Power Conversion Efficiency of Airborne Parametric Array

Transcription

1 Power Converion Efficiency of Airborne Parametric Array JIANFENG CHEN Laboratorie for Information Technology, Heng Mui Keng Terrace, Singapore Abtract: - In thi paper the power converion efficiency of airborne parametric array i tudied and an explicit expreion of the relationhip between the power converion efficiency and the primary wave power, primary wave frequency, and econdary frequency i deduced from Berktay ound preure function. The reult obtained are ued to evaluate ome parameter election criteria and the performance of thi technology ued in the air a a uper directional loudpeaker. It how that the power converion efficiency i approximately proportional to the quare of the ratio of the differential frequency to the carrier frequency and alo proportional to the ound power of original primary wave. The carrier frequency i uggeted to be within 3kHz to 7kHz to guarantee a good converion efficiency. Saturation phenomenon, trong reflection and afety iue are dicued to how the limitation of thi technology. Keyword: - Parametric Array, Ultraonic, Audio Beam Introduction Parametric array i a term firtly introduced by Wetervelt [] in 96. It exploited an effect known a elf-demodulation, i.e., uing nonlinear interaction of high frequency ound to generate uper directional low frequency ound ource, which would otherwie require an huge array of audible frequency tranducer. Since the early 96, except a few initial attempt which ued air a medium [], the work on thi ubject ha been focued mainly on underwater application [3]-[5]. Until recent year, a the requirement on high quality audio reproduction and ound pace control increaing, many practical reearch paper and experimental report appeared on the ue of airborne parametric array for audible ound beam generation [6]-[]. By applying the parametric array principle in the air, one may create a uper directional virtual loudpeaker through a condened ultraonic tranducer array, intead of a cumberome loudpeaker array or a big parabolic reflector. The virtual loudpeaker bundle ound like a beam of light and focue it accurately onto a very mall urface. Beide high directivity, the virtual loudpeaker i alo poible to reproduce whole audio frequency band ignal by uing ingle tranmitter, rather than combining woofer, midrange driver, and tweeter together. A a reult of uing ultraound a carrier, the audible ound beam may have a trong reflection effect, enabling ome pecial effect, uch a virtual ound, to be realized much eaier. In addition, there i no magnet part in thi type of loudpeaker. The technology contain great commercial opportunitie in many field, uch a entertainment, communication, tranport and military application, etc., and conequently attract the attention of ome famou corporation and reearch center. In 998, F. Joeph Pompei, from MIT Media Lab, preented hi report [9] of hi prototype and a real ite demontration wa hown a well to prove the performance he aerted. In 999, Sennheier wa preented with the 999 German Indutry Innovation Award for Medium-Sized Companie for it development of a ound beam ytem. In, American Technology Corporation (ATC) publihed their white paper on the work in thi area. In the ame year their commercial ytem wa intalled aboard USS WINSTON Churchill. However, all thee achievement do not ugget that the technology were perfect. On the contrary, everal difficultie, uch a the limitation of the

2 tranducer bandwidth, technique for reducing harmonic ditortion, power converion efficiency, etc., till have not been olved well. In thi paper we are going to tudy the converion efficiency, which i a very important iue of thi ubject. Although people have found ome rule on thi topic, little work ha been reported in open publication which may provide an analytical reult on power converion efficiency. In thi paper we will tart from the reult of Berktay on econdary ound wavefield ditribution and deduce an analytic relationhip between the power converion efficiency and everal important parameter of parameter array. We will ee that the reulted relationhip expreion may help u on the parameter election and the experiment analyi. where i the gradient operator, p and denote the ound preure of primary and econdary wave, β i the coefficient of nonlinearity, ρ i the ambient denity of the air, and i the mall ignal ound velocity. In thi paper c the component of interet i the differential frequency ω, and the high frequency ω component ω + ω will be ignored, a it i inaudible and will be attenuated relatively rapidly. By applying the uperpoition integral of the Green function and the virtual econd ource [ee right ide of Eq. (a)], the olution of p for equation () may be expreed a p The organization of thi paper i a follow. The previou work on parametric array i briefly reviewed in Section. Then the converion efficiency i defined and an explicit expreion i further deduced in Section 3. Several iue regarding the technology limitation and parameter election are dicued in Section 4. Some concluding remark are provided in Section 5. Sound Preure of Secondary Wave When two well collimated inuoidal ultraound wave (primary wave), with finite amplitude and different frequencie ω and ω, are radiated in the air from a parametric array, they would interact each other in the medium and new ound wave (econdary wave) whoe frequencie correpond to the um and the difference of the primary wave, i.e., ω + ω and ω ω (audible ound), may be produced a the reult. Thi phenomenon wa firtly analyzed by Wetervelt in 96 and wa nominated a nonlinear interaction of ound wave. An inhomogeneou wave equation, which i atified by the ound preure of the econdary wave, wa derived by Wetervelt [] in the form p p c t β q ρ c 4 t p q ρ t ρ p q, t 4π r v r r t c r r d r () where r i the obervation point poition vector, r i the ource poition vector, and v i the nonlinear interaction pace. Let u further aume that the two primary wave are with the ame power W / and the ame initial ound preure p. The cro ection of the (a) number ( k ω / c ), R i the ditance from the array, γ i the ratio of pecific heat and i (b) defined a γ α+ α α with α, α, and α being the aborption coefficient of the ound at collimated primary wave i circular with effective radiu a. To implify the decription, we denote the difference frequency ω ω a ω and it correponding wavelength a λ. Then the reult obtained by Berktay [] for the amplitude of the differential preure at a point ( R, θ ) in the far field, where R a λ, can be given a / α R 4 γ k 4 πρcr ka β pse ω θ J( kain θ) p( R, θ) + 4 in 4 inθ where S (3) denote the effective cro ection area of the primary beam ( S π a ), k i the wave

3 frequencie, ω and ω α α ω. Becaue α i much maller than and, it will be omitted in the following. By imply ubtituting θ in Eq. (3), we obtain the ound preure amplitude of the econdary wave along the ymmetry axi of the ound beam β ps ω p ( R, ) (4) 4 4πρ crγ By integrating the intenity of differential ound I for the hemiphere, we can calculate the ound power W in the form π R W I πr inθdθ p inθ π / π / ρ c dθ (8) In Eq. (3), conidering the beam angle i very mall, it i reaonable to approximate Conequently the directivity function may be J( kain θ ) expreed a ka inθ then p( R, θ) J( kain θ) D( θ ) p (, ) R k kainθ + θ (5) in γ ps p β ω 4k in θ 4 γ 4 + 4πρcR 3 Power Converion Efficiency Power converion efficiency of the parametric array i defined a the ratio of the audible ound power W generated (econdary wave) to the original ultraound power W (primary wave). The power converion efficiency reflect the percentage of the audible ound power uccefully tranferred from ultraound during the nonlinear interaction. It will be proved below that the power converion efficiency i determined by the radiated ultraound power, audible ound frequency and ultraound frequency at the ame time. According to the Berktay aumption, the econdary wave wa aumed to radiate from a erie of continuou circular pan and in thi cae the intenity of the differential ound, I, i defined a p I (6) ρ c Subtitute for p in Eq. (8), we obtain π / inθdθ k θ (9) arctg in 4 θ γk 4k in γ γ + k arctg π γk γ () γk Note that in Eq. () we ued the approximation k π atan γ ince γ k. Uing Eq. (6), (8) and (9) in Eq. (7), we have W 6 ρc π ωβw 3 γ () According to the definition mentioned at the beginning of thi ection, the power converion efficiency can be repreented a A the primary wave are aumed to be plane and collimated, the ound power of the primary wave at frequency and are given by ω ω ps W W W (7) ρ c W ωβ W 6 W 3ρc γ () It i worth noting that under certain condition the ound aborption coefficient α in the air i proportional to the ound frequency quare, i.e.,

4 α A (3) f A may change with humidity and temperature of the air. The aborption coefficient α i from.3/m to /m for ultraonic frequencie from khz to khz, which correpond to a characteritic array length adjutable from L 33 to m. Since, ω are much higher than ω ω, for their aborption coefficient we have α α ( α+ α) / α α (4) Hence where γ f ( f + f α Af (5) )/. Uing Eq. (4) in Eq. (), the power converion efficiency can be rewritten a W π β f 6 W 6ρcA f Similarly, by uing Eq. (5) in Eq. (4), we have W πβ ps f p ( R,) 4 AρcR f (6) (7) From Eq. (6) we can ee clearly that the power converion efficiency of the parametric array i proportional to (f /f o ) and the ound power of original primary wave W. For the ound preure along the ymmetry axi of the array, it ha the imilar concluion except that it i additionally proportional to the area of the effective radiation cro ection. Another obervation i that the parametric array obtain the high directional mall ize low frequency ound ource by dramatically penalizing the power converion efficiency. 4 Dicuion 4. Carrier Frequency Selection When we ue the amplitude modulation ignal to replace the two inuoidal o a to generate more complex audible ound, we encounter the problem of election the proper carrier frequency. In [3], the author compared the ound generated by 5kHz and 3kHz primarie with the ound generated by 5kHz and 55kHz primarie. They found the difference tone lightly tronger from the lower frequency primarie in the far field. Thi phenomenon now can be eaily interpreted by the Eq. (6) or Eq. (7), a the power converion efficiency i inverely proportional to the quare of the carrier frequency. It eem that the carrier frequency need to be a low a poible to improve the power converion efficiency of the ytem. Unfortunately the evidence how that the radiation ultraound with lower frequency ha more eriou effect to human body than that of high frequency. Certainly, to reproduce the ound with enough bandwidth it i impoible to reduce the carrier frequency too much a well. On the other hand, the ound beam preading would occur in cae of low carrier frequency and reult in the degradation of the directivity conequently. It alo hould be mentioned that in practice tranducer with wide bandwidth in low frequency band i quite difficult to be produced. Therefore the well-accepted compromie frequency hould be between 3kHz to 7kHz. 4. Repone of Audible Sound Once the carrier frequency i determined, we will find that the power converion efficiency and the ound preure are proportional to the quare of differential frequency. It mean the ideal frequency repone of the tranducer hould be exponentially decreaing againt the frequency and decay -db per octave, rather than be flat. In fact, it hould be. In other word, it i eaier to generate high frequency component than low frequency component. Since the audio range i octave wide, by imple calculation we may notice that if we hope to produce the Hz component to a certain level, the 5kHz component hould be attenuated up to db! We will get to know next that the driven power cannot be increaed a high a we expect to give the deired audio output level, which i due to the aturation phenomenon of the air. In light of thi, Croft and Norri [] ugget that if we hould give up trying to generate equal level ignal below 5Hz, for example, we would have a much better chance of

5 providing good output over the ret of the range. Keep thi in mind we can undertand eaily why the audio generated from the ultraonic tranducer array i hardly to be comparative with the woofer. 4.3 Sound Intenity and Effect of Saturation It hould be emphaized that the bai of the nonlinearity i that the ultraound radiated in the local pace hould be trong enough, ay db or more. The audible ound arien from nonlinearity will be enhanced in the air alone the ymmetry axe until the level of ultraound reduce to ome extend when there i no nonlinearity occur any more. From Eq. (6) it i found that the power of the primary wave i another important factor in power converion efficiency. One may naturally expect to obtain tronger audible ound and converion efficiency by imply increaing the power of original primary wave. Thi approach i by no mean applicable. In fact, when the ound preure level of the ultraound exceed certain value, the air column in front of the emitter would be aturated and much of the energy would be diipated a heat, rather than turning it into ound. Phyically peaking, when a ound wave, for example, a inuoidal wave, i over intenive, the waveform would turn into aw-toothed wave due to the characteritic of medium. On the other hand the attenuation coefficient would be increaed with the power of the wave. Therefore the maximum power of the aw-toothed wave i limited olely by the media itelf and i irrelevant to the ound ource itelf. What even wore i that once the ound i aturated in the axi of the beam, the beam width would be widened and the idelobe level would be raied if the primary wave continue increaing. That i becaue that when the primary wave fall into aturation, the peak value of the mainlobe will remain unchanged, while the idelobe and the offet of the mainlobe will till be expanding. In thi cae, one may increae the ize of the emitter to increae the audible ound level. Thi i, however, undeirable in the application ometime. 4.4 Limitation of Encloure Application When the ultraonic parametric array i ued in a normal room, trong reflection will eriouly affect it directivity and reduce the overall performance. On the other hand, trong ultraonic radiation would definitely do harm to the human being, jut like the over trong audible ound doe. According to the Canadian Healthy Regulation, the afety level of ultraound i below db for 4kHz and db for 6kHz. While in order to generate the audible ound up to 8dB, the ultraound level will be much higher than db, ay 4dB or even higher. Therefore, in practice ome pecial device hould be employed to reduce uch effect. 5 Concluion In thi paper an explicit relationhip between the power converion efficiency and the primary wave power, primary wave frequency, and econdary frequency i derived. It how that the power converion efficiency i proportional to the quare of the ratio of differential frequency to carrier frequency and i proportional to the ound power of original primary wave. The carrier frequency i uggeted to be within 3kHz to 7kHz to guarantee a good converion efficiency. Saturation phenomenon, trong reflection and afety iue are dicued a thee are crucial in the application of thi technology. 6 Reference [] P. J. Wetervelt, Parametric acoutic array, J. Acout. Soc. Am., Vol. 35, pp , Apr. 963 [] M. B. Bennett, D. T. Blacktock, Parametric array in air, J. Acout. Soc. Am., Vol. 57, pp , 975 [3] H. O. Berktay, Poible exploitation of nonlinear acoutic in underwater tranmitting application, J. Sound Vibr., Vol. (4), pp , 965 [4] H. O. Berktay, Parametric amplification by the ue of acoutic non-linearitie and ome poible application, J. Sound Vib., (4), 965, pp [5] J. N. Tjφtta, S. Tjφtta, Nonlinear equation of acoutic, with application to parametric acoutic array, J. Acout. Soc. Am., Vol. 69, pp , June 98 [6] M. Yoneyama, J. Fujimoto, et al, The audio potlight: An application of nonlinear interaction of ound wave to a new type of

6 loudpeaker deign, J. Acout. Soc. Am., Vol. 73, pp , May 983 [7] D. T. Blacktock, Audio application of the parametric array, J. Acout. Soc. Am., Vol., pp. 36(A) 997 [8] T. D. Kite, J. T. Pot, M. F. Hamilton, Parametric array in air: Ditortion reduction by preproceing, in Proc. 6 th Int. Cong. Acout., Vol., P. K. Kuhl and L. A. Crum, Ed. (Acoutic Society of America, New York, 998, pp. 9-9 [9] A.L. Butler and J.L. Butler, The Rebirth of a Ribbon Tweeter, J. Acout. Soc. Am., 37 (A) (997), Preented at the 33rd meeting of the Acoutical Society of America (June, 997) Penn State Univerity [] F. J. Pompei, The ue of airborne ultraonic for generating audible ound beam, J. Audio Eng. Soc., Vol. 47, No. 9, Sept. 999, pp [] J. J. Croft, J. O. Norri, Theory, Hitory, and the advancement of parametric loudpeaker: A technology overview, White Paper, American Technology Corporation, [] H. E. Ba, L. C. Sutherland, A. J. Zuckerwar, D. T. Blacktock, D. M., Heter, Atmopheric aborption of ound: Further development, J. Acout. Soc. Am. 97, (997) [3] A parametric loudpeaker Applied example, Electronic and Communication in Japan, Part 3, Vol. 77, No., 994, pp64-73 [4] Steven, Davi, Hearing, It Pychology and Phyiology, John Wiley & Son, Inc., 938