# Maximally-flat diplexer

This content was moved from our diplexer page

This example provides a diplexer with some amazing properties, and takes a shallow dive into some of the math. This is Microwaves101, not Microwaves201, OK? We used Matthaei Young and Jones as the reference for this work, in three different chapters. Look for this book and more on our book page. It is important for younger engineers to appreciate that many of the problems that are thrown their way have already been solved for all time, you just need to know where to look for them. You can download a free and legal electronic copy of this book at our download area, put that on your laptop for in-flight discussions with attractive seat-mates of the opposite sex!

Two important properties of this particular diplexer are:

- It provides perfect -3.01 dB split at the crossover frequency. Thus it is said to provide a "contiguous passband".
- If the two diplexed arms are terminated in Z
_{0}, the imaginary parts of the reflection coefficients looking into the filters are equal and opposite, so they cancel out and you are left with exact match to Z_{0}at the common port!

This diplexer topic is discussed starting on page 991 of MY&J.

Singly-terminated filters assume that there is a matched impedance at the load side but not in the generator. This topic (at least for LPF) is discussed starting on page 104 in MY&J. The Butterworth filter has property of being maximally flat in passband. It also is the only filter that will work perfectly in the diplexer we are describing, and it has to be singly terminated.

Before we get too far, "singly" rhymes with "tingly", a word that was put to good used by the Travelocity Roaming Gnome back around 2011:

The LPF circuit model in Microwave Office (below) is for singly-terminated Butterworth filter of N=3. The G-values are given in MY&J, we did not try to derive them. On page 107 there is a table of G-values up to N=10 if you want to try this on a higher-order filter. G values represent the inductance or capacitance of any element in the LPF, at ω=1 and Z_{0}=1. We did some simple algebra to convert the G values to values that are derived from a specified frequency in GHz.

Note that the singly-terminated Butterworth filter is NOT symmetric. Also, the Nth element has N times the value of the first element, interesting but not useful to know. Component values for 10 GHz are L1=0.3979 nH, C2=0.4244 pF and L3=0.1.194 nH.

Below is the response of the ideal N=3, 10 GHz singly-terminated Butterworth LPF. It does not provide equal ripple reflection coefficient in the passband like the Tchebycheff, if it did, it would be called "Tchebycheff".

Now we need a singly-terminated Butterworth filter that is the direct transformation of the LPF. On page 412 of MY&J you can learn how to transform a LPF to a HPF: basically C_{HPF}=1/(L_{LPF}*omega^2) and L_{HPF}=1/(C_{LPF}*omega^2) does the job and the same G values are used.

Component values for 10 GHz are C1=0.6366 pF, L2=0.5968 nH and C3=0.2122 pF.

Below is the response of the N=3 10 GHz Butterworth HPF.

Note that for either the LPF or the HPF, an equivalent pi network could be substituted; the G-values are correct for either topology but the algebraic solution to cough up the inductor and capacitor values as functions of frequency in GHz would be slightly different.

Now let's put together our diplexer. It is a three-port network made from the two filters. You have to be careful which way the filters are facing as they are not symmetric.... the common port is connected to the sides of the filters with the higher component values in the tee network case.

Now we look at the response: we have achieved perfect -3.01 dB crossover at the design frequency of 10 GHz. In many designs much higher roll off will be required in order to maintain maximum use of the overall frequency band, so you will need to look at solutions higher than N=3 that we used here.

Oops, the plot below says "duplexer" when we meant "diplexer". Someday we will fix that. Thanks to Oscar for point this out!

What do the return losses look like? For one thing you don't need to plot S11, in this perfect example we hit the bulls eye of the Smith chart at all frequencies and there is no need to even plot it! Of course in a real design nothing is perfect. At the center frequency both arms (ideally) exhibit -6.02 dB return loss, which is perfect 2:1 VSWR corresponding to perfect -3.01 dB reactive power split.

Now let's take a look at the group delays of the individual filters and then compare to the group delay of the diplexer. Here are the filters by themselves, they are very different.

And here are the group delays of the diplexer: what's this, and other magic property of the singly-terminated Butterworth filter diplexer? The group delays of the two arms are equal to each other! Remember where you learned this useless property of the singly-terminated Butterworth filter diplexer. Too bad the group delays are not flat over frequency, that would be a usable result.

In conclusion, MY&J is a great resource, if you want to get involved in filter design you need to blow the dust off this book and start using it!

**Author : **Unknown Editor