Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by:

Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  04-December-2002

 

Content

 

This lecture is intended to explain a little more about earcushions and how their basic parameters can be described as an equivalent circuit.  In doing so I hope to give a better insight into how simple equivalent circuits are formulated.

 

I will also show a method for measuring and calculating the real mechanical compliance and resistance values for a standard type of foam earcushion.  I hope that the methods I adopt provoke discussion and also inspire the class to look for other ways into doing these measurements and even other ways of analysing the performance.

 

 

On with the Lecture

 

In lecture 7, which was more of a presentation, we saw how the mathematical model shown below was demonstrated on an Excel spreadsheet.

 

 

Figure 8.1

 

 

The model shows that one of the key features of a hearing protector is the ear-cushion, it plays two important roles.

 

Interfaces the earshell to the ear

 The earcushion  is the mechanical part of the hearing protector that is directly in contact with the skin, so its physical feel and pressure distribution are very important in determining the wearers perception of comfort.

 

Determines attenuation performance

The earcushion  plays a significant role in the overall attenuation performance of the hearing protector. This role is very much determined by the mechanical Compliance and Resistance of the earcushion.

 

I have underlined two very important features of a hearing protector, namely ‘Comfort and Attenuation Performance’. The ideal would be to have a hearing protector with both a high level of comfort and a high level of attenuation. Unfortunately life is never that easy, the reality is that both comfort and performance pull in opposite directions, the best that can be achieved is a compromise between the two.  A headset with optimum attenuation performance is usually at the expense of comfort, in the case of the earcushions the more rigid they are the better the attenuation will be. However, a rigid earcushion tends to be very uncomfortable when worn around the ears.  Making a set of earcushions comfortable for prolonged periods of use involves making them as soft and compliant as possible, this has the affect of worsening the attenuation performance. The area of the audio spectrum where this compromise is very evident is in the Low to mid frequency range (20Hz to 2kHz).

 

I will now try and explain the affects of single and multi-foam cushions.

 

Firstly let us consider the simple case of a single foam cushion with a very thin compliant membrane. For this example I will assume that the membrane and cushion platform have no significant affect on the overall performance of the earcushion,.  Figure 8.2 shows the cross section of the earcushion:

 

 

 

 

 

 

Figure 8.2

 

Now let us consider what happens when we apply a force to the top of the cushion platform.

 

 

Figure 8.3

 

The equivalent Mechanical system is shown below in figure 8.4:

 

Note the applied Force ‘F’ has to overcome the restoring Forces of the cushion spring Compliance  ‘F_C’ and Resistance ‘F_R’ . I have also assumed that the affects of Gravity are not relevant to the system.

 

 

 

 

Figure 8.4

 

The differential equation describing the motion of this system is:

 

If I rewrite this equation in terms of the system velocity, we have:

 

 

Thus the equivalent electrical circuit is easily shown below.

 

Figure 8.5

 

Note that both the spring and damper are moving with the same velocity.

 

In order to increase the attenuation performance of a hearing protector we need to maximize the impedance of the system shown in figure 8.5.

 

Quite clearly we can do this in two ways:

 

1.     Increase the Cushion Resistance. (Dampen the cushion movement).

2.     Reduce the Cushion Compliance. (Stiffen the cushion).

 

If we want to increase the comfort of the cushion we do the opposite to the above:

 

1.     Reduce the Cushion Resistance.  (Less Damped movement)

2.     Increase the cushion Compliance. (Make the cushion more springy)

 

Now lets consider another common earcushion configuration where there are two different foams sitting on top of each other surrounded by the cushion membrane. See figure 8.6 below:

 

 

Figure 8.6

 

As in the case of the single foam cushion let us consider what happens when we apply a force to the top of the cushion platform.

 

 

Figure  8.7

 

The equivalent Mechanical system looks like:

 

 

Figure 8.8

 

We can derive the differential equations for this system by simply looking at the action of the forces at each of the two nodes ‘A’ and ‘B’ shown in figure 8.8.  First lets investigate node ‘A’.

 

re-expressing this equation in terms of velocity we have:

 

 

Now for Node ‘B’

 

 

Again re-expressing in terms of velocities we have:

 

 

Equations [8.4] and [8.6] fully describe the motion of the system.

 

In order to draw the equivalent electrical circuit we need to treat the mechanical velocities like electrical currents.  When we have a situation where the Force is a function of two different velocities then we must apply  Kirchhoff,s law and treat the difference in velocities as being the result of one velocity entering a node and then splitting into two velocities at the node.  Remember that Kirchhoff’s law states that the sum of all currents entering a node must equal the sum of all currents leaving the node.

 

If we simply apply Kirchhoff’s law to the two equations [8.4] and [8.6] we can easily draw the following conclusion:

 

 

Now equation [8.4] contains a velocity that is a function of the difference in displacements (x₂ – x₁), and this velocity is common to both the Compliance C₂ and Resistance R₂, thus  C₂ and R₂ must be in series and form one of the node branches.

 

Likewise equation [8.6] shows that C₁ and R₁ must also be in series because they share the same velocity, also this velocity is a function of x₁ only. Thus C₁ and R₁ must form the other branch of the node.

 

The equivalent electrical circuit looks like:

 

Figure 8.9

In order to maximize the impedance of the above circuit, we need to make the lowest impedance component as high as possible.  Like any parallel impedance circuit, the impedance is always lower than the lowest branch impedance. Thus for two foams on top of each other, (positioned in series) the compliance will be dominated by the piece of foam with the highest compliance, and by the piece of foam with the lowest resistance.

 

Note that the foam has an inherent compliance and resistance, thus the total impedance of a piece of foam is both reactive (Compliance) and resistive (Resistance). You can’t change one without affecting the other, The combination of these two forms the total impedance of a branch in the equivalent circuit.

 

Figure 8.10

 

Quite often you see hearing protectors with two pieces of foam in the cushion, often one piece is very compliant, virtually making the other piece redundant.

 

We will now consider the less familiar situation where the two foams inside the cushion are side by side, see figure 8.11 below:

 

Figure 8.11

 

Applying a force to the cushion:

 

Figure 8.12

 

The Equivalent mechanical Circuit:

 

Figure 8.13

The differential equation at Node ‘A’:

 

 

Expressing in terms of the velocity:

 

 

A little rearranging:

 

 

Note that the velocity is now the same for both types of foam, so their compliances and resistances must all be in series.

 

Thus the equivalent electrical circuit looks like:

 

 

Figure 8.14

 

Which can be simplified to:

 

Figure 8.15

 

In this case the least compliant and most resistive components tend to dominate the impedance.

 

Measuring The Compliance and Resistance of a simple foam  Earcushion.

 

Compliance

This is without doubt the easier of the two parameters to measure.  It is simply a case of measuring cushion deflections for given applied forces and then plotting the result on a graph. The graph should obey Hooke’s law (Robert Hooke 1635 to 1703), with a gradient in the linear region equal to the stiffness constant for the cushion. From this the compliance can be calculated:

 

Figure 8.16

 

Resistance

This is not so easy.  We must remember what resistance does in a reactive system,  it controls the damping of the system. This gives us a clue as to the affect the  Resistance has  on an earcushion. We need to measure this affect.

 

Here is my approach to this problem:

 

Consider that we have a cushion and we apply a known Force F₀  so that it produces a known deflection in its linear region, we will call this initial deflection x₀. See figure 8.17 below:

 

 

Figure 8.17

 

The equation of motion for the displaced cushion is:

 

 

Now if we remove the mass the internal forces within the cushion will try and restore the cushion back to its original equilibrium position. The equation for this motion is the homogenous expression:

 

Applying the following set of logical manipulations:

 

 

 

Applying the initial condition at time t = 0 , x = x_o

 

Thus:

Thus:

Thus:

Our important final expression:

 

 

Expression [8.20] puts us most of the way towards finding the cushion Resistance.  The expression shows all the characteristics of a system discharging its stored energy.  If we take a special case where we let the time  t = RC, then the expression becomes very simple indeed.

 

 

This simply means that when the time elapsed from the point of removing the force from the cushion reaches the value t=RC, the cushion would have recovered to within  x_oe^-1 of its original rest position (x=0).

 

 

Figure 8.18

 

Measuring the time taken for the cushion to reach x_oe^-1 gives the value of RC, so with C already calculated it should be easy to establish a value for R.

 

 

 

End of Lecture