A History of Crystal Filters
Proceedings of the 1998 IEEE International Frequency Control Symposium © 1998 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists or to reuse any copyrighted component of this work in other works must be obtained from IEEE.
Robert G. Kinsman
The first use of quartz crystals as filter elements was suggested by Walter Cady in 1922 in his paper “The Piezo-Electric Resonator”, although in the initial conception a single crystal only was used as a very narrow bandwidth coupling element. The use of multiple crystals in ladder and lattice configurations was proposed in the late 1920’s and in 1934 Walter P. Mason published the results of his work on crystal filters which were developed for use in frequency-division-multiplex telephone systems. In Mason’s paper he describes a “narrowband” design which is useful for bandwidths up to about 0.4% of center frequency and a “wideband” design for bandwidths of about 2% to 6% of center frequency. The wideband designs had very limited applications because extremely low impedance crystals were required which were only available at certain frequencies. However, the narrowband design concepts were used with very little modification for the next 20 years. In the mid 1950’s new designs were developed for “intermediate-bandwidth” filters which covered the bandwidth region between the earlier narrowband and wideband designs. In the early 1960’s the more general filter design process based on “insertion loss” theory was adapted for use with crystal filters and made possible the fabrication of filters with true Chebyshev, Butterworth, and other traditional filter characteristics. In 1962 a paper was presented describing the first practical monolithic crystal filter elements and included a cascaded 6-pole design. This was a forerunner of the 2-pole filter elements in wide use today.
This paper will discuss the progression of these developments in the crystal filter art and the accompanying work on filter crystals which was essential to their success. A short discussion of measuring and modeling problems is included and the final section covers the applications for crystal filters that drove their technology.
The history of the development in crystal filter technology, from the initial concepts of Cady to the current wide range of products, provides a fascinating chapter in the development of today’s highly complex electronic products. The crystal filter has been a particularly critical element in the development of narrowband communications systems. The desire to send multiple voice messages on a single telephone line resulted in the introduction of carrier telephone systems in 1916. These early systems used LC filters in the 10 to 40 kHz frequency range. However, the bandwidth limitations caused by realizable coil Q’s were quickly recognized. In 1929, W.P. Mason of Bell Laboratories developed methods for incorporating crystals into LC lattice filter networks. This work resulted in the development of a 60 to 108 kHz basic group-band filter set used to frequency multiplex 12 voice channels. This work is described in Mason’s 1934 paper which was the basis for essentially all crystal filter designs generated during the next 20 years . During this period the major application for crystal filters was in carrier telephone equipment. However, in the mid-1950s, newer narrowband radio communications systems were developed both for military and commercial use which required high-frequency, stable, narrow-bandwidth filters. In most cases, crystal filters were the answer for these filtering applications and a new manufacturing industry was formed to supply these needs. Other applications quickly followed in navigation and radar equipment, in new fire-control systems, and missile control systems. This increased level of activity resulted in new filter design procedures and substantial improvement in the quality of high-frequency filter crystals.
In the 1960s another major technology step occurred with the development of monolithic crystal filters. Through the 1970s and ’80s evolutionary improvements were made with the development of multi-pole monolithic filters and the extension of the high frequency limits through continued process improvements. Significant theoretical work was also accomplished in this period in the area of device modeling. High frequency limits are still being pushed today with blank etching techniques and improved photo-lithography.
Professor Walter Cady, who carried out much of the original development work on quartz crystals, was apparently the first to suggest the use of crystals as filter elements . In his 1922 paper he shows single-mode crystals with divided electrodes which could be used as coupling elements between adjacent circuits. However, in this configuration, only very narrow bandwidths are achievable and the filter would be useful only as a carrier-frequency filter. In 1927 patent disclosures were filed by L. Espenschied of AT&T and C. Hansell of RCA on the use of crystals in a filter structure [3,4]. The Espenschied patent shows the use of crystals in a ladder filter structure in a variety of configurations. He also shows the use of inductors in series or in shunt with the crystals to widen the filter bandwidth. In Hansell’s patent the use of a bridge circuit is shown using all capacitors or a center-tapped transformer to balance out the crystal shunt capacitance. This is essentially the hybrid-lattice configuration which is commonly used in discrete crystal filters. He also proposed a method for widening the bandwidth by placing several crystals with slightly different frequencies in parallel. The challenge of providing filters with useful bandwidths was apparent from the earliest days and many techniques (usually unsuccessful, including Hansell’s) were attempted. In his patent Hansell makes the interesting observation that “The crystal filter has the advantage of being so sharply selective that the necessity for more than a single section probably will never arise.”
The ladder filters proposed by Espenschied, although highly selective, were too narrow for voice bandwidths in the 60 kHz carrier frequency region. The solution to this problem was found by W.P. Mason in 1929 when he developed a new type of filter using the full lattice configuration. Each arm of the lattice contained a crystal in series with an inductor. In the final configuration the series inductors were removed from the lattice arms and placed in series with the terminations. Mason expanded on this work in a landmark paper published in 1934 . In this paper he presented two basic filter configurations as shown in Figure 1. The first configuration contains only crystals and capacitors and is know as a “narrowband” design. The second circuit combines the selectivity of both crystals and LC resonant circuits and is known as a “wideband” design. As shown in, the lattice arms are designed so that the zeros of one arm are aligned with the poles of the other arm and vice-versa. The passband edges are defined by the outermost singularities of the lattice arms. Resistive termination values are selected which match the filter’s image impedance at its center frequency. Note: An excellent historical summary of the early work at Bell Telephone on both LC and crystal filters is given in a 1937 paper by O.E. Buckley .
The general design approach used by Mason is based on the “image-parameter” filter design technique proposed by Zobel in 1923 . A problem with this design approach is that the image impedance of the filter is not a constant across the passband. As a result, when the filter is terminated in a fixed resistance, passband ripple occurs towards the band edges. This condition is a minor problem for single-section filters. However, in higher order filters, where identical filter sections are cascaded, the ripple problem can be severe and resistive isolation sections were often added to minimize the problem at the expense of an increased insertion loss.
The full-lattice filter was convenient for use with balanced lines as used in telephone equipment, but is not as attractive for other applications which normally have a common ground between the input and output. As a result, most filters utilize one of the hybrid-lattice equivalent circuits shown in Figure 2. These circuits provide the desired unbalanced terminations and in addition cut the number of resonators in half. These basic filter building blocks are used in nearly all discrete crystal filter realizations.
The narrowband and wideband design approaches developed by Mason proved to be the design standards for the next 20 years. Except for the telephone channel-bank filters, the wideband designs were rarely used because they were limited to frequency ranges where low impedance crystals with low capacitance ratios were available (60 – 300 kHz with extensional X-cuts and later 10 – 30 MHz with AT-cuts). The minimum bandwidth of a wideband filter is limited by inductor Q to about 2% of its center frequency. The maximum bandwidth is a factor of the shunt capacitance of the crystals which set the inductance limits in the LC resonator circuits. Filters in the 60 – 300kHz range can be built with bandwidths up to 6% of their center frequency. In the 10 to 30 MHz range bandwith is limited to about 4% of center frequency. The narrowband designs worked very well but were limited in bandwidth by the capacitance ratio of the crystal. In examining the equivalent circuit of a crystal, as shown in Figure 3, we find a series and a parallel resonance frequency where the separation between these two frequencies is determined by the crystal’s capacitance ratio (C0/C1). An examination of the narrowband filter illustration in Figure 1a indicates that the filter bandwidth is limited to approximately twice the separation between the series and parallel resonance frequencies of the filter crystals. The maximum bandwidth achievable for a narrowband filter falls in the range of 0.2 to 0.4% of the center frequency for commonly used filter crystals.
Efforts were frequently made to extend the bandwidth of the narrowband designs by adding inductors or tuned circuits to the filter circuit, but these filters were usually touchy to align and very sensitive to temperature. In 1956 a new design procedure, using low-Q tuned circuits across the terminations, was developed by L. Storch . This procedure could be used to design filters in the narrowband range and also in the “intermediate-bandwidth” range covering the region between the narrowband and wideband designs of Mason. In this procedure some of the singularities of the lattice arms were placed in the stopband as shown in Figure 4. The passband edges were defined by series resonances and the resultant filters had very tightly controlled passband characteristics, but at the expense of more crystals than an equivalent narrowband design.
The image parameter theory as invented by Zobel in 1923 was an extremely significant development in the design of multiple element filters . Prior to that time filters consisted primarily of single reactive elements or resonators or at most a cascade of simple identical sections. From this work a filter with an unlimited number of reactive elements could be designed providing excellent stopband selectivity with reasonable end-load matching. The full significance of Zobel’s work can be appreciated when we recognize that image parameter theory was the only technique available for the design of complex filters until about 1940 and was used into the 1950’s for general LC filter design and into the early 1960’s for crystal filters. The goal of synthesizing filters to meet a prescribed mathematical function proved to be a daunting task. Synthesis of reactive one-port networks was accomplished by Foster and Cauer in the 1920’s, but the synthesis of two-port networks with realizable transfer functions was not accomplished until 1939 by Darlington and by Cauer in 1940. However, application of these new design techniques was seriously hampered by the significant computation required, particularly for bandpass filters, and they did not come into general use until the early to mid 1950’s . With the advent of reasonably priced computers in the early 1950’s, tables were developed by Weinberg, Dishal, and others for a wide variety of “modern network theory” filter designs which quickly became the standards in the LC filter industry [9,10].
In the crystal filter field, image-parameter theory was used as the primary filter design tool until the early 1960’s. In this procedure, identical filter sections were cascaded to achieve the desired selectivity, and optimized matching sections were used to minimize passband ripple. This technique was highly refined for LC filters with the use of “constant-k” and “m-derived” filter sections to provide highly selective filter realizations. These techniques were carried into the crystal filter domain with somewhat less success. The cascading of identical filter sections tends to create excessive passband ripple. For example, a 4-section, 8-pole, narrowband filter has a theoretical passband ripple in excess of 3.5 dB which is unacceptable for most applications. With optimized matching sections, these effects could be minimized but unfortunately, the narrow bandwidths of crystal filters would require elements with Q values which are unobtainable. Various design tricks using resistive isolation were utilized to minimize this problem at the expense of added insertion loss. Also, no methods were available to control the phase response to improve phase linearity or time response. These problems were finally solved by applying “modern network theory” of “insertion loss” methods to the design of crystal filters. The publication of tables for symmetrical ladder filter networks which could be used to generate filters with Butterworth, Chebyshev, and Bessel type responses was accomplished in the early and mid 1950’s. These networks were first adapted to single section crystal filters in the late 1950’s and to cascaded designs in the early 1960’s [11,12]. Today these techniques are used for the design of a wide variety of narrowband, intermediate-bandwidth, and wideband crystal filters .
The initial crystals used by Mason in his channel-bank filters were extensional mode X-cuts. These crystals use a rectangular blank with their width along the z-axis and length along the y-axis. The X-cut crystal has the lowest capacitance ratio (C0/C1 = 125) of all quartz crystal types making it very attractive for filter use. One problem with this crystal design is a strong coupling to a shear mode at a frequency approximately 40% higher than the extensional mode. Mason found that by rotating the crystal blank -18.5o off the mechanical axis (y-axis) this coupling could be eliminated at the expense of a small increase in the capacitance ration (C0/C1 = 137). A second problem with this crystal is that it has a very steep frequency/temperature characteristic (25 ppm/oC). Although this amount of frequency drift was acceptable for the channel bank filters, it would be intolerable for narrow bandwidth filters. In the years following Mason’s initial work a variety of crystal designs were developed across the frequency range from audio to approximately 1 MHz. The design goals for each of these crystal types was to provide good temperature stability in a reasonable size with a minimum of unwanted (spurious) modes. Crystal types such as the J, NT, E, DT, and CT were commonly used for filter applications and classified as low-frequency elements.
In the 1950’s the demand for higher frequency filters increased dramatically with the development of new communications, radar, and navigation equipment. Single-sideband filters in the 1.5 to 6 MHz range were needed, IF filters for new communications receivers were required at 10.7, 11.5, and 21.4 MHz and 30 MHz IF filters were needed for radar receivers. Unfortunately the AT-cut, thickness-shear crystals which had been developed for oscillator applications at these frequencies had serious problems with spurious modes. The primary concern in the design of oscillator crystals was to control the crystal resistance and to minimize frequency discontinuities across an operating temperature range. Attenuation of spurious modes by 3 to 6 dB was adequate for most applications and usually was not a serious problem for crystal designers. A substantial amount of work was carried out by many crystal designers to improve the spurious performance of the AT-cut crystals. Each developed their own “bag of tricks” but often found that a design which worked well at one frequency did not translate to other frequencies.
Over a period of time, contoured blank designs were optimized in the 1 to 6 MHz region. However, good designs for higher frequency crystals using flat blanks proved much more elusive. Designs were optimized at a particular frequency by varying the electrode geometry and metal thickness in a cut-and-try procedure. In 1961 Bechmann proposed design guidelines, for crystals using flat blanks, which specified electrode and blank diameters as a function of blank thickness . These guidelines, based on empirical data, were designed to achieve a minimum of 40 dB suppression of spurious modes. He also discussed the effects of beveling the edges of the blank. In 1963 Shockley, Curran, and Koneval presented their “energy trapping” theory to explain the behavior of AT-cut resonators . This theory relates spurious mode frequencies to the crystal’s electrode geometry and metal thickness. During this same time period work was being carried out by Professor Mindlin and his students in an effort to develop a mathematical foundation for describing the motion of the crystal resonator. Since that time extensive work has been done by Tiersten, Dworsky, and others to refine the energy trapping formulas for overtone operation and to include the effects of various electrode and blank geometries.
The earliest monolithic crystal filter could logically be traced to the low-frequency, split electrode crystal used by Cady as a coupling element . In the 1930’s the split-electrode X-cut crystal was widely used to replace two single crystals in the telephone channel-bank filters. However, these devices used a single mode of vibration and thus were essentally one-pole devices. Split-electrode thickness-shear devices were investigated by Sykes and others in the 1930’s, but were typically considered to be single-mode devices .
The concept of building a filter with dual-mode, acoustically-coupled, resonators on a single AT-cut blank was successfully tested as early as 1946 . However, the first commercially feasible filters did not appear until 16 years later. In 1962, Y. Nakazawa disclosed a 2-pole monolithic filter at 10.7 MHz which was mounted in a 3-pin, HC-18 holder. He also presented data on a 6-pole filter containing three cascaded 2-pole devices . In Nakazawa’s original design a fixed electrode pattern was used and the coupled bandwidth was altered by adjusting the diameter of the crystal blank. Later, energy trapping theory was applied to the design of monolithic filters by Onoe and Jumonji in 1965 and by Onoe, et al and by Beaver and Sykes in 1966[19,20,21]. These efforts resulted in design methods in which the coupled bandwidth is determined by the electrode geometry and metal thickness. The term monolithic crystal filter (MCF) was apparently coined by Sykes in the 1966 paper. Subsequent work by Teirsten established accurate design models for overtone mode devices as well as improved fundamental-mode models .
After its introduction by Nakazawa in 1962, the MCF was developed for commercial use and by 1965 the Japanese offered 4-pole and 6-pole cascaded designs for sale. By 1967 a standard line of 10.7 MHz devices with several bandwidths and up to 10-pole performance was offered for sale throughout the U.S. and Europe as well as Japan . In the U.S. initial progress was slow in adapting the new MCF technology. However, by 1970 new mobile radio and telephone equipments were designed incorporating 2-pole and cascaded MCF filters and their usage increased rapidly from that point on [24,25].
The monolithic crystal filter is now a commodity product which is widely used to provide IF selectivity in communications and paging receivers. Essentially all of the units manufactured today are of the 2-pole type. These devices are easily cascaded using insertion loss design techniques to provide filters with a variety of frequency response functions . Two notable exceptions exist to the 2-pole type monolithic filter. The first is the 8-pole, A6 channel-bank filter developed by Bell Laboratories. These filters operating near 8 MHz performed the same function in Bell’s frequency-division-multiplex (FDM) equipment as the original channel bank filters developed by Mason. These filters were all manufactured in a dedicated manufacturing facility with machinery developed specifically for this product . The second filter of note is the 4-pole monolithic unit developed by Motorola for use in VHF paging receivers. These filters are manufactured across the VHF band from 132 to 174 MHz. The filter units also contain an offset oscillator resonator for the receiver local oscillator which automatically temperature tracks the very narrow bandwidth filter. These units are fabricated using photolithographic techniques for electrode pattern generation and are frequency tuned by gaseous anodization .
Measuring and Modeling
Today it is hard to imagine the difficulties encountered in testing narrow bandwidth filters in the pre-frequency synthesizer/network analyzer era. Early filter developers had only signal generators and RF voltmeters available for the measurement of a filter’s response with typical signal generators having a frequency accuracy of two or three orders of magnitude. This accuracy was adequate for the relatively wide bandwidth filters used in telephone systems. However, as higher frequency filters were developed for radio applications in the 1950s, accuracies of 1 – 10 ppm were needed. Fortunately, digital frequency counters were developed in the early 1950s. Prior to this time, the most common frequency measurement instrument was a meter that was tuned manually and included a large manual of correction factors to be used in each frequency range. Good-quality, manually-tuned, RF signal generators were available, but sensitive RF voltmeters were not. Typically, filters were measured manually with a test station including a signal generator, frequency counter and RF voltmeter. At frequencies above 5 or 6 MHz, broadband RF voltmeters only permitted measurement over about a 40 dB range. If more sensitive measurements were needed they were accomplished by using a radio receiver as the RF detector. Needless to say, a very time-consuming process. The measurement of phase shift was very tedious with early instruments incorporating a combination of fixed and variable delay lines which were manually tuned for each measurement point.
About 1960, sweep frequency test stations were made possible by the availability of a motor-driven sweep drive manufactured by The General Radio Company. This sweep drive was connected to the tuning knob of a sweep generator and provided a somewhat limited range sweep length and speed. The unit also provided a sawtooth voltage for the x-axis input on an oscilloscope. By adding a logarithmic RF amplifier and detector a very usable test station was realized which substantially reduced the time required for filter alignment and testing. Early efforts by instrument manufacturers to develop electronic sweep generators were not very encouraging as the sideband noise levels were much too high for the testing of crystal filters and the mechanical stations were used for quite a number of years. A problem with this arrangement was that the bearings would wear out rather quickly on the RF generators as they were not designed for this type of continuous use.
As synthesizers with improved sideband noise were developed, the mechanically-driven signal generators were replaced and by the mid 1980’s network analyzers became available which provided a complete test station in one instrument.
Two other instruments of note:
1. The Boonton Radio Co. R-X meter, developed in the late 1950’s was the first general impedance meter which contained its own variable frequency source and read out directly in ohms and pico Farads. Earlier impedance bridges required an external signal source and detector and provided only normalized impedance values. This instrument was invaluable in the design of impedance matching networks and transformers.
2. The Hewlett-Packard Vector Voltmeter, developed in the mid 1960’s made possible the direct measurement of phase and provided an 80 dB dynamic range for amplitude measurements. This instrument covered the frequency range of 1 – 1000 MHz, which was ideal for high frequency filter work. The Vector Voltmeter was also used as an integral part of the crystal “pi” network measuring station.
Measurement problems in addition to the direct measurement of filter characteristics also existed in the characterization of the filter’s component parts. A particular challenge was the accurate measurement of the filter crystal’s motional parameters, particularly at higher frequencies. Various methods were used to measure the crystal motional capacitance, which usually involved measuring a frequency shift with a load capacitance in series with the crystal. This method was sensitive to test fixture stray capacitances and also stray inductances at higher frequencies. The development of a “standard” pi network provided a useful testing tool which included a switchable load capacitor. However, the pi network test method was primarily useful for fundamental mode crystals and was essentially unusable for crystals above 50 MHz. Any system using a test network is sensitive to the network values and is difficult to correlate from one test site to another. Attempts to measure using impedance bridges were awkward at best and generally unreliable. The advent of high quality network analyzers made possible accurate measurement of a crystal’s impedance in the vicinity of resonance and a standard test method has been developed for the complete characterization of all types of filter crystals . A method has also been developed for the characterization of 2-pole monolithic filter elements using an extension of the single crystal procedure .
A somewhat related issue is the problem of intermodulation distortion in crystal filters caused primarily by the non-linearity of the crystal elements as a function of drive level. This behavior can cause problems in 2-way radio receivers where high attenuation of adjacent channel signals is needed . In a recent paper a model has been proposed relating intermodulation distortion to particles loosely attached to the active surface of the vibrating resonator. The model is shown to predict not only intermodulation distortion but also the phenomenon of “starting resistance” well known to oscillator crystal designers .
A history of crystal filter development would not be complete without considering the applications and equipment needs which dictated their performance. As is the case with most new product development, new opportunities were recognized and these provided the incentives to push the state of the art. These applications also provided the incentives needed to start up new development operations and manufacturing facilities. Until the mid 1950’s the major manufacturers and users of crystal filters were the telephone companies. In 1954 a development group was formed at Hughes Aircraft Co. to develop a channel filter bank for a new airborne radio. In 1956 a new company, Hycon-Eastern, was started to supply packaged filters to the electronics industry. This company’s design team, headed by D. Kosowsky, generated a collection of standard filters for a variety of applications . By 1960, many crystal companies had joined the parade and were able to design and manufacture a wide variety of filters.
The needs for highly selective filters for carrier telephone use provided the background for Mason’s original work. These filters went into production in 1938 and with some evolutionary changes were manufactured until the 1970’s . It was reported in 1958 that production volume of channel bank filters at that time was of the order of 100,000 units per year at Bell Telephone. The need to reduce size and cost and improve reliability resulted in the development of the monolithic A6 channel bank filter by Bell. This development program went through several evolutionary steps resulting in a complete 8-pole filter on a single quartz blank with no additional components . A parallel program at Lenkurt resulted in the development of a cascaded or polylithic filter for use in their FDM equipment . Similar filters were also used extensively in microwave FDM systems.
Crystal filters are widely used today to provide the IF selectivity in communications receivers. As the use of two-way radio equipment increased, improved frequency utilization was a necessity. For example, the VHF band (132 – 174 MHz) was originally established for 120 kHz channel spacing for commercial 2-way FM radio use. This was later split to 60 kHz and again to 30 kHz . With each channel split, narrower bandwidth, more stable selectivity was required. By the early 1960’s most crystal filter manufacturers offered standard six and eight-pole discrete filters at 10.7 MHz with bandwidths of 15 and 30 MHz for these specific applications. In approximately 1962 the US. Army Signal Corps’ VRC-12 and PRC-25 two-way FM radios were placed in production. Both of these radios used crystal filters at 11.5 MHz with bandwidths of 32 and 36 kHz. The PRC-25 also used a crystal discriminator. The projected needs for these filters provided the impetus for the start up of many new crystal filter companies.
One of the early problems in promoting the use of crystal filters to radio receiver designers was the fact that most receiver IF amplifier sections had several gain stages with the selectivity distributed among these stages. The typical receiver with vacuum tube amplifiers and later with transistor amplifier stages used narrow-band, double-tuned circuits (IF cans) as coupling elements between the gain stages. The use of a single crystal filter to supply all of the selectivity did not fit well with this design concept. The development of the solid-state, IC amplifier changed all this and essentially forced the development of packaged filters . The first commercial product to benefit from these technologies was the Heath Company’s AR-15 stereo broadcast FM receiver which was produced in the late 1960’s. This receiver used two RCA IC amplifiers and two CTS Knights 4-pole, wideband, 10.7 MHz crystal filters. Over a period of two years approximately 40,000 filters were used in this program. This same general design philosophy is still in use today although the crystal filters in broadcast FM receivers have been replaced with lower cost ceramic filters.
In 1970 Motorola announced a new line of mobile FM radios which incorporated cascaded monolithic 2-pole filters to provide IF selectivity. In these receivers the monolithic filters were used as individual components on the IF circuit board thus eliminating the need for packaged assembled filters. The bandwidth and center frequency needs of 2-way FM radio are a perfect match for the capabilities of monolithic filters and this industry is the major user of these devices .
Another application where crystal filters are widely used is in single-sideband systems both for generating the signal in the transmitter exciter and for selectivity in the receiver. For voice applications an asymmetrical 4-pole or a symmetrical 6-pole design is usually adequate. However, in SSB data transmission systems, much tighter selectivity is needed. Under the sponsorship of the US Army Signal Corps some very complex filters were developed for these applications in the late 1950’s. One model used 18 crystals and others required a very involved alignment process [39,40].
It would be impossible to cover all of the applications which have been filled by crystal filters in navigation and fire-control systems, instruments and other equipment. However, I hope that this brief history gives a flavor of the excitement which I have enjoyed in watching and participating in the development of this fascinating technology.
I would like to gratefully acknowledge the suggestions and references supplied by Bob Smythe and Bill Horton of Piezo Technology Inc. which were a great help to me in preparing this paper.
9. L. Weinberg, Network Design by Use of Modern Synthesis Techniques and Tables, Hughes Aircraft Co., Culver City, CA, Technical Memorandum N.427 (April 1956) Also see Network Analysis and Synthesis, McGraw Hill, New York (1962)
19. M. Onoe and H. Jumonji, Analysis of Piezoelectric Resonators Vibrating in Trapped-Energy Modes, Jour. Inst. Elec. Comm. Engrs., Japan, Vol. 48 (Sept. 1965) English translation in Electronics and Comm. in Japan, Vol. 48 (Sept. 1965)
32. L. Dworsky and R.G. Kinsman, A Simple Single Model for Quartz Crystal Resonator Low Level Drive Sensitivity and Monolithic Filter Intermodulation, IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, Vol 41,(March 1994)