Faraday's Bucket. Faraday's Ice Pail .. Check Math Please  - Page 1
Forum section: electronics / projects

author: LaserSteve
author_rank: Super Contributor
timestamp: March 26, 2021, 04:15:29 pm
content: At work we are working on charging non-woven polymer fiber sheets as Electrets. You put the sheet in an oven, between two highly charged (20kV) plates and heat it. Done correctly, and at a temperature close to melting, the material becomes an Electret and retains its charge. This has some very unique uses in filtration.The goal is to quantify that charge and see how well it is retained. A stainless steel "prep pan" or Bain Marie (see pic) is mounted on an ultra-high resistance aluminosilicate glass tube used as an insulator. This is inside a Faraday Cage. A DC motor lowers the sample into the pan slowly, while the charge is measured by the electrometer. They plot the charge induced vs distance and the final charge when the material is suspended inside the Bucket.An ancient EE set them up with Faraday's Ice Pail. I had to really rework it for better shielding, grounding, noise immunity, low triboelectric cable, and a better insulator. I'm working on setting up a Kelvin Probe, Aka Vibrating Probe Field Meter, but they love the Pail.The grad students are obtaining very inconsistent math results with the Ice Pail. I've been accused of creating a negative capacitance, so I'm wondering about the math they are using.So could someone who is at home with the wordsMaxwellandCurltake a look at the attached PDF.Chemical Engineers receive a semester long basic RCL circuits course their junior year on their undergrad degree. So while they have the math skills, they may not have the theory.Any help would be appreciated!Steve

author: Nominal Animal
author_rank: Super Contributor
timestamp: March 26, 2021, 08:50:36 pm
content: The voltage \$V\$ between the plates of a parallel plate capacitor with charge \$Q\$, area \$A\$, separation \$d\$ and dielectric permittivity \$\epsilon\$ of the separating material or vacuum (\$\epsilon = \epsilon_r \epsilon_0\$ where \$\epsilon_r\$ is relative permittivity), is$$V = \frac{Q d}{\epsilon A} \tag{1a}\label{NA1a}$$Capacitance \$C\$ is defined as$$C = \frac{Q}{V}$$so we can describe the capacitance of the aforementioned parallel plate capacitor as$$C = \frac{\epsilon A}{d} \tag{1b}\label{NA1b}$$It is useful to realize that the totalenergy\$W\$ stored in the capacitor is$$W = \frac{1}{2} C V^2 = \frac{1}{2} V Q = \frac{Q^2}{2 C}\tag{1c}\label{NA1c}$$which hints at a different method of measuring the charge stored in the electret material, if one can measure both the voltage and the current or power (via a thermally monitored insulated resistance) precisely.Mathematically, it does not matter if one of the plates is grounded or not, because the voltage \$V\$ is measured between the plates. In the real world, grounding establishes a hopefully static reference frame, so that a measurement can be made; essentially, that instead of charge flowing into the capacitor from outside the capacitor when something moves out/comes into contact with the test setup, such excess/noise is conducted safely into ground.Multiple capacitors in parallel have a capacitance the sum of their individual capacitances, \$C = \sum C_i\$. Here, the capacitance is a sum of the plate capacitance \$C_p\$ and the measurement device capacitance \$C_m\$,$$C = C_p + C_m \tag{2}\label{NA2}$$This is also logically obvious, since using two identical plate capacitors in parallel is equivalent to a single plate capacitor with twice the plate area.The measurement circuit can be modeled as an RC circuit, where \$R\$ is the total resistance in the measurement leads and the leakage through air/support structure et cetera. (The resistance can also be separated into multiple parallel parts, each with much higher resistance than the total; for example the leakage can be measured by taking a measurement, detaching the measurement device from the circuit, wait a specific interval, and then reconnecting the measurement device and taking another measurement. The voltage drop due to the leakage during that interval only contains the leakage resistance, since the measurement device was not connected to the circuit.)The time-dependent behaviour of an RC circuit (where the circuit stays unchanged; including the properties of both the resistors and capacitors!) is described as$$C \frac{d V(t)}{d t} + \frac{V(t)}{R} = 0 \tag{3a}\label{NA3a}$$which is a linear differential equation with a well-known solution,$$V(t) = V_0 e^{-\frac{t}{RC}} \tag{3b}\label{NA3b}$$where \$V_0\$ has no deeper physical meaning; it is just the voltage at time \$t = 0\$.There is no reason to add complexity to the situation by adding a variable describing the "ground potential" to the mix, because the entire reason for "grounding" the test setup is to get a stable reference for the measurements. Just don't do it.If the measurement curve does not follow the exponential decay in \$\eqref{NA3b}\$, it means the RC circuit does not model the situation well. A particular case is when the properties of the resistors or the capacitor change during the measurement. Slowly inserting the electret material to the Ice Pail does affect the circuit: the charge in the Ice Pail capacitor plate is increased when the electret material comes to contact with the ice pail; but also the electret material itself and the Ice Pail is another capacitor, and the electret material and the grounded containment vessel another. By keeping the electret material static with respect to the Ice Pail the measured voltage should follow the exponential decay, and any bumps or distortions are indications that the circuit model is incorrect: there are additional sources or sinks for charge to flow to/from. (As LaserSteve mentioned, triboelectric effects are one.)

author: Nominal Animal
author_rank: Super Contributor
timestamp: March 27, 2021, 06:33:17 am
content: (Note, I tried to write the above post in a form that helps others follow your question without opening the PDF, LaserSteve. I am not really trying to answer the stated question, because I personally use CASes like Maxima, Maple, and Sagemath to do the low-level math work for me; although, I do believe your notes are correct, except for the unnecessary inclusion of the ground potential \$V_g\$ that changes nothing. I am trying to make it easier for others to consider the question at hand, and hopefully offer their notes too.)

author: LaserSteve
author_rank: Super Contributor
timestamp: March 27, 2021, 05:02:07 pm
content: Thanks, That is the Prof's notes...I'm going over this now.Third party review always helps and is appreciated.Steve

author: Nominal Animal
author_rank: Super Contributor
timestamp: March 28, 2021, 01:09:07 am
content: Quote from: LaserSteve on March 27, 2021, 05:02:07 pmNever be afraid to point out inconsistencies or errors to professors, just remember to always explainwhy. That's how science, and peer review, works.(Because you are working on their experiment setup, it is theirjobto consider you a peer in this case, as they lack your domain of expertise. All professors I know that have worked on a real-world experiment ("real" or "educational"), have valued the engineers that help build and maintain the experiments. Good lab engineers are valued, if the department is at all sane. But do not confuse frustration with insults: many scientists aren't that socially adept, so their frustration may come across wrong in many social situations. It is best to always ask, directly, with explanation/basis on as to why the question needs to be answered.)In this case, I understand perfectly well why the prof was thinking about the ground potential here; it just isn't relevant in the math.Doing a test where one keeps the electret material unmoving wrt. the ice pail, would tell a lot.If the voltage between the ice pail and its surrounding groundincreases, it means either the pail is acquiring additional charge from whatever it is connected to, or the ground surrounding the pail is losing charge to somewhere else.Consider a situation where the surrounding ground is actually isolated, and has an initial charge with the same sign as the electret material. During measurements, something causes some of the charge in the surrounding ground to leak somewhere else. That will seem like the charge in the electret material increasing; like the setup had a negative capacitance.There are many real-world reasons why this can occur. One of the most annoying ones is local fluctuation in the "ground" potential; say from large local switchmode supplies, motors, et cetera, when the resistance to true ground potential is non-neglible (say, a thinnish cable, some oxidation in the ground connectors, et cetera).Even static charges in the air (dust) can make a difference here. I'm pretty sure the professor who wrote the notes was thinking the same thing.