Practical Papers, Articles and Application Notes

In this issue you will find three practical papers that should be of interest to the EMC community. The first, entitled "Reciprocity and EMC Measurements" by Jasper Goedbloed was first presented at the 2003 EMC Zurich Symposium and has been reprinted by permission. In the introduction, Jasper quotes radio pioneer Stuart Ballantine who said: "Among the tools of thought and artifices by which man forces his mind to give him more service, perhaps the most intensely useful are the simple mathematical rules of inversion known as reciprocity theorems." If you read the paper, I think that you will understand why this statement was made. I think that you will also find that the paper is worth the investment of time needed to read it. The second paper is entitled, "Overview of the Threat of Intentional Electromagnetic Interference (IEMI)" and has been written by William Radasky and Manuel Wik. This was first presented at the recent IEEE EMC Symposium in Istanbul and has been reprinted with permission. In our age, we have become simultaneously more dependent upon electromagnetic waves for communication and concerned about the threat of global terrorism. As specialists in EMC, we need to understand the threat of intentionally generated electromagnetic interference. I think that you will find this paper a good introduction to the subject. Finally, in response to a recent article by Jose Perini, Brandão Faria has written a short paper designed to clarify some of the points about the polarization ellipse. It is a useful complement to Professor Perini's paper.

The purpose of this section is to disseminate practical information to the EMC community. In some cases the material is entirely original. In others, the material is not new but has been made either more understandable or accessible to the community. In others, the material has been previously presented at a conference but has been deemed especially worthy of wider dissemination. Readers wishing to share such information with colleagues in the EMC community are encouraged to submit papers or application notes for this section of the Newsletter. Click here for my e-mail. While all material will be reviewed prior to acceptance, the criteria are different from those of Transactions papers. Specifically, while it is not necessary that the paper be archival, it is necessary that the paper be useful and of interest to readers of the Newsletter.

Comments from readers concerning these papers are welcome, either as a letter (or e-mail) to the Technical Editor or directly to the authors.


Please click here for the full article "Reciprocity and EMC Measurements"


Jasper J. Goedbloed received the M.Sc. degree in experimental physics from Amsterdam University in 1967 and the Ph.D. degree from the Technical University of Eindhoven in 1973. From 1954 to 1978 he worked at Philips Research, Eindhoven, in various fields, such as electro-mechanical coupling systems, noise in IMPATT-diode microwave oscillators and low-noise avalanche photo-diodes. In the period 1963-1967 he investigated effects of gamma irradiation of silicon at the Physics Laboratories of Amsterdam University.


From 1978 until 1997 when he retired, he was with the EMC Department of Philips Research as Supervisor and Senior Scientist. Until 2002 he was an active member of CISPR/A, concentrating on uncertainties in standardized compliance testing.


EMC education is his great interest. He is the author of a textbook on EMC, has presented many EMC courses and founded the Post-Graduate Course on EMC in the Netherlands in 1984. He has also participated several times in the Experiment Demonstration Sessions of the IEEE EMC Symposia.


Dr. Goedbloed is a Member of the Dutch Physical Society (NNV) and the Dutch Society for Radio and Electronics (NERG). He is a Senior Member of the IEEE EMC Society and an Honorary Life Member of the Dutch EMC/ESD Society. EMC


Overview of the Threat of Intentional Electromagnetic Interference (IEMI)




Abstract: The new threat of intentional electromagnetic interference (IEMI) has become more important today given the reliance of society on sophisticated electronics and the threat of global terrorism. This paper defines important terms, describes the different types of electromagnetic threats, summarizes the current understanding of equipment susceptibility, provides several approaches for protection of buildings, and summarizes the new work in international standardisation.


Keywords: High Power Electromagnetics, Intentional EMI, Transients, Standardisation


What is Intentional EMI?
The term Intentional Electromagnetic Interference or IEMI has been used in recent years to describe all categories of interference caused to electronic equipment whether it is by terrorists, criminals or hackers. While the motives of the attacker may be different, the results can be the same on society. The scientific community has been working to understand and define this threat in a more precise manner.
In February 1999 at a workshop held at the Zurich EMC Symposium a widely accepted definition was suggested: "Intentional malicious generation of electromagnetic energy introducing noise or signals into electric and electronic systems, thus disrupting, confusing or damaging these systems for terrorist or criminal purposes." Note that hackers are not mentioned explicitly in this definition, although in most countries of the world, an attack on commercial interests for "entertainment" is against the law.


The first question one might ask is whether there really is any reason for society to be concerned about this problem. In fact there are many as indicated below:

  • Terrorist threats are increasingly of concern world-wide
  • Intentional EMI is a new threat dimension
  • Attractiveness of covert operation outside physical barriers
  • Technological advances in higher energy RF sources and antennas
  • Increasing proliferation of IEMI sources and knowledge world-wide
  • Increasing dependence on information and on automated mission-critical and safety-critical electronic systems
  • Increasing EM susceptibility for new high density IT systems working at higher frequencies and lower voltages

In August of 1999 this problem was also addressed by the Union of Radio Scientists International (URSI) during a special session that resulted in an official URSI resolution. The URSI Resolution of Criminal Activities using Electromagnetic Tools was intended to make people aware of:

  • The existence of criminal activities using electromagnetic tools and associated phenomena
  • The fact that criminal activities using electromagnetic tools can be undertaken covertly and anonymously and that physical boundaries such as fences and walls can be penetrated by electromagnetic fields
  • The potential serious nature of the effects of criminal activities using electromagnetic tools on the infrastructure and important functions in society such as transportation, communication, security, and medicine
  • That in consequence, the possible disruption on the life, health and economic activities of nations could have a major consequence

The URSI Council resolved that URSI should recommend to the scientific community in general and the EMC community in particular to take into account this threat and to undertake the following actions:

  • Perform additional research pertaining to criminal activities using electromagnetic tools in order to establish appropriate levels of vulnerability
  • Investigate techniques for appropriate protection against criminal activities using electromagnetic tools and to provide methods that can be used to protect the public from the damage that can be done to the infrastructure by terrorists
  • Develop high-quality testing and assessment methods to evaluate system performance in these special electromagnetic environments
  • Provide reasonable data regarding the formulation of standards of protection and support the standardization work which is in progress

Types of IEMI Threat Environments
In order to understand the threats to electronic equipment, it is necessary to understand the different types of electromagnetic environments that can be produced and that can create operational problems for exposed equipment. There are two major categories of EM environments of concern: narrowband and wideband. There are also two major ways for this energy to be delivered to a system: radiated and conducted.


A narrowband waveform is nearly a single frequency (typically a bandwidth of less than 1% of the center frequency) of power delivered over a fixed time frame (often on the order of microseconds). For experiments performed on equipment where vulnerabilities have been noted due to radiated fields, frequencies between 0.3 and 3 GHz seem to be of most concern. Of course higher and lower frequencies may also cause problems with system performance, especially if a system resonance is found. Also some environments in this category include modulation of the sine waves, shifting frequencies and repetitive applications. This category of radiated threat is often referred to as high power microwaves (HPM), although this term is used loosely to include frequencies outside of the microwave range.


A wideband waveform is usually one in which a time domain pulse is delivered, often in a repetitive fashion. The term "wideband" indicates that the energy in the waveform is produced over a substantial frequency range relative to the "center frequency". Of course many pulse waveforms do not have an explicit center frequency; and more precise definitions are being considered at this time to divide the wideband category into several subcategories. One of the common terms used for radiated fields includes the ultrawideband pulse or UWB, and it typically rises in 0.1 nanoseconds and decays in approximately 1 nanosecond.


In terms of creating system vulnerabilities, the narrowband threat is usually one of very high power, since the electrical energy is delivered in a narrow frequency band. It is fairly easy to deliver fields on the order of thousands of volts/meter at a single frequency. Of course each system under test may have a vulnerable frequency that is different from the next. Often the malfunctions observed in testing equipment with narrowband waveforms are those of permanent damage.


The wideband threat is somewhat different in this respect. Since a time domain pulse produces energy over many frequencies at the same time, the energy density at single frequencies are much less. This means that damage is not as likely as in the narrowband case; however, it is easier to find a system's vulnerability since many frequencies are exposed at the same time. Sources that have been built in the past typically produce repetitive pulses that can continue for many seconds or minutes thereby increasing the probability of producing a system upset.


While the waveform characteristics are defined above, there are two primary ways that they may be delivered to a system. One is through the application of radiated fields, and the other is through conduction along cables and wires. These two methods of delivery are consistent with the general treatment of electromagnetic disturbances in the field of electromagnetic compatibility (EMC) where nearly all environments and tests are defined in terms of radiated and conducted environments (e.g., IEC 61000-2-5).
The above discussion clearly indicates that there are four general cases to be considered for IEMI: radiated narrowband, radiated wideband, conducted narrowband, and conducted wideband.


For the two radiated cases, it seems clear that frequencies above 100 MHz are of primary concern in that they are able to penetrate unshielded or poorly protected buildings very well and yet couple efficiently to the equipment inside of the building. In addition, they have the advantage that antennas designed to radiate efficiently at these frequencies are small. Tests performed in the IEC using standard 61000-4-3 indicate that most equipment sold commercially in Europe should be immune to radiated narrowband fields of 3 to 10 V/m (between 80 MHz and 2.5 GHz), depending on the equipment type. For wideband radiated threats, the IEC equivalent is the electrostatic discharge test (61000-4-2), which produces a peak electric field on the order of 1 kV/m near the ESD arc; this pulse has a rise time of 0.7 ns and a decay time of approximately 30 ns. Although this field is produced by a discharge on or near an equipment enclosure, the EM fields propagating away from the arc couple well to and propagate through small apertures in the enclosure walls. This is very similar to the exposure from a radiated field.


For the two conducted cases, there are some differences in terms of the frequency range of interest. It is well known that if conducted signals are injected into the power supply or telecom cables outside of a building, that frequencies below 10 MHz (and pulse widths wider than 50 ns) propagate more efficiently than higher frequencies. In fact it is known that power line disturbances caused by lightning pulses (see IEC 61000-4-5) and electric fast transients (IEC 61000-4-4) propagate well enough from the outside to the inside of a building to provide peak pulse voltages as high as 2 kV at the wall plugs. These pulses have rise times between 5 ns and 10 microseconds with pulse widths typically a factor of 10 longer than the rise. While these lower frequency waveforms are not ultrawideband, they satisfy the general wideband definition. Of course narrowband conducted signals also can be injected into a building's wiring, including its grounding system at frequencies even lower than 50 Hz. Experiments by Parfenov et. al. have shown that even these low frequency narrowband waveforms can disrupt the operation of equipment inside a building [1].


In completing the discussion about the threat environments, it must be realized that there are two aspects of interest in terms of deciding which types of waveforms are the most important. For buildings with wooden construction or with dielectric windows, the radiated waveforms (narrowband and wideband) will be most dangerous at frequencies above 100 MHz. These fields will easily penetrate the outer walls of a building and expose the equipment and its wiring in offices near the outer walls. For energy to penetrate further without attenuation into the building there will be a need to have a straight clear path into inner portions of the building. This is because reflections and diffraction will reduce the strength of the fields as they penetrate deeper inside a building. For the conducted threat that is injected on the outside of cables entering a building, frequencies below 10 MHz (narrowband or wideband) tend to be more important if the threat is to propagate beyond the interface. These insights will be useful when it comes to designing mitigation methods to deal with this threat.


Susceptibility Levels of Commercial Equipment
Over the past five years there have been significant experiments that have tested the response of commercial equipment to narrowband and wideband threats. In general this experience can be summarized in the following way:


Modern computers and other types of equipment using microprocessors appear to be vulnerable to malfunction from radiated narrowband fields above 100 V/m [2]. Of course there are large variations in the responses of equipment due to the specific experiment setups and the quality of the equipment enclosures that are used. In addition, tests performed in the range of 1 - 10 GHz seem to indicate that malfunctions occur at lower field levels at the lower frequencies [3]. Unfortunately there have not been many experimental results published that have covered frequencies below 1 GHz, so it is not clear if this trend continues to lower frequencies. There is less experience with the use of wideband radiated field testing, however, there are some indications that peak pulse field levels of 1 kV/m will produce malfunctions for a 0.1/1 nanosecond pulse, that is repetitively pulsed.


One should note that these experiments are usually performed by directly exposing the equipment under test within line of sight of a radiating antenna. Of course if the equipment is within a building or in a room without a window, there will be a reduction of the incident field from outside to inside. Also most experiments have not really examined the polarization and angle of incidence aspect thoroughly, and therefore most of the effects noted during testing will likely occur at lower field levels when an optimum coupling geometry is applied. Of course what is really needed is to understand the average vulnerability of equipment since the orientations of equipment and the polarization of the threats are likely to be extremely variable.


For the conducted threats, it seems clear that if access to external telecom or power cables is not prevented, it is fairly easy to inject harmful signals into a building. Experiments have shown that narrowband voltages injected into the grounding system of a building can cause significant equipment malfunctions inside. Frequencies below 100 Hz and levels below 100 volts have been known to cause problems [4]. For wideband waveforms, it appears that pulse widths on the order of 100 microseconds can create damage to equipment power supplies and with interface circuit boards at levels of 2 - 4 kV [1].


While these values may seem to be low, they should not really be a surprise. When one examines the EMC test requirements for immunity in Europe, it is unusual to see a narrowband radiated field level requirement above 10 V/m (for frequencies above 80 MHz). This is the required level for medical devices that are needed to support life (IEC 60601-1-2). Higher levels are not recommended because of the expense of providing the increased protection. For narrowband voltages induced on cables connected to equipment, 10 V is the upper level required in most cases (IEC 61000-4-6). The frequencies of application are below 80 MHz. Clearly the immunity levels required for commercial equipment are well below the levels that can be produced by commercially available IEMI sources today.


In the area of wideband conducted transients, most of the lightning and electric fast transient tests are performed for levels up to 2 kV. Only in special cases, such as for equipment in a power generating facility or a substation will the levels be higher. The typical EMC wideband test waveforms have rise times as fast as 5 ns and pulse widths as slow as 700 microseconds.


There is one area in which a wideband threat has higher levels to consider - the high altitude electromagnetic pulse. HEMP is generated from a high altitude nuclear detonation. HEMP standards developed by the IEC required radiated field testing for fully exposed equipment to a standard level of 50 kV/m with a 2.5/25 ns wideband pulse. There are corresponding wideband conducted waveforms that are created during the coupling process. These induced voltages may reach levels of hundred of kilovolts on an external power line. Of course we cannot expect that commercial equipment connected to these lines are immune to HEMP threats, and therefore it is clear that the intentional EMI threats clearly exceed the levels that equipment are protected to using standard EMC practices.


Protection Approach
When it comes to protecting a building and its internal equipment from the threat of IEMI, there are several aspects of protection to keep in mind:

Security Approach

  • Develop a "keep out" or buffer zone around your building
  • Prevent unauthorized access to all power and communications entering your building
  • Keep important internal equipment away from the outer walls of the building
  • Use redundancy and diverse routing for important wiring inside the building
  • Make sure backup power is available for critical operations

Electromagnetic Approach

  • Monitor the building and its wiring for unusual transients-provide alarms to security personnel
  • Provide EM shielding for critical equipment
  • Provide surge protection and filters for cables used in critical operations
  • Use non-metallic fiber optic cables when possible
  • Ensure that the grounding system is properly designed
  • Develop a verification program to periodically test the immunity of the facility

For buildings where nearly all equipment and functions inside are considered critical, it is likely that an approach similar to that taken to protect from HEMP should be considered. This involves a well-shielded (from EM fields) building with all penetrating conductors (including those from external antennas) to be protected with high level filters and surge protectors. Unfortunately this approach can be very expensive especially if it is not used when the building is constructed. It is clear that the HEMP protection approach with some enhancements to cover frequencies above 100 MHz has value for critical facilities.


There are many details to be added with respect to the guidelines provided here. It is expected that over the next few years, additional details will emerge from the technical community especially due to the active work ongoing in this area in the International Electrotechnical Commission (IEC).


The International Electrotechnical Commission
The International Electrotechnical Commission (IEC) is headquartered in Geneva, Switzerland, and it is responsible for preparing voluntary standards for electrical and electronic equipment worldwide. It has over 60 nations participating actively in its work including all of the major industrial nations of the world. It is the worldwide leader in the development of electromagnetic compatibility (EMC) standards.


In the late 1980s, the IEC began the development of environment, protection and test standards for commercial equipment that might be exposed to the electromagnetic fields produced from a high altitude nuclear burst. These fields are generally known as the HEMP (high altitude electromagnetic pulse), although in actuality the term represents a series of pulses with different frequency content that need to be considered. These range from nanosecond pulses to pulses with rise and fall times of seconds.
When the active work on HEMP began it was assigned to Subcommittee 77C (SC 77C) under the parent committee TC 77 (EMC). In June 1999 after 10 years of intense activity, the scope of the work in SC 77C was expanded to include all high-power EM transient threats, including those from Intentional EMI. The formal scope of the subcommittee is indicated in the following paragraph.
Technical Committee No 77: Electromagnetic Compatibility
Subcommittee SC 77C: High power transient phenomena
Scope: Standardization in the field of electromagnetic compatibility to protect civilian equipment, systems and installations from threats by man-made high power phenomena including the electromagnetic fields produced by nuclear detonations at high altitude.
(Note: High power conditions are achieved when the incident electric field exceeds 100 V/m).
As of January 2003, SC 77C has 17 documents in its program of work, including 14 that have been published. The work accomplished to date is described in an accompanying paper; however, the work has followed the basic approach used in TC 77 of the IEC.

  • Evidence of the importance of the threat is documented
  • Threat environments are defined (both radiated and conducted)
  • Radiated environments are established based on published information
  • Conducted environments are developed using available EMC coupling models and experimental data
  • Immunity test levels are defined, depending on equipment location
  • Immunity tests for equipment use available IEC EMC test methods as much as possible
  • Test facilities are surveyed and documented
  • Protection guidance is provided for facility managers
  • Protective device specifications and test methods rely on existing EMC methods

While all of the documents completed so far have dealt with HEMP, there are now two documents under development that deal exclusively with the problem of high power electromagnetic (HPEM) threats. The two projects involved were approved in 2000, and the second committee drafts were circulated in late 2002.

  • HPEM effects on civil systems (IEC 61000-1-5)
  • HPEM environments (IEC 61000-2-13)

After the HPEM environment document receives clear support from the National Committees of the IEC, then work will begin on the protection methods to deal with these threats.


In this overview of the problem of Intentional EMI, we have defined the problem and the terminology involved in simple terms. We have further described the basic types of threat environments involved and have summarized our current understanding of the importance of the different types of threats. In addition, some of the basic concepts have been laid out for protection from this new threat, although it is clear that more work remains to be done. Finally the work of the IEC is briefly mentioned in order to assure the reader that there are strong international efforts underway to understand and to standardize ways to protect our society from these threats.


[1] V. Fortov, V. Loborev, Yu. Parfenov, V. Sizranov, B. Yankovskii, W. Radasky, "Estimation of Pulse Electromagnetic Disturbances Penetrating into Computers Through Building Power and Earthing Circuits," Metatech Corporation, Meta-R-176, December 2000.
[2] J. LoVetri, A. Wilbers, A. Zwamborn, "Microwave interaction with a personal computer: Experiment and modeling," 13th International Zurich Symposium and Technical Exhibition on EMC, February 1999, pp. 203-206.
[3] M. Bäckström, "HPM testing of a Car: A Representative Example of the Susceptibility of Civil Systems," 13th International Zurich Symposium Supplement, February 1999, pp. 189-190.
[4] V. Fortov, Yu. Parfenov, L. Zdoukhov, R. Borisov, S. Petrov, L. Siniy, "A computer code for estimating pulsed electromagnetic disturbances penetrating into building power and earthing connections," 14th International Zurich Symposium and Technical Exhibition on EMC, February 2001.

Dr. William A. Radasky's current interests include studies to understand the threat of Intentional EMI and to develop mitigation and monitoring methods to protect equipment from this new threat. He is Chairman of IEC Subcommittee 77C, which is developing high-power electromagnetic environment and test standards for civil systems. He is also the Chairman of TC-5 (High Power EM) for the IEEE EMC Society. In addition, he is the Chairman of the IEC Advisory Committee on EMC (ACEC), which is tasked to coordinate all EMC standardization work for the IEC. During his 35-year career, he has published over 250 technical papers and reports dealing with electromagnetic interference (EMI) and protection.



Manuel W. Wik has been a Senior Member of IEEE until his retirement by the end of year 2001 and a valued IEEE member for 25 years. He was born in Stockholm, Sweden, in 1936 and became Master of Science in Electrical Engineering from The Royal Institute of Technology in Stockholm in 1962.

He became Chief Engineer and the first person to receive the title Strategic Specialist at the Defence Materiel Administration (FMV) in Stockholm in 1990. He is still active at FMV after his retirement. He was previously head of procurement at the Swedish Armed Forces Telecommunication Transmission Network Section at FMV. Earlier on he worked as research engineer at The Research Institute of National Defence (FOA) responsible for protection against electromagnetic effects from low and high altitude nuclear explosions. He contributed to the International Council of Scientific Union, Scientific Committee on Problems of the Environment project on Environmental Consequences of Nuclear War (ICSU SCOPE ENUWAR) resulting in a series of books. He has written numerous articles, contributed to books, government reports and conference proceedings, and given presentations in the fields of Nuclear Electromagnetic Pulse effects (EMP), Electromagnetic compatibility, Intentional electromagnetic interference, Electronic warfare, Threats to telecommunications, and National strategy for enhanced IT-security and protection against information operations. His present interest as Strategic Specialist includes emerging technologies such as High Power Electromagnetics (HPEM), and systems and methods for Network Centric Warfare (NCW, in Sweden named Network Based Defence).

Mr. Wik is Fellow of the Royal Swedish Academy of War Sciences and Secretary of its Division of military technology, Member of the Swedish National Committee of the International Union of Radio Science (URSI), Secretary of the International Electrotechnical Committee (IEC) standardisation Sub-Committee 77C and recognised as an EMP Fellow by the US Summa Foundation. He has been awarded the Carolus XIV medal Ingenio et Fortitudine from the King of Sweden. EMC

The Polarization Ellipsoid Revisited

Many problems of EMC and EMI require a detailed knowledge of the polarization status of emitted radiation by devices or interfering radiation on devices. The topic of wave polarization is a classic one. Despite that, erroneous ideas still can be found in the literature. Moreover, the way the topic is traditionally treated leaves some loose ends -the question being not what is told but what is omitted. In this tutorial article a concise but complete discussion of the topic of field polarization is provided.
Let us consider the description of a sinusoidal time-varying electric field vector E(t) in a fixed point in space, using a Cartesian reference frame. If the field vector is described by one single component, for instance along x, we write


where w is the angular frequency, Ex is the field magnitude, ax is the phase angle, and ux is the unit vector along the x-direction. Both Ex and ax are time-invariant scalar quantities. In this case, the field is said to be linearly polarized in the x-direction, [1-4].
Consider now a two-dimensional representation for E, using for instance its x and y components,


In this case, for an arbitrary combination of values of Ex, Ey, ax and ay the field is said to be elliptically polarized [1-4] in the x-y plane. If, in addition, Ex=Ey and ax-ayp/2 then a circularly polarized field is obtained [1-4].
All the above considerations can be found in most modern textbooks on Electromagnetics [2-4]. However very seldom [5] can one find the topic of polarization discussed for the case of a three-dimensional representation of E(t).
In a recent article [1] appearing in the EMC Society Newsletter, the theme of the polarization status of a field vector characterized by three simultaneous components has been addressed. There, it is claimed that in the general case of E fields given by


the tip of the vector moves on the surface of a three-axial ellipsoid, which would then be the generalization of the polarization ellipse found for E fields described by only two Cartesian components.
Well, this idea seems to make sense and sounds appealing, but, unfortunately, it is a mistake. In fact, as we are going to show, even for the 3D representation in (3), the most general polarization status one may find is the elliptical polarization on a plane.
For analysis purposes we employ the phasor representation of sinusoidal time-varying scalar and vector quantities. Let us begin with the scalar components of the E field. For k = x, y, z we have

where (4)

where the phasors Ex, Ey, and Ez are complex time-invariant scalar quantities.
Substitution of (4) into (3) yields


The complex quantity in parenthesis is the so-called phasor vector representation of E(t), which can be broken down into its real and imaginary parts

Both vectors E1 and E2 are time-invariant vectors defined in the three-dimensional real space, with fixed magnitudes and fixed directions, E1=E2u1, E2=E2u2 , .
Substitution of (6) into (5) then yields


According to (7) we finally see that, in general, the vector field E(t) is the result of the superposition of just two linear polarization states (not necessarily orthogonal) oscillating with a phase lag of p/2; the result of which is not an ellipsoid but instead an ordinary ellipse belonging to the plane defined by the directions u1 and u2, which make an angle q,


The sense of rotation of the tip of vector E(t) is from E2 to E1.
The major and minor axes of the polarization ellipse are not, in general, coincident with the directions u1 and u2. To determine the sizes and directions of the ellipse principal axes we should use (7) to find E2(t)=E.E. With the help of basic trigonometry results we obtain




Now, using (9) and (7), we conclude that the following orthogonal vectors define the major and minor axes of the polarization ellipse





The effective or rms value of the time-varying electric field E(t) can be directly obtained from its general definition

Then, using (9), we find

For arbitrary combinations of values of Ex, Ey, Ez, ax, ay and az the ellipse axial ratio AR = Emax/Emin can take any value from 1 to ∞ . AR = 1 means the field is circularly polarized, and AR = ∞ it is linearly polarized.

In this article we addressed the issue of the polarization of harmonic electric field vectors E in the general case (ordinarily not discussed) when the fields are given by 3D expressions with all its spatial components explicit. It was shown that for arbitrary field components, the tip of vector E(t) does not describe, as it may be believed, a three-axial ellipsoid, but, instead, a simple ordinary ellipse, whose characterization has been made.

[1] J. Perini, "What is a Phasor Anyway?" IEEE EMC Society Newsletter, no. 195, pp. 23-25, 2002.
[2] C. A. Balanis, Advanced Engineering Electromagnetics, New York: Wiley, 1989.
[3] J. A. Kong, Electromagnetic Wave Theory, New York: Wiley, 1986.
[4] S. Ramo, J. Whinnery, T. Duzer, Fields and Waves in Communication Electronics, 2nd Ed. New York: Wiley, 1984.
[5] M. Born, E. Wolf, Principles of Optics, Cambridge UK: Pergamon Press, 1980.

Professor Brandão Faria was born in Portugal in 1952. He received his degrees in Electrical Engineering from the Instituto Superior Técnico (IST) of the Technical University of Lisbon/Portugal, including the Ph.D in 1986 and the Aggregate title in 1992. He has been teaching since 1975 at the Department of Electrical Engineering of the IST, where he is currently a Full Professor. From 1994 to 2000 he served as President of the Centro de Electrotecnia Teórica e Medidas Eléctricas -a University Center for Research & Development in Lisbon. His areas of interest include wave propagation phenomena in multiconductor transmission-line structures, optical fibers, and electrooptic devices. Professor Faria has authored two textbooks and contributed over 80 papers to refereed journals and conferences. He is a Member of the Optical Society of America and a Senior Member of the EMC Society, MTT Society, PE Society, and Education Society of the Institute of Electrical and Electronics Engineers. EMC

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