Flin- HEWLETT~~ PACKARD

RF & MicrowaveMeasurement Symposiumand ExhibitionElectrical Characterizationof Quartz Crystal UnitsThrough High PerformanceVector Network Analysis

Ken Voelker - Author

Lake Stevens Instrument Division8600 Soper Hill RoadEverett, Washington98205-1298

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ELECTRICAL CHARACTERIZATION OF QUARTZ CRYSTAL UNITSTHROUGH HIGH PERFORMANCE VECTOR NETWORK ANALYSIS

OUTUNE:

• The Measurement Problem: Crystal Parameters

• The Measurement Solution: Network Analysis

• Practical Techniques

ne Measurement Problem

Crystal Parameters:

Co Shunt Capacitance

R'n C1 Motional Capacitance

~;~f'R1 Motional Resistance

L1 Motional Inductance

fa Series Resonant Frequency

fp Parallel Resonant Frequency

Q Agure of Merit

This is a seminar on how modern network analysistechniques can be applied to the problem of makingquartz crystal measurements.

The equivalent circuit diagram for a crystalincludes a series resonant circuit (Rl, Cl andLl - the motional components) with Co in shunt.Crystal characterization involves finding thelumped element values for these parts, as wellas the associated resonant frequencies and figureof merit. These determinations must be made withaccuracy and repeatability.

Crystal Impedance

1\1/:'\.

\ I

I ,....... ....... L.I /

\

• •

ZI

e

These values will be derived from a plot of thefrequency dependant vector impedance between thetwo ports (pins) of the crystal device. A reviewof the mathematical relationships between thisplot and the desired measurements follows.

fa

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Series Resonant Frequency

[!;]

• determined by L1, C1

• minimum Impedance

• zero reactance (zero phase)1

• fs = 2 1T 'Y'L1C1

Motional Resistance[ill]

• residual Impedance at f s

• can be read directly from plot

Motional Inductance

[b1]

The series resonant frequency fs is determined bythe motional reactances LI and CI, as shown. Itis recognizable on the impedance plot as thefrequency of minimum impedance and zero phaseangle.

with the reactances of LI and CI cancelling eachother out at fs, the remaining impedance iscontributed entirely by RI, whose value can be readdirectly from the impedance plot.

The motional inductance is directly related to therate of change of crystal reactance with frequencyin the vicinity of fs, as shown.

•

•

calculated from "slope"of reactance at fs

L1=1 dx41f ctf

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••

Motional Capacitance

[gI]

calculated from fs , L1

Given the series resonant frequency (fs), and themotional inductance, CI can then be calculated fromthe familiar series resonance equation.

Shunt Capacitance

~

• determines fp (paranel, or anti-resonance)

• spreads fm• fr • f.

To this point, shunt capacitance Co has beenignored. While this may safely be done in somecases (particularly lower frequency crystals), itis important to understand its effects on themeasurements.

Parallel Resonant Frequency

~•

•

•

•

determined by Co ' C1, L1

maximum Impedance

zero reactance (zero phase)

First, the shunt capacitance provides a secondfrequency of resonance, or zero phase, in this casean impedance maximum. This parallel resonancefrequency (fp) will be related to Ll and Cl as wellas Co.

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SpreadIng of fm• \- • f.

-.v-;

V /IZI II

"'- /i'...

'" / /V ........

• v ~

""J + .....,

Influence of Stray Capacitance

• crystal behaves as ifCstray were part of Co

• large effect on fp

• lesser effect on f s

The other effect can be seen on this plot ofcrystal impedance, taken over a very narrowfrequency span centered on fs. In the presence ofCo, fr (zero phase) and fm (minimum impedance), nolonger coincide, as previously indicated.Therefore, fs (series resonance) cannot becorrelated to any specific point on the plot.

Further, any stray capacitance across the crystal(fixturing, etc.) will appear in parallel with,and indistinguishable from Co. It will have asignificant effect on parallel resonant frequencyand can also lead to inaccurate values for thelumped motional elements. Means for overcomingthese problems will be discussed later.

•

•

Measurement of Co

find Xo at non-resonant frequency

Co = 27Tf Xo

Co can be easily found by measuring the crystalimpedance a few percent away from series resonance.At such frequencies, the motional arm presents ahigh impedance in parallel with Co, and the totaldevice impedance is simply the capacitive reactanceof Co at the measurement frequency. Co can alsobe found analytically using a technique to bepresented later.

• or. find It analytically.....

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Resonator Q

• Calculate from motional parameters

Q = 27T f. L1IR1

• calculate from phase slope at f s

a = -tg 7T fs • where

-~360 df

The Measurement Solutio.

Network Analysis

~IDevice Under Test

Source

• furnishes stimulus. determinesmeasurement frequency

• desirable characteristics:frequency synthesisfrequency sweepvariable amplitude

The last crystal parameter to be derived isresonator Q. This can be calculated from themotional parameters already found, or from therate of change of phase with frequency. Thelatter can often be measured directly as "groupdelay" (tg).

The crystal measurement problem is therefore oneof very precise impedance measurements. Today'sbest answer is state-of-the-art network analysis.An appropriate network analyzer can be conceptu­alized as having three fairly simple functionalblocks.

The source provides sine wave energy to thedevice under test and determines the frequency atwhich the measurement is made. The hi Q natureof crystal devices will require that it havesynthesizer accuracy and stability, and it shouldbe able to sweep over a variety of frequencies .

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Receiver

FROMLO.

INPUT

• quantifies test device response

• desirable characteristics:vector measurementsnarrowband

Display

• presents measurement results

• desirable characteristics:

- computational ability

- graphical output

Network MeasurementsS - Parameters

JEi -----,

I 521 II 511 522:• I: I! 512 i: :Laaa ... J

511 - input reflection coefficient

521 - forward gain

512 - reverse gain

522 - output reflection coefficient

The receiver is the portion of the networkanalyzer that receives and quantifies the responsefrom the device under test. It is desirable thatit be able to respond to both the real and imaginaryportions of the input signal, allowing both ampli­tude and phase measurements. Dynamic range require­ments necessitate a narrowband detector, which inturn requires that the receiver and source share acommon local oscillator, to provide tracking.

Finally, the display section of the analyzercalculates and displays the measurement resultsin a usable format, beginning from the raw datasupplied by the receiver.

S-parameters are commonly used to describe thecharacteristics of single or mUlti-port RF devices.They are based on the (vector) ratios of powerreflected from, or transferred between, the variousports under conditions of perfect source and/orload match. As these are the measurements mosteasily made with a network analyzer, they will beused as the basis for the following measurements.

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---~------------

Test Setup - S21

D

A typical 821 measurement (forward gain orfrequency response) would be made in this manner.The internal source is split and applied to boththe device under test (OUT) and one of the analy­zer's receiver inputs. The output of the OUT ismonitored by receiver input B.

BR

~------<c] "'"~ U"'~ T".PowerSpntter

Source

Typical S21 Measurement A typical measurement result is shown as magnitudeand phase versus frequency. Notice that bydisplaying the OUT output (B) in ratio with theactual input (R), any imperfections in sourceflatness automatically cancel out, because theyappear in both terms. Results are in decibels orother relative units.

_Magnitude SIR

_Phase SIR

~. can -us.•~-1\

J \1/

'\11

LII A~

I • 1\ 1\ 1\r--

.... LKftL IDlY .... I. 7GD DDD.~

....~ ao.~ tMCtal) 40".~

.....~ .a.~ ~ aD 7aD GClD.aaatta

CDlTlIIt ID 1m) 0ClC.~ .,.,. ICID CXlD.DCDW",..TD lL~

D

Test Setup - 811 Crystal measurements focus on 811, the ratio ofpower reflected from a OUT relative to that appliedto it. To the previous measurement setup has beenadded a directional coupler, capable of separatingthe reflected power from the incident and routingit to receiver A. 811 is therefore equal to theratio AIR, and will take on a value between 0(perfect impedance match, all power absorbed) and 1(perfect reflection, no power absorbed by OUT) atany phase angle.

Test Port

Directional~ Device Under TestCoupler ~

R A Bo

PowerSplitter

Source

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Conversion of 811 to Impedance

z=Zo [1+ S11 ]1- S11

The familiar transmission line equation allows anyvalue of 511 to be converted to a correspondingvalue of impedance Z. The practical limit of thistechnique is about two orders of magnitude aroundZo, the system characteristic impedance.

S11 = ~

Z 0 = System characteristic impedance

Measurement range: Z 0 ± 2 decades

S - Parameter Test Set A measurement accessory available for many networkanalyzers is an 5-parameter test set. It providesthe interconnection of the power splitter,directional coupler, and switching functions in asingle unit. The user need only connect his one ortwo port OUT to the appropriate measurement port(s)on the test set.

Test Ports

:=D -or-

0'''''' U'd'~T.J

R A B

D

TestSet

NetvJork

Analyzer L--f--f--"";I-+..J

Connecting Crystal to Test Set

02

Measurementports

!. II :.......__.._.._----'

Crystals can be measured using either one or twoport techniques. The latter offers the additionalinformation of case-to-pin capacitances Cll andC22, at the expense of somewhat more complex math.

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F1xturlng

• r.peaiabllity

• compatlbllty

Measurement Calibration

Required beca.... of:

• cabling losses, III shlft• stray L. C• Instrument errora

Uncallbrated Meaaurement

I

I ,~

~ -.-

- I. .-~ II'

\\II

fa fp fp"pulled" .ctue'

Fixturing for a crystal measurement will involveproblems that are often more mechanical than theyare electrical. It is generally more importantthat stray impedances be highly repeatable thanthat they be held to an absolute minimum. Inaddition, the calibration standards to be used willhave to be adapted for use directly at the crystalpi.n socket.

Most of the sources of measurement inaccuracy areactually external to the network analyzer. strayfixture impedances will appear in parallel with thecrystal and change its electrical characteristics.Even the phase shift and losses of the interconnectcables will appear superimposed on the deviceresponse.

In this example, the shunt capacitance of thetest fixture has "pulled", or lowered the crystal'sparallel resonant frequency. The solid line showsthe actual, uncalibrated measurement that hasresulted, as compared to the actual value (dashedline) .

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Calibration -Reference Plane-

Definition:

the point In space to which all amplitudeand phase data are referenced

Internal CallbraUon F1nnw~re

NE PO T

....... .ruo

_ 1STUI "'-0 "I 0

I •• ....I.

Calibration creates a measurement "reference plane",the physical point from which all measured valuesare referenced. Impedances within the referenceplane are effectively removed, allowing thoseexternal to it (i.e. the OUT) to be measuredindependantly.

Calibration firmware within the HP 3577A networkanalyzer provides step by step operator instructionswhereby standard open, short and fifty ohmterminations are attached to the reference planeand measured. Data from these known devices arethen used to correct future measurements.

Calibrated Measurement.." L.8YD. ~IV ~ • _ ...,........

J"~ J"~ IMACLIDI") e,."'"

/\V r-.

VI--

1/\~,

CIlIlfTIIII so 000 ...,.. ao~ PAN. 000.~~TD 0.0.-.

Shown is a measurement following correction for allof the error-causing effects previously mentioned .The calibration procedure has allowed the crystalto be measured as if it was in complete isolationfrom the external world.

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General Measurement Sequence

'tI, I: Instrument Setup

• one-port connections

• analyzer settings

Network Analyzer Initial Settings

The following slides will provide a practicalsequence of steps whereby these measurements mightbe made on an HP 3577A Network Analyzer.

The first step is to set-up the instrument with theproper connections and front panel settings.

INPUT

INPUT

DISPLAYFUNCTION

IFREQUENCY I

IAMPUTUDE I

1+ S11 or UDF 'F4'1- S11

trace 1 - linear magnitudetrace 2 - phase

center = faspan = 5 fs IQ

as required

UDF "F4" identifies user defined function F4,which is pre-defined to be impedance, as calculatedfrom SIlo

DISPLAY FUNCTION

Trace 1 - linear magnitude (in ohms)Trace 2 - phase

FREQUENCY

Center frequency - as appropriateFrequency span - approx. 5 times the Q bandwidth

AMPLITUDE

As required

Ite, 2: Calibrate Don't forget to calibrate!

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= motional arm impedance

Bte, 3: Refine Instrument Settings

• frequency

• scafing

• bandwidth

• averaging

• sweep time

Ita, c: Removal of Co

...:-------1 permits accurate determination of fs •r--'"-, motional parameters

iZtotal = Zm x Z eoi Zm + Zeo • where:

II

<;---

0+ j Xeo

• define display as Z m

leo x Ztot

Zeo - Ztot

• adjust Zeo until f m = f r = f 8

Refinement of the initial instrument settings maybe desirable for optimum measurement performance.Very high performance devices may, for example,require use of signal averaging or a narrowerbandwidth in order to maximize dynamic range.Re-calibration is a must after any of these steps,or after any change in frequency span.

Finally, the effects of Co can be mathematicallyremoved from the impedance plot, allowing themotional parameters (RI, CI, LI) and seriesresonant frequency to be conveniently found. Bymodelling the crystal as two parallel impedances,one of which is the reactance of Co, it is possibleto solve for the motional impedance given anyassumed value for Co.

starting from an arbitrarily chosen value, Zco (orXCo) is adjusted until the resultant plot shows thecharacteristics of a simple series resonant circuit(i.e. fm=fr). This identifies both fs and Co.Alternatively, if Co has been separately measured,it's reactance can be calculated and "plugged into"the equation at this point.

• Co = 211" f Xeo

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Measurement Results: f 5 • R1..,. UY&L 'lit'll ~ ...... "'.-.0H8D. 000 L CIOOO ....c (l.C)II') N. 1."Dr. oe.. so. 000.... toWIb<P .... _'7. SOOHa

This is the resulting impedance plot of a typicalovertone crystal unit. Fm and fr coincide because278.1 ohms of capacitive reactance have been removedfrom in parallel with the network. This correspondsto a shunt Co of about 6.8 pF.

Fs is read directly as about 83.498 MHz. Motionalresistance Rl is read directly as 36.18 ohms.

R1 = 36.18ohms

~H CL.IJIII") -o.07SGI.

r\\ ./

I"- /1/ /

'\. VI'-.V Vv- I""--. ~

/......HT'K" III ••••17.1()QIota .... 'llllll._TDa.._

...-

Xeo= -j278.1

Co=6.78 pF

fs 83.498.367.5 Hz

Measurement Results: U. ct

1\\. V

\. ./.,/ /

'\..V/ I'... V

v ........... ,/

.... LI:Va. .IIIrv

.. DOD &.000II-10._ 'D.QOD

.....-1It ..... ..,. aoaHIt.,...CCl..lDIn -'1..-cwrrKT zoo. DCX:IWJ -.c: Q.GF') II'"

t>X = 11.967ohms

L1 = 4.762mH

Ll is found by displaying the imaginary (reactive)portion of the device impedance, and using thedisplay markers to determine it's slope. Inthis case, with a 12 ohm change over a 200 Hzspan, Ll is about 4.8 millihenrys.

Cl is then calculated from the values for fs andLl, and is equal to about .76 fF.

CCNTPt I. 4H ~. SQDIoM .,.AH I DOD. OOCHII......TO 0.0.-.

C1 =.7629fF

t>F = 200 Hz

Measurement Results: Q The Q of the device can be determined withouttaking a separate measurement. After selectingthe group delay display function, the value atseries resonance is found to be -262 microseconds.Q is therefore about 68.7K.

a =68.71<

tg =-262usee

-

1\\ \.

I\. I'\.. /

/\.

I'... 1/

........... ...... ,/

...... LIYIIL .III IV MAJ8(P I .... M'7.~N. DOO a. 0000 MAC (1JOlJ") N. II.-as. oo".c aD. ooo".c MAltMl" .. ••• lie? SQOIW;

, DELAYCUOn zez J11t&IEC

CPlTP U ... Z17. SOOHa .AJoII I ODQ. DDDH.""TQ 0.0.- ~., ~ ea.~

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Conclusions

• Quartz crystal electrical parameters are readilyobtained from common network measurements.

• Today's network analyzers provide a more completesolution than ever before by:

simplifying measurement setupself-calibratingperforming complex data manipulationdisplaying results in more usable forms

Network analysis is gaining rapid acceptance asthe preferred method for crystal devicecharacterization. Excellent measurement perfor­mance and extensive mathematical capabilitiesmakes the HP 3577A an ideal solution in eitherstand-alone or ATE applications.

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