3.8. Conversion from admittance-type analogies to impedance-type analogies

In the preceding parts, we showed that electromagnetic and electrostatic transducers require two different types of analogy if they are to be represented by the networks shown in Table 3.1. A further need for two types of analogy is apparent from the standpoint of ease of drawing an analogous circuit by inspection. The admittance type of analogy is better for mechanical systems and the impedance type for acoustic systems. The circuits we shall use, however, will frequently contain electrical, mechanical, and acoustical elements. Because analogies cannot be mixed in a given circuit, we must have a simple means for converting from one to the other.
We may readily derive one analogy from the other if we recognize that:
  • Elements in series in the circuit of one analogy correspond to elements in parallel in the other.
  • Resistance-type elements become conductance-type elements, capacitance-type elements become inductance-type elements, and inductance-type elements become capacitance-type elements.
  • The sum of the drops across the series elements in a mesh of one analogy corresponds to the sum of the currents at a branch point of the other analogy.
This is equivalent to saying that one analogy is the dual of the other. In electrical circuit theory one learns that the quantities that “flow” in one circuit are the same as the “drops” in the dual of that circuit. This is also true here.
To facilitate the conversion from one type of analogy to another, a method that we shall dub the “dot” method is used [9]. Assume that we have the admittance-type analog of Fig. 3.17 and that we wish to convert it to an impedance-type analog. The procedure is as follows (see Fig. 3.41):
image
Figure 3.41 Preparation by the “dot” method for taking the dual of Fig. 3.17.
  • Place a dot at the center of each mesh of the circuit and one dot outside all meshes. Number these dots consecutively.
  • Connect the dots together with lines so that there is a line through each element and so that no line passes through more than one element.
  • Draw a new circuit such that each line connecting two dots now contains an element that is the inverse of that in the original circuit. The inverse of any given element may be seen by comparing corresponding columns for admittance-type analogies and impedance-type analogies of Table 3.3. The complete inversion (dual) of Fig. 3.41 is shown in Fig. 3.42.
  • Solving for the velocities or the forces in the two circuits using the rules of Table 3.1 will readily reveal that they give the same results.
After completing the formation of an analogous circuit, it is always profitable to ask concerning each element. If this element becomes very small or very large, does the circuit behave in the same way the device itself would behave? If the circuit behaves properly in the extremes, it is probably correct.
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