SW40+ I.F. Crystal Filter This note proceeds through the design of a Cohn-type crystal ladder filter. Starting with fixed crystal specs and desired bandwidth, a simple generic filter is designed . The generic filter is then modified to accommodate source and load impedances found in the SW40+ receiver. Component values work out close to those in the rig.
3-xtal Cohn filter

The simplest form of the ladder-type Cohn filter requires at least two crystals, but any number can be cascaded. Between pairs of crystals and ground must appear a coupling reactance. Source resistance and load resistance must also be defined. A two-crystal filter would result in inadequate sideband suppression. Four or more crystals require extreme care in selecting crystal frequencies, coupling capacitors and termination resistances.
Crystal Model
From the Handbook you should review the electrical equivalent of a quartz crystal. By far, the two most important crystal parameters are the motional inductance Ls and motional capacitance Cs. From my junk-box a number of 4.0 Mhz. HC-18 microprocessor crystals were characterized. These should be hopefully similar to those in the SW40+ kit:

Fs = 3999165 Hz. Series resonant frequency
Ls = 0.1971 H Motional Inductance
Cs = 8.03554x10-15 F Motional Capacitance
Rs = 48 ohms Motional Resistance
Cp = 3.5 pF Parallel plate capacitance

    Besides crystal parameters, we must choose a desired filter passband width (BW). I choose 300 Hz. because I know from experience that component values will come out about right.
    We can also choose from Butterworth, Chebychev, Gaussian, or Bessel responses. Each is optimized for different applications. One has minimum ringing, another has best shape factor. Butterworth response is reasonable for listening to CW.
Generic Ladder Design
    I have followed the ladder design process as described in Handbook of Filter Synthesis by Zverev². His ladder designs are exhaustively complete and unfortunately complex. The result is shown in the schematic "Generic Cohn 3-crystal Filter" below.
All Butterworth filters share coupling coefficients which determine the reactances of C1 and C2 of the generic filter above. For a Butterworth response, Rsource and Rload are scaled by another coefficient called "q" which is related to loaded filter Q (Fs / BW). Here is a table of Butterworth coefficients for ladder filters of N crystals:

N	q	k₁₂	k₂₃	k₃₄	k₄₅
2	1.4141	0.70711	-	-	-
3	1.0	0.70711	0.70711	-	-
4	0.7654	0.8409	0.4512	0.8409	-
5	0.618	1.0	0.5559	0.5559	1.0

For our 3-crystal generic filter, both coupling capacitors (C1 and C2) will have the same value, because k₁₂and k₂₃ have the value of 0.70711. Their capacitance is roughly:
C1 = Cm x Fo / (BW x k₁₂)    =    8.03554x10^-15 x 4000000 / (300 x 0.70711)    =    152 pF
C2 = Cm x Fo / (BW x k₂₃)    =    8.03554x10^-15 x 4000000 / (300 x 0.70711)    =    152 pF
Since each of these capacitors sees Cp from two adjacent crystals, we should subtract off 2 x Cp from these values leaving us with 145pF. In any case, these are close to the values Dave chose for C13 and C14 in the SW40+ schematic.
   Now let's try a simplified equation for Rsource and Rload. Our generic filter is symmetrical, so these will have the same value:
Rend = 2PI x Ls x BW / q    =    6.2832 x 0.1971 x 300 / 1.0    =    372 ohms
   These simplified values are close to those derived from the complex methods of Zverev².

Generic 3-xtal PSPICE schematic That's it - the complete generic filter. R14 and R15 (both 1gigohm) were added to satisfy PSPICE's requirement that all nodes not "float". There is one more aspect of this filter that we haven't addressed: frequency matching of the three crystals. For proper tuning, the two end crystals should have a series-resonant frequency 106 Hz. higher than the center crystal (from Zverev's design process). In the Generic Cohn filter above, C13 and C15 have been decreased to reflect this frequency offset. In practice, we cannot modify our sealed crystals, and must execute the offset a different way so that three crystals of identical Fo can be used.
The frequency response of this filter is plotted below (Generic). Were the filter lossless, output voltage would be 500 mv.

Frequency response: PSPICE
     This generic filter will now be adapted to make it work using three identical crystals. Since the SA612 source and load resistances are different from the generic design, matching networks must be added as well.
    Compensating for 106 Hz. Offset
    We can raise the series resonant frequency of the two end crystals by adding a capacitor in series with each. Currently, series resonance is:
Fs = 1 / (2 * PI * SQRT ( 0.1971 * 8.03554x10^-15))    =    3999164.7 Hz.
What capacitance would raise Fs to (Fs + 106) ?
Cs(new) = 1 / ((2PI x (Fs+106))² x 0.1971)    =    8.035114x10^-15 farad

Cs(new) is the total capacitance of two in series: our original crystal motional capacitance (Cs) of 8.03554 ff and our externally added modifying capacitor. Knowing Cs and Cs(new), we can find the value of the external capacitor:
Cexternal = (Cs * Cs(new)) / (Cs - Cs(new))    =    151.56 pF.
    By adding this capacitance in series with Xtal 1 and Xtal 3, the 106 Hz. offset is accommodated - and all three crystals can be frequency-matched to within about 30 Hz. These capacitors appear in Dave's SW40+ schematic as C12 and C15. Is it coincidental that the coupling capacitors are the same as these frequency-tuning capacitors? For this filter (and only for 3-crystal types) this will always be the case.
Matching to SA612 source and load
    Driving the filter is a SA612 chip (U1), with 1500 ohm source impedance. The filters' load is also a SA612 (U3) whose input impedance is also 1500 ohms. Our filter would like to see terminations of 354 ohms.
    Dave has chosen to simply load down the filter's output with a 470 ohm resistor (R1). This resistor, in parallel with the chip's 1500 ohms works out very close to the required 354 ohms.
    At the filter's input, Dave has chosen to add an L-matching network consisting of a 22uH choke (RFC1) and a 47pf capacitor (C11). Let's see how well these component values do in matching the required 354 ohms to SA612's 1500 ohm source:
    At 4 MHz., the choke's reactance is +j553 ohms. C11's reactance (added to the SA612's 3pf output capacitance) is -j796 ohms. What's the parallel equivalent of 354+j553?
It is 1218 ohms in parallel with inductive reactance of 780 ohms. C11's reactance nulls out this equivalent parallel reactance quite well, leaving 1218 ohms resistance - nearly matching to SA612's 1500 ohms.
Here's the PSPICE circuit showing the SA612 equivalents at both ends of the filter:
PSPICE schematic of SW40+ filter
    This filter's response is overlaid (green) in the frequency plot above.
    Let's take a quick look at filter attenuation. Each crystal has a series resistance of 48 ohms. For the generic filter, if we assume that all filter reactances cancel somewhere close to the filter's center frequency, we have left 3 * 48 ohms series resistance dumping signal into a 354 ohm load. That's a simple voltage divider that delivers 0.713 of input voltage to the output.
    For the SW40+ filter, additional losses are incurred because of the 470 ohm loading at the output. That's why the amplitude response is lower than the generic filter. The input side is matched, the output isn't.

References:

1 Hayward, Wes A Unified Approach to the Design of Crystal Ladder Filters QST May, 1982.

2 Zverev A.I. Handbook of Filter Synthesis Wiley 1967.