The Radio Link--A tutorial--Part III

The Range Equation

Earlier we stated that under ideal conditions, and assuming an isotropic transmitting antenna, the power received at the output of the receiving antenna is given by

This relationship assumes that all of the power supplied to a transmitting antenna’s input terminals is radiated and all of the power incident on a receiving antenna’s effective aperture is captured and available at its output terminals. If we now utilize a directional transmitting antenna with gain G_t, then the power received in the direction of the transmitting antenna main lobe is given as

The term effective isotropic radiated power  (EIRP) is often used instead of P_tG_t. EIRP is the power that would have to be supplied to an isotropic transmitting antenna to provide the same received power as that from a directive antenna. Note that a receiver cannot tell whether a transmitting antenna is directive or isotropic. The only distinction is that when a directive antenna is used, the transmitter radiates very little power in directions other than toward the receiver, whereas if an isotropic antenna is used, the transmitter radiates equally in all directions.

Using the definition of EIRP, we can write Equation (2.10) as

Since the gain and effective aperture of an antenna are related by Equation (2.6), and since the gain and effective aperture are properties of the antenna that do not depend on whether the antenna is used for transmitting or receiving, it is not necessary to specify both parameters. Antenna manufacturers frequently list the antenna gain on specification sheets. If we replace Aer in Equation (2.11) using Equation (2.6) we obtain

In applying Equation (2.6) we have set the loss parameter η =  1 for simplicity. We will include a term later on to account for antenna losses as well as losses of other kinds. It is instructive to write Equation (2.12) in the following way:

This form suggests that the received power can be obtained by starting with the transmitted power and multiplying by three “gain” terms. Since the reciprocal of gain is “loss,” we can interpret the term (4π)²  (d/λ)²  as a loss term. Let us define the free-space path loss L_path  as

It should be observed that the free-space path loss L_path  varies directly with the square of the distance and inversely with the square of the wavelength. We can interpret the ratio d/λ  in Equation (2.14) to mean that path loss is a function of the distance between transmitting and receiving antennas measured in wavelengths. It is important to note that the path loss is not an ohmic or resistive loss that converts electrical power into heat. Rather, it represents the reduction in power density due to a fixed amount of transmitted power spreading over an increasingly greater surface area as the waves propagate farther from the transmitting antenna; that is, just as G  represents focusing and not actual “gain,” L_path  represents spreading and not actual “loss.” Equation (2.12), the so-called range equation , can be written in any of the alternate forms

The first of these forms is known commonly as the Friis transmission equation in honor of its developer, Harald T. Friis³  (pronounced “freese”). Subsequently we will see that there is a minimum received power that a given receiver requires to provide the user with an adequate quality of service. We call that minimum power the receiver sensitivity p_sens. Given the receiver sensitivity, we can use Equation (2.15) to calculate the maximum range of the data link for free-space propagation, that is,

In our analysis so far we have ignored any sources of real loss, that is, system elements that turn electrical power into heat. In a realistic communication system there are a number of such elements. The transmission lines that connect the transmitter to the antenna and the antenna to the receiver are slightly lossy. Connectors that connect segments of transmission line and connect transmission lines to antennas and other components provide slight impedance mismatches that lead to small losses. Although the loss caused by one connector is usually not significant, there may be a number of connectors in the transmission system. We have mentioned above that antennas may produce small amounts of ohmic loss. There may also be losses in propagation, where signals pass through the walls of buildings or into or out of automobiles. In mobile radio systems there can be signal absorption by the person holding the mobile unit. And so on. Fortunately it is easy to amend the range equation to include these losses. If L₁ ,L₂ , … , L_n  represent  real-world losses, then we can modify Equation (2.15) to read

where in the last form we have lumped all of the real-world losses together into a single term L_sys, which we call “system losses.” In typical applications, signal levels range over many orders of magnitude. In such situations it is often very convenient to represent signal levels in terms of decibels. Recall that a decibel is defined in terms of a power ratio as

where P₁ and P₂  represent power levels, and the logarithm is computed to the base 10. For example, if the signal power is 1 W at 1 mile, and 10 mW at 10 miles, then the ratio in decibels is

Absolute power levels (as opposed to ratios) can also be expressed in decibels, but a reference power is needed for P₁ in Equation (2.18). Reference values of 1 W and 1 mW are common. When a 1 W reference is used, the units are written “dBW” to reflect that fact; when a 1 mW reference is used, the units are written as “dBm.” Expressing quantities in decibels is particularly useful in calculating received power and in any analysis involving the range equation. This is because the logarithm in the decibel definition converts multiplication into addition, and as a result the various gain and loss terms can be added and subtracted. Converting Equation (2.17) to decibels gives

Notice particularly the way in which the 1 mW reference power is handled in Equation (2.20). The reference power is needed as a denominator to convert the transmitted power P_t  into decibels. The gain and loss terms are all ratios of powers or ratios of power densities and can be converted to decibels directly, without need for additional reference powers. To make the dependence of received signal power on range explicit, let us use Equation (2.14) to substitute for L_path  in Equation (2.20). We have

The exponent of distance d in Equation (2.21) is called the path-loss exponent. We see that the path-loss exponent has the value of 2 for free-space propagation. The path-loss exponent will play an important role in the models for propagation in real-world environments that will be discussed in detail in Chapter 3. Converting Equation (2.21) to decibels gives

We see that in free space, the path loss increases by 20 dB for every decade change in range. Substituting Equation (2.22) into Equation (2.20) gives

where all of the constant terms have been lumped together as K. Note that Equation (2.23) is a straight line when plotted on semilog scale.

³. H. T. Friis, “A Note on a Simple Transmission Formula,” Proceedings of the Institute of Radio Engineers 34 (1946): 254–56.

Next:  Thermal Noise and Receiver Analysis

Introduction to Wireless Systems By Bruce A. Black, Philip S. DiPiazza, Bruce A. Ferguson, David R. Voltmer, Frederick C. Berry, Published Jun 7, 2011 by Prentice Hall, is reprinted with permission by Pearson Publishing.