Radar Range Equation
2.2 BASIC RADAR RANGE EQUATION
2.2.1 Derivation of Es
2.2.1.1 The Transmitter
We begin the derivation at the transmitter output and go through the waveguide and antenna out into space (see Figure 2.1). For now, assume the radar is in free space. We can account for the effects of the atmosphere in the loss term, L. We assume the transmitter generates a single pulse with a rectangular envelope that has a width of τp. Figure 2.2 contains a simplified representation of this pulse. In this example, the pulse is modulated with a constant frequency of fc, the carrier frequency.3
Fig ure 2.1 Transmit section of a radar.
Fig ure 2.2 Depiction of a transmit pulse.
The average transmit power in the signal over the duration of the pulse, is termed the peak transmit power and is denoted as PT. We term this power the peak transmit power because later we will consider the transmit power averaged over many pulses.
The waveguide in Figure 2.1 carries the signal from the transmitter to the antenna feed input. The waveguide’s only feature of interest in the radar range equation is that it is a lossy device that attenuates the signal. Although we only refer to the “waveguide” here, there are several devices included between the transmitter and antenna feed of a practical radar (see Chapter 5).
Because it is a lossy device, we characterize the waveguide in terms of its loss, which we denote as Lt and term transmit loss. Since Lt is a loss, it is greater than unity. With this, the power at the input to the antenna feed takes the form
Generally, the feed and other components of the antenna attenuate the signal further. If we consolidate all these losses into an antenna loss term, Lant, the radar finally radiates the power
Since the pulse envelope width is τp, the energy radiated by the antenna is
2.2.1.2 The Antenna
The purpose of the radar antenna is to concentrate, or focus, the radiated energy in a small angular sector of space. As an analogy, the radar antenna works much like the reflector in a flashlight. Like a flashlight, a radar antenna does not perfectly focus the beam. For now, however, we will assume it does. Later, we will account for imperfect focusing by using a scaling term.
Given the purpose above, we assume all the radiated energy is concentrated in the area, Abeam, indicated in Figure 2.3. With this, the energy density over Abeam is
To extend (2.5) to the next step, we need an equation for Abeam. Given lengths for the major and minor axes of the ellipse in Figure 2.3 of RθA and RθB, we can write the area of the ellipse:
We recognize that the energy is not uniformly distributed across Aellipse and that some of the energy will “spill” out of the area Aellipse (i.e., the antenna does not focus the energy perfectly, as indicated earlier). We account for this by replacing π/4 with a scale factor KA. Further discussion of KA will follow shortly. We can write Abeam, then, as follows:
Substituting (2.7) into (2.5) produces the following:
We now define a term, GT, the transmit antenna directivity, or directive gain, as
Using (2.9) to rewrite (2.8), we get
Fig ure 2.3 Radiation sphere with antenna beam.
We reiterate: the form of antenna directivity given in (2.9) depends upon the assumption that Lant captures the losses associated with the feed and other components of the antenna. Some analysts combine the feed and antenna losses with the transmit antenna directivity and term the result the power gain, or simply gain, of the antenna [10]. We will avoid doing so here, owing to the confusion it produces when using (2.9) and the difficulties associated with another form of directivity, presented shortly.
The form of GT in (2.10) and the radar range equation tacitly assume an antenna pointed directly at the target. If the antenna is not pointed at the target, we must modify GT to account for this. We do this by means of an antenna pattern, which is a function that gives the value of GT at the target, relative to the antenna’s pointing direction.
2.2.1.3 Effective Radiated Power
We temporarily interrupt our derivation to define the quantity termed effective radiated
power. To do so, we ask the question: What power would we need at the output of an isotropic radiator to produce an energy density of SR at all points on a sphere of radius R? An isotropic radiator (ideal point source) is a hypothetical antenna that does not focus energy but instead distributes it uniformly over the surface of a sphere centered on the antenna. Though it cannot exist in the real world, the isotropic radiator serves a mathematical and conceptual function in radar theory, not unlike that of the impulse function in mathematical theory.
By denoting the effective radiated power as Peff and recalling the surface area of a sphere of radius R is 4πR2, we can write the energy density on the surface of the sphere (assuming lossless propagation) as
If we equate (2.10) and (2.11) and solve for Peff, we obtain
as the effective radiated power (ERP).
We emphasize that Peff is not the power at the output of the antenna. The power at the output of the antenna is PT/LtLant. The antenna’s purpose is to focus this power over a relatively small angular sector.
2.2.1.4 Antenna Directivity
We turn next to the factor KA in (2.9). As we indicated, KA accounts for the properties of the antenna. Specifically, it accounts for two facts:
• The energy is not uniformly distributed over the ellipse.
• Not all of the energy is concentrated in the antenna beam (the ellipse of Figure 2.3). Some energy “spills” out the ellipse into what we term the antenna sidelobes.
The value 1.65 is a somewhat common value for KA [11, p. 143]. Using this figure, we can write the antenna directivity as
We term the quantities θA and θB the antenna beamwidths, which have the units of radians. In many applications, θA and θB are specified in degrees. In this case, we can write the directivity as
where the two beamwidths in the denominator are in degrees. The derivation of (2.14) is straightforward and left as an exercise.
While (2.14) uses a numerator of 25,000, various authors provide alternative approximations, accounting for factors such as antenna type, beamshape, sidelobe characteristics, and so on. For example, some authors use 41,253, which would apply to a rectangular beam pattern with no sidelobes and would be indicative of an ideal antenna with maximum directivity [7]. Similarly, some authors use 32,383 for a rectangular aperture with uniform illumination and 33,709 for circular apertures with uniform illumination [10]. As still another variant, some authors prefer 26,000 over 25,000 [10]. It has been the authors’
experience that 25,000 or 26,000 apply well to antennas that use some type of weighting to reduce sidelobes. As a note, the different approximations correspond to different values of KA.
To visualize the concept of beamwidth, consider Figure 2.4, which is a plot of GT(α,ε) versus α for ε = 0. As discussed in Chapter 12, GT(α,ε) is a means of representing antenna directivity as a function of target location relative to antenna pointing angles. If α = 0 and ε = 0, the beam is pointed directly at the target and the directivity is maximum. As illustrated in Chapter 12, α and ε are orthogonal angles roughly related to azimuth and elevation, respectively.
The unit of measurement on the vertical axis of Figure 2.4 is dBi, or decibels relative to isotropic (see Chapter 1), the common unit of measurement for GT in radar applications. We define the antenna beamwidth as the distance between the 3-dB points4 of Figure 2.4. These 3-dB points are the angles where GT(α,ε) is 3 dB below its maximum value. With this, we find the antenna represented in Figure 2.4 has a beamwidth of 2°. We might call this θB, of Figure 2.3 and (2.14). If we were to plot GT (α,ε) versus ε for α = 0, and find a distance between the 3-dB points of 2.5°. We would then say the beamwidth, θA, was 2.5° The antenna directivity would be computed as
Fig ure 2.4 Sample antenna pattern.
In subsequent sections, we drop the notation dBi and use dB.
The humps on either side of the central antenna beam depicted in Figure 2.4 are the antenna sidelobes discussed above.
2.2.1.5 The Target and Radar Cross Section
To return to our derivation, we have an equation for SR, the energy density at the location of the target. As the electromagnetic wave passes the target, the target captures some of its energy and reradiates it toward the radar. More accurately, the electromagnetic wave induces currents on the target, and the currents generate another electromagnetic wave that propagates away from the target. Analysts occasionally designate this as energy reflection, a technically incorrect term. The process of capturing and reradiating energy is very complicated and the subject of much research. For now, we take a simplified approach to the process by using the concept of radar cross section (or RCS). We note that SR has the units of W-s/m2. Therefore, if we were to multiply SR by an area, we would convert it to an energy. This is what we do with RCS, which we denote by σ and ascribe the units of m2, or dBsm if represented in dB units.
Hence, we represent the energy captured and reradiated by the target as
To continue our idealized assumption, we posit the target acts as an isotropic radiator and radiates Etgt uniformly in all directions. The target, in fact, behaves much like an actual antenna and radiates energy with different amplitudes in different directions.
Given the assumption that Etgt is the energy radiated by a target and the target acts as an isotropic antenna, we can represent the energy density at the radar as
or, by substituting (2.10) into (2.16) and the result into (2.17),
2.2.1.6 Antenna Again
As the electromagnetic wave from the target passes the radar, the radar antenna captures a part of this wave and sends it to the radar receiver. If we extend the logic we applied to the target, we can formulate the energy at the output of the antenna feed as
where Ae denotes the effective area of the antenna and is a measure describing the antenna’s ability to capture the returned electromagnetic energy and convert it into usable power. A more common term for Ae is effective aperture of the antenna.
The effective aperture is related to the physical area of the antenna. That is,
where Aant is the area of the antenna projected onto a plane placed directly in front of the antenna and ρant denotes the antenna efficiency. We make this clarification of area because we do not want to confuse it with the actual surface area of the antenna. For example, if the antenna is a parabola of revolution (a paraboloid), a common type of antenna, the actual area of the antenna would be the area of the paraboloidal surface of the antenna, whereas Aant is the area of the disc defined by the front rim of the antenna. In most phased array antennas (flat-face phased array antennas), Aant is the area of the part of the antenna containing the array elements.
While the antenna efficiency can take on any value between 0 and 1, it is seldom below 0.5 or above 0.8 [12]. A rule of thumb for the antenna efficiency value is ρant = 0.6.
Substituting (2.18) into (2.19) yields
2.2.1.7 Antenna Directivity Again
Equation (2.21) is not very easy to use because of the Ae term. We can characterize the antenna more conveniently by using directivity, much as we did on transmit. According to antenna theory, we can relate antenna directivity to effective aperture by the equation [13, p. 61; 8, p. 6]
Substituting (2.22) into (2.21) produces the following:
We next need to propagate the signal through the receiver. We do this by including a gain term, G, which accounts for all of the receiver components up to the point where we measure SNR. With this, we get
2.2.1.8 Losses
As a final step in this part of the development, we need to account for losses we have ignored thus far. There are many losses that we will need to account for (see Chapter 5). For now, we will consolidate all these losses with LtLant and denote them by L. Using this approach, we say the signal energy in the radar is given by
which is Erec, with the additional losses included.
We said ES denotes the signal energy in the radar, although we did not say where in the radar. We will defer this discussion for now and turn our attention to the noise energy term, EN.