,By Isidor Straus
Curtis-Straus LLC
Since we’ve got a focus article on shielding in this issue, it seems timely to review the relationship between source configuration, operating frequency, measurement distance, and wave impedance. Wave impedance is the ratio of the electric field magnitude, E, to that of the magnetic field magnitude, H. It characterizes an electromagnetic field with the same units - ohms -- as the circuit concept impedance does. The ratio of E, which is in volts per meter, to H, in Amperes per meter, is simply ohms. In shielding, this ratio plays an important part in the mechanisms of reflection and absorption, which play a key part in EMI/EMC control.
In this article, we’ll show that the wave impedance in free space is determined by two things:
Statics First
For our purposes, circuits that cause fields can be sorted into four basic classes:
1. Electrostatic
2. Magnetostatic
3. Electric, time-variant
4. Magnetic, time-variant
The electrostatic class is simply a fixed distribution of charges. Since the charges do not move, no current flows. A simple familiar (but useful) case is the charge dipole, where two equal and opposite charges are spaced some distance apart. Another would be one or wires held at some fixed voltage or voltages. Under these conditions, there is an electric field which does not vary with time (E is constant in time). There is no magnetic field (H is zero). The concept of wave impedance isn’t useful in electrostatics because the ratio of E to H is infinite.
For a charge dipole, the electric field falls away as the cube of the distance. The inverse cube law is derived from the near cancellation of fields by the opposite charges. Each charge has an field which falls off as 1/r2, but because the charges are of opposite signs, the resulting field falls off more rapidly.
Magnetostatic circuits consist of DC current loops. This is the dual to the electrostatic case. There is a constant magnetic field H which is time invariant and falls off with the cube of distance, but no electric field (E is zero). Hence wave impedance isn’t of interest here either - it would be zero. In fact, for both the electrostatic and magnetostatic cases, wave impedance is meaningless, because there are no waves-nothing is changing, and field information doesn’t propagate.
Varying the charges and currents
Once we have variation in either charge or current distribution, we will have changing fields. Maxwell’s equations tell us that we can’t change an electric field at any point without there being a corresponding change in the magnetic field, and that information on the change (i. e., changes in the fields at more distant points) will propagate away at the speed of light in the surrounding medium.
We can reconcile the concept of wave propagation with those of electro- and magnetostatics by looking at the way electric and magnetic time-variant circuits behave.
The third class of circuit mentioned above is a time-variant electric circuit. We could start by thinking of a slowly varying electrostatic circuit. Consider these more or less equivalent cases:
1. a charge dipole where the charges vary sinusoidally
2. a current element where current flows back and forth sinusoidally along a line (charges would build up and reverse at the ends, so this is equivalent to the previous example)
3. any assemblage of open-ended wires driven by voltage sources, including dipole and whip antennas, as well as low speed leads exiting circuit boards driven by common mode voltages
A short sinusoidally varying current element is known as a "Hertzian dipole." "Short" means small in comparison with a wavelength at the drive frequency, so the current is uniform over the wire at any instant. To understand the relation ship between the charge dipole and the time-varying current element, think of the charges as varying with time. If each charge at the end of the dipole has the value
,
then a current equal to the rate of change of the charge,
will flow.
A mathematically exact description of the electric and magnetic fields generated by the infinitesimal Hertzian dipole is shown in Fig. 1.
Figure 1: A sinusoidal current element
produces large electric fields
in the near field because of the 1/r3 term.
Note the following:
1. Both electric and magnetic fields are present.
2. The electric field contains components which vary as
1/(ßr)3, 1/(ßr)2, and 1/ ßr, where ß = (2
/
), and the wavelength =
(c/f), where c is the velocity of light and f is the frequency.
3. The magnetic field contains components which vary as 1/(ßr)2, and 1/ ßr.
4. The quantity (ßr) can therefore be rewritten as
.
It is unity at a distance of ro
= /2. For r
<< ro, 1/(ßr) is the dominant term. This is what we call the near field,
and is electric. For r >> ro, 1/ ßr is the dominant term. This is the
far field. Note that in the far field, E and H are related by the characteristic impedance
of the medium (120 or 377 ohms for a vacuum). In the vicinity of ro,
there is a transition region where all terms are of the same order of magnitude.
What does this tell us? For one thing, the terms
"slow", "fast", "near" and "far" are all relative.
A "slowly" varying Hertzian dipole is one that we are "near" to in the
sense of being much closer than /2 when we measure it. Since wavelength and frequency
are inversely related, low frequencies have long wavelengths and vice-versa.
The dual to the Hertzian dipole is our fourth case, a sinusoidally excited current loop. A current loop is characterized by its "moment," which is the product of the current it carries times its amplitude. Looking at figure 2, we see that the electric and magnetic fields for a sinusoidally driven infinitesimal current loop mirror those for the Hertzian dipole. Here, the near field magnetic field exhibits 1/r3 behavior, while the near-field electric strength falls off as 1/r2. In the far field, both E and H exhibit 1/r behavior. Their ratio, which is the wave impedance, is the characteristic impedance of the surrounding medium, just as it is for the Hertzian dipole.
Figure 2: A sinusoidal current loop produces
large magnetic fields in
the near field. In the formulae, the term "dm" stands for the magnetic moment,
given by the product of current and loop area.
In the far field, both sources "look" the same.
That is, given measuring equipment-antennas and a field strength meter-we would have no
way of knowing whether the electric and magnetic fields we measured were created by a
dipolar or current loop source. This is the situation for measurements taken above 30 MHz
for FCC and CISPR measurements. The dividing line between near and far field at 30 MHz, /2, is
only 1.6 meters. Therefore, we expect 1/r behavior, or inverse scaling, for high frequency
measurements. Deviations from true 1/r behavior usually have to do with the fact that the
equipment under test is not a point source, and due to site and groundplane reflections.
At low frequencies, however, the situation is quite different. Consider a magnetic source-a driven loop - operating at 60 kHz. This source could be the horizontal drive of a CRT monitor, or a switching power supply, or the field coil in a magnetic security system. The dividing line between near and far field at this frequency would be 800 meters. Any practical measurements would be taken in the near field at a much closer distance of the magnetic fields with a loop antenna. It would be easy to demonstrate inverse cube law dependence of the field strength with distance. Electric field strengths would be relatively low in the near field for a loop-like structure. Low E and high H mean low wave impedance (see figure 3).
Figure 3: The wave impedance measures the
relative strength of electric and magnetic
fields. It is a function of source structure.
Conversely, a dipolar source measured in the near field would exhibit strong electric fields and weak magnetic fields. High E and low H mean high wave impedance. In both cases, as the distance increases to the far field, the wave impedance approaches that of free space.
The FCC and CISPR ITE standards mandate the use of
magnetic field antennas (shielded loop) below 30 MHz and linearly polarized electric field
antennas (dipoles, biconicals, log periodics) at higher frequencies. This essentially
assumes that only magnetic fields from loop-like sources are of interest at the lower
frequencies (say below 5 to 10 MHz). This is by no means a universal regulatory policy.
Both military (MIL-STD-461/2) and Bellcore (GR-1089) include the requirement to also
measure low frequencies with an active 1 meter long whip antenna, which is electric field
sensitive.
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