Antenna length and sound frequency of a theremin.

Hello thereminists !

I am doing a research on "How does the length of the antenna of a theremin affect its sound frequency (trying to keep all the other variables constant)" 

From previous questions I know that the higher the antenna, the larger its area, and the larger its area, the greater its capacitance. However I struggle to demonstrate how does the capacitance between the hand of the player and the antenna affect the sound frequency of the theremin. Overall I have to demonstrate two main relationships using formulas:

1. How does the area of the plates increase the capacitance between them?

2. How does the capacitance between the two plates of the theremin(the antenna and the hand) affect the sound frequency of the     theremin.

Thank you very much for your help!

"...using formulas"  - feliperodrigosek

Closed form equations are notoriously difficult to come by in the inductance and capacitance realms.  Sometimes there are trivial solutions for toy situations that have a lot of symmetry, but for real situations you end up relying on FEA (finite element analysis) or on real-world data obtained via experiment.

First order, capacitance is proportional to the plate area and the inverse distance between them.  Inductance is proportional to wire length. Beyond that things get hairy.  Capacitance as seen by the Theremin is a combination of variable intrinsic (antenna and the universe) and variable mutual (antenna and the hand).  If you do FEA you'll see that the intrinsic decreases and the mutual increases as the hand approaches (as more and more lines of force from the antenna switch from landing at infinity to landing on the hand) with the net result a rather advantageously musical functional response (roughly exponential heterodyned frequency).

If you are using a simple LC oscillator you can work backwards from the resonance equation F = 1 / [2 * pi * sqrt(L * C)], where the C is the parallel combination of any physical component C and the antenna intrinsic + mutual.  The antenna intrinsic + mutual is the hard part, though to a first order you can use the inverse distance times a constant plus an offset.

If you take everything into account via various rules of thumb and such you may end up with a 90% correct answer, or even 99%, but probably not 100%.  It's definitely a diminishing returns type thing, where the more work you put into it the less you get back.

Thank you very much for your answer dewster!!! It will help me a lot.

However there has to be a relationship between the capacitance and the frequency of the sound. I mean, the mutual does increase when the hand approaches the antenna, increasing the capacitance and consequently the frequency. If that happens then the capacitance and the frequency have to be directly proportional. This is what I am trying to demonstrate with formulas.

Thank you very much for your reply!!!

My collected real-world antenna C data: http://www.mediafire.com/file/co61z6z527l7r4e/Analog_Digital_2014-12-19.xls

My collected simulation antenna C data: http://www.mediafire.com/file/zpln5ccdort7ykr/plate_capacitance_2016-02-24.xls

When the hand is not too close nor too far the 1/d proportional relationship is quite strong.  When the hand is near (<0.1m) the geometry of the antenna tends to dominate: antennas with larger dimensions having proportionally less less sensitivity to change here.  This makes sense, as the hand in this region is closely interacting with less of the antenna area.  For common cases (plates and rods) one could probably come up with an approximating integral which collapses to some algebraic equation here.  When the hand is quite far it often merges with the player's body, which causes a similar drop in proportional sensitivity, which also makes sense.

I'm at the point where I mostly care about this stuff in terms of how best to obtain a linear response.  The academic side still interests me, but I've got more immediate fish that need frying.

Wow, thank you for that data, It will definitely help me a lot in my research!

When you say 1/d what do you mean?

Thank you!

"When you say 1/d what do you mean?"

The inverse of the distance between the hand and the antenna.  Look up capacitance on Wikipedia and you'll see the simplified 2 circular plate equation with 1/d running the show.

 "Look up capacitance on Wikipedia" - dewster

The "1/d" law is a spherical cow. My experiments show "the power of 1.7" law (1/(d)^1.7 )  *****

________________
***** [EDIT-2019] it's not true for wide distance range. For more precize formula refer this thread.

Thanks!

"The "1/d" law is a spherical cow. My experiments show "the power of 1.7" law (1/(d)^1.7 )"  - ILYA

LOL!  Yeah, a first-order thing.  Though the heterodyned response is an exponential approximation that also breaks down in the near and far fields - it works well enough when all you have is stone knives and bear skins (tubes and such) but there are better solutions to be found in the latter 20th century.  The simple solutions are all going to be approximations of one sort or another.  (It's interesting that one of the better inductor formulas also uses non-integer powers.)

Physicists are really vague when it comes to whatever the hell they are talking about when they say the word "capacitance".  When they give the self-C of a sphere in space it's intrinsic C.  For two spheres or plates it's mutual C.  Go to any physics forum and watch the big brains argue and talk over each other because they aren't taking the basics into account in the same ways.  It can be a very disorienting and disappointing experience for one seeking a modicum of clarity.

Anyway, for the Theremin it's a combination, where the hand "plate" is grounded, and the antenna "plate" experiences the total combination of its own intrinsic C plus the mutual C with the hand.  The intrinsic isn't a constant as one might naively imagine, and the mutual C isn't simple either.  The only way I'm aware of "knowing" this is to do FEA, where intrinsic and mutual are separately and clearly presented in the solution matrix.  Lab experiments will only provide us with the total C as seen by the antenna - which is what we want, thank goodness - but the raw data doesn't reveal the underlying mechanism behind the response.

Plate antennas are different than pole antennas.

The frequency range is one octave below a piano to one octave above a piano.

Capacitance is not between the antennas, but between the antennas and ground (earth) - your hand affects that value.

"Antenna" is a marking term since radios were very popular with the Theremin were introduced. It does not transmit and if it receives, you get radio coming through your music, so it really is not purposed as a radio antenna.

The choice of a rod for the plate has made theremins very difficult to master.

Which suit me fine.