Impedance Matching Part 13. How to Design Binomial (Butterworth) Multi-Section Impedance Transformer
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2 Common Designs of Multistage Transformer
Binomial (Butterworth):
This design has the “flattest” response at the center frequency.
Chebyshev:
This design has a constant ripple over the bandwidth.
This design has the largest bandwidth for a given acceptable level of reflection coefficient.
The maximally-flat transformer is a multi-section stepped-impedance transformer, where the sections are all one quarter-wavelength at center frequency.
In the context of the max flat solution or the binomial approximation, the bandwidth you aim to achieve isn't a parameter you can optimize directly. Rather, it's determined by the number of sections and the ratio of the two impedances you're attempting to match. Essentially, you're constrained by the available space or the tolerable amount of loss.
The binomial transformer offers the exact solution for N=1 (which is trivial) & N=2 (which is useful). However, for higher-order transformers, it only provides an approximation. This approximation deteriorates rapidly, especially when the mismatch between the source and load exceeds 2:1 and therefore, it should not be relied upon. It's important to note that this is only an approximation and should be avoided unless you are unconcerned about degraded performance. The mathematical requirements for understanding this approximation are quite simple.