Chapter Three

Electromechanoacoustical circuits

Abstract

The greatest advance in the designing of electroacoustic devices has been the development of electrical-equivalent circuits that permit the combination of electrical, mechanical, and acoustical elements into a combined circuit. Following a lengthy introduction, the physical and mathematical meanings of the elements in these types of circuits are tabulated. Admittance-type and impedance-type circuits are derived and the conversion of one to the other is demonstrated (“dual” conversion). It is then followed by mechanical elements, acoustical elements, levers, and lengthy examples. The final part deals with transducers—the convertors from forces and velocities to voltages and currents. These lead into energy and power relations and combinations of transducers.

Keywords

Acoustical circuits; Admittance; Electromechanoacoustics; Impedance; Mass; Mechanical circuits; Transducers

Part VI: Mechanical circuits

3.1. Introduction

The subject of electromechanoacoustics (sometimes called dynamical analogies) is the application of electrical circuit theory to the solution of mechanical and acoustical problems. In classical mechanics, vibrational phenomena are represented entirely by differential equations. This situation existed early in the history of telephony and radio as well. As telephone and radio communication developed, it became obvious that a schematic representation of the elements and their interconnections was valuable. Unlike a mechanical drawing, a schematic representation simply shows how the individual circuit elements are connected, or their topology, rather than where they are physically located. One of the most celebrated examples outside the field of engineering is the map of the London Underground, which was designed by an electrical engineer [1] who realized that passengers simply wanted a clear diagram of how to get from one place to another without the geographical details of the route. Schematic diagrams made it possible for engineers to visualize the performance of a circuit without laboriously solving its equations. Such a study would have been hopelessly difficult if only the equations of the system were available.
There is another important advantage of a schematic diagram besides its usefulness in visualizing the system. Often one has a piece of equipment for which one desires the differential equations. The schematic diagram may then be drawn from visual inspection of the equipment. Following this, the differential equations may be formed directly from the schematic diagrams. Most engineers are trained to follow this procedure rather than to attempt to formulate the differential equations directly.
Schematic diagrams have their simplest applications in circuits that contain lumped elements, i.e., where the only independent variable is time. Such elements are valid when the wavelength greatly exceeds the dimensions of the component. In other words, lumped element models are models with zero space dimensions. In distributed systems, which are common in acoustics, there may be as many as three space variables and a time variable. Here, a schematic diagram becomes more complicated to visualize than the differential equations, and the classical theory comes into its own again. There are many problems in acoustics, however, in which the elements are lumped and the schematic diagram may be used to good advantage.
Four principal requirements are fulfilled by the methods used in this text to establish schematic representations for acoustic and mechanical devices. They are as follows.
  • 1. The methods must permit the formation of schematic diagrams from visual inspection of devices.
  • 2. They must be capable of such manipulation as will make possible the combination of electrical, mechanical, and acoustical elements into one schematic diagram.
  • 3. They must preserve the identity of each element in combined circuits so that one can recognize immediately a force, voltage, mass, inductance, and so on.
  • 4. They must use the familiar symbols and the rules of manipulation for electrical circuits.
Several methods that have been devised fulfill one or two of the above four requirements, but not all four. A purpose of this chapter is to present a new method for handling combined electrical, mechanical, and acoustic systems. It incorporates the good features of previous theories and also fulfills the above four requirements. The symbols used conform with those of earlier texts wherever possible [2–6].
Note that a simple procedure for conversion of admittance-type circuits to impedance-type circuits is given in Part IX, Section. 3.8.
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