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Analysis and Optimal Condition of the Rear-Sound-Aided Control Source in Active Noise Control

Analysis and Optimal Condition of the Rear-Sound-Aided Control Source in Active Noise Control Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 697108, 9 pages doi:10.1155/2011/697108 Research Article Analysis and Optimal Condition of the Rear-Sound-Aided Control Source in Active Noise Control 1 2 Karel Kreuter and Yasuhide Kobayashi Department of Electrical Engineering, Technical University of Darmstadt, 64283 Darmstadt, Germany Department of Mechanical Engineering, Faculty of Engineering, Nagaoka University of Technology, Nagaoka, Niigata 940-2188, Japan Correspondence should be addressed to Yasuhide Kobayashi, kobayasi@vos.nagaokaut.ac.jp Received 14 March 2011; Revised 3 May 2011; Accepted 18 May 2011 Academic Editor: Arnaud Deraemaeker Copyright © 2011 K. Kreuter and Y. Kobayashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 1. Introduction the upstream sound at the mount of the upstream source is eliminated. The advantage in the control performance can be In the recent past the concept of active noise control (ANC) explained by the extension of the time period by which the has arrest more attention. Due to the falling prices for digital sound emitted by the control source travels to the reference signal processors (DSPs), ANC applications become more microphone. It has been proven that the Swinbanks’ source popular everyday. The Swinbanks’ (unidirectional) source introduces an additional time delay corresponding with [1] has frequently been applied to control noise scenarios to twice the distance between control source and duct end show its advantages over a standard bidirectional case with [5, 6]. Previous research pointed out on empirical basis that one control source. The implementation has shown to be of the closed-loop performance is improved when the actuator lower cost and at the same time better control performance. (control source) and the sensor (reference microphone) are This has been confirmed by experiments in both adaptive of further distance [7, 8]. and robust control setups [2, 3]. A welcome side effect was In addition to the Swinbanks’ source, the rear-sound- a lower controller gain and driving signals for the used aided source has been proposed to achieve similar advantage of Swinbanks’ source by using one loudspeaker [9]: the loudspeakers [3, 4], which enables the usage of lower power source is composed of a loudspeaker and a subduct which loudspeakers. The advantage of the Swinbanks’ source has connects the rear side of the loudspeaker to an upstream been theoretically proven for a practical setting on the duct junction in the main duct in order to attenuate the upstream length and loudspeaker locations [5, 6]. The intention of the Swinbanks’ source is cancelling of sound by interfering the rear sound as shown in Figure 1. the upstream sound, which is emitted by the downstream Experiments demonstrated that the proposed source has an loudspeaker. In order to accomplish this, the upstream advantage. It achieves a lower amplitude of error microphone signal by a smaller amplitude of control input over the source is driven with the same amplitude as the downstream loudspeaker but opposite phase and proper time delay. Thus, conventional bidirectional source. 2 Advances in Acoustics and Vibration 3cm 8cm 71 cm 158 cm 121 cm Error mic. SPK2 SPK3 Ref. mic. SPK1 88 cm Pow. Pre- LPF A/D LPF D/A AMP AMP u w Pow. Pow. LPF D/A LPF PC D/A AMP AMP Pre- LPF A/D AMP Figure 1: Experimental apparatus used in [9]. However, the following problems remain for the pro- in numerical simulations to directly visualize the additional posed source. time delay. (i) The first principle model derived in [9]has notbeen 2. First Principle Model well consistent with the experimental result of the frequency response. In this section, the damped wave equation is used to derive frequency response functions in order to obtain a finite (ii) No advantage of the proposed source has been amplitude response at resonance frequencies. The damped shown theoretically: the additional time delay in wave equation is the wave equation with an additional the feedback-path transfer function has not been term of the first partial derivative of pressure in time (see, analyzed. e.g., [10]). The differences to the existing result of [9] In this paper, we concentrate on the second problem, will be explained, since the existing result is based on the because the novel contribution of showing an advantage in a (undamped) wave equation. Note that the derivation process theoretical way is also necessary to solve for the first problem of the damped wave equation will be described in this in the future. section, since the intermediate equations ((1)and (2)below) In order to show the advantage of the proposed source, are necessary to exploit the existing result of the transmission feasibility of the additional phase lag in the frequency line theory. response and the additional time delay in the time response The damped wave equation can be derived by two are both investigated in this paper based on the frequency physical laws, the conservation of mass (1) and momentum response analysis by the first principle model and the (2). Theacousticvelocityisgiven by u, p represents the time-response simulation by the numerical calculation. In pressure. Note that an additional term du is included in addition, the consistency between the frequency response the conservation of momentum to take friction effects into analysis and the time response simulation will be shown to account, where d is a damping coefficient, validate an assumption involved in the simulation model 1 ∂p ∂u due to the junction of the proposed source, which is not + ρ = 0 ,(1) c ∂t ∂x necessary for more simpler structures of the conventional sources. To show consistency, the frequency response is ∂u ∂p numerically calculated by injecting a sinusoidal sound, where (2) ρ + + du = 0. ∂t ∂x the damped wave equation is used to consider energy dissipation. Herewith, the steady-state response does not If the damping is neglected, the two equations are well diverge for resonance frequencies. known in acoustic literature (see, e.g., [11]) and are used to This paper is outlined as follows. Firstly, the first princi- derive the wave equation without consideration of damping. ple model [9] is derived based on the damped wave equation However, to the best of our knowledge, (2)with d =0has which later enables comparison of the frequency response not been used to derive a physical model in the field of of the first principle model with the numerical simulation. active noise control. Usually, the damped wave equation In Section 3, the damped wave equation will be directly (3) is solved without considering damping. The resulting implemented in a numerical simulation by using finite undamped solution is used to approximate damping effect differences. We will demonstrate that the frequency response by modifying the wave number k := ω/c (real number) to calculated by numerical implementation is consistent with k := (ω/c )+ jα (complex number) [12], which technique the one given by the first principle model. Thus, the model has also been used in [9]. The damped wave equation can be will be available to examine both the additional phase lag and obtained by the derivation of (1)withrespect to t and (2) time delay in the feedback-path transfer function. Finally, the with respect to x and combining both optimal condition achieving the longest time delay on the 2 2 ∂ p ∂p ∂ p feedback-path transfer function will be given based on the 1 d + = . (3) 2 2 2 2 first principle model in Section 4. The condition will be used c ∂t ρ c ∂t ∂x 0 0 0 Advances in Acoustics and Vibration 3 On the other hand, more general partial differential equa- Likewise, a similar lemma holds for the sound propaga- tions and the solution with damping consideration are tion in a straight duct. common in the transmission line theory. It is well known for Lemma 2. Suppose that p(x, t) and u(x, t) are in steady state instantaneous voltage v(x, t)and current i(x, t)atalocation x with the relation given as on a lossy transmission line that the following relations hold: jωt jωt ∂v ∂i ∂i ∂v p(x, t) := Re p(x)e , u(x, t) := Re u(x)e . (4) =−Ri − Li , =−Gv − C , ∂x ∂t ∂x ∂t (11) where R, L, C,and G are resistance, inductance, capacitance, Then, p(x) and u(x) at arbitrary position x is related with and conductance per unit length of the line. One can verify the values at origin, p(0) and u(0),by that (4)equals(1)and (2) when using the following relations: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ cosh γx −Z sinh γx p(x) p(0) v(x, t) = p(x, t), i(x, t) = u(x, t), R := d, ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ , u(x) − sinh γx cosh γx u(0) (5) L := ρ , G := 0, C := . ρ c (12) 0 0 For harmonic signals Z := ρ c 1+ , γ := α + jβ, (13) 0 0 0 jωρ jωt jωt v(x, t) := Re V (x)e , i(x, t) := Re I(x)e ,(6) 1  d the partial differential equation of transmission line theory  2 2 α := ω ω + − ω , 2c ρ 0 0 can be rewritten as (14) dV dI =−ZI, =−YV , (7) 1 d dx dx 2 2 β := ω ω + + ω , 2c ρ where Z := R + jωL and Y := G + jωC are impedance and admittance per unit length. The general solution of (7)is where Z is called the acoustic characteristic impedance of the known as duct. −γx γx −γx γx (8) Proof. It is trivial from Lemma 1 and the relations (5). V = Ae + Be , I = (Ae + Be ), The next lemma is used to derive first principle model where A and B are complex-valued integral constants. γ is so that series connections of ducts are simplified in the given by derivation process, which is an extended result to the undamped case in [9]. γ := ZY := α + jβ, Lemma 3. Define a matrix T(l) with arbitrary complex numbers γ and Z by 2 2 2 2 2 2 2 α := R + ω L G + ω C − (ω LC − RG), (9) ⎡ ⎤ cosh γl −Z sinh γl ⎢ ⎥ 2 2 2 2 2 2 2 T(l) := ⎣ ⎦ . (15) ( ) 1 β := R + ω L G + ω C + ω LC − RG . − sinh γl cosh γl Lemma 1. In transmission line, suppose that v(x, t) and i(x, t) Then, for any positive real number x and y, the following holds: are in steady state with the relations given in (6).Then, V (x) and I(x) at arbitrary position x is related with the values at T(x)T y = T x + y . (16) origin, V (0) and I(0),by Proof. It can be verified by multiplication and application of ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ cosh γx −Z sinh γx the theorems for sinh and cosh. V (x) V (0) ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ , (10) I(x) I(0) − sinh γx cosh γx The relationship of pressure p and acoustic velocity u between different points in space will serve as the basis for deriving a unifying transfer function which includes where Z := Z/Y is the characteristic impedance of the the proposed control source. We consider the duct system transmission line. in Figure 2 which is the same as in [9], where SPK1 is a Proof. By substituting x = 0 into (8), A and B are written in noise source, SPK2 and SPK3 are used as a control source, V (0) and I(0). Then, the elimination of the constants A and controller is connected between y and u to drive the control B leads to (10). source based on the information of the reference microphone 4 Advances in Acoustics and Vibration Ll l l l 0 z u v y (p , u ) (p , u)(p , u)(p , u )(p , u ) (p , u)(p , u )(p , u ) 5 5 z z 4 4 3 3 2 2 1 1 y y 0 0 (p , u ) (p , u ) s S S 6 6 Error mic. Ref. mic. SPK2 SPK1 (p˜ , u˜ ) S S uy w z SPK3 Figure 2: Theoretical apparatus. 70 35 40 25 −720 −1440 −2160 −2880 −45 −3600 −4320 −90 1 2 3 10 10 10 10 Frequency (Hz) Frequency (Hz) Present model Present model Modelof[9] Modelof[9] (a) Overview (b) Zoom on low frequencies Figure 3: Comparison of G (Case A) with [9]. zw output y, and the error microphone output z is used to (iii) Case C—rear sound aided source (proposed one): evaluate control performance. Suppose that SPK1, Ref.Mic., The rear sound of SPK2 is redirected by a subduct and SPK3, SPK2, and Err.Mic. are located at x = 0, l , l , l ,and attached in a way that the rear sound will interfere y v u l , respectively, in the main duct of length L. with the front sound at the junction of ducts. Note There are three scenarios for the control source. here that the assumption is made, that the rear sound travelling in the subduct can be simulated by a subduct of length L = l − l with an additional S u v (i) Case A—Bidirectional source: SPK3 is not used; that loudspeaker (SPK3) to cover the rear sound. The is, v(t) = 0or H = 0 and no subduct installed (L = amplitude of SPK3 can be adjusted by the variable H; 0). This can be described as bidirectional case since that is, u(t) = H · v(t), where H is treated as negative SPK2 emits waves in both directions. real number, whose magnitude is less than or equal (ii) Case B—Swinbanks’ source: SPK3 is used to cancel to 1, in this paper. (H is set to −1 when the strength of the rear sound is as the same to the front one.) the upstream sound produced by SPK2 without subduct (L = 0). It is, therefore, driven with the The realizability of this setting is under consideration, opposite signal, but delayed v(t) =−u(t−(l −l )/c ). however, one possibility of this is to cover the inside u v 0 wall of the subduct by glass wool to weaken the rear This can be represented by the transfer function s((l −l )/c ) u v 0 H(s) =−e . sound. Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Advances in Acoustics and Vibration 5 In Figure 2,(p , u ) represents the pressure and the According to Lemma 2, the derivation process of the • • particle velocity at each position in the duct. For the first principle model is eventually similar to the undamped modelling process, the input signals w, u,and v are assumed case [9], where the derivation of the transfer function has to be u , u and u , respectively, neglecting the loudspeaker been explained in detail. Hence, only the resultant transfer 0 S S dynamics. The outputs z and y are assumed to be p and functions are shown as follows: p , respectively, neglecting the microphone dynamics. Thus, u := w, u := u, u := v, p := z, p := y. 0 S S z y Z sinh γ(L − l ) cosh γL + sinh γ(L − l ) · H 0 u s v G = , 0u sinh γ(L − l ) sinh γL cosh γl +cosh γL cosh γL v s v s G = G · cosh γl , yu 0u y sinh γ(l − l ) z v G =−Z sinh γ(l − l ) + · H zu 0 z u cosh γL + G sinh γ(l − l ) tanh γL cosh γl +cosh γl , 0u z v s v z (17) ( ) Z sinh γ L − l tanh γL sinh γl + sinh γL 0 v s v G = , 0w sinh γ(L − l ) tanh γL cosh γl +cosh γL v s v G = cosh γl G − Z sinh γl , yw y 0w 0 y G = sinh γ(l − l ) tanh γL cosh γl +cosh γl G zw z v s v z 0w − Z sinh γ(l − l ) tanh γl sinh γl + sinh γl . 0 z v s v z It is obvious that for undamped case the derived functions overall bode characteristics did not change, some minor and the ones given by [9] ought to be the same. This can differences appear due to the different approach of taking be confirmed by setting the damping coefficient d = 0and damping into account in low frequencies as it is shown in using relationships between hyperbolic and trigonometric Figure 3(b). functions. Indeed, consistency has been confirmed for the We here emphasize that the derived model in this paper undamped case. The major difference for the damped case is is more relevant than that of [9], since the frequency that the damping and acoustic impedance Z considered in response of the model can be directly compared with the one [9] are constant for all frequencies, while those in the derived calculated by numerical implementation in the next section. functions above are frequency dependent: the imaginary part of the complex wave number in [9], α of k = (ω/c ) − 3. Numerical Implementation jα, does not depend on the angular frequency ω, while the one in the corresponding parameter −jγ = β − jα The solution of the damped wave equation (3) using the depends on ω. Specifically, the difference mainly arises in low transmission line theory shall be confirmed by comparing frequency range, since α given by (14) might be regarded as to an alternative approach, that is, numerical simulation by a constant number in high-frequency range, whose limit is implementing a finite difference method, in which forward given by lim α = d/(2ρ c ). Figure 3 shows the frequency ω→∞ 0 0 and backward difference are used to calculate the frequency response of Case A (only G is shown in the figure due zw response of the system. to the limitation of the space) for the both considerations In this section, the numerical algorithm for Case A will of damping, where the same parameters in [9]are used as first be derived. Based on this, the numerical algorithm for follows: Case C will be explained. L = 3.61, l = 3.53, l = 2.32, l = 0.03, z u y 3.1. Case A. The effect of a loudspeaker in duct can be c = 340, ρ = 1.21. 0 0 considered in the wave equation by using a spacial delta (18) function [8]. Thus, the damped wave equation with a loudspeaker can be described as follows: The damping coefficient is set as d = 72.6 which is obtained from the relationship d/(2ρ c ) = 30/340 ≈ 0.09 (0.09 is 0 0 2 2 ∂ p ∂p ∂ p givenin[9]) so that both considerations of damping result 1 d + = + ρ u˙δ(x − l ). (19) 0 u 2 2 2 2 similar damping effect in high-frequency range. Although c ∂t ρ c ∂t ∂x 0 0 0 6 Advances in Acoustics and Vibration For numerical analysis, time t and space x are discretized as Table 1: Numerical approximation settings. follows: Parameter Notation Value Unit Spatial step width Δx 0.01 m t = t := i · Δt, ∀i ∈{0, 1,... , N}, Simulation time T 1s x = x := j · Δx, ∀j ∈{0, 1,... , M}, j Time steps N 40000 (20) T L l N := , M := , j := , Δt Δx Δx holds with a pressure at a fictitious location x =−Δx =: x . −1 This relation can be used in (23)for j = 0 leading to where T is a simulation period. After applying forward and backward difference on the damped wave equation (19), one obtains p(t , x ) = p(t , x ) · [2 − s − r] + p(t , x ) · [s − 1] i+1 0 i 0 i−1 0 + p(t , x ) · [r]. i 1 p t , x − 2p t , x + p t , x i+1 j i j i−1 j (26) Δt p t , x − p t , x i j i−1 j d By using the relations above, frequency response of the feedback-path system G can be numerically calculated by yu ρ Δt injecting the source input as a sinusoidal signal u(t ) = −U cos ωt : p t , x − 2p t , x + p t , x i i j+1 i j i j−1 δ jj 2 2 = c + c ρ u˙ , 0 0 Δx Δx (1) initial condition: p(t , x ) = p(t , x ) = 0for all j, (21) 0 j 1 j (2) for i = 1, 2,... , N − 1 repeat the following: where δ denotes the Kronecker delta to approximate the jj original property of integration being 1 by summation as follows: (a) at the open end: p(t , x ) = 0, i 0 M−1 L ⎨ 1, i = j , (b) at the closed end: (26), jj 1 = δ(x − l )dx ≈ Δx, δ = u ij Δx (c) for j = 1, 2,... , M − 1repeat(23). (only 0 i= j . j=0 at the source position j = j , the last term (22) rΔxρ u˙ (t )δ appears as rΔxρ U sin t ). 0 i jj 0 i Rearranging the recurrence relation for future time value and introducing auxiliary variables leads to For the stability of the numerical algorithm above, parameters in Table 1 are set. The steady state pressure p t , x = p t , x · [2 − s − 2r] + p t , x · [s − 1] i+1 j i j i−1 j variation at the microphone position p(t , x )isevaluated i j to get gain and phase from the source input u to the [ ] [ ] + p t , x · r + p t , x · r i j+1 i j−1 microphone output y,where j is defined as j := l /Δx. y y y The other frequency responses for G , G ,and G can be zu zw yw + rΔxρ u˙δ , 0 jj calculated in a similar way. For each transfer function 2000 (23) logarithmically distributed frequencies have been simulated. 3.2. Case C. As Case C consists of the main duct and an d c · Δt additional subduct, the previous derivation has to be adapted s = Δt · , r = . (24) ρ Δx accordingly. First, similar to the main duct, an additional z axis is introduced so that SPK3 is located at z = 0 and the Since the duct has two ends, boundary conditions have to be subduct is connected at z = L as shown in Figure 2,by set accordingly. As it has been done in the analytical process, which the instantaneous pressure in the subduct is denoted the pressure at the open end is set to zero; that is, p(t , x ) = i M as p (t, z). Then, t and z are discretized in a similar manner 0for all i. For the closed end at x = 0, the pressure gradient as Case A has to be zero. Thus, the relation (27) ∂p(t, x) z = z := k · Δx ∀k ∈{0, 1,... , M }, M := , k S S Δx ∂x x=0 where common spacing Δx is used. The input for SPK3 is p(t , x ) − p(t , x ) i 0 i −1 ≈ = 0 ⇐⇒ p(t , x ) = p(t , x ) i −1 i 0 specified as v(t ) = Hu(t ). The boundary condition for i i Δx the closed end of the attached subduct is added like before. (25) Advances in Acoustics and Vibration 7 360 360 0 0 −360 −360 −720 −720 −1080 −1080 −1440 −1440 −1800 −1800 −2160 −2160 −2520 −2520 −2880 −2880 1 2 3 1 2 3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Analytical Analytical Numerical Numerical (a) Case A (b) Case C Figure 4: Comparison of numerical and analytical solution, transfer function G , H =−1. yu Furthermore, the pressure at the junction is assumed as the 70 average of all surrounding pressure points 60 p t , x i+1 j = p t , z S i+1 M p t , x + p t , x + p t , z i+1 j −1 i+1 j +1 S i+1 M −1 v v S = , (28) −720 −1440 where j := l /Δx. The relation above might now be only −2160 v v −2880 an assumption; however, the validity will be shown later. −3600 The numerical algorithm in detail is omitted from the paper, −4320 since it is parallel to Case A based on the additional setting 1 2 3 10 10 10 for Case C. So far, two solutions of the frequency response Frequency (Hz) based on the damped wave equation (3) have been obtained. H =−1 H =−0.8 In the remaining of this section, the results will be compared. H =−0.95 H =−0.5 Figure 4 shows frequency response of the feedback-path system for Cases A and C. Figure 5: Comparison of transfer function G of Case C for yu Apparently, both solutions match almost perfect as it different rear-sound strengths H. 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 compared to itself with several settings for H,asitcan be seen A did not occur. In both cases, the total phase lag is about in Figure 5. 2520 degree. A possible cause for the missing phase lag could Evidently, a lower value of H will lead to the desired be a different setting of the rear-sound strength H,which effect and, thus, shows an advantage over the bidirectional leads to different bode diagrams for G as pointed in [9]. yu scenario. The reason for that behaviour can be found in the transfer function for G ,sinceitispartof G .For 0u yu 4. Optimal Condition for the Longest Time the explanation, the undamped equation will be used as it simplifies the next steps and did not show much influence on Delay in Feedback-Path System phase characteristics. The derivation will be shown for the In order to examine the effect of a different setting of the transfer function of Case A, the same method can later be rear-sound strength H, the analytic solution is modified and applied to Case C. Firstly, G will be simplified for Case A by 0u Phase (deg) Magnitude (dB) Phase (deg) Phase (deg) Magnitude (dB) Magnitude (dB) 8 Advances in Acoustics and Vibration −s(l /c ) u 0 −− −10 −s(2L−l /c ) −s(2L/c ) u 0 0 e e −20 Figure 6: Signal flow chart in frequency domain. −30 −40 −720 setting L = 0, H = 0 and rewriting as exponential functions. −1440 After making the equation causal by multiplying with the −2160 −jkL −2880 longest time delay e to both denominator and numerator, −3600 and replacing jk by s/c , the following is obtained: −4320 −5040 −5760 1 2 3 −s(l /c ) −s(2L−l )/c u 0 u 0 10 10 10 e − e . (29) G = ρ c 0u 0 0 Frequency (Hz) −s(2L/c ) 1+ e Case (A) Case (C) ) H =−0.5 This transfer function corresponds with a causal system Case (A) modified depicted in Figure 6, where each signal block represents a time delay. Figure 7: Comparison of phase characteristics for Case A, Case A It can be observed that the shortest time delay is given by with additional time delay, and Case C (H =−0.5). −s(l /c ) u 0 e . As mentioned in the introduction, the value of the shortest time delay is critical for the control performance. Similar for Case C, the same procedure is applied and some variables for better overview are introduced. Since the denominator is not of relevance and long, it has been neglected here and the modified transfer function G can 0.012 0u be derived as 0.01 −s(3l −2l )/c −s(2L−l )/c u v 0 u 0 G = 4ρ c e − e 0.008 0u 0 0 0.006 −s(l /c ) −s(2L−2l +l )/c u 0 v u 0 +[(1/2) + H] e − e . (30) 0.004 0.002 3.5 From this equation, it becomes clear that half rear-sound 2.5 1.5 0 1 amplitude will eliminate the shortest time delay given by 0.5 −s(l /c ) −s(3l −2l )/c u 0 u v 0 e and leads to e as the shortest time delay. Figure 8: Numerictimeresponse, Case A. Compared to the previous result for Case A (29), an −s(2l −2l )/c u v 0 additional time delay of e could be achieved. This corresponds to twice the length of the subduct and can be seen in Figure 5. The time delay can also be verified by multiplying the transfer function G of Case A by an yu −40 additional, fictitious time delay. In Figure 7,itcan be seen 0.012 that this modified Case A and Case C with a rear-sound strength of −0.5 produce indeed a very similar phase lag. 0.01 The numerical implementation reveals the additional time delay in feedback-path system by displaying the prop- 0.008 agation of the incident waves. In Figures 8 and 9, the wave 0.006 propagation for Case A and Case C is shown. The y-axis depicts the acoustic pressure distribution for different points 0.004 in time, the x-axis represents the position in the main duct, where the closed end is located at x = 0. One complete 0.002 3.5 period with a frequency of 1000 Hz is excited by the control 2.5 1.5 loudspeaker at l = 2.32, and the propagation through the 1 0.5 duct is observed. Note that the first upwards travelling wave is extinguished by the rear sound in Case C and previously Figure 9: Numerictimeresponse, Case C. derived time delay is obtained. Phase (deg) p(x, t) Magnitude (dB) p(x, t) Advances in Acoustics and Vibration 9 5. Conclusion sound sources,” Transactions of the Japan Society of Mechanical Engineers. C, vol. 67, no. 656, pp. 52–57, 2001 (Japanese). In this paper, as an additional approach in active noise [3] Y. Kobayashi and H. Fujioka, “Active noise cancellation for control next to the Swinbanks’ source, the proposed rear- ventilation ducts using a pair of loudspeakers by sampled-data sound-aided source has been analytically examined. The H∞ optimization,” Advances in Acoustics and Vibration, vol. advantage of additional phase lag and time delay in the 2008, Article ID 253948, 8 pages, 2008. feedback-path transfer function of the proposed control [4] J. Winkler and S. J. Elliott, “Adaptive control of broadband source has been analyzed. To accomplish this, the damped sound in ducts using a pair of loudspeakers,” Acustica, vol. 81, no. 5, pp. 475–488, 1995. wave equation has been used and based on this: a first [5] Y. Kobayashi and H. Fujioka, “Analysis for robust active principle model has been derived by the application of noise control systems of ducts with a pair of loudspeakers,” transmission line theory. The model has been confirmed by in Proceedings of the International Congress on Sound and verifying that Vibration (ICSV ’08), Daejeon, Korea, 2008. (i) the previous model is included as the special case of [6] Y. Kobayashi and H. Fujioka, “Robust stability analysis for an undamped scenario, active noise control systems of ducts with a pair of loudspeak- ers,” in Proceedings of the 37th Symposium on Control Theory (ii) the frequency response is consistent with the one (SICE ’08), pp. 21–247, 2008. obtained by a numerical simulation based on a [7] D. S. Bernstein, “What makes some control problems hard?” finite difference implementation of the damped wave IEEE Control Systems Magazine, vol. 22, no. 4, pp. 8–19, 2002. equation. [8] J. Hong and D. S. Bernstein, “Bode integral constraints, It has been shown, that the different consideration of colocation, and spillover in active noise and vibration control,” damping induced some minor changes in low-frequency IEEE Transactions on Control Systems Technology, vol. 6, no. 1, pp. 111–120, 1998. characteristics. Characteristics in middle and high-frequency range were consistent to the previous model, which implies [9] Y. Kobayashi, H. Fujioka, and N. Jinbo, “A control source structure of single loudspeaker and rear sound interference validity of the previous results in [9]: The proposed control for inexpensive active noise control,” Advances in Acoustics and source has shown additional phase lag by assuming a weaker Vibration, vol. 2010, Article ID 730813, 9 pages, 2010. rear sound. The advantage has also been examined from [10] W. Barten, A. Manera, and R. Macian-Juan, “One- and two- a theoretical point of view in order to determine under dimensional standing pressure waves and one-dimensional which conditions an additional time delay is achieved. It travelling pulses using the US-NRC nuclear systems analysis has been demonstrated that the shortest time delay in code TRACE,” Nuclear Engineering and Design, vol. 238, no. the feedback-path transfer function is extended by 2L /c , S 0 10, pp. 2568–2582, 2008. when H =−0.5, where L resembles the length of the [11] P. A. Nelson andS.J.Elliott, Active Control of Sound,Academic subduct and c the speed of sound. This time delay has been Press, New York, NY, USA, 1993. also confirmed by comparison with the phase characteristic [12] S. J. Elliott, Signal Processing for Active Control, Volume in the for the bidirectional source with additional time delay. In Signal Processing and Its Applications Series, Academic Press, addition, the time delay has been directly confirmed in the New York, NY, USA, 2009. time response simulation of the pressure variation by an incident wave. 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 H. This ends up showing that the original setting of H =−1 is not suitable for the proposed source to maximize the additional time delay, which might be inconsistent with the result in [9], since the similar phase characteristic of Swinbanks’ source has been achieved by H =−1in the experiment. To tackle this problem, the time response simulation in this paper will be used for comparison in detail with a measured time response by experiments. Further effect of the rear-sound amplitude on control performance will be examined. References [1] M. A. Swinbanks, “The active control of sound propagating in long ducts,” Journal of Sound and Vibration, vol. 27, pp. 411– 436, 1973. [2] S. Kijimoto, H. Tanaka, Y. Kanemitsu, and K. Matsuda, “Howling cancellation for active noise control with two International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2010 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Acoustics and Vibration Hindawi Publishing Corporation

Analysis and Optimal Condition of the Rear-Sound-Aided Control Source in Active Noise Control

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Copyright © 2011 Karel Kreuter and Yasuhide Kobayashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 697108, 9 pages doi:10.1155/2011/697108 Research Article Analysis and Optimal Condition of the Rear-Sound-Aided Control Source in Active Noise Control 1 2 Karel Kreuter and Yasuhide Kobayashi Department of Electrical Engineering, Technical University of Darmstadt, 64283 Darmstadt, Germany Department of Mechanical Engineering, Faculty of Engineering, Nagaoka University of Technology, Nagaoka, Niigata 940-2188, Japan Correspondence should be addressed to Yasuhide Kobayashi, kobayasi@vos.nagaokaut.ac.jp Received 14 March 2011; Revised 3 May 2011; Accepted 18 May 2011 Academic Editor: Arnaud Deraemaeker Copyright © 2011 K. Kreuter and Y. Kobayashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 1. Introduction the upstream sound at the mount of the upstream source is eliminated. The advantage in the control performance can be In the recent past the concept of active noise control (ANC) explained by the extension of the time period by which the has arrest more attention. Due to the falling prices for digital sound emitted by the control source travels to the reference signal processors (DSPs), ANC applications become more microphone. It has been proven that the Swinbanks’ source popular everyday. The Swinbanks’ (unidirectional) source introduces an additional time delay corresponding with [1] has frequently been applied to control noise scenarios to twice the distance between control source and duct end show its advantages over a standard bidirectional case with [5, 6]. Previous research pointed out on empirical basis that one control source. The implementation has shown to be of the closed-loop performance is improved when the actuator lower cost and at the same time better control performance. (control source) and the sensor (reference microphone) are This has been confirmed by experiments in both adaptive of further distance [7, 8]. and robust control setups [2, 3]. A welcome side effect was In addition to the Swinbanks’ source, the rear-sound- a lower controller gain and driving signals for the used aided source has been proposed to achieve similar advantage of Swinbanks’ source by using one loudspeaker [9]: the loudspeakers [3, 4], which enables the usage of lower power source is composed of a loudspeaker and a subduct which loudspeakers. The advantage of the Swinbanks’ source has connects the rear side of the loudspeaker to an upstream been theoretically proven for a practical setting on the duct junction in the main duct in order to attenuate the upstream length and loudspeaker locations [5, 6]. The intention of the Swinbanks’ source is cancelling of sound by interfering the rear sound as shown in Figure 1. the upstream sound, which is emitted by the downstream Experiments demonstrated that the proposed source has an loudspeaker. In order to accomplish this, the upstream advantage. It achieves a lower amplitude of error microphone signal by a smaller amplitude of control input over the source is driven with the same amplitude as the downstream loudspeaker but opposite phase and proper time delay. Thus, conventional bidirectional source. 2 Advances in Acoustics and Vibration 3cm 8cm 71 cm 158 cm 121 cm Error mic. SPK2 SPK3 Ref. mic. SPK1 88 cm Pow. Pre- LPF A/D LPF D/A AMP AMP u w Pow. Pow. LPF D/A LPF PC D/A AMP AMP Pre- LPF A/D AMP Figure 1: Experimental apparatus used in [9]. However, the following problems remain for the pro- in numerical simulations to directly visualize the additional posed source. time delay. (i) The first principle model derived in [9]has notbeen 2. First Principle Model well consistent with the experimental result of the frequency response. In this section, the damped wave equation is used to derive frequency response functions in order to obtain a finite (ii) No advantage of the proposed source has been amplitude response at resonance frequencies. The damped shown theoretically: the additional time delay in wave equation is the wave equation with an additional the feedback-path transfer function has not been term of the first partial derivative of pressure in time (see, analyzed. e.g., [10]). The differences to the existing result of [9] In this paper, we concentrate on the second problem, will be explained, since the existing result is based on the because the novel contribution of showing an advantage in a (undamped) wave equation. Note that the derivation process theoretical way is also necessary to solve for the first problem of the damped wave equation will be described in this in the future. section, since the intermediate equations ((1)and (2)below) In order to show the advantage of the proposed source, are necessary to exploit the existing result of the transmission feasibility of the additional phase lag in the frequency line theory. response and the additional time delay in the time response The damped wave equation can be derived by two are both investigated in this paper based on the frequency physical laws, the conservation of mass (1) and momentum response analysis by the first principle model and the (2). Theacousticvelocityisgiven by u, p represents the time-response simulation by the numerical calculation. In pressure. Note that an additional term du is included in addition, the consistency between the frequency response the conservation of momentum to take friction effects into analysis and the time response simulation will be shown to account, where d is a damping coefficient, validate an assumption involved in the simulation model 1 ∂p ∂u due to the junction of the proposed source, which is not + ρ = 0 ,(1) c ∂t ∂x necessary for more simpler structures of the conventional sources. To show consistency, the frequency response is ∂u ∂p numerically calculated by injecting a sinusoidal sound, where (2) ρ + + du = 0. ∂t ∂x the damped wave equation is used to consider energy dissipation. Herewith, the steady-state response does not If the damping is neglected, the two equations are well diverge for resonance frequencies. known in acoustic literature (see, e.g., [11]) and are used to This paper is outlined as follows. Firstly, the first princi- derive the wave equation without consideration of damping. ple model [9] is derived based on the damped wave equation However, to the best of our knowledge, (2)with d =0has which later enables comparison of the frequency response not been used to derive a physical model in the field of of the first principle model with the numerical simulation. active noise control. Usually, the damped wave equation In Section 3, the damped wave equation will be directly (3) is solved without considering damping. The resulting implemented in a numerical simulation by using finite undamped solution is used to approximate damping effect differences. We will demonstrate that the frequency response by modifying the wave number k := ω/c (real number) to calculated by numerical implementation is consistent with k := (ω/c )+ jα (complex number) [12], which technique the one given by the first principle model. Thus, the model has also been used in [9]. The damped wave equation can be will be available to examine both the additional phase lag and obtained by the derivation of (1)withrespect to t and (2) time delay in the feedback-path transfer function. Finally, the with respect to x and combining both optimal condition achieving the longest time delay on the 2 2 ∂ p ∂p ∂ p feedback-path transfer function will be given based on the 1 d + = . (3) 2 2 2 2 first principle model in Section 4. The condition will be used c ∂t ρ c ∂t ∂x 0 0 0 Advances in Acoustics and Vibration 3 On the other hand, more general partial differential equa- Likewise, a similar lemma holds for the sound propaga- tions and the solution with damping consideration are tion in a straight duct. common in the transmission line theory. It is well known for Lemma 2. Suppose that p(x, t) and u(x, t) are in steady state instantaneous voltage v(x, t)and current i(x, t)atalocation x with the relation given as on a lossy transmission line that the following relations hold: jωt jωt ∂v ∂i ∂i ∂v p(x, t) := Re p(x)e , u(x, t) := Re u(x)e . (4) =−Ri − Li , =−Gv − C , ∂x ∂t ∂x ∂t (11) where R, L, C,and G are resistance, inductance, capacitance, Then, p(x) and u(x) at arbitrary position x is related with and conductance per unit length of the line. One can verify the values at origin, p(0) and u(0),by that (4)equals(1)and (2) when using the following relations: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ cosh γx −Z sinh γx p(x) p(0) v(x, t) = p(x, t), i(x, t) = u(x, t), R := d, ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ , u(x) − sinh γx cosh γx u(0) (5) L := ρ , G := 0, C := . ρ c (12) 0 0 For harmonic signals Z := ρ c 1+ , γ := α + jβ, (13) 0 0 0 jωρ jωt jωt v(x, t) := Re V (x)e , i(x, t) := Re I(x)e ,(6) 1  d the partial differential equation of transmission line theory  2 2 α := ω ω + − ω , 2c ρ 0 0 can be rewritten as (14) dV dI =−ZI, =−YV , (7) 1 d dx dx 2 2 β := ω ω + + ω , 2c ρ where Z := R + jωL and Y := G + jωC are impedance and admittance per unit length. The general solution of (7)is where Z is called the acoustic characteristic impedance of the known as duct. −γx γx −γx γx (8) Proof. It is trivial from Lemma 1 and the relations (5). V = Ae + Be , I = (Ae + Be ), The next lemma is used to derive first principle model where A and B are complex-valued integral constants. γ is so that series connections of ducts are simplified in the given by derivation process, which is an extended result to the undamped case in [9]. γ := ZY := α + jβ, Lemma 3. Define a matrix T(l) with arbitrary complex numbers γ and Z by 2 2 2 2 2 2 2 α := R + ω L G + ω C − (ω LC − RG), (9) ⎡ ⎤ cosh γl −Z sinh γl ⎢ ⎥ 2 2 2 2 2 2 2 T(l) := ⎣ ⎦ . (15) ( ) 1 β := R + ω L G + ω C + ω LC − RG . − sinh γl cosh γl Lemma 1. In transmission line, suppose that v(x, t) and i(x, t) Then, for any positive real number x and y, the following holds: are in steady state with the relations given in (6).Then, V (x) and I(x) at arbitrary position x is related with the values at T(x)T y = T x + y . (16) origin, V (0) and I(0),by Proof. It can be verified by multiplication and application of ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ cosh γx −Z sinh γx the theorems for sinh and cosh. V (x) V (0) ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ , (10) I(x) I(0) − sinh γx cosh γx The relationship of pressure p and acoustic velocity u between different points in space will serve as the basis for deriving a unifying transfer function which includes where Z := Z/Y is the characteristic impedance of the the proposed control source. We consider the duct system transmission line. in Figure 2 which is the same as in [9], where SPK1 is a Proof. By substituting x = 0 into (8), A and B are written in noise source, SPK2 and SPK3 are used as a control source, V (0) and I(0). Then, the elimination of the constants A and controller is connected between y and u to drive the control B leads to (10). source based on the information of the reference microphone 4 Advances in Acoustics and Vibration Ll l l l 0 z u v y (p , u ) (p , u)(p , u)(p , u )(p , u ) (p , u)(p , u )(p , u ) 5 5 z z 4 4 3 3 2 2 1 1 y y 0 0 (p , u ) (p , u ) s S S 6 6 Error mic. Ref. mic. SPK2 SPK1 (p˜ , u˜ ) S S uy w z SPK3 Figure 2: Theoretical apparatus. 70 35 40 25 −720 −1440 −2160 −2880 −45 −3600 −4320 −90 1 2 3 10 10 10 10 Frequency (Hz) Frequency (Hz) Present model Present model Modelof[9] Modelof[9] (a) Overview (b) Zoom on low frequencies Figure 3: Comparison of G (Case A) with [9]. zw output y, and the error microphone output z is used to (iii) Case C—rear sound aided source (proposed one): evaluate control performance. Suppose that SPK1, Ref.Mic., The rear sound of SPK2 is redirected by a subduct and SPK3, SPK2, and Err.Mic. are located at x = 0, l , l , l ,and attached in a way that the rear sound will interfere y v u l , respectively, in the main duct of length L. with the front sound at the junction of ducts. Note There are three scenarios for the control source. here that the assumption is made, that the rear sound travelling in the subduct can be simulated by a subduct of length L = l − l with an additional S u v (i) Case A—Bidirectional source: SPK3 is not used; that loudspeaker (SPK3) to cover the rear sound. The is, v(t) = 0or H = 0 and no subduct installed (L = amplitude of SPK3 can be adjusted by the variable H; 0). This can be described as bidirectional case since that is, u(t) = H · v(t), where H is treated as negative SPK2 emits waves in both directions. real number, whose magnitude is less than or equal (ii) Case B—Swinbanks’ source: SPK3 is used to cancel to 1, in this paper. (H is set to −1 when the strength of the rear sound is as the same to the front one.) the upstream sound produced by SPK2 without subduct (L = 0). It is, therefore, driven with the The realizability of this setting is under consideration, opposite signal, but delayed v(t) =−u(t−(l −l )/c ). however, one possibility of this is to cover the inside u v 0 wall of the subduct by glass wool to weaken the rear This can be represented by the transfer function s((l −l )/c ) u v 0 H(s) =−e . sound. Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Advances in Acoustics and Vibration 5 In Figure 2,(p , u ) represents the pressure and the According to Lemma 2, the derivation process of the • • particle velocity at each position in the duct. For the first principle model is eventually similar to the undamped modelling process, the input signals w, u,and v are assumed case [9], where the derivation of the transfer function has to be u , u and u , respectively, neglecting the loudspeaker been explained in detail. Hence, only the resultant transfer 0 S S dynamics. The outputs z and y are assumed to be p and functions are shown as follows: p , respectively, neglecting the microphone dynamics. Thus, u := w, u := u, u := v, p := z, p := y. 0 S S z y Z sinh γ(L − l ) cosh γL + sinh γ(L − l ) · H 0 u s v G = , 0u sinh γ(L − l ) sinh γL cosh γl +cosh γL cosh γL v s v s G = G · cosh γl , yu 0u y sinh γ(l − l ) z v G =−Z sinh γ(l − l ) + · H zu 0 z u cosh γL + G sinh γ(l − l ) tanh γL cosh γl +cosh γl , 0u z v s v z (17) ( ) Z sinh γ L − l tanh γL sinh γl + sinh γL 0 v s v G = , 0w sinh γ(L − l ) tanh γL cosh γl +cosh γL v s v G = cosh γl G − Z sinh γl , yw y 0w 0 y G = sinh γ(l − l ) tanh γL cosh γl +cosh γl G zw z v s v z 0w − Z sinh γ(l − l ) tanh γl sinh γl + sinh γl . 0 z v s v z It is obvious that for undamped case the derived functions overall bode characteristics did not change, some minor and the ones given by [9] ought to be the same. This can differences appear due to the different approach of taking be confirmed by setting the damping coefficient d = 0and damping into account in low frequencies as it is shown in using relationships between hyperbolic and trigonometric Figure 3(b). functions. Indeed, consistency has been confirmed for the We here emphasize that the derived model in this paper undamped case. The major difference for the damped case is is more relevant than that of [9], since the frequency that the damping and acoustic impedance Z considered in response of the model can be directly compared with the one [9] are constant for all frequencies, while those in the derived calculated by numerical implementation in the next section. functions above are frequency dependent: the imaginary part of the complex wave number in [9], α of k = (ω/c ) − 3. Numerical Implementation jα, does not depend on the angular frequency ω, while the one in the corresponding parameter −jγ = β − jα The solution of the damped wave equation (3) using the depends on ω. Specifically, the difference mainly arises in low transmission line theory shall be confirmed by comparing frequency range, since α given by (14) might be regarded as to an alternative approach, that is, numerical simulation by a constant number in high-frequency range, whose limit is implementing a finite difference method, in which forward given by lim α = d/(2ρ c ). Figure 3 shows the frequency ω→∞ 0 0 and backward difference are used to calculate the frequency response of Case A (only G is shown in the figure due zw response of the system. to the limitation of the space) for the both considerations In this section, the numerical algorithm for Case A will of damping, where the same parameters in [9]are used as first be derived. Based on this, the numerical algorithm for follows: Case C will be explained. L = 3.61, l = 3.53, l = 2.32, l = 0.03, z u y 3.1. Case A. The effect of a loudspeaker in duct can be c = 340, ρ = 1.21. 0 0 considered in the wave equation by using a spacial delta (18) function [8]. Thus, the damped wave equation with a loudspeaker can be described as follows: The damping coefficient is set as d = 72.6 which is obtained from the relationship d/(2ρ c ) = 30/340 ≈ 0.09 (0.09 is 0 0 2 2 ∂ p ∂p ∂ p givenin[9]) so that both considerations of damping result 1 d + = + ρ u˙δ(x − l ). (19) 0 u 2 2 2 2 similar damping effect in high-frequency range. Although c ∂t ρ c ∂t ∂x 0 0 0 6 Advances in Acoustics and Vibration For numerical analysis, time t and space x are discretized as Table 1: Numerical approximation settings. follows: Parameter Notation Value Unit Spatial step width Δx 0.01 m t = t := i · Δt, ∀i ∈{0, 1,... , N}, Simulation time T 1s x = x := j · Δx, ∀j ∈{0, 1,... , M}, j Time steps N 40000 (20) T L l N := , M := , j := , Δt Δx Δx holds with a pressure at a fictitious location x =−Δx =: x . −1 This relation can be used in (23)for j = 0 leading to where T is a simulation period. After applying forward and backward difference on the damped wave equation (19), one obtains p(t , x ) = p(t , x ) · [2 − s − r] + p(t , x ) · [s − 1] i+1 0 i 0 i−1 0 + p(t , x ) · [r]. i 1 p t , x − 2p t , x + p t , x i+1 j i j i−1 j (26) Δt p t , x − p t , x i j i−1 j d By using the relations above, frequency response of the feedback-path system G can be numerically calculated by yu ρ Δt injecting the source input as a sinusoidal signal u(t ) = −U cos ωt : p t , x − 2p t , x + p t , x i i j+1 i j i j−1 δ jj 2 2 = c + c ρ u˙ , 0 0 Δx Δx (1) initial condition: p(t , x ) = p(t , x ) = 0for all j, (21) 0 j 1 j (2) for i = 1, 2,... , N − 1 repeat the following: where δ denotes the Kronecker delta to approximate the jj original property of integration being 1 by summation as follows: (a) at the open end: p(t , x ) = 0, i 0 M−1 L ⎨ 1, i = j , (b) at the closed end: (26), jj 1 = δ(x − l )dx ≈ Δx, δ = u ij Δx (c) for j = 1, 2,... , M − 1repeat(23). (only 0 i= j . j=0 at the source position j = j , the last term (22) rΔxρ u˙ (t )δ appears as rΔxρ U sin t ). 0 i jj 0 i Rearranging the recurrence relation for future time value and introducing auxiliary variables leads to For the stability of the numerical algorithm above, parameters in Table 1 are set. The steady state pressure p t , x = p t , x · [2 − s − 2r] + p t , x · [s − 1] i+1 j i j i−1 j variation at the microphone position p(t , x )isevaluated i j to get gain and phase from the source input u to the [ ] [ ] + p t , x · r + p t , x · r i j+1 i j−1 microphone output y,where j is defined as j := l /Δx. y y y The other frequency responses for G , G ,and G can be zu zw yw + rΔxρ u˙δ , 0 jj calculated in a similar way. For each transfer function 2000 (23) logarithmically distributed frequencies have been simulated. 3.2. Case C. As Case C consists of the main duct and an d c · Δt additional subduct, the previous derivation has to be adapted s = Δt · , r = . (24) ρ Δx accordingly. First, similar to the main duct, an additional z axis is introduced so that SPK3 is located at z = 0 and the Since the duct has two ends, boundary conditions have to be subduct is connected at z = L as shown in Figure 2,by set accordingly. As it has been done in the analytical process, which the instantaneous pressure in the subduct is denoted the pressure at the open end is set to zero; that is, p(t , x ) = i M as p (t, z). Then, t and z are discretized in a similar manner 0for all i. For the closed end at x = 0, the pressure gradient as Case A has to be zero. Thus, the relation (27) ∂p(t, x) z = z := k · Δx ∀k ∈{0, 1,... , M }, M := , k S S Δx ∂x x=0 where common spacing Δx is used. The input for SPK3 is p(t , x ) − p(t , x ) i 0 i −1 ≈ = 0 ⇐⇒ p(t , x ) = p(t , x ) i −1 i 0 specified as v(t ) = Hu(t ). The boundary condition for i i Δx the closed end of the attached subduct is added like before. (25) Advances in Acoustics and Vibration 7 360 360 0 0 −360 −360 −720 −720 −1080 −1080 −1440 −1440 −1800 −1800 −2160 −2160 −2520 −2520 −2880 −2880 1 2 3 1 2 3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Analytical Analytical Numerical Numerical (a) Case A (b) Case C Figure 4: Comparison of numerical and analytical solution, transfer function G , H =−1. yu Furthermore, the pressure at the junction is assumed as the 70 average of all surrounding pressure points 60 p t , x i+1 j = p t , z S i+1 M p t , x + p t , x + p t , z i+1 j −1 i+1 j +1 S i+1 M −1 v v S = , (28) −720 −1440 where j := l /Δx. The relation above might now be only −2160 v v −2880 an assumption; however, the validity will be shown later. −3600 The numerical algorithm in detail is omitted from the paper, −4320 since it is parallel to Case A based on the additional setting 1 2 3 10 10 10 for Case C. So far, two solutions of the frequency response Frequency (Hz) based on the damped wave equation (3) have been obtained. H =−1 H =−0.8 In the remaining of this section, the results will be compared. H =−0.95 H =−0.5 Figure 4 shows frequency response of the feedback-path system for Cases A and C. Figure 5: Comparison of transfer function G of Case C for yu Apparently, both solutions match almost perfect as it different rear-sound strengths H. 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 compared to itself with several settings for H,asitcan be seen A did not occur. In both cases, the total phase lag is about in Figure 5. 2520 degree. A possible cause for the missing phase lag could Evidently, a lower value of H will lead to the desired be a different setting of the rear-sound strength H,which effect and, thus, shows an advantage over the bidirectional leads to different bode diagrams for G as pointed in [9]. yu scenario. The reason for that behaviour can be found in the transfer function for G ,sinceitispartof G .For 0u yu 4. Optimal Condition for the Longest Time the explanation, the undamped equation will be used as it simplifies the next steps and did not show much influence on Delay in Feedback-Path System phase characteristics. The derivation will be shown for the In order to examine the effect of a different setting of the transfer function of Case A, the same method can later be rear-sound strength H, the analytic solution is modified and applied to Case C. Firstly, G will be simplified for Case A by 0u Phase (deg) Magnitude (dB) Phase (deg) Phase (deg) Magnitude (dB) Magnitude (dB) 8 Advances in Acoustics and Vibration −s(l /c ) u 0 −− −10 −s(2L−l /c ) −s(2L/c ) u 0 0 e e −20 Figure 6: Signal flow chart in frequency domain. −30 −40 −720 setting L = 0, H = 0 and rewriting as exponential functions. −1440 After making the equation causal by multiplying with the −2160 −jkL −2880 longest time delay e to both denominator and numerator, −3600 and replacing jk by s/c , the following is obtained: −4320 −5040 −5760 1 2 3 −s(l /c ) −s(2L−l )/c u 0 u 0 10 10 10 e − e . (29) G = ρ c 0u 0 0 Frequency (Hz) −s(2L/c ) 1+ e Case (A) Case (C) ) H =−0.5 This transfer function corresponds with a causal system Case (A) modified depicted in Figure 6, where each signal block represents a time delay. Figure 7: Comparison of phase characteristics for Case A, Case A It can be observed that the shortest time delay is given by with additional time delay, and Case C (H =−0.5). −s(l /c ) u 0 e . As mentioned in the introduction, the value of the shortest time delay is critical for the control performance. Similar for Case C, the same procedure is applied and some variables for better overview are introduced. Since the denominator is not of relevance and long, it has been neglected here and the modified transfer function G can 0.012 0u be derived as 0.01 −s(3l −2l )/c −s(2L−l )/c u v 0 u 0 G = 4ρ c e − e 0.008 0u 0 0 0.006 −s(l /c ) −s(2L−2l +l )/c u 0 v u 0 +[(1/2) + H] e − e . (30) 0.004 0.002 3.5 From this equation, it becomes clear that half rear-sound 2.5 1.5 0 1 amplitude will eliminate the shortest time delay given by 0.5 −s(l /c ) −s(3l −2l )/c u 0 u v 0 e and leads to e as the shortest time delay. Figure 8: Numerictimeresponse, Case A. Compared to the previous result for Case A (29), an −s(2l −2l )/c u v 0 additional time delay of e could be achieved. This corresponds to twice the length of the subduct and can be seen in Figure 5. The time delay can also be verified by multiplying the transfer function G of Case A by an yu −40 additional, fictitious time delay. In Figure 7,itcan be seen 0.012 that this modified Case A and Case C with a rear-sound strength of −0.5 produce indeed a very similar phase lag. 0.01 The numerical implementation reveals the additional time delay in feedback-path system by displaying the prop- 0.008 agation of the incident waves. In Figures 8 and 9, the wave 0.006 propagation for Case A and Case C is shown. The y-axis depicts the acoustic pressure distribution for different points 0.004 in time, the x-axis represents the position in the main duct, where the closed end is located at x = 0. One complete 0.002 3.5 period with a frequency of 1000 Hz is excited by the control 2.5 1.5 loudspeaker at l = 2.32, and the propagation through the 1 0.5 duct is observed. Note that the first upwards travelling wave is extinguished by the rear sound in Case C and previously Figure 9: Numerictimeresponse, Case C. derived time delay is obtained. Phase (deg) p(x, t) Magnitude (dB) p(x, t) Advances in Acoustics and Vibration 9 5. Conclusion sound sources,” Transactions of the Japan Society of Mechanical Engineers. C, vol. 67, no. 656, pp. 52–57, 2001 (Japanese). In this paper, as an additional approach in active noise [3] Y. Kobayashi and H. Fujioka, “Active noise cancellation for control next to the Swinbanks’ source, the proposed rear- ventilation ducts using a pair of loudspeakers by sampled-data sound-aided source has been analytically examined. The H∞ optimization,” Advances in Acoustics and Vibration, vol. advantage of additional phase lag and time delay in the 2008, Article ID 253948, 8 pages, 2008. feedback-path transfer function of the proposed control [4] J. Winkler and S. J. Elliott, “Adaptive control of broadband source has been analyzed. To accomplish this, the damped sound in ducts using a pair of loudspeakers,” Acustica, vol. 81, no. 5, pp. 475–488, 1995. wave equation has been used and based on this: a first [5] Y. Kobayashi and H. Fujioka, “Analysis for robust active principle model has been derived by the application of noise control systems of ducts with a pair of loudspeakers,” transmission line theory. The model has been confirmed by in Proceedings of the International Congress on Sound and verifying that Vibration (ICSV ’08), Daejeon, Korea, 2008. (i) the previous model is included as the special case of [6] Y. Kobayashi and H. Fujioka, “Robust stability analysis for an undamped scenario, active noise control systems of ducts with a pair of loudspeak- ers,” in Proceedings of the 37th Symposium on Control Theory (ii) the frequency response is consistent with the one (SICE ’08), pp. 21–247, 2008. obtained by a numerical simulation based on a [7] D. S. Bernstein, “What makes some control problems hard?” finite difference implementation of the damped wave IEEE Control Systems Magazine, vol. 22, no. 4, pp. 8–19, 2002. equation. [8] J. Hong and D. S. Bernstein, “Bode integral constraints, It has been shown, that the different consideration of colocation, and spillover in active noise and vibration control,” damping induced some minor changes in low-frequency IEEE Transactions on Control Systems Technology, vol. 6, no. 1, pp. 111–120, 1998. characteristics. Characteristics in middle and high-frequency range were consistent to the previous model, which implies [9] Y. Kobayashi, H. Fujioka, and N. Jinbo, “A control source structure of single loudspeaker and rear sound interference validity of the previous results in [9]: The proposed control for inexpensive active noise control,” Advances in Acoustics and source has shown additional phase lag by assuming a weaker Vibration, vol. 2010, Article ID 730813, 9 pages, 2010. rear sound. The advantage has also been examined from [10] W. Barten, A. Manera, and R. Macian-Juan, “One- and two- a theoretical point of view in order to determine under dimensional standing pressure waves and one-dimensional which conditions an additional time delay is achieved. It travelling pulses using the US-NRC nuclear systems analysis has been demonstrated that the shortest time delay in code TRACE,” Nuclear Engineering and Design, vol. 238, no. the feedback-path transfer function is extended by 2L /c , S 0 10, pp. 2568–2582, 2008. when H =−0.5, where L resembles the length of the [11] P. A. Nelson andS.J.Elliott, Active Control of Sound,Academic subduct and c the speed of sound. This time delay has been Press, New York, NY, USA, 1993. also confirmed by comparison with the phase characteristic [12] S. J. Elliott, Signal Processing for Active Control, Volume in the for the bidirectional source with additional time delay. In Signal Processing and Its Applications Series, Academic Press, addition, the time delay has been directly confirmed in the New York, NY, USA, 2009. time response simulation of the pressure variation by an incident wave. 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 H. This ends up showing that the original setting of H =−1 is not suitable for the proposed source to maximize the additional time delay, which might be inconsistent with the result in [9], since the similar phase characteristic of Swinbanks’ source has been achieved by H =−1in the experiment. To tackle this problem, the time response simulation in this paper will be used for comparison in detail with a measured time response by experiments. Further effect of the rear-sound amplitude on control performance will be examined. References [1] M. A. Swinbanks, “The active control of sound propagating in long ducts,” Journal of Sound and Vibration, vol. 27, pp. 411– 436, 1973. [2] S. Kijimoto, H. Tanaka, Y. Kanemitsu, and K. 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