A recent online discussion contained an analysis of the radiation efficiency of a vertical monopole over real ground.

The poster dismissed the values calculated by 4NEC2 and proposed his own formula \(RadiationEfficiency=\frac{35.6}{\mathbb{R}Z_f}\) where 35.6 is the radiation resistance Rr of a quarter wave monopole over a perfectly conducting earth (PCE).

The reasoning seems to depend on Rr being independent of the ground type, but that is quite flawed.

## NEC insight

Let us look at an example of a quarter wave monopole with 120 shallow buried radials, average ground, at 3.8MHz.

Above is the model geometry.

Some key values directly from the NEC report file and hand calcs:

- - - ANTENNA INPUT PARAMETERS - - - TAG SEG. VOLTAGE (VOLTS) CURRENT (AMPS) IMPEDANCE (OHMS) ADMITTANCE (MHOS) POWER NO. NO. REAL IMAG. REAL IMAG. REAL IMAG. REAL IMAG. (WATTS) 900 3721 1.00000E+00 0.00000E+00 2.77741E-02-4.47034E-04 3.59954E+01 5.79359E-01 2.77741E-02-4.47034E-04 1.38871E-02

Input impedance is found above, it is 35.9+j0.58Ω

AVERAGE POWER GAIN= 7.00031E-01 SOLID ANGLE USED IN AVERAGING=( 2.0000)*PI STERADIANS.

Recalculating that for 4*pi steradians (ie the whole sphere), we get average power gain is 0.350=-4.56dBi. Average power gain is RadiationEfficiency, 35%.

Scanning the gain table, maximum gain is 0.53dBi, so we can calculate \(Directivity=Gain_{max}-Gain_{avg}=0.53–4.56=5.09dB\).

Above is the summary form from 4NEC2. It is consistent with the hand calcs from the NEC report file.

If we used the flawed expression \(RadiationEfficiency=\frac{35.6}{\mathbb{R}Z_f}\) we would arrive at \(RadiationEfficiency=\frac{35.6}{\mathbb{R}Z_f}=\frac{35.6}{35.9}=99.1\%\). It is simply unbelievable that an antenna in such close proximity to natural ground would lose less than 1% of the input power to heating of the soil.

In some ground scenarios, feed point R may be less than 35.6 which would imply more than 100% RadiationEfficiency.

Above is the gain plot from 4NEC2, and because of the antenna symmetry, it is approximately symmetric in rotation about the Z axis. The calculated average gain of 35% or -4.56dBi is believable.

## So, what is Rr?

We can allocate the feed point resistance according to the radiation efficiency, and find that \(R_r=0.35 \cdot 35.9=12.57\Omega\).

The plain English explanation is that more current is needed than with a PCE to obtain the same total power in the far field of \(I^2R\) so since I is increased, R is decreased.

## Is 120 buried radials equivalent to PCE?

No. The gain pattern is substantially different. Different in shape and magnitude.

Above, the gain pattern for the same monopole over PCE. The maximum gain is substantially higher, and average gain is almost 0dBi (some very small loss in the monopole conductor).

## References / links

- IEEE. 2013. IEEE standard for definition of terms for antennas IEEE 2013.