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29 April 2007 - Designing a Cohn Crystal
Ladder Filter
Let's design a Cohn-style crystal filter. A Cohn filter is
also known as a "minimum loss" filter and can be implemented with LC components
or crystals (which, of course, are mechanical analogs of LC components.) The
characteristics of a Cohn crystal filter are:
- Simple design—all coupling elements (normally
capacitors) are equal in value
- Low insertion loss (that's why it's known as a "minimum
loss" filter, after all.)
- Flat top response can be achieved.
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Our filter will use ECS-80-S-1X 8.0 8 MHz crystals, the
same crystals used in the Z90's Gaussian crystal filter. Rather than go through
the math of designing the filter, we'll use the excellent (and free) filter
design program AADE Filter
Design, version 4.2.1. Our design bandwidth is 3.0 KHz, but we know
from experience that the actually achieved bandwidth will be less, due to the
finite Q of the crystals we use. Our filter will be order 4 (uses four
crystals). Speaking of crystals, the motional parameters of the
ECS-80-S-1X 8.0 crystal are given below. This data is the mean of 65
samples I measured using an HP87510A VNA in automatic crystal parameter
calculation mode.
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I'll go quickly over how one uses AADE Filter Design. Select
Design | Crystal Ladder | Classic | Min-Loss (Cohn) and enter the desired design
data into the screens that you are presented with. Use the mean parameters above
for the crystal data. (I have used AADE FIlter Design's save custom crystal
feature to save these parameters so they may be entered at the press of a
button.) If you have entered the correct data, your
automatic design should match the following schematic. |
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R is 243 ohms and all C values are 82 pF, after rounding.
The predicted power effective gain shows a reasonably flat top, but fairly
significant asymmetry in the skirts. This is not unexpected. As the frequency
increases, the crystal arms reach parallel resonance, which causes a major
increase in filter rejection, i.e., much better flank selectivity. On the low
frequency side, there is no similar effect; the passband is set by the series
resonant parameters of the crystals and below series resonance, the crystals
look like a capacitor (the holder capacitance, mostly). Hence, we have a multi
section RC filter as we move far away from crystal resonance. |
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Now we match the crystal filter design to our target impedance, which, in this
case, will be 50 ohms. We could have used AADE Filter Design's automatic
matching network design feature, which adds a tapped LC matching network at
either end of the filter. However, we can achieve match with a simpler technique
and also make the resulting design more amenable to bandwidth switching via
shunt capacitance changes.
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First, we wish to remove the series capacitors, substituting
instead shunt capacitors. We do this by noting that at any series network
may be transformed into an equivalent parallel network. Where the network
contains frequency sensitive elements (inductance or capacitance) the transform
is valid only at the frequency for which it is computed.
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In our filter, Rs = 243Ω and Cs=82pF. The frequency for the transform is, of
course, 8 MHz. At 8 MHz, the capacitive reactance of 82pF is -j243Ω. (It's not
an accident that Xc = R in the Cohn design.)
The equations relating the series and parallel values are
shown below. X can be either inductive or capacitive.
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The relationship between capacitive reactance XC
and inductive reactance XLand frequency (f) is, of course, given by
the following equations: |
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Applying the two two equations, we compute:
RP = 486Ω
XP= 486Ω
Solving the relationship between capacitive reactance,
frequency and capacitance, we find  so,
CP = 41pF
As should be clear, this transform is valid only at the
frequency for which it is computed. At this point, our crystal filter design is
shown below. |
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Now we have a simple task remaining, to transform the 486
ohms input and output impedance to 50 ohms. We do this via an L network. I'm not
going to discuss how to compute the L network parameters, as this is well
covered in the ARRL's Radio Amateur's Handbook, Chapter 25 in recent editions.
Or, you can use one of the programs available to automatically compute L and Pi
matching networks. (I use RFSim99, which produces the results shown below.)
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Applying the L network, we see that C7 and C5 may be combined
in a single 162 pF capacitor. |
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The complete filter, matched to 50 ohms is presented below
(substituting 2.7uH inductors that I had on hand for the calculated 2.9uH
parts.) |
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As built, I reduced all capacitor values by 15pF and paralleled them with 5-30
pF trimmers, to tweak the filter into optimum shape.
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After adjusting the trimmers, the resulting filter has a flat
response and a 3 dB bandwidth of 2.24 KHz. The filter has low loss, 1.6 dB.
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Looked at over a wider frequency range, we see some asymmetry, but the filter's
poor skirt selectivity masks the worst of the low frequency skew.
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At this point, you should see how one might make an
adjustable bandwidth crystal filter—replace the constant shunt capacitors with
relay or diode or MOSFET switched capacitors, or a set of varactor diodes,
perhaps controlled by D/A converters. Of course, as the bandwidth changes, the
filter impedance also changes, so the matching network must also be switched or
variable.
Of course, since these crystals cost about $0.50 each even
in small quantities, it would be cheaper in most cases to build multiple
filters. |
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