current "flowing" into the bottom of the coil? Scroll down to the bottom of this page.

A transmission line has a distributed inductance and distributed capacitance that causes a delay through it. A real-world loading coil has a distributed inductance and distributed capacitance that causes a delay through it. At a single frequency, this delay can be specified in degrees. Let's use the same assumptions as Kraus. Consider the following loading coil with all four currents having equal magnitudes, i.e. |If1|=|If2|=|Ir1|=|Ir2|.

In reality, the magnitudes of If1, If2, Ir1, and Ir2 are not equal so this exercise is completely accurate only for thin-wire antennas. However, the ballpark conclusions from that assumption are apparently good enough for John D. Kraus.

The moral to this exercise is to avoid using a lumped circuit analysis on a distributed network problem. That includes all problems where forward and reflected waves exist as they do on standing-wave antennas and transmission lines with an SWR greater than 1:1.

The following graphic explains why the current is different at the top and bottom of a loading coil. The magnitudes and phases of the currents at each end of the coil depend simply upon its physical location within the standing wave environment.

**Moral: There is no useful phase information contained in the standing wave current phase
measurement.** Therefore, the standing wave current phase measurement alone cannot be used
to determine the percentage of a wavelength that is occupied by the loading coil. Loading
coils occupy tens of degrees of a wavelength but measuring that length is quite a
technical challenge. The estimated number of degrees occupied by the coil in the above
examples is estimated to be ~60 degrees since the self-resonant frequency of the coil
is approximately 9 MHz. A very rough estimate of the electrical length of the coil can
be obtained using an arc-cosine function on the standing wave current amplitudes. **Hint: The only
phase information in a standing wave is embedded in its amplitude, not in its phase.**

Download the EZNEC files for the above antennas: Download zipped test316.EZ. Download zipped test316c.EZ.

There are two very interesting web pages that shed light on this issue. Please pay close attention to the limitations of the lumped circuit model when applied to large loading coils:

RF Coils, Helical Resonators and Voltage Magnification by Coherent Spactial Modes

Class Notes: Tesla Coils and the Failure of Lumped-Element Circuit Theory

Download the EZNEC file for the above antenna: Download zipped test316y.EZ.

A free demo version of EZNEC is available at http://www.eznec.com