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I'm reading through a tutorial on filter design and Bartlett's bisection theorem. It gives an example of a filter with 2.909k terminations:

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Then, the right side of the filter is scaled by a factor of 2.2:

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I think I understand the significance of R2: this may represent the antenna impedance for example. The calculated filter response is only valid for the given load impedance.

But what of R1? I've not seen filters with actual resistors in them: this seems like a waste of perfectly good transmitter power. And I may not know the output impedance of the transmitter either, though I do know it's important that looking into the filter, the transmitter sees 50 ohms. It's unclear to me how this requirement is satisfied when performing the filter design.

And for that matter, is characterizing the load impedance as a resistor really accurate? For example, a half-wave dipole may have a known resistive impedance at resonance, but if the filter is intended to suppress spurious emissions then it must work at other frequencies where the dipole will present a potentially very different impedance. What can be said about the performance of the filter under these conditions? Will it still effectively suppress harmonics?

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And I may not know the output impedance of the transmitter either, though I do know it's important that looking into the filter, the transmitter sees 50 ohms. It's unclear to me how this requirement is satisfied when performing the filter design.

The input impedance of the filter is typically specified so as to match the output impedance of the signal source. This allows maximum power transfer from the source generator.

The input impedance of the filter circuit can be validated by doing series/parallel circuit analysis starting from the load resistor and working backward toward the source. The voltage of the source should be shorted out for this analysis leaving the source resistance as the last circuit element to be factored into the series/parallel analysis.

As mentioned by Kevin Ried AG6YO, R1 and V1 in your schematic represent the signal source with R1 as the output impedance of the source. R1 does not waste output power of the signal source, but only represents the source impedance. In amateur radio, the transmitter output power is specified into a specific load impedance, typically 50 ohms. This is not necessarily the output impedance of the transmitter as the transmitter's output impedance is time variant. But it is the impedance at which the amplifier will develop its specified output so this is an acceptable proxy for output impedance of the signal source and the desired input impedance of the filter in this case.

And for that matter, is characterizing the load impedance as a resistor really accurate? For example, a half-wave dipole may have a known resistive impedance at resonance, but if the filter is intended to suppress spurious emissions then it must work at other frequencies where the dipole will present a potentially very different impedance. What can be said about the performance of the filter under these conditions? Will it still effectively suppress harmonics?

You are correct to question the validity of the filter characteristics when connected to a load impedance that varies with frequency, such as a typical antenna where reactance is added in series with the feedpoint resistance as the frequency moves away from resonance. The frequency response of the filter must be analyzed at each frequency in question with the appropriate complex load at that frequency. This will typically result in a different characteristic curve than found with a simple resistive load. The affect of this change can generally be minimized adding an additional filter pole.

A load that changes its complex impedance with frequency also presents the conundrum that the signal source (a transmitter, for example), does not see the desired impedance looking into the filter. That is to say that the input impedance of the filter is dependent on the impedance at the output of the filter. So the changing impedance of the antenna with a change in frequency changes the load impedance of the filter which changes the input impedance of the filter which is the load the transmitter sees. This results in a reduction in power of the transmitter which is typically accompanied by a change in harmonic content. So to avoid this multi-variable situation in lab tests or simulation, it is best to analyze each in isolation and mathematically combine the results.

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  • $\begingroup$ Hm...If I take an ordinary prototype butterworth low-pass filter, appropriately scaled for 50 ohms as the source and load impedance, the impedance seen by the voltage source is (for most of the passband) 100 ohms, not 50. Easy to see, since the shunt capacitors have reactance approaching infinity while the shunt inductors have reactance approaching zero for decreasing frequency, so for a low frequency in the passband the voltage source is looking at two series 50 ohm resistors. So is this the kind of filter I'd want to use to present a 50 ohm load to the transmitter? $\endgroup$
    – Phil Frost - W8II
    Feb 18 '18 at 16:51
  • $\begingroup$ Also in practical terms, given the example in the question, really what's happened by scaling the right half of the filter? Has it matched a 6.4k ohm load to a source expecting 2.909k? Simulations seem to suggest no, though I could be doing something wrong. $\endgroup$
    – Phil Frost - W8II
    Feb 18 '18 at 17:45
  • $\begingroup$ @PhilFrost-W8II It sounds like you are doing something wrong with either the filter design or the analysis. Properly designed, the filter can do impedance matching. $\endgroup$
    – Glenn W9IQ
    Feb 19 '18 at 1:09
  • $\begingroup$ Yeah...playing with it more I think I have a handle on what this particular transformation does: it preserves the filter response. However, the impedance seen by the load is not preserved. $\endgroup$
    – Phil Frost - W8II
    Feb 19 '18 at 1:24
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This seems slightly implausible due to the particularly high value of 2.909 kΩ, but taking the circuit diagram literally, I would expect V1 to be an ideal voltage source and R1 to be the modeled source impedance, not part of the filter.

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    $\begingroup$ Sure, but the question is about practical significance, where the antenna impedance is frequency dependent, and the transmitter output impedance not known. Why is this a good model, and how does it differ from reality, and what impact does that have on performance? $\endgroup$
    – Phil Frost - W8II
    Feb 17 '18 at 18:54
  • $\begingroup$ @PhilFrost-W8II It does make sense to inspect antenna impedance at the transmitted frequency (where 99.999% power exists). That's where you want efficient power transfer from filter input to output. Chances are fairly good that an antenna will radiate power at 2x, 3x its design frequency inefficiently compared to a good match at 1x. $\endgroup$
    – glen_geek
    Feb 23 '18 at 22:48

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