Klopfenstein, circa 1956
impedance taper was first described by R. W. Klopfenstein
in a paper titled A
Transmission Line Taper of Improved Design, published
in the Proceedings of the IRE, page 31-35, January 1956. It's
just as useful today, more than fifty years later. We don't
post IEEE papers on line because that would violate their
copyright, but if you're clever at Googling you can probably
fetch yourself an original copy of the paper. Probably even
Jethro Bodine could do that we reckon. Hint: go
For September 2010
we've been in discussions with Retired Major Peter Klopfenstein,
the son of the man that created the taper, who provided us
with the image to the left. Klopfenstein now appears in the
Hall of Fame!
New for June 2013! Thanks to David, we now know that there was a correction to Klopfenstein's math which was published by IEEE MTT in May 1973 by Darko Kajfez and Jame Prewit. The title was '"Correction to "A Transmission Line Taper of Improved Design"'. Here is some background info sent by David:
I have been using a Mathcad worksheet that I found many years ago to design Klopfenstein tapers. If you are interested and have access to Mathcad it can be found at http://www.gb.nrao.edu/~rnorrod. (The author of the spreadsheet is R. Norrod (now deceased) from Green Bank observatory.) The problem with Mathcad is that not that many people have access to it, so my colleagues haven’t been able to take advantage of the Klopfenstein.
After downloading the (Microwaves101) spreadsheet and sorting out the user interface, I used it to design a 10Ω to 50Ω transformer with minimum 20 dB return loss and compared the results to the Mathcad worksheet. The results were close, but not dead-on like I expected. I believe that is because the spreadsheet is based solely on Klopfenstein’s original 1956 paper and does not take into account a correction that was published in 1973 by Kajfez and Prewitt. I dug deeper into the spreadsheet and implemented the Kajfez & Prewitt correction. This involved modifying the formula for calculating LN[Z(x)] in columns D and J on the Imp tab of the spreadsheet. After the correction was implemented the spreadsheet and the Mathcad worksheet results agree to within about 2 decimal places, e.g., the spreadsheet predicts 45.253 for the last section and the Mathcad worksheet predicts 45.242.
I also made one other change to the spreadsheet. The tab titled “Freq res” suggests that the length of the tapered line is arbitrarily defined by the user. This is not correct. The taper must have a minimum length at the lowest frequency of interest in order to exhibit the desired return loss at that frequency. This is implicit in equation (10) in Klopfenstein’s paper. In order for (10) to produce a real (in the sense of real vs imaginary) result the expression under the square root sign has to be positive, which requires that βl≥A. I modified the Freq res tab to allow the user to input the desired Fmin and calculate the required minimum length in terms of wavelengths at Fmin. For the 10Ω to 50Ω transformer with a minimum 20 dB return loss and a Fmin of 2 GHz, this comes out to 3.81 cm. The revised frequency response data and plot show that the desired return loss is achieved someplace in between 1.92 and 2.04 GHz.
here to go to our download area and get the Klopfenstein transformer
Ralph Walter Klopfenstein
R. W. Klopfenstein was brilliant
but perhaps a difficult man to work with, according to his son,
Retired Major Peter Klopfenstein:
The picture (above) is probably
slightly before 1956. He spent most of his career working at the
David Sarnoff Research Center (where he was a Senior Research
Fellow) in Princeton, NJ, with a short stint as Director of the
Iowa State University Computing Center, in the early sixties.
After his Antenna work, he went on to get an MS in EE degree and
a PhD in Applied Mathematics. I am not sure how modest he was,
he was very intense and the most focused person I ever knew. He
had an extreme reverence for science and its power to reveal.
Him and his WWII generation of scientists, had a dedication and
commitment to science that is almost unknown today. I didn’t understand
it then, but he and the others of that generation brought a vision
and clarity to the World that I sorely miss today.
Klopfenstein sponsored an annual
math prize through his estate with the MAA. Its been awarded annually
for over 15 years now. It's called the Merten
Hasse Prize for Best Expository Work of a Mathematical Topic.
It was named after a high school math teacher he had, who greatly
influenced his career choices.
Here's why this taper is important,
according to the inventor in his original article:
"The performance of
the Dolph-Tchebycheff transmission-line taper treated here is
optimum in the sense that it has minimum reflection coefficient
magnitude in the pass band for a specified length of taper, and,
likewise, for a specified maximum magnitude reflection coefficient
in the pass band, the Dolph-Tchebycheff taper has minimum length."
Klopfenstein didn't try to name
the taper after himself. He called it the Dolph Tchebycheff transformer!
The timeless relevance of his
work is indicated by recent IEEE papers that leverage the work from
Tapered Transmission Lines
with a Controlled Ripple Response, John P. Mahon and Robert
S. Elliot, 1990
Formulaton of the Klopfenstein
Tapered Line Analysis from Generalized Nonuniform Line Theory,
G. Razmafrouz, G. R. Branner and B. P. Kumar, 1997
The Design Charts by Waveguide
Model and Mode-Matching Techniques of Microstrip Line Taper Shapes
for Klopfenstein and Hecken Type Microstripline Tapers, Hirokimi
Effects of Klopfenstein
Tapered Feedlines on the Frequency and Time Domain Performance
of Planar Monopole UWB Antennas, G. Ruvio and M. J. Ammann,
C-Band Spatially Power Combiner
Using Klopfenstein Tapered Slot Antenna Arrays in Waveguide,
Wu Jin, OuYang SiHua, Li YanKui, Yan YuePeng, Liu XinYu
Klopfenstein Taper, Part 1
Thanks to Chip, we now offer
a downloadable Excel spreadsheet that performs the math behind Klopfenstein's
taper (go to download area).
The math is pretty ugly, think about a transcendental equation involving
a Bessel function while enjoying a cool malted beverage at the local
watering hole. You never know, there might be some hot chicks you
could meet that want to help you with the math. If you meet one,
have her send us a resume!
The figure numbers that Chip
refers to are recreations of the figures from the original paper.
Commence to ciphering!
A taper is a high pass structure
and will work well at all higher frequencies. The limit on
the low end turns out to be how long the taper is compared
to a wavelength. From figure 2 you can see that for the taper
to work to a -20 dB level (20 dB reduction in maximum reflection
coefficient) the length needs to be roughly half wave length
at the lowest frequency. If you want to work down to 2 GHz
the length has to be half wave at 2 GHz. If you want the taper
to work down to 1 GHz it has to be twice as long.
The length in figure 4 is normalized
to L/2 (+/-). The Y axis is the impedance along the line. Here we're
matching 50 ohms to 75 ohms, which is the classic problem that Klopfenstein
was trying to solve way back over fifty years ago.
Klopfenstein taper, Part 2
Here we'll update the analysis
to make it more understandable for applying this component.
First, let's be up front about
the effort to applying a taper properly. Calculating the impedance
versus length is actually the easy part! Later you'll have to realize
the taper in a physical geometry. Klopfenstein used coax as an example,
this is perhaps the easiest medium to translate impedance characteristics
to because it comes down to simple closed-form equations. If you
apply it to microstrip, CPW or stripline, you'll have to break the
impedance taper down to a discrete number of points (perhaps at
least 10, but maybe 50 would be better...) then calculate the line
widths (or gaps in the case of CPW) at these points, then blend
them together in a CAD tool. Microstrip has an additional hazard,
the speed of light along the taper will chance with line width,
we'll have to see how that messes up the result. Also, even though
Klopfenstein described a perfect high-pass element, eventually at
some higher frequency the transmission line will start to pass additional
spurious modes and then all bets are off!
So here's the plan...
- Requirements are: match 25
ohms to 50 ohms, starting at 10 GHz, with 20 dB return loss up
to 40 GHz. Maximum return loss is to be -40 dB, we'll find the
minimum length that will do that on microstrip on 10 mil alumina
(ER=9.8). 10 mil height will support 40 GHz performance without
- Use a modified version of
Chip's spreadsheet to calculate 10 impedance steps.
- Use this websites microstrip
calculator to determine the width of the required lines
- Use"real" EDA software
to look at the performance.
- Use FraudoCAD
to create a cheesy layout of the microstrip taper.
Using the Microwaves101 microstrip
calculator, we see that 50 ohms on 10 mil alumina corresponds to
9.6 mils line width, and the effective dielectric constant is 6.60.
For 25 ohms, we calculate 31 mils line width and 7.39 for the effective
The effective dielectric constant
drives the length of the taper, this effect was not treated in Klopfenstein's
paper. For the purposes of this design exercise we'll use the average
of the 50 and 25 ohm values (7.0).
The modified spreadsheet show
that 300 mils of taper is required.
Here's the impedance taper versus
the position in inches:
And here's the frequency response:
The plot makes a lot more sense
when the X-axis is frequency, not Lambda/L, don't you agree? FYI, the spreadsheet that created these plots is located in our download area and is yours for free so long as you are not a terrorist and the CIA sees that you are interested in microwave technology...
Now we'll pick off 21 discrete
points along the curve:
Position (inches) Impedance (ohms)
Before we convert all of those
impedances to microstrip, let's look at the response of the taper
using ideal T-line segments. Below is the schematic created with
Agilent's ADS. Click on it for a better view!
Here's the predicted frequency
response. Note that we don't quite get the 40 dB return loss we
designed for, the slight difference is due to rounding errors and
the use of discrete constant-impedance sections instead of a true
taper. Be happy with 37 dB!
Here's the predicted group delay.
This circuit provides very flat group delay response.
Next: calculating the line widths
for microstrip... we did this using the Microwaves101 microstrip
calculator (painfully solving for line widths for the target
impedances, the microstrip calculator is just one-way) and arrived
at this schematic (click for larger image). Note that we ignored
the change in phase velocity for different microstrip impedances,
the 20 sections are each 15 mils long.
Here's the frequency response:
Kind of disappointing, it degrades
before 40 GHz and keeps getting worse. What's going on? Here's a
1. Our microstrip calculator
is so inaccurate that it messes up the response
2. We made a "keypunch"
error (you don't hear that phrase very often anymore...)
3. Our approximation that the
phase velocities of the lines don't change with frequency so each
segment can be the same length just bit us
4. Dispersion in microstrip is
What do YOU think?
Layout of microstrip Klopfenstein
We used the width values and
FraudoCAD to create a layout of the
example taper. We made two versions, one where we segmented the
transmission line into twenty 15-mil sections according to line
widths we calculated, and another where we used the widths to create
a piecewise continuous taper. Here's the first version. In this
case we exaggerated the Y-axis scale so you can see the beauty of
the taper, which resembles a wine bottle. Like John Keats said,
of beauty is a joy forever, right? You can't exaggerate
the Y-axis in AutoCAD, another example of the superior powers of
Here's the version where we "smoothed
out" the steps. Which one would work better? Trust us, they're
both just as good.
Now lets look at the Y-axis to
scale and you'll see why we exaggerated it. Here you can't really
get an appreciation of the taper function, it almost looks linear.
Say, wouldn't it be really cool
to combine the taper spreadsheet, the microstrip
calculator and FraudoCAD to create
an all-in-one Klopfenstein taper design tool? Here's the problem...
the microstrip calculator is one-way only. It calculates impedances
based on line widths. The equations can't be solved in reverse in
a closed form. But what about using linear
interpolation from a calculated table of impedance versus width?
Check back soon, we might just take that challenge!