Polar map, Apex and distorted polar plot

Polar map, polar plot, contour plot

Heavily used at AudioHorn, the polar map or in fact the contour plot in our case, is a graphic representation of the energy radiation provided by a device, in dB according to degrees.

He is obtained by measuring the speaker every 5 or 10° on an axis by turning the speaker on himself and not moving the mic, at a precise distance, windowed, etc…

The contour plot has the advantage to have all the frequencies visibles in one graph:

Where the -6 dB, here the transition from red to yellow by convention, define the opening angle of the horn or waveguide. As it’s flat we can also said that the directivity is constant, we use VituixCAD for visualization here.

But turning the speaker on itself means precisely using an axe, and to define where he is?
It’s why we will talk about Apex or apparent Apex.

What is Apex ?

The apex is a point where the horn or waveguide profiles axes virtually meet, if the profile is curved, which is systematically the case today and in our profile, it’s where the opening axes used to calculate the profile meet.

This is not the source of emission although by choice some designers have made the two coincide. It’s often not the same between horizontal and vertical, here point A is the Apex when B is considered as the baffle:

This point allows to properly characterize a “guided” transducer (waveguide and horns but also coaxial driver) to have real directivity measurement.

To measure a direct-radiating loudspeaker or with a very short/small WG, it doesn’t matter and rotating around the baffle is sufficient.

In the case of polar measurement of a single element, we will place the mic at the height of the targeted element and rotate around the apparent apex.

Rotating around something other than the apex for a horn or waveguide will introduce a “distorted polar plot”: Closer we measure, the larger the mouth and deeper the apparent apex is, the more crucial this point is.

Distorted polar plot

The distorted polar plot is not a distortion in the usual sense in audio, it should be seen as a trigonometric error between the position of the microphone, the apex and the width of the horn mouth, The shape of the directivity will generally be correct but the opening given by the angle at -6dB, will be in the wrong position, sometimes to a large extent, we are talking about a difference of the order of 10°. The entire energy distribution is distorted.

Here the green line is aligned with the opening of the horn and also shows how to find the Apex and the red lines are the equivalent by rotating around the baffle, but this is valid at any point.

Note: A horn is never measured free-air without full return or ISO baffle, if not a lot of midrange narrowing will appear.

We can see on this diagram that the further we go, the smaller the difference between rotating to the baffle compared to rotating on the APEX is.

So in close measurement and large horn, we must be close to the Apex horn.

Hence the fact that a complete speaker is measured in anechoic chamber around 3m, as the apex being not the same between the different elements of the speaker.

We can then ask ourselves what to do when measuring a complete speaker fairly closely when we don’t have these perfect conditions, two cases are available to us:

The guided part is small and does not go low so much, a point equidistant between the mid-woofer Apex and the “loaded” part, will suffice.
The guided part goes down (let’s say 700Hz), it covers a big part of the reverberant field, that is after the Schroeder distance, below it’s the modal field.
This is where directivity control is the most important because it will have a very audible impact in the reverberant field more than in the modal field.
In this case, we will directly choose the Apex of the top guided part.

In other words, the closer you are, the more interest there is in using the apparent apex to be in the “truth”.

Window your measurements

Even if you measure at 1m of the Apex you will have a floor/ceiling rebound that will impact your measurement in the low part, you can visual it with a tool in VituixCAD to see when (in time), according to distance and height position of elements, you will have it.

The way to do is: In your measurement tool, usually REW, you have to look at the Impulse, check where the accident due to surfaces occurs in time and then Window your measurement just before the first accident as shown here:

Here we put:

Lef Winfow: 1ms
Right Window: 4.8ms, just before floor reflection

Be also sure that Ref is on the impulse of the driver because at 75 or 90° the wall reflection can be more important in volume than the driver itself due to the fact that we turn the speaker or horn. So the measurement tool can in some cases choose the wrong Ref and put it on a reflection, so we have to move the ref to the real driver impulse, to the left.

Sources

A message from Kolbrek about HornResp talking about it:

I believe Hornresp uses a Far-field approximation model for directivity, i.e. that the directivity pattern is calculated as it would appear if you measured it at a very large distance, but the level is scaled back to a 1m distance. Otherwise, you would have to specify a measurement distance and point of rotation, and there would be a large variation in the pattern depending on those values.

If the measurement point is very far away, these variations become insignificant, and the actual rotation point does not matter.

What is “very far away” depends on the size of the source (horn mouth) and also on the “apparent apex” or center of curvature of the far field wave front.

In addition, there are near-field effects. These typically happen when the distance is shorter than (Source area)/(wavelength), this distance is called the Rayleigh distance. At larger distances, the pressure varies as 1/distance, but closer to the source the variations do not follow this law, and have peaks and dips you wouldn’t see at a greater distance.

These effects are usually not a problem when measuring small devices like direct radiators and small horns, but most horns are large enough that they become noticeable, especially at the standard distance of 1m.

When measuring horns, it is usually recommended to rotate them around the “apparent apex”. This will avoid distortion of the directivity pattern at short distances compared to the far-field pattern.

Also note that Hornresp uses one-dimensional horn models, so the directivity models are only approximations.