Free Space Path Loss

The Ripple Effect

Drop a stone in a pond. Ripples spread outward in all directions.

Near the stone:

• Ripples are tall and strong
• Lots of energy in a small area

Far from the stone:

• Ripples are shallow and weak
• Same energy spread over a much larger area

The energy isn’t disappearing. It’s spreading out over a larger and larger area.

This is exactly what happens to radio signals.

Why Signals Get Weaker

When an antenna transmits, the signal radiates outward like an expanding sphere.

Here’s the key insight:

ALL of the transmitted power spreads across the surface of this sphere. Every single watt.

As the sphere gets bigger, that same power is spread over a larger surface area.

The math is brutal:

Same total power. Bigger sphere. Less power at any single point.

This weakening is called path loss.

The surface area of a sphere grows with distance squared ($r^2$).

So power density shrinks with $\frac{1}{r^2}$.

Double the distance → quarter the power density. This is the inverse square law.

Free Space Path Loss

Free Space Path Loss (FSPL) is the ideal case:

• Signal traveling through empty space (no air, no objects, nothing)
• Nothing in the way
• No reflections, no obstacles, no interference

Pure geometric spreading. The minimum loss you’ll ever get.

The formula:

$L_p = 20\log_{10}\left(\frac{4\pi d}{\lambda}\right)$

Don’t panic. Let’s break it down:

• $L_p$ = path loss (in dB) — how much weaker the signal gets
• $d$ = distance — how far the signal travels
• $\lambda$ = wavelength — the “length” of each wave

The Distance Factor

How does distance affect path loss?

Path loss grows with the square of distance. Go 10x further → signal is 100x weaker.

The Frequency Factor

Here’s something that surprises people:

Higher frequency = more path loss at the same distance.

But wait. The signal spreads out the same way regardless of frequency. So why does frequency matter?

The answer: It’s about the receiving antenna.

Think of your antenna as a bucket catching rain.

• The rain (signal) is falling everywhere
• Your bucket (antenna) can only catch what falls into it
• A bigger bucket catches more rain

Here’s the key insight:

An antenna’s effective size depends on the wavelength it’s designed for.

At higher frequencies (shorter wavelengths), an antenna’s effective “catching area” gets smaller.

Same signal spreading out. But the antenna catches less of it.

Higher frequency → smaller effective antenna → less signal captured → more “loss”.

Real-world impact:

This is why 5G needs more cell towers than 4G. Higher frequency = shorter range.

The Alternative Formula

Sometimes you know frequency instead of wavelength. No problem:

$L_p = 20\log_{10}\left(\frac{4\pi d f}{c}\right)$

• $f$ = frequency (in Hz)
• $c$ = speed of light ($3 \times 10^8$ m/s)

Since $\lambda = \frac{c}{f}$, both formulas give the same answer.

Use whichever is more convenient for your situation.

Practical Example

Scenario: WiFi router at 2.4 GHz, receiver 10 meters away.

Plugging into the formula:

$L_p = 20\log_{10}\left(\frac{4\pi \times 10 \times 2.4 \times 10^9}{3 \times 10^8}\right)$

$L_p \approx 60 \text{ dB}$

What does 60 dB mean?

That’s one million times weaker than the transmitted signal.

And this is the ideal case. No walls. No interference. No obstacles.

Reality is always worse.

The Catch

Free space path loss assumes perfect conditions.

In the real world, you also deal with:

• Obstacles: walls, buildings, trees absorb signal
• Reflections: ground, ceilings, metal bounce signal
• Atmospheric absorption: rain, humidity attenuate signal
• Interference: other signals compete with yours

FSPL is the minimum loss. Everything else adds on top.