Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by:

Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  08-January-2003

 

Content

 

In lecture 10 we established a very simple mathematical expression describing the closed loop system response needed to achieve ANR at the face of the microphone.  This lecture will look at the system equation necessary to achieve ANR at some point located forward of the microphone.  I will also explain the difference between partitioned and un-partitioned ANR techniques, there will also be an explanation of the all important Gain/Phase plots which we use to view the degree of compensation required to achieve stable ANR.

 

On with the Lecture

 

ANR at the face of the microphone

In Lecture 10 I was able to formulate an ideal expression for ANR at the face of the monitor microphone_.

 

 

If we are able to achieve the ideal condition then in theory wide band ANR with unlimited bandwidth is predicted.

 

The condition being:

 

From this condition we end up with a system model that looks like that shown in figure 11.1.  

 

 

Figure 11.1

 

Now let us consider another situation where we want to locate and determine the conditions to achieve ANR at some point away from the face of the monitor microphone. In particular we may want to determine and optimize the ANR at the entrance to the ear canal.

 

 

Figure 11.2

 

 

The equivalent system block diagram

 

 

Figure 11.3

 

Using the same symbolic representation as in lecture 10, I can draw the system block diagram as shown in figure 11.4.  Note that the monitor microphone located at the localized zone ‘Z’ is only there to measure the acoustic pressure at ‘Z’ and to help in establishing the transfer function R(s).

 

 

Figure 11.4

 

To make a simplification to this model we can remove the awkward noise deviation function DP₀(s) and represent the deviation as a transfer function d(s).

 

 

Figure 11.5

 

We can now start putting together the system closed loop transfer function as follows:

 

First lets start by working our way around the first loop comprising K, G(s), E(s), T(s) and M(s).

 

 

Figure 11.6

 

We have already seen from lecture 10 that this will give a transfer function which takes the form:

 

 

Where H(s) is the open loop transfer function  M(s)T(s)E(s)

 

Now let us breakdown the components that make up the  loop containing d(s), T(s) and R(s).

 

Firstly we have two summing nodes, so lets deal with each one individually.

We will start with the node at the face of the monitor mic located at zone ‘Z’

 

Thus we have:

 

 

Rearranging  [11.4]

 

 

At the summing node located at the face of the ANR microphone we have:

 

 

Substituting for P_a(s):

 

 

Now going back to equation [11.3] and re-expressing as:

 

Thus:

 

 

Substituting into [11.7] gives:

 

 

Thus separating out P₀(s) and f(s) we have:

We now have to consider what d(s) actually is. d(s) represents a transfer function which creates the difference between the noise pressure reaching the ANR microphone and the Noise pressure reaching the localized zone ‘Z’.  We can approximate d(s) as the ratio of the two acoustic transfer functions R(s) and T(s):

 

Such that:

 

 

This now provides for a much simpler transfer function:

 

 

For this situation, once again the optimization occurs when G(s)=H(s)^-1 leading to the ideal expression:

 

 

Assuming that k>>1 the modulus of this expression reduces to:

 

 

Unlike equation [11.1] the modulus of this expression is frequency dependant and very much influenced by the acoustic transfer function T(s) and R(s).

 

Partitioned and Un-partitioned ANR systems

 

Figures 11.7 and 11.8 show typical examples of the partitioned and un-partitioned headset ANR systems:

 

Partitioned ANR

 

Figure 11.7

 

The partitioned ANR system is the most common approach to achieving ANR in Military headsets. Companies such as BOSE, Technofirst, ELNO, HISL, TELEX and others have adopted this technique to achieve ANR.  The partition literally splits the earshell volume into a front and back cavity, with only a small controlled bleed hole linking the two cavity chambers together. The main consequence of this approach is that the ear is no longer acoustically coupled to the full acoustic volume of the earshell.

 

The partition makes a significant difference to the shape of the open loop transfer function H(s), giving it a much shorter impulse response making it a much easier function to compensate for in terms of gain and Phase. Relatively high peak levels of ANR are achievable with this method, however it is significantly at the expense of the low frequency passive attenuation.  Remember from lectures 6 and 7 that the low frequency passive attenuation is a function of the earshell cavity volume exposed to the wearer’s ear.

 

Un-Partitioned ANR

 

 

Figure 11.8

 

The un-partitioned ANR is an approach used by Thales Acoustics (formerly Racal Acoustics). This approach was adopted to achieve optimum passive and Active attenuation.  The drawback is that the larger backshell volume presents an open loop transfer function H(s) with a much longer impulse response.  This makes compensation far more difficult to achieve. However when optimization is achieved the combined Passive and Active Attenuation performance generally out-performs the partitioned headsets.

 

Earshell Gain Phase plots

 

Gain phase plots are the most useful  and understandable way of interpreting the open loop transfer function H(s) and the resulting system transfer function.  The affect of the compensation achieved by G(s) can easily be seen along with the stability aspects of the system such as gain and phase margin.

 

A typical gain Phase plot of H(s) for a Non-Partitioned system is shown in figure 11.9.

 

H(s) for a non-partitioned earshell

 

 

Figure 11.9

 

Figure 11.11 shows the Gain-Phase plot for the compensated open loop system. This is when G(s) and Gain K have been set so that the ANR system remains stable.

 

The Open Loop response of the entire compensated system is measured as shown in figure 11.10 below:

 

 

Figure 11.10

 

The Gain-Phase plot for the compensated system looks something like that shown in figure 11.11.

 

 

Figure 11.11

The Gain and Phase margins shown in figure 11.11 are important criteria for determining the margin of system stability.  There definitions are as follows:

 

Gain Margin (GM)

Additional Gain needed to make the system marginally stable.  Marginal stability occurs when the system gain is 0dB and the relative phase angle is  –180°

 

Thus for system phase lag of -180° and a corresponding system gain of -5dB the gain margin is:

 

GM=0dB-(-5dB) =5dB

 

Phase Margin (PM)

Additional phase lag needed to make the system marginally stable. 

Thus for a system gain of 0dB and a corresponding phase lag of -140° the phase margin is:

PM=180°-140° =40°

 

Further examples of gain-phase plots and system stability will be explained in future lectures:

 

 

End of Lecture