EN 60444-5:1997

SLOVENSKI STANDARD
01-september-2002
Measurement of quartz crystal unit parameters - Part 5: Methods for the
determination of equivalent analyser techniques and error correction (IEC 60444-
5:1995)
Measurement of quartz crystal unit parameters -- Part 5: Methods for the determination
of equivalent electrical parameters using automatic network analyzer techniques and
error correction
Messung von Schwingquarz-Kennwerten -- Teil 5: Meßverfahren zur Bestimmung der
elektrischen Ersatzschaltungsparameter von Schwingquarzen mit automatischer
Netzwerkanalysatortechnik und Fehlerkorrektur
Mesure des paramètres des résonateurs à quartz -- Partie 5: Méthodes pour la
détermination des paramètres électriques équivalents utilisant des analyseurs
automatiques de réseaux et correction des erreurs
Ta slovenski standard je istoveten z: EN 60444-5:1997
ICS:
31.140 3LH]RHOHNWULþQHLQ Piezoelectric and dielectric
GLHOHNWULþQHQDSUDYH devices
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

NORME CEI
IEC
INTERNATIONALE
444-5
INTERNATIONAL
Première édition
STANDARD
First edition
1995-03
Mesure des paramètres des résonateurs
à quartz
Partie 5:
Méthodes pour la détermination des paramètres
électriques équivalents utilisant des analyseurs
automatiques de réseaux et correction des erreurs
Measurement of quartz crystal unit parameters —
Part 5:
Methods for the determination of equivalent
electrical parameters using automatic network
analyzer techniques and error correction
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444-5 ©IEC:1995 – 3 –
CONTENTS
Page
FOREWORD 5
Clause
1 Scope 7
2 Introduction 7
2.1 General 7
2.2 Methods of admittance measurement 9
2.3 Admittance analysis and estimation of equivalent circuit parameters 13
2.4 Normative references 13
3 Measurement procedures 15
3.1 General 15
3.2 Environmental control 15
3.3 Calibration 15
3.4 Level of drive 15
3.5 Co measurements 15
3.6 Choice of measurement frequencies 17
3.7 Data collection 17
3.8 Data correction 19
3.9 Admittance calculation 19
3.10 Admittance analysis an
d estimation of the equivalent circuit parameters 19
4 Choice of admittance measurement method 19
4.1 General 19
4.2 Advantages and disadvantages of the one-port S-parameter reflection method 19
4.3 Advantages and disadvantages of the two-port S-parameter tr ansmission method 21
4.4 Advantages and disadvantages of the direct tr ansmission method 21
5 Calibration techniques 23
5.1 S-parameter method 23
5.2 Direct transmission method 23
Verification of calibration 5.3 23
6 Low-frequency measurements 25
7 Admittance analysis and estimation of the equivalent circuit parameters 25
7.1 General least-squares fitting method 25
7.2 Linear least-squares fitting procedure 27
7.3 Circle-fitting method 33
7.4 Two-point iterative method 37
8 Measurement errors, instrumentation and test fixtures 41
8.1 General comments 41
8.2 Measurement conditions 41
8.3 Reproducibility 43
d
8.4 Measurement an test fixtures 43
Figures 49
Annexes
A Calibration 71
B Low-frequency measurement 93
C Bibliography 100
444-5 ©IEC:1995 – 5
INTERNATIONAL ELECTROTECHNICAL COMMISSION
MEASUREMENT OF QUARTZ CRYSTAL UNIT PARAMETERS —
Part 5: Methods for the determination of equivalent electrical parameters
using automatic network analyzer techniques and error correction
FOREWORD
1)
The MC (Internati
onal Electrotechnical Commission) is a world-wide organization for standardization comprising all na tional
electrotechnical committees (IEC National Committees). The object of
the EEC is to promote international cooperation on all ques
tions
concerning standardization in the electrical and
electronic fields. To this end and in addition to other activities, the IEC publishes
International Stan
dards. Their preparation is entrusted to technical committees; any IEC Na tional Committee interested in the
subject
dealt with may participate in this preparatory work. Inte
rnational, governmental and non-governmental organizations liaising with
the
IEC also participate in this preparation. The IEC collaborates closely with the Internati
onal Organization for Standardization (ISO) in
accordance with conditions determined by agreement between the
two organizations.
2) The formal decisions or agreements of the
IEC on technical matters, prepared by technical committees on which all the ti
Na onal
Committees having a special interest therein are
represented, express, as nearly as possible, an international consensus of opinion on
the subjects dealt with.
3) They have the
form of recommendations for inte rnational use published in the form of standards, technical
reports or guides and they
are accepted by
the National Committees in that sense.
4)
In order to promote international unification, IEC National Committees undertake to apply IEC Inte
rnational Standards transparently
to the maximum extent possible in their na tional an
d regional standards. Any divergence between the IEC Standard and the
corresponding national or regional stan
dard shall be clearly indicated in the latter.
International Standard IEC 444-5 has been prepared by IEC technical committee 49: Piezoelectric and dielectric
devices for frequency control and selection.
It forms Part
5 of a series of publications dealing with the measurements of piezoelectric quartz crystal unit
parameters.
Part 1: Basic method for the measurement of resonance frequency and resonance resistance of quartz crystal
units by zero phase technique in a tc network, is issued as IEC 444-1.
Part
2: Phase offset method for measurement of motional capacitance of quartz crystal units,
is issued as
IEC 444-2.
Part 3: Basic method for the measurement of two-terminal parameters of quartz crystal units up to 200 MHz by
phase technique in a iv-network with compensation of parallel capacitance Co, is issued as IEC 444-3.
Part 4: Method for the measurement of the load resonance frequency fv load resonance resistance, R L and the
calculation of other derived values of quartz crystal units, up to 30 MHz,
is issued as IEC 444-4.
Part 6: Measurement of drive level dependence (DLD),
is issued as IEC 444-6.
The text of this standard is based on the following documents:
DIS Report on voting
49(CO)248 49(CO)268
Full information on the voting for the approval of this standard can be found in the repo
rt on voting indicated in
the above table.
Annex A forms an integral part of this standard.
Annexes B and C are for information only.

444-5 ©IEC:1995 –
MEASUREMENT OF QUARTZ CRYSTAL UNIT PARAMETERS —
Part 5: Methods for the determination of equivalent electrical parameters
using automatic network analyzer techniques and error correction
1 Scope
The objective of this Inte rn
ational Standard is to give methods for determining the best representations of
modes in quartz crystal resonators by linear equivalent circuits. Circuit representations are based on electrical
parameters measured with vector network analyzer equipment using automatic error correction. Determination
of the equivalent parameters by the method of this st dard is based on the measurement of device immittance
an
in the vicinity of series resonance. The further problem of characterizing the device for operation with a series
load capacitance has not been directly addressed, although it is recognized that some applications require such
characterization. The same measuring equipment, an rt of measurement, provides
d fundamentally the same so
the means to characterize completely the test load capacity fixture as well as the series combination of load
capacity fixture and crystal unit.
2 Introduction
2.1 General
2.1.1 This st
andard describes methods for determining the values of the electrical parameters of piezoelectric
quartz crystal units using automated vector network analyzer equipment. The recommended procedures for S-
parameter systems use shielded open-circuit, short-circuit, an resistive terminations, and (in the case of
d
transmission methods) thru-line connections. Coaxial open-circuit, short-circuit and resistive terminations
designed for 50 SI systems are readily available, and can be calibrated in terms of national st andards of
impedance over very wide frequency r anges. At the present time thru-line connections suitable for calibrating
the test fixtures must be calibrated by the user or supplier; however, the techniques for doing this are quite
well known. Non-coaxial standard resistors for use in the direct tr ansmission (n-network) method are
commercially available as well, but are not as easily traceable to National Standards. Further guidance on the
application of this standard may be found in IEC 1080.
2.1.2 The procedure involves the measurement of crystal resonator admittance at prescribed frequency points
by one of a number of methods followed by data interpretation and evaluation of the equivalent circuit
parameters (figure 1).
2.1.3 The measurement methods described are intended to provide reference values for the electrical
equivalent circuit parameters. Manufacturers and users may employ other methods of measurement, but the
values thus obtained shall be correlated with those obtained by the reference method.
2.1.4 This standard is only concerned with the representation of quartz crystal resonators by linear equivalent
circuits which are valid over a narrow frequency b and covering at most a small percentage of the resonance
frequency.
2.1.5 In general, some degree of non-linearity will be present and the circuit parameters may have a
noticeable dependence on d rive level. If non-linear effects are very large then the accepted circuit
representations may be unusable.

444-5 © IEC:1995 – 9
2.1.6 Normally, the
equivalent circuit will be used to represent an
isolated mode of vibration, but occasionally
additional modes may occur extremely close to
the main response; a more complex circuit representation may
then be used, and consideration of this problem is included in this standard. See reference [1]*.
2.2
Methods of admittance measurement
2.2.1 The following terminology will be used to describe the circuit elements. See figures 3 and 4:
Co is the static capacitance (for the one-port model)
C01
is the electrode to can capacit ance
CO2 is the static capacitance (for the simplified two-po
rt model)
CO3 is the electrode to can capacitance
G0 is the conductance associated with C0
G01 is the conductance associated with
C01
G02
is the conductance associated with
CO2
G03 is the conductance associated with
CO3
R 1 is the motional resistance
L 1 is the motional inductance
C1
is the motional capacitance
(.0 = series resonance frequency (rads/s)
1/2
1 C1)
(L
The transfer admittance function, th
Y,12 for e equivalent circuits shown in figures 3 and 4, describes a circular
locus in the complex admittance plane, as
depicted in figure 5a. The transform of this locus to the impedance (Z
= 1/Y) plane is also a circle as
in figure 5b. There are six characteristic frequencies associated with such a circuit:
fs is the series resonance frequency
fm is the frequency of maximum admittance (minimum impedance)
fr is the resonance frequency (zero ph ase)
fa is the anti-resonance frequency (zero ph ase)
4 is the parallel resonance frequency (lossless)
fn is the frequency of minimum admittance (maximum impedance)
Of these, the series resonance frequency alone is essentially independent of the
value of static capacitance, and is
therefore the parameter of choice for purposes of specification as
it will be little influenced by strays. The
relationships of the characteristic frequencies to fs may be found in IEC 302, or in EIA St
andard 512 (1985).
2.2.2 Three basic methods of measurement are described (see figure 2a):
a) Single-port
reflection method; the crystal resonator is characterized as a one-po rt device with one
electrode driven and all other electrodes and
the crystal enclosure earthed.
In a reflection measurement the admittance Y can be calculated from the measured value of S11.
1-S11
(2.1)
ROY 1 +S 11
NOTE - R o is the value of the standard termination used in calibration of the system.
* The figures in square brackets refer to the bibliography in annex C.

444-5 ©IEC:1995 – 11 –
b) Two-port transmission method; the crystal resonator is treated as a two-po rt device with two driven
electrodes, all other electrodes and the crystal enclosure being earthed.
In transmission the transfer admittance of the two-po
rt circuit in figure 4 is:
r
+ jc0 Co2 (2.2)
Y12 = 1G02
R 1 + jcoLl +1/ jwC1J
Using the relationships between admittance and scattering parameters we may define, as above, a
quantity Y
^ 1
R Y = R + jwCO2 + I
G02
0 o
Rt
+ j00L1 + 1 /
jwC1J
(2.3)
_ 2S 12
(1
+S11)(1+
l
S22)— S21S12
which is easily calculated from the measured S-parameters.
c) Direct amplitude/phase transmission method. The crystal resonator is treated as a two-terminal
device according to figure 3 in a transmission fixture using nominally resistive elements, as in IEC 444.
The impedance of the device is determined from the amplitude and phase of the signal across the
fixture. Using standard equations, this impedance is converted to an admittance.
Y = Go + + (2.4)
jc,)CO
+ +1/ jwCl
R 1 jü)L 1
2.2.3 The S-parameter reflection measurement is potentially the most accurate, only coaxial traceable
as
standard impedances are used for calibration. The S-parameter two-po rt measurement provides most
information about the device, while the direct transmission determines the transimpedance of the device.
2.2.4
Restrictions to the validity of the models
The following approximations are implicit in these equivalent circuits.
a) It is assumed that a lumped circuit representation is valid.
b) It is assumed that the device closely approximates an ideal lossless component, and hence that all
significant resonances have high Q factors.
However, over narrow bandwidths, spanning at most a small percentage of the resonance frequency, the
circuits of figures 6 and 7 provide a very good representation of the resonances in the majority of c ases.
2.2.5 Accuracy and traceability
The accuracy and traceability of the measurements are directly related to the calibration components, and are
largely independent of the particular network analyzer system. However, the system shall conform to the
theoretical models described in annex A, and shall therefore be a linear vector detector system capable of high
accuracy in the ratio mode (AIR etc.); beyond this, no detailed specification need be given. A frequency source
traceable to a national st an
dard of frequency is also required.

444-5 ©IEC:1995 – 13 –
2.2.6 Equipment
The use of computer controlled instruments is essential for data collection, error correction, admittance
calculations and for the estimation of the crystal parameters. Error correction a rises from the need to
characterize the measurement circuit. This is achieved by means of a calibration using known st andards of
impedance. Data collection of large numbers of measurement points is required for subsequent admittance
calculations.
2.2.7 Application to other devices
This standard is specifically concerned with single resonator measurements. However, many of the techniques
are directly applicable to more complex devices such as bipoles and monolithic filters; these generalizations
are indicated where appropriate.
2.3 Admittance analysis and estimation of equivalent circuit parameters
There are
four methods (see figure 2b) of admittance analysis leading to the crystal equivalent circuit. If the
crystal being measured is described by the nonnal equivalent circuit and, for instance, the device behaves
linearly, then all these methods are equivalent.
a) General least-squares method
This is the more general non-linear technique which can be applied in all situations. The method is
capable of measuring multiple resonances, such as inharmonics.
b) Linear least-squares method
This method minimizes the sum of the squares of weighted differences between the measured and
theoretical admittan an
ces d is applicable to models with a single resonance.
c) Circle-fitting method
This method fits a circle to an odd number of equally spaced points on the right-hand side of the
admittance circle of the crystal resonator (±45°).
d) Two point iterative method
This is potentially the fastest method and could be used for production. It involves obtaining two
frequencies which lie approximately ±45° on the admittance circle of the resonator. The calculated
crystal parameters are used to re-estimate better values of these two frequencies. Iterations continue
until the estimate is within a given tolerance.
2.4 Normative references
The following normative documents contain provisions which, through reference in this text, constitute
provisions of this part of IEC 444. At the time of publication, the editions indicated were valid. All normative
documents are subject to revision, and parties to agreements based on this part of IEC 444 are encouraged to
investigate the possibility of applying the most recent editions of the normative documents indicated below.
Members of IEC and ISO maintain registers of currently valid Inte rnational Standards.
IEC 302: 1969, Standard definitions and methods of measurement for piezoelect r
ic vibrators operating over
the frequency range up to 30 MHz

444-5 ©IEC:1995 – 15 –
IBC 1080: 1991,
Guide to the measurement of equivalent electrical parameters of quartz crystal units
EIA 512: 1985,
Standard methods for measurement of equivalent electrical parameters of quartz crystal units,
1 kHz to 1 GHz
3 Measurement procedures
3.1 General
The procedure for determination of the equivalent circuit parameters of a quartz crystal unit is shown in
figure 8. It is recommended that the crystal enclosure is grounded. In the case of a glass enclosure, the unit is
fitted with a grounded shield cover.
3.2 Environmental control
All crystal devices are influenced to at least some degree by temperature, the rate of ch ange of temperature,
and by the level of drive. It is therefore necessary to protect the device from temperature ch ange during the
period of measurement, and to determine as closely as possible its actual temperature at the time of
measurement, so that measured values can be corrected for differences in temperature between two
measurements. Care shall also be taken to ensure that the d rive level applied during measurement is that
specified for the device. Another possible source of error, when measuring low-frequency, high-Q devices
especially, is unavoidable small temperature drift during the course of the measurement, as relatively long wait
and last data are recorded will cause distortion
times are required. Such slow drifts between the time the first
of the admittance locus. By a first approximation, this can be avoided if alternate points are recorded with
increasing frequency d then the remaining values obtained with decreasing frequency. The least squares
an
estimation methods will effectively average the data so obtained, and thus compensate to a degree for the
differences due to temperature drift.
3.3 Calibration
This is described for each method in clause 5.
3.4 Level of drive
The drive level at fs for the ensuing measurement must be specified either in power or current for that crystal
type. This requires that the output level of the generator be set. If a reasonable estimate of the R t
of the crystal
is known, then this can be used to calculate this level. Alternatively, a quick estimate of the R 1 can be
determined from an initial sweep through the resonance.
3.5 C o measurements
3.5.1 For a one-port measurement at low frequencies the impedance of Co may be much greater than 50 S2.
This results in low sensitivity, and hence poor estimation of the static capacit ance. Sensitivity may be
improved by making a separate measurement at some higher frequency well away from resonance; however, it
is possible that the effective static capacit ance at this frequency will differ from that at resonance. For most
crystal unit types in common use. Co will remain essentially const ant over the frequency range below about
100 MHz. At higher frequencies, it is advisable to measure Co and the other static parameters at frequencies
within a small percentage of the resonance frequency. If the unit will influence the behaviour of wide-band
circuits, it may be necessary to determine the static parameters over a wide frequency r ange.

444-5 ©IEC:1995 – 17 –
3.5.2 The measurement of the direct pin-to-pin capacitance Co may be made using one of two methods:
a) For crystals up to 30 MHz, the measurement is to be made at five frequencies slightly above
30 MHz (i.e. 30,1, 30,2, 30,3, 30,4 and 30,5 MHz) and the average of the three values nearest the mean
of the five should be used as the best estimate.
b) For crystals above 30 MHz, it is recommended that three pairs of measurements be made, each pair
being equidistant from the series resonance frequency, fs. i.e. (1 ± 0,05), fs (1 ± 0,06) and fs
fs
and higher frequency is
(1 ± 0,07). For each pair, the mean of the Co values determined from the lower
to be calculated, and then the best estimate taken as the mean of the two such values which are closest
together.
Prior to either measurement a) or b) above, it should be confinned that no spurious responses of the crystal
unit exist at the measurement frequency.
3.5.3 The measurement of the two pin-to-c ase capacitances (usually designated C 13 and C23) when required
by the specification, or to aid in modelling of the transmission fixture, should be made by a separate
measurement with a guarded capacitance bridge.
3.6 Choice of measurement frequencies
Two alternative methods for obtaining admittance data on a crystal resonance can be used. The first is a
multifrequency method used both by the least squares fitting procedure described in 7.1 and 7.2 and the
admittance circle fit described in 7.3.
Here, it is recommended that a total of nine frequency points be used, chosen so that the transadmittance
points determined lie within the right-hand half circle of the Y-pl ane locus (see figure 5a), which implies that
several frequencies should lie within the "Q-bandwidth" centred on the series resonance frequency. The
measurement program must therefore perform a preliminary search for the frequency of series resonance, and
then establish the array of frequencies at which measurements are to be made, based either on an estimate of
the Q furnished by the operator, or from examination of the data. The measurement frequencies may be
equispaced for convenience, and all points should be within a r ange of (ff/2Q) of fs. Additional points outside
this r
ange may also be used, but add little infonnation about the motional parameters of the unit. Closer
ange can be useful if the detection of weak
spacing of the measurement frequencies and coverage of a broader r
unwanted modes is desired. For general purpose use, 9 to 15 data points are adequate, and result in rapid
measurements. For highest accuracy. however. 20 to 30 measurement points are preferred, together with a
stable environment.
The second method is the two-point iterative method described in 7.4.
3.7 Data collection
A c.w. mode of measurement is recommended (rather than swept), with adequate settling time calculated from
g
the estimated Q. There are two reasons for this – first, with most si nal sources available, frequency accuracy
in a slow sweep is degraded and, second, the response of the high-Q crystal devices requires a finite time to
reach equilibrium after the excitation frequency is applied. There are three distinct delay times involved in
such a stepped-frequency mode: a finite time is required for the frequency source to stabilize after being

444-5 ©IEC:1995 – 19 –
programmed to a specific value T ins, a finite time for the receiver to stabilize to that new frequency (Tree).
There is also a finite delay while the previously applied frequency response of the crystal decays and the new
response builds to equilibrium value (Tr). At low frequencies with high-Q crystals, Tr may be several hundred
milliseconds, or even seconds.
transmission fixture shows that the new
Analysis of the equivalent circuit of the crystal installed in a two-po rt
response will have settled within 0,1% of its final value in a time of about 2,5 (Qeffifs) after application of the
drive signal, where is the loaded Q-factor of the crystal test fixture. The phase transient will decay to
Qeff
within about 0,1 degree of its final value in the same period. For most purposes, this degree of precision is
adequate; however, for highest precision, it is recommended that this interval be extended to at least
so that no significant distortion of the data will be caused by transient phenomena. Thus, the
3,5 (Qeff/fs),
Td= Tins + Tr
program should set the frequency source to a specific frequency, then wait for a period before
and the average value
measuring the system response. At this time, several measurements should be made
recorded, so that random deviations caused by electrical noise and digitizer errors will be minimized – the
actual number of readings to be averaged can be made a function of the magnitude of the response, as relative
noise level increases as the signal decreases.
Alternatively, the rate of frequency stepping can be determined from repetitive measurements at each
frequency until the phase response has settled down. Averaging of several readings is also recommended as
this minimizes the effects of random and quantization noise.
3.8 Data correction
In the case of the S-parameter methods, the data needs to be corrected by means of error matrices. This is
discussed in annex A. This procedure is not required for direct transmission methods.
3.9 Admittance calculation
For S-parameter methods, this is given by the st andard S-parameter to Y-parameter conversion equations. For
direct transmission methods this procedure is outlined in annex A.
3.10 Admittance analysis and estimation of the equivalent circuit parameters
For the various methods, refer to clause 7.
4 Choice of admittance measurement method
4.1
General
For the vast majority of resonators either one-po rt or two-port characterization is quite satisfactory. As it
provides a more complete desc ription, the two-port transmission method is to be considered as the prime
reference standard. However, in ce rtain situations, the one-port technique may provide information under
conditions more nearly approximating end-use conditions.
4.2 Advantages and disadvantages of the one port S-parameter reflection method
a) This method gives better accuracy and traceability because only coaxial reference impedances are
needed for calibration.
444-5 ©IEC:1995 – 21 –
b) This method is more sensitive than transmission for low R 1 crystals, and will give greater
measurement speed than two-port S-parameter transmission measurements as the error correction
routines are simpler and
fewer responses are measured.
c) It is not very satisfactory for crystals with very high although values up to 1 kS2 can be
R 1,
accommodated.
d) this may not be adequate in some
It characterizes the device as a two-terminal component, and
applications.
e)
Less accurate at frequencies below 100 kHz.
4.3
Advantages and disadvantages of the two-port S-parameter transmission method
a) The crystal is evaluated as a three-terminal device, and more information is available.
b) High R crystals are easily measured.
c) Measurement and calibration are more complex, and some additional uncertainties in the calibration
may exist.
d) The method is less sensitive for low R1.
e) If the measurement frequency r ange includes the anti-resonance frequency then very accurate values
of the static capacitance may be obtained.
f)
Less accurate at frequencies below 100 kHz.
4.4 Advantages and disadvantages of the direct transmission method
riginal IEC 444-1 method, but includes full calibration and
a) This method is an extension of the o
automation. The basic equipment should already be available.
b) Large R 1 crystals are easily measured.
c) The method is less sensitive for small R1.
d) Calibration uses special non-coaxial reference impedances.
e) Electrode-to-can capacitance C01 and CO3 have to be measured independently if the can is to be
earthed, or if the can is floating these are included in CO.
f) The method is fast an is suitable for production.
d
g) Accurate at low frequencies.

444-5 ©IEC:1995 – 23 –
5 Calibration techniques
5.1 S-parameter method
5.1.1 The underlying principle of the calibration procedure lies in a hypothetical error model of the RF
andards [3]. When
measurement system, whose parameters may be determined by measurements on known st
the calibration is complete, and the error parameters are known, system errors in actual device measurements
may be removed by computation.
traceability of the proposed crystal measurement techniques ultimately rely on the
5.1.2 The accuracy and
validity of the automatic error correction procedure; annex A gives the equations of the error model and
describes suitable calibration procedures.
5.1.3 Any network analyzer capable of performing two-port measurements, as described in this section, is
also suitable for one-port measurements; although the converse of this statement is not necessarily true.
ge will depend upon the equipment being used and the application. It
5.1.4 The calibration frequency ran
ge of frequencies within which measurements are anticipated. The error terms
should include at least the ran
are generally smooth, slowly varying functions of frequency, and it is only necessary to obtain data at
sufficiently closely spaced frequency points to permit accurate interpolation of the functions at any frequency
within the range. Given the desired frequency limits, the program should be structured to select automatically
the frequencies needed for adequate calibration. In addition to establishing the frequency list, the program
shall set up the appropriate instrument settings, including source level, sweep rate, detector bandwidth, number
of readings to be averaged (if c.w. mode is selected).
5.2 Direct transmission method
The underlying principle of calibration lies in calculating the impedance values of the crystal measurement
fixture. This is accomplished using known impedances such as a nominal sho rt, open and standard resistor
(25 SI or 50 0). Details are described in annex A.
5.3 Verification calibration
of
t function which should be part of the software system is the capability of making
5.3.1 The third importan
ange of calibration, to verify that the
measurements of known components over the entire frequency r
an accuracy of the calibration terms are within required tolerances.
performance of the instrumentation d
Figure 9 is the flowchart of such a program for the S-parameter methods. A similar algorithm can be applied to
the
the direct transmission method. Input infonnation to this program should include the frequency range and
number of points for measurement, the drive level at which the measurements are to be made, and the
parameters to be output, i.e. transimpedance, transadmittance, input impedance, etc. This program would use
the system error terms previously stored by the "calibration" program, and the error correction and parameter
conversion routine used by the "measurement" program. The components to be used must be connected to the
system at the measurement reference pl ane.
5.3.2 The verification can be done by measuring the terminations used in the calibration procedure, the short
circuits used in calibration or by use of specially prepared components such as R-C networks housed in the
same type of enclosures as used for the crystals being measured. With the calibration devices, the measured
parameters would be converted to input and output impedances, Z and Z22, as the transmission with these
devices is essentially zero. A device having a series connected R-C between the active pins of a crystal
enclosure would provide the Y12 (or Z 1/Y ) output. Such components provide a very convenient
444-5 © IEC:1995 –
25 –
verification, but are generally best suited for internal use, as they must be initially calibrated by several
measurements made with a well-verified system.
5.3.3 For example, if the verification program is used to measure the standard termination devices used to
calibrate the instrument, which would have an impedance of very nearly 50
n, and be accurately known, it
would be expected that the Z11 and Z22
values obtained would be within 0,2 % of the correct value, and would
have a reactive component less than 0,2 % of the
resistive term. If the standard sho rt circuits are measured,
Z11
and Z22 would be expected to give both real and reactive impedance components less than 0,1 f2.
5.3.4 It is desirable that verification measurements shall not be made at exactly the same frequencies at which
calibration measurements were made, so that the accuracy of the interpolation algorithms will also be checked.
5.3.5 Implementations of software systems which provide these functions are commercially available for some
of the network analyzer systems offered by instrument manufacturers.
6 Low-frequency measurements
6.1 Measurements at frequencies below 100 kHz present two particular difficulties; first,
the impedance level
of the resonator may be very high, producing an
excessive insertion loss in a 50 L 1 system, and second, the
reflectometers in the network analyzer test set may have inadequate directivity at low frequencies.
6.2 It is recommended therefore that the amplitude/ph
ase transmission method be used. For high R 1 devices,
it is also recommended that a fixture which presents a high impedance to
the crystal terminations be used. To
avoid low instrumentation levels, an impedance matching transformer may be required. Should
the use of
S-parameter methods be desired, annex B outlines some b
asic details.
7 Admittance analysis and estimation of the equivalent circuit parameters
7.1
General least-squares fitting method
7.1.1 As a general criterion, the values of the equivalent circuit parameters will be defined
as those which
minimize the function
E Wi'Yi
– YiA =1W. ((Gi – Gi ) 2 + (B – Bi
2 (7.1)
i ) )
where Yi = Gi + jBi and Wi is a weighting factor.
Yi denotes the theoretical value of the admittance, Y, computed at the measurement frequency co
i, and expressed
in terms of the parameters G0, G02 , R 1 , L 1 and C1 ; represents the measured value of Y at
YA inferred from the
wi
error-corrected S-parameters. Y is essentially the same function for both reflection and transmission
measurements, and from now on its parameters will be referred to as Go, CO,
R 1 , L 1 , C1. Various possible
choices of W
i may be used; if Co and G0 are determined by separate off-resonance measurements then
the choice
Wi = 1 is usually most appropriate; if C 0 is to be estimated from measurements in th
e vicinity of resonance and it
is necessary to ensure that
the anti-resonance point is accurately described, then the choice w
i = I Y; ` r 1 should
be used.
444-5 © 27 –
IEC:1995 –
7.1.2 The function E has the following properties
1,.A A2
-
lyi
a) As W. Y Y' 1
/
2 = Wi i
YA
i
the relative error in Y is weighted to give greatest significance to the region close to series resonance
(where 11' is a maximum).
b) Both the real and imaginary parts of Y are used.
the
c) If Wi is chosen to be I 11'11 close agreement between the theoretical and experimental values of
measurement points toi.
anti-resonance frequency is assured, provided it is within the range of the
e use of a single- mode
7.1.3 On occasions, an unwanted mode may appear so close to the main response that th
equivalent circuit is questionable; in this case, the more general circuits of figures 6 and 7, containing several
E as before, but
resonant arms, may be used. The equivalent circuit parameters can be determined by minimizing
the minimization is now with respect to an increased number of variables. A simultaneous minimization of this
only entirely satisfactory way to treat this problem.
form is the
7.2 Linear least-squares fitting procedure
and easily generalized, but the process of
7.2.1 The recommended method of data fitting is versatile
minimization can only be performed by an iterative procedure. It is recognized that a mathematical
implementation of this method may be impractical for some users, and for this reason certain alternative
non-iterative procedures are given.
7.2.2 All of non-iterative (linear) methods use a combination of at least two fitting operations, and they are
the
the more
not readily generalized to more complex problems, such as two resonance fits, which require
sophisticated non-linear approach. In order to maintain overall consistency, it is advisable to choose linear
as possible to the general non-linear criterion.
methods which correspond as closely
7.2.3 To an excellent approximation, in the vicinity of resonance,
(7.2)
«C = o Co = B0
tos
)
2(w- (7.3)
CO L1 - w C –
L1
where
ws
L 1 C1
cos
(7.4)
Bo - 2(to-tos) R. G
Therefore B0 +^(
B =
Ll –
Rj
4(0) - t)s )2 a)
where G = R1 I (R? +
ance Go is neglected.
is the real component of transadmittance when any shunt conduct

444-5 ©IEC:1995 – 29 –
The expression for E
may therefore be rewritten
\21
L 1 )
+ Bo - 2(ar -co s) Gi - BA E = ^ W i (Gi - GA )2 (7.5
R1
L ,J
7.2.4 To obtain a suitable expression, it may be assumed that the difference between Gi and GiA can be
neglected, so if Gi = Gi
^ \2
L
then = (A)— Gi - (7.6)
E
7.2.5 Defining three independent variables
L1
p l = -47t R
2 ws L1 4rtfs Ll
(7.7)
P 2 = –
R 1 R1
P3 = Bo
E becomes + p2) + (7.8)
^ W i Gi p3 - BA)2
E = ((P1 .Î
The conditions for a minimum
aEaE DE
— = —=o
_
a Pl a P2 a p3
give the linear equations
)21
L
+ Wi f iGi Wi f iGi Bi
Gi (Gi ^ = 1,
E Wi (f i )2 Pl + I Wi f i P2 P3
J J ^ J i Li i
p BA
Wi fi(GA )2 Wi (GA )2 I I Wi Gl J 3 = Wi Gi (7.9)
LE l + P2 + ^
L^ J
i
P
+ ^
Wi f i Gi ^Wi GA Î p2 WiJ
P1 + I I P3 = 1 WiBA
I
L i J i
i J
From the point of view of numerical conditioning, it is advisable to use a frequency scale measured from some
reference value fairly close to fs when solving these equations.
L
th exact expression for R1
7.2.6 If the values of cos and 1 just determined are substituted into e E, then Co and
R1
may be obtained by another minimization of E.

444-5 © IEC:1995 – 31 –
i L
2\
i
If T=
co-^s x
c.^
^ i R 1
(7.10)
=
and U and V TU
1 + T 2
U V
then G= Go +— B=wCo - — (7.11)
R 1
R1
2 )21
I
U; Vi
E= + - + (7.12)
y, W;) Go G;' — BA ^
(0);c0
R r R1
(-
Minimizing with respect to
Go, Co and R 1 gives
F 1 F 1 1
Wi + W Wi
i Go i U i I x—_ Gt
LI r ^1 i J R1 i
f 1 F 1 1
[y,Wiq I Co - LI, WicoiVi I x — = / Wit,hBi (7.13)
J r r J R1 i
Wi(UiGi - ViBi )
^ wiui Go - = E
WiwiVi Co + LWiUiI x —
^i ^ i i R1
E;
7.2.7 This procedure does not, of course, determine the true minimum of two parameters are found by
minimizing an approximate expression for E. and then the exact function is minimized with respect to the
other parameters keeping the values of the first two fixed. The steps in the operation may be summarized as
follows.
7.2.8
Recommended linear procedure
a) Form approximate expressions for E by setting Gi = Gi .
b) Minimize approximate expression to determine values for:
L1
Bo. —  c,.>;
R1
L
c) With fixed values for 1 and cos minimize the exact expression for E to determine Go, an Ri.
Co d
RI
7.2.9 The above sequence of operations can be iterated by returnin g to step a), and instead of approximating
G; by using the parameter values already calculated to approximate G;. Steps a), b) and c) can then be
Gi
repeated until the parameter values agree on successive iterations to within prescribed limits. If Co has been
determined by off-resonance measurements then its value may be held const
ant in the above expressions, and
the minimizations performed only with respect to the other va riables.

444-5 © IEC:1995
— 33 —
7.3 Circle-fitting method
7.3.1 General
An alternative linear procedure, which is probably the simplest and most efficient way of data fitting, is
referred to as the circle-fitting method.
From the fully corrected S-parameters or amplitude/ph ase data measured at several frequencies, the —Y21
st-fit circular locus
admittance vectors are calculated. These values are used to establish the equation of the be
in the Y-plane.
It can be shown that
+ (B - B0)2 — (7.14)
G - Go-
2R1, 4 R 2
i
so that the points (G;, B;), being the real and imaginary components of -Y lie on a circle of radius 2 RI
and centred (Go + ,B0).
2R1
If this expression is expanded we obtain
-2BB (7.15)
Gô+Bp+R° I - G I 2Ga +R1 + G2 +B2 0 = 0
If parameters p2 and p3 are defined by
pl
Go )
[G +B 2
+— = 2 G + —, = 2B0
Pi = - p3
o
R ^
R1
then - (B^-Bo) (7.16)
1 - GA- G 1
o -
2R1
4Ri
2 (B )2)2
- A
GA )
(p1 p2 BA (GA
p3
p l , p2
If this expression is minimized with respect to p l . p2 and p3, then a linear set of equations for and p3 is
obtained, and
-1/2
\
+ 122 + B p
Bo = ^3,R1=—
pl
^ l ,
(7.17)
1^
— p2 -
Go
2 ^ R1
and the radius of the circle is given by
\ 1/2
P
p 1 + + (7.18)
Rad = — = B^
2 R1
i
444-5 ©IEC:1995 – 35 –
7.3.2 Estimation of C O3 Go
The static shunt capacitance can be detennined from
C = B o
CO
w
It is easily determined whether or not a zero-ph ase condition is possible - if Rad < Bo, the circle will not
ion ph
intersect the real axis in the Y-plane, and therefore the crystal can never produce a zero inse rt ase. Go is
the shunt conductance and may or may not be included in the equivalent circuit. Certainly if Go < 0,0001 x
Rad, the
errors due to neglecting the presence of Go are not significant.
7.3.3 Estimation ofR 1 , C 1 L 1 ,

, fs
The geometric measurements of admittance circle do not directly provide information about frequency or
the
motional induct the Z-plane.
ance. These parameters are found from examination of the reactance functions in
First, the Y-pl the measured -Y21
ane circle is used to "smooth" the measured data in t
...