Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by: Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  13-November-2002

 

Content

 

Lectures 1 to 5 were intended to provide the necessary mathematical tools required to model a simplistic hearing protector headset. 

 

This lecture will attempt to put together a mathematical model for a simple symmetrical earshell fitted with cushions and having a possible leak path under the cushion.

 

Lecture 6 is intended to set the groundwork for lecture 7 which will be a combined lecture and visual presentation of the mathematical model simulated in excel.

 

 

 On with the Lecture

 

Firstly I want to outline what I mean by a simple hearing protector and the assumptions made to formulate the model.

 

A hearing protector is a device which shields the entrance to the ear canal from an externally generated scalar noise field.  The Noise field could be anything from a noisy factory to the inside of a main battle tank.

 

I shall only consider a single earshell located on the side of a human head. The earshell will comprise and earcup made from a fairly rigid and dense material giving it a physical Mass M_S .  Attached to the earshell is an earcushion comprising a Foam type material surrounded by some form of flexible membrane, The cushion will have a resulting Compliance C_C , and Resistance R_C.  The vector area of the earcup will be A_S. while the Vector area between the inner walls of the earcushion will be A_e. 

The Earcup inner Volume will be V_S while the Volume between the earcushion will be V_C.

 

This arrangement is shown in figure 6.1 below:

 

 

Figure 6.1

 

My assumptions for this simple representation are:

 

1.     The earcup is a straightforward hemispherical shape with symmetry through its central axis.

2.     The Mass of the earcushion is negligible compared to the Mass of the earcup.

3.     The earcushion is rigidly attached to the earcup.

4.     The earcup Material is rigid with no significant Compliance or Mechanical Resistance.

 

I will now attach to our simple earcup a representation of the human head.  The interface between the earcushion and the head is human flesh which has its own Mechanical Compliance C_SK and Resistance  R_SK.

 

The human ear sits within the inner volume of the earcushions and is represented by a tubular cavity.

 

Figure 6.2 illustrates the situation:

 

Figure 6.2

 

My assumptions for the above representation are:

 

a.      The Skin Compliance and the Earcushion Compliance exhibit linear stiffness only. (We are not considering non-linear behaviour).

b.     The ear canal volume is negligible compared to the volume between the cushion.

 

Now that we have our earshell attached to a head we can look at how the attenuation is performed from the mechanical and acoustic elements of the system. 

 

Firstly we must define what we mean by acoustic attenuation. Clearly if we are dealing with hearing protection in a noisy environment we have to reduce the level of external noise pressure P₀ getting to the ear drum. The use of the hearing protector will do this for us, however rather than measuring the actual pressure at the eardrum P_d (Very tricky) we measure the acoustic pressure at the entrance to the ear canal P_e.  So for our hearing protector we define its attenuation performance as being.

 

 

 

Figure 6.3 shows the headset with the pressures added:

 

 

 

Figure 6.3

 

The External Pressure P₀ will exert a force on the ear cup over the vector Area of the earshell. 

 

The Resulting force F₀ will cause the earshell to be displaced in sympathy with the fluctuations in external pressure. 

 

The movement of the earshell will cause the Volume between the earcushions to change (i.e as the earcushion is compressed the Volume V_e will reduce. The result of this volume change is that the Pressure at the ear P_e will also change at the same rate.  The rate of change of pressure is then transmitted down the ear canal and causes the eardrum to move giving the human perception of Sound. (in this case unwanted Noise). The intensity of the sound at the eardrum  P_d² will determine the level of perceived loudness. 

 

Note I have added the respective compliances for the earcup cavity C_V and the earcushion cavity C_e.

 

Objective

For this simple hearing protector our objective is the make the attenuation described by equation [6.1] as large as possible.

 

To do this we need to create a mathematical model for the hearing protector so that we can see how each of the physical parameters affects the overall attenuation performance.

 

So here we go…

 

Firstly I will start at the noisy end and transform the external Pressure P₀ into a resultant Force F₀.  This is simply a case of applying the transformer representation described during Lecture 1.

 

Figure 6.4

 

Note that the Transformer scaling factor is the Vector Area of the earcup.

I have represented the Acoustic Volume Velocity and Mechanical velocity in the dot notation. Thus one dot above the variable represents the first order differential of that variable, two dots will represent the second order differential etc….

 

We can now build up the mechanical side of the circuit by adding in the Earshell Mass M_S , Cushion Compliance C_C and Cushion Resistance R_C. The first thing to note is that the earcup is rigidly attached to the front end of the earcushion , so the attached end of the earcushion will move with the same mechanical velocity as the earcup.

 

Represented Mechanically it would look something like figure 6.5

 

 

Figure 6.5

 

Thus our Mathematical Model can be improved to:

 

Figure 6.6

 

Now we have to consider the affect of the Skin Compliance C_SK  and the Skin Resistance R_SK . It is best to view this situation as a mechanical model, it should be fairly easy to see that in a mechanical model the Skin parameters are acting in series with the cushion.  Thus:

 

 

Figure 6.7

 

We must now tread carefully and understand what is happening to the velocities.  What we have in figure 6.7 is the mechanics of the cushion and skin acting in series with each other. We basically have two springs of different compliance connected in series with each other. The point at which the two springs are joined will move with a different velocity to the velocity of the Mass.  There is a velocity difference across the earcushion, this difference equates to the new velocity through the cushion.

 

In the same way that an electrical current splits down separate branches of a circuit, so does the velocity split down separate branches of the mechanical domain circuit.  Thus in the mechanical domain circuit the Skin parameters act in parallel with the cushion parameters. Thus our Mathematical model can be further improved to:

 

 

Figure 6.8

 

We have now completed the mechanical domain Model and must complete the picture by transforming back into the Acoustic domain.  Remember we are interested in relating the external noise pressure to the pressure at the entrance to the ear canal. To do this we must use a transformer to relate what is happening to the acoustics within the earshell. We are interested in the pressure developed in the cavity between the earcuhion, this is where the ear canal is located when the ear-defender is worn. It should not take too much effort to see that our Mechanical to Acoustic transformation factor is the vector Area between the cushions A_e. Thus we can improve our mathematical model to look like figure 6.9.

 

 

Figure 6.9

 

Now that we have transformed into the earshell we can look at what the Acoustic volume velocities are doing with respect to the two Cavities, namely the cavity between the cushion V_e and the main cavity in the earcup

V_S.

 

 

Figure 6.10

 

I have shown the ear canal in this representation, however we can assume that it is acting like a long narrow tube and not like a Volume cavity. We can assume for this example that the size of the ear canal is negligible compared to that of the volume in the earshell and between the cushion. The ear canal can be considered as not having a significant influence in determining the value of P_e ,  However it is the main path to the eardrum and there will be a  volume velocity Ú₁ down its length to the eardrum. To add reality to the model I have shown this Volume velocity.

 

Figure 6.10 shows the Volume velocity movement in the two ear-defender cavities when the mechanical parts of the headset are moving. This can be represented as a distribution of Volume currents in much the same way that we would use Kirchoffs law to show current distributions in an electrical circuit.

 

Figure 6.11

 

 

Adding this circuit to our model gives:

 

 

Figure 6.12

 

We now have our basic mathematical model for a passive hearing protector.  This model is based on the following assumptions.

 

1.     There are no acoustic leaks between the outside and inside of the earshell.

2.     There is no sound transmission path directly through the earcup material or the earcushion material.

3.     There are no other restraints on the movement of the earcup such as a neck band attachment, boom arm or cable entry.

 

Converting to One Domain

 

In order for the model in figure 6.12 to be easily analyzed we need to convert all of the components into one domain.   For this type of circuit I am going to use the mechanical domain.

 

The reason for not transforming everything into the acoustic domain at this stage is that I would have to evaluate the scaling affect of all the impedances in order to represent F₀ in terms of F_e. This would be a necessary requirement in establishing the transfer function in the acoustic domain.

 

The transformation of both acoustic domains to the  mechanical domain is as follows. 

 

Firstly I will transform the Acoustic cavities into the mechanical domain using the impedance scaling factor of the transformer A_e². I will also show how the acoustic Volume velocities Ú₁ and Ú₂ transform into the mechanical domain.  

 

 

 

Figure 6.13

 

Now transforming the external Acoustic pressure into the mechanical domain using the earcup vector area as the scaling factor A_S.

 

Figure 6.14

 

Figure 6.14 almost completes our simplistic model, however I say almost because we I now need to add in an important feature that is very influential on the hearing protector performance.  In reality there is always the likelihood that an acoustic leak could occur between the skin and the earcushion. The most common reason for such a leak is down to the variability in human head profiles, in some cases a near perfect seal can be achieved and in others the undulations in the skin make a leak an almost certainty.  A leak will greatly influence the attenuation performance of any circum aural hearing protector and must be included into the model to give it an element of realism.

 

The Acoustic Leak Path

The leak path under the cushion links the outside scalar pressure field P₀ to the inside of the earshell via the Inertance and Resistance of the leak path.

In most cases a cushion leak can be treated like a small tube or slit between the skin and the cushion, the leak path may be isolated to just a small area of the cushion in contact with the skin.  Because we are working in the mechanical domain our acoustic leak needs to be transformed into that domain.  We can do this by using a couple of transformers.  The first transformer converts the external pressure P₀ into a mechanical force pushing a the mass of air through the leak tube. (The mass  of air in the leak can be treated a mechanical mass). The scaling factor is the cross sectional area of the leak path A_L .  The second transformer converts the mechanical output of the leak into the acoustics of the earcushion cavity, with a scaling factor A_e.  Thus the leak alone can be modeled as follows:

 

 

Figure 6.15

 

Transforming into the mechanical domain we have:

 

Figure 6.16

 

I now need to find a way of integrating this leak model into the headset model. We can use the volume velocity through the leak path as a means of injecting extra velocity into the mechanical system.

 

 

Figure 6.17

 

To make the circuit easier to look at I have also stopped showing the velocity currents in the transformed earshell volumes.

 

I now have to deal with the input to the leak circuit,  Ideally I would like to find a way of tying the leak force F_L to the input force F₀ shown in the headset model.

 

From Figure 6.17 we see that:

 

…………..[6.1]

 

We can modify this expression as follows:

 

……..[6.2]

 

We can show this arrangement as a transformer coupled system in the mechanical domain. With a scaling factor of  1:(A_L/A_S ).

 

Thus:

 

Figure 6.18

 

I will now make an assumption about the above arrangement.

 

The combined volume Compliances C_e and  C_V produce an impedance that is significantly smaller than that of the cushion leak path. Therefore I can transform M_L and  R_L into the F₀ zone without significantly affecting the rest of the circuit:

 

Figure 6.19

 

Figure 6.19 can be put into a more general form by combining the cavity compliances and embedding the area scalars into the components.

 

Thus:

 

 

Figure 6.20

 

The transfer function for the circuit shown in figure 6.20 above can be represented as shown below:

 

 

Figure 6.21

 

I will leave it to Lecture 7 to show the mathematical model in action and how all the parameters affect the overall attenuation performance.

 

Beyond the Simple Model

The model depicted in figures 6.19, 6.20 and 6.21, cater for the elements that have the most significant impact on the attenuation performance. However in reality there are a lot of other factors that could be added to the model and which have a smaller influence on the overall performance. These factors include:

 

1.        Neckband attachment:  Providing extra loading to the earshell.

2.        Bone conduction path due to the earcushion contact with the head.

3.        Direct bone conduction path due to sound pressure impinging onto the head.

4.        Sound Conduction through the cushion Material.

5.        Sound Conduction through the earcup material.

 

All of these factors add to the mathematical model as shown:

 

 

Figure 6.21

 

 

 

End of Lecture