Capacitance (intended and parasitic)

[fog37]

TL;DR Summary

understanding the concept of capacitance C

Hello,

My understanding is that capacitance $C$ is an electrical property that is possessed by devices called capacitors whose basic construction involves two conductors separated by a dielectric. The two conductors involved in a capacitor don't need to be identical either in shape or size (see cylindrical or spherical capacitor where the conductors in the pair are not identical). To define capacitance, we assume that the same amount of charge $Q$ is on both conductors, regardless of their shape and size. Connecting the conductors to a voltage source $\Delta_V$, the system's capacitance becomes $$C=\frac {Q}{\Delta V}$$

In general, all capacitors are electrically neutral since $|Q_1|\neq |Q_2|$.

Question 1: How would we (and could we?) define the capacitance $C$ of a system composed of two differently shaped and different in size conductors carrying different amounts of charge, say $Q_1 \neq Q_2$ of opposite sign, separated by air? The two metal objects are not connected to anything, just separated and carrying electric charge. Is there a way to measure the capacitance of a system like that?

What if the two charges $Q_1$ and $Q_2$ were different in magnitude but equal in sign? Would it be even more challenging to define/measure the capacitance?

Question 2: every electrical component exhibits some unintended parasitic capacitance between its leads (air between them). How is that capacitance measured? Is the capacitance measured indirectly from measuring the capacitive impedance $X_C$ of the electric component?

Thanks!

 

[alan123hk]

Q1 :
Can we think of it as two capacitors connected together, one with mutual capacitance and the other with self-capacitance?

Q2 :
The capacitance of the leads is usually much smaller than the nominal capacitance at low frequencies. However, at high frequencies, capacitors need to be modeled by lumped passive components, including R, C, and L.

 

[Baluncore]

Capacitance is a function of the geometrical distribution of space or dielectric in the volume between two conductive electrode surfaces. Capacitance is NOT the conductive electrodes or terminals. A voltage difference between the electrodes results in an equal and opposite charge imbalance between the electrodes. C = Q / V

You might think of the conduction electrons flooding the negative electrode, a charge being pushed out onto the surface, while remaining tied to the conductor. The positive electrode is reduced by the same charge, the electrons being pulled into the electrode and removed along the wire, via the external circuit, for delivery to the negative electrode.

Electrons in the dielectric are tied to that insulator and are not free to flow. Those electrons in the dielectric are displaced electrostatically, all in the same direction, storing energy in the “elasticity” of the dielectric, making a capacitor. E = ½·C·V²

Capacitance is measured by changing the voltage on the capacitor, while monitoring the current, and so change of charge. C = Q / V; Q = I·t

 

[fog37]

Thank you Baluncore and alan123hk.

I am still perplexed in the situation of two isolated arbitrarily shaped conductors with different charges $Q_1`$ and $Q2$. Each conductor is an equipotential body of potential $V_1$ and $V_2$ relative to, say, infinity. So we have a potential difference $\Delta V= V_1 - V_2$ and a system of two conductors hosting charge that cannot jump due to the dielectric air in between them. There is also an electric field between the two conductors so energy is stored in the system. It seems to have all the requisites of a capacitor but I am not sure how its capacitance would be found...

alan123hk mentions two capacitors (in series or in parallel?) connected together...Why? I am mentally working on that hint...

 

[Baluncore]

I am still perplexed in the situation of two isolated arbitrarily shaped conductors with different charges Q1‘Q1‘Q_1` and Q2Q2Q2.

That is impossible. The charge is stored as electrostatic stress between the plates, not on them. That stress is a force that must have an equal and opposite reaction. The current into one side of a capacitor must be equal to that removed from the other.

 

[alan123hk]

alan123hk mentions two capacitors (in series or in parallel?) connected together...Why? I am mentally working on that hint...

I'm still not sure if my idea is correct, maybe I am wrong.

Please refer to the wiki in the link : - https://en.wikipedia.org/wiki/Capacitance#Self_capacitance

Quote "However, for an isolated conductor, there also exists a property called self capacitance, which is the amount of electric charge that must be added to an isolated conductor to raise its electric potential by one unit (i.e. one volt, in most measurement systems).[2] The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, with the conductor centered inside this sphere. "

About whether it is series or parallel, I think it depends...

 

[alan123hk]

Cm and Cs1 are connected in series between points A and B.
Cm is in parallel with two capacitors Cs1 and Cs2 connected in series.

Is this at least approximately reasonable?
Or is there a fundamental error?

Besides, I agree that unless intentionally made in a laboratory, unbalanced charges of a capacitor is usually not possible.

I am still perplexed in the situation of two isolated arbitrarily shaped conductors with different charges Q1‘Q1‘Q_1` and Q2

In practice, as an electronic circuit designer, I would not be too perplexing in this case. I don't rule out that, although the chance is very small, this may happen for unknown reasons, but the unbalanced electric charges of a capacitor should be at very low level. I am not worried, as long as it does not have the danger of electric shock and does not affect the operation of the circuit. Sooner or later the excess charge on the conductor will be neutralized. The capacitance of the capacitor is still the nominal value, the excess charges in one of the two conductors of the capacitor should be considered to be related to parasitic capacitance to nearby objects or self-capacitance.

 

[anorlunda]

am still perplexed in the situation of two isolated arbitrarily shaped conductors with different charges Q1‘Q1‘Q_1` and Q2Q2Q2.

To truly understand this case, make it as simple as possible. Reduce the size of those two isolated conductors to zero, and reduce the number of charges to the minimum. What remains is a single isolated charge (say an electron) on one side and two electrons on the other side. Now you can write the equations for the Coulomb force at any point in 3D space considering all three electrons. From the forces, comes the electric field lines and the equipotential lines.

Any case with nonzero size conductors, and more than 3 electrons follows the same principles.

Drawing equivalent circuits for the 3 electron case makes it so complex that it can't be understood. Reject everything not necessary and what remains is simple enough to solve.

 

[fog37]

Thanks everyone. I will explore these concepts of mutual capacitance and self-capacitance. I think inductance has also these two versions, mutual and self-.

This is the simple situation I envisioned: two conductors with charge on them.

There is clearly an electric field and a net Coulomb force between the two conductors carrying different charges and I wondered if we can assign a capacitance to this system in virtue of the fact that it stores energy in the electric field, like a capacitor does.

 

[anorlunda]

Thanks everyone. I will explore these concepts of mutual capacitance and self-capacitance. I think inductance has also these two versions, mutual and self-.

This is the simple situation I envisioned: two conductors with charge on them.

View attachment 258991

There is clearly an electric field and a net Coulomb force between the two conductors carrying different charges and I wondered if we can assign a capacitance to this system in virtue of the fact that it stores energy in the electric field, like a capacitor does.

Yes you can. You can use the classical approach of calculating the work required to bring those charges in from infinity to the positions in your diagram. (I personally hate that approach, but it gives the correct answer.)

If you reduce it to just two point charges, you can find solutions to that energy calculation already done in most textbooks on electrostatics.

 

[fog37]

Thanks!

It appears that self and mutual capacitance are topics well developed in capacitive sensing technologies...

Self-capacitance for a single isolated conductor is simply the total charge on the conductor's surface divided by the equipotential $V$: $$C=\frac {Q}{V_(surface)}$$ at the surface. Example: for a sphere of radius $R$, the potential at the surface $V= \frac {Q} {4\pi \epsilon_0 R}$, considering the zero reference potential at $R= \infty$, and the self-capacitance is $4\pi \epsilon_0 R$.

So, regardless of its shape, the self capacitance of a conductor seems simple to calculate once we know the expression for the surface potential, which is the same at every point on the conductor...

Now, if we take two isolated conductors close to each other, their self-capacitance may change since their surface potential will change, the electric field will change, etc.

I guess the system now has a mutual capacitance which is different from the original self capacitance of either conductors...

 

[alan123hk]

I have been thinking about your questions and trying to find more information on the internet.

Now, if we take two isolated conductors close to each other, their self-capacitance may change since their surface potential will change, the electric field will change, etc.

Yes, the self-capacitance should change.

I guess the system now has a mutual capacitance which is different from the original self capacitance of either conductors...

Yes, mutual capacitance and self-capacitance are two kinds of variables with different meanings when analyzing the capacitance of a multi-conductor system.

I think it is difficult to perform a complete analysis of the electrostatic capacitance of the multi-conductor system. Anyway, after more study, I believe that there are two methods to model the capacitances in the system.

Let's use a simplest example, there are only two arbitrary shape conductors A1, A2, and a common reference ground.

Method 1 :
We may model the system with a lumped model with discrete capacitors like the one shown in the image I posted.

Method 2 :
The other method is to represent it as a system of equations (capacitance matrix) as follows : -

Q1 = C11*V1+C12*V2 (C11 is the self-capacitance of A1)
Q2 = C21*V1+C22*V2 (C22 is the self-capacitance of A2)

Q1 = charge in A1
Q2 = charge in A2
V1 = potential of A1 relative the reference ground
V2 = potential of A2 relative the reference ground

Note that the capacitances C11,C12... in the capacitance matrix are different from the capacitances in the lumped model.

Thus the most difficult task is finding out the values of C11, C12..in the capacitance matrix or the capacitance values in lumped model. For a multi-conductor system with arbitrary shape conductors, simulation is an effective method to solve this problem.

 

[fog37]

Thank you alan123hk. I am processing your answer...

By the way, I found this explanation online:

Capacitance has nothing to do with actual charge stored on plates...Any two conductor plates facing each other will have capacitance between them, whether charged or not...OK

...In case different charges Q1 and Q2 are given to two parallel plates, the inside (facing each other) surfaces get a charge by redistribution, equal to (Q1-Q2)/2 in magnitude and opposite in sign. This is charge which decides the field inside the dielectric, or the space in between...

What do you think? It sounds correct...

 

[zoki85]

By the way, I found this explanation online:

Capacitance has nothing to do with actual charge stored on plates...Any two conductor plates facing each other will have capacitance between them, whether charged or not...OK

Capacitance has to do with change or transfer of charges between insulated conductors.
That's how it is defined:
C=ΔQ/ΔV

 

[alan123hk]

Capacitance has nothing to do with actual charge stored on plates...

I am unable to comment on these words.
It's like saying that conductance has nothing to do with current.
For capacitors, VC = Q and for conductors, VG = I
In any case, the mathematical relations between them are clearly described by the formulas.

By the way, the link below clearly illustrates the relation between the mutual capacitance matrix and the Maxwell capacitance matrix. The mutual capacitance matrix represents each capacitance in the lumped model with discrete capacitors.

https://www.comsol.com/blogs/how-to-calculate-a-capacitance-matrix-in-comsol-multiphysics/

 

[alan123hk]

Summary:: understanding the concept of capacitance C
How would we (and could we?) define the capacitance CCC of a system composed of two differently shaped and different in size conductors carrying different amounts of charge, say Q1≠Q2Q1≠Q2Q_1 \neq Q_2 of opposite sign, separated by air? The two metal objects are not connected to anything, just separated and carrying electric charge. Is there a way to measure the capacitance of a system like that?
What if the two charges Q1Q1Q_1 and Q2Q2Q_2 were different in magnitude but equal in sign? Would it be even more challenging to define/measure the capacitance?

Back to the questions on #1

For any charge Q1 in conductors 1, and any charge Q2 in conductor 2, regardless their amounts and signs, we can define the capacitance in two ways, either by Maxwell capacitance matrix or by mutual capacitance matrix.

We can find out the actual charges stored in Cm12, Cm11and Cm22 in the lumped equivalent capacitor circuit.

Step 1
Find the values of C11, C22 and C12 by measurement, analysis or simulation

Step 2
Find the values of Cm12, Cm11 and Cm22 using the relations below
Cm11 = C11 + C12
Cm22 = C22 + C21
Cm12 = -C12
Cm21 = -C21

Step 3
Use the elastance matrix to calculate the voltages of V1 and V2
V1=P11*Q1+P12*Q2
V2=P21*Q1+P22*Q2

Step 4
Find the stored charges in the mutual capacitors
Q12 = (V1-V2)*Cm12
Q11 = V1*Cm11
Q22 = V2*Cm22

I believe we can expect Q1=Q12+Q11 and Q2=Q21+Q22, where Q1 and Q2 are the charges we intentionally add to the conductor 1 and conductor 2, respectively.

 

[Baluncore]

There seems little point considering such an undetermined hypothetical situation.
Knowing the different charges on the two conductors tells us there must be additional unspecified circuit present. As a minimum we must add a ground reference, with capacitance to both conductors. That gives us three unknown capacitors, then we need is to know the voltage on the two conductors relative to ground, so we can solve for the capacitor values.

 

[alan123hk]

Knowing the different charges on the two conductors tells us there must be additional unspecified circuit present. As a minimum we must add a ground reference, with capacitance to both conductors. That gives us three unknown capacitors, then we need is to know the voltage on the two conductors relative to ground, so we can solve for the capacitor values

I agree with that.

There must be a common reference. Therefore, more specifically, the voltages V1, V2 in the equations are relative to the ground. Cm11 represents the mutual capacitance between conductor 1 and the ground, Cm22 represents the mutual capacitance between conductor 2 and the ground..etc

I do think that this is a rather complex calculation method, and it may be mainly used in electromagnetic simulation software.