Link budget — path loss, EIRP, G - T, Eb - N0: Foundations

Before you can read a single line of the parent note, you need to know what each little symbol means as a picture. This page builds every one of them from nothing. Read top to bottom — each idea is a brick for the next.

1. Energy, power, and the unit "watt" ( $P$ , W)

Picture a garden hose. The water pressure at the nozzle is like transmitter power: how much "push" you start with. A radio transmitter's power $P_{t}$ (the subscript $t$ means transmit) is typically a few watts to tens of watts on a spacecraft — tiny, because electrical power in space is scarce.

Why the topic needs it: everything in a link budget is ultimately a comparison of powers — the power you send versus the power that arrives versus the power of the noise.

2. Spreading over a sphere — the number $4 π r^{2}$

Here is the single most important picture in the whole topic.

Figure — Link budget — path loss, EIRP, G - T, Eb - N0

Let us earn every symbol in $4 π r^{2}$ :

$r$ is the radius — the distance from the source, in metres. Look at the red arrow in the figure.
$π$ (pi) $\approx 3.14159$ is a fixed number defined as a circle's circumference divided by its diameter ( $π = circumference / diameter$ ); it is the same for every circle.
$4 π r^{2}$ is the surface area of a sphere. It grows with the square of distance: double $r$ , and the area becomes four times as big.

Why the topic needs it: this sphere is why signals get weak with distance, and it is the origin of both path loss and the definition of EIRP (both defined below).

3. Intensity / power density — $I$ or $S$ (W/m²)

Take the total power and divide by the bubble's area: $S = \frac{P}{4 π r ^{2}}$

Look at the figure: the same four rays of "power" pass through a small square up close, but at double the distance they are spread across a square four times larger — each patch gets one-quarter as much.

Why the topic needs it: the receiving antenna is a bucket that catches whatever power density is falling on it. $S$ is the rainfall; the antenna catches a puddle's worth.

4. Antenna gain — $G$ (a pure ratio, then dBi)

Real antennas do not radiate equally in all directions. A dish focuses the beam, exactly like a flashlight reflector focuses a bulb.

$G = \frac{power density in the main beam}{power density an isotropic antenna would give}$

$G_{t}$ = gain of the transmit antenna.
$G_{r}$ = gain of the receive antenna.
Gain of 1 means "no focusing at all" (isotropic). Gain of 300 means the beam is 300× more intense straight ahead — but weaker off to the sides. No antenna creates power; it only redirects it.

Intuition Why bigger dishes have more gain (the "why" on this page)

Gain comes from squeezing the same power into a narrower cone. Total power is fixed, so if a dish sends it out through a tiny solid angle $Ω$ (a small patch of directions) instead of the whole sphere ( $4 π$ worth of directions), the density straight ahead is boosted by roughly $G \approx 4 π /Ω$ . A bigger dish makes a narrower beam (smaller $Ω$ ), hence higher gain. Because a narrow beam also acts as a wide "collecting bucket" on receive, gain is directly tied to the antenna's effective aperture $A_{e}$ (its effective catch-area in m²) by $A_{e} = G λ^{2} / (4 π)$ — the $λ$ enters because a shorter wave "fits" a smaller catch-area for the same gain. We use this exact relation in §8. The full geometric proof is in Antenna Gain and Effective Aperture.

Why the topic needs it: gain is the "free help" in the budget — it concentrates your precious watts toward Earth instead of wasting them into empty space.

5. EIRP — combining power and gain into one number

Now that we have power $P_{t}$ and gain $G_{t}$ , we can build the first compound quantity of the topic.

Multiply the two ingredients: $EIRP = P_{t} \cdot G_{t} (watts)$

Why the topic needs it: EIRP is the single "how loud do we shout?" number at the transmitting end of the budget.

6. Decibels — turning multiplication into addition ( $10 lo g_{10}$ , dB)

The numbers in a link budget span from tiny (a received power of $0.000000000000001$ W) to enormous (a path loss factor of $1 0^{27}$ ). Multiplying such numbers by hand is misery. Engineers use a trick.

Here $lo g_{10}$ (base-10 logarithm) answers the question "10 to what power gives this number?" For example $lo g_{10} (1000) = 3$ , because $1 0^{3} = 1000$ .

Reference-flavoured decibels (the reference $P_{ref}$ is baked into the name):

Recall Quick dB self-test

What is 100 W in dBW? ::: $10 lo g_{10} (100) = 20$ dBW. Why do we add gains in dB instead of multiplying? ::: Because $lo g$ turns multiplication into addition.

7. Wavelength and frequency — $λ$ , $f$ , $c$

Radio signals are waves. Two numbers describe a wave.

They are locked together by: $c = f λ ⟺ λ = \frac{c}{f}$

This is the symbol promised back in §3: the far-field condition " $r$ much bigger than a wave" is now simply $r ≫ λ$ .

Why the topic needs it: path loss secretly depends on $λ$ — a shorter wavelength means a tiny receiving antenna "sees" a smaller effective area, so higher frequencies lose more per metre unless you also focus them with gain.

8. Path loss — $L_{p}$ , the range penalty

We are now ready to combine the sphere ( $4 π r^{2}$ ) with the wavelength ( $λ$ ) into the topic's biggest loss.

Where the $4 π r / λ$ factor comes from (the WHY, worked on this page):

Transmit side: a source of EIRP produces power density $S = \frac{EIRP}{4 π r ^{2}}$ at distance $r$ — that is the first factor of $4 π r^{2}$ (the spreading sphere from §2–§3).
Receive side: an antenna does not catch all of $S$ ; it catches only the power falling on its effective aperture $A_{e}$ (its catch-area in m², introduced in §4), so received power $P_{r} = S \cdot A_{e}$ . From §4 we already have $A_{e} = \frac{G _{r} λ ^{2}}{4 π}$ — that is the second factor of $4 π$ and where $λ^{2}$ enters.
Put both together: $P_{r} = \frac{EIRP}{4 π r ^{2}} \cdot \frac{G _{r} λ ^{2}}{4 π} = EIRP \cdot G_{r} \cdot \frac{λ ^{2}}{( 4 π r ) ^{2}}$ . Strip out the antenna gains $G_{t}, G_{r}$ ; what is left over — the pure distance-and-wavelength penalty — is $L_{p} = (\frac{4 π r}{λ})^{2} .$

Why the topic needs it: $L_{p}$ is the single "how much the journey costs" number — the main thing EIRP and antenna gains must overcome.

9. Noise and temperature — $T_{sys}$ (Kelvin), $k$

Even with the transmitter switched off, a receiver hears a faint random hiss. This is noise, produced by the jiggling of warm electrons everywhere — in the sky, the ground, the amplifier.

$N_{0}$ (read "N-nought") is the noise power spectral density — noise power per hertz, in W/Hz. In decibels $k$ becomes $- 228.6$ dBW/K/Hz (a constant you will see added in every $C / N_{0}$ line, defined in §10).

See Noise Temperature and Noise Figure for the full story of where $T_{sys}$ comes from.

Why the topic needs it: the signal only "survives" if it arrives louder than this hiss. Noise is the enemy the whole budget fights.

10. Received carrier power and $C / N_{0}$

Why the topic needs it: $C / N_{0}$ is where the transmit side (EIRP), the journey ( $L_{p}$ ) and the receive side (gain and noise) finally meet in one number.

11. Putting receiver quality in one number — $G / T$

Why the topic needs it: $G / T$ is the single "figure of merit" that lets you compare two ground stations at a glance, without re-listing gain and temperature separately. It slots straight into the $C / N_{0}$ equation: $\frac{C}{N _{0}} = EIRP - L_{p} + \frac{G}{T} - 10 lo g_{10} (k)$ .

The unit of energy; roughly the energy to lift a small apple one metre.

What does a watt measure, and how does it relate to the joule?

Power — energy per second;

1 W = 1 J/s

Define

π

precisely.

A circle's circumference divided by its diameter, the same for every circle (

\approx 3.14159

Why does

4 π r^{2}

appear everywhere in this topic?

It is the surface area of the expanding spherical wavefront the power spreads over.

State the inverse-square law in one sentence.

Power density falls as

1/ r^{2}

: double the distance, one-quarter the strength.

What does "isotropic" mean?

Radiating equally in every direction.

What is antenna gain, in plain words?

How many times more power density an antenna makes in its best direction versus an isotropic radiator using the same total power.

Why does a bigger dish have more gain?

It squeezes the same power into a narrower beam (smaller solid angle), so

G \approx 4 π /Ω

Define EIRP and give its formula.

The isotropic power that would match your focused antenna's forward power density;

EIRP = P_{t} G_{t}

Convert 10 W to dBW.

10 lo g_{10} (10) = 10

dBW.

Why do engineers use decibels?

Logarithms turn multiplying huge/tiny numbers into simple addition.

What is the reference for dB/K and for dB-Hz?

1 kelvin of temperature, and 1 hertz of bandwidth, respectively.

Where does the

4 π r / λ

in path loss come from?

One

4 π r

from the transmit sphere and a

λ

from the receive antenna's aperture; squared because power depends on it at both ends.

Why does path loss use

20 lo g_{10}

not

10 lo g_{10}

Because the ratio is squared, and

lo g (x^{2}) = 2 lo g (x)

What does the far-field assumption

r ≫ λ

let us do?

Use the clean

1/ r^{2}

inverse-square law and the free-space path-loss formula.

Relate

λ

f

, and

c

c = f λ

, so

λ = c / f

What is carrier power

C

The signal power arriving at the receiver:

C = EIRP - L_{p} + G_{r}

in dB.

What does

C / N_{0}

compare, and in what units?

Arriving carrier power against noise density per hertz; units of hertz, quoted in dB-Hz.

What does noise temperature

T_{sys}

describe?

How much noise the whole receiving system generates, in Kelvin — lower is quieter.

What is Boltzmann's constant used for here?

Converting temperature to noise power density:

N_{0} = k T_{sys}

, with

k = 1.38 \times 1 0^{- 23}

J/K.

Give the G/T formula in dB.

G / T (dB/K) = G_{r} (dBi) - 10 lo g_{10} (T_{sys})

Why divide

G

T

To reward both grabbing signal and staying quiet in one figure of merit.

Why measure energy per bit rather than total power?

Faster data spreads power over more bits, giving each bit less energy and more errors;

E_{b} / N_{0}

captures this honestly.

Foundations — Link budget — path loss, EIRP, G - T, Eb - N0