New for October 2025. This discussion is important in the world of power amplifier design and we would not have attempted it without the help of Dr. Mike Roberg who is one of Qorvo's most prolific power amplifier designers. The Bode-Fano limit states that the more energy that is stored reactively in a load, the smaller the bandwidth you can match it to [1],[2] This was the subject of Robert Fano's doctoral thesis at MIT in 1948. His paper is available at MIT's DSpace site, for unlimited distribution. We downloaded a local copy, you can get in our download area. Robert Fano is now in our Microwaves Hall of Fame!
Fano's paper describes the limitations of matching to a shunt RC in exquisitely simple terms:
Fano gives full credit to Bode but takes the discussion a lot further. Read the paper if you fancy yourself as a microwave designer!
Mike Roberg has written several IEEE MTT-S papers that illustrate how the Bode Fano limit affects high-power, wideband GaN power amplifier architectures [3],[4]. He is a Distinguished Microwave Lecturer for IEEE MTT-S from 2025-2027 (DML's serve a three-year term). Catch him when he's in your neighborhood, we did! Here's an intro to Mike's DML talk:
Power Without Pain: High Power MMIC PA Design, the Pitfalls and how to Avoid Them by Dr. Michael Roberg
Among other power amplifier topics, Mike's lecture[5] discusses how the Bode-Fano limitation affects the bandwidth of a power amplifier output matching network. He presents a very useful version of the Bode-Fano limitation on RC circuits, for those of us who like to think in decibels:
Note that for the units in the equations, capacitance in picoFarads works out to frequency in terahertz. With a little algebra you can convert Fano's equation to Roberg's, we did this within a single page of scribbling, which we promptly recycled. Note that the equations represent an inequality, i.e. you had better accept that you are not going to achieve the theoretical limit in real life.
What does this mean for power amplifiers? A GaN or GaAs FET equivalent circuit shows that the output of the device can be reduced approximately to a shunt RC circuit. The output capacitance and resistance are related to transistor periphery (size), as they are expressed in ohm-mm and pF/mm. Let's say a GaN process has 50 ohm-mm and 0.3pF/mm. If you choose to match a 1mm FET, you get a 50 ohm resistor in parallel with 0.3 pF. If you want 9dB more power, choose an 8mm FET and you will have to match to 6.25 ohms and 2.4 pF.
In both cases, if you want 20 dB match, just do the math and you will see that you can get a maximum of 14.5 GHz of bandwidth. You will still be stuck with the real part of the impedance; for the FET you end up conveniently at 50 ohms, for 8mm you will be stuck at 5mm. The next step would be to add a transformer to get to 50 ohms, which is not a trivial exercise.
Let's make a matching network. Where do you begin? You do what AI does so well, you re-use something someone else has already developed. Thanks again, Mike! The schematic below contains a ladder network of L's and C's, with provisions for injecting the DC current and blocking the drain voltage. All of the lumped elements scale with "Alpha" which is the FET periphery in millimeters. Note that the termination impedance at the port is scaled by 1/Alpha, so it is set to 6.25 ohms when matching to an 8mm transistor. Regardless of the value of Alpha, you get the same result.
An optimizer was let loose to try to get it to match to 20 dB across 14.5 GHz of bandwidth. We did not reach the goal but we can say with confidence that if we added infinite elements we would find out that Bode and Fano were correct!
But there is a big problem for the 8mm version. The output is now matched to 6.25 ohms real impedance. A sporty impedance transformer is needed, and this is a big complication.
A simple-minded (stupid?) way of doing that impedance transformation is to use a multi-section quarter-wave transformer. You can download a spreadsheet that calculates the impedances for a Chebyshev transformer right here at Microwaves101. Here is the schematic for a five section network:
Note that the impedances are as low as 8 ohms, and the transformer is going to be extremely long if you intend to put it on a MMIC. You will be fired if you insist on doing this, but it is of interest just the same.
Here is the response of that transformer, it hits around 18 dB match in the frequency band. You will soon see that 18 dB is not good enough.
Now let's cascade that transformer with the output matching network:
Here is the composite response of the Bode-Fano matching network with the impedance transformer. The blue line with triangles is the Bode-Fano network as seen in a 6.25 ohm system. The purple line (squares) is the multi-section transformer taking 6.25 ohms to 50 ohms. The brown (diamond) line is the composite response in 50 ohms. Yes, the composite response is ugly over the theoretical 14.5 GHz of bandwidth. This is just one of the many reasons that wideband power amplifier design is hard to do.
Only a fool would use a multi-section transformer on a MMIC. The two transformers you need to study are Ruthroff and tri-filar, we'll leave that discussion for another day. Can you ever get to 20 dB match? Nope, but you could do a lot better than the above example.
Now, let's continue designing an actual GaN power amp MMIC with 8mm output periphery. No one is going to use a single 8mm FET, but they might use eight 1mm FETs in parallel. That's what is pictured below. Here, Alpha is 1 instead of eight, so all of the inductors will be larger and all of the capacitors will be smaller by a factor of eight compared to the single-FET design. It still predicts the same performance as the original network, and it is still matched to 6.25 ohms.
Woah, our schematic is starting to look like the output matching network of a real eight-way-combined amplifier, look at the photo below[6]. Wait, where is the impedance transformer? This is a narrow-band design, and the designer built in the transformation into the output network. If you want to see a design that includes an actual transformer, see [7].
The next steps will entail combining some of the capacitors and transmission lines to make a network that can be laid out. If you want to see this continued, send us an email!
References
- H. W. Bode, "Network Analysis and Feedback Amplifier Design", Sec. 16.3, Van Nostrand, New York, 1945.
- R. M. Fano, “Theoretical Limitations on the Broadband Matching of Arbitrary Impedances," Technical Report No. 41, Research Laboratory of Electronics, MIT, 1948.
- M. Roberg, M. Pilla, S. Schafer, T. R. Mya Kywe, R. Flynt and N. Chu, "A Compact 10W 2-20 GHz GaN MMIC Power Amplifier Using a Decade Bandwidth Output Impedance Transformer," 2020 IEEE/MTT-S International Microwave Symposium (IMS), Los Angeles, CA, USA, 2020, pp. 261-264
- M. Roberg, M. Pilla, T. R. Mya Kywe, R. Flynt and N. Chu, "A 20W 2-20 GHz GaN MMIC Power Amplifier Using a Decade Bandwidth Transformer-Based Power Combiner," 2020 IEEE/MTT-S International Microwave Symposium (IMS), Los Angeles, CA, USA, 2020, pp. 1287-1290
- M. Roberg, "Power Without Pain: High Power MMIC PA Design, the Pitfalls and how to Avoid Them", presentation at University of Arizona on 9 October 2025.
- A. P. de Hek, G. van der Bent and F. E. van Vliet, "400-Watt S-band Power Amplifier MMIC," 2021 16th European Microwave Integrated Circuits Conference (EuMIC), London, United Kingdom, 2022, pp. 160-163
- C. F. Campbell, M. D. Roberg, J. Fain and S. Nayak, "A 1–8GHz Gallium Nitride distributed power amplifier MMIC utilizing a trifilar transformer," 2016 11th European Microwave Integrated Circuits Conference (EuMIC), London, UK, 2016, pp. 217-220