Message	Date	From	Subject
1777	2008-07-04 21:29:10	Wes Hayward	Quartz Crystals
Crystal thoughts, July 4, 2008 Hi All, The recent crystal discussion is especially interesting to me. It's one of ever continuing interest, for quartz crystals are at the center of much that we do. I thought I'd post some info on this, for it may expand our ability to make filters, oscillators, and synthesizers. Also, this is one of those areas where we can apply the scientific method to actually do some science as amateurs. About the only physics text that I know about on this subject is Virgil E. Bottom, "Introduction to Quartz Crystal Unit Design," Van Nostrand Reinhold, 1982. It's an interesting book about a rather specialized segment of RF technology. The book is quite mathematical, dealing with the matrix relationships that are used to describe any crystal, be it quartz or a semiconductor or simple rock salt. It quickly specializes with the quartz crystals that we use for frequency control. We have all probably grown crystals. I remember doing it under a small toy microscope where I heated some water on a glass slide by placing the slide near a light bulb. I would then dissolve a salt that was included with the microscope. Simple table salt works, as I recall, but some of the colored salts were better. I'd let them cool while watching under the microscope. You could see regions that I assumed to be single crystals form from a uniform liquid. It was fun quasi-science for a kid in junior high school. The quartz crystals are things that appear in nature and can be mined. The crystals can also be grown. The crystal can then be cut or sliced along one of many possible crystal planes. This is akin to cutting a diamond to make jewelry or the perfect cutting tool. One can make models of chemical compounds including crystal material. This starts with a bunch of Styrofoam balls and sticks. The balls represent the atoms of Silicon and Oxygen. Quartz is SiO2. When the model is built, the atoms appear in a regular pattern that is periodic along many different lines that one can find in the structure. This regular structure is called the crystal lattice. We can cut through the model with a large saw in the same way that we can cut or saw through the real crystal along desired planes. Quartz is a piezoelectric material. That means that an applied electric field in one particular direction with regard to the crystal lattice will cause a mechanical force to occur, usually in another direction. It is easy to create an electric field in a crystal. One merely grows a metal electrode on the slabs of crystal and applies a voltage to it. Indeed, the old FT-243 type crystals that some of us used for our early transmitters didn't even have electrodes that were plated onto the crystal. Rather, they just had a pair of metal pieces with a slab of quartz mechanically clamped between them. The quartz with metal forms a capacitor. This is actually a pretty darn good capacitor, for SiO2 is a good insulator with a well defined dielectric constant, low thermal drift, and low RF loss. But for all intents and purposes, it is just a capacitor at most frequencies. But let's get back to the piezoelectric properties. As we mentioned above, a field in one direction generates a mechanic force. The force can then cause a displacement. This might be nothing more than a strain or slight motion within the crystal lattice. Conversely, if a mechanical motion or displacement occurs in the right direction with regard to the lattice structure, an electric field is generated. This could then lead to the appearance of a voltage at the capacitor electrodes. Some disc ceramic capacitors have a dielectric with piezoelectric characteristics, making them micro phonic. Sound waves, or other vibrations, generate a mechanical motion that then produces a voltage. A material with intimately related mechanical displacements and voltages sounds like the kind of environment that might lead to waves of one sort of another. Think about the wave equations that we see strewn through the fabric of physics. This is the kind of thing that would encourage us to examine the quartz crystal with an AC signal. That is, instead of merely measuring our capacitor at one frequency in order to determine a value of C, we investigate the crystal at a wide range of frequencies. When we do this, we discover strong resonances. These are related to the mechanical boundaries of our crystal. Specifically, the resonant frequency is inversely proportional to the thickness of the crystal slab. If we grind away some of the surface, the frequency increases. But if we load the surface with a material, such as the metal film that we attach as an electrode, it decreases the frequency. This has the added affect of decreasing the Q of the resonance. That is, the bandwidth increases. Electrical measurements of this resonance consist of merely examining the properties of this "thing" with two wires. We drive it and look at the current. The element behaves just like a simple capacitor, usually with a small value of a few pF, at most frequencies. At the resonant frequency, the current increases dramatically. The frequency of highest current (lowest impedance) is that where the current is almost in phase with the driving voltage. (It would be exact if we didn't have that C0.) This suggests that we have a series resonant RLC circuit and this becomes our basic model. At series resonance, the current is determined mainly by the resistive component, the "equivalent series resistance," or ESR. We measure the current (in concept) and watch the way it varies with frequency. We can do this with simple scalar instruments to get the essential details, which is the approach taken for my May 1982 QST paper. Essentially, the bandwidth is measured in a well controlled low impedance environment. This then allows calculation of the motional L and C. C0 can be measured at a lower frequency with an instrument like the AADE LC Meter, or a homebrew equivalent. We can extract more information if phase can be added to the measurements. When we use a vector network analyzer to measure impedance versus frequency, around resonance, all three reactive components can be inferred. Let's review what we know about the crystal before we extend it. The quartz material is a stable, low loss insulator that can be used to form simple capacitors. If the crystalline material is properly "cut" to apply an electric RF field at the right lattice plane, the crystal has piezoelectric properties. This leads to it behaving like a series resonant RLC circuit. The presence of a parallel capacitance across the series resonator also produces a parallel resonance. Parallel resonance is a frequency where the current becomes extremely small for the two terminal circuit. The series resonant frequency is inversely related to crystal thickness. Doubling thickness halves frequency. Simple, but modern crystals are round with a diameter of perhaps a quarter to half an inch, or smaller. They are coated with a metal over the entire area of each side of the disc. A typical thickness might be 20 mils for a 10 MHz fundamental crystal. (I'm guessing at that I need to measure one sometime.) The metallization is much less than 1 mil thick. The presence of the metal degrades Q. The thicker the metal, the lower the Q drops. This mass loading also decreases the frequency. Generally, the crystal is very sensitive to things happening at the surface. It is interesting to perform a very wide band sweep of the crystals to try to find all of the resonances. Neglecting some spurious responses that are usually minor, we find that there are several dominant responses. One is the fundamental operation that we have just described. Another is a the so called third overtone, which is close to 3X the fundamental frequency. The 5th overtone is also active and so on, but only with odd orders. There is no response at the even overtones. I'll not go into the details, but this behavior supports a specific mechanical motion as the crystal oscillates. The surfaces are moving back and forth at right angles to the axis of the disc. The applied E field is parallel to the axis of the disc. There exists a region half way through the crystal thickness that has no motion for a fundamental mode oscillation. Two nulls exist with the 3rd overtone, and so on. Although not exact, the motional inductance that models a crystal is constant with overtone number. Hence, the motional capacitance moves to determine resonance, and moves as the square of the overtone number. It might be worthwhile to throw in some numbers. I've loaded a figure called rocks_10.jpg that shows three 10 MHz crystals. One is a metal can job while the other two are glass units. They are all at the same nominal frequency, but the glass crystals are third overtone units. That is, they have a 3.3 MHz fundamental. The metal crystal has a measured motional inductance of 20.6 millihenry and a parallel capacitance of 3.1 pF. You can calculate the motional C from the stated motional L and the series resonant frequency of 9.9981 MHz. The Q for this crystal was pretty good at 262,000. The ratio of parallel to motional C is 252. The crystal physics presented by Bottom suggests a proper value is from 200 to 220. The difference is attributed to stray C. If we take 200 as the proper value, we would predict a parallel C of 2.46. The difference of 0.64 pF is the stray related to the container and support leads. The glass crystals were much different. I only did measurements at 10 MHz, but will take a look at the fundamental at a later time. Both of the glass crystals were about the same, so only one will have data presented. The series resonant frequency was 9.9999887 MHz with a parallel C of 5.80 pF. Q comes in at 760,000. This is not your garden variety microprocessor crystal. The motional L is 0.11143 H, but the motional C is only 2.273 fF. One fF is 1/1000 pF and the fF stands for femptofarad. The ratio of parallel to motional C is now 2552. But this is an overtone crystal, with motional C that is diminished by a factor of 9 over what we would expect for a fundamental. The physical size of the glass crystal is much larger than the metal can job. Also, the glass enclosure is evacuated, creating a vacuum, although probably not much of one. All of the measurements reported here were done with a vector network analyzer of the N2PK type. Just Google N2PK on the web to find out more about this instrument. The crystals were manufactured by Colorado Crystal. (They don't make ham crystals, so please don't bug 'em. The crystals I have are surplus.) The approximately fixed ratio relationship between motional C and parallel C suggests a means for controlling parameters. We gain control over the motional parameters by picking the fraction of the crystal that contains plating. Note the glass crystals have about half of the area metalized. Although not clear in the photo, the region under the plating and for a bit larger diameter is transparent to light. The outer region is translucent, probably the result of thickness beveling. The third overtone crystals had the higher Q. I've seen 10 MHz 3rd overtone crystals with a Q of a bit over 1,000,000. That was a while ago, so they may have improved over the last years. (Who knows?) We can now do an interesting "thought experiment." Note that our crystals have a frequency that is related to the thickness, but not to other dimensions. So consider a 10 MHz crystal that has metallization applied over the total rock area. We attach some wires and measure it to confirm performance. Next, we remove the wires and carefully snap the crystal blank into two chunks of approximately equal size. If we now attach wires to the two sections and measure each, we still measure 10 MHz, for the thickness has not changed. But the parallel capacitance will have decreased by 2. We find a motional C that is also down by 2. The motional L is, accordingly, up by 2. This experiment can be continued with similar results until we arrive at the crystals that we find in some of the really small holders. They can still be used to make filters, but the lower motional C values will lead to correspondingly smaller coupling capacitors. The parasitic capacitors will not generally drop at the same rate, so the miniature sized crystals will have a higher proportion of stray to "proper" parallel C. One of the questions that has been posed in recent weeks has to do with the construction of filters with crystals that do not have identical motional L. The software that I've written as well as that by Neil at AADE assumes identical Lm. But this is only an approximate requirement. As this discussion may suggest, there is going to be a slight variation in Lm, even for a given manufacturing process. The same thing happens when we build LC filters. We wind the toroids, or whatever, to have the same L, but they don't always come out exactly the same. If we have a +/- 10 % variation in L, we are in great shape so long as each mesh, or node, is properly tuned. We tweak the LC filters with a trimmer cap. A crystal that has Lm away from the nominal value will still be close to resonance, for we have selected it to be thus. The tuning capacitors only have to account for the departures from resonance introduced by the coupling capacitors that are different from mesh to mesh. I have always been able to make predictable filters by fully characterizing only a few of the samples within a batch. The rest of the work is done with the frequency matching. On the other hand, I would not try to use the two extremely different crystals that we have used in this discussion. A filter could be designed with a mixture of them, but the synthesis procedure would take us beyond the simple methods in the programs. In the same vein, we can design coupled resonator LC filters with drastically different L values. Zverev deals with this. Such variations are sometimes done with crystal filters as a means for mixing the spurious responses so that they don't all line up. Anyway, the holiday is about finished and it's time to button this one up and post it. I hope that this has illuminated some of the questions presented in recent postings. 73, Wes w7zoi
1778	2008-07-04 22:26:17	chuck adams	Re: Quartz Crystals
On Friday 04 July 2008 21:29:08 Wes Hayward wrote: > Crystal thoughts, July 4, 2008 > > Hi All, > > The recent crystal discussion is especially interesting to me. It's > one of ever continuing interest, for quartz crystals are at the > center of much that we do. I thought I'd post some info on this, > for it may expand our ability to make filters, oscillators, and > synthesizers. Also, this is one of those areas where we can apply > the scientific method to actually do some science as amateurs. > > About the only physics text that I know about on this subject is > Virgil E. Bottom, "Introduction to Quartz Crystal Unit Design," Van > Nostrand Reinhold, 1982. It's an interesting book about a rather > specialized segment of RF technology. The book is quite > mathematical, dealing with the matrix relationships that are used to > describe any crystal, be it quartz or a semiconductor or simple rock > salt. It quickly specializes with the quartz crystals that we use > for frequency control. > ...snip snip... > > 73, Wes > w7zoi Thanks for the posting Wes. I worked with Dr Bottom while I was an undergraduate physics major at McMurry College from 1961 to 1964. I transferred my Sr year to Texas Tech when Dr Bottom went to Brazil for a year on a Fulbright Fellowship. A lot of the data that I took and others worked on is in the book. Probably my number 1 or 2 book in my library. It cost me $80 and well worth every penny. Hard to find. If you Google for Virgil E. Bottom, you will find a number of good hits. He wrote a paper for the IEEE on the history of the quart crystal. See http://www.4timing.com/history.htm for a pointer to his paper and others (one on ladder filters) that may be of interest to others. Other hits point to patents, etc. I miss those days of access to a lab with all the top HP and Collins lab equipment and drawers and drawers full of the best Brazilian quartz you could possibly use...... I ground a lot of crystal blanks to different surface curvatures for measurements. I did the Millikan oil drop experiment for a physics lab in the middle of the room surrounded by quartz and got a 1/3e measurement. Wish I could reproduce that one. :-) FYI -- chuck adams, k7qo k7qo@commspeed.net http://www.k7qo.net
1779	2008-07-04 22:43:40	k5nwa	Re: Quartz Crystals
At 12:26 AM 7/5/2008, you wrote: >If you Google for Virgil E. Bottom, you will find a number of good >hits. He wrote a paper for the IEEE on the history of the quart >crystal. See > ><http://www.4timing.com/history.htm>http://www.4timing.com/history.htm > >-- >chuck adams, k7qo That article is incredibly fascinating, it's interesting that hams were at the foreront of crystals prior to the war. Cecil K5NWA www.softrockradio.org www.qrpradio.com "Blessed are the cracked, for they shall let in the light."
1781	2008-07-04 23:21:44	kerrypwr	Re: Quartz Crystals
A great overview; thanks Wes. The crystal filters chapter in Zverev is very interesting. I have Zverev on loan for a few weeks and the scanner is working overtime!!
1782	2008-07-05 11:27:16	Graham Haddock	Re: Quartz Crystals
Here are some current references by John R. Vig. http://www.gpstime.com/files/Vig-tutorial_Jan_2007.ppt http://www.ieee-uffc.org/freqcontrol/quartz/vig/vigtoc.htm http://www.ieee-uffc.org/freqcontrol/VigBallato/fcdevices.PDF An interesting site in its own right: http://www.gpstime.com/ Once you understand quartz crystals, the next thing you are going to want to do is have high stability, accuracy and low phase noise. --- Graham / KE9H
1783	2008-07-05 19:29:52	Dave	Re: Quartz Crystals

1784	2008-07-05 20:46:01	Russell Shaw	Re: Quartz Crystals
Dave wrote: >
1785	2008-07-08 10:50:35	Wes Hayward	Re: Quartz Crystals
Hi Chuck, and group. Many thanks for that ieee paper on the history. Really great stuff. One of the things that comes from any connection that I've even had with the crystal industry is that the work that Bottom did is highly revered. He was the central figure for it all for a long time, it seems. Thanks to the rest of the folks who responded on this thread too. Lots of good stuff. 73, Wes w7zoi
1807	2008-07-12 08:33:19	wimmie262000	Re: Quartz Crystals

Crystal thoughts, July 4, 2008

Hi All,

The recent crystal discussion is especially interesting to me. It's
one of ever continuing interest, for quartz crystals are at the
center of much that we do. I thought I'd post some info on this,
for it may expand our ability to make filters, oscillators, and
synthesizers. Also, this is one of those areas where we can apply
the scientific method to actually do some science as amateurs.

About the only physics text that I know about on this subject is
Virgil E. Bottom, "Introduction to Quartz Crystal Unit Design," Van
Nostrand Reinhold, 1982. It's an interesting book about a rather
specialized segment of RF technology. The book is quite
mathematical, dealing with the matrix relationships that are used to
describe any crystal, be it quartz or a semiconductor or simple rock
salt. It quickly specializes with the quartz crystals that we use
for frequency control.

We have all probably grown crystals. I remember doing it under a
small toy microscope where I heated some water on a glass slide by
placing the slide near a light bulb. I would then dissolve a salt
that was included with the microscope. Simple table salt works, as
I recall, but some of the colored salts were better. I'd let them
cool while watching under the microscope. You could see regions that
I assumed to be single crystals form from a uniform liquid. It was
fun quasi-science for a kid in junior high school.

The quartz crystals are things that appear in nature and can be
mined. The crystals can also be grown. The crystal can then be cut
or sliced along one of many possible crystal planes. This is akin
to cutting a diamond to make jewelry or the perfect cutting tool.

One can make models of chemical compounds including crystal
material. This starts with a bunch of Styrofoam balls and sticks.
The balls represent the atoms of Silicon and Oxygen. Quartz is
SiO2. When the model is built, the atoms appear in a regular
pattern that is periodic along many different lines that one can find
in the structure. This regular structure is called the crystal
lattice. We can cut through the model with a large saw in the
same way that we can cut or saw through the real crystal along
desired planes.

Quartz is a piezoelectric material. That means that an applied
electric field in one particular direction with regard to the crystal
lattice will cause a mechanical force to occur, usually in another
direction. It is easy to create an electric field in a crystal.
One merely grows a metal electrode on the slabs of crystal and
applies a voltage to it. Indeed, the old FT-243 type crystals that
some of us used for our early transmitters didn't even have
electrodes that were plated onto the crystal. Rather, they just had
a pair of metal pieces with a slab of quartz mechanically clamped
between them.

The quartz with metal forms a capacitor. This is actually a pretty
darn good capacitor, for SiO2 is a good insulator with a well defined
dielectric constant, low thermal drift, and low RF loss. But for
all intents and purposes, it is just a capacitor at most
frequencies.

But let's get back to the piezoelectric properties. As we mentioned
above, a field in one direction generates a mechanic force. The
force can then cause a displacement. This might be nothing more than
a strain or slight motion within the crystal lattice. Conversely,
if a mechanical motion or displacement occurs in the right direction
with regard to the lattice structure, an electric field is
generated. This could then lead to the appearance of a voltage at
the capacitor electrodes. Some disc ceramic capacitors have a
dielectric with piezoelectric characteristics, making them micro
phonic. Sound waves, or other vibrations, generate a mechanical
motion that then produces a voltage.

A material with intimately related mechanical displacements and
voltages sounds like the kind of environment that might lead to waves
of one sort of another. Think about the wave equations that we see
strewn through the fabric of physics. This is the kind of thing
that would encourage us to examine the quartz crystal with an AC
signal. That is, instead of merely measuring our capacitor at one
frequency in order to determine a value of C, we investigate the
crystal at a wide range of frequencies. When we do this, we
discover strong resonances. These are related to the mechanical
boundaries of our crystal. Specifically, the resonant frequency is
inversely proportional to the thickness of the crystal slab. If we
grind away some of the surface, the frequency increases. But if we
load the surface with a material, such as the metal film that we
attach as an electrode, it decreases the frequency. This has the
added affect of decreasing the Q of the resonance. That is, the
bandwidth increases.

Electrical measurements of this resonance consist of merely examining
the properties of this "thing" with two wires. We drive it and
look at the current. The element behaves just like a simple
capacitor, usually with a small value of a few pF, at most
frequencies. At the resonant frequency, the current increases
dramatically. The frequency of highest current (lowest impedance)
is that where the current is almost in phase with the driving
voltage. (It would be exact if we didn't have that C0.) This
suggests that we have a series resonant RLC circuit and this becomes
our basic model. At series resonance, the current is determined
mainly by the resistive component, the "equivalent series
resistance," or ESR. We measure the current (in concept) and watch
the way it varies with frequency. We can do this with simple
scalar instruments to get the essential details, which is the
approach taken for my May 1982 QST paper. Essentially, the
bandwidth is measured in a well controlled low impedance
environment. This then allows calculation of the motional L and
C. C0 can be measured at a lower frequency with an instrument
like the AADE LC Meter, or a homebrew equivalent.

We can extract more information if phase can be added to the
measurements. When we use a vector network analyzer to measure
impedance versus frequency, around resonance, all three reactive
components can be inferred.

Let's review what we know about the crystal before we extend it.
The quartz material is a stable, low loss insulator that can be used
to form simple capacitors. If the crystalline material is
properly "cut" to apply an electric RF field at the right lattice
plane, the crystal has piezoelectric properties. This leads to it
behaving like a series resonant RLC circuit. The presence of a
parallel capacitance across the series resonator also produces a
parallel resonance. Parallel resonance is a frequency where the
current becomes extremely small for the two terminal circuit. The
series resonant frequency is inversely related to crystal
thickness. Doubling thickness halves frequency.

Simple, but modern crystals are round with a diameter of perhaps a
quarter to half an inch, or smaller. They are coated with a metal
over the entire area of each side of the disc. A typical thickness
might be 20 mils for a 10 MHz fundamental crystal. (I'm guessing at
that I need to measure one sometime.) The metallization is much
less than 1 mil thick. The presence of the metal degrades Q. The
thicker the metal, the lower the Q drops. This mass loading also
decreases the frequency. Generally, the crystal is very sensitive
to things happening at the surface.

It is interesting to perform a very wide band sweep of the crystals
to try to find all of the resonances. Neglecting some spurious
responses that are usually minor, we find that there are several
dominant responses. One is the fundamental operation that we have
just described. Another is a the so called third overtone, which is
close to 3X the fundamental frequency. The 5th overtone is also
active and so on, but only with odd orders. There is no response at
the even overtones. I'll not go into the details, but this
behavior supports a specific mechanical motion as the crystal
oscillates. The surfaces are moving back and forth at right angles
to the axis of the disc. The applied E field is parallel to the
axis of the disc. There exists a region half way through the
crystal thickness that has no motion for a fundamental mode
oscillation. Two nulls exist with the 3rd overtone, and so on.

Although not exact, the motional inductance that models a crystal is
constant with overtone number. Hence, the motional capacitance
moves to determine resonance, and moves as the square of the overtone
number.

It might be worthwhile to throw in some numbers. I've loaded a
figure called rocks_10.jpg that shows three 10 MHz crystals. One is
a metal can job while the other two are glass units. They are all
at the same nominal frequency, but the glass crystals are third
overtone units. That is, they have a 3.3 MHz fundamental. The
metal crystal has a measured motional inductance of 20.6 millihenry
and a parallel capacitance of 3.1 pF. You can calculate the
motional C from the stated motional L and the series resonant
frequency of 9.9981 MHz. The Q for this crystal was pretty good at
262,000. The ratio of parallel to motional C is 252. The
crystal physics presented by Bottom suggests a proper value is from
200 to 220. The difference is attributed to stray C. If we take
200 as the proper value, we would predict a parallel C of 2.46. The
difference of 0.64 pF is the stray related to the container and
support leads.

The glass crystals were much different. I only did measurements at
10 MHz, but will take a look at the fundamental at a later time.
Both of the glass crystals were about the same, so only one will have
data presented. The series resonant frequency was 9.9999887 MHz
with a parallel C of 5.80 pF. Q comes in at 760,000. This is not
your garden variety microprocessor crystal. The motional L is
0.11143 H, but the motional C is only 2.273 fF. One fF is 1/1000
pF and the fF stands for femptofarad. The ratio of parallel to
motional C is now 2552. But this is an overtone crystal, with
motional C that is diminished by a factor of 9 over what we would
expect for a fundamental. The physical size of the glass crystal is
much larger than the metal can job. Also, the glass enclosure is
evacuated, creating a vacuum, although probably not much of one.

All of the measurements reported here were done with a vector network
analyzer of the N2PK type. Just Google N2PK on the web to find out
more about this instrument. The crystals were manufactured by
Colorado Crystal. (They don't make ham crystals, so please don't
bug 'em. The crystals I have are surplus.)

The approximately fixed ratio relationship between motional C and
parallel C suggests a means for controlling parameters. We gain
control over the motional parameters by picking the fraction of the
crystal that contains plating. Note the glass crystals have about
half of the area metalized. Although not clear in the photo, the
region under the plating and for a bit larger diameter is transparent
to light. The outer region is translucent, probably the result of
thickness beveling.

The third overtone crystals had the higher Q. I've seen 10 MHz 3rd
overtone crystals with a Q of a bit over 1,000,000. That was a
while ago, so they may have improved over the last years. (Who
knows?)

We can now do an interesting "thought experiment." Note that our
crystals have a frequency that is related to the thickness, but not
to other dimensions. So consider a 10 MHz crystal that has
metallization applied over the total rock area. We attach some
wires and measure it to confirm performance. Next, we remove the
wires and carefully snap the crystal blank into two chunks of
approximately equal size. If we now attach wires to the two
sections and measure each, we still measure 10 MHz, for the thickness
has not changed. But the parallel capacitance will have decreased
by 2. We find a motional C that is also down by 2. The motional
L is, accordingly, up by 2. This experiment can be continued with
similar results until we arrive at the crystals that we find in some
of the really small holders. They can still be used to make
filters, but the lower motional C values will lead to correspondingly
smaller coupling capacitors. The parasitic capacitors will not
generally drop at the same rate, so the miniature sized crystals will
have a higher proportion of stray to "proper" parallel C.

One of the questions that has been posed in recent weeks has to do
with the construction of filters with crystals that do not have
identical motional L. The software that I've written as well as
that by Neil at AADE assumes identical Lm. But this is only an
approximate requirement. As this discussion may suggest, there is
going to be a slight variation in Lm, even for a given manufacturing
process. The same thing happens when we build LC filters. We
wind the toroids, or whatever, to have the same L, but they don't
always come out exactly the same. If we have a +/- 10 % variation
in L, we are in great shape so long as each mesh, or node, is
properly tuned. We tweak the LC filters with a trimmer cap. A
crystal that has Lm away from the nominal value will still be close
to resonance, for we have selected it to be thus. The tuning
capacitors only have to account for the departures from resonance
introduced by the coupling capacitors that are different from mesh to
mesh.

I have always been able to make predictable filters by fully
characterizing only a few of the samples within a batch. The rest
of the work is done with the frequency matching. On the other
hand, I would not try to use the two extremely different crystals
that we have used in this discussion. A filter could be designed
with a mixture of them, but the synthesis procedure would take us
beyond the simple methods in the programs. In the same vein, we
can design coupled resonator LC filters with drastically different L
values. Zverev deals with this. Such variations are sometimes
done with crystal filters as a means for mixing the spurious
responses so that they don't all line up.

Anyway, the holiday is about finished and it's time to button this
one up and post it. I hope that this has illuminated some of the
questions presented in recent postings.

73, Wes
w7zoi