Gain and Directivity

The Lightbulb vs The Flashlight

Imagine you have a lightbulb and a flashlight, both using the same battery.

The lightbulb spreads light in all directions.

Stand anywhere around it, and you see some glow.

The flashlight focuses light in one direction.

• Stand in front of it, and it’s blinding
• Stand behind it, and there’s nothing

Same power. Different distribution.

The flashlight isn’t creating more light. It’s concentrating the same light into a smaller area.

This is exactly how directional antennas work.

The Isotropic Antenna

In theory, there’s a perfect “lightbulb antenna” that radiates equally in all directions.

We call it an isotropic antenna.

Here’s the thing:

It doesn’t actually exist. You can’t build one.

But it’s incredibly useful as a reference point for comparing real antennas.

Directivity

Directivity measures how much an antenna focuses its power compared to an isotropic antenna.

If an antenna has directivity of 10, it means:

In its strongest direction, the power density is 10 times what an isotropic antenna would produce.

Think of it like water pressure.

A garden hose and a pressure washer might use the same water flow. But the pressure washer focuses it through a tiny nozzle.

Same input. More concentrated output.

From Directivity to Gain

Directivity is the theoretical best case. It assumes a perfect antenna with no losses.

Real antennas aren’t perfect.

Some power gets absorbed by:

• Antenna materials (metal resistance)
• Cables (signal loss)
• Connectors (impedance mismatches)

This lost power turns into heat.

Antenna efficiency tells us how much of the input power actually gets radiated:

$\eta = \frac{P_{\text{radiated}}}{P_{\text{input}}}$

• $P_{\text{input}}$ = power you feed into the antenna
• $P_{\text{radiated}}$ = power that actually leaves as radio waves
• The difference is lost as heat

An efficiency of 0.8? That’s 80% radiated, 20% wasted.

Gain

Gain is what you actually get in practice.

$G = D \times \eta$

Gain equals directivity times efficiency.

If directivity is what could happen, gain is what does happen.

The Unit: dBi

Gain numbers can get huge. A satellite dish might have a gain of 10,000.

Writing “10,000x” everywhere gets awkward. So we use decibels (dB) to compress big numbers into manageable ones.

How it works:

The logarithm asks: “10 to what power gives me this number?”

• $\log_{10}(10) = 1$ because $10^1 = 10$
• $\log_{10}(100) = 2$ because $10^2 = 100$
• $\log_{10}(1000) = 3$ because $10^3 = 1000$

Decibels turn multiplication into addition. A gain of 1,000,000 becomes just 60 dB.

$G_{\text{dBi}} = 10 \log_{10}(G)$

The “i” in dBi means relative to isotropic. It’s our reference point.

The pattern: Every +3 dB doubles power. Every +10 dB multiplies by 10.

dBm: Decibels for Power

dBi compares gain to an isotropic antenna.

dBm measures absolute power relative to 1 milliwatt.

$P_{\text{dBm}} = 10 \log_{10}\left(\frac{P}{1 \text{ mW}}\right)$

Converting dBm back to Watts:

$P_{\text{watts}} = 10^{(P_{\text{dBm}} - 30) / 10}$

Example:

A 5G signal has received power $P_r = -72$ dBm. What is this in Watts?

Solution:

$P = 10^{(-72 - 30) / 10} = 10^{-10.2} \approx 6.31 \times 10^{-11} \text{ W}$

That’s 63 picowatts. Receivers are incredibly sensitive.

The Tradeoff

High directivity is a double-edged sword:

• Great if you know exactly where your receiver is
• Terrible if you need coverage in all directions

There’s no free lunch. Focusing power somewhere means taking it away from everywhere else.