The sound field at desired zone of quiet could be expressed as a linear superposition of eigenfunctions that satisfy the boundary condition of the sound field. If one wants to control the noise of desired quiet zone, then one must have sufficient number of sensors and control sources so that the active control system is observable and controllable. The number of sensors and control sources for controlling noise at desired quiet zone has to be incorporated with the number of eigenfunctions which one wants to control. However, when the number of eigenfunctions is relatively large, it is impractical to utilize a control system that consists of such a large number of sensors and control sources. These considerations motivated the present study that considers a control system which has less sensors and control sources than required.
In this work, a study on the control performance of the active noise control system that consists of smaller number of sensors and control sources than the number of eigenfunctions, which is necessary to represent the sound field, has been carried out. For this purpose, equations of residual acoustic potential energy at desired quiet zone were derived as a function of sound field variables, quiet zone variables and control system variables. The unknown primary sound fields are considered as the sound field variables, and the set of primary sound fields that can be produced at desired quiet zone is modeled by using uncertainty principle. As the quiet zone variables, the size and location of desired quiet zone are considered. As the control system variables, the number and location of control sources and error microphones are considered.
The control performances of the control system were theoretically investigated in terms of sound field variables, quiet zone variables and control system variables. The possible maximum and minimum value, mean and variance of residual acoustic potential energy are derived for the set of primary sound fields. Also, the condition for implementing the best-oriented control system that could maximize the control performance, by using given number of control sources and sensors, is obtained.
As an application example, active control system for the reduction of radiated duct noise is considered. The control system which has one error microphone and one control source for reducing the radiation power that consists of three propagating modes is considered for its simplicity. In the experiment, the best-oriented control system was installed in laboratory, and the control performance was investigated in terms of sound field variables. The experimental results demonstrate that it is possible to control the radiation power that consists of higher modes, by using appropriately placed one error microphone and one control source. The experimental results also demonstrate that the control performance of the best-oriented control system is better than other control systems which do not satisfy the condition for the best-oriented control system.
