Part XVIII: Combination microphones

5.7. Electrical combination of pressure and pressure-gradient transducers

One possible way of producing a directivity pattern that has a single maximum (so-called unidirectional characteristic) is to combine electrically the outputs of a pressure and a pressure-gradient microphone. The two units must be located as near to each other in space as possible so that the resulting directional characteristic will be substantially independent of frequency.
Microphones with unidirectional, or cardioid, characteristics are used primarily in broadcast or public-address applications where it is desired to suppress unwanted sounds that are situated, with respect to the microphone, about 180   degrees from wanted sounds. With respect to impedance and sensitivity, this type of cardioid microphone is similar to a ribbon or to a moving-coil microphone when suitable impedance-matching transformers are used.
The equation for the magnitude of the open-circuit output voltage of a pressure microphone in the frequency range where its response is “flat” is
e˜0=Ap˜.
image (5.59)
The equation for the open-circuit output voltage of a magnetic or ribbon type of pressure-gradient microphone in the same frequency range is
e˜′0=Cp˜cosθ.
image (5.60)
Adding Eqs. (5.59) and (5.60) and letting C/A   =   B gives
e˜0=Ap˜(1+Bcosθ).
image (5.61)
B will be a real positive number only if e˜0 image and e˜′0 image have the same phase.
The directional characteristic for a microphone obeying Eq. (5.61) will depend on the value of B. For B   =   0, the microphone is a nondirectional type; for B   =   1, the microphone is a cardioid type; for B   =   ∞, the microphone is a figure 8 type. The value of B can also take on other values that are optimized for particular characteristics. For example, we may wish to maximize the rejection of ambient or nondirect sound. The directivity factor Q as defined in Eq. (3.142) is given by
Q(B)=2D2(0)∫0πD2(θ)sinθdθ=3(1+B)23+B2,
image (5.62)
where the directivity function is given by
D(θ)=1+Bcosθ1+B
image (5.63)
and on-axis D(0)   =   1. The condition for maximum off-axis rejection is
∂∂BQ(B)=6(1+B)(3−B)(3+B2)2=0,
image (5.64)
which is met when B   =   3 and Q(3)   =   4. This gives what is known as the hypercardioid pattern. Alternatively, we may wish to maximize the ratio of the sound captured from the front to that received from the rear, where the rear is defined as anything at an angle of greater than 90   degrees. Let us define the function P such that
P(B)=∫0π/2D2(θ)sinθdθ∫π/2πD2(θ)sinθdθ=3+3B+B23−3B+B2.
image (5.65)
The condition for maximum front-to-rear ratio is
∂∂BP(B)=6(3−B2)(3−3B+B2)2=0,
image (5.66)
which is met when B   =     3 image and P(3)=(2+3)/(2−3)=13.9 image . This gives what is known as the supercardioid pattern. In Fig. 5.30 directional characteristics for six values of B are shown.
image
Figure 5.30 Graphs of the expression R   =   20 log10((1   +   B cos θ)/(1   +   B)) as a function of θ for B   =   0, ½, 1, 3 image , 3, and ∞.
The voltage e˜′0 image is a function of kr, as discussed in Section 5.3, so that the voltage e˜o image as given by Eq. (5.61) will vary as a function of frequency for small values of ωr/c, where r is the distance between the microphone and a small source of sound. Here, as is the case for a pressure-gradient microphone, a “bassy” quality is imparted to a person's voice if he stands very close to the microphone.
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