There are many methods of measuring the gain of an antenna, most of them call for a reference antenna of known gain. This method requires three antennas and does not require knowledge of the gain of any of them, but will find the gain of each of them.

## Explanation

Harald Friis gave us the familiar transmission equation: \(\frac{P_r}{P_t}=\frac{A_r A_t}{r^2 \lambda^2}\\\).

More common usage is to use Gain instead of aperture: \(\frac{P_r}{P_t}=G_r G_t (\frac{\lambda}{4 \pi r})^2\\\).

Let’s convert to dB and use frequency instead of wavelength: \(PdB_r-PdB_t=GdB_r + GdB_t +20 log \frac{c_0}{4 \pi f r}\\\).

We can express path gain as \(G_p=GdB_r + GdB_t +G_{fs}\), Gfs is the free space component of the path, \(G_{fs}=20 log \frac{c_0}{4 \pi f r}\\\).

Rearranging, \(GdB_r + GdB_t =G_{p}-G_{fs}\) and making \(GdB_r + GdB_t =G_{rt}\) we can write \(G_{rt} =G_{p}-G_{fs}\).

This leads to three simultaneous equations in three unknowns.

\(1G_1+1G_2+0G_3=G_{p_{12}}-G_{fs}\\0G_1+1G_2+1G_3=G_{p_{23}}-G_{fs}\\

1G_1+0G_2+1G_3=G_{p_{31}}-G_{fs}\\\)

We can write the three simultaneous equations out in matrix form and calculate the solution.

\(\begin{vmatrix}G_1\\G_2\\G_3 \end{vmatrix}=\begin{vmatrix}1 & 1 & 0\\0 & 1 & 1 \\ 1 & 0 & 1\end{vmatrix}^{-1} \times\begin{vmatrix}G_{p_{12}}-G_{fs}\\G_{p_{23}}-G_{fs}\\G_{p_{31}}-G_{fs}\end{vmatrix}\\

\)

So, we have calculated the three gain figures, G1, G2, and G3.

The online calculator Antenna gain calculator – three antenna method is a convenient way to document the measurements and calculate the gain of each antenna.

### Gauss-Jordan elimination

This case is pretty trivial to solve by elimination.

\(1G_1+1G_2+0G_3=G_{p_{12}}-G_{fs} \tag 1 \label{eq:eq1}\)

\(0G_1+1G_2+1G_3=G_{p_{23}}-G_{fs} \tag 2 \label{eq:eq2}\)

\(1G_1+0G_2+1G_3=G_{p_{31}}-G_{fs} \tag 3 \label{eq:eq3}\)

Taking \(\eqref{eq:eq1}-\eqref{eq:eq2}+\eqref{eq:eq3}\) we get \(G_1=\frac{G_{p_{12}}+G_{p_{31}}-G_{p_{23}}-G_{fs}}{2}\) and so on.

Note that if you load the calculator with frequency, distance and set \(s21dB_{12}=s21dB_{23}=s21dB_{31}=0\) then \(G_{fs}=-2G_1\), a quick way to evaluate G_{fs} to check feasibility of the proposed test range and instruments.

## Summary

- Since the method depends on the Friis transmission equation, all the conditions for its validity must apply.
- An antenna test range is not a trivial exercise, read widely before you rush into measurements that might be disturbed by multi-path / ground reflection etc.
- This article does not address measurement uncertainty, but it is very important to the outcome.