Since Dewster has been bringing up the subject of RF emissions and radiated power on a couple occasions, I thought that a separate discussion topic might be a good idea. Here is his question from another thread:
Hello Dominique,Off topic: So you're an RF person? Do you know how to calculate the RF emission (power) of a 0.5m long x 10mm diameter Theremin antenna swinging 200Vp-p @ ~1MHz? I tried to figure this out but hams speak a different EE dialect of impedance mismatch, SWR, etc.
For the case of a 3D antenna I would think a sphere antenna would give the highest intrinsic capacitance for the lowest RF emission, and for the 2D case I believe this would be a circular plate antenna - with both driven from the center (I think the drive point is important?).
If a person is using an air-core inductor vs. a ferrite core inductor, do you think there would be significant RF emitted from the inductor? I was told this was true (long ago by member livio).
I ask the above because the LC energy lost to RF places an upper limit on LC tank resonance Q, and there is always radiated compliance to worry about (no Theremin builders / manufacturers seem to address this - though there is a compliance sticker on the bottom of the Theremini - go figure!).
And it's just one of those things I'd like nailed down in my understanding of this stuff. If you could just point me to an equation or two that would work - Dewster
Regarding Dewster's question:
NOTE: the asterisk in " antenna* " refers to the theremin elements commonly called antennas which aren't really
1) A theremin operating with a base-loaded lambda/600 antenna* is for all practical purposes not radiating any RF power. The antenna* is acting as one element of a capacitor that is swinging a large voltage at nearly zero current (it's open ended, hence current has to be nearly zero). Power can be delivered by the driver source through a theremin antenna* and into a resistive grounded object brought into close proximity, but that's really more of a result of the displacement current passing through the theremin antenna*/capacitor-plate that also passes though the grounded resistive material. It's a power transfer that occurs wirelessly, but that doesn't mean it should be considered to be caused by radiation.
2) If your concern is FCC/CISPR22 compliance, it would be easier to calculate with certainty that your theremin is safely under a radiated power limit than it would be to calculate an actual number. Just measuring (at resonance) the feed point current and voltage and relative phase so that you can calculate real power should, I would think, give you the total power covering both internal losses and any trace of radiated power. Separating the Q power loss from the radiated is a different matter (see 4)
3) If number 2) above is your concern, I will add that I'm not sure if a theremin is classed by the FCC as an Unintentional or Intentional radiator, and I have only had to test to, and comply with Class A and B levels for Unintentional radiators. If it is Unintentional I think you could be safe because radiated testing goes no lower than 30MHz (or that was the limit back when) as measured with a biconical antenna. Conducted emissions are tested lower, but that is generally easy for a simple devices with external power. If the theremin is considered an Intentional radiator, I believe that there are radiated power limits that are band dependent, but I don't know offhand what they are. I still don't think you would have a problem from the theremin antennas*. You could still have issues with any emissions from the high-speed digital FPGA hardware, but my guess is that your exposure here is minimal.
4) If you want to calculate an actual number for your total radiated power without any concern over field pattern variation, I would think (because this is way out of my frequency range) that you could begin with a total power measurement as in 2) above. Then from there remove the antenna and replace it with a low-loss equivalent capacitance, like an air-dielectric variable cap, to obtain the same resonant frequency, and then remeasure the real power. Since you have taken away (most of) any potentially radiating element, any difference could be attributed to radiation loss. My guess is that in a normal pitch field of air you won't see much difference, since I think that any radiation losses will be low.
For more info, here is an excerpt from something else I was writing about theremin antennas* a while back - some related, some not
On Theremin Antennas* versus Real Antennas
First. let me preface this by warning that I was an RF/microwave engineer that never designed antennas, although I worked alongside a few that did. I did manage to pick up a few things over the wall (literally) by osmosis, and although our antennas were very specialized for the frequencies and operational requirements that we worked with, the general principals apply at lower frequencies too. I have not given a moment's thought to this subject for at least 13 years, so my descriptions may be a little sloppy, but it's the thought that counts...
I think that most who study theremins a bit are aware that even though we are pretty loose about throwing around the term antenna* when referring to theremin pitch rods, volume loops, or plates, they are not radiating antennas in the conventional sense. They can practically be treated as purely voltage elements - rods, loops, plates, or other forms. The fields surrounding these elements are predominantly electric fields that drop off rapidly in magnitude. The voltage swing at the theremin antenna* is quite high and yet the current is so low as to be nearly immeasurable with conventional EMI test equipment, and this extremely high voltage-to-current ratio represents such a high source impedance mismatch to the lower impedance of free space that no energy will be propagated for any distance.
Now you can argue that point a little, in much the same way that the fringe-lunatic audiophiles and high-end audio cable manufacturers will argue that speaker cables must be viewed as RF transmission lines. Antenna effects, and transmission line effects, don't just disappear at some low frequency -the rules apply down to DC. Theoreticians worry the problems down to DC, but engineers get to decide when certain effects become irrelevant, and ignore them in order to get on with other business.
That said, if you are asking the question "How much radiated power do you get from a theremin antenna*", you can't accurately answer "zero". There is some, as observed by the Q reduction of resonant circuits when a hand or any other resistive material enters the near field of the capacitive portion of the antenna* resonant circuits. Lossy magnetic materials would have a Q-reducing effect on genuine antennas too, but we will soon see that the theremin antenna* has very little magnetic field, and hence lossy magnetics will have imperceptible effects unless they enter the fields of the series inductors, and this is not usually possible.
A real transmitting antenna can be thought of as sort of a coupling transformer to convert conducted power of one impedance into radiated power at a different impedance. Its purpose is to effectively couple a conducted voltage and current of impedance = V/I into radiated electric (E) fields and magnetic (H) fields with relative strengths such that the ratio (mag)E/(mag)H attempts to match the impedance of free space (Zo=377 ohms in a vacuum, about the same in air). It works the other way too; the rule of reciprocity says that a good transmitting antenna is also an effective receiving antenna.
It you want to consider the prospect of RF radiated power from a theremin pitch or volume antenna*, however small it may be, it would probably be best to compare the antenna* element and its series inductance to a specific form of practical antenna - the base-loaded, quarter-wave vertical monopole.
First consider a full length vertical monopole that is not base-loaded. A conductor that is a quarter-wave long at a specific frequency and is open at one end and grounded or held to a low impedance at the other end is resonant at odd multiples of 1/4 wave (1/4, 3/4, 5/4, etc.). At resonance, the driven quarter-wave conductor will support a standing wave with currents and voltages that are compatible with the boundary conditions at the shorted and open ends. The open end will have a voltage maximum with zero current, and the shorted end will have a current maximum with zero voltage. Surrounding the voltage maximum will be an E-field maximum. Likewise, surrounding the shorted end with the current maximum will be an H-field maximum.
On circuit boards, and transmission lines as well, you have conducted currents that have surrounding H fields, and voltages that have surrounding E fields. Circuit boards (at least the well-designed ones) will have traces with instantaneous currents in one direction (with their corresponding magnetic fields) that are accompanied by return currents that have opposite magnetic fields. If located near each other, the magnetic fields will cancel. Likewise, voltages on circuit traces have surrounding E-fields that ideally will terminate on a conductor with an opposite polarity E-field. Twisted pairs of wires, and in fact coaxial cables and all other forms of transmission lines use this same concept - magnetic and electric fields cancel each other so that a distant observer detects nothing radiated from the lines.
Now if you take a section of a transmission line and pull away one conductor over some length close to a quarter-wavelength, you leave the other conductor with magnetic and electric fields that are no longer balanced or cancelled by the opposing conductor. These E and H fields radiated outward from the single remaining conductor, and if the E/H ratio is not totally mismatched from the 377 ohm free-space impedance, an antenna is born. Rejoice if that was your goal, or deploy the copper tape and ferrite beads if it wasn't.
Now at lower frequencies, a quarter-wave long antenna can be physically too long to be practical. A 27MHz CB radio antenna would be about 9 feet long, a bit large for a hand-held radio. So a common practice was to base-load a shorter antenna to maintain the same resonant frequency and good efficiency with a shorter physical length. Base loading is accomplished by replacing a portion of the physical length near the low-Z driven end of the antenna (this length also equates to a number of degrees of electrical length) with a lumped element inductor, so that the same resonant frequency is maintained. Those familiar with impedance matching and the use of Smith charts will recognize that replacement of a portion of the electrical and physical length by a series inductance has a small and usually acceptable effect on the behavior around resonance, including Q and antenna efficiency.
So...... How does this apply to theremins? Well, pitch rods, volume loops, or whatever, with their associated inductors if used, can be considered base-loaded antennas that went too far. Way too far to be considered radiating antennas, except under the closest scrutiny. The base-loading inductance grew larger and larger, and the antenna length grew smaller and smaller in an effort to maintain the same f=1/{2*pi*sqrt(L*C)}. For a 300kHz theremin a quarter-wave long antenna (a real antenna) would be something like 250 meters in length. That has been replaced by a huge lumped-element inductance (large L) in series with the very tip section of the original antenna (very small C).
This creates a series LC circuit/antenna* whose resonant frequency is extremely sensitive to small changes in capacitance from your body - the fundamental goal for theremin applications. But at the same time it's now a painfully bad radiating antenna, which is also a good thing. As antennas go, it's "all hat and no cattle" - nearly all unterminated E field and no significant H field - representing too high of an impedance to source power to the now-much-lower 377 ohm E/H ratio necessary for propagation in free space.
(excerpt cut.. that's plenty for now)
