Klopfenstein's taper

Updated February 18, 2009

Click here to go to our download area and get the Klopfenstein transformer spreadsheet

Here's why this taper is important, according to Klopfenstein,

"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!

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.

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...

1. 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 moding problems.
2. Use a modified version of Chip's spreadsheet to calculate 10 impedance steps.
3. Use this websites microstrip calculator to determine the width of the required lines
4. Use"real" EDA software to look at the performance.
5. 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 dielectric constant.

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? As us nicely and we'll let you have an advanced copy of the modified spreadsheet, otherwise wait a week or two and we'll post it!

Now we'll pick off 21 discrete points along the curve:

Position (inches) Impedance (ohms)

-0.151 25.5
-0.136 25.8
-0.120 26.2
-0.105 26.8
-0.090 27.5
-0.075 28.4
-0.060 29.5
-0.045 30.8
-0.030 32.3
-0.015 33.9
0.000 35.7
0.015 37.6
0.030 39.5
0.045 41.4
0.060 43.2
0.075 44.9
0.090 46.4
0.105 47.7
0.120 48.7
0.136 49.5
0.151 50.0

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 few possibilities:

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 the culprit...

What do YOU think?

Layout of microstrip Klopfenstein taper

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, a thing of beauty is a joy forever, right? You can't exaggerate the Y-axis in AutoCAD, another example of the superior powers of FraudoCAD...

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!