If you have been following my pages on impedance measurements, I've been using a 17.1 foot length of RG-213 terminated with a 25.2 Ohm load as a generator of complex impedance values between 2 and 29 MHz. The complex impedance values at the input to the cable are representative of values frequently seen in antenna work. The resistance swings between 25 and 96 Ohms, and the reactance swings between -36 and +36 Ohms.
I used a program to compute the expected input impedance values across the frequency range. I compared these against several contemporary antenna analyzers, and two traditional impedance bridges. The utility and importance of that comparison will be tied to the accuracy of the computed data. The computed data was based upon knowing the exact electrical length of the cable, and the value of the termination resistor. The resistance value is relatively easy to measure, the cable was the challenge.
When I first started to make measurements, I computed the frequency where the cable was 180 degrees long. That value was 18.90 MHz. All computed impedance transformations were based upon that frequency. While working on an unrelated page on transmission line Q, I had the opportunity to measure the length of the cable with a different and probably more accurate technique. Those measurements ended up suggesting that a more accurate 180 degree long frequency was 18.99 MHz, an increase of 90 KHz.
Since the electrical length of the cable changed, the computed input impedance values would also change. I was concerned that the difference might impact the comparison between computed data and measured data. I knew that sooner or later I would have to revisit the computations of input impedance, and see what difference that made in my comparisons. That's what this page is all about.
What impact would the change in 90 KHz have on the computed impedance data? I wanted to make a table with two sets of impedance values, the data from the 18.90 MHz estimate, and the 18.99 MHz estimate. The table would allow a comparison between the old and the new data.
Impedance transformations should be computed by a calculation, usually buried in a program, that is based upon a lossy transmission line model. In other words, it is designed to be accurate, not ideal. Loss is expressed as a set of coefficients which are part of the equations. When I first computed expected values, I used the Lowband software, available from ON4UN, and the TLA software, which was part of certain editions of the ARRL Antenna Book.
In using these programs, I had to consult various loss tables, select my RG-213 cable, and enter data into the program. There is room for error in that multiple step approach, and it's not clear that my interpretation of error matches the expectation of the program.
For the calculations on this page, I've switched to the TLDetails program by Dan, AC6LA. This program has a convenient user interface, and I believe that the author has a very good understanding of the loss mechanism in transmission lines, and has built that into the program. The coax type can be selected from a list. This means that any expression of loss will be consistent within the program.
When using any program, I would prefer to specify the cable as a physical length, not an electrical length. If it's a physical length, then I can simply set the frequency, and read the resulting impedance at the input of the cable. If it's an electrical length, I would need to recompute the length for each frequency, since electrical length is a function of frequency. The problem is that I know that the cable is 180 electrical degrees at 18.99 MHz, how do I convert that into a physical length?
My approach was to adjust the physical length until the reactance was zero Ohms at the target frequency. Since the load is a pure resistance, without reactance, the reactance will again reach zero when the cable is 180 degrees long.
When using the initial frequency of 18.90 MHz, I found that the cable was 206.1 inches long, which is 17.175 feet. With the new frequency of 18.99 MHz, the length dropped to 205.11 inches, which is 17.095 feet. By the way, the cable is physically 17.1 feet long. Assuming that the velocity factor and characteristic impedance of the cable were exactly as specified, the lengths would be identical. Clearly, they are very close, and, the net difference is about 1 inch out of 206 inches, less than 0.5 percent. While that seems darn small, I wanted to run the numbers to see what difference there would be.
I set the cable to be 206.1 inches in TLDetails, and specified RG-213. The load was 25.2 Ohms of resistance, and 0 Ohms of reactance. I stepped the frequency from 2 through 29 MHz, with a 1 MHz step size. I recorded the input impedance values. I repeated the measurements after resetting the length to be 205.11 inches. The data is shown in the next table.
|TLDetails Computed Input Impedance, 18.90 MHz Versus 18.99 MHz Transmission LIne Reference|
Initial: 18.90 MHz, 206.1 Inches
|Updated: 18.99 MHz, 205.11 Inches|
|Freq. (MHz)||R (Ohms)||X (Ohms)||R (Ohms)||X (Ohms)|
Cells where there was a difference of between 0.5 and 1 Ohm are highlighted in green. When the difference is between 1 and 1.5 Ohms, the cell is highlighted in yellow. Above 1.5 Ohms difference, the cell is highlighted in red, which looks like salmon. The highlighting is only in the updated data columns. Obviously the same differences would appear in the initial data columns to the left as well, but I chose not to highlight those.
I found it very interesting to note that by a small change of 1 inch in a cable that is over 17 feet long it was possible to see almost 2 Ohms of difference, although it was at the upper end of the frequency range. That is not the typical difference, however. The lesson here is that an inch can matter in making certain measurements, even at shortwave frequencies, which are considered to have long wavelengths. As frequency rises, length matters more, as straight wires become measurable inductors.
My intent in producing the previous table was to get a sense if the change in length showed up in the expected input impedance - the reference impedance values. For me, there is a difference, but it's not earth shaking. Of course the comparison graphs I have been constructing up to this point on other web pages were all based upon data computed by the ON4UN Lowband software. So, there will be some difference between those values, and the TLDetails values based upon the updated cable length. Here is a table comparing the Lowband data to the most recent TLDetails data. Cells are marked with colors using the previous description.
|Previous Lowband Computed Impedance Data versus Updated TLDetails Computed Data|
Initial: Lowband Computed Impedance
|Updated: 18.99 MHz, 205.11 Inches|
|Freq. (MHz)||R (Ohms)||X (Ohms)||R (Ohms)||X (Ohms)|
These differences are the important ones, since these are the reference values used in comparisons against measured data. Here is a graph comparing these two data sets.
Updated Impedance Reference Versus Initial Values
Even though there is some divergence at the higher frequencies, the initial and updated data are still very close. The (small) error visible in this comparison does not suggest that if I redraw my previous comparison graphs the measured data will appear to be more or less accurate. So, for now, I'm going to claim that the change in cable length does not demand a total revisiting of the comparisons made with it.
I've not attempted to calibrate any of my impedance measuring devices. It's clear from their measured data curves that they are closely following the expected resistance and reactance. It's a question or getting the measured and expected data to overlap, and not just be close to each other. Often, this suggests that adjustment will bring the device in calibration. If the measured and expected curves are way off, it may suggest a device which just does not really measure what it should, and no amount of calibration will clean it up. If they are far enough apart, perhaps it's simply broken.
I would like to look into calibrating whatever devices allows calibration, but I'm holding off until I feel as if I have a good sense of what being calibrated means.
When making measurements, it's a good idea to ask yourself: where is the reference plane? By reference plane, I mean the precise physical location when the device under test is being measured. With the popular antenna analyzer, the reference plane is probably located right at the edge of the case inside the coaxial connector. As you move away from the reference plane, the accuracy of the measurements will suffer, since they are made at the reference plane, that's why it's called the reference plane.
When making many measurements, we must use test leads. In this case, it's best if the reference plane can be extended to the end of the test leads. This basically happens with the traditional impedance bridge, since you can make the initial null (or balance) with the test leads in place. When doing that, the manuals strongly suggest that you not move the test leads into new positions, since that will upset the extended reference plane.
The antenna analyzers allow us to measure components such as inductors and capacitors. Since they have a microprocessor in there, it's easy to convert reactance into capacitance and inductance. The problem is that they have no mechanism to extend their reference plane. The best that we can do is to connect the parts we want to measure to the analyzer with very short wires. If possible, no wires. I started out with a few inches of wire, and I soon learned that they were as much a source of error as anything within the analyzer itself.
I've learned one thing, if you want to make accurate impedance measurements, you can't be sloppy. You've got to figure out where the reference plane is, and get the device to be measured connected to that point with the shortest possible wires.
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