# How can the design of an RF filter also encompass impedance matching?

I'm installing an antenna which will require a matching network. (A fixed network: the load will not change.)

I'm also building some low-pass filters to reduce transmitted harmonics.

It would be a simple exercise to design an ordinary 50Ω filter through normalized tables (which assume equal input and output impedances), and follow that with a matching network. Maybe a pair of inductors or capacitors can be combined, but not much is saved. Furthermore the characteristic of the filter is perturbed, as the reactance of the matching network will change over frequency in a way which was not considered in the filter design.

But I wonder: is there a more intelligent way to design a filter for unequal input and output impedances? Perhaps a way which involves fewer components, or affords additional design flexibility, for example to steer component values to standard values, or minimize loss? How?

So, to answer your: "Is it possible?" question: yes.

Yes, and in fact, it's more inevitable than possible!

As you know, one of the major challenges in RF design is that not only do you need to match components and transmission lines at a single frequency (or wavelength, depending on how you look at it), but usually across a whole bandwidth! Generally, the larger that bandwidth becomes, the harder it is to sustain a good match.

Now, what happens outside that bandwidth where you've successfully matched? Right, signal energy gets reflected at the impedance discontinuity, and hence, doesn't "move forward" through the signal chain.

That's exactly what a filter is!

In fact, any passive filter can be understood as frequency-selective matching network. Consider the simple RC low-pass: at low frequencies, its complex impedance is very much dominated by the real-valued resistance, but as soon as you get close to the the cutoff frequency, you see the phase of transmission increase – which reflects the fact that the imaginary part of the impedance now reaches the magnitude of the real part. From the (hopefully) purely real-valued wave impedance of the transmission line, you now see a complex impedance, not even with the same absolute value. You get reflections (which basically means you get higher SWR).

So what one can do here: Read about the problems of making high-bandwidth matching circuits, and just do the opposite to get a filter.

Or really, just design a filter whose impedance at passband works to match in- to output :) The RC example actually works pretty nicely here: you actually have two degrees of freedom when designing an RC filter, one is the resistance, and the second the capacitance; as the we assume that $f_c = \frac1{2\pi RC}$ is fixed, you'll find that actually, you can't just pick any capacitance – the ratio of $R$ to $C$ is fixed by that requirement!

Thus, pick your $R$ so that at the frequencies you want to pass, your complex impedance is usable as matcher.

• So for a more realistic example, say a 5th order Chebyshev low-pass filter that might be found for harmonic suppression at the output of an amplifier, how might this be done in practice? – Phil Frost - W8II Feb 19 '18 at 0:42
• Look for "Modified ladder topologies"; Wikipedia actually is a nice starting point there: en.wikipedia.org/wiki/… en.wikipedia.org/wiki/M-derived_filter – Marcus Müller Feb 19 '18 at 0:48

It is quite possible to combine filtering and impedance matching. In fact, this essentially a design requirement for any type of filter where the input and output impedances are not equal. Practicality and balance enter into the equation, however, when it comes to complexity, efficiency, practical circuit values, etc.

In the amateur radio world we see this balance play out. Our transmitter contains the necessary harmonic filtering to meet industry practices/expectations. The filtering effectiveness is only for a specified load (typically 50 ohms resistive). If our antenna is not a 50 ohm resistive load, we might use an antenna tuner to convert the antenna impedance (or more precisely to convert the impedance of the feed line at the transmitter end) to 50 ohms. We do this to maximize radiated power without any consideration of filtering effectiveness.

The antenna tuner is typically an L network consisting of a capacitor and inductor that the operator adjusts until the desired match is made. It is interesting to note that the adjustments made by the operator also affect the high or low pass characteristics of the LC filter that the tuner forms but the operator doesn't care because the necessary filtering has been dealt with in the transmitter. The operator is only concerned with its impedance matching function.

As the design engineer of the transmitter and the tuner, I could reason that I will save costs and component count by combining the LC values in the tuner with the LC values in my output filter of the transmitter. But the challenge is now that as the operator adjusts component values to match the load, he or she is also adjusting the harmonic suppression characteristics of the transmitter. I must therefore provide additional indicators or methods to the operator to ensure the effectiveness of the filter for any particular load setting. So the design complexity has actually increased and the operator has a presumably more difficult procedure that must be followed compared to isolated filter and impedance matching circuits.

We even see this balance issue in modern transmitters that contain an internal, automatic antenna tuner - all under a common control circuit. Typically there are still separate harmonic filters for each band in addition to the LC antenna tuner function. The seemingly obvious approach of combining these circuit values and functions does not always lower the cost or complexity of the design.

• Doesn't the addition of the antenna tuner alter the filter response whether it's combined with the filter or not? I guess I didn't state it explicitly in the question, but I was also hoping for a hint how the matching might be accomplished if the load is known. Of course I could design an L network and tack it on the end, and perhaps get to combine a pair of capacitors or inductors. But is there a better approach? – Phil Frost - W8II Feb 19 '18 at 0:17
• @PhilFrost-W8II The source and load impedance of the filter are fundamental terms in the design of the filter. A little experimentation with Elsie or an on-line calculator should make this clear. Keep the topology and the passband terms the same and only alter the load impedance to see the effect on component values. – Glenn W9IQ Feb 19 '18 at 1:52
• I understand that much. So when the load isn't already matched to the source, what do filter designers do? Naively tack on an L matching section and give up on realizing the desired filter? Or are there techniques which consider the filter and the matching together? – Phil Frost - W8II Feb 19 '18 at 1:58
• @PhilFrost-W8II Nearly any LC filter can simultaneously have filtering and impedance transformation characteristics. Many textbooks and app notes show equal input and output impedances but that is not a requirement by any means. The equivalency is typically depicted to simplify the math. – Glenn W9IQ Feb 19 '18 at 2:02
• I guessed the problem in the question can be solved. Could you please explain how? – Phil Frost - W8II Feb 19 '18 at 2:10