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Equivalent Circuit Analysis of Mechano-Acoustic Structures

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Equivalent Circuit Analysis of Mechano-Acoustic Structures Web: http://www.pearl-hifi.com 86008, 2106 33 Ave. SW, Calgary, AB; CAN T2T 1Z6 E-mail: [email protected] Ph: +.1.403.244.4434 Fx: +.1.403.245.4456 Inc. Perkins Electro-Acoustic Research Lab, Inc. ❦ Engineering and Intuition Serving the Soul of Music Please note that the links in the PEARL logotype above are “live” and can be used to direct your web browser to our site or to open an e-mail message window addressed to ourselves. To view our item listings on eBay, click here. To see the feedback we have left for our customers, click here. This document has been prepared as a public service . Any and all trademarks and logotypes used herein are the property of their owners. It is our intent to provide this document in accordance with the stipulations with respect to “fair use” as delineated in Copyrights - Chapter 1: Subject Matter and Scope of Copyright; Sec. 107. Limitations on exclusive rights: Fair Use. Public access to copy of this document is provided on the website of Cornell Law School at http://www4.law.cornell.edu/uscode/17/107.html and is here reproduced below: Sec. 107. - Limitations on exclusive rights: Fair Use Notwithstanding the provisions of sections 106 and 106A, the fair use of a copyrighted work, includ- ing such use by reproduction in copies or phono records or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for class- room use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include: 1 - the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes; 2 - the nature of the copyrighted work; 3 - the amount and substantiality of the portion used in relation to the copy- righted work as a whole; and 4 - the effect of the use upon the potential market for or value of the copy- righted work. The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors ♦ PDF Cover Page ♦ ♦ Verso Filler Page ♦ 112 TRANSACTIONS OF THE I.R.E. EQUIVALENT CIRCUIT ANALYSIS OF MECHANO· ACOUSTIC STRUCTURES* B. B. Bauer Shure Brothers, Inc. Chicago, Illinois 1. INTROD("CTION To comprehend the new, the unknown, we often fall creasing tendency to turn to the analogous electrical upon the old, the known. Thus, in the time of Volta and circuits for solution of mechanical, acoustical and other Ohm (circa 1800) many of the conceptions of electricity problems. were based upon similarity with the older arts, i.e., After having been introduced to a subject as old and hydraulics, mechanics and heat. Volta was among the venerable as this, the reader might wonder what there is first to recognize that the phenomena �ouped under the new to be said about it. Surprisingly enough, the study name of galvanism were a manifestation of "electricity in of analogies has had a recent spurt in activity. New motion" - as contrasted with the older electrical phe­ uses as well as limitations of analogies are being dis­ nomena which re�sented "electricity in tension." covered almost daily. Complex physical organizations Since there was no e'vldence of accumulation of electricity are being put into form manageable by analogies. Defini­ at any point in a circuit it followed that the current could tions and terminology are being brought in line with the be represented figuratively by the flow of an incompres­ new developments. The object of this paper is to attempt sible fluid along rigid and inelLtensible pipes. Ohm used to give the reader a broad view of the subject to �cquaint Fourier's analyses of the conduction of heat to derive him with some of the new thinking regarding analogies. electrical laws and he was instrumental in developing the TIlE EFP AND THE IFP ANALOGIES concepts of "current" and "electromotive force. '71 Thus n. since the earliest days of electrical theory, electro-motive The reader should be cautioned at this time that the force became endowed with the attributes akin to a traditional concept of similarity between voltage-force­ mechanical force of hydraulic pressure, and electric pressure and current-velocity-fluid velocity is not the current has been thought of as being of similar nature as only one upon which a system of analogies can be based. mechanical velocity or the velocity of fluid flow. Un­ During the past quarter century, through the efforts of doubtedly, these classical concepts form the historical Firestone! and others, it has become known that it is basis for equivalent circuit analysis as it is known today. possible to establish another consistent system of The early analogies became especially important analogies based upon certain mathematical similarities during the end of the 19th century when ac electricity was between electrical current, force, and pressure on the one still in its infancy while the theory of vibrations and hand, and voltage, velocity, and fluid flow on the other. sound had been already highly developed by Rayleigh and The older "classical" analogy is currently being spoken others. It was c\iscovered that certain differential equa­ of as the "voltage-force-pressur.e" analogy while the tions developed for use with vibrating mechanical bodies newer analogy (orfginally called "mobility" analogy by were equally applicable to electrical quantities. llayleigh ·Firestone) is often referred to nowadays as the "current­ was among the first to bring the subject of alternating force-pressure" analogy. For brevity we call them the current electricity within the scope of acoustics and to EFP and the IFP analogies, respectively. prove that similar mathematical principles were applicable It seems feasible to apply either analogy about to both phenomena.2 equally well to the solution of various problems, although Much has happened during the half century which it is generally recognized that the analogy is the followed Hay leigh' s writings to change the relative BFP more advantageous with respect to acoustic devices and technological positions of electricity, acoustics and electrostatic (condenser, piezoelectric) transducers, mechan ics. The improvement in electrical circuit ele­ while the IFP analogy is the more adaptable to mechanic­ ments, oscillators, amplifiers, oscillographs and meters al devices and electromagnetic (magnetic, magnetostric­ - and indeed the perfection of the "analogy" computer - tive) transducers. (Incidentally, equivalent circuits for have made of the electrical network a most useful and transducers are a field in themselves and they are not flexible analytical tool. 'Therefore, we find an ever-in- treated here,) Almost everyone doing much equivalent circuit eventually becomes conversant with both ... i\'lanuscript received Ju ne I, 1954. work analogies. The casual user will avoid confusion by I W. D ampier, "A History of Science," The Mac\1illan Corr.pany, choosing a single analogy. N. Y., p. 232, etc.; 1944. Lord "The 1 Hayleigh, Theory of Sound," Dover Publications, I Firestone � "A new analogy between mechanical and electrical r-:. r., vol. I, p. 433, etc. ; 1945. system:;;," Jour. Acoust. Soc. Amer., vol. 4, pp. 239-2607; 1933. PROFESSIONAL GROUP ON AUDIO 113 In the writer's OpInIOn, the classical EFP analogy slits, etc. Flow of air through a pipe is accompanied by ' is to be preferred because electrically-trained people an increase in kinetic energy because of the acceleration find it the easiest to comprehend and because it has of the mass of air as it flows from the . volume into the decided advantages in connection with acoustic devices pipe and vice-versa. Therefore, a pipe or aperture de­ which, after all, are the principal domain of the audio fines a mass-type or inductive element. Acoustic resist­ technologist. However, there are differencl1s of opinion ance or damping is obtained through the use of capillaries, on this ma tter, and some authors prefer to use the IFP slits, or crevices, such as the interstices between the analogy. threads of cloth through which the air is caused to flow, The relationships which are the basis of the EFP with the accompaniment of viscous friction. Additionally, and the IFP analogies are shown in Table r. In this table when the openings connect to the atmosphere, sound is it is assumed that all circuit elements are constant and received or radiated by the acoustic structure and the all variables are steady-state RMS values. In each box impedance of the medium becomes a part of the acoustic under the Electrical, �1echanical and Acoustic elements struc�ure. TABLE I ELECTRICAf.- QUANTITY ACOUSTICAL QUANTITY MECHAN ICAL QUANTITY EFP ANALOGY IFP ANALOG Y Sound Pressure (p) Force (F) Voltage (El Curren t (I) microbar dyne newton per sq. ·m newton volt ampere Volume velocit y (U) Velocity (v) Current (/) Voltage (E) cu. cm per sec cm. per sec. cu. m per sec. m per sec. ampere volt Volume displacement (V) Displacement (D) Charge (Q) Impulse cu. cm cm cu. m m coulomb volt-sec. Acoustic Resistanfe (RA) Mechanical Resistance Resistance (R) Conductance (G) - (Rr) rayl (dyne-sec-cm 5 mech. ohm (dyne-sec-cm-1 ) - mks-rayl (newton-sec-m ) mks-mech. ohm (newton-sec-m-l) ohm ohm Inertance (MA) Mass (M) I nductanc e (L) Capacitance (C) gram-cm-� gram kg-m-' kg henry farad Acoustic Compliance (CA) Mechanical Compliance (CM) Capacitance (C) Inductance (L) cmS_dyne_l cm per dyne m5-newton-1 m per newton farad henry Acoustic Impedance Mechanical Impedance Impedance Admittance (Z A) (ZM ) (Z) (Y) Z a piU � = Ell Y - liE A ZM Flv Z are shown the term and its commonly used symbol, name Impedance values for acoustic network elements have of the unit (if available) and the unit in cgs and mks been treated by various writers4 and derivations will not systems.
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