Free Space Path Loss and Formula or Equation

This radio path loss and link budget tutorial is split into several pages each of which address different aspects of radio path loss and link budget:

    [1] Understanding signal path loss
    [2] Free space path loss
    [3] Link budget

The free space path loss is used in many areas for predicting radio signal strengths that may be expected in a radio system. Although it does not hold for most terrestrial situations as there are several situations in which it can be used and it is also useful as the basis for understanding many real life radio propagation situations.

Despite this, the free space path loss is an essential basic parameter for many RF calculations. It can often be used as a first approximation for many short range calculations. Alternatively it can be used as a first approximation for a number of areas where there are few obstructions. As such it is a valuable tool for many people dealing with radio communications systems.

In addition to this, these calculations can be used in wireless survey tools. With the growing requirements to be able to analyse wireless or radio coverage, wireless survey tools are being sued increasingly to enable coverage to be predicted at the early stages of design. Accordingly these wireless survey tools are being used increasingly in the development and installation of radio and wireless systems.

Free space path loss basics

The free space path loss, also known as FSPL is the loss in signal strength that occurs when an electromagnetic wave travels over a line of sight path in free space. In these circumstances there are no obstacles that might cause the signal to be reflected refracted, or that might cause additional attenuation.

The free space path loss calculations only look at the loss of the path itself and do not contain any factors relating to the transmitter power, antenna gains or the receiver sensitivity levels. These factors are normally address when calculating a link budget and these will also be used within radio and wireless survey tools and software.

To understand the reasons for the free space path loss, it is possible to imagine a signal spreading out from a transmitter. It will move away from the source spreading out in the form of a sphere. As it does so, the surface area of the sphere increases. As this will follow the law of the conservation of energy, as the surface area of the sphere increases, so the intensity of the signal must decrease.

As a result of this it is found that the signal decreases in a way that is inversely proportional to the square of the distance from the source of the radio signal.

Signal       =	1
	distance2

Free space path loss formula

The free space path loss formula or free space path loss equation is quite simple to use. Not only is the path loss proportional to the square of the distance between the transmitter and receiver, but the signal level is also proportional to the square of the frequency in use for other reasons explained in a section below.

	FSPL = (   4   π   d / λ   ) 2                 = (   4   π   d   f   / c   ) 2

Where:
FSPL is the Free space path loss
d is the distance of the receiver from the transmitter (metres)
λ is the signal wavelength (metres)
f is the signal frequency (Hertz)
c is the speed of light in a vacuum (metres per second)

The speed of light is 2.99792458 x 10⁸ metres per second, although for most practical purposes, this is taken to be 3 x 10⁸ metres per second.

The free space path loss formula is applicable to situations where only the electromagnetic wave is present, i.e. for far field situations. It does not hold true for near field situations.

Free space loss formula frequency dependency

Although the free space loss equation given above seems to indicate that the loss is frequency dependent. The attenuation provided by the distance travelled in space is not dependent upon the frequency. This is constant.

The reason for the frequency dependence is that the equation contains two effects:

1. The first results from the spreading out of the energy as the sphere over which the energy is spread increases in area. This is described by the inverse square law.

2. The second effect results from the antenna aperture change. This affects the way in which any antenna can pick up signals and this term is frequency dependent.

As one constituent of the path loss equation is frequency dependent, this means that there is a frequency dependency within the complete equation.

Decibel version of free space path loss equation

Most RF comparisons and measurements are performed in decibels. This gives an easy and consistent method to compare the signal levels present at various points. Accordingly it is very convenient to express the free space path loss formula, FSPL, in terms of decibels. It is easy to take the basic free space path loss equation and manipulate into a form that can be expressed in a logarithmic format.

FSPL (dB) = 20 log₁₀ (d) + 20 log₁₀ (f) + 32.44

Where:
d is the distance of the receiver from the transmitter (km)
f is the signal frequency (MHz)

Affect of antenna gain on path loss equation

The equation above does not include any component for antenna gains. It is assumed that the antenna gain is unity for both the transmitter. In reality, though, all antennas will have a certain amount of gain and this will affect the overall affect. Any antenna gain will reduce the "loss" when compared to a unity gain system. The figures for antenna gain are relative to an isotropic source, i.e. an antenna that radiates equally in all directions.

FSPL (dB) = 20 log₁₀ (d) + 20 log₁₀ (f) + 32.44 -Gtx - Grx

Where:
Gtx is the gain of the transmitter antenna relative to an isotropic source (dBi)
Grx is the gain of the receiver antenna relative to an isotropic source (dBi)

The free space path loss equation or formula given above, is an essential tool that is required when making calculations for radio and wireless systems either manually or within applications such as wireless survey tools, etc. By using the free space path loss equation, it is possible to determine the signal strengths that may be expected in many scenarios. While the free space path loss formula is not fully applicable where there are other interactions, e.g. reflection, refraction, etc as are present in most real life applications, the equation can nevertheless be used to give an indication of what may be expected. It is obviously fully applicable to satellite systems where the paths conform closely to the totally free space scenarios. .......

Link Budget

This radio path loss and link budget tutorial is split into several pages each of which address different aspects of radio path loss and link budget:

    [1] Understanding signal path loss

    [2] Free space path loss

    [3] Link budget

When designing a complete, i.e. end to end radio communications system, it is necessary to calculate what is termed the link budget. The link budget enables factors such as the required antennas gain levels, radio transmitter power levels, and receiver sensitivity figures to be determined. By assessing the link budget, it is possible to design the system so that it meets its requirements and performs correctly without being over designed at extra cost.

Link budgets are often used for satellite systems. In these situations it is crucial that the required signal levels are maintained to ensure that the received signal levels are sufficiently high above the noise level to ensure that signal to noise levels or bit error rates are within the required limits. However larger antennas, high transmitter

power levels that required add considerably to the cost, so it is necessary to balance these to minimise the cost of the system while still maintaining performance.

In addition to satellite systems, link budgets are also used in many other radio communications systems. For example, link budget calculations are used for calculating the power levels required for cellular communications systems, and for investigating the base station coverage.

Link budget style calculations are also used within wireless survey tools. These wireless survey tools will not only look at the way radio signals propagate, but also the power levels, antennas and receiver sensitivity levels required to provide the required link quality.

What is link budget?

As the name implies, a link budget is an accounting of all the gains and losses in a transmission system. The link budget looks at the elements that will determine the signal strength arriving at the receiver. The link budget may include the following items:

•             Transmitter power.

•             Antenna gains (receiver and transmitter).

•             Antenna feeder losses (receiver and transmitter).

•             Path losses.

•             Receiver sensitivity (although this is not part of the actual link budget, it is necessary to know this to enable any pass fail criteria to be applied.

Where the losses may vary with time, e.g. fading, and allowance must be made within the link budget for this - often the worst case may be taken, or alternatively an acceptance of periods of increased bit error rate (for digital signals) or degraded signal to noise ratio for analogue systems.

In essence the link budget will take the form of the equation below:

Received power (dBm)   =   Transmitted power (dBm)   +   gains (db)   -   losses (dB)

The basic calculation to determine the link budget is quite straightforward. It is mainly a matter of accounting for all the different losses and gains between the transmitter and the receiver.

Link budget equation

In order to devise a link budget equation, it is necessary to investigate all the areas where gains and losses may occur between the transmitter and the receiver. Although guidelines and suggestions can be made regarding the possible areas for losses and gains, each link has to be analysed on its own merits..

A typical link budget equation for a radio communications system may look like the following:

PRX   =   PTX   +   PTX   +   GTX   +   GRX   -   LTX   -   LFS   -   LP   -   LRX

Where:

     PRX  = received power (dBm)

     PTX  = transmitter output power (dBm)

     GTX  = transmitter antenna gain (dBi)

     GRX  = receiver antenna gain (dBi)

     LTX  = transmit feeder and associated losses (feeder, connectors, etc.) (dB)

     LFS  = free space loss or path loss (dB)

    LP  = miscellaneous signal propagation losses (these include fading margin, polarization mismatch, losses  associated with medium through which signal is travelling, other losses...) (dB)

     LRX  = receiver feeder and associated losses (feeder, connectors, etc.) (d)B

    NB for the sake of showing losses in the link budget equation is "minus" actual loss figures, e.g. LTX or LFS, etc should be taken as the modulus of the loss.

Antenna gain and link budget

The basic link budget equation where no levels of antenna gain are included assumes that the power spreads out equally in all directions from the source. In other words the antenna is an isotropic source, radiating equally in all directions.

This assumption is good for theoretical calculations, but in reality all antennas radiate more in some directions than others. In addition to this it is often necessary to use antennas with gain to enable interference from other directions to be reduced at the receiver, and at the transmitter to focus the available transmitter power in the required direction.

In view of this it is necessary to accommodate these gains into the link budget equation as they have been in the equation above because they will affect the signal levels - increasing them by levels of the antenna gain, assuming the gain is in the direction of the required link.When quoting gain levels for antennas it is necessary to ensure they are gains when compared to an isotropic source, i.e. the basic type of antenna assumed in the equation when no gain levels are incorporated. The gain figures relative to an isotropic source are quoted as dBi, i.e. dB relative to an isotropic source. Often gain levels given for an antenna may be the gain relative to a dipole where the figures may be quoted as dBd, i.e. dB relative to a dipole. However a dipole has gain relative to an isotropic source, so the dipole gain of 2.1 dBi needs to be accommodated if figures relative to a dipole are quoted for an antenna gain..

Effect of multipath propagation

For true free space propagation such as that encountered for satellites there will be no noticeable reflections and there will only be one major path. However for terrestrial systems, the signal may reach the receiver via a number of different paths as a result of reflections, etc that will occur as a result of the objects around the path. Buildings, trees, objects around the office and home can all cause reflections that will result in the signal variations.

The multipath propagation will cause variations of the signal strength when compared to that calculated from the free space path loss. If the signals arrive in phase with the direct signal, then the reflected signals will tend to reinforce the direct signal. If they are out of phase, then they will tend to cancel the signal. If either the transmitter or receiver moves, then the signal strength will be seen to vary as the relative strengths and phases of the different signals change.

In order to allow for this in a link budget, a link margin is added into the equation to allow for this.

Link budget calculations are an essential step in the deign of a radio communications system. The link budget calculation enables the losses and gains to be seen, and devising a link budget enables the apportionment of losses, gains and power levels to be made if changes need to be made to enable the radio communications system to meet its operational requirements. Only by performing a link budget analysis is this possible.

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