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An active noise control scenario of simple ducts is considered. The previously suggested technique of using an single loudspeaker and its rear sound to cancel the upstream sound is further examined and compared to the bidirectional solution in order to give theoretical proof of its advantage. Firstly, a model with a new approach for taking damping effects into account is derived based on the electrical transmission line theory. By comparison with the old model, the new approach is validated, and occurring differences are discussed. Moreover, a numerical application with the consideration of damping is implemented for confirmation. The influence of the rear sound strength on the feedback-path system is investigated, and the optimal condition is determined. Finally, it is proven that the proposed source has an advantage of an extended phase lag and a time delay in the feedback-path system by both frequency-response analysis and numerical calculation of the time response.

In the recent past the concept of active noise control (ANC) has arrest more attention. Due to the falling prices for digital signal processors (DSPs), ANC applications become more popular everyday. The Swinbanks’ (unidirectional) source [

The intention of the Swinbanks’ source is cancelling of the upstream sound, which is emitted by the downstream loudspeaker. In order to accomplish this, the upstream source is driven with the same amplitude as the downstream loudspeaker but opposite phase and proper time delay. Thus, the upstream sound at the mount of the upstream source is eliminated. The advantage in the control performance can be explained by the extension of the time period by which the sound emitted by the control source travels to the reference microphone. It has been proven that the Swinbanks’ source introduces an additional time delay corresponding with twice the distance between control source and duct end [

In addition to the Swinbanks’ source, the rear-sound-aided source has been proposed to achieve similar advantage of Swinbanks’ source by using one loudspeaker [

Experimental apparatus used in [

However, the following problems remain for the proposed source.

The first principle model derived in [

No advantage of the proposed source has been shown theoretically: the additional time delay in the feedback-path transfer function has not been analyzed.

In this paper, we concentrate on the second problem, because the novel contribution of showing an advantage in a theoretical way is also necessary to solve for the first problem in the future.

In order to show the advantage of the proposed source, feasibility of the additional phase lag in the frequency response and the additional time delay in the time response are both investigated in this paper based on the frequency response analysis by the first principle model and the time-response simulation by the numerical calculation. In addition, the consistency between the frequency response analysis and the time response simulation will be shown to validate an assumption involved in the simulation model due to the junction of the proposed source, which is not necessary for more simpler structures of the conventional sources. To show consistency, the frequency response is numerically calculated by injecting a sinusoidal sound, where the damped wave equation is used to consider energy dissipation. Herewith, the steady-state response does not diverge for resonance frequencies.

This paper is outlined as follows. Firstly, the first principle model [

In this section, the damped wave equation is used to derive frequency response functions in order to obtain a finite amplitude response at resonance frequencies. The damped wave equation is the wave equation with an additional term of the first partial derivative of pressure in time (see, e.g., [

The damped wave equation can be derived by two physical laws, the conservation of mass (

In transmission line, suppose that

By substituting

Likewise, a similar lemma holds for the sound propagation in a straight duct.

Suppose that

Then,

It is trivial from Lemma

The next lemma is used to derive first principle model so that series connections of ducts are simplified in the derivation process, which is an extended result to the undamped case in [

Define a matrix

It can be verified by multiplication and application of the theorems for

The relationship of pressure

Theoretical apparatus.

There are three scenarios for the control source.

Case A—Bidirectional source: SPK3 is not used; that is,

Case B—Swinbanks’ source: SPK3 is used to cancel the upstream sound produced by SPK2 without subduct (

Case C—rear sound aided source (proposed one): The rear sound of SPK2 is redirected by a subduct and attached in a way that the rear sound will interfere with the front sound at the junction of ducts. Note here that the assumption is made, that the rear sound travelling in the subduct can be simulated by a subduct of length

In Figure

According to Lemma

Comparison of

Overview

Zoom on low frequencies

We here emphasize that the derived model in this paper is more relevant than that of [

The solution of the damped wave equation (

In this section, the numerical algorithm for Case A will first be derived. Based on this, the numerical algorithm for Case C will be explained.

The effect of a loudspeaker in duct can be considered in the wave equation by using a spacial delta function [

By using the relations above, frequency response of the feedback-path system

initial condition:

for

at the open end:

at the closed end: (

for

For the stability of the numerical algorithm above, parameters in Table

Numerical approximation settings.

Parameter | Notation | Value | Unit |
---|---|---|---|

Spatial step width | 0.01 | m | |

Simulation time | 1 | s | |

Time steps | 40000 |

As Case C consists of the main duct and an additional subduct, the previous derivation has to be adapted accordingly. First, similar to the main duct, an additional

Comparison of numerical and analytical solution, transfer function

Case A

Case C

Apparently, both solutions match almost perfect as it can be seen in above figures. This testifies the application of transmission line theory and numerical simulation via finite difference method with the applied assumptions. However, the desired additional phase lag of Case C compared to Case A did not occur. In both cases, the total phase lag is about 2520 degree. A possible cause for the missing phase lag could be a different setting of the rear-sound strength

In order to examine the effect of a different setting of the rear-sound strength

Comparison of transfer function

Evidently, a lower value of

Signal flow chart in frequency domain.

It can be observed that the shortest time delay is given by

Comparison of phase characteristics for Case A, Case A with additional time delay, and Case C (

The numerical implementation reveals the additional time delay in feedback-path system by displaying the propagation of the incident waves. In Figures

Numeric time response, Case A.

Numeric time response, Case C.

In this paper, as an additional approach in active noise control next to the Swinbanks' source, the proposed rear-sound-aided source has been analytically examined. The advantage of additional phase lag and time delay in the feedback-path transfer function of the proposed control source has been analyzed. To accomplish this, the damped wave equation has been used and based on this: a first principle model has been derived by the application of transmission line theory. The model has been confirmed by verifying that

the previous model is included as the special case of an undamped scenario,

the frequency response is consistent with the one obtained by a numerical simulation based on a finite difference implementation of the damped wave equation.

It has been shown, that the different consideration of damping induced some minor changes in low-frequency characteristics. Characteristics in middle and high-frequency range were consistent to the previous model, which implies validity of the previous results in [

The presented results have analytically proved that the proposed control source induces additional time delay in the feedback-path transfer function as compared to the conventional bidirectional source by a suitable choice of