here to go to our main page on quarterwave tricks
here to go to our page on tapered transformers
here to go to our page on the Klopfensten taper
here to go to our download area and get an Excel file that will
calculate multi-section transformers
here to go to our page on maximally-flat transformers
Click here to learn how to calculate Z0 from measured data using transformer theory
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Suppose you want to match a device
of a different impedance than your system impedance. A simple quarter-wave
transformer can do this for you, with bandwidth somewhat inversely
proportional to the relative mismatch you are trying to overcome.
For a single-stage quarter-wave transformer, the correct transformer
impedance is the geometric mean between the impedances of the load
and the source:
We will deal only with purely
real impedances here, but transformers can be used successfully
to impedance match loads with reactive components as well. The problem
is that the math gets ugly quickly. It is doable inside an Excel
spreadsheet using complex math. We promise we'll get back to this
and post the equations soon (feel free to remind us!)
There is a natural bandwidth
limitation of quarter-wave transformers, because you only get exactly
a quarter-wave at one frequency; lower frequencies "see"
less than a quarter-wave, higher frequencies see more. However,
a second trick is to keep piling quarter-wave transformers in a
series, so that the impedance mismatch that each transformer is
"correcting" becomes less and less. How do you arrive
at the intermediate impedances of the multistage transformer when
there are infinite solutions to the problem?
Responses such as Chebyshev (equi-ripple)
and maximally flat are possible for multi-section transformers;
luckily we have already done the math for you!
First, consider that each each
transformer brings you to an intermediate impedance. We can select
the intermediate impedances, then solve for the transformer impedances.
Below we have transformed 25 ohms to 50 ohms in N=1, N=2, N=3 and
N=4 transformers. In each case we have used intermediate impedances
in an arithmetic series. For example, for N=4 there are three intermediate
impedances. An arithmetic series the steps are equal, so
the impedances are 31.25 ohms, 37.5 ohms, 43.75 ohms. Solving for
the transformers yields Z1=27.951, Z2=34.233, Z3=40.505 and Z4=46.771
ohms. Return loss of N=1, N=2, N=3 and N=4 transformers matching
50 ohms to 25 ohms are shown below (S11 is for N=1, S22 is for N=2,
etc.), in a plot generated using Agilent's Advanced Design System
EDA software. The graph illustrates well that
bandwidth gets better the more sections we add. However, it is clearly
not the best solution to the problem,we would like to see more bandwidth
with a -20 dB match, at the expense of the bandwidth that falls
below -30 dB.
Our second "simple solution"
to multi-section transformers involves a geometric series from impedance
ZL to impedance ZS. Here the impedance from
one section to the next adjacent section is always a constant ratio.
Plotted below for the same parameters, we like it better than the
arithmetic series. Instead of presenting the math here, we offer
an Excel file that will calculate up to ten-section transformers
for you, it's in the Microwaves101 download
area. We didn't even bother to lock the formulas!
So what does a ten-section transformer
buy you? Below we have plotted the response of a geometric transformer,
10 sections, matching from 5 to 50 ohms. Here it is apparent that
we have failed to provide the familiar Chebyshev equal-ripple response,
but our transformer ain't half-bad. What did you want for free?
Maximally flat transformers
This topic now has its own
The Chebychev transformer produces
equal ripple in passband of your choosing. The maximally-flat and
exponential transformers don't allow frequency band as an independent
More to come!
The Professor's multisection
transformer spreadsheet is available for free on our download
area. It can compute Chebychev, maximally flat and exponential
Tapered transformers (which are
not necessarily quarter-wavelength) can
be found on this page.