Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by: Dave L Moulton

 

 

 

 

                                    Location:  Thales Acoustics Harrow UK

                           Date:  23-October-2002

 

Content

 

This fourth lecture starts to look at the three main system domains encountered in audio electro-acoustics. These domains are namely the Electrical, Mechanical and Acoustic.  By analysing equivalent systems within these domains I hope to establish the analogies between the components, and system variables.

 

Electro-acoustic engineers need a level of understanding such that they are able to freely move between domains. The typical example being the design of an earphone where the input is in the Electrical Domain, the earphone responds in the Mechanical Domain and produces the desired output in the Acoustic Domain. The reverse process is true for a microphone.  In the case of a hearing protector headset only two domains are used, these being the external noise level in the Acoustic Domain resulting in the headset earshell vibrating in the Mechanical Domain, followed by a pressure change at the entrance to the ear canal in the Acoustic Domain.

 

I will be introducing second order differential equations to help show the similarity between equivalent oscillatory systems in each Domain.

 

Note:  Do not be afraid of the mathematics, the realization is that the differential equations are just the summation of very simple physical relationships. Each of these simple  relationships should have been covered at O-level and A-level physics.

 

On With the Lecture

 

The Mechanical Domain

 

I thought that it would be best to start by analysing a simple Mass-Spring-Damper system, since it is very easy to visualize and interpret the system motion.

 

The diagram 4.1 below shows the Mechanical system consisting of a Mass M_m connected to spring of compliance C_m and a damper of Resistance R_m.  Both the Spring and Damper are rigidly fixed to the Surface S.

 

 

Figure 4.1

 

The Damper with mechanical resistance R_m represents the friction between the Mass and the surface on which it sits.

 

Now consider the situation when a force F is applied to the Mass M_m  causing the mass to be displaced by an amount x from its rest position. This is shown in figure 4.2

 

Figure 4.2

In order for the Force F to displace the Mass M by an amount x it must overcome the Inertial Force of the Mass F_M  , the Stiffness force of the Spring F_C  and the Resistive force of the Damper F_R.  

 

Thus the Force equation for the system can be very simply expressed as:

 

 

Now lets consider the expression for each of the component forces, starting with the Inertial Force of the Mass.

 

From very simple A-level physics the inertial Force of a Mass is:

 

 

Now for the restoring Force in the Spring, again from Hooke’s law:

 

 

Finally the Frictional Resistive Force can be expressed as:

 

 

Applying our individual expressions to the Force Equation Described in [4.1] we have:

 

 

Equation [4.5] represents the complete equation of motion for the mechanical system described in figure 4.2.

 

 

 

I will come back to this equation later in the lecture.  Now lets consider an electrical system:

 

Electrical Domain

 

Consider the simple series electrical circuit consisting of an Inductor L,  Resistor R and Capacitor C. As shown in figure 4.3 below:

 

Figure 4.3

 

Now consider the situation in figure 4.4 where an ac Voltage  V is applied across the system, giving rise to a current flow i.

 

Figure 4.4

Thus the Voltage equation for the system can be very simply expressed as:

 

 

Now lets consider the expression for each of the component Voltages, starting with the Back-emf in the Inductor.

 

Again from very simple A-level physics the Back-emf from the inductor can be defined in terms of current i and charge q as follows:

 

 

Now for the Voltage across the Capacitor in terms of current i and charge q.

 

Finally the Voltage drop across the Resistor.

 

Applying our individual expressions to the Voltage Equation Described in [4.6] we have:

 

 

Equation [4.10] represents the complete equation of charge flow for the electrical system described in figure 4.4.

 

Once again like with the mechanical system equation I will come back to it later on during the lecture.  Now lets consider an acoustic system:

 

 

 

 

Acoustic Domain

Rather than dive straight in to the acoustic parameters for this system I will firstly draw a typical cavity system, then analyze it for its mechanical properties and then convert to the Acoustic properties. 

 

This may seem like an odd approach, however for myself, visualizing acoustic motion is not quite as straight forward as visualizing the equivalent mechanical motion. You may have your own views on this and be quite happy to visualize the acoustic system only.

 

Firstly lets draw the acoustic system at rest. This will consist of a cavity filled with air attached to a neck with an open end, see figure 4.5.

 

Figure 4.5

 

If we now consider the action of a pressure at the open end of the neck, giving rise to a force acting on the Mass M_m of air in the neck.

 

When dealing with standard audio acoustics we must always observe the law for the  conservation of Mass.

 

The Mass M_m can be considered to move as a block of Mass similar to what happens in the Mechanical system.

 

The Volume V_m of air trapped in the cavity will be compressed and act like a spring attached to the mass in the neck. This spring will have a mechanical Compliance C_m.

 

The movement of air in the neck in contact with the walls of the neck will experience a viscous drag. This drag is equivalent to the Mechanical Frictional Resistance experience by a Mass moving over a surface in the Mechanical Domain.

 

Redrawing the cavity diagram showing the applied pressure, we have :

 

Figure 4.6

 

Thus the Force equation for the system can be very simply expressed as:

 

 

Now lets consider the expression for each of the component Forces and then convert each one to a Pressure related component. Remembering that Pressure ‘P’ = Force ‘F’ per unit Area ‘A’. 

Starting with the Mass of air in the neck we have:

 

 

In terms of Acoustic Pressure:

 

 

Introducing Volume Displacement

 

 

Where M_a is the acoustic Mass, also known as the Inertance

 

Using the same reasoning as for equation [4.14], we can re-express the Compliance Force in the Cavity Volume as:

 

 

Where C_a is the Acoustic Compliance.

 

Finally treating the Mechanical Resistance in the same way:

 

 

Re Writing [4.11] as a Pressure Equation we have:

 

 

We can now add in our individual component blocks to form the second order differential equation:

 

 

Equation [4.20] represents the complete equation of Volume displacement for the Acoustic system described in figure 4.6.

 

Notice that the Acoustic elements are functions of mechanical elements:

 

Acoustic Inertance  

 

Acoustic Compliance 

 

Acoustic resistance 

 

Analogies

 

Comparison of the three differential equations [4.5], [4.10] and [4.20] gives a direct insight into the Analogies between the three Domains. The table below shows all of the Analogies.

 

Equivalent Circuits

 

Often in Acoustics and Mechanics we want to analyze complex systems. To do this it is much better to represent the Acoustic and Mechanical Systems as an equivalent electrical circuit. We use the electrical symbols, and represent their values by the analogue equivalent in the Mechanical or Acoustic Domain.

 

The simple Mechanical and Acoustic Systems described in this  lecture can be easily represented as equivalent circuits:

 

Mechanical Equivalent Circuit:

 

Figure 4.7

 

Acoustic Equivalent Circuit:

 

Figure 4.8

 

Natural Frequency

If we were to remove the resistive elements from differential equations [4.5], [4.10] and [4.20], and then remove the driving mechanism, Force, Pressure, voltage. The systems will become self sustaining, this means that they will resonate at their own natural frequency. In the ideal case the oscillations will continue indefinitely. We can draw the analogy between the Adiabatic process that keeps a sound wave propagating to the potential and kinetic energy exchange of a simple pendulum and the charge exchange for a simple tuned Inductor and capacitor.

 

This oscillation is known as Simple Harmonic Motion SHM and can be described by the following differential equation:

 

 

Where  w=2pf₀

 

f₀  is the natural resonance frequency of the Harmonic system.

 

Lets have a look at the reduced equations.

 

 

 

 

Mechanical

   

 

Thus                            

 

And                              

 

Electrical

 

Thus                           

 

And                           

 

 

Acoustic

 

 

Thus                             

 

And                          

 

 

 

 

The Helmholtz Resonator

 

A Helmholtz resonator is basically an acoustic cavity represented as shown:

 

The cavity neck has a cross sectional area ‘A’

The neck has a length ‘L’

The Cavity has a volume ‘V’

The density of air is ‘r’

The speed of sound is ‘c’

The Bulk Modulus of air is ‘B’

 

See figure 4.9 below:

 

Figure 4.9

 

Using equation [4.32] we can find an expression for the natural frequency of the resonator in terms of its mechanical dimensions:

 

Thus:

 

 

Where:

 

 

 

Substituting these expressions into equation [4.33] gives:

 

 

This expression is commonly used in acoustics for working out the resonance frequency of individual simple cavities at the front and back of microphones and earphones.

 

End of Lecture