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Product Tech Highlight Crystals & Clock Oscillators

Frequency Pullability in AT Cut Crystals

Applications to Phase Locked Loops
by Louis Bradshaw, Fox Electronics
the series resonant frequency must be calculated in order which in the present case is 16.01pF.
lthough symmetrical frequency pullability is typi- to determine pullability. Therefore, Equation 2.0 must be Be aware that the new value of load capacitance is not

A cally the goal of the PLL designer, it is difficult to

achieve. There is no completely practical way to
achieve symmetrical frequency pullability. However, this
modified to account for the load capacitance: to be substituted for the nominal value in the specifica-
tion. Rather, the new value of load capacitance is to be
substituted for the nominal in the actual circuit.
article will discuss a theoretical method. It is important to note that the result of this calculation
Typically, crystal units to be used in phase locked loop is accurate only insofar as the actual values of capacitance
(PLL) applications are specified with three values of load are identical to the stated values. Since this is usually not
capacitance: a nominal (center) value; a value higher than the case, given typical manufacturing tolerances, one must
nominal; and a value lower than nominal. The operating Where fs is the series resonant frequency in MHz, fload also consider the impact of another parameter relating fre-
frequency of the crystal is that found at the nominal load is the load resonant frequency in MHz, C1 is the motional quency to load capacitance: trim sensitivity.
capacitance value. Frequency pullability is defined as the capacitance in pF, C0 is the static or shunt capacitance in
resulting change in frequency when the load capacitance pF and Cload is the load capacitance in pF. By applying Impact of Trim Sensitivity on
is switched from the nominal value to either end point. Equation 3.0 (above), using the values given for the speci- Symmetrical Pullability
(While pullability is sometimes specified as the change in fied crystal, we arrive at a series resonant frequency (fs) of The previous equations describe the significant change
frequency that results from switching from one end point the following: in frequency resulting from a significant change in load
load capacitance to the other, this is uncommon and will capacitance. “Trim sensitivity” refers to changes in fre-
not be addressed in this article). 18.423622 MHz quency resulting from a small change in load capacitance.
The values of the end point load capacitances are typi- Usually designated as “S”, it is calculated by Equation 5.0:
cally specified to be symmetrical about the nominal value. After calculating the series resonant frequency, we may
However, symmetrical changes in load capacitance do not now calculate the pullability of a crystal with the values
result in symmetrical changes in frequency. This is often defined above by using Equation 3.0. The change in fre-
disappointing to the PLL designer in search of symmetri- quency needs to be determined from the (calculated) series
cal frequency pullability, and we are often asked if there is resonance frequency to the frequency at the nominal load
a way to achieve symmetrical pullability. To our knowl- capacitance value, which for the specified crystal is 8378
edge, there is no completely practical way; however, a theo- Hz or 454.8 PPM. Also, it will require the pullability from Where S is the trim sensitivity in PPM/pF, C1 is the
retical method is decribed below. The PLL designer must the series resonance frequency to the lowest load capaci- motional capacitance in pF, C0 is the static or shunt capac-
determine the worth of this method on a case-by-case tance and from the series resonance frequency to the high- itance in pF and Cload is the load capacitance in pF.
basis. est load capacitance. In the present case, pullability from From the equation, one can see that the value of trim
series to 10.0 pF is 13957 Hz or 758 PPM; from series to sensitivity (usually given in PPM/pF) changes, possibly
Calculation of Pullability 30.0pF, pullability is 5982 Hz or 325 PPM. dramatically, depending on the load capacitance. For a
Before describing the theoretical method of achieving Note that Equation 3.0 includes both the shunt capaci- crystal used in PLL applications, different trim sensitivity
symmetrical pullability, the equations need to be present- tance and the load capacitance. Note also that the change will be found at each of the three defined load capacitance
ed describing frequency pullability itself. The pullability of in frequency will be greater at lower values of load capaci- values. Any variance in load capacitance from specified
a crystal at a specific frequency is given by Equation 1.0: tance than at higher values. To illustrate, we have plotted values will impact the calculation of pullability.
in frequency changes through a range of load capacitance Trim sensitivity should not be confused with pullability,
values in Figure 1. From Figure 1, it clearly can be seen though there is a tendency to do so. The calculated values
that a symmetrical change in load capacitance about a for trim sensitivity are applicable only to small changes in
central (nominal) value will not lead to a symmetrical load capacitance, and must not be extrapolated to calcu-
change in frequency. late pullability.

Where fa is the anti-resonant frequency in MHz, fs is Achieving Theoretical Symmetrical Editor’s Note
the series resonant frequency in MHz, C1 is the motional Frequency Pullability While the equations presented here are commonplace
capacitance in pF and C0 is the static or shunt capacitance As noted above, the equation for frequency pullability in the frequency control industry, it is essential to remem-
in pF. includes values for both the shunt capacitance and the ber that the company is only applying math to theoretical-
Note that the calculation is based on the frequency load capacitance. The value for shunt capacitance is fixed ly perfect quartz. Since perfect quartz is not found in the
change from series resonance to the natural anti-resonant during manufacture of the crystal and can thus be consid- real world, actual values may, and often do, differ, some-
(parallel) frequency of the crystal. Since crystal units used ered invariant. This leaves the load capacitance as the times significantly. PLL designers are strongly advised to
in PLL applications are invariably specified as load reso- only value that can be varied. By using Equation 4.0, one discuss their needs with crystal vendors early in the
nant (parallel) crystals, one must first calculate the series can calculate a theoretical value of load capacitance which, design process to determine the range of possible solu-
resonant frequency. The series resonant frequency is calcu- if substituted for the nominal (center) capacitance value, tions. One should not substitute the use of these equations
lated as follows: for such discussions, as one’s desired values of crystal
parameters may well be unavailable. To be as clear as pos-
sible, use these equations only for your own edification. Do
not attempt to design your own crystal.

Where fs is the series resonant frequency in MHz, fa is will yield symmetrical pullability. Louis Bradshaw is a Technical Consultant for Fox
the load resonant frequency in MHz, C1 is the motional Where Cload is the load capacitance in pF required to Electronics. He attended Texas Technological College,
capacitance in pF, C0 is the static or shunt capacitance in place a crystal at a specified frequency, C1 is the motional Lubbock, TX, for two years and has over 25 years expe-
pF and Cload is the load capacitance in pF. Obviously, capacitance in pF, C0 is the static or shunt capacitance in rience in the quartz crystal industry, having worked in
Equation 2.0 (above) would be used in cases where only pf and ⌬f is the change in frequency (in MHz) from series engineering, quality control, manufacturing and sales.
the anti-resonant frequency is known. resonance to the specified load resonant frequency in He can be reached at: c/o Fox Electronics, 5570
Equations 1.0 and 2.0 are valid as long as the crystal is MHz. Enterprise Parkway, Fort Myers, FL 33905;
properly designed and manufactured and free from To apply Equation 4.0 (above) to the present case, one (888) GET-2-FOX; Fax: (941) 693-1554 or via
defects, which should be assumed in this case. To use must recall the pullability values from series to 10.0pF e-mail at tech_support@foxonline.com.
Equation 2.0 in a modified form, one must also make some and from series to 30.0pF. These values are as follows:
assumptions about the values specified for the load reso- Write in 5213 or www.ecnmag.com/info
nant crystal: Series to 10.0 pF: 13957 Hz
Operating frequency: fo = 18.432000 MHz Series to 30.0 pF: 5982 Hz EDITORIAL EVALUATION
Nominal load capacitance: Cload0 = 20.0pF
Lowest load capacitance: Cload1 = 10.0pF The median value of these two frequency deviations is Write in Number or Reply Online
I found this article:
Highest load capacitance: Cload2 = 30.0pF 9970 Hz or .009970 MHz. The median value is added to
Shunt capacitance: C0 = 5.0pF the (calculated) series resonant frequency of 18.423622 Very Useful Useful Not Useful
5214 5215 5216
Motional capacitance: C1 = 0.022727pF (22.727fP) MHz for a new frequency of 18.433592 MHz. A symmetri-
cal pullability about this new frequency will be achieved
Since fo is specified as a load resonant frequency (fload), by using the load capacitance derived by Equation 4.0,