Van de Graaff generator voltage measurement

[haversin]

The setup to measure the top voltage of my Van de Graaff generator is shown
below.


The max sphere gap distance for spark breakdown was found to be 13.5 cm. Using the equations from the paper

“Peak Voltage Measurements Using Standard Sphere Gap Method”
 by Constantin Ungureanu and Lacramioara Mihaela Nemtoi

A peak voltage of about 310 kV was obtained.
This voltage is a little higher than I expected given that the dimensions of the top terminal is 8 inches wide and 5 inches tall. Supposedly the sphere gap should not exceed the sphere radius. The 13.5 cm is larger than the radius by less than a cm.
For a second method of voltage measurement as a comparison I'm thinking of trying to measure the force between the two spheres at a distance a little greater than 13.5 cm.

[Uspring]

This voltage is a little higher than I expected given that the dimensions of the top terminal is 8 inches wide and 5 inches tall. Supposedly the sphere gap should not exceed the sphere radius. The 13.5 cm is larger than the radius by less than a cm.

I've tried to calculate corona currents and came up with I=3/2*π*ε0*μ*V^2/r with some approximations. This is for a pointed electrode, spherical symmetry, r the distance to ground and μ the mobility of ions in air. For 312 kV and 1 m distance to ground that would be about 1 mA, probably more than your VDG can deliver. In the case of a spherical top electrode with radius r0 you can subtract roughly E0*r0 from V before using the equation, where E0 is the breakdown field of air of about 30kV/cm. That can reduce the current significantly.

Your setup is gorgeous. 

[haversin]

I don't quite follow you what corona current am I measuring to get top terminal voltage? The output current of my  VDG is about 5 microamps on an average day. It's a about 6.5 microamps after I clean the belt, pulleys and column but this only lasts a few days.

Your setup is gorgeous. 

Thanks!

[Uspring]

The idea is, that corona current is the main limiting factor to VDG voltage. Current begins, if the field at the electrode boundary exceeds 30kV/cm. The voltage for a sphere would be r*30kV/cm. The equation I gave calculates the current for voltages above this limit.
Since your VDG supplies only 5uA, your voltage can go only slightly above this limit. That is what I thought you meant in your first post. I didn't realize, that VDG currents were so minute.
When I derived this equation, I had in mind also CW multipliers, which can supply a lot more current and sometimes don't have nice big spherical top loads.

[davekni]

Have you tried the weighted disk lifting measurement on this generator?  (Thank you for correcting my formula error.)  I think that method would provide a reasonable estimate.  Measured value could be a bit high if disk isn't centered or a bit low if there is significant density of air ions above the disk reducing field strength at surface of your top sphere.

[haversin]

No I haven't.
I will try that, good idea thanks!
From what I have read the most common way to measure the voltage of a VDG is with a field mill. The field mill is placed a safe distance from the VDG and A calibration voltage is applied to the top terminal. From the field mill reading a calibration factor is found. This factor is then applied to field mill reading while the VDG is running. I have tried this in the past but never trusted my results. The field mill reading would steadily increase as the VDG was running. I'm assuming because the growing ion cloud from the top terminal and objects around the room were getting charged. I will try this method again with a higher calibration voltage and a quicker reading of the field mill.

[davekni]

I'm assuming because the growing ion cloud from the top terminal and objects around the room were getting charged.

Makes sense.  I've been amazed at the degree to which nearby walls and ceiling get charged.  I would have expected the surfaces of typical painted drywall construction to be insufficiently insulating for that.  I have a low-power CW multiplier with 300M (30 x 10M 10kV) resistance in series to a 250mm diameter top ball, which I use for typical VG-style demonstrations such as hair standing on end.  As walls and ceiling charge, I have to increase voltage to get the same effect.

[Uspring]

It might be worthwhile to use your field mill outside. The ground there likely won't hold much charge. The ion cloud is not very dense, about the order of I*ε0/(μ*Q), where I is the corona current, μ the ion mobility and Q the VDG top charge. For 5uA current and about 3 uC top charge, this is about 0.07 uC/m^3. This is for spherical symmetry.

In principle that could be a large charge for a big volume, but in reality we don't have spherical symmetry and the ground plane will cause the ions to move downward following the field lines, so that the charged volume is probably not more than a few m^3.

Ion speed is about 600 m/s for 30kV/cm field near the top load. It is still 6 m/s at maybe 1 m distance, so I would expect the field mill to show a more or less stationary result outside, because the cloud will be in place and settled before you get a chance to measure the field.

[haversin]

so I would expect the field mill to show a more or less stationary result outside, because the cloud will be in place and settled before you get a chance to measure the field.

Interesting prediction! I will try to make the voltage measurement outside.

[haversin]

I did some experimenting with the lifting disk method suggested by David to measure the voltage of my VDG. Circles of Aluminum foil of course lifted off the terminal easily but circles of thicker aluminum would almost always tilt up on one side and when the lifted edge was more than a few mm above the terminal, corona leakage off this edge would kill the VDG voltage. I could could tell this by the change in the crackling noise coming off the terminal. Next I tried a platter from a hard disk, it was 9.5 cm in diameter with a 2.5 cm diameter center hole and was 0.66 mm thick. The mass was 23.2 grams. Using the lifting disk equation gives a voltage of 279 kV to lift the platter. The platter didn't move. The crackling sound coming of the terminal sounded normal and the sphere gap test showed no reduction in voltage with the platter on the terminal. Finally a 5 cm diameter circle of Aluminum foil with some thin pieces wood on top (mass = 4.78 grams) would slowly slide off to the edge of the terminal giving a voltage of 230 kV. This is much lower that the 310 kV measured using the sphere gap.

[davekni]

but circles of thicker aluminum would almost always tilt up on one side and when the lifted edge was more than a few mm above the terminal, corona leakage off this edge would kill the VDG voltage.

Is there any reasonable way to form the thicker disks to be curved with a radius at or slightly less than your top sphere?  Or even a large O-ring or some such structure under the disk edge.  That way, as soon as one side lifts, that would be a valid voltage reading, as it would be lifting around a pivot point of the opposite disk edge, which should have almost no force if the disk is well-centered.  (Perhaps this is a wasted comment of mine.  Just realized that your final test is roughly what I was trying to suggest here.)

Next I tried a platter from a hard disk, it was 9.5 cm in diameter with a 2.5 cm diameter center hole and was 0.66 mm thick. The mass was 23.2 grams. Using the lifting disk equation gives a voltage of 279 kV to lift the platter. The platter didn't move.

Perhaps I'm making another calculation mistake, but I'm getting 349kV for your platter (3.5166 kg/m^2 on top of a 0.125m radius sphere).  Also, I wonder if 9.5cm diameter is too large for a disk not curved to your 12.5cm radius.  Edges might be high enough to have significant corona losses.  Although your overall voltage didn't drop, I wonder if the corona would charge adjacent air enough to affect net lift.

Finally a 5 cm diameter circle of Aluminum foil with some thin pieces wood on top (mass = 4.78 grams) would slowly slide off to the edge of the terminal giving a voltage of 230 kV. This is much lower that the 310 kV measured using the sphere gap.

I'm calculating 2.4344 kg/m^2 for 291kV at 0.125m radius sphere.

I'm using my formula after correcting it per your post:
V = (1.49MV * r_in_meters) * sqrt(kg / m^2)
from thread
https://highvoltageforum.net/index.php?topic=2017.msg15087#msg15087
Did I make a mistake in this corrected formula?  Or do I have the sphere radius wrong for your test?

[haversin]

Did I make a mistake in this corrected formula?  Or do I have the sphere radius wrong for your test?

Your calculations are correct, I used r = .1m and you used r = .125m so my numbers are four fifths of yours. Looking at your numbers maybe be I should be using r = .125m ! The width of my terminal is 8 inches so I used r = .1m. The shape of my terminal is far from a sphere and has a flat section on top. Your lifting disk voltage equation assumes E = V/r which is only true for an isolated sphere. I'm thinking of trying to model my terminal and ground plane in FEMM to get a more accurate E field and force calculation. I haven't used FEMM for electrostatics but supposedly it has the capability. 

[davekni]

Your calculations are correct, I used r = .1m and you used r = .125m so my numbers are four fifths of yours. Looking at your numbers maybe be I should be using r = .125m ! The width of my terminal is 8 inches so I used r = .1m. The shape of my terminal is far from a sphere and has a flat section on top. Your lifting disk voltage equation assumes E = V/r which is only true for an isolated sphere. I'm thinking of trying to model my terminal and ground plane in FEMM to get a more accurate E field and force calculation. I haven't used FEMM for electrostatics but supposedly it has the capability.

I'd guessed that you were putting the disks on top of the 10" sphere you were using for your spark tests.  Any reason not to do that?  Would avoid the need for FEMM modeling.  If you use the flat-top shape for disk lifting, it definitely requires modeling.  The field will be lower on the flat portion.  I don't know of any way to determine how much lower without a model.

[haversin]

I'd guessed that you were putting the disks on top of the 10" sphere you were using for your spark tests.  Any reason not to do that?

I was worried about field distortion at the top of the sphere caused by the VDG terminal and prime conductor. I was less worried about this distortion for the sphere gap because the second sphere is directly behind the first sphere and assumed some shielding is provided by the first sphere. This is pure guess work on my part.

[davekni]

I was worried about field distortion at the top of the sphere caused by the VDG terminal and prime conductor. I was less worried about this distortion for the sphere gap because the second sphere is directly behind the first sphere and assumed some shielding is provided by the first sphere. This is pure guess work on my part.

How about placing the sphere on top of your VDG for this test?
If you were leaving the prime conductor and sphere in place to simultaneously run spark testing, I'd expect those to distort the field around the VDG top as much as the VDG top and prime conductor would distort the field around the sphere.

[haversin]

Following david's suggestion, I placed the 10 inch diameter sphere on top of the VDG. An aluminum soda can was used to provide some separation between the VDG terminal and sphere.  The total mass of the 5 cm diameter foil circle and thin wood piece was now up to 5.94 grams to slowly slide off the sphere. This gives a voltage of 324 kV which is real close to the 310 kV from the sphere gap. The setup is shown below and the small wood piece can be seen on top of the sphere. The foil circle is there but can't be seen. I'm going to try the outside field mill voltage measurement of the VDG next.

[davekni]

Nice result!  You have reasonably-close agreement between two independent measurements.  Gives reasonable confidence in your VDG's voltage capability.
You could repeat this lifting experiment outside too.  Nearby objects will have a small effect on field strength, increasing field if not charged and decreasing field if charged by VDG ions.
Centering is important.  When using this technique, I'd note which direction the disk slid, then place it slightly farther from that direction for the next test, adjusting placement until slide direction was reasonably random.  Of course, centering error will result in an optimistic estimate of voltage, as sliding occurs before quite all of gravitational force is negated by electrostatic force.

[haversin]

  It's been hot and windy so I decided to do some field mill voltage measurements in the garage. The field mill reading was jumping around between about 3.6 and 4.3 volts and got to this level within a second of turning on the VDG. Not the steady increase that I remembered. It occurred to me that the measurement I was remembering was with a much smaller VDG and a slightly different field mill.
The field mill I'm using now is setup for fair weather atmospheric field measurements and is very sensitive and has a very long power and signal cable. The first thing I tried was shielding this cable. No luck the reading was still jumping around. Next I tried shielding the AC power and house ground cable going to the VDG with a separate Earth ground. No difference. There is a tenth of second time constant low pass filter on the field mill signal output and the current output of the VDG is very steady. This made No sense! I started to run out of ideas and began randomly trying stuff. This went on for a few days until I put a 25 cm stainless sphere on top of the VDG terminal. Suddenly the readings steadied and only moved around by about a tenth of a volt every few seconds. The only theory I have for this is that the extra capacitance of the sphere helps smooth out voltage variations on the terminal caused by random corona/streamer breakout.
  With the 25 cm sphere on top of the VDG terminal the field mill was placed so it read about -5 volts with the VDG running. From linearity tests I done the field mill is fairly linear in the range of -8 to +8 volts. The zero field reading is -0.1 volts (caused by a small misalignment in the synchronous rectifier) The actual reading with the VDG running was -4.9 volts and correcting for the zero field reading gives -4.8 volts (The VDG charges negative).
 To compensate for the ion cloud, objects in the garage charging and the VDG column charging I quickly grounded the terminal after turning off the VDG and noted the field mill reading.
I was expecting a value a little more negative that -0.1 zero field value, but the value was close to zero! A value of zero means a net positive field and that makes no sense so no attempt was made to make this compensation.
  Field mill calibration was done by first making sure the VDG and field mill did not move and anything else in the garage did not change. I tried to keep my body as far away for the VGD and field mill as possible during both the calibration and VDG running readings. One terminal from a 15kV by 30ma NST was half wave rectified to a 225 pF HV capacitor. The voltage on this capacitor was controlled by a variac feeding the NST. The variac was adjusted until the current thru a 1 Gohm HV resistor read 10 uA indicating a voltage of 10 kV. This voltage was then applied to the VDG terminal via a 71 Mohm HV resistor in series with an electrode see photo below the resistor is inside the white tubing and can't be seen. It was found that the field mill was reading about -0.2 volts before even touching the electrode to the VDG terminal. To compensate for this the electrode was touched to the terminal then quickly taken off the terminal and placed on the ground and the variac was switched off. The field mill reading was then taken as quickly as possible. The field mill reading was dropping about 0.02 volts every few seconds so I have good confidence in this reading. After doing this calibration about five times all the values were close to -0.27 volts and correcting for the zero field value gives -.17 volts giving a calibration factor of 10kV/.17V = 58.8 kV/V. Applying this to the running VDG reading of -4.8 volts gives a voltage reading of  -282 kV. A little lower than all the other measurements.

Sphere gap: -310 kV
lifting disk:  -324 kV
field mill:     -282 kV

[Uspring]

Congratulations on your successful field mill measurement.
The strongly varying field values without the additional sphere are strange. I would have expected corona discharges to be more continuous. Perhaps the fact, that the field around the top exceeds the breakdown limit only slightly, leads to a scarcity of avalanches. Anyway, you found a fix for that and the reasoning behind it sounds plausible.
It is my impression, that corona discharges limit the voltage of your VDG. In your measurements you have used differing configurations of spheres and conductors and the differing fields generated by them and the corresponding corona currents might account for the differences in your voltages.

I've tried to model the field of your original (oblate) top electrode by adding a quadrupole term to the potential. I haven't been able to reproduce an equipotential surface looking quite like your top, but my general impression was, that the field on the flat top is quite a bit smaller than around the more curved sides. That would explain the low voltages in your first disk lifting measurement.
On the other hand, the strong field at the sides probably produce a lot of coroma current, limiting top voltage. A bigger top load might increase voltage significantly.

[haversin]

It is my impression, that corona discharges limit the voltage of your VDG.

Yes, I think so too.

In your measurements you have used differing configurations of spheres and conductors and the differing fields generated by them and the corresponding corona currents might account for the differences in your voltages.

Yes possibly. I should really take like five or ten measurements with each method to see what the means and standard deviations look like.

I've tried to model the field of your original (oblate) top electrode by adding a quadrupole term to the potential. I haven't been able to reproduce an equipotential surface looking quite like your top, but my general impression was, that the field on the flat top is quite a bit smaller than around the more curved sides. That would explain the low voltages in your first disk lifting measurement.

I'm working on a FEMM model of the terminal and some very preliminary results show a lower field on the flat top also. I think this could indeed explain the low voltages for the first lifting disk measurement.

P.S. I found the source of the positive field reading when I grounded the terminal after turning off the VDG. I feel kind of stupid now. For my self excited negative charging VDG, the negative roller is on top inside the terminal and the positive roller is on the bottom and not shielded. After turning off the VDG and grounding the terminal it shielded the negative roller but left the positive roller still unshielded giving a net positive field. I confirmed this by putting some grounded shielding around the bottom roller and the positive field disappeared.