Moulton Lectures
On
Electro-Acoustics
|
Lecture
31 Fundamental Operation of Noise Cancelling Microphones |
Presented
by:
Dave
L Moulton
Location:
Racal Acoustics Limited
Harrow UK
Date: 21-July-2004
Content
In this Lecture I want to give an explanation of how
noise cancelling microphones work. Over the years I have often been asked the
following questions. Why does a noise cancelling microphone only work well at
low frequencies and why does it cancel sound from distant noise sources and not
close up sources such as voice?
In order to answer these questions I will use the
following approach:
·
Give a simple outline of a noise cancelling
microphone.
·
Give a simple reminder of the Planar and Spherical
Wave Equations.
·
Introduce the concept of a pressure gradient across an
exposed diaphragm .
·
Develop the pressure gradient ratio model.
·
Investigate the model to help answer the above
question.
On With The Lecture
We have already seen in lecture 30 that there is a
class of microphone that we call a Bi-Directional Pressure Gradient Microphone.
In its simplest form this type of microphone comprises a diaphragm that is
exposed on both its front and back faces.
Figure
31.1
We will assume that our diaphragm is rigidly attached around its outer edge, and the diaphragm material has elastic properties. Thus it has a restoring force Fr when deflected.
In order for the diaphragm to move there must be difference of Force acting on the front and back faces. The diaphragm will move in the direction of the greater force. The movement of the diaphragm by an external Force will be limited by its own inherent stiffness. This is shown in figure 31.2 below:
Figure 31.2
Clearly we can see from figure 31.2 that the diaphragm will only be displaced if there is a difference of force across it.
We can now go back to basics and conclude that a difference in Force is also directly related to a difference in pressure either side of the diaphragm.
Where P is the pressure (scalar) and A is the Area (vector).
We can describe a sound wave as a pressure wave which varies with both time and propagation displacement. So let us now consider the displacement effect on the exposed diaphragm when a sound wave passes through the vicinity. We will consider the effect of a near field sound source (Spherical Wave) and a far field sound source (planar wave). Firstly let us remind ourselves of the general wave equation expressions and solutions for a one dimensional Spherical wave and a one dimensional Planar wave:
One Dimensional Spherical Wave:
Solution in the positive x direction:
One Dimensional Planar Wave:
Solution in the positive x direction:
We are interested in the difference in pressure seen either side of the exposed diaphragm when either of the above wave types passes through the vicinity. We know that a one dimensional sound wave will propagate along a one dimensional axis. In this case it is the x axis and we have only consider the wave equation solution in the positive x direction. The reality is that waves tend to be three dimensional, however for the purposes of this lecture I intend to keep the analysis as simple as possible.
We will start by looking at the effect of a Spherical wave passing through the vicinity of the diaphragm. The first thing to consider is that the diaphragm is housed in a structure, which in this case is given a front to back path length d, we may also refer to this as the depth of the microphone. See figure 31.3.
Figure 31.3
The effect of the Spherical wave on the diaphragm will be considered as the wave front passes from the front face of the housing structure, designated x=0, through to the back face designated x=d. Clearly as the wave propagates over a distance d we are interested in the resulting changes in pressure either side of the microphone diaphragm.
The change in pressure over the depth of the housing is known as the Pressure Gradient. The pressure gradient function directly relates to the difference in Forces seen across the diaphragm and consequently the diaphragm displacement.
A mathematical expression for the pressure gradient created by our one dimensional spherical wave can easily be found by differentiating the wave solution with respect to x, thus:
Applying the well known rules for differentiating a product, we have:
Thus:
Simplifying we get:
Remember I was generally asked two questions, one relating to the frequency range of noise cancellation and the other to the near field and far field discrimination. To answer both of these questions I am going to derive an expression known as the pressure gradient ratio. This is ratio of the Spherical wave pressure gradient to the Planar wave pressure gradient across the depth of the microphone. So rather than stopping to analyse expression [31.10] in isolation, I will instead start with a diagram and then move straight on to formulating an expression for the planar wave pressure gradient, thus:
Figure 31.4
Thus
This differentiation is very easy:
We can now form the pressure gradient ratio as follows:
Simplifying:
In order make the pressure gradient ratio meaningful, it is important that the Spherical wave front and Planar wave front are assumed to have the same amplitude at the front face of the microphone housing. In this case at x =0. The simplest way of doing this is to assume that the planar wave amplitude Ppo has the same amplitude as the spherical wave, thus.
This assumption is only valid if we assume that the dimensions of the microphone housing are significantly smaller than the shortest wavelength in the acoustic wave. As a rule of thumb the dimensions should be at least 1/10 of the shortest wavelength. From previous lectures we know that for a plane wave the amplitude can be considered independent of propagation distance. However as we also know, this is not true for a spherical wave. We can now further simplify expression [31.14] down to:
Leading to:
Finally we get:
The amplitude of this function, gives us the expression we are looking for, thus:
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we can be more specific and write x as the depth dimension of our microphone and re express the wave number k in terms of frequency.
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Equation [31.20] tells us a lot about the how the microphone diaphragm responds to the two different wave types. We can view the pressure gradient ratio as a means of determining the effect of close speech (Spherical Wave Fronts) and distant background noise (Planar Wave Fronts). The greater the value of the ratio, the better the discrimination between a spherical wave and a planar wave. This can be related to the acoustic noise cancelling performance of the microphone, the bigger the ratio the better the noise cancellation.
The following points can be deduced from equation [31.20].
· As the frequency increases the pressure gradient ratio reduces, tending toward unity as the frequency becomes very large. Thus for a given microphone dimension d, the noise cancelling properties are far better at the lower frequencies than at high frequencies.
· The noise cancelling performance is also determined by the depth of the microphone, the smaller the depth the better the discrimination.
· With this type of exposed diaphragm the noise cancelling performance reduces with increasing frequency as a first order slope, until a point is reached where the response has no discrimination. This point is normally called the cross over frequency.
The analysis done in this lecture explains the fundamental principle behind why noise cancelling microphones are able to discriminate between noise and speech, and the effect of the housing depth dimension, also known as path length. However, it must be pointed out that in reality microphones are not completely open cavities with diaphragms that are equally exposed on the front and back. Usually the housing is also acting as an acoustic filter creating an acoustic mismatch. These often require some effort to match the front and back response of the microphone. Some of the larger microphones have diaphragms attached to coils and other mechanical devices, the resulting mass loading will also have the affect of reducing the pressure gradient ratio. For a single diaphragm microphone the mechanical and acoustic loading effects tend to reduce the cross over point.
Typical microphone cross over points can range from 500Hz to 3kHz. Small electret microphones tend to perform the best with cross over points in the range 1kHz to 3kHz. Larger dynamic noise cancelling microphones tend to have cross over points in the range 500Hz to 1.5kHz .
The Figures below show a typical near field (10mm) and far field (1m) frequency response profile for several Racal Acoustics noise cancelling microphones:
Figure 31.5
Figure 31.6
A noise cancelling microphone can be turned into a non-noise cancelling microphone by simply blocking the back face. A simple way of subjectively assessing how well a noise cancelling microphone is performing is to cover the back face with your finger and listen for the change in acoustic background noise. A good noise canceller will give a significant perceived change in a low frequency noise environment.
Noise cancelling microphones are of no real use for background noise with a spectrum that is near to and beyond the cross over frequency of the microphone. Other means such as adaptive noise stripping need to be used to deal with these high frequency noise profiles.
In this lecture I have only considered the case where the Planar and Spherical waves are incident directly onto the front face of the microphone. A more complicated analysis would look at the sound waves being incident at an angle to the face of the diaphragm. Mapping the noise cancelling effect through 360˚ around microphone would create a polar response for the microphone and reveal its cardioid nature. The exposed diaphragm model used in this lecture would reveal a figure of eight shaped response with the maximum noise cancellation occurring when the planar wave meets the diaphragm face on. The noise cancellation would reduce to zero when the planar wave is incident side on to the diaphragm.
My Final Statements are a summary that answers the two most often asked questions stated earlier.
Noise Cancelling Microphones are able to discriminate
between background noise and close speech because of the different way in which
they react to a spherical wave front (Speech) and a Planar wave front (Noise).
These discrimination properties are frequency dependant
and reduce with increasing frequency.
The degree of Noise cancellation is also a function of
the housing shape and the acoustic mismatch between the front and back of the
microphone.
End of Lecture