here to go to our main page on matching networks
here to go to our page on quarterwave tricks
to go to our page on VSWR
This page will discuss exponential
and Klopfenstein (and possibly other) tapered transformers. Transformers
are used to match one impedance to another (from Z1 to Z2). In most
cases, the impedances are purely real (not inductive or capacitive).
By tapering a transmission line, a very broadband impedance match
(low VSWR) can be realized over a wide bandwidth, the longer the
taper, the wider the frequency band.
In the exponential taper, the
natural logarithm of the line's characteristic impedance varies
linearly from Z1 to Z2:
Where z is the distance from
the Z0 side of a transformer that has length L.
Perhaps a better way to look
at this is that the line impedance increases exponentially according
Here's a plot of the line impedance
of a tapered transformer of length 10 (units are arbitrary) that
is being used to match ZL/Z0=10.
We'll add some more
math here so you can see what the impedance looks like over frequency.
This topic was contributed by
someone at Endwave who prefers not to have his name attached to
it. Thanks, Mark! We plan to expand on it in the future, after we
look up the 51 year old article.
In the January 1956 Proceedings
of the IRE, R. W. Klopfenstein presented equations which can be
used to design transmission line tapers. "...the Dolph-Tchebycheff
taper is optimum in the sense that it has minimum reflection coefficient
magnitude in the passband for a specified length of taper".
One drawback of the Klopenstein taper is that an impedance discontinuity
or step occurs at the ends of the taper.
A Mathcad routine for synthesizing
the dimensions can be found at:
More to come!