Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by:

Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  11-December-2002

 

Content

 

In this lecture I want to show how common filters such as low pass, high pass, band pass and band stop filters can be realized in the acoustic domain. The same filtering effects achievable with electronic circuits using combinations of  passive Resistors, Capacitors and Inductors can also be achieved with combinations of cavities and tubes.

 

This is a very important area in the field of electroacoustic design, since at the end of the day the prime objective of an electroacoustics engineer is to manipulate sound pressures to achieve a desired performance.

 

The conversion from the acoustic domain to the electrical is not always obvious.  There are particular situations where a cavity and a transmission duct can have acoustic compliance, yet in some cases it is not immediately obvious if the compliance is acting as a series element or as a parallel shunt element.  There are also situations where it is very difficult to represent the acoustic system as a mechanical equivalent.

 

The only true way of expressing the nature of the electroacoustic circuit is to derive all of its system equations, multiple differential equations, each one describing a particular mesh within an electrical circuit.  For this lecture I do not intend going down that route, mainly due to the amount of rigorous mathematics involved and the fact that to get through it all I would have to extend the lecture duration to many hours.  I will try and use the direction and distributions of Volume Velocities to try and establish the equivalent circuits.

 

On with the Lecture

 

In lecture 4 we looked at electroacoustic analogies and how simple cavity systems such as the Helmholtz resonator could be translated in an equivalent electrical circuit.  The Helmholtz resonator was represented as series impedance circuit, where the impedance reached a minimum value at a particular excitation frequency. At the excitation frequency the Volume velocity of the mass of air in the neck of the tube was at its peak value. This is in fact the foundation of an acoustic filter made up of a cavity and a tube. We were able to determine the excitation frequency as a function of the dimensions of the Helmholtz system.

 

What I want to do now is introduce the acoustic equivalents for what is probably the four most common filter types in any domain.

 

Acoustic Low Pass Filter

Consider the acoustic system shown in figure 9.1 below:

 

 

Figure 9.1

Where U_V represent volume velocity. M_a, C_a, and R_a are the respective Acoustic Inertance, Compliance and Resistance components of the system.

 

The simple open cavity system above is very similar in form to a Helmholtz resonator, except I have introduced an exit hole into the end wall of the cavity.  I will make the assumption that the hole has walls of negligible thickness compared to its diameter, this allows me to ignore any Inertance and acoustic resistance effects at the exit hole.

The equivalent Mechanical circuit is shown below in figure 9.1a.

 

Figure 9.1a

NOTE: we have to make an assumption that the end of the spring is in fact attached to a load, in this case it should be the equivalent mechanical load of air.  If we don’t make this assumption then we cannot assume that the two ends of the spring will move different velocities. Remember to represent acoustic compliance this needs to be an ideal spring with no mass and no resistance.

 

Based on the methodology used in lecture 8 we can easily draw the electrical equivalent circuit for this system. See figure 9.2 below.  Remember that in this case the there is a difference in velocity through the mechanical spring and therefore the same applies to the acoustic compliance, hence the spit in current.

 

 

Figure 9.2

 

This should be familiar to most engineers as the common second order Low Pass Filter Circuit.

 

The transfer function for this circuit expressed in Laplace notation assuming that at time t=0 all initial conditions are equal zero is quite simply:

 

Which takes the general form:

 

 

Where w₀ is the natural angular frequency of the system and z is the damping coefficient:

 

The frequency response of this circuit looks like:

 

 

Figure 9.3

 

Higher orders of low Pass Filter can be achieved by cascading cavities and tubes:

 

 

Figure 9.4

 

 

Figure 9.4a

 

Figure 9.5

Acoustic High Pass Filter

Consider the acoustic system shown in figure 9.6 below:

 

Figure 9.6

 

This arrangement assumes that the through path (or transmission duct) for the pressure wave has a diameter that is much greater than its length. Thus allowing us to neglect any Inertance and Resistance affect.

 

Unfortunately there is no equivalent mechanical circuit for this arrangement. See the notes at the end of this lecture.

 

Using the same reasoning as for the low pass filter the equivalent electrical circuit is:

 

 

Figure 9.7

 

Again this circuit is the familiar Second Order High Pass Filter.

 

The transfer function for this circuit expressed in Laplace notation assuming that at time t=0 all initial conditions are equal zero is quite simply:

 

 

Which takes the general form:

 

 

The frequency response of this circuit looks like:

 

 

Figure 9.8

 

 

 

Higher orders of High Pass Filter can be achieved by cascading cavities and tubes:

Figure 9.9

 

 

Figure 9.10

 

Acoustic Band Pass Filter

Consider the acoustic system shown in figure 9.11 below:

 

 

Figure 9.11

 

Unfortunately there is no equivalent mechanical circuit for this arrangement. See the notes at the end of this lecture.

 

A little bit of careful thinking is now required to put together the equivalent electrical circuit. Remember that the velocity through C_a is the difference between the velocity entering the system and the velocities leaving through the tube and open port.

 

 

Figure 9.12

 

 

The transfer function for this circuit is:

 

 

 

 

 

The frequency response of this circuit looks like:

 

 

Figure 9.13

 

Cascaded band pass filters all with equivalent filters can be used to provide either a very sharp tuned filter or it the filter values are different they can overlap and provide a much wider pass band.

 

 

 

Figure 9.14

 

 

Figure 9.15

 

Acoustic Band Stop Filter (Also known as a Notch Filter)

 

Consider the acoustic system shown in figure 9.16 below:

 

 

 

Figure 9.16

 

 

Unfortunately there is no equivalent mechanical circuit for this arrangement. See the notes at the end of this lecture.

 

The equivalent electrical circuit is:

 

 

Figure 9.17

 

The transfer function for this circuit is:

 

 

 

 

The frequency response of this circuit looks like:

 

 

Figure 9.18

 

Again we can cascade band stop filters to either produce a very sharp Notch  Response or to have a wider band stop response:

 

 

Figure 9.19

 

 

Figure 9.20

 

An explanation why some acoustic systems have no simple mechanical equivalent.

 

Many Acoustic systems have no equivalent Mechanical system.  The reason being that each end of a mechanical spring can only move with one degree of freedom and that is backwards and forwards.  Whereas a compliant cavity can distribute air into numerous ports and outlets directly connected to the cavity, the resulting velocity movement of air in each port or tube can be different depending on their dimensions. For an acoustic system the Volume velocity entering the system is equal to the sum of all the Volume velocities leaving the system.  The end of a mechanical spring on the other hand can only move the object or objects that are directly connected to it with the same velocity.

 

Multi ported acoustic systems share more in common with a real electrical system for the simple reason that electric ac current and air volume velocity can pass straight through an electrical Capacitor or Acoustic compliance.  For example in an electrical circuit the ac current entering a series capacitor will also leave the capacitor. The same can be said for a volume velocity entering a cavity completely open at both ends (i.e no Inertance or Resistance at either end) .  In the case of a mechanical system the only velocity that passes through the spring is the differential velocity between the two ends of the spring.

 

Equivalent systems

 

 

Figure 9.21

 

More examples of this subtlety will be found in future lectures.

 

There are a few rules that can be applied to help with deciding if a compliance is represented as a series or shunt element.

 

Rule 1.  Sound moving from a tube to a cavity. The compliance acts as a shunt.

Figure 9.22

 

Rule 2.  Sound moving through an open duct.  The compliance acts in series.

 

Figure 9.23

 

Rule 3.  Sound moving through an open duct to a tube. The compliance acts as a shunt.

 

Figure 9.24

 

Rule 4.  Sound entering a contained cavity.  The compliance acts as a shunt.

 

Figure 9.25

 

The message I want to get across is that you have to think very carefully when converting between domain models.  Not all models are realizable in all domains.

 

 

End of Lecture