Mechanical filter

At the input and output of the filter, transducers convert the electrical signal into, and then back from, these mechanical vibrations.

It is only necessary to set the mechanical components to appropriate values to produce a filter with an identical response to the electrical counterpart.

For example, filtering of audio frequency response in the design of loudspeaker cabinets can be achieved with mechanical components.

A representative selection of the wide variety of component forms and topologies for mechanical filters are presented in this article.

By the 1950s mechanical filters were being manufactured as self-contained components for applications in radio transmitters and high-end receivers.

Contemporary researchers are working on microelectromechanical filters, the mechanical devices corresponding to electronic integrated circuits.

The elements of a passive linear electrical network consist of inductors, capacitors and resistors which have the properties of inductance, elastance (inverse capacitance) and resistance, respectively.

Resistances are not present in a theoretical filter composed of ideal components and only arise in practical designs as unwanted parasitic elements.

Likewise, a mechanical filter would ideally consist only of components with the properties of mass and stiffness, but in reality some damping is present as well.

[1] The mechanical counterparts of voltage and electric current in this type of analysis are, respectively, force (F) and velocity (v) and represent the signal waveforms.

Equivalent circuits produced by this scheme are similar, but are the dual impedance forms whereby series elements become parallel, capacitors become inductors, and so on.

It is still possible to represent inductors and capacitors as individual lumped elements in a mechanical implementation by minimising (but never quite eliminating) the unwanted property.

[10][11] Versions of the harmonic telegraph were developed by Elisha Gray, Alexander Graham Bell, Ernest Mercadier[b] and others.

[10][11] Once the basics of electrical network analysis began to be established, it was not long before the ideas of complex impedance and filter design theories were carried over into mechanics by analogy.

Once these ideas were in place, engineers were able to extend electrical theory into the mechanical domain and analyse an electromechanical system as a unified whole.

The horn of the phonograph is represented as a transmission line, and is a resistive load for the rest of the circuit, while all the mechanical and acoustic parts—from the pickup needle through to the horn—are translated into lumped components according to the impedance analogy.

The resulting phonograph has a flat frequency response in its passband and is free of the resonances previously experienced.

[1] Norton's mechanical design predates the paper by Butterworth who is usually credited as the first to describe the electronic maximally flat filter.

[21] Another unusual feature of Norton's filter design arises from the series capacitor, which represents the stiffness of the diaphragm.

[25] The idea was taken up by Collins Radio Company who started the first volume production of mechanical filters from the 1950s onwards.

These were originally designed for telephone frequency-division multiplex applications where there is commercial advantage in using high quality filters.

It is possible to dispense with the magnets if the biasing is taken care of on the electronic side by providing a d.c. current superimposed on the signal, but this approach would detract from the generality of the filter design.

It is common practice to add a capacitor in parallel with the coil so that an additional resonator is formed which can be incorporated into the filter design.

For some types of resonator, this can provide a convenient place to make a mechanical attachment for structural support.

[44] The frequency response behaviour of all mechanical filters can be expressed as an equivalent electrical circuit using the impedance analogy described above.

As with the electrical counterpart, the more elements that are used, the closer the approximation approaches the ideal, however, for practical reasons the number of resonators does not normally exceed eight.

The frequency at which the transition from lumped to distributed modeling takes place is much lower for mechanical filters than it is for their electrical counterparts.

A common component used for radio frequency filtering (and MEMS applications generally), is the cantilever resonator.

Experimental complete filters with an operating frequency of 30 GHz have been produced using cantilever varactors as the resonator elements.

[52] Cantilever resonators are typically applied at frequencies below 200 MHz, but other structures, such as micro-machined cavities, can be used in the microwave bands.

photograph
Figure 1. A mechanical filter made by the Kokusai Electric Company intended for selecting the narrow 2 kHz bandwidth signals in SSB radio receivers. It operates at 455 kHz , a common IF for these receivers, and is dimensioned 45 mm × 15 mm × 15 mm (1.77 in × 0.59 in × 0.59 in).
Mechanical filter from a telephone carrier system using torsional resonator elements
Details of a mechanical filter using torsional resonator elements
Details of a mechanical filter using disc flexural resonators
A diagram of a phonograph mechanism with the mechanical impedance of each part shown in an equivalent circuit diagram. The identified elements are; needle point, needle elasticity (shunt), needle arm transformer, needle arm mass (series), pivot elasticity (series), needle arm elasticity (shunt), elasticity of attachment point of arm to spider (series), spider mass (series), diaphragm edge elasticity (series), air chamber elasticity (shunt), air chamber transformer, horn to air impedance (shunt).
Figure 2. Harrison's phonograph mechanism and its electrical equivalent circuit.
A diagram of a phonograph mechanism with the mechanical impedance of each part shown in an equivalent circuit diagram. The correspondence between mechanical parts and equivalent circuit elements is not indicated on the diagram (this is explained in the text of Norton's patent). A second circuit diagram is shown with the circuit transformed to a form more convenient for analysis as explained in the article text.
Figure 3. Norton's mechanical filter together with its electrical equivalent circuit.
Three different transducers are depicted. (a) A metal rod with one end passing through a cylinder of magnetostrictive material on which is wound a coil of enamelled copper wire. (b) An upright cylindrical resonator in which there is sandwiched a horizontal layer of piezoelectric material. The Piezoelectric layer has an embedded electrode from which is leading an enamelled copper wire. (c) An upright cylindrical resonator in which there is sandwiched a vertical layer of piezoelectric material with an electrode as in (b).
Figure 4. Mechanical filter transducers. a magnetostrictive transducer. b Langevin type piezoelectric transducer. c torsional piezoelectric transducer.
Various resonator shapes are depicted with the vibration direction indicated with arrows and the vibration nodes indicated with broken lines.
Figure 5. Some possible vibrational modes of resonators; The modes shown are (5a) the second longitudinal mode fixed at one end, (5b) the first torsional mode, (5c) the second torsional mode, (5d) the second flexural mode, (5e) the first radial expansion mode, and (5f) the first radially symmetric drumhead mode. [ 36 ]
A chain of four disc resonators coupled together with metal rods at the edges at the edges of the discs. Transducers at either end are of the magnetostrictive type with small bias permanent magnets near each. The transducers are coupled to the centre of the first and last resonator respectively with a metal rod
Figure 6 . A mechanical filter using disc flexural resonators and magnetostrictive transducers
A chain of six thin cylindrical resonators with piezoelectric transducers at either end. The resonators are arranged in a compact zigzag pattern. Two coupling rods are attached to one end of each resonator, except for the first and last which have only one each. The other end of these two rods are attached to the resonator either side. Transducers on the first and last resonator are of the type in figure 4b.
Figure 7. A filter using longitudinal resonators and Langevin type transducers
A chain of five upright cylindrical resonators. They are coupled together with two horizontal rods both attached to the same side of the resonators. The input transducer is of the type in figure 4c and the output transducer is of the type in figure 4a. This last has a small bias magnet nearby.
Figure 8a. A filter using torsional resonators. The input is shown with a torsional piezoelectric transducer and the output has a magnetostrictive transducer.
A ladder topology filter circuit diagram consisting of five series LC circuits interspersed with four shunt capacitors
Figure 8b. Equivalent circuit of the torsional resonator circuit above
A chain of five disc-shaped resonators coupled together with a cylindrical bar through their centres. Attached to either end of the central bar are transducers of the type shown in figure 4a. Each of these has a small bias magnet nearby.
Figure 9. A filter using disc drumhead resonators
Three thin disc resonators are coupled together with long rods towards the edge of the discs. Transducers on the first and second disc are coupled with rods connected on the opposite edge of the disc. The transducers are of the type shown in figure 4a and each has a small bias magnet nearby. A pair of pivots are shown on each disc at the 90° positions relative to the coupling rods.
Figure 10a. A semi-lumped design using disc flexural resonators and ⁠ λ / 2 ⁠ coupling wires
Circuit diagram depicting a ladder topology filter. The series branches consist of series LC circuits (three total) and the shunt branches consist of shunt LC circuits (two total).
Figure 10b. Equivalent circuit of the semi-lumped circuit above
A generic filter is depicted, consisting of a chain of coupled resonators, in four different bridging configurations. Filter (a) has no bridging wires, (b) has briding between resonators 3 and 5, (c) has bridging between resonators 2 and 5, and (d) has briding between resonators 2 and 6 and resonators 3 and 5. The bandpass frequency response of each is shown and is described in the article text.
Figure 11. Schematic bridging arrangements and their effect on frequency response.
A photomicrograph of a vibrating cantilever. Cursors on the picture indicate that the peak-to-peak amplitude of the vibration is 17.57μm.
Figure 12. MEMS cantilever resonator. The device can be seen to be vibrating in this picture.