How we measure the HPBW of an antenna

The HPBW (Half-Power Beam Width) is the width of the main beam of the antenna's radiation pattern, measured at the points of half maximum power. The knowledge of the HPBW is important, because it also tells us how well the antenna dish is illuminated by the feed antenna, thus how efficient the antenna is, and how high the angular resolution ist, i.e. what minimum angular diameter a radio source must have in order to be identifiable as a single object. Ideally the HPBW is determined by measuring the power of a point source (whose angular size is smaller than the HPBW) which is moved across the front of the antenna.

The theoretical radiation pattern of a circular dish of diameter D which is evenly illuminated is given by the power transmitted to (or received from) the direction angle phi; from the central axis of the dish: p(φ) = [2λ/(πD) * J1((π D /λ) sin(φ)) / sin(φ) ]² with the wavelength λ and the first-order Bessel function J1(x). The main lobe has the width HPBW = 59°/(D/λ) - with D and λ given in the same units, e.g. metres. False colour map of the theoretical radiation pattern of a circular dish of 8 m diameter at 1.3 GHz.

For practical purposes, it is best to use the Sun on its promenade across the sky, as it is at all frequencies the brightest radio source in the sky, and the perfectly regular rotation of the Earth lets it move in the sky in a precisely predictable manner. We take a drift scan: we place the antenna to a position where the Sun will be some time from now, and then measure the power while the Sun passes over the beam.

However, there is a small problem: The Sun with its angular diameter of about 0.5° is not a perfect point source. Therefore, our method will depend on the width of the beam:
(a) For beams that are much wider than about 0.5° we may consider the Sun as sufficiently small bright spot in the sky, and thus the curve measured by the drift scan gives directly how the antenna's sensitivity depends on the angle from the beam centre.
(b) If the beam is narrower than or comparable to the size of the Sun's disc, the curve from the drift scan shows how the brightness of the Sun varies across its face. But as at the edge of the Sun the brightness makes a sharp jump from bright to dark, we may use the shape of the solar rims to provide information for the beam width.

Wide beam

Making a full drift scan requires an accurate positioning system of the antenna. We place the antenna to a position, where the Sun will pass some time later. Since the sky moves with 1° in 4 min, you might choose for a 3° wide beam a position of 9° ahead or 36 min. With the antenna at that position, you record the measured power. After the maximum, wait until the signal power has come down again to a constant value ... this is the empty sky. Perhaps you might pick up the side lobes ... thus it is a good idea to move the antenna horizontally towards the East and measure the empty sky here. You will need this value! Interpretation: If the data are in dB, convert into linear power values (p = 10dB/10). The measured powers are the sum of the solar noise and the noise from the Earth atmosphere through which the radio waves must pass from the Sun to us. Get the average value for the power from the empty sky noise, which constitutes the zero for the solar powers. Then subtract this value from all measured powers. Now look for the two points in the flank of the bell-shaped curve, where the power is half of the maximum power. Convert the time difference between the two points into angles: HPBW = (t2-t1) * 0.25°/min * cos(decsun) where decsun is the declination of the Sun (viz. the angle above/below the celestial equator). The reason for this correction with the declination is that an object at the celestial equator moves across the sky with the full 0.25°/min, but a source at the pole does not appear to move, because it lies on the rotational axis of the Earth.

This method takes only half the time. Furthermore it is also useful, if the positioning system of the antenna is not accurate or not stable to allow placing the antenna to a position. We simply point the antenna to the Sun, until we get the highest signal. Then we fix the antenna, and start recording the measured powers, until the power comes down and remains stable at a low value, which would be the noise from the empty sky. Perhaps you pick up also the sidelobes. In any case, it is a good idea to move the antenna at the same elevation, well away from the Sun and measure the empty sky. You will need this value! Interpretation: If the data are in dB, convert into linear power values (p = 10dB/10). Get the average value for the power from the empy sky noise. Then subtract this value from all measured powers. Now look for the point in the flank of the curve, where the power has dropped to half of the maximum power. Convert the time between this point and the start time into angles: HPBW = 2 * (t-t0) * 0.25°/min * cos(decsun) where decsun is the declination of the Sun (viz. the angle above/below the celestial equator).

Narrow beam

The Sun very closely resembles a uniformly bright disc (black rectangular curve). Viewed by a narrow antenna beam (HPBW < 0.05°) we see this flat-topped brightness profile. But when viewed with a beam much larger 0.5°, the resulting drift scan curve is a bell-shaped curve, whose shape is determined by the sensitivity response of the antenna's main lobe, and its width measures directly the antenna HPBW - this is the trick which permits us to measure the width of wide beams, as used above. When the HPBW is comparable to the Sun's angular size, the drift scan curve is somewhere between these limiting cases, and is more complicated to interpret. Luckily, the most prominent change appears at the solar rims: the sharp drop from maximum to zero intensity is washed out by the antenna beam. The edge is smoothed out into a gentle decline. Since a wider beam gives a wider profile, we can use the width of the rim to measure the HPBW. This is done by measuring the time difference between two levels - for example at 80% and 20%, of the maximum intensity. Then we use the Earth rotation to get the rim's angular width rimWidth = (t20 -t80) * 0.25°/min * cos(decsun) (here for the outgoing (right) rim). The Table below allows us to obtain the HPBW from the rim width and the chosen flank's level values.

This is a half drift scan of the Sun taken with the 3.7m dish on 24 GHz in 2014. We first search for the maximum signal from the Sun, then move the antenna a little bit to the West and stop the tracking to let the Sun drift over the antenna beam. The blue lines indicate the intensities of the Sun and the empty sky (which gives the zero solar intensity), the green lines mark the 15% and 85% levels. The rim width of 0.28° gives a HPBW of 0.38°. Using the AntennaCalc tool we find that a uniformly illuminated 3.7 m dish would have a HPBW of 0.2°. This implies that the antenna has an aperture efficiency of only 28%, or that it has an effective diameter of only 2 m. Measurements with the Moon gave the same very disappointing values.

There is a long story about the 3.7m dish on 24 GHz. In 2014 the HPBW was measured as 0.38°. This implies that the antenna has an aperture efficiency of only 28%, or an effective diameter of only 2 m. Also, the Sun was about 10 dB over the empty sky, and the Moon 0.7 dB. Obviously this was not something to be content with ... Eventually, a new feed was installed and other improvements were made ... The measurements in 2021 – it may suffice to show only the Moon scan – revealed substantial improvements: The Sun is now 13.5 dB over the empty sky, the Moon 1.4 dB. Both Sun and Moon give rim widths of 0.22°, which means 0.26° HPBW. The aperture efficiency is 60%, the effective diameter 2.9 m. The gain is 57.2 dBi, a great improvement over 53.8 dBi in 2014. Unfortunately, a temperature sensitive amplifier for the long cable in the i.f. chain caused an annoying gain drift. Here, we analyze only one rim.