Comparing Quartz Crystal Measurement Methods

Wes Hayward, w7zoi
                Copyright, 27July09     Major update on 13April2010, added comments 5March11.

Abstract:
The motional parameters of a quartz crystal are determined with two different test oscillators and a vector network analyzer.    The results are compared, showing good correlation between methods.  The simple oscillator measurements appear adequate for the design of crystal filters.

Introduction

Homebrew crystal filters were common in the early days of single sideband when the radio amateur was first beginning to experiment with the mode.    These early filters were built at low frequencies (e.g., 455 kHz) where surplus crystals were available.   A later paper, but still in "the day," was published by Ben Vester, W3TLN, “Surplus-Crystal High-Frequency Filters,” QST, Jan, 1959.   This was one of those cornerstone QST papers for me, providing information to get me started with the construction of some functioning crystal filters.    The filters that Vester presented were of the cascade half-lattice form that functioned in the HF spectrum of several MHz.    Vester’s paper was strictly an experimenter’s guideline with no information on modeling or synthesis.   

The scene changed in the late 1960s with the introduction of effective imported 9 MHz crystal filters from KVG in Germany.   Other filter vendors also appeared on the scene at that time.   Radio amateurs continued to build gear, but most of them used commercial filters.   KVG eventually discontinued their line of filters, causing hams to become serious about building their own crystal filters.    An early pioneer in this effort was Hardcastle from the UK.   His first paper appeared in RadCom, December 76, and was adapted and reprinted in QST in December 78.   He was the first (that I recall) to have presented the lower sideband ladder form of crystal filter to the amateur radio community.

My first serious efforts with crystal ladder filters (QST, 1982) were aimed at some simple methods for crystal characterization.    I used these crystal models to generate some simple filters.    A later paper (QST, 1987) presented a different simplification afforded by the so called minimum loss topology of Cohn, leading to filters that can be built with little if any design effort.  

All of the crystal filter efforts depend upon our knowledge of the crystals to be used in the filter.   The crystal is modeled by the circuit of Fig 1.
   Fig 1.

The crystal is essentially a series tuned circuit consisting of a very large (compared to the discrete elements we usually associate with RF filters) motional inductor, LM, that is resonated by a motional capacitor, CM.   A loss element is modeled as motional resistance RM.    Physically, the crystal consists of a wafer of quartz that is coated by a metal film on the two surfaces.   This structure is one that creates an electric field within the quartz, required to excite the piezoelectric effect. Wafer thickness determines resonant frequency.   There is a parasitic element associated with the structure, the result of the parallel capacitor formed by the plated  surfaces.    This is C0 within the model.     To effectively design crystal filters, we need to know all model parameters.      

A typical 5 MHz crystal has the following parameters:   Lm=0.1 H.   (Yea, Henry.)  This is resonant with motional capacitance at 5 MHz, so Cm=.010 pF.   Parallel capacitance is C0=3 pF, which will include some stray C related to the mounting of the crystal within a package.     Not all 5 MHz crystals will have the same parameters.   They can be varied by controlling the amount of the quartz surface that is plated with metal.     The 5 MHz crystals that I have in my junk box have an ESR (equivalent series resistance) of about 15 Ohms for a Qu of 200,000.  

Measurement Methods

There are several schemes that can be applied to determine the parameters in the model of Fig 1.   The first measurements that I did were based upon an over simplified model for the crystal where the parallel capacitance, C0, was ignored, shown in Fig 2.    
   Fig 2.  Scheme used for crystal measurements.  C0 was ignored during analysis after measurements.

The method depended upon two measurements.   First, the insertion loss of the crystal versus that when the crystal was replaced with a “through” path was evaluated.   We assumed that C0 was zero.   Hence, the insertion loss at resonance, which is where the response is largest, allows one to calculate RM.      The second measurement was realized by tuning the signal source and noting the frequency where the response was 3 dB below the peak.    Knowing these, a bandwidth could be calculate, allowing a loaded Q to be evaluated.     This then allowed one to calculate the motional L and C at resonance and to extract the unloaded Q and, hence, RM.   

The scheme of Fig 2 works, although it requires a frequency stable source of RF for bandwidth measurement.     It is, of course, invalid to assume that C0=0.   The assumption does not produce major errors, especially if transformers are used to move to system characteristic resistance, R0, to values well below the usual 50 Ohms.      C0 can be measured with an independent bridge that operates well away from any crystal resonance.   The ubiquitous AADE L/C meter works great for this determination, although care is required during meter zeroing.   More will be said about this later.

The methods of Fig 2 were presented in QST for May of 1982.   Shortly after that paper was published, I received letters from Dr. David Gordon-Smith, G3UUR.   In that correspondence, he suggested that one could infer the motional L and C from observation of the crystal operating in an oscillator.   The frequency would be shifted by inserting a known capacitance in series with the crystal.   The basic scheme is presented in Fig 3.
 Fig 3.   Oscillator scheme suggested by Gordon Smith, G3UUR.  The oscillation frequency is measured with the switch open and closed.  The values for ΔC and C0 are determined with independent measurements.

I did not  implement this suggestion for a long time, for I was able to get the data I needed from the method of Fig 2.   But several years later the scheme was implemented.   Two oscillator circuits were eventually used, shown in Fig 4.  Biasing and grounding are omitted for clarity in this presentation.  
 Fig 4.  Two variations of the G3UUR measurement scheme.   The left circuit uses a Colpitts-like oscillator.   It is assumed that CP (the Colpitts capacitors) are much larger than ΔC.   The right hand variation uses a bifilar wound wideband transformer in a Hartley like circuit.   It functions to lower frequencies, allowing many low frequency ceramic resonators to be evaluated.  A practical variation is given in Fig 9 below.

The method presented in Fig 4 has been used extensively in my lab, although I always wondered just how accurate it was.   Which oscillator topology is better and how do the results compare with measurements performed with a high quality vector network analyzer?    I believe that VNA  methods are  used for virtually all non amateur applications these days.    A VNA measurement is shown in Fig 5.
  Fig 5.  The ideal method for crystal characterization uses a vector network analyzer with a suitable bridge.   Measurement at four known frequencies, ideally near the crystal resonances, will allow the four unknown parameters to be calculated.
The VNA measurement of Fig 5 uses a bridge, shown here as a classic return loss bridge.   Many other bridges or directional couplers are suitable.   A signal is applied to the bridge input and the signal at the detector is observed.   Both amplitude and phase of the detector port voltage is recorded where the reference for the phase measurement is the source used to excite the bridge.   It is because we observe both amplitude and phase that we call this a vector network analyzer.  The voltage at the detector with suitable normalization is termed the voltage reflection coefficient.    Well known analytic transformations can be applied to calculate the impedance of the unknown related to this reflection coefficient.     

The formal method used to determine the model components is simple:   Four frequencies are picked, usually close to the crystal series resonance.   The impedance is measured at each of these frequencies to produce four equations in four unknowns.   They are solved for the four model parameters.

Actually, there is a plethora of data, for each measurement will produce two results in the form of a reflection coefficient amplitude and phase, or an impedance real  and imaginary part.     Accuracy is enhanced by picking the right frequencies, but the concept is well established from fundamental considerations.

There are ancillary measurements that we can do to determine the model parameters with a network analyzer that will make us feel good about the results generated by the canned software that comes with the VNA.     These measurements result from careful analysis of the crystal model.  Some of these details are:

1)   Analysis shows that the parallel capacitance C0 will produce an impedance that is unrelated to the series resonance of the motional components.   Specifically, we can measure this capacitance at a frequency much lower than the series resonance.    (The operating frequency of the AADE L/C meter is usually around 500 kHz.)
2)   If the impedance of the model is estimated for typical component values, we see that the impedance near series resonance is effected very little by C0.   It is thus valid to measure complex impedance near resonance in order to infer motional parameters without any knowledge of C0.  
3)   The work that I did in 1982 QST started with a determination of  series resistance.   I then measured the loaded Q and jumped through a hoop or two to be sure that this was close to an unloaded Q.  From that, I calculated the crystal's motional inductance.     But this is not required.   At series resonance, the real part of the impedance (the ESR or Rm) is essentially a constant.    In Smith Chart terms, this just says that the changing position on the chart is essentially along a circle of constant resistance.      Consider an ideal series tuned circuit with no series resistance and no parallel capacitance.    Calculate the reactance as a function of frequency, X(f).   Now differentiate this with respect to frequency.    Solve the resulting equation for inductance to find that
       
What this means is that we can obtain the motional inductance merely by numerically evaluating two reactances near resonance and then applying this equation.      Several figures are devoted to this calculation in another discussion on this web page.   See http://w7zoi.net/Lm_calcs/motional_induct_calcs.html.    There is no need for any resistance determination.

A Series of Experiments

VNA Measurements:

This section examines results obtain with a VNA of the type designed by Paul Kiciak, N2PK.    One of the software programs with Paul's analyzer is XTAL2.EXE.    This program starts with user supplied frequencies that bracket the series and parallel resonant frequencies.   The program then searches for the resonances.   Once found, it does detailed measurements near the series resonance, which is the frequency where the reactance of Lm is canceled by that of Cm.    The program then calculates and outputs all four model values as well as the unloaded Q.   Series and parallel resonant crystal frequencies are also included.

It was time to perform the important comparison experiment.   So I fired up the VNA and searched my junk box.    A crystal was in the box with some other goodies, all left from earlier experiments.    The crystal was made by FOX and was in an HC-49 can (full sized!) and was marked with a frequency of 11.0592 MHz.  This rock was from a batch of 100 from a local surplus source.   

The N2PK VNA is a fundamental instrument that has a RF output port with a power of about 0 dBm available and a detector port.    A computer controls the instrument, saves measurement data, and performs calculations as needed.    A measurement is almost always preceded by a calibration.   In this case, an OSL (open, short, load) calibration is used with the return loss bridge.   The bridge is a homebrew unit in need of further characterization.    I ask that the program do 10 measurements for each one indicated, which allows some averaging of data.

VNA Results for the Test Crystal:
F-series = 11.0594411 MHz
F-par = 11.0826015 MHz
C0=4.53 pF
LM = 0.01089184 H
CM = 19.016041 fF
RM = 7.43 Ohms
Qu = 101850

 
Fig 6.  Crystal being measured with the VNA.   The crystal leads are soldered to a SMA connector.

I see from the photo above that the full output of the VNA is available to the return loss bridge.     Since that measurement I have been using a 14 dB pad in the VNA output.   This drops the power available to the crystal to -20 dBm where better linearity is expected.

Parallel Capacitance Measurements

The next crystal measurements following those with the VNA will use simple oscillators.    These measurements require independent determination of the parallel capacitance, which can be done with an AADE L/C meter.   The oscillator measurements will require that the crystal leads be exposed to fit in my oscillators.   Hence, I wanted to measure C0 before the SMA connector would be removed from the crystal.  

I started by turning on the AADE meter, switching to C and pushing the zero button.   This eliminated the residual reading of almost 3 pF, producing 0 on the scale.   I then attached the small clips to the wires on the crystal.   This produced an inflated value of 5.6 pF.   Recall that the VNA result was 4.5 pF.   The alligator clips on the wire ends were much closer than they had been during calibration.    

A more exacting procedure was needed.   I rearranged the wires so the clips had a lower C overlap during the zero process with closer spacing.   This is shown in Fig 7.
 
Fig 7.  Test clip leads arranged “end to end” and clipped to a strip of paper for zeroing.

The clip leads were then attached to the crystal for a measurement result of 5.17 pF, shown in Fig 8 below.  This is still higher than the VNA measurement, but the clips are closer to each other than they were with the zero operation.    Clearly, this is a place where more exacting measurement practices might be useful.
 
Fig 8.   Measurement of C0 with the AADE L/C meter.

Following the measurement shown in Fig 8, the crystal was unsoldered from the SMA connector and measured, producing a result of 4.11 pF.   This is the value we will use for later calculations, although this is probably low owing to strays.   Our VNA calibration standards are all on SMA connectors.

Measurements with the Wideband Hartley Oscillator.

The next step in the process was to measure the motional parameters with the oscillators.   My test oscillator was initially configured with the Wideband Hartley circuit.   The two circuits are show in Fig 9 below.
 Fig 9.  Two forms of the test oscillator are built into one box.   Only one is functional at a time.   The unused oscillator is disconnected from the crystal socket and the output is disconnected from Q4, which is an analog to TTL converter needed for my Radio Shack frequency counter in the highest resolution mode.  ΔC = 36.9 pF, which results from a 33 pF capacitor and the capacitance of the open switch.   

The lower frequency was 11.057796 MHz while the upper value was 11.060615 MHz.  Note that the lower value is below the series resonance measured with the VNA.   (Corrections have been applied to account for slight frequency offsets between instrument clocks.)   The frequency difference is 2819 Hz.    From this we calculate that the motional inductance is 0.0099089 H.   This is below the VNA result by 9%.

Measurements with the Colpitts Oscillator.

The Hartley variation was disconnected and the Colpitts circuit was connected to the crystal socket and switch and counter buffer.   The circuit was built with Cp=470 pF.  The two oscillation frequencies were 11.059474 and 11.062005 MHz.  The lower is now above the series resonance measured with the VNA.   The frequency difference is 2531 Hz, resulting in Lm=11.0385 mH, which is above the VNA result by 1.3 %.    

The expression used to calculate motional capacitance for both of the above oscillator measurements is

    
This is the corrected expression presented on the web. (w7zoi.net and click on EMRFD errata.)    Cs  is the switched capacitance in pF while C0 is the crystal parallel capacitance.   Cm, Cs, and C0 are all in pF.    Once Cm is known, Lm is calculated from resonance.   Lm and Cm are not independent, but are linked to each other through resonance.


Measurement of Q

An independent measurement is required to determine Q when the VNA is NOT being used.   The scheme that I used is that from EMRFD, Chapter 7.     The crystal is attached as a shunt element in a 50 Ohm system, shown in Figures 10 and 11 below.    At resonance, the attenuation was 13.5 dB for this crystal.   The result was Qu=114490 using Eq. 7.6 from EMRFD.   
  Fig 10. Method used for Q measurement.   This is a simple transmission measurement rather than a reflection determination with a bridge.

  Fig 11. Test fixture used to measure crystal Q.  This is normally used with the N2PK VNA, although it can also be used with a stable signal source.   Accuracy depends upon having an accurate 50 Ohm load.

An extremely interesting, but subtle detail emerged with the later measurements.   The Q of some crystals was compromised by as much as a factor of two by excessive power.   Better results were obtained with a power applied to the bridge of -14 dBm or less.  I encountered this phenomenon in the past in connection with crystal filters used in commercial spectrum analyzers.  

Additional Data

Having measured just one crystal in detail left questions.   What would the Colpitts oscillator do at other frequencies?    Additional crystals were resurrected from the junk box and measured with the VNA and then with the Colpitts characterization oscillator.   The following results were obtained:

Freq.               xtal #           Lm (VNA)      Lm(osc wrt VNA)          Q(VNA)        
4.000 MHz,     7                0.13834 H       +1.0 %                            180K
5.000            624               0.09936          +10.3%                            116K
10.00            393              0.01901            -1.3%                             223K
5.000            446              0.098467          +8.2%                            147K
5.000            107              0.102118          +2.7%                            139K
6.002             1                .065648             +1%                               247K

Conclusions and Extensions

The vector network analyzer is the preferred instrument for crystal characterization.   However, it is slow to use and is missing from the measurement arsenal for most amateur experimenters.    The correlation between the VNA and the oscillator is very encouraging, suggesting that oscillator measurements are good enough.     

Chris Trask has built a crystal characterization oscillator based upon the Butler topology.   His circuit isolates the crystal from reactive circuit elements and should be ideal for LM determination.    This design is found on his web site, http://www.home.earthlink.net/~christrask/Crystal%20Test%20Set.pdf.

My May 1982 QST paper put a lot of emphasis on measuring crystal Q.   I now feel that this is less important, at least for crystal filter applications.   It is useful to sample a small fraction of the crystals within a batch (10% is usually more than enough) to be sure that the typical Qu exceeds a reasonable minimum.   If the desired filter is a very narrow one and is to be built with poor quality crystals, it may be necessary to test all units before use.   It is not as critical with wider filters.     As an example, I built an 8th order SSB filter at 11 MHz with a bandwidth of 2.4 kHz.   Crystals from the same batch used for this study were used. Simulation (with GPLA) suggested an insertion loss of about 3.3 dB when Qu=100K for the crystals.    The filter was built and measured, yielding a bandwidth within a few percent of the design value and an insertion loss under 4 dB.   An 8 crystal CW filter would, however, be stressed with these parts.    Simulations with a typical value for Qu usually provides enough information.   The 8th order SSB crystal filter is shown in Fig 12 with a schematic in Fig 13 and measured response in Fig 14.
  Fig 12, 8th order filter.   Shielding of terminating transformers improved stopband attenuation.
 
Fig 13.   Schematic for crystal filter of Fig 12.

 

Fig 14.  Measured frequency response for the 11.06 MHz crystal filter of Fig 12.   A small ripple is shown in the passband.   This ripple was also in the simulations.  It resulted from the use of standard capacitor values rather than parts picked to exactly fit the values dictated by the design process.   The ripple does not appear to be an error related to the crystal characterization.    Incidentally, the degraded stopband attenuation on the high side is real and seems related to the construction of the filter.   It does not appear in the simulations.   The more common observation is that of steeper skirts on the high side of a filter.