Mechanical filter

A mechanical filter is a signal processing filter usually used in place of an electronic filter at radio frequencies. At the input and output of the filter there are transducers which convert the electrical signal into, and then back from, a mechanical vibration.

The components of a mechanical filter are all directly analogous to the various elements of an electrical circuit. It is therefore possible to apply electrical network analysis and filter design techniques to mechanical filters. This aids electrical engineers when they are designing such circuits. With the right choice of component values it is possible to produce any of the classic frequency responses. Steel and nickel-iron alloys are common materials for mechanical filter components with nickel sometimes being used for the input an output couplings.

While the usual meaning of mechanical filter is one that is used in an electromechanical role, it is perfectly possible to use a mechanical design to filter mechanical vibrations or sound waves (which are also essentially mechanical in nature) directly. For example, consideration of the effect on frequency response of mechanical components arises during the design of loudspeaker cabinets.

Mechanical filters find many applications in radio transmitters and high-end receivers, particularly the intermediate frequency (IF) stage, mostly because mechanical resonators can be manufactured with a "quality factor" Q far higher than an all-electrical LC circuit can produce. Mechanical filters can consequently be built with excellent selectivity characteristics, an important performance measure in radio.

A mechanical filter using disc flexural resonators and magnetostrictive transducers.

Elements

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. The mechanical counterparts of these properties are, respectively, mass, stiffness and damping. In most designs of electronic filter, only inductor and capacitor elements are used in the body of the filter (although the filter may well be terminated with resistors at the input and output). 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 also.

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. From this, a mechanical impedance can be defined which entirely follows the electrical analogy.

Mechanical element	Formula (in one dimension)	Mechanical impedance	Electrical counterpart
Stiffness, S	$S={F \over x}$	$Z={S \over j\omega }$	Elastance, ( = 1/C)
Mass, M	$M={\frac {F}{dv/dt}}={F \over a}$	$Z=j\omega M\,$	Inductance, L
Damping, D	$D={F \over v}$	$Z=D\,$	Resistance, R

• Note: the mechanical quantity compliance can be used instead of stiffness to give a more direct correspondence to capacitance, but stiffness is used in the table as the more familiar quantity.

A minority of sources (but including Rockwell Collins Inc., a principal manufacturer of mechanical filters) would have force corresponding to current and velocity corresponding to voltage. This has equally valid results, but is at variance with the above table where now M → C, S → 1/L, D → G and G is electrical conductance. Equivalent circuits produced by this scheme are similar, but are the dual impedance forms whereby parallel capacitors become series inductors and vice versa and so on. The scheme presented in the table is called the impedance analogy and is readily understood by electrical engineers. The dual scheme is called the mobility analogy and is not so intuitive for electrical engineers. However the dual analogy places the components on the circuit diagram in much the same topology as the mechanical components are physically laid out. This makes it more readily understood by mechanical engineers.

A solid mechanical component will unavoidably possess both mass and stiffness. This translates in electrical terms to an LC circuit, hence mechanical components are resonators and are often used as such. 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. Capacitors may be made of thin, but stiff rods, that is, the mass is minimised and the stiffness is maximised. Inductors, on the other hand, may be made of large squat shaped pieces which maximise the mass in comparison to the elasticity of the piece.

Mechanical parts act as a transmission line for mechanical vibrations. If the wavelength is short in comparison to the part then a lumped element model as described above is no longer adequate. Just as electronic components in an analogous situation need to be described by distributed element models and filters designed as distributed element filters, so too do mechanical filters, and the elements of their equivalent electrical circuit will be distributed elements.

Transducers

Fig 1. Mechanical filter transducers. a magnetostrictive transducer. b Langevin type piezoelectric transducer. c torsional piezoelectric transducer.

There are two general types of transducer used with mechanical filters. These are magnetostrictive and piezoelectric types. Piezoelectric is favoured in more recent designs since the piezoelectric material can also be used as one of the resonators of the filter, thus saving a component and space. They also avoid the susceptibility to extraneous magnetic fields from which the magnetostrictive type suffers.

Magnetostrictive transducer

A magnetostrictive material is one which changes shape when a magnetic field is applied. In reverse, it produces a magnetic field when distorted. The magnetostrictive transducer requires a coil around the magnetostrictive material. The coil either induces a magnetic field in the transducer and sets it in motion or else picks up an induced current from the motion of the transducer at the filter output. It is also usually necessary to have a small magnet to bias the magnetostrictive material into its operating range. 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.

The usual magnetostrictive materials used for the transducer are either ferrite or compressed powdered iron. Mechanical filter designs often have the resonators coupled with steel or nickel-iron wires, but on some designs, especially older ones, nickel wire may be used for the input and output rods. This is because is possible to wind the transducer coil directly on to a nickel coupling wire since nickel is slightly magnetostrictive. However, it is not strongly so and coupling to the electrical circuit is weak. This scheme also has the disadvantage that there are no measures taken to prevent eddy currents, a problem not encountered if ferrites are used.

The coil of the transducer adds some inductance on the electrical side of the filter. 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. While this will not improve performance to the extent that an additional mechanical resonator would, there is some benefit and the coil has to be there in any case.

Piezoelectric transducer

A piezoelectric material is one which changes shape when an electric field is applied. In reverse, it produces an electric field when it is distorted. A piezoelectric transducer, in essence, is made simply by plating electrodes on to the piezoelectric material. Common piezoelectric materials are quartz and some ceramics, particularly lead zirconate titanate (abbreviated PZT).

One type of piezoelectric transducer is the Langevin type, named after a transducer used by Paul Langevin in early sonar research. This is good for longitudinal modes of vibration. It can be used on resonators with other modes of vibration also if the motion can be mechanically converted into a longitudinal motion. The transducer consists of a layer of piezoelectric material sandwiched transversally into a coupling rod or resonator.

Another kind of piezoelectric transducer has the piezoelectric material sandwiched in longitudinally, usually into the resonator itself. This kind is good for torsional vibration modes and is called a torsional transducer.

Resonators

Material	Q-factor
Nickel	~100
Steel	~1000
Aluminium	~10,000
Nickel-iron alloy	10,000 to 25,000 depending on composition

It is possible to achieve an extremely high Q with mechanical resonators. A torsional resonator made from nickel-iron alloy can have a Q as high as 25,000, and 10,000 is not atypical for mechanical resonators generally. This is an unreasonably high figure to achieve with LC circuits, whose Q is limited by the resistance of the inductor coils.

Early designs in the 1940s and 1950s started by using steel as a resonator material. This has given way to nickel-iron alloys; primarily to maximise the Q since this is often the primary attraction of mechanical filters rather than price. Some of the metals that have been used for mechanical filter resonators and their Q are shown in the table.

Piezo-electric crystals are also sometimes used in mechanical filter designs. This is especially true for resonators that are also acting as transducers for inputs and outputs.

One advantage of mechanical filters is that they can be made very stable. Figures of 1.5 parts per billion (ppb) over the operating temperature range and 1 part per million (ppm) drift with time have been achieved. This stability with temperature is another cause for using nickel-iron as the resonator material. Additional small amounts of certain other elements in the alloy can produce a material with a Young's modulus that changes sign from negative through zero to positive with temperature. Such a material will have a zero coefficient of temperature with resonant frequency around that point. It is possible to adjust the point of zero temperature coefficient to a desired position by heat treatment of the alloy.

Resonator modes

Fig 2. Some possible vibrational modes of resonators.

It is usually possible for a mechanical part to vibrate in a number of different modes, however the design will be based on a particular vibrational mode and the designer will take steps to try to restrict the resonance to this mode. As well as the straightforward longitudinal mode some others which are used include flexural mode, torsional mode, radial mode and drumhead mode.

When the vibration is in one of the higher modes, there will be multiple nodes on the resonator where there is no motion. For some types of resonator, this can provide a convenient place to make a mechanical attachment for structural support. Wires attached at nodes will have no effect on the vibration of the resonator or the overall filter response. In figure 2, some possible anchor points are shown as wires attached at the nodes. The modes shown are (2a) the second longitudinal mode fixed at one end, (2b) the first torsional mode, (2c) the second torsional mode, (2d) the second flexural mode, (2e) first radial expansion mode and (2f) first radially symmetric drumhead mode.

Circuit designs

Fig. 3. A filter using longitudinal resonators and Langevin type transducers.

There are a great many combinations of resonators and transducers that can be used to construct a mechanical filter. A selection of some of these is shown in the diagrams. The picture in the introduction of the article shows a filter using disc flexural resonators and magnetostrictive transducers. The transducer drives the centre of the first resonator setting it vibrating, the edges of the disc moving in antiphase to the centre when the driving signal is at, or close to, resonance and the signal is transmitted through the connecting rods to the next resonator. When the driving signal is not close to resonance, there is little movement at the edges, and the filter rejects (does not pass) the signal. Figure 3 shows a similar idea with longitudinal resonators connected together in a chain by connecting rods. In this diagram, the filter is driven by piezoelectric transducers. It could equally well have used magnetostrictive transducers. Figure 4 shows a filter using torsional resonators. In this diagram, the input has a torsional piezoelectric transducer and the output has a magnetostrictive transducer. This would be quite unusual in a real design, both input and output usually have the same type of transducer. The magnetostrictive transducer is only shown here to demonstrate how longitudinal vibrations may be converted to torsional vibrations and vice versa. Figure 5 shows a filter using drumhead mode resonators. The edges of the discs are fixed to the casing of the filter (not shown in the diagram) so the vibration of the disc is in the same modes as the membrane of a drum. This type of filter is called a disc wire filter by Collins.

Fig. 4a. A filter using torsional resonators. The input is shown with a torsional piezoelectric transducer and the output has a magnetostrictive transducer.

Fig. 4b. Equivalent circuit of the torsional resonator circuit above.

The various types of resonator are all particularly suited to different frequency bands. Mechanical filters with lumped elements of all kinds cover frequencies from about 5 to 700 kHz although mechanical filters down as low as a few kHz are rare. The lower part of this range, below 100 kHz, is best covered with bar flexural resonators. The upper part is better done with torsional resonators. Drumhead disc resonators are in the middle, covering the range from around 100 to 300 kHz.

Fig 5. A filter using disc drumhead resonators.

The frequency response behaviour of all mechanical filters can be expressed as an equivalent electrical circuit using the impedance analogy described above. An example of this is shown in figure 4b which is the equivalent circuit of the mechanical filter of figure 4a. Elements on the electrical side, such as the inductance of the magnetostrictive transducer, are omitted but would be taken into account in a complete design. The series resonant circuits on the circuit diagram represent the torsional resonators, and the shunt capacitors represent the coupling wires. The component values of the electrical equivalent circuit can be adjusted, more or less at will, by modifying the dimensions of the mechanical components. In this way, all the theoretical tools of electrical analysis and filter design can be brought to bear on the mechanical design. Any filter realisable in electrical theory can, in principle also be realised as a mechanical filter. In particular, the popular finite element approximations to an ideal filter response of the Butterworth and Chebyshev filters can both readily be realised. 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.

Semi-lumped designs

Fig. 6a. A semi-lumped design using disc flexural resonators and λ/2 coupling wires.

Fig. 6b. Equivalent circuit of the semi-lumped circuit above.

Frequencies of the order of MHz are above the usual range for mechanical filters. The components start to become very small, or alternatively the components are large compared to the signal wavelength. The lumped element model described above starts to break down and the components must be considered as distributed elements. The frequency where the transition from lumped to distributed models takes place is a lot lower for mechanical filters than it is for their electrical counterparts. This is because mechanical vibrations travel at the speed of sound for the material the component is composed of. For solid components, this is many times (x15 for nickel-iron) the speed of sound in air but still considerably less than the speed of electromagnetic waves. Consequently, mechanical wavelengths are much shorter than electrical wavelengths for the same frequency. Advantage can be taken of these effects by deliberately designing components to be distributed elements, and the components and techniques used in electrical distributed element filters can be brought to bear. The equivalents of stubs and impedance transformers are both achievable. Designs which use a mixture of lumped and distributed elements are referred to in electronics as semi-lumped.

An example of such a design is shown in figure 6a. The resonators are disc flexural resonators similar to those shown in the top diagram, except that these are energised from an edge, leading to vibration in the fundamental flexural mode with a node in the centre, whereas the top diagram design is energised in the centre leading to vibration in the second flexural mode at resonance. The resonators are mechanically attached to the housing by pivots at right angles to the coupling wires. The pivots are to ensure free turning of the resonator and minimise losses. The resonators are treated as lumped elements; however, the coupling wires are made exactly one half wavelength (λ/2) long and are equivalent to a λ/2 open circuit stub in the electrical equivalent circuit. For a narrow-band filter, a stub of this sort has the approximate equivalent circuit of a parallel shunt tuned circuit as shown in figure 6b. Consequently, the connecting wires are being used in this design to add additional resonators into the circuit and will have a better response than one with just the lumped resonators and short couplings.

For even higher frequencies, microelectromechanical techniques can be used as described below.

Bridging wires

Fig.7. Schematic bridging arrangements and their effect on frequency response.

Bridging wires are rods that couple together resonators that are not adjacent. They can be used to produce poles of attenuation in the stopband. This has the benefit of increasing the stopband rejection. When the pole is placed near the passband edge, it also has the benefit of increasing roll-off and narrowing the transition band. The typical effects of some of these on filter frequency response are shown in figure 7. Bridging across a single resonator (figure 7b) can produce a pole of attenuation in the high stopband. Bridging across two resonators (figure 7c) can produce a pole of attenuation in both the high and the low stopband. Using multiple bridges (figure 7d) will result in multiple poles of attenuation. In this way, the attenuation of the stopbands can be deepened over a broad frequency range.

Microelectromechanical filters

Fig. 8. MEMS cantilever resonator. The device can be seen to be vibrating in this picture.

A new technology emerging in mechanical filtering is microelectromechanical systems (MEMS). These systems are mostly fabricated from silicon (Si), silicon nitride (Si₃N₄), or polymers. A common component used for radio frequency filtering (and MEMS applications generally), is the cantilever resonator. Cantilevers are a simple mechanical component to manufacture by techniques of masking, photolithography and etching used by the semiconductor industry, with a final undercutting etch to separate the cantilever from the substrate. The technology has great promise since cantilevers can be produced in large numbers on a single substrate – much as large numbers of transistors are currently contained on a single silicon chip.

The resonator shown in figure 8 is around 120 μm in length. Experimental complete filters with an operating frequency of 30 GHz have been produced using cantilever varactors as the resonator elements. The size of this filter is around 4×3.5 mm. 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. Extremely high Q resonators can be made with this technology, flexural mode resonators with a Q in excess of 80,000 at 8 MHz are reported.

History

Harmonic telegraph

Mechanical filter design was developed by carrying over into mechanics the discoveries made in electrical filter theory. However, a very early (1870s) example of acoustic filtering was the "harmonic telegraph", which arose precisely because electrical resonance was poorly understood but mechanical, in particular, acoustic, resonance was very familiar to engineers. This situation was not to last for long, electrical resonance had been known to science for some time before this, and it was not long before engineers started to produce all-electric designs for filters. In its time, though, the harmonic telegraph was of some importance. The idea was to combine several telegraph signals on to one telegraph line by what would now be called frequency division multiplexing thus saving enormously on line costs. The key of each operator activated a vibrating electromechanical reed which converted this vibration into an electrical signal. Filtering at the receiving operator was achieved by a similar reed tuned to precisely the same frequency, which would only vibrate and produce a sound from transmissions by the operator with the identical tuning.

Versions of the harmonic telegraph are due to Elisha Gray, Alexander Graham Bell (1870s), Ernest Mercadier and others. Its ability to act as sound transducers to and from the electrical domain was to inspire the invention of the telephone.

Mechanical equivalent circuits

Once the basics of electrical network analysis began to be established, it was not long before the ideas of complex impedance, and later on, filter design theories, were carried over into mechanics by analogy. Kennelly, who was also responsible for introducing complex impedance, and Webster were the first to extend the concept of impedance into mechanical systems (1920). Mechanical admittance and the associated mobility analogy came much later and are due to Firestone (1932).

It was not enough to just develop a mechanical analogy. This could be applied to problems that were entirely in the mechanical domain, but for mechanical filters with an electrical application it is necessary to include the transducer in the analogy as well. Poincaré (1907) was the first to describe a transducer as a pair of linear algebraic equations relating electrical variables (voltage and current) to mechanical variables (force and velocity). These equations can be expressed as a matrix relationship in much the same way as the z-parameters of a two-port network in electrical theory, to which this is entirely analogous,

${\begin{bmatrix}V\\F\end{bmatrix}}={\begin{bmatrix}z_{11}&z_{12}\\z_{21}&z_{22}\end{bmatrix}}{\begin{bmatrix}I\\v\end{bmatrix}}$

Wegel (1921) was the first to express these equations in terms of mechanical impedance as well as electrical impedance. The element $z_{22}\,$ is the open circuit mechanical impedance, that is, the impedance presented by the mechanical side of the transducer when no current is entering the electrical side. The element $z_{11}\,$, conversely, is the clamped electrical impedance, that is, the impedance presented to the electrical side when the mechanical side is clamped and prevented from moving (velocity is zero). The remaining two elements, $z_{21}\,$ and $z_{12}\,$, describe the transducer forward and reverse transfer functions respectively. Once these ideas were in place, engineers were able to extend electrical theory through into the mechanical domain and analyse an electromechanical system as a unified whole.

Sound reproduction

Fig. 9. Harrison's phonograph mechanism and its electrical equivalent circuit.

An early application of these new theoretical tools was to phonographic sound reproduction. A recurring problem with early phonograph designs was that mechanical resonances in the pickup and sound transmission mechanism caused wild peaks and troughs in the frequency response, resulting in poor quality sound. Harrison (1923) of the Western Electric Company filed a patent for a phonograph in which the mechanical design is entirely represented as an electrical circuit. 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 according to the impedance analogy into lumped components. The circuit arrived at is a ladder topology of series resonant circuits coupled by shunt capacitors. This can be viewed as a bandpass filter circuit. Harrison designed the component values of this filter to have a specific passband corresponding to the desired audio passband (in this case 100 Hz to 6 kHz) and a flat response. Translating these electrical element values back into mechanical quantities provided specifications for the mechanical components in terms of mass and stiffness which can be translated into physical dimensions for their manufacture. The resulting phonograph has a flat frequency response in its passband and is free of the resonances previously experienced. Shortly after this, Harrison filed another patent using the same methodology on telephone transmit and receive transducers.

Fig. 10. Norton's mechanical filter together with its electrical equivalent circuit.

The filter theory used by Harrison was the image filter theory of Campbell which was the most advanced theory available at the time. In this theory, the problem is viewed essentially as an impedance matching problem. More advanced filter theory was brought to bear on this problem by Norton (1929) at Bell Labs. Norton followed the same general approach though he later described to Darlington the filter he designed as being "maximally flat". This is interesting because Norton's mechanical design predates the paper by Butterworth who is usually credited as the first to describe the electronic maximally flat filter. The equations Norton gives for his filter correspond to a singly terminated Butterworth filter, that is, one driven by an ideal voltage source with no impedance, whereas the form more usually given in texts is for the doubly terminated filter with resistors at both ends, making it hard to recognise the design for what it is. Another interesting feature of Norton's filter design arises from the series capacitor, which represents the stiffness of the diaphragm. This is the only series capacitor in Norton's representation, and without it, the filter could be analysed as a low-pass prototype. Norton moves the capacitor out of the body of the filter to the input at the expense of introducing a transformer into the equivalent circuit (Norton's figure 4). Norton has used here the "turning round the L" impedance transform to achieve this.

The definitive description of the subject from this period is Maxfield and Harrison's 1926 paper. There, they describe not only how mechanical bandpass filters can be applied to sound reproduction systems, but also apply the same principles to recording systems and describe a much improved disc cutting head.

Mass production

The first volume production of mechanical filters was undertaken by Collins Radio Company beginning in the 1950s. These were originally designed for telephone frequency-division multiplex applications where there is commercial advantage in using high quality filters. Precision and steepness of the transition band leads to a reduced width of guard band, which in turn leads to the ability to squeeze more telephone channels into the same cable. However, they quickly also found popularity in VHF/UHF radio IF stages of the high end (transmitters, military, marine, amateur radio and the like) radio sets manufactured by Collins. They were favoured in the radio application because they could achieve much higher Q-factor than the equivalent LC filter. They also had an advantage in stability over both LC filters and monolithic crystal filters. The most popular design for radio applications was torsional resonators because radio IF typically lies in the 100 to 500 kHz band.

See also

• Ceramic resonator
• Surface acoustic wave
• Crystal oscillator

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