Basic Technology of Quartz Crystal Filters

1. Monolithic Crystal Filters vs. Discrete Crystal Filters

There are two types of filter design technologies based on quartz crystal: Monolithic Crystal Filter (MCF) and Discrete Crystal filter.  MCF is a filter that realizes the characteristic of filter by mounting electrodes on a piezoelectric substrate (such as quartz blank) and using a mechanical combination occurring between the electrodes.  MCFs are usually smaller, more reliable and more cost effective than discrete crystal filters because they use fewer components and have fewer interconnections.  The majority of MCFs are symmetrical band pass filters.  In addition, in the VHF range, the MCF approach allows bandwidths on overtones that can only be realized with more costly fundamental mode resonators if a discrete crystal filter design is used.  On the other hand, discrete crystal resonator filters have better power handling capabilities than MCFs.  They may be a better design choice for very narrow or very wide bandwidth because of their more flexibility in the network topology.  This allows the design of networks that have sharply asymmetrical single sideband performance, which is difficult to achieve with monolithic designs.  MCF is first developed by Toyocom in Japan in 1962, and since then many kinds of high reliability filters in a wide frequency range in small packages have been developed.  Today, MCF is one of the most widely used frequency control components in radio communication equipments.

2. Crystal Filter Bandwidth Categories

Because of bandwidth limitations imposed by the material properties of quartz (represented in the resonator equivalent circuit by the ratio of static to motional capacitance C₀/C₁), crystal filters, both monolithic and discrete, are commonly categorized according to fractional bandwidth as follows:

Narrow Band Crystal Filters

A narrow band filter is one in which the network can be designed so that the crystal static capacitances can be accommodated without the use of inductors.  The capacitance ratio C₀/C₁ of the resonator equivalent circuit determines the maximum bandwidth for the inductorless narrow band crystal filters.  For filters using fundamental mode AT-cut quartz resonators, under ideal conditions this maximum bandwidth is approximately 0.32% of the center frequency.  If the required bandwidth exceeds this limit, the network design must be changed to incorporate inductors.

Intermediate Band Crystal Filters

Intermediate band filters use inductors to remove excess capacitance presented by the resonator C₀ plus unavoidable stray capacitance.  Most intermediate band designs use discrete resonators but they may incorporate monolithic dual resonators. Bandwidths are between 0.3% and 1.0% for fundamental mode resonators.  Spurious responses will sometimes be present in the filter passband, as well as in the transition region, and will usually be fairly strong in the filter stopband.  Those spurious responses can be controlled by proper design tactics to achieve required performance.

Wide Band Crystal Filters

Wide band crystal filters provide the final link between crystal and LC filters by using inductors to contribute poles to the filter response, while at the same time accommodating the resonator static capacitance.  For this reason, the filter response is quite sensitive to inductor and Q values and requires precise temperature compensation of inductors if performance is to be maintained.  Bandwidths are between 1% and 10% of center frequency.  Discrete resonators are generally used. When AT-cut crystals are used, spurious responses will often be present in the filter passband and transition region.

3. Attenuation Specifications

Attenuation is the power loss (in dB) incurred by a signal in passing through a two-port network, such as a filter.  Absolute Attenuation may be defined as the attenuation relative to either a direct connection of load to source or the available source power.  For conjugate or resistive and equal source and load impedances, the two definitions are equivalent.  Relative Attenuation is the attenuation measured relative to a reference, usually the point of minimum loss (maximum transmission).  Guaranteed Attenuation is the maximal guaranteed attenuation at the specified frequency. 

4. Response Characteristics

The type of filter design depends on the desired response characteristics. The complexity of the filter design and the number of crystals used in a filter depends on the required shape factor and the attenuation.  Common design types are discussed below.

Bessel or Linear Phase :  The transfer function of the filter is derived from a Bessel polynomial.  It produces filters with a flat delay around center frequency. The more poles used, the wider the flat region extends. The roll-off rate is poor.  This type of filter is close to a Gaussian filter.  It has poor VSWR and loses its maximally flat delay properties at wider bandwidths.

Butterworth: The transfer function of the filter offers maximally flat amplitude.  Selectivity is better than Gaussian or Bessel filters, but at the expense of delay and phase linearity.  For most bandpass designs, the VSWR at center frequency is extremely good.  Butterworth filters are usually the least sensitive to changes in element values.

Chebyshev:  The transfer function of the filter is derived from a Chebychev equal ripple function in the passband only.  These filters offer performance between that of Elliptic function filters and Butterworth filters.  For the majority of applications, this is the preferred filter type since they offer improved selectivity, and the networks obtained by this approximation are the most easily realized.

Elliptic Function:   The passband ripple is similar to the Chebyshev but with greatly improved stopband selectivity due to the addition of finite attenuation peaks.  The network complexity is increased over the Butterworth or Chebyshev, but it still yields practical realizations over nearly the entire operating region.

Gaussian:  The transfer function of the filter is derived from a Gaussian function.  The step and impulse response of a Gaussian filter has zero overshoot.  Rise times and delay are the lowest of the traditional transfer functions.  These characteristics are obtained at the expensive of poor selectivity, high element sensitivity, and a very wide spread of element values.  Gaussian filter is very similar to the Bessel except that the delay has a slight "hump" at center frequency and the rate of roll-off is slower.  Because of the delay response, the ringing characteristics are better than the Bessel.  Realization restrictions also apply to these filters.

Synchronously Tuned:   These filters have the same advantages and disadvantages as the Bessel and Gaussian except that the ringing response is the best of all design types and the roll-off is even slower that the Gaussian. As with the other two types, some realization restrictions apply.

Chebyshev Phase Error:   In this approximation the Chebyshev approximating technique is applied to the phase (delay) over the passband region.  It produces the bell shaped amplitude response similar to a Gaussian or Bessel design and an equiripple phase and delay response.  The selectivity is better than the Bessel or Gaussian.

Gaussian to 6 (or 12) dB:   This approximation has a passband response that follows the Gaussian shape and, at either the 6 or 12 dB point, the response changes and follows the Butterworth characteristic.  The phase, or delay, response is somewhat improved over a strict Butterworth and the attenuation is better than the pure Gaussian and so it is a true compromise type of approximation.  As with all of the filters where there is an attempt to control the phase response, the realization becomes more difficult and so its operating region is slightly restricted.

5. Stability of Crystal Temperature Filters

There are some crystals filters built below 1 MHz, but today, most are above and use AT-cut crystals.  While coils or other factors can influence the stability, the crystals are the main controlling components, and in some filter such as MCF the only ones.  Therefore, the stability can be linked to the stability of the AT-cut crystals.  The frequency change vs. temperature follows a cubic characteristic as shown below.  The variation of the family of curves is controlled by slight changes in the angle at which the individual crystal blank is cut from the crystal bar.  Each curve is offset from the adjacent one by a change of only two minutes of arc.  The designer will select the proper curve (angle) to give the minimum deviation over the specified temperature range.

6. Frequency and Bandwidth of Crystal Filters

Crystal filters can be built with center frequencies ranging from a few kilo Hertz to several hundred Mega Hertz, but the best operating regions fall where the dimensions and operating parameters of the crystals are near their optimum.

The best operating range for "Low" frequency filters is from about 100 kHz to 800 kHz and with bandwidths that lie within the following chart.

The best operating range for "Mid" frequency crystals falls between about 2 MHz and 50 MHz.  It is noted that this leaves a band of frequencies between 800 kHz and 2 MHz not covered.  This does not mean that crystal filters cannot be built within this range; it only means that it is more difficult, and therefore more costly.

Crystal filters with center frequencies above 30 MHz can be built using either overtone mode crystals or high frequency fundamentals.  The overtones have much higher Qs and are suitable for the narrower bandwidths, while the fundamentals have much lower impedance and are used for the wider bandwidths.  However, all crystals have spurious responses, and the spurs normally occur on the high side of the passband.  This spurious characteristic limits the maximum bandwidth that a filter can be achieved.

7. Center Frequency and Nominal Frequency

Center frequency is a given frequency in the specification, to which other frequencies may be referred, while nominal frequency is the nominal value of center frequency and is used as the reference frequency for specifying relative levels of attenuation.  In bandpass and bandstop filters F_on denotes the nominal center frequency; F_o denotes the actual or measured center frequency of an individual filter and is usually defined as:

                      F_o = (f_l × f_u)^½

Where f_l and f_u are measured lower and upper passband limits, usually the 3 dB attenuation frequencies.  Sometimes it is more convenient to specify frequency relative to the actual or measured filter center frequency.  The value of F_o will, of course, vary from unit to unit and within the same unit as function of temperature and time.  Therefore, there must be a tolerance associated with F_o, making allowance for temperature, aging, and manufacturing tolerances.

8. Passband, Stopband & Bandwidth

Passband is the frequency range in which a filter is intended to pass signals.  It is expressed as a range of frequencies attenuated less than the specified value, typically specified at 3 dB.

Stopband is a band of frequencies in which the relative attenuation of a filter is equal or great than specified values.  It is expressed as a range of frequencies attenuated by more than some specified minimum, such as 60 dB.

For a bandpass or bandstop filter, the width (frequency difference) between lower and upper points having a specified attenuation, such as the 3 dB bandwidth or the 80 dB bandwidth.  For a lowpass filter, bandwidth is simply the frequency at which the attenuation has the specified value.

9. Ripple and Passband Ripple

Generally referring to the wavelike variations in the amplitude response of a filter with frequency.  Ideal Chebychev and elliptic function filters, for example, have equal-ripple characteristics, which means that the differences in peaks and valleys of the amplitude response in the pass band are equal.  Butterworth, Gaussian, and Bessel functions, on the other hand have no ripple.  Ripple is usually measured in dB.

The pass band ripple is defined as the difference between the maximum and minimum attenuations within a pass band.

10. Shape Factor

Shape factor is the ratio of the stopband bandwidth to the passband bandwidth, typically the ratio of 60 dB bandwidth to the 3 dB bandwidth.  It is a critical parameter that determines the number of poles and complexity required to meet the specification.

11. Insertion Loss

The frequency response of filters is always considered as relative to the attenuation occurring at a particular reference.  The actual attenuation at this reference is commonly called insertion loss.  It is referenced at the minimum attenuation point within the pass band.  Insertion loss can be defined as the logarithmic ratio of the power delivered to the load impedance before insertion of the filter to the power delivered to the load impedance after insertion of the filter.  In other words, it is the decrease in power delivered to the load when a filter is inserted between the source and the load.  The insertion loss is given by:

                    IL_dB = 10log (P_L1 / P_L2)

Where P_L1 is the power delivered to the load with filter bypassed and P_L2 is the output power with filter inserted into the circuit.  The equation above can also be expressed in terms of a voltage ratio as:

                    IL_dB = 20log (V_L1 / V_L2)

This allows insertion loss to be measured directly in terms of output voltage.

12. Insertion Loss Linearity

The insertion loss of a filter may change with drive level.  At high power levels, quartz resonators become non-linear causing the filter loss to increase, and this phenomenon is primarily determined by properties of the quartz, not by processing of the filters.  However, at low drive levels resonator processing becomes critical in maintaining constant insertion loss.  With the application of proper design, stringent processing and rigid controls, filters have been being produced with no more than ± 0.005 dB change in insertion loss for a 40 dB change in drive level.

13. Spurious Responses

All resonators, whether they are LC tuned circuits, cavity resonators or crystal resonators have unwanted resonance modes.  Quartz crystals have anharmonic resonance normally occurring just above the desired resonance as well as near- harmonic overtone responses.  Consequently, almost all crystal filters will exhibit unwanted responses in their amplitude and phase characteristics. The deviations are often, but not always, of narrow bandwidth.  Normally they occur in the filter stopband and appear as narrow, unwanted regions of reduced attenuation.  Spurious response usually appears at a higher frequency than the center frequency.  Occasionally in wider bandwidth filters a spurious response may occur in the filter passband, causing undesirable ripple.

The AT- cut crystal resonator, which is most commonly used for filters, has a family of unwanted anharmonic responses at frequencies slightly above the desired resonance and harmonic (overtone) responses at approximately odd integer multiples of the fundamental resonance.  The location of the overtones and the major anharmonics can be calculated in advance.  The overtone responses can be suppressed by additional LC filtering which, given adequate package dimensions, can be accommodated inside the filter package if required.  The near-by anharmonic responses cannot normally be suppressed by LC filtering.  Here suppression of spurious responses is accomplished by a combination of resonator design, resonator processing and filter circuit design.

As the crystal resonator electrode area is increased, more unwanted anharmonic responses will be excited (assuming a constant operating frequency) and the motional inductance will decrease.  In order to reduce insertion loss and/or retain a narrow band design, it may be necessary to increase the electrode dimensions at wider bandwidths. Therefore, wider bandwidth filters can be expected to have more and stronger spurious responses.  However, one can always take advantage of narrow band design by operating the crystal filter at a higher frequency with the reduced percentage bandwidth, such that the spurious response will be improved for a given bandwidth requirements.

14. Group Delay Distortion

Group delay, also called envelope delay, is the time taken for a narrow-band signal to pass from the input to the output of a device.  Group delay distortion is the difference between the maximum and the minimum group delay within a specified pass band region or at two specific frequencies.  For most bandpass filters, the delay response will have a peak close to each passband edge, where the filter attenuation begins to increase rapidly.  Filter delay and attenuation characteristics are interdependent.  The more rapidly the filter attenuates, the larger the delay peaks.  In general, large delay peaks are associated with filters having many poles or filters that have close-in stopband poles (such as elliptic function filters).  On the other hand, the MCFs have a very small group delay distortion, typically less than 10 ms.

15. Inter-modulation (IM)

Inter-modulation occurs when a filter acts in a nonlinear manner causing incident signals to mix.  The new frequencies that result from this mixing are called inter-modulation products, and they are normally third-order products, which means that a one dB increase in the incident signal levels produces a 3 dB increase in IM.  The IM can be classified in the following two types:

Out-of-band inter-modulation occurs when two incident signals (typically -20 to -30 dBm) in the filter stopband produce an IM product in the filter passband.  This phenomenon is most prevalent in receiver applications when signals are present simultaneously in the first and second adjacent channels.  This IM performance of crystal filters at low signal levels is primarily determined by surface defects associated with the resonator manufacturing process and is not subject to analytical prediction. 

In-band inter modulation occurs when two closely spaced signals within the filter passband produce IM products that are also within the filter passband.  It is most prevalent in transmit applications where signal levels are high (typically -10 dBm and +10 dBm).  This IM performance at high signal levels is a function of both the resonator manufacturing process and the nonlinear elastic properties of quartz.  The latter is dominant at higher signal levels, and can be analyzed.

16. Phase Shift and Minimum Phase Transfer Function

The change in phase of a signal as it passes through a filter. A delay in time of the signal is referred to as phase lag and in normal networks, phase lag increases with frequency, producing a positive envelope delay.

The great majority of crystal filters are minimum phase shift filters.  Mathematically, this means that there is a functional relationship between the attenuation characteristic and the phase characteristic of the filter.  The transfer function of such a two-port network is said to have the minimum phase shift property, which means that its total phase shift from zero to infinite frequency is the minimum physically possible for the number of poles that it possesses.

17. Terminating Impedance

This is the required impedance to be seen on input and load side of the filter to maintain a good characteristic response.  Terminating impedance is typically specified as a series resistance with a parallel capacitance that should also include the stray capacitance of the circuit board.

18. Power Handling

Power handling is usually specified as the maximum input power. In design, it is closely related to the factors determining in- band inter-modulation performance. Given the bandwidth, insertion loss and spurious response requirements, the power handling capability of a filter can be estimated.

19. Vibration-induced Sidebands

Vibration-induced sidebands may appear on a crystal filter output signal when the filter is subjected to mechanical vibration.  Vibration produces acceleration forces on the crystal resonators, causing their resonance frequencies to change slightly -- typically a few parts per billion for one G acceleration.  For sinusoidal vibration, the resonance frequency is modulated at the frequency of vibration, and the peak deviation is determined by the acceleration sensitivity of the crystal resonator and the amplitude of vibration.  Viewed on a spectrum analyzer, the filter output will have sidebands offset from the carrier by the frequency of vibration.  For most filters, the vibration-induced sidebands are quite small and of no concern. However, narrowband spectrum cleanup filters may require special attention.  Vibration-induced sidebands are minimized by minimizing resonator acceleration sensitivity and by control of mechanical resonance within the filter structure.

20. Settling Time and Rise Time

Settling time is the time it takes for the output signal to settle within a specified overshoot percentage after the input has been subjected to a step response, pulse, impulse, or ramp.

Rise time is often defined as the time required for the output of a filter to move from 10% to 90% of its steady state value on the initial rise.  While the exact value of rise time can readily be calculated or determined from filter handbooks, the following rule of thumb relating rise time to bandwidth provides an useful estimates.

                    T_r = 0.35 / f_c

Where Tr is the rise time in seconds and  f_c is the 3 dB cut off frequency in hertz.