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8.2 - DIRECTIVITY

8.2.4 - THE MATHEMATICAL MODEL FOR DIRECTIVITY

The greatest danger with the all powerful mathematical model is when we actually believe in it! The best attitude is one of distant mistrust - the mathematical model is just a convenient tool, nothing more! It would certainly be an error to think that we can actually analyse our perception of the microphone sound as if we were hearing the two constituent parts of the model – the omni and the figure-of-eight . That would be similar to saying that when we see yellow, we actually see green and red! But there again analogies are also dangerous animals.

The mathematical model for directivity combines an omnidirectional directivity response with a bi-directional or figure-of-eight response, in varying proportions for each directivity pattern. This mathematical concept has a real physical realisation in unidirectional microphones manufactured around the 40s and 50s. A typical example from that period was the ST&C 4033, where two microphones (an omnidirectional moving coil microphone and a bi-directional ribbon microphone) were combined within the same casing, together with a suitable switchable matrixing circuit to produce either of the two basic microphone directivities and also their combined response as a cardioid. Mathematically we use the same process to produce a model for the different directivity patterns. The directivity response is given by the formula:


The coefficient ‘A’ gives the percentage of omnidirectional response, and the coefficient ‘B’ gives the percentage of bi-directional response, the bi-directional response being represented by the term ‘cos(α)’ where ‘α’ is the angle of incidence of the sound wave. The representation of bi-directional directivity with the function cos(α) in the mathematical model which is positive from 0° to 90° (and 270° to 360°) and negative from 90° to 270° corresponds to the inversion of polarity between the front and back of the real bi-directional microphone.

Figure 41 in section 8.1 shows to what extent this mathematical model corresponds to the real directivity response of a cardioid condenser microphone. The dashed line in this figure represents the theoretical response as defined by the mathematical model, which can be compared with the measured response for a cardioid microphone. Up to about 120° the correspondence is reasonably near. Towards the rear of the microphone the divergence is considerable, simply because none of the techniques used to produce directional microphones are able to maintain their characteristics throughout the whole of the audible frequency range.

However the mathematical model is sufficiently near the real response of a microphone to be valid for all but the very high and very low frequencies, and for all angles up to about 120° from the microphone directivity axis. This model will be used for all the calculations concerning the operational characteristics of the stereophonic and multichannel microphone array systems.

In almost all the calculations of these characteristics, the mathematical model is used for the directivity response at angles of no more than 90° off axis, so we can consider that in this case the mathematical model for the directivity characteristics of small diaphragm microphones is a reliable representation of the real microphone directivity response. The model, of course, highlights the possibility of many different first-order directivity patterns, many more than are in fact manufactured at present. We can calculate an infinity of directivity patterns simply by varying the matrixing coefficients ‘A’ and ‘B’. Table F gives an illustration of these different directivity patterns by varying the coefficients ‘A’ and ‘B’.


This presentation of directivity as a function of the coefficients ‘A’ and ‘B’ is rather cumbersome - in practice it is more convenient to read the back attenuation value from the polar diagram for each microphone published by the manufacturer, rather than calculate the values for each coefficient in the mathematical directivity model. The next table - Table G - therefore expresses the directivity pattern with respect to the attenuation in response of the microphone at 180°. For both hypercardioid and hypocardioid microphones, there are always corresponding directivities with the same back attenuation value.


The terms ‘directivity factor (DF)’ and ‘directivity index (DI)’ are the accepted way in which to define the total directional discrimination of a specific microphone directivity pattern^[10]. Another term that is sometimes used to describe this same directional discrimination function is ‘Random Energy Efficiency (REE)’. The directivity factor is defined as a ratio, whereas the directivity index is the same energy ratio but expressed in decibels. The directivity factor is the ratio of the microphone’s response (output level) to diffuse or reverberation sound coming from all directions around the microphone and with equal energy distribution, with respect to the response of a truly omnidirectional microphone of the same axial sensitivity and in the same acoustic environment. A cardioid microphone has a directivity factor of about 0.333, the directivity index is therefore 10*log₁₀(0.333) or –4.77dB. Table G shows the corresponding DF and DI for each of the specified directivity patterns.

The case of one particular hypocardioid directivity is of particular interest. As we have seen there is always an inevitable off-axis high frequency loss determined by the diameter of the diaphragm and the microphone body size. However with careful design, it is possible to integrate this change in directivity pattern into the overall directivity pattern of the microphone. If the microphone is designed with only pressure acoustic coupling in the high frequencies and pressure-gradient coupling for the rest of the audible frequency range, then the directivity pattern in the medium and low frequencies range can be made to correspond almost exactly with the natural directivity pattern at high frequencies - this equivalent to designing the acoustic labyrinth as an acoustic low pass filter circuit. This is obviously different for each diaphragm diameter, but in the case of a small diaphragm condenser microphone, the high frequency directivity with pressure acoustic coupling corresponds to a hypocardioid directivity pattern with about 10dB back attenuation. Schoeps have manufactured a microphone capsule (MK21) using this design approach. The result is a directivity pattern that in fact remains remarkably constant up to about 12kHz as shown in Figure 46.

As we will see later, this directivity also has another advantage in that it has a better low frequency response compared to a cardioid microphone. Schoeps have given this directivity pattern the name ‘Infra-cardioid’ or ‘Wide Angled Cardioid’, I prefer the generic term 'hypocardioid' with a specification of the back attenuation value (example - hypo10).


Figure 46 – Polar Diagram for a Schoeps MK21 (Hypocardioid)
at 250Hz, 500Hz, 1kHz, 2kHz, 4kHz, 8kHz & 16kHz
(Back Attenuation 10dB)
© frequency response curves published by courtesy of Schoeps

The hypercardioid and supercardioid directivity patterns have received particular attention in the audio industry because they have specific directional discrimination characteristics. The hypercardioid has the minimum Directivity Factor value - this corresponds to the maximum rejection of indirect or reverberant sound with respect to the on-axis response as shown in Figure 47.

Figure 47 – Directivity Factors as a function of Directivity
The Supercardioid, on the other hand, has the minimum ratio (i.e. the minimum numerical value of the ratio) of back hemisphere response to front hemisphere responses^[11] as shown in Figure 48.


Figure 48 – Directivity Index as a function of Directivity
These two figures (47 & 48) are calculated using the mathematical model for directivity. It is debatable whether the precision shown in these two figures can be justified in the real world. As we have seen previously the directivity response for a Cardioid is very close to the theoretical response in the front hemisphere, but can show considerable divergence at angles exceeding about 130° to the directivity axis. Similarly the directivity pattern for hypercardioids and supercardioids is usually very irregular around the region of maximum attenuation between 100° to 140° (and 220° to 260°). The numerical value of the directivity factor (DF) for these directivities will therefore be somewhat higher than the theoretical value, dependent on specific irregularities in directivity response in these regions with respect to each microphone. However the difference in DF will in practice be more significant between the omnidirectional, the “infracardioid” and the cardioid, rather than between the cardioid and the members of the hypercardioid family.

[11] 1941 - "A New Microphone Providing Uniform Directivity over an Extended Frequency Range" by R.N.Marshall and W.R.Harry Journal of the Acoustical Society of America, Vol 12 (April) p483