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You can buy a book on this topic by Matthew Morgan, see our book page to order it from Amazon.
Matt provides a shout-out to Microwaves101 on page XII:
Thanks, Matt!
Most filters provide rejection by reflecting signals back outside of the passband. This can sometimes cause a problem, especially when you cascade two filters and you are counting on seeing the rejection combine. It does not typically work like that, rejection of each filter depends on that filter seeing a matched impedance.
If you had filters that provide good impedance match in their stop band(s), you could cascade them and get good results. Some recent publications from Matt Morgan at NRAO explain how to synthesize reflectionless filters. Here's a title of one paper that is out on the ol' web:
"Synthesis of a New Class of Reflectionless Filter Prototypes" by Matthew A. Morgan and Tod A. Boyd
You can find other papers on the topic by searching on IEEE Explore.
Before you go off and copy this idea, bear in mind that it has been submitted for US patent protection.
We glanced over the referenced paper and decided to skip the math and use the infinite monkeys technique (random optimization) to try to tune up some reflectionless filters. Note that the actual math behind these structures suggests that impedance match can be perfect, but we accepted 20 dB return loss. These are the simplest structures you can make, you can increase rejection by cascading the resulting filters. Unfortunately it is always a problem trying to make decent graphics for a web page from plots that were first dumped into PowerPoint. So the figures below all stink, but they are no worse than the figures in the pdf paper we were looking at. We'll try to fix the graphics later; drop us a line and we will provide you with the PowerPoint where you can read everything if you promise not to pretend it is your own work and give credit to Matt Morgan. We included the element values in the text since you can't read them.
Although we only examined a lumped element filter, it is also possible to create such a filter using transmission lines. Check out this patent application.
High pass filter
All of the capacitors are valued at C1 (0.66 pF), and all the inductors are L1 (1.36 nH); there are only two variables. The resistors are 50 ohms for a 50-ohm system. Captain Obvious tells us that the two capacitors in series can be replaced with one capacitor of half the value.
Low pass filter
To make a low pass design, you have to change inductors to capacitors, and vice-versa, and then re-optimize. Below the element values are 0.30 pF and 0.92 nH.
Band pass filter
In this case you have four variables. We used the same nomenclature that Matt did. Cx is 0.24 pF, Cs is 0.56 pf, Lx is 0.62 nH and Ls is 1.36 nH.
Band stop filter
Again, you have four variables. Note that the parallel LCs and series LCs have been swapped in the schematic compared to the band pass filter. After re-optimization, Cx is 0.26 pF, Cs is 0.33 pf, Lx is 0.72 nH and Ls is 0.92 nH.
We admit our treatment of this topic was limited, but at least you now know that reflectionless filters exist, and you know where to find more information. Later we will show you what happens when you cascade these filters, compared to "normal" reactive-element-only filters.