# Variable dielectric capacitors

1. Oct 10, 2011

### JohnielWhite

Good day All:
How can I prove that a linear relationship exist between the displacement (d) and the capacitance(C) of a Variable dielectric capacitor?

I know that the equation for a capacitor's capacitance is:
C= εA/d

2. Oct 11, 2011

### MATLABdude

I'm not aware of variable dielectric capacitors--are you sure you don't mean a variable capacitor in which the distance between plates is adjusted?

If so, you may have a problem: the capacitance is inversely proportional to the distance between the plates (as per your formula)!

EDIT: Variable capacitor article at Wikipedia:
http://en.wikipedia.org/wiki/Variable_capacitor

3. Oct 11, 2011

### JohnielWhite

@ MATLABdude Thanks for your response but I read in a book "Industrial Control Handbook" that there are four fundamentals of variable capacitance displacement transducers.
i. Capacitor basics
ii. Variable Seperation
iii.Variable area
iV. Variable Dielectric
From the formula of a parallel plate capacitance:
C=EA/d
It is observed that each term of the formula can be varied to form a displacement transducer.
That is the seperation, area and dielectric. So what I meant by variable dielectric was
when the dielectric moves between the plates causing the permittivity to alter. So from such behavior how can I prove that a linear relationship exist between the capacitance and displacement.

4. Oct 11, 2011

### cmb

Can you provide further clarification for you question? You seem to be asking 'how do I prove a distance dependence by changing the dielectric'?

That sounds like it would be similar to asking how you count how many apples you've got, once you know how many bananas you don't? Or how to determine the length of a piece of string by knowing what it is made of.

5. Oct 11, 2011

### jim hardy

""So what I meant by variable dielectric was
when the dielectric moves between the plates causing the permittivity to alter. ""

i'm trying to think of a real world example. Hewlett Packard used something like that for a level sensor in some surveyor's instruments around 1970.... sorta an electronic version of the bubble you see in a carpenter's level.

anyhow wouldn't you just differentiate the equation for capacitance wrt d ?
if neither area nor epsilon is a f(d) it seems to me you're there...

6. Oct 11, 2011

### JohnielWhite

7. Jul 23, 2012

### michelle15g

I just tried to solve this problem. Instead of dividing by d straight away, I integrated d/epsilon and divided the answer by that. I think that's a safe way to handle it. But as you can see, I'm searching the net for confirmation :)
Hope this idea works for you.
Michelle

8. Jul 23, 2012

### vk6kro

If you start off with a simple example, you can derive a relationship from it.

Take a pair of rectangular metal plates, 1 cm apart measuring 20 cm by 10 cm, initially with an air dielectric between them.

The formula for the capacitance of this is 0.0885 * dielectric constant * Area (sq cm) / spacing (cm)
So, for this example C (in pF) = 0.0885 * 1 * (20 * 10) / 1 = 17.7 pF.

Now gradually introduce a glass dielectric with a dielectric constant of 5, 10 cm wide and 1 cm thick, into the area between the plates from one of the narrow ends.

There are now two capacitors in parallel. The air dielectric one is reducing in area and capacitance, and the glass dielectric one is increasing in area and capacitance.

For example, when there is 5 cm of glass introduced, there will be 13.275 pF of air dielectric capacitor and 22.125 pF of glass dielectric capacitor giving a total capacitance of 35.4 pF

ie C = 0.0885 * 1 * (15 * 10) / 1 + 0.0885 * 5 * (5 * 10) / 1 = 35.4 pF

So, you get a situation like this Excel chart:
http://dl.dropbox.com/u/4222062/dielectric%20tuned%20capacitor.PNG [Broken]

The yellow trace is the total capacitance and the black and purple ones are the air and glass dielectric sections respectively.

It looks linear with an initial offset of 17.7 pF and rising to 88.5 pF but you can derive a formula for this if you like and this should establish whether the relationship is linear or not.

Last edited by a moderator: May 6, 2017