# Acoustics/Active Control

## Introduction

The principle of active control of noise, is to create destructive interferences using a secondary source of noise. Thus, any noise can theoretically disappear. But as we will see in the following sections, only low frequencies noises can be reduced for usual applications, since the amount of secondary sources required increases very quickly with frequency. Moreover, predictable noises are much easier to control than unpredictable ones. The reduction can reach up to 20dB for the best cases. But since good reduction can only be reached for low frequencies, the perception we have of the resulting sound is not necessarily as good as the theoretical reduction. This is due to psychoacoustics considerations, which will be discussed later on.

## Fundamentals of active control of noise

### Control of a monopole by another monopole

Even for the free space propagation of an acoustic wave created by a punctual source it is difficult to reduce noise in a large area, using active noise control, as we will see in the section.

In the case of an acoustic wave created by a monopolar source, the Helmholtz equation becomes:

${\displaystyle \Delta p+k^{2}p=-j\omega \rho _{0}q}$

where q is the flow of the noise sources.

The solution for this equation at any M point is:

${\displaystyle p_{p}(M)={\frac {j\omega \rho _{0}q_{p}}{4\pi }}{\frac {e^{-jkr_{p}}}{r_{p}}}}$

where the p mark refers to the primary source.

Let us introduce a secondary source in order to perform active control of noise. The acoustic pressure at that same M point is now:

${\displaystyle {\rm {p(M)=}}{\frac {{\rm {j}}\omega \rho _{\rm {0}}q_{p}}{4\pi }}{\frac {e^{-jkr_{p}}}{r_{p}}}+{\frac {{\rm {j}}\omega \rho _{\rm {0}}q_{s}}{4\pi }}{\frac {e^{-jkr_{s}}}{r_{s}}}}$

It is now obvious that if we chose ${\displaystyle q_{s}=-q_{p}{\frac {r_{s}}{r_{p}}}e^{-jk(r_{p}-r_{s})}}$ there is no more noise at the M point. This is the most simple example of active control of noise. But it is also obvious that if the pressure is zero in M, there is no reason why it should also be zero at any other N point. This solution only allows to reduce noise in one very small area.

However, it is possible to reduce noise in a larger area far from the source, as we will see in this section. In fact the expression for acoustic pressure far from the primary source can be approximated by:

${\displaystyle p(M)={\frac {j\omega \rho _{0}}{4\pi }}{\frac {e^{-jkr_{p}}}{r_{p}}}(q_{p}+q_{s}e^{-jkD\cos \theta })}$
Control of a monopole by another monopole

As shown in the previous section we can adjust the secondary source in order to get no noise in M. In that case, the acoustic pressure in any other N point of the space remains low if the primary and secondary sources are close enough. More precisely, it is possible to have a pressure close to zero in the whole space if the M point is equally distant from the two sources and if: ${\displaystyle D<\lambda /6}$ where D is the distance between the primary and secondary sources. As we will see later on, it is easier to perform active control of noise with more than on source controlling the primary source, but it is of course much more expensive.

A commonly admitted estimation of the number of secondary sources which are necessary to reduce noise in an R radius sphere, at a frequency f is:

${\displaystyle N={\frac {36\pi R^{2}f^{2}}{c^{2}}}}$

This means that if you want no noise in a one meter diameter sphere at a frequency below 340Hz, you will need 30 secondary sources. This is the reason why active control of noise works better at low frequencies.

### Active control for waves propagation in ducts and enclosures

This section requires from the reader to know the basis of modal propagation theory, which will not be explained in this article.

#### Ducts

For an infinite and straight duct with a constant section, the pressure in areas without sources can be written as an infinite sum of propagation modes:

${\displaystyle p(x,y,z,\omega )=\sum \limits _{n=1}^{N}{a_{n}(\omega )\phi _{n}(x,y)e^{-jk_{n}z}}}$

where ${\displaystyle \phi }$ are the eigen functions of the Helmoltz equation and a represent the amplitudes of the modes.

The eigen functions can either be obtained analytically, for some specific shapes of the duct, or numerically. By putting pressure sensors in the duct and using the previous equation, we get a relation between the pressure matrix P (pressure for the various frequencies) and the A matrix of the amplitudes of the modes. Furthermore, for linear sources, there is a relation between the A matrix and the U matrix of the signal sent to the secondary sources: ${\displaystyle A_{s}=KU}$ and hence: ${\displaystyle A=A_{p}+A_{s}=A_{p}+KU}$.

Our purpose is to get: A=0, which means: ${\displaystyle A_{p}+KU=0}$. This is possible every time the rank of the K matrix is bigger than the number of the propagation modes in the duct.

Thus, it is theoretically possible to have no noise in the duct in a very large area not too close from the primary sources if the there are more secondary sources than propagation modes in the duct. Therefore, it is obvious that active noise control is more appropriate for low frequencies. In fact the more the frequency is low, the less propagation modes there will be in the duct. Experiences show that it is in fact possible to reduce the noise from over 60dB.

#### Enclosures

The principle is rather similar to the one described above, except the resonance phenomenon has a major influence on acoustic pressure in the cavity. In fact, every mode that is not resonant in the considered frequency range can be neglected. In a cavity or enclosure, the number of these modes rise very quickly as frequency rises, so once again, low frequencies are more appropriate. Above a critical frequency, the acoustic field can be considered as diffuse. In that case, active control of noise is still possible, but it is theoretically much more complicated to set up.

### Active control and psychoacoustics

As we have seen, it is possible to reduce noise with a finite number of secondary sources. Unfortunately, the perception of sound of our ears does not only depend on the acoustic pressure (or the decibels). In fact, it sometimes happen that even though the number of decibels has been reduced, the perception that we have is not really better than without active control.

## Active control systems

Since the noise that has to be reduced can never be predicted exactly, a system for active control of noise requires an auto adaptable algorithm. We have to consider two different ways of setting up the system for active control of noise depending on whether it is possible or not to detect the noise from the primary source before it reaches the secondary sources. If this is possible, a feed forward technique will be used (aircraft engine for example). If not a feed back technique will be preferred.

### Feedforward

In the case of a feed forward, two sensors and one secondary source are required. The sensors measure the sound pressure at the primary source (detector) and at the place we want noise to be reduced (control sensor). Furthermore, we should have an idea of what the noise from the primary source will become as he reaches the control sensor. Thus we approximately know what correction should be made, before the sound wave reaches the control sensor (forward). The control sensor will only correct an eventual or residual error. The feedforward technique allows to reduce one specific noise (aircraft engine for example) without reducing every other sound (conversations, …). The main issue for this technique is that the location of the primary source has to be known, and we have to be sure that this sound will be detected beforehand. Therefore portative systems based on feed forward are impossible since it would require having sensors all around the head.

Feedforward System

### Feedback

In that case, we do not exactly know where the sound comes from; hence there is only one sensor. The sensor and the secondary source are very close from each other and the correction is done in real time: as soon as the sensor gets the information the signal is treated by a filter which sends the corrected signal to the secondary source. The main issue with feedback is that every noise is reduced and it is even theoretically impossible to have a standard conversation.

Feedback System

## Applications

Usual headphones become useless when the frequency gets too low. As we have just seen active noise cancelling headphones require the feedback technique since the primary sources can be located all around the head. This active control of noise is not really efficient at high frequencies since it is limited by the Larsen effect. Noise can be reduced up to 30dB at a frequency range between 30Hz and 500Hz.

### Active control for cars

Noise reduction inside cars can have a significant impact on the comfort of the driver. There are three major sources of noise in a car: the motor, the contact of tires on the road, and the aerodynamic noise created by the air flow around the car. In this section, active control for each of those sources will be briefly discussed.

#### Motor noise

This noise is rather predictable since it a consequence of the rotation of the pistons in the motor. Its frequency is not exactly the motor’s rotational speed though. However, the frequency of this noise is in between 20Hz and 200Hz, which means that an active control is theoretically possible. The following pictures show the result of an active control, both for low and high regime.

Low regime

Even though these results show a significant reduction of the acoustic pressure, the perception inside the car is not really better with this active control system, mainly for psychoacoustics reasons which were mentioned above. Moreover such a system is rather expensive and thus are not used in commercial cars.

#### Tires noise

This noise is created by the contact between the tires and the road. It is a broadband noise which is rather unpredictable since the mechanisms are very complex. For example, the different types of roads can have a significant impact on the resulting noise. Furthermore, there is a cavity around the tires, which generate a resonance phenomenon. The first frequency is usually around 200Hz. Considering the multiple causes for that noise and its unpredictability, even low frequencies become hard to reduce. But since this noise is broadband, reducing low frequencies is not enough to reduce the overall noise. In fact an active control system would mainly be useful in the case of an unfortunate amplification of a specific mode.

#### Aerodynamic noise

This noise is a consequence of the interaction between the air flow around the car and the different appendixes such as the rear views for example. Once again, it is an unpredictable broadband noise, which makes it difficult to reduce with an active control system. However, this solution can become interesting in the case an annoying predictable resonance would appear.

### Active control for aeronautics

The noise of aircraft propellers is highly predictable since the frequency is quite exactly the rotational frequency multiplied by the number of blades. Usually this frequency is around some hundreds of Hz. Hence, an active control system using the feedforward technique provides very satisfying noise reductions. The main issues are the cost and the weigh of such a system. The fan noise on aircraft engines can be reduced in the same manner.