It is accepted that the change in magnitude vs. frequency is represented by the plot of the frequency response (SPL) of a typical driver. However the radiation pattern is less familiar. Most amateurs are aware of a polar response but because it has to be represented as discreet slices in frequency, they have a difficult time visualizing how the change progression The radiation profile changes drastically with commencement of lobbing and it is especially interesting to define this region.


The main difficulty in representing the radiation patterns is the number of variables involved, frequency, horizontal and vertical directivity as well as magnitude, phase. The limitation is that in 3D representation we have to choose which we’ll use and thereby create a blind spot. I shall start by attempting to define the radiation pattern of a dome or a cone that is treated as a point source and has a hemispherical pattern, then take a look at a ribbon which has a cylindrical pattern. Possibly look at a line array as made from discreet drivers. All of the plots exclude the edge diffraction effects because they complicate the analysis of the response. For the dome an example of diffraction is given to complete a more realistic picture of radiation response.


Radiation Directivity Patterns

Dome Directivity

Fig.1. Dome Frequency/Phase Response of the radiation wavefront
The plot shows the magnitude and phase response for a model of a dome driver. The off-axis response is for 85,45deg. vs. the convention of 60, 30, because the commencement of lobbing is @ 90deg.
Note that the phase wraps around @ the null which corresponds to the onset of lobbing.
This plot also shows that the lobbing commences @ about 16kHz and @ ~ 18kHz. the main lobe would be a pencil beam.
The lobbing patterns and cone/dome geometry are discussed in Appendix A.
This examines the lobbing frequency vs. cone diameter.

Fig.2 Polar Response.
The polar plot shows the frequency slices @ 10,15 and 17kHz. At 15kHz we can see the shrinkage of magnitude @ 90deg. and @ 17kHz. the main lobe is ~ 30 deg.
It is these various views that I would like to tie together into a comprehensive representation. A possible format would be the contour projection tied to the conical projection plots.

Fig.3. Cylindrical Projection

The radiation wavefront can be viewed in 3D as the Horizontal angle vs. magnitude vs. frequency, as a cylindrical projection. This shows the narrowing of the directivity and the repeating of the pattern. This also shows the limitation of the 3D view, since the lobbing and phase change are not represented.
Before we go into the complexities of the 3D views, it may be good to step back and familiarize ourselves with the radiation pattern of a dome.
A typical dome directivity pattern symmetrical in Horizontal and Vertical axis. At 10kHz. the off-axis attenuation is only ~ 4dB and there is no lobbing.
The smooth curve results is an ideal representation as the inclusion of edge diffraction changes the hemisphere drastically, see the last plot pg.3 for edge diffraction effect.
Fig.4 Horizontal Directivity


Fig.4. Dome Radiation @ 10kHz.

Over an angle of about 60 deg. the radiation profile is hemispherical. The directivity is symmetrical around the X Y axis.

Lobbing Onset.

Fig.5 Horizontal Directivity

As we approach the narrowing in the directivity with frequency, the main lobe narrows to ~ 60 deg and side lobes appear @ the 90 deg. angles.
The polar and directivity patterns are more thoroughly covered in Appendix A.
Note that the 2 axis symmetry of fig.4 has changed to a 4 axis and the radiation is more squarish than hemispherical. Since this is the same type of pattern seen when baffle edge diffraction effects are seen, the hemispherical radiation is limited to an ideal case
Fig. 6 Dome Radiation: Lobbing Onset

Dome Directivity @ 20kHz.
At 20kHz we see the correspondence between the polar and directivity plots and the 3D radiation pattern representation. This clearly shows the 4 axis symmetry.

Fig.7 Directivity @ 20kHz.


Fig.8. Polar Response @ 20kHz.


Fig.9 Radiation Pattern @ 20kHz.


Baffle Edge Diffraction



Fig.10 Baffle Diffraction Frequency Response.

Frequency Response with Baffle Edge Diffraction Baffle 15 x 35cm with the dome tweeter centered.
Response for 0, 30, 60 deg. off-axis By adding the edge diffraction effects the Horizontal and Vertical directivity patterns are no longer symmetrical, as shown in fig.11.
The horizontal axis = blue, vertical axis = red.
The contour lines of fig.12 suggest that in the vertical axis we have in effect 3 sources of radiation. This is similar to the Appendix A polar response where the radiation wavelength is smaller than the dome diameter.
Dome Diffraction @ 10kHz
Note that @ 0deg on axis the magnitude is ~2dB lower than for 5-10deg. off-axis.

Fig.11 Dome Directivity @ 3kHz.


Fig.12 Radiation Pattern with Edge Diffraction.


Fig. 13 Dome Directivity @ 10kHz.


Fig.14 Radiation Pattern with Edge Diffraction.

Note that with increasing frequency the diffraction magnitude is decreasing and the basic radiation shape becomes more prominent,mark
Dome Diffraction @ 18kHz

Fig.15 Dome Directivity @ 18kHz.


Fig.16 Radiation Pattern with Edge Diffraction.

Ribbon (0.5 x 15cm) Wavefront Patterns.
The ribbon model is for a 0.5x15cm planar driver with a 3.5cm length conical horn. The fundamental problem in the simulation of the ribbon is the accuracy of the model. This is documented in the Appendix B section.

Fig.17 Ribbon Horizontal Axis Response.

Unlike the dome, the ribbon has distinctly different responses for the horizontal and vertical axes. The response is defined by the geometry of the horn as the ribbon is primarily the driver at the horn’s throat.

Fig.18 Ribbon Vertical Axis Response.

The vertical directivity is substantially narrower off-axis as shown in fig.19 and 23.

Directivity @ 5kHz
The vertical directivity is ~ 30 deg. and the attenuation is ~ 25 dB @ 60 deg. relative to the on axis(ribbon length).

Fig.19 Ribbon Directivity @ 5kHz.


Fig.20 Cylindrical Radiation Pattern.


Directivity @ 10kHz

Fig.21 Ribbon Directivity @ 10kHz.


Fig.22 Cylindrical Radiation Pattern.


Directivity @ 15kHz

Fig.23 Ribbon Directivity @ 15kHz.


Fig.24 Cylindrical Radiation Pattern.

Directivity @ 20kHz

Fig.25 Ribbon Directivity @ 20kHz.


Fig.26 Cylindrical Radiation Pattern.


Lobbing vs. Directivity Angle and Frequency
Appendix A plotted the development of lobbing via polar plot required discreet frequency slices. I would like to extend the view to the cylinder projection, first for the dome, then for the ribbon’s cylindrical wavefront.

Fig.30 On-Axis Frequency Profile

[X,Y,Z] matrix size is 100x100 as defined by imported data file. If the row array is linked to frequency, then lobbing can be represented as magnitude variation within the row array.
Frequency data file (fd00.mat) fig.30 is an array 100x3, from 300Hz to 50kHz. The resolution at 18kHz is 300Hz. (i=60) The nulls correspond to the frequencies where the radiation becomes a pencil beam. The lobbing start @ ~ 16.5kHz.
Fig.31 represents the 3D on-axis wavefront showing the repeat of the lobbing pattern. The lobbing directivity would be inside the outer magnitude shell, starting @ about 16.5kHz.
Fig.31 On-Axis Wavefront Pattern


Fig.32 On End View

View(-180,0)

The on end view shows the off-axis magnitude vs frequency as color. The lobbing magnitude would be a closed shape within the end view.
Polar Response

Ribbon Simulation Calibration

Last updated: October 5, 2005 9:28 PM