This image shows a down-the-guide view of a 10GHz sinusoid propagating in a WG-16 waveguide.  This simulation was performed with a vs FDTD code using PEC boundary conditions and an incident wave from the lower left face of the guide.  Red and yellow portray the positive portions of the sinusoid while purple and blue portray the negative.  A WG-16 guide should be roughly 2.29cm (long wall) by 1.02cm (short wall).
While certainly possible to locate in the nearest EM textbook, it is valuable to maintain the notation here for consistency across the site pages and example problems. Through the years of text resources, the reader will find many varied notational styles.

### Differential Form

\\\begin{aligned} \\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac{1}{v_p}\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{v_p}\\vec{\\mathbf{j}} \\\\ \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\ \\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac{1}{v_p}\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\ \\nabla \\cdot \\vec{\\mathbf{B}} & = 0 \\end{aligned} \ where:
• \$$\\vec{\\mathbf{E}}\$$ is the vector instantaneous electric field in the time domain
• \$$v_p\$$ is the velocity of propagation in the local medium, \$$c_0 \\approx 3\\times 10^8 \\frac{m}{s}\$$ in essentially free-space conditions

### Light Speed

While commonly estimated as \$$3 \\times 10^8 \\frac{m}{s} \$$, the speed of light is defined as exactly: \$$c_0 = 299,792,458 \\frac{m}{s} \$$.  From this, the other constants of \$$\\mu_0\$$ and \$$\\epsilon_0\$$ may be defined:

﻿\\\begin{aligned} c_0 & = 2.99792458e8 \\frac{m}{s}\\\\ \\mu_0 & = 4\\pi \\times 10^{-7} \\approx 12.566370614e-7 \\frac{N}{A^2} \\\\ \\epsilon_0 & = \\frac{1}{ \\mu_0 c_0^2 } \\approx 8.854187817e-12 \\frac{F}{m} \\\\ Z_0 & = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}} \\approx 376.73031346958504 \\,\\Omega \\end{aligned} \

From here we can define a spatially local velocity of propagation:

\\\begin{aligned} v_p & = \\frac{1}{\\sqrt{\\mu\\epsilon}} \\\\ \\mu & = \\mu_r \\mu_0 \\\\ \\epsilon & = \\epsilon_r \\epsilon_0 \\end{aligned} \ where \$$\\mu_r \$$ and \$$\\epsilon_r\$$ are the relative permeability and permittivity, respectively, spatially local to the region of interest.

### References

While at the outset this may solely appear as a mine detector, this system is quite general and capable of detecting numerous buried anomalies as defined by a difference in the electrical properties of the object buried in the earth background.  GeoModel, Inc. is one company producing systems for buried detection of objects. This would include pipes, rocks and gradations in earth denoting possible old structures now buried (archaeology).  Each represents a possible change in the permittivity, permeability and conductivity versus the background and would affect the received signal strength. The question then becomes how to optimize and interpret the interactions of the electromagnetic fields.  Several examples of operation are shown here which begin to elucidate the operation of this device. An understanding of the detection mechanisms will help improve the design.

One of the first simulations should be the detector design over an unfilled background earth.  The earth is simulated with a relative permittivity of 2.5 which is roughly comparable to dry loamy soil.  A 790 MHz CW source was used for the transmitter. This shows the detector in operation without an anomaly and will be considered the control for further analysis.

As seen in the control simulation, most of the energy is directed into the earth.  This is good because more energy will be available for detection of anomalies below the earth surface.  Note from the outset that some energy is escaping at the left side of the detector where the transmit cavity is.  Note also that significant portions of the energy radiating into the earth are not directed towards the detection cavity.  This becomes an issue when multiple anomalies are present.  In essence, the transmit cavity and transmitter are not directive enough to determine where the anomaly is only that it exists.  This is the extreme, of course as the detector does detect location as will be shown below.  The key here is that energy flowing to the left will couple with objects on the left and then possibly reradiate to the receive cavity which could muddle the received signal from the anomaly directly beneath the cavity.  This effect could yield detection flaws.

The next simulation is based off a rectangular block of dielectric buried beneath the earth surface.  This anomaly has a width of 11.811 inches, a height of 3.1496 inches and a relative permittivity of 8.0.  The anomaly was centered 5.118107 inches below the surface of the earth and centered relative to the detector center.  The numbers were chosen to match a particular physical experiment.

Clearly, the signal strength at the receive (right) cavity has increased which would indicate the presence of an object.  This detection will be clarified a bit later, but for now consider the animation.  Locate the lensing effect at the upper left corner of the anomaly and also locate the pulses which travel along the upper surface of the anomaly.  Follow the pulses along the top of the anomaly and watch as they reradiate at the far right top edge of the anomaly and then couple into the receive cavity.  In this scenario, the reradiating appears to be the mechanism for detection.

The next simulation shows a simple modification of the detector was chosen to illustrate detector design issues.  The outer walls of the transmit and receive cavities were extended to the earth surface.  This could be accomplished via a light metal screen or a set of non-ferrous metal wires.  Non-ferrous would avoid some metal detection devices in a mine trigger device.  This, of course, begins to raise the issue of overall detector design.  The emitted signal must be of low field strength in order to avoid detection by the anomaly device.  However, the lower emitted signal strength will diminish the capability to penetrate the earth for detection.

Note that this simple design modification did not appear to effect a change in the received signal strength.  The dominant coupling mechanism remains the lensing and surface guided waves through the anomaly.  The surface guided waves then reradiate upon termination of the anomaly geometry.

A slightly different structure shows the detector design works for pipes as well as rectangular objects.  These results could be compared to the control.  The center PVC (relative permittivity 2.8) pipe is 10 inches in diameter, 10 inches underground, centered relative to the detector and filled with water (relative permittivity 78.0).  The left metal pipe is 8 inches in diameter, 7 inches underground, 7 inches left of detector center and filled with jet fuel (relative permittivity 2.08).  The right PVC pipe is 4 inches in diameter, 5 inches underground, 7 inches right of detector center and filled with jet fuel (relative permittivity 2.08).

The left and center pipes clearly increase the receive signal strength.  The right pipe is barely perturbing the fields and would likely be undetected in the current system.

The next simulation illustrates the effect of raising the detector too high above the earth surface.  Only the height has changed to 7 inches from 3.1496 inches in the previous cases.  Clearly the receive signal has increased yet this is not a beneficial increase.

The coupling is clearly traveling through the air gap between the earth surface and the metal septum of the detector.  This corroborates the results in hayes91:_trans_line_matrix_tlm_method hayes90:_trans_line_matrix_model where detector height sensitivity was studied.

TDLO methods also offer the capability to study scattered fields.  By comparing the actual fields in the control and rectangular homogeneous dielectric data before the images were made, a scattered field result may be calculated, that is, the fields which have changed due to the presence of the anomaly.

This analysis shows this scattered field result which is the effect of the anomaly on the default background.  The lensing effect in the upper left corner of the anomaly and the top surface guided wave are clearly shown.  A scattered field result may also be calculated for the simple modification of cavity side walls extending to the ground surface as shown below.

Ideally, the most prominent change should be in the receive cavity indicating an increased receive signal strength.  Thus, the simple geometry modification was not very effective.

Anomalous numerical dispersion is an issue arising out of limited computational resources and the time-domain method currently in use.  The effect could easily be traced back to Nyquist sampling of waveforms for accurate representation.  If enough'' samples are present, then a waveform could be regenerated from the samples.  Enough'' could be qualified as at least two sample points per wavelength of the signal being sampled.  In practice, more samples would provide a more accurate sampling from which a more accurate regeneration would be possible.

As an example, a 2D FDTD code was written to propagate a pulse which could reflect off a boundary.  As a test case, a sample problem of cell size 0.03m by 0.03m with a total mesh size of 256x256 was created.  This lead to a physical problem size of 256 times 0.03m per side of the mesh of 7.68m.  As can be seen in the animation, the pulse begins to propagate fairly well and encounters the PEC wall at the left of the mesh.

The boundaries are a combination of second-order Mur boundaries along the walls and first-order Mur boundaries at the corners.  Notice that some reflections occur.  The magnitudes of the reflected electric field do appear higher at the corners and when the incident energy is non-normal to the wall.  The reflections are quite low when the incident waveform is normal to the walls.  As the animations continue, the scale of the plot is dynamically shifted to the largest electric field magnitude (either positive or negative).  Since the excitation was a static pulse of electric field, no further energy should enter the system after time 0.0s.  Given a constant in energy and a spreading wave front, one would expect the electric and magnetic field amplitudes to diminish.  The overall energy should remain exactly the same until the wave front begins to exit the mesh with the Mur absorbing boundary conditions.  Once the pulse encounters the PEC barrier in the left side of the mesh, the animation has begun to exhibit a ringing'' phenomena behind the original wave front.  This could be called anomalous numerical dispersion, numerical dispersion, phase distortion or one of several other bad names.  The overall idea, though, is that it is NOT real in the system under study.  One way to confirm the dispersion is NOT real is to decrease the size of the cell.  In order to model the same physical size system, the overall mesh must be increased accordingly.

In this example, the cell size was decreased from 0.03 to 0.01 meters which increased the overall mesh size to 768x768 cells.  Compare this animation to the previous at 0.03 meter cell size and notice that the dispersion has been alleviated.    It is important in that the issue has been alleviated but not removed.  This could be considered a failure mode'' of this 2D FDTD implementation and method.  In other words, the simulation did not fail exactly, but the solution will degrade as the physical cell size increases and the solution will improve as the cell size decreases.  One of the beneficial aspects of time-domain local-operator methods is the ability to run in a `degraded'' mode to test boundary conditions and initial problem setup.  Once these issues have been confirmed suitably correct, a much higher resolution (smaller cell size and larger overall mesh size) simulation may be run.  This aids in rapid debugging of user input and code modifications.

The Kreh Antenna Library is a grouping of numerous NEC antenna systems gathered by an avid CEM enthusiast Hartmut Hans and provided recently (2004) by Ursel Kreh.  In 2001, these were also provided to Bill Walker and available on a few NEC sites as the  DL5SAY Collection.  Hartmut Hans, DL5SAY, died of cancer in January 2001, aged 54.  He was a radio amateur with much experience, and in his profession he was a software development engineer for telecommunication devices.  During his disease he worked with the NEC (MoM) program, and developed many graphical applications including GNECX.  He was very proud of GNECX and the antenna models.  Some weeks before he died, he told Ursel that he wanted to finish these programs,  but the cancer was stronger.  Included herein are the original GNEC files along with the original NEC files from the wide assortment gathered by Hartmut Hans.  The antenna files and updated TCL files will also appear elsewhere on the site.

In the words of Ursel Kreh, "In international QSOs, he normally called himself Hart because of better understanding.  In March 1965, aged 19, he got his first callsign DL8ON.  Nine years later, still being a student, he gave back his callsign as there was no time and little opportunity for radio amateur activities. The callsign has since been given to an other radio amateur.  In August 1981, Hart applied for a license again obtaining the callsign DL5SAY (which he kept until his death)."  It was during this time that Ursel also participated.

"Hart\\''s enthusiasm concerning amateur radio was the technical aspect.  He bought good devices from Yaesu, and then he started to build further devices himself, e.g. a PA with 500 W and many small devices for the improvement of transmission etc.  The other passion concerning this hobby was to experiment with antennas.  At home he had a normal ground plane and also a butter nut he ordered from USA.  On many vacation trips the amateur radio equipment was part of our baggage, and each time he had new ideas in creating antennas with some "simple wires".  He had a very good feeling for antennas and resulting from this a great knowledge.  The biggest success was our vacation trip to Guadeloupe in October 1992 (with 15 kg equipment) where he threw a wire over the orange trees and where he got many pile ups."

"As Hart\\''s antenna at home had a big range, he had QSOs with people in the whole world, often with American radio amateurs, of course.  He liked to participate in contests and to distribute points.  Very often he only listened to the radio amateur traffic."

"In July 1997, we got the diagnosis of cancer.  In the following 3 1/2 years, we fought together for his life. These years have been full of hope and full of despair.  In this time he needed new purposes in life, and one was to work on antennas.  He spent much time on this.  In parallel, he trained for running, and with the cancer and with chemotherapy, he ran two marathons and several half marathons.  Hartmut died on 31st of January 2001. He was a wonderful person!"

"His whole amateur radio equipment including miles of antenna cables and lots of electrical components etc.  I have given to a small radio amateur club in our town. There are many young people, and the club has not much money, there was no club station, no antenna, etc. These people have been enthusiastic about the comprehensive and good equipment.  I am sure that Hartmut would be glad about my decision."

This antenna library page is dedicated to Hartmut Hans and Ursel Kreh and the extraordinary work and devotion they have both shown to the community.

krehLibrary.tgz (tar gzipped)

GNECX (gnecx1 or gnecx2) is written in TCL and needs the BLT library available on the Internet and possibly in your Linux/Unix distribution.  Several snapshots show this as a useful tool for analysis of NEC output files, which does not provide a graphical interface, results.

The results, animations, and analysis from the krehLibrary model files are included in the NEC Antenna Library section.