11.3.2 Received Power And The Effective Aperture

Equation (11.8) shows the density of power arriving at the antenna—that is, the watts per square meter. The power received (in watts) is calculated by multiplying the power density of the received signal by the effective receiving area of the antenna (called the effective aperture). The power collected by the antenna is therefore:

P_{a n t} = P_{d e n} A_{e} (W)

(11.9)

where A_e is the effective aperture. Gor an antenna with gain G_r at wavelength λ:

A_{e} = \frac{λ^{2}}{4 π} G_{r}

(11.10)

Substituting Equations (11.8) and (11.10):

P_{a n t} = \underset{P_{d e n}}{\underset{⏟}{P_{a m p} \frac{L_{t} G_{t}}{4 π d^{2}} L_{a}}} \underset{A_{e}}{\underset{⏟}{\frac{λ^{2} G_{r}}{4 π}}} (W)

(11.11)

This equation is often rewritten to be:

P_{a n t} = P_{a m p} {(\frac{λ}{4 π d})}^{2} L_{t} G_{t} L_{a} G_{r} (W)

(11.12)

since the factor (λ/4πd)² is known as the free-space loss (FSL) factor, which we discuss in more detail shortly.

This equation now represents the power received by the receive antenna. We still need to deliver this power to the receiver, but the signal once again must suffer attenuation due to the coupling and mismatch losses in the cable from the receive antenna to the receiver. The total collected power at the receiver's terminals is therefore reduced by the coupling losses in the receiving antenna system (L_r) so that:

P_{r} = P_{a m p} (F S L) L_{t} G_{t} L_{a} G_{r} L_{r} (W)

(11.13)

In its most simple form (when both antennas are assumed to be lossless and no atmospheric losses are present), the equation can be simplified to the form:

P_{r} = P_{a m p} {(\frac{λ}{4 π d})}^{2} G_{t} G_{r} (W)

(11.14)

which is known as the Friis formula. This formula is useful because it provides a quick assessment of the power received over a particular path. Of course, the actual power received is less than this value but, since free-space loss is the largest loss, the Friis formula produces reasonable results for quick assessments. Certainly, if the Friis formula predicts insufficient receive power then precise analysis is not necessary.

So we now have an understanding of what proportion of the power sent from the terminals of the transmitter amplifier (P_amp) arrives at the receiver front end (P_r) after travelling over the transmission path. It is important to note that, for a given transmitting and receiving antenna and a fixed propagation path, the amount of received power varies only with the wavelength. This bears a little more consideration. The received power increases proportional to the square of the wavelength—so for a fixed system (with fixed antenna gains), better results can be obtained if the wavelength is increased (or the frequency is lowered), or, in other words, longer ranges are available from lower frequencies. However, life isn’t that simple and we need to consider a few other factors, which we discuss in the rest of this section and the next.

What we need to know is what proportion of the transmitted power will reach the receiver. This quantity is called the transmission path loss (TPL), which is the ratio of the power received at the receiver’s input to that generated by the transmitter’s amplifier such that:

T P L = \frac{P_{r}}{P_{a m p}} = (F S L) L_{t} G_{t} L_{a} G_{r} L_{r}

(11.15)

This equation can be rewritten, because the combination of free-space loss (FSL) and path propagation losses (L_a) is called the radio path loss (RPL) since they represent the losses in the path (as opposed to the losses in the equipment at either end). That is:

T P L = (R P L) L_{t} G_{t} G_{r} L_{r}

(11.16)

The maximum transmission path loss (TPL_max) is a design measure nominated for a given system and represents the maximum loss that can be accommodated between the transmitter amplifier and the receiver in a communications system before the path becomes unworkable. The antenna gains and losses are normally constant for a given system configuration, and they can be grouped together as the system value, S_v, so that:

S_{v} = L_{t} G_{t} G_{r} L_{r}

(11.17)

and therefore:

{T P L}_{m a x} = ({R P L}_{m a x}) S_{v}

(11.18)

When deploying communications systems, the maximum RPL is called the radio path capability (RPC) such that:

R P C = {R P L}_{m a x}

(11.19)

A radio system can continue to work while the losses on the path (the RPL) are lower than the system RPC (the maximum RPL acceptable). Whether the radio path is viable or not can be ascertained, therefore, by calculating the RPL and determining whether or not it is lower than the system RPC. The RPL is calculated by determining the extent of the losses in the path, which can arise from a number of sources. Each of these is dealt with in more detail in the following sections but, briefly, they are free-space loss (FSL), reflection loss (RL), diffraction loss (DL), clutter loss (CL), and atmospheric loss (AL). In decibel format:

R P L = F S L + R L + D L + C L + A L (d B)

(11.20)

where each of the loss factors is a negative decibel value.

The above calculations have been conducted by examining the transmitted and received powers. A similar process can be conducted using electric field or magnetic field. The time-average power density (strictly speaking, the time-average magnitude of the Poynting vector) at a particular point in space is related to the electric and magnetic field by the following relationships:

P_{d e n} = | E_{r m s} | . | H_{r m s} | = 120 π {| H_{r m s} |}^{2} = \frac{{| E_{r m s} |}^{2}}{120 π} (W m^{-2})

(11.21)

Equation (11.21) applies to fields in free space. More generically, the constant of proportionality is the intrinsic impedance for the medium through which the fields are passing (in free space, $η_{0} \approx 120 π$ ):

P_{d e n} = | E_{r m s} | . | H_{r m s} | = η {| H_{r m s} |}^{2} = \frac{{| E_{r m s} |}^{2}}{η} (W m^{-2})

(11.22)

We have already encountered this relationship when we considered the propagation of the fields along various transmission media in Section 10.1.1 and we showed the approximate velocities.