the antenna to the power delivered to the antenna from the transmission line. It can be expressed as follows:

3.6.6 Effective Area and Antenna Gain

The antenna has an effective area A_e by which the power density is multiplied to obtain power delivered to the load. The effective area is also referred to as the capture area of the antenna. The effective area is related to the physical size and shape of the antenna.

From Eqs. (3.32) and (3.22), the antenna gain is expressed as

and

Therefore

where A_e = LW.

The effective area and antenna gain are therefore related as

The following examples are used to illustrate the above mentioned antenna parameters.

Example 3.1 By approximating the antenna pattern as a rectangular area, with θ_3dB = φ_3dB = 1° and an average efficiency of 70%, the antenna gain G can be computed as follows.

From Eq. (3.32), one obtains

The area of the antenna radiation pattern can be computed as follows, from Fig. 3.11:

FIGURE 3.11 Antenna radiation pattern approximated as a rectangular area.

Therefore

For a small angle θ, sin θ θ and sin φ φ. Therefore

Given θ_3dB = φ_3dB = 1°, one obtains

For antenna efficiency equal to 70%, one obtains

Example 3.2 By approximating the antenna radiation pattern as an elliptical area with θ_3dB = φ_3dB = 1°, and an average efficiency of 55%, the antenna gain G can be computed using Eq. (3.32):

From Fig. 3.12, the area of ellipse can be computed as

FIGURE 3.12 Antenna radiation pattern approximated as an elliptical area.

The antenna gain is then computed as

For small values of θ_3dB, one can approximate sin θ_3dB θ_3dB and sin φ_3dB φ_3dB. Therefore

Given the values θ_3dB = φ_3dB = 1°, one obtains

With efficiency equal to 55%, the antenna gain becomes

3.6.7 Polarization

The orientation of the fields in the wave is called polarization. For a dipole antenna, the original electric field is oriented along the axis of the antenna, and therefore the induced magnetic field will be perpendicular to both the electric field and the direction of travel. As the wave propagates outward, the electric and magnetic fields will remain perpendicular to each other; they will also be perpendicular to the direction of propagation (Fig. 3.13).

FIGURE 3.13 Polarized fields.

When the wave propagates outward and the field remains in a particular direction, as in the case of waves from the dipole antenna, the wave is considered to be linearly polarized. The field direction will be aligned to the antenna. A vertical antenna will create a vertically, linearly polarized electromagnetic wave. A receiving antenna that is aligned with the polarization will have the greatest sensitivity.

In the case of circular polarization, the electric field rotates as it travels along. If the rotation is clockwise as seen when looking in the direction of propagation, it is called right-hand circular polarization (RHCP). The other possibility is left-hand circular polarization (LHCP). Circular polarization is often used in satellite communications because it eliminates the need to match the receiving antenna to the orientation of the satellite's antenna.

3.7 COMMONLY USED ANTENNA GEOMETRIES

3.7.1 Single-Element Radiators

Many antennas are designed with a single-element radiator. A half-wavelength dipole antenna is an example of a single-element radiator. The circular loop antenna is another example of a wired single-element radiator. Although this type of antenna is easy to implement, its dimension is wavelength-dependent. Therefore, it has some restrictions on high-frequency applications—typically a few GHz.

An open-ended waveguide is inefficient in radiating energy because of a mismatch of impedance at its mouth. By flaring out the sides into a horn to match the intrinsic impedance of free space, the open-ended waveguide becomes a horn antenna. Horns flared in the E or H plane are called sectoral horns and if flared in both E and H plane, pyramidal horns.

The parabolic antenna normally consists of a parabolic reflector and a small feed antenna. The reflector is a metallic surface having a focal point. The small feed antenna is typically a half-wavelength dipole or a horn-based antenna.

Figure 3.14 displays the typical shapes of a circular loop antenna, linear polarized horn antenna, and the parabolic antenna. There are many other well-known antennas and their radiation patterns can easily be found in various books or journal publications on antennas, and will not be repeated here.

FIGURE 3.14 Popular antennas: (a) circular loop antenna; (b) linear polarized horn antenna; (c) parabolic antenna.

3.7.2 Microstrip Antennas and Antenna Array

There are many types of microstrip antennas; and the most common one is the printed patch antenna, consisting of thin metallic patches etched on a dielectric substrate, with a ground on the other side of the substrate. The dimensions of the patch are usually a fraction of the wavelength, while the thickness of dielectric substrate is normally less than a few percent of the wavelength. Because of its size, the microstrip antenna is usually used for high-frequency applications in the range from a few gigahertz (GHz) to 100 GHz.

The feeding signal to the microstrip patch antenna is an important issue in designing a high-performance antenna. There are four popular techniques for feeding the signal to a microstrip patch antenna: (1) the microstrip transmission line, (2) coaxial probe, (3) aperture coupling, and (4) proximity coupling. Methods 1 and 2 utilize a direct-contact approach, while methods 3 and 4 are noncontacting with the patch antenna. A printed patch antenna with a coaxial probe feed is shown in Fig. 3.15. Figure 3.15a is a 3D view of the circuit; Fig. 3.15b is a side view of the microstrip antenna. The size of the patch, the method of current feed, the thickness and characteristics of the substrate, and the geometry of the circuit will decide both the operating frequency and the radiation pattern of the microstrip antenna.

The radiation pattern of a microstrip antenna is normally shown in both the E and H planes. The E plane is the plane in parallel with the E field of the antenna, while the H plane is the plane perpendicular to the E field.

An antenna array consists of multiple (usually identical) radiating elements. Arranging the radiating elements in an array achieves unique radiation characteristics, which cannot be obtained through a single element. Phased array is the antenna array with multiple radiating elements. It changes the radiation pattern electronically by carefully choosing and controlling the phase shift and the magnitude of the signal fed to each radiating element.

FIGURE 3.15 Printed patch antenna.

FIGURE 3.16 Configuration of a 4-dipole linear array.

A planar circuit-based dipole can be used to implement the phased antenna array. Figure 3.16 shows a simple configuration of a linear phased array based on four units of planar half-wavelength dipole, each spaced λ/2 apart from other units and fed with in-phase signal to each of the 4 dipoles. Sean Hum and Rob Randall designed such an array in 2003. [ENEL 619.50 Project: Self-Phased Array System; available at http://www.enel.ucalgary.ca/People/Okoniewski/ENEL619-50/Projects2003/SPAS; accessed on 1/10/07.] The metal (copper) strip–based dipole has one layer of substrate and an integrated planar balun. For λ/2 dipole array with N = 4, the antenna length L = 4 × λ/2 = 2λ; therefore the 3-dB beamwidth can be calculated as θ = λ/L = 0.5 in radians or 28.64°. A measured beamwidth of 28.6° along the x axis matches the theoretical result quite closely.

To design a 2D planar circuit-based antenna array, the signal line must be aligned with equal distance to provide an in-phase signal to each radiating element. One possible implementation of a 4 × 4 antenna with half-wavelength dipole is shown in Fig. 3.17. Figure 3.17a is a front–back view of the planar circuit-based dipole array; Fig. 3.17b shows the configuration of a 2D antenna array.

FIGURE 3.17 Half-wavelength dipole-based 2D antenna array.