Theremin Hand Capacitance Approximation Needed

[theremins_r_us]

I've been working on a Theremin Excel simulation for the past couple of months. For those who don't know what a Theremin is, it was one of the very first electronic instruments to be invented, and has two "antennas" that independently change the pitch and amplitude of a tone via hand capacitance. The capacitive changes "seen" by the "antennas" during play are in the femto Farad range so it's quite remarkable that it works at all!

The pitch antenna is vertical, ~500mm in height, ~10mm in diameter, and the player's hand moves horizontally nearer to and farther from the antenna on a line roughly perpendicular to the midpoint.

I have an equation that gives me the free space capacitance of the pitch antenna and it seems to jibe with lab measurements I've made. But I don't have a good equation that will estimate the capacitance of a human hand near the antenna.

For modeling purposes I'm thinking the antenna rod could be represented by a 2D rectangular metal plate, and the player's hand represented by a 2D square metal plate facing the antenna, moving along the playing centerline. Inputs to the equation would be antenna height, antenna width, hand height & width (the same since it's a square), and distance between the plates. Output would be capacitance.

I've spend a couple of days pawing the web and trying to make sense of my old college texts, but can't find this problem or one like it solved. Could anyone please help me?

 

[Drmarshall]

I've been working on a Theremin Excel simulation for the past couple of months. For those who don't know what a Theremin is, it was one of the very first electronic instruments to be invented, and has two "antennas" that independently change the pitch and amplitude of a tone via hand capacitance. The capacitive changes "seen" by the "antennas" during play are in the femto Farad range so it's quite remarkable that it works at all!

The pitch antenna is vertical, ~500mm in height, ~10mm in diameter, and the player's hand moves horizontally nearer to and farther from the antenna on a line roughly perpendicular to the midpoint.

I have an equation that gives me the free space capacitance of the pitch antenna and it seems to jibe with lab measurements I've made. But I don't have a good equation that will estimate the capacitance of a human hand near the antenna.

For modeling purposes I'm thinking the antenna rod could be represented by a 2D rectangular metal plate, and the player's hand represented by a 2D square metal plate facing the antenna, moving along the playing centerline. Inputs to the equation would be antenna height, antenna width, hand height & width (the same since it's a square), and distance between the plates. Output would be capacitance.

I've spend a couple of days pawing the web and trying to make sense of my old college texts, but can't find this problem or one like it solved. Could anyone please help me?

It might help you to know that
At theremin frequencies the dielectric constant of the human hand is about 80
The ISOLATED human hand might be a bit like a fist-shaped sphere
BUT
It is connected to a large body (conductor) capacitively-coupled to ground.
So it depends on which floor of the building the player is on and how near wiring and reinforcement metal construction mesh.

You could MEASURE this using a digital voltmeter as sensitive microammeter.

 

[theremins_r_us]

It might help you to know that
At theremin frequencies the dielectric constant of the human hand is about 80
The ISOLATED human hand might be a bit like a fist-shaped sphere
BUT
It is connected to a large body (conductor) capacitively-coupled to ground.
So it depends on which floor of the building the player is on and how near wiring and reinforcement metal construction mesh.

You could MEASURE this using a digital voltmeter as sensitive microammeter.

Thanks Drmarshall,

I was thinking sphere for the hand but maybe that makes the math more difficult? I assume the solution will be a double integral, but don't know the exact approach. My physics book says to assume the antenna and hand are oppositely charged with +/- Q, then find V between them, then use C = Q / V. Even though it's a second year book it's kind of sketchy as to how to do that second step.

It's true that grounding can very much influence the response of the Theremin, but I'm just looking for a parameterized formula that describes the capacitance formed by the hand | antenna better than the equation I'm currently using.

My source for these formulas is "Physics of the Theremin" by Kenneth D. Skeldon, Lindsay M. Reid, Viviene McInally, Brendan Dougan, and Craig Fulton, 1998:

Antenna capacitance above a ground plane (this gives pretty much what I can measure):

Cant = 2*pi*epsilon0*L / (ln(2*L/D)-k)

where:
L = length of antenna in meters
D = diameter of antenna in meters
epsilon0 = permittivity of free space = 8.85E-12 Fm^-1
k is a constant depending on how far above the ground it is mounted
k = 0.4 for almost at ground level.Capacitance due to a hand in proximity (this underestimates capacitance near the antenna):

Chand = pi*epsilon0*L / (10*ln(4*x/D))

where:
L = length of antenna in meters
D = diameter of antenna in meters
x = distance from hand to antenna in meters
epsilon0 = permittivity of free space = 8.85E-12 Fm^-1I have measured antenna and hand capacitance for three different antennas, but my desire is to have a good enough model of things so that I can do more of the optimization up front and easily on my PC, rather than painfully, slowly, and expensively in the lab. I'd also like the reassurance that my measurements aren't out in the weeds as these are exceedingly tiny capacitances. I can do the heavy lifting but am in need of some hand holding and guidance.

Can anyone point me to articles or texts that clearly describe the process and maybe give a few examples? What is the best physics text out there for this (that maybe doesn't use vector math for everything, I'm an MSEE and never really got the hang of curl and such, and I kind of don't like the text I was taught with)?

 

[Drmarshall]

Thanks Drmarshall,
I was thinking sphere for the hand but maybe that makes the math more difficult?
ψ No, easier. The self-capacitance of a sphere in pF is 1.1 times its radius in cm.

I assume the solution will be a double integral, but don't know the exact approach.
ψ Model it on your computer and thus understand what matters by SEEING what happens if you change your assumptions.

My physics book says to assume the antenna and hand are oppositely charged with +/- Q, then find V between them, then use C = Q / V. Even though it's a second year book it's kind of sketchy as to how to do that second step.
ψ Agreed, useless book

It's true that grounding can very much influence the response of the Theremin, but I'm just looking for a parameterized formula that describes the capacitance formed by the hand | antenna better than the equation I'm currently using.
ψ Well how bad is the equation you are using!
Maybe it really is so bad that you MIGHT do less bad.
But a better way to look at this is you should find the capacitance!
Your theremin book tell you the rf frequency (MHz) and how far the performer typically stands away
This gives a 1 kHz difference in frequency to ONE OF the rf oscillators.
(The "antenna" is part of the TUNED LC CIRCUIT that determines the rf frequency and may represent a bit more L or C depending on its length n wavelengths and whether the far end is shorted or open)
For any ASSUMED circuit diagram find the capacitance pF that DOES cause this frequency change when how many pF is placed where in the circuit (say between antenna input and earth)

My source for these formulas is "Physics of the Theremin" by Kenneth D. Skeldon, Lindsay M. Reid, Viviene McInally, Brendan Dougan, and Craig Fulton, 1998:
ψ Try writing to them. Some authors are actually INTERESTED in their books

Antenna capacitance above a ground plane (this gives pretty much what I can measure):

Cant = 2*pi*epsilon0*L / (ln(2*L/D)-k)

where:
L = length of antenna in meters
D = diameter of antenna in meters
epsilon0 = permittivity of free space = 8.85E-12 Fm^-1
k is a constant depending on how far above the ground it is mounted
k = 0.4 for almost at ground level.
ψ Yes a good assumed circuit diagram would place the pF in shunt (parallel) with this.
But on the computer simulation (see above) you coulds SEE what happens WHEREVER on the antenna you place the pf to earth.


Capacitance due to a hand in proximity (this underestimates capacitance near the antenna):

Chand = pi*epsilon0*L / (10*ln(4*x/D))

where:
L = length of antenna in meters
D = diameter of antenna in meters
x = distance from hand to antenna in meters
epsilon0 = permittivity of free space = 8.85E-12 Fm^-1
ψ The man has pF to Earth of say 100pF
In SERIES with this is the air-spaced capacitor between the antenna wire and the hand (maybe as your formula: probably a few pF DIVIDED by distance in cm)
The hand capacitance, being far smaller, can be taken as the total of the two in series.


I have measured antenna and hand capacitance for three different antennas, but my desire is to have a good enough model of things so that I can do more of the optimization up front and easily on my PC, rather than painfully, slowly, and expensively in the lab. I'd also like the reassurance that my measurements aren't out in the weeds as these are exceedingly tiny capacitances. I can do the heavy lifting but am in need of some hand holding and guidance.

Can anyone point me to articles or texts that clearly describe the process and maybe give a few examples? What is the best physics text out there for this (that maybe doesn't use vector math for everything, I'm an MSEE and never really got the hang of curl and such,
ψ Curl is where the fun starts! If you need any help with the computer programming, write back
and I kind of don't like the text I was taught with)?

See my entries above.

Excellent books are
Langford Smith : Radio Designer's Handbook
Reference Data for Radio Engineers: publisher Howard Sams

 

[theremins_r_us]

ψ No, easier. The self-capacitance of a sphere in pF is 1.1 times its radius in cm.

Hmm, I'm not exactly seeing how the self-capacitance plays into capacitance between two objects?

ψ Well how bad is the equation you are using!
Maybe it really is so bad that you MIGHT do less bad.
But a better way to look at this is you should find the capacitance!
Your theremin book tell you the rf frequency (MHz) and how far the performer typically stands away
This gives a 1 kHz difference in frequency to ONE OF the rf oscillators.
(The "antenna" is part of the TUNED LC CIRCUIT that determines the rf frequency and may represent a bit more L or C depending on its length n wavelengths and whether the far end is shorted or open)
For any ASSUMED circuit diagram find the capacitance pF that DOES cause this frequency change when how many pF is placed where in the circuit (say between antenna input and earth)

The equation I'm using for hand capacitance seems to be way off (underestimated) in the near and mid field of the hand to the antenna (compared to capacitance data I've taken). Far field seems maybe OK.

My spreadsheet sims are quite flexible in terms of oscillator circuit topology, so all I need is a decent hand/antenna inter capacitance formula to examine perceived linearity between hand position and pitch.

ψ Try writing to them. Some authors are actually INTERESTED in their books

It's a student paper, sorry I should have been clearer.

ψ The man has pF to Earth of say 100pF
In SERIES with this is the air-spaced capacitor between the antenna wire and the hand (maybe as your formula: probably a few pF DIVIDED by distance in cm)
The hand capacitance, being far smaller, can be taken as the total of the two in series.

The antenna often has a few hundred pF in series for various reasons as well, and the spreadsheet sim takes this into account.

I guess the hand model not taking the attached arm and body of the player into account is a rather large simplification, but probably anything will be better than what I'm using now.

ψ Curl is where the fun starts! If you need any help with the computer programming, write back

Ha ha! Thanks!

Excellent books are
Langford Smith : Radio Designer's Handbook
Reference Data for Radio Engineers: publisher Howard Sams

Thanks for this as well, will check them out.

 

[Drmarshall]

Hmm, I'm not exactly seeing how the self-capacitance plays into capacitance between two objects?The equation I'm using for hand capacitance seems to be way off (underestimated) in the near and mid field of the hand to the antenna (compared to capacitance data I've taken). Far field seems maybe OK.

My spreadsheet sims are quite flexible in terms of oscillator circuit topology, so all I need is a decent hand/antenna inter capacitance formula to examine perceived linearity between hand position and pitch.It's a student paper, sorry I should have been clearer.The antenna often has a few hundred pF in series for various reasons as well, and the spreadsheet sim takes this into account.

I guess the hand model not taking the attached arm and body of the player into account is a rather large simplification, but probably anything will be better than what I'm using now.Ha ha! Thanks!Thanks for this as well, will check them out.

The capacitance of ANY capacitor depends on the area of each plate and the distance apart.

The potential v depends on the charge Q per unit area.
Thus the self capacity of a sphere gives you its area and the potential V falls off as 1/r
All you need to do is add the Q/V for the other plate of the capacitor - the "antenna" - for which the formula you give may be accurate enough

Yes the midfield region always gives trouble, for here the field strength does not build RADIALLY (as for a sphere, 1/r) nor cylindrically (as from a straigt wire - logs).
BUt for any situation you consider LINEAR, the result of A + B is simply the sum of A and b if tasken alone. This is very powerful - but of course few things are linear despite what schoolteachers tell us!

It is easy to write a computer prog that DOES cater for the midfield also - use R V Southwell's "Relaxation of Constraints"

If your simulation is good enough use THAT to tell you how much hand capacitance changes the oscillator frequency 1%

The "few hundred pF in series TUNES the (impedance of) the antenna.
A straight wire length L, has an impedance of say 300 ohms times tangent (2Pi times its length in wavelengths) . So a wire open at its far end (L) looks like a short circuit at its other end where we desperately TRY to feed power in.
This is VERY difficult! Resonance helps no end
Short antennae are capacitive and those longer that 1/4 wavelength are inductive (until L=1/2 wavelength - and so on)

So the series few hundred pF CHANGES the impedance SEEN by the energy source.
We'd best MATCH it to the "internal impedance" of that source!
The few pF of "hand-capacitance detunes this resonance - more than just "leaking away current".

My capacitance (total to ground) is so BIG (100 pf) and in SERIES with your hand capacitance from antenna to my hand, that ONLY that hand capacitance really counts!
(For capacitors in series 1/C=1/c1+1/c2 where c2 much greater than c1)

 

[theremins_r_us]

The capacitance of ANY capacitor depends on the area of each plate and the distance apart.

The potential v depends on the charge Q per unit area.
Thus the self capacity of a sphere gives you its area and the potential V falls off as 1/r
All you need to do is add the Q/V for the other plate of the capacitor - the "antenna" - for which the formula you give may be accurate enough

OK, self-capacitance:

For a sphere, my physics book says:
V = Q / ( 4 * pi * epsilon0 * radius )

so:

C = Q / V = 4 * pi * epsilon0 * radius

For the antenna, that paper says:
C = 2 * pi * epsilon0 * Length / ( ln( 2 * Length / Diameter ) - 0.4 )

How do I combine these linearly so that I get mutual capacitance, and the distance between them is a factor?

If your simulation is good enough use THAT to tell you how much hand capacitance changes the oscillator frequency 1%

My sim is definitely good enough for that and that's what I'm using it for.

Short antennae are capacitive and those longer that 1/4 wavelength are inductive (until L=1/2 wavelength - and so on)

This is an exceedingly short antenna, nowhere near 1/2 wave, which is why I put "antenna" in quotes. It's really just a way to sense capacitance via change in the resonant frequency of an LC tank.

If anyone is interested, my spreadsheet is here (along most of the project files including verilog - I am designing a mostly digital Theremin that doesn't use heterodyning per se): http://www.mediafire.com/?w36b3brqyg2g3

Snag the latest one that has the file name "Theremin_simulation_*.xls. The phase criterion for the "Linearity" worksheet is a bit off, I've got a better one in the works but am holding off until this hand & antenna capacitance model is improved.

 

[Drmarshall]

OK, self-capacitance:

For a sphere, my physics book says:
V = Q / ( 4 * pi * epsilon0 * radius )

so:

C = Q / V = 4 * pi * epsilon0 * radius

For the antenna, that paper says:
C = 2 * pi * epsilon0 * Length / ( ln( 2 * Length / Diameter ) - 0.4 )

How do I combine these linearly so that I get mutual capacitance, and the distance between them is a factor?




My sim is definitely good enough for that and that's what I'm using it for.


This is an exceedingly short antenna, nowhere near 1/2 wave, which is why I put "antenna" in quotes. It's really just a way to sense capacitance via change in the resonant frequency of an LC tank.

If anyone is interested, my spreadsheet is here (along most of the project files including verilog - I am designing a mostly digital Theremin that doesn't use heterodyning per se): http://www.mediafire.com/?w36b3brqyg2g3

Snag the latest one that has the file name "Theremin_simulation_*.xls. The phase criterion for the "Linearity" worksheet is a bit off, I've got a better one in the works but am holding off until this hand & antenna capacitance model is improved.

The field due to two things ( Flux density, potential and charge) is merely the sum of the field of A PLUS that of B

 

[theremins_r_us]

The field due to two things ( Flux density, potential and charge) is merely the sum of the field of A PLUS that of B

I'm confused. If I simply add the two self-capacitances there is no distance-between-them factor (mutual capacitance should increase with decreasing distance between them). Could you be more explicit or perhaps show me the formula you're thinking of?

 

[Andy Resnick]

<snip>
I have an equation that gives me the free space capacitance of the pitch antenna and it seems to jibe with lab measurements I've made. But I don't have a good equation that will estimate the capacitance of a human hand near the antenna.

Here's an alternate method, making use of a circuit diagram which you may be able to easily alter for your specific circuit.

The output tone is generated by subtracting the resonant frequency of an LC circuit from a ‘reference’ LC circuit operating at 173.4 kHz. The tunable LC circuit has L = 598 uH, C1 = 1.41 nF and C2 is variable, based on the hand position. C1 and C2 are in parallel. When your hand is far from the antenna, fvar = fref. When your hand is very close to the antenna, fvar = 171.6 kHz. This tells you the capacitance contribution from your hand/antenna.

 

[theremins_r_us]

Here's an alternate method, making use of a circuit diagram which you may be able to easily alter for your specific circuit.

The output tone is generated by subtracting the resonant frequency of an LC circuit from a ‘reference’ LC circuit operating at 173.4 kHz. The tunable LC circuit has L = 598 uH, C1 = 1.41 nF and C2 is variable, based on the hand position. C1 and C2 are in parallel. When your hand is far from the antenna, fvar = fref. When your hand is very close to the antenna, fvar = 171.6 kHz. This tells you the capacitance contribution from your hand/antenna.

Thanks Andy!

I've already measured hand | antenna mutual capacitance using this method for three different length antennas, so I have some data. Now I'm looking for a mathematical function for hand | antenna mutual capacitance that takes distance between them, the physical dimensions of the antenna (length & diameter) and perhaps hand size as inputs, and gives me capacitance. It can be a rough approximation. I just need it to be good enough to have some predictive power.