### Electro-mechanic analogy

Especially in the course of the development of electroacoustic transducers, the topic of electro-mechanic analogies has early become a classic. In many cases an electro-mechanic transducer, e.g., the magnet-plus-electric-coil driver system of a loudspeaker, can be symbolically represented by a kind of electro-mechanical circuit, i.e., a plane structure that is composed of both electrical and mechanical elements. Often it is desirable to represent such a mixed circuit by a homogeneous one, i.e., a system that exclusively is composed of either electrical or mechanical elements. As it turns out, it is rarely - if ever - helpful to represent an electrical system by a mechanical one. Ordinarily, the goal is to represent a mechanical system by an equivalent electrical one. The benefit of such a transformation is, that an electro-mechanical transducer can be entirely represented by an electrical network. Another benefit is that, in fact, representation of a mechanical system by an equivalent electrical network renders it easier to analyze and to compute the mechanical system's dynamic behavior (and this is true not only for electronic engineers). A favorable aspect of all this is that the systems to be considered in electroacoustics are linear - or at least can with sufficient approximation be treated as linear.

So, to be able to represent an inhomogenous (i.e., mixed) system such as an electro-mechanical transducer, by an equivalent homogenous (purely electrical) one, one must work out the fundamentals of transforming its mechanical portion into an equivalent electrical network. The basis of that transformation is provided by the formal equivalence of the pertinent fundamental equations that describe the dynamic behavior of the elements of which the mechanical and electrical systems are composed. That type of equivalence is particularly simple and clear when one describes the mechanical dynamics in terms of force and velocity, and that of the electrical network in terms of voltage and electrical current.

Essentially, then, one can envisage two kinds of analogy, namely

• the one that sets mechanical force analogous to electrical voltage, implying that velocity is analogous to electrical current (analogy I);
• the one which uses the other alternative, i.e., force analogous to current, velocity analogous to electrical voltage (analogy II).

In the literature on the topic, the advantages and disadvantages of the respective analogies were extensively discussed. Prevalent was the view that both types of analogy are consistent. It was immediately noticed that the second analogy has the advantage that the structure of the equivalent electrical circuit such obtained is almost identical with that of the original plane mechanical system. (By and large, one has to replace every mechanical frictional resistance element by an electrical resistor, every spring by an inductance, and every mass by a condenser, while the mechanical connections of the elements are interpreted as electrical wire connections.)

When I became concerned with the topic, I acknowledged that view, and I focused on the problem to understand what are the fundamentals of the miraculous metamorphose by which a plane mechanical structure (composed of frictional resistances, springs, masses, and levers) changes into an electrical network (composed of resistors, condensers, inductances, and transformers) . Due to the simple structural relationship that for analogy II holds between a mechanical and the analogous electrical system, the second analogy appeared to be of particular value.

The result was, that such understanding can be achieved only when every mechanical element that has two mechanical "gates", i.e., frictional resistance and spring, is represented by an electrical elementary circuit that has two gates, as well. (Note that a mechanical "gate" is just a mechanical connection, i.e., to another element; by contrast, an electrical gate has two connections.) So, the major message of the paper  was, that representation of, e.g., a mechanical resistance merely by an electrical one, is not adequate; likewise, for the spring. Another message was, that a mass (that is treated as a point mass) in the electrical domain must be represented by a simple element (an element with only one gate), i.e., either an inductance (analogy I) or a capacitance (analogy II). This notion was made with respect to the widely distributed view that a mass could be conceived as having two mecanical "gates", i.e., an input gate (where the accelerating force acts), and an output gate (through which the mass somehow rests on an environmental reference system, i.e., to account for the force required for acceleration).

These notions were not made just to do hairsplitting. Firstly, that approach is the only way to the design of a formal method - an algorithm - by which the transformation of a mechanical plane structure into an equivalent electrical network can be achieved without requiring ingenious intuitive faculties. Secondly, I had not at all been happy about the way the mass often had been treated, i.e., in an unpleasant conflict with Newtonian mechanics. And, last not least, the topic of electroacoustic transducers is difficult anyhow, and teaching to students such a difficult matter with a method that in some respects is inconsistent, is just unfair.

On the basis of the above results I believed I was prepared to teach the topic to my students in a clean and consistent manner. I aimed at preferrably teaching a method that takes advantage of the aforementioned simplicity of analogy II. However, I stumbled over another inconsistency. When various types of electro-mechanic transducers were theoretically investigated, it became apparent that something must be wrong with the second analogy. Whenever the transducer was of a type that depends on a magnetic field, i.e., electrodynamic, or magnetic, its treatment with analogy II yielded correct results by absolute magnitude; however, often with the wrong sign.

An analysis of this problem revealed that it originates from setting voltage analogous to mechanical velocity, and current to force, because in certain physical aspects those respective entities in fact are not analogous ,  p. 63-64. With respect to how a force acts on the elements of a mechanical system, and how an electrical current flows through an electrical system, there is a fundamental difference; likewise, for velocity and voltage, respectively. The difference is, that the sign of both velocity and current include the information about in what direction within the system energy is instantaneously transmitted. Neither force nor voltage include that information. Both force and voltage are potential-type entities, while velocity and current are flow-type entities. Consequently, when by analogy II velocity is represented by voltage, the directional information carried by velocity gets lost. When, and if, the system is of the reciprocal type, that error does not matter, because a reciprocal transducer's basic behavior is independent of the direction of energy flow through its two gates. So, for reciprocal systems, the problem does not become apparent in the results of calculations.

However, when electro-mechanic transduction (and likewise, mechanic-electric transduction) is physically dependent on a magnetic field, the transducer is non-reciprocal, due to the gyrator effect. Its basic behavior is not the same for the two directions of signal transmission, and this is why it is important to preserve description of the direction of energy flow. In fact, the magnetic-field transducers (i.e., the electrodynamic and the magnetic ones), by their physical principle of transduction, relate a potential-type signal (e.g., voltage) to a flow-type signal (velocity), and vice versa. When, by analogy II, in the transducer's mechanical part velocity is represented by voltage and force by current, the resulting electrical network - that is meant to represent the entire transducer - no longer includes the gyrator effect; it merely is wrong. The network correctly represents the transducer's behavior only for one direction of signal transmission. When the network is employed for describing the transducer's behavior in the opposite direction (e.g., microphone instead of loudspeaker), one gets the error of sign.

So it turns out that analogy II is not tenable because it confuses the information on the direction of energy flow through the mechanical system that it is meant to represent. And that confusion emerges from the mixed assignment by analogy II of a potential-type signal to a flow-type signal, and vice versa. So it became apparent that there is only one type of analogy that can consistently be employed, i.e., analogy I.

This is the message of the paper . Remarkably, the distinction between potential-type entities such as force and voltage on the one hand, and flow-type entities such as velociy and electrical current on the other, is not new. That distinction has traditionally been made in the distinction between impedances and admittances. Ordinarily, an impedance is defined as the ratio of a potential-type signal divided by a flow-type signal; and the reciprocal thereof is called an admittance. One may suspect that the distinction between potential-type signals and flow-type signals implied in this tradition has been made just intuitively. Intuition, however, may occasionally be misleading. Indeed, one may observe in one and the same textbook that mechanical force is conceptualized as a flow-type entity while the ratio of force by velocity is called a mechanical impedance.

I am inclined to believe that the aformentioned respective features of potential-type and flow-type entities provide an unambiguous criterion for identifying the type of any given physical signal: If its sign carries the information about direction of instantaneous flow of energy, it is flow-type; otherwise, potential-type.

Author: Ernst Terhardt terhardt@ei.tum.de - Mar 10 2000

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