Electromagnetic Waves, Antennas, and RF: Friis Transmission Equation, Free-Space Path Loss, and RF Link Budgets

Electromagnetic Waves, Antennas, and RF:
Friis Transmission Equation, Free-Space Path Loss, and RF Link Budgets

EM24 established the receive relation

(1)

and the reciprocity result

(2)

EM25 now combines those results with spherical electromagnetic power spreading to derive the Friis transmission equation, the free-space path-loss equation, and the logarithmic bookkeeping used in RF link budgets [1, 2, 3, 4].

The central derivation chain is

(3)

The result will be

(4)

under the assumptions developed below.

This equation is often memorized. The purpose of this article is instead to make every factor physically visible.

1 Reference planes and assumptions

A link equation is only unambiguous when the power reference planes are stated. In the basic Friis derivation used here,

• P_t is power accepted by the transmitting antenna;
• G_t is the transmitting antenna gain in the receiver direction;
• G_r is the receiving antenna gain in the transmitter direction;
• the antennas are in one another’s far fields;
• the propagation region is homogeneous free space;
• the polarization states are matched;
• the receiving antenna is conjugately matched when A_e is interpreted as maximum available effective aperture;
• multipath, atmospheric absorption, feeder loss, pointing loss, and other nonideal effects are initially omitted.

These assumptions will later be relaxed by adding explicit efficiency or loss factors.

2 Step 1: directional transmitted power

For an isotropic radiator with total power P_t, the power density at radius r is

(5)

A real antenna redistributes the radiated power angularly. If its gain toward the receiver is G_t, then the far-field power density in that direction is

(6)

The numerator

(7)

is the equivalent isotropically radiated power in that direction.

Thus

(8)

The 1∕r² dependence is geometric spreading. In ideal free space, power is not being converted to heat by this term. The same total outward power crosses larger and larger spherical surfaces.

3 Step 2: receiving effective aperture

EM24 defined effective aperture by

(9)

For a reciprocal receiving antenna with gain G_r,

(10)

Substituting the transmitted power density gives

Pr	=	(11)
	= PtGtGr.	(12)

Therefore

(13)

This is the Friis transmission equation for the ideal free-space case [1].

4 What each Friis factor means

The equation can be read as a sequence of physical transformations:

(14)

The factor

(15)

is therefore not an arbitrary communications-engineering correction. It appears because a transmitted wave spreads over an area proportional to r², while an antenna of fixed receive gain has effective aperture proportional to λ².

5 Free-space path loss

For isotropic antennas,

(16)

Then Friis becomes

(17)

Define the dimensionless free-space path loss as the reciprocal power ratio,

(18)

Since

(19)

we may also write

(20)

This is the free-space path-loss equation.

6 Free-space path loss is not dissipative attenuation

The word loss can be misleading. EM20 introduced material attenuation through an amplitude factor such as

(21)

where electromagnetic energy is transferred to the medium through dissipative mechanisms such as J ⋅ E.

Free-space path loss is different. In ideal vacuum,

(22)

The power crossing a complete sphere remains constant, but the power density falls as

(23)

Thus free-space path loss describes the coupling between a finite receiving aperture and a geometrically expanding wavefront. It is not absorption by empty space.

7 The frequency dependence of path loss

At fixed antenna gains,

(24)

Therefore doubling frequency while keeping r, G_t, and G_r fixed increases free-space path loss by a factor of four.

It is important to understand why. For fixed gain,

(25)

so higher frequency means smaller effective aperture.

The geometric spreading term

(26)

contains no frequency. A wave of higher frequency does not geometrically “spread faster” in free space. The f² dependence enters the gain-normalized Friis relation through the receive-aperture factor.

For an aperture antenna of fixed physical area A_phys and fixed aperture efficiency,

(27)

so the gain increases as 1∕λ². The frequency dependence of a complete link therefore depends on what physical quantities are being held fixed.

8 Decibel form of free-space path loss

Taking 10 log ₁₀ of the power ratio gives

LFS,dB	= 10 log 10	(28)
	= 20 log 10.	(29)

Therefore

(30)

The factor of 20 appears because the underlying dimensionless ratio is squared before the logarithm is taken.

Useful unit-specific forms follow from inserting the appropriate scale factors. With distance in kilometers and frequency in megahertz,

(31)

With distance in kilometers and frequency in gigahertz,

(32)

These constants are not new physics; they only encode the unit conversions and the factor 4π∕c.

9 Scaling rules worth remembering

From the logarithmic form,

(33)

we obtain several immediate rules:

• doubling distance adds

  (34)

• doubling frequency at fixed gains also adds approximately 6.02 dB;
• multiplying distance by ten adds 20 dB;
• multiplying frequency by ten at fixed gains adds 20 dB.

10 Example: an isotropic 2.4 GHz link

Let

(35)

with

(36)

The free-space path loss is

LFS,dB	= 20 log 10	(37)
	≈ 80.05 dB.	(38)

Since

(39)

we obtain

(40)

In linear units,

(41)

11 Friis in decibel form

For the ideal link,

(42)

Taking logarithms gives

(43)

This additive form is the foundation of an RF link budget.

The units should be tracked carefully:

• dBW and dBm describe absolute power levels;
• dBi describes antenna gain relative to an isotropic radiator;
• dB describes dimensionless ratios such as path loss, cable loss, or mismatch loss.

Also,

(44)

12 Adding real link losses

Let L_i ≥ 1 denote multiplicative power-loss factors. A generalized linear link may be written

(45)

In decibels,

(46)

Typical additional losses include

• transmit feeder or cable loss;
• receive feeder loss;
• polarization mismatch;
• antenna pointing loss;
• atmospheric or rain attenuation;
• radome loss;
• implementation or installation loss.

A clean link budget keeps each physical mechanism separate until there is a good reason to combine them.

13 Polarization mismatch

If the incident and receiving polarization unit vectors are e_t and e_r, define the polarization loss factor

(47)

Then

(48)

The equivalent polarization loss in decibels is

(49)

For two linear polarizations separated by angle ψ,

(50)

At 45^∘,

(51)

so the polarization loss is approximately

(52)

14 Impedance mismatch and realized gain

If the antenna terminal reflection coefficient is Γ_A, the fraction of incident terminal power accepted by the antenna is

(53)

The corresponding mismatch loss is

(54)

If realized gain is used instead of ordinary gain, terminal mismatch is already incorporated. A link budget must not subtract the same mismatch twice.

This is another reason reference-plane definitions are essential.

15 Example: a complete 5.8 GHz link budget

Consider an illustrative line-of-sight link with

Pt	= 10.0 dBW,	(55)
Ltx feed	= 1.0 dB,	(56)
Gt	= 12.0 dBi,	(57)
f	= 5.8 GHz,	(58)
r	= 5.0 km,	(59)
Gr	= 8.0 dBi,	(60)
Lpol	= 0.5 dB,	(61)
Lother prop	= 0.5 dB,	(62)
Lrx feed	= 1.5 dB.	(63)

The free-space path loss is

(64)

The EIRP after transmitter feed loss is

EIRPdBW	= 10.0 − 1.0 + 12.0	(65)
	= 21.0 dBW.	(66)

The final receive power is

Pr,dBW	= 21.0 − 121.70 + 8.0 − 0.5 − 0.5 − 1.5	(67)
	≈−95.20 dBW.	(68)

Therefore

(69)

The important point is not this specific number. It is the auditable chain of reference planes and physical mechanisms that produced it.

16 Link margin

If a receiver requires at least P_req at the chosen reference plane, define the link margin as

(70)

A positive margin means the predicted receive power exceeds the stated requirement by that amount.

At this stage P_req is simply an externally specified threshold. EM26 will replace this black-box threshold with receiver-noise physics: thermal noise, N₀ = kT, noise temperature, noise figure, G∕T, and ultimately C∕N₀.

17 Illustrative GNSS-like free-space link

To prepare for that transition, consider an illustrative L-band link with

(71)

(72)

and directional transmit EIRP

(73)

For an ideal 0 dBi receiving antenna, the free-space path loss is

(74)

If all additional receive and propagation losses total 2 dB, then

Pr	= 27 − 182.50 − 2	(75)
	≈−157.50 dBW.	(76)

Thus

(77)

This is an intentionally simplified GNSS-like example, not a specification for a particular satellite or receiver. Its purpose is to show why satellite-navigation signals arrive at very low power levels and why the next step must be a noise-density calculation rather than power alone.

18 A second derivation using field strength

EM24 gave the free-space RMS electric field associated with EIRP,

(78)

The corresponding incident power density is

(79)

Using η₀ ≈ 120π Ω,

S	=	(80)
	= .	(81)

Multiplying by

(82)

again gives Friis. This alternative route verifies that the link equation is consistent with the electromagnetic field-strength formulation.

19 Conditions for using Friis

Friis is powerful because it compresses a field problem into a scalar power relation, but it is not universal. The standard form should be used only when its approximations are appropriate.

Important checks include

• Far field: each antenna should lie in the radiation far field of the other for the gain description to be valid in its usual form.
• Free-space-like propagation: strong reflections, diffraction, waveguiding, scattering, or multipath require a more complete channel model.
• Known polarization: polarization mismatch must either be negligible or included explicitly.
• Defined gain convention: distinguish gain, realized gain, and directivity.
• Consistent power reference planes: avoid mixing transmitter output power, antenna accepted power, and EIRP without accounting for feeder and mismatch losses.
• Narrowband interpretation: for broadband signals, frequency-dependent gains and channel response may require integration over frequency rather than a single-number link equation.

20 Why the Friis factor can appear greater than one outside its domain

The isotropic free-space factor is

(83)

If one formally chooses a sufficiently small r, this factor can exceed one. That does not imply passive antennas create power. It means the far-field aperture model has been extrapolated outside the regime where it applies.

Near-field energy exchange must be analyzed with the actual fields and mutual coupling, not with the far-field Friis equation.

21 From Friis to noise and interference

EM25 has now completed the deterministic power-delivery chain

(84)

But received carrier power alone does not determine receiver performance. The next questions are

(85)

and later

(86)

The next stages of the series will therefore introduce

(87)

(88)

and eventually jammer-to-signal quantities such as J∕S.

22 Summary

A transmitting antenna with accepted power P_t and directional gain G_t produces far-field power density

(89)

A receiving antenna with gain G_r has effective aperture

(90)

Multiplying the two yields Friis:

(91)

The corresponding free-space path loss is

(92)

or

(93)

A practical logarithmic link budget is then

(94)

The key physical distinction is that free-space path loss is geometric coupling loss, not dissipative attenuation of electromagnetic energy in vacuum.

References

References