% zak smith
% ece220 capacitor thing
% this is the full writeup

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% begin document
\vskip 2in
\centerline{\bigbig Investigation into the Construction}
\centerline{\bigbig of Parallel Plate Capacitors}
\vskip .2in
\centerline{\big ECE 220, Prof. Booske, Dec. 20, 1995}
\centerline{\big Zak Smith}
\vskip .2in
\centerline{\big Group Members}
\centerline{Stefan Gerhardt}
\centerline{Joel Kellogg}
\centerline{Ben Jauquet}
\vskip 1in

\section { Abstract }

We designed and built two capacitors based on the parallel
plate model: one consisting of two parallel disks separated
by paper; and another consisting of many dielectric
and conducting layers.
We were able to achieve capacitances
of 1.21nF and 61.7nF respectively.  Modifying our
dielectric by adding ``Inductor Oil'' further increased
each roughly by a factor of 2 to 2.15nF and 121.8nF.
Our results confirmed the predicted $C = \E A / d$.

\section { Theory of Parallel Plate Capacitors }

Using Gauss's Law, it is easy to derive an expression for the
capacitance per unit area for two infinite conducting parallel
plates, separated by a dielectric, 
$${C \over A} = {\E \over d} \hskip .3in \hbox{from} \hskip .3in 
C = { { \E A } \over d }$$
where $d$ is
the distance between the two plates and $\E$ is the dielectric constant
of the material.  Since it is quite difficult to get {\it infinite\ }
plates in real life, it is useful to notice what the difference is between
a set of infinite plates and a set of finite plates.  In the case of
two infinite plates, the electric field flux lines are normal to each
surface, going from one to the other.  In the case of the finite plates,
the flux lines near the edges will be curved outwards; all the flux will
not pass through the area under the plate.  These are called the {\it fringing
fields\ } and this is the important difference between the infinite case
and the finite case.  

If we consider a finite portion of the infinite plates, we get the
capacitance $ C = { \E A \over d }$, where $A$ is the surface area
of the portion of the plates in our portion.  This expression will
be a good approximation for the finite plate capacitor if 
there is
little fringing flux, that is --- when both the length and width 
dimensions of the plates are much greater than the separation
distance, $d$.  Our TA Ben was nice enough to graph
the percent error vs. the $l/d$ ratio for a two--dimensional
case, where $l$ is the length of a plate.  It indicates
that if $l = 10 d$, the percent error will be $50\%$, and
if $l = 100 d$, the percent error will be only $10\%$.  When
$l > 10^3 d$, the error is virtually zero.


\section { First Design }

There were several issues constraining our first design.  First,
we wanted to minimize the fringing fields; second, we obviously
wanted to be able to construct it without too much trouble; and finally,
we wanted to maximize the capacitance.

In order to minimize the fringing fields, several conditions
must be met. 
The first is that the separation distance must be small relative to
a length dimension of the plate.  This is easy to satisfy by
using a very thin dielectric layer and a plate on the order
of several centimeters long.   
The second condition is that the amount of edge length
must be minimized for the given area.  This was accomplished
by using circular plates.

The last two issues deal with charge distribution on the plates.  In
the ideal infinite plate case, the charge distribution is uniform.  
Charges tend to be more concentrated on sharp edges of conductors,
so in order to resemble the infinite plate case by reducing
fringing fields, we want to reduce the
``sharpness'' of the edges.  On the other hand, since
electric flux lines are always normal to a conducting surface, we wanted
to minimize the edge surface area so as to minimize the fringing fields.
It is hard to satisfy both of these last two conditions at once, because
a thin plate tends to have sharper edges.

After searching through tables of dielectric constants of materials
that we could use, we decided that since all the possible safe dielectrics
we could get our hands on had $\E_r$ in the $1$ to $4$
range, it would probably be more effective to choose a thin material
over a thicker one with a higher $\E_r$.  This is confirmed by
$C = \E A / d$.  We planned to experiment with common substances
such as paper, high voltage paper from Plasma Physics, plastic,
 and Reynolds Wrap.

\section { Construction of First Design }

After discussion of these issues, we decided to use two round plates of
copper, with a layer of some thin dielectric between them.  We would
then put a phenolic plate on either side of the copper plates and
compress the assembly with C--clamps.  Finally, we decided to solder
two leads coming off the disks on the edge to connect to the rest
of the circuit.

We in part chose these materials because Joel and Stefan work
at Plasma Physics in Chamberlin Hall, and had access to
scrap material and tools there.

The dimensions for our plates were determined by the size of
circular drill bit we had access to.  Our copper plates had
radius $4.8$cm.  We tried to round off the
edges of the plates with a file to reduce fringing fields
and make them less likely to cut someone.  

\section { Measurement }

Originally, we proposed a method of measuring the capacitance
by using our capacitor in a RC circuit.  We would put our capacitor
in series with a resistor and voltage source, and then
measure the time it takes for the voltage across the capacitor
to rise to $63\%$ of the power supply voltage.  Then it
is easy to calculate $C = \tau / R$.  This equipment is available
in the ECE270 Laboratory.

The Plasma Physics shops in Chamberlin had, among lots
of other interesting hardware, a Capacitance Meter.  We checked
the values it reported against some labeled capacitors to make 
sure it was accurate and then used this for our subsequent measurements.

\section { Results of First Design }

First, we tried regular copier paper as the dielectric.  With a micrometer from
the lab, we measured the thickness of the paper we were using to
be $0.08$mm when compressed.
We sandwiched the paper between
the two copper plates, surrounded those by the phenolic blocks, and then
tightened the C--clamp down on the assembly.  The capacitance meter reported
$1.21$nF.

The CRC handbook reported the $\E_r$ for paper to be somewhere between
$2$ and $4$.  For the sake of calculations, let's use $2$, which yields
a theoretical capacitance 
$$C = { \E_o \E_r A \over d } = { (8.854\x10^{-12}) (2) (\pi 0.048^2) \over 
           0.08\x10^{-3} } = 1.6\hbox{nF}$$

Our actual and theoretical values differ by $27\%$, but that isn't 
really very helpful because we had no way to measure the $\E_r$ of paper, 
besides knowing that it was probably between $2$ and $4$.

We also tried other other dielectrics, but found that either they
were too thin and caused a short when compressed (tissue and Reynolds Wrap),
or that they were too thick and yielded
capacitances much less than $1$nF (high voltage paper and plastic bag material).

\section { Second Design }

We weren't satisfied with a measely $1.21$nF, we wanted more!

We had already searched for and tried all the possible sheet--type
dielectrics that we could find in the Plasma Lab in our original
model, so we decided to increase the capacitance by using
several layers of larger plates, separated by dielectric layers.   
Each alternate conducting layer goes to a different lead.  So
``even'' layers go to one lead and ``odd'' layers go to the
other.
This whole unit can be viewed as 7 capacitors in parallel.

We also thought that maybe we could increase the capacitance
by somehow increasing $\E_r$ of the dielectric.  There was an
old can of ``Inductor Oil'' which we decided to try soaking the paper
in to test its effects.  We found later this oil is called ``Diallyl.''

\section { Construction of Second Design }

We decided to use $20\x26$cm plates, because that was the largest
we could make our surrounding phenolic blocks from the material we had.
It would have been a little better in terms of fringing fields
to have round plates, but we didn't want to reduce the
size of our plates at all.  The reduction of area would have reduced
the capacitance more than the losses due to the fringing fields using rectangular
plates.

We decided to use 8 layers of aluminum foil, which was
the only conducting material we could get enough of.  We also chose
to use regular copier paper as in the first design.  Our design was
8 layers of
aluminum foil, interspersed with 7 layers of paper.   We had to
use very small pieces of Scotch Tape to hold the layers in place because
the paper and aluminum foil pieces were very slippery.
We then
put the phenolic blocks on both sides and compressed the unit with
3 C--clamps. 

The aluminum foil pieces each had an extra 0.75" square tab
on one corner to clip the alligator
clips onto.

\section { Results of Second Design }

When we tried the whole assembly with no oil, we measured
a capacitance of $C = 0.0617\mu$F.  The theoretical
capacitance of this, assuming again $\E_r = 2$,
$$C = { n \E_o \E_r A \over d } = {  (7) (8.854\x10^{-12}) (2)
      (0.20)(0.26) \over 0.08\x10^{-3} } = 80.6\hbox{nF}$$

Our actual value differs from this by 27\% again.  Notice that for
our first design, the measured $C$ was also 27\% low.  Does this
mean we were ``just wrong'' twice?  I think it means that the
$\E_r$ for {\it our} paper was actually lower than the range 2 -- 4
given in the CRC handbook.  If we, for the moment, assume as given
our measured $C$ values, and we re-calculate the $\E_r$ of paper
from that data, we get
$$ {1.6 \over 2 } = { 1.21 \over \E_{r1} },  \hbox{\hskip .3in} \E_{r1} = 1.51$$
and
$$ {80.6 \over 2 } = { 61.7 \over \E_{r2} }, \hbox{\hskip .3in} \E_{r2} = 1.53$$

Notice that there is very good agreement between these two calculated $\E_r$
values for both the small single layered capacitor and the large multilayered
capacitor.

Before we wanted to make a mess trying
the big capacitor with the oil, we put the small round capacitor
together using a piece of oil-soaked paper as the dielectric to see
if it would increase the capacitance at all.  It did, yielding a
measured capacitance of $C = 2.15$nF, almost 2 times the value we got
for regular paper.

So we disassembled our large capacitor and soaked each piece of paper
in the ``Inductor Oil.''  We then assembled the layers on the
phenolic block and clamped it together.  This time
we didn't have to use tape to hold the layers together
because the oil's surface tension stuck them together for us.

The Cap Meter reported ``Error --- Short.''  So we disassembled the 
layers and found that a piece of grit had penetrated 4 layers when
we tightened the clamps and had shorted the layers.  To fix this, we
replaced the punctured dielectric layers and reassembled, more carefully
this time.

We slowly tightened the C--clamps while measuring $C$.  When no pressure
was applied, $C = 0.0947\mu$F.  As we tightened, it finally rose
to $C = 0.1218\mu$F after we had applied a large amount of pressure
by tightened each C--clamp with a short hollow pipe to get more 
leverage and torque.  This increase in capacitence lends credence
to the $1/d$ relationship.

We had no way to measure the $\E_r$ for this oil, but if we
assume the measured $C$ is correct, we can go back and
find $\E_r$ for both the small and large capacitors with
oil by comparing them to the dry case, using $\E_r$ for paper
to be the average of 1.51 and 1.53, $\E_{rdry} = 1.52$,
$$ {2.15 \over 1.21 } = { \E_{r1oil} \over 1.52 },  \hbox{\hskip .3in}
        \E_{r1oil} = 2.7 $$
and
$$ {121.8 \over 61.7 } = { \E_{r2oil} \over 1.52 }, \hbox{\hskip .3in} 
        \E_{r2oil} =  3.0 $$

These don't correspond quite as nicely as the values for the first
part, but there is still only a 10\% difference between 2.7 and 3.0.

\section { Conclusions }

The increase in $C$ as we tighted the clamps suggests that 
the $1/d$ relationship is correct, although we didn't design our
experiment to test this.  If we wanted to conduct such a test, we
would measure the capacitence of our little capacitor with different
numbers of dielectric layers. In each case, we would use the same pressure to try
to keep $d$ constant.
Then we could plot $C$ as a function of $1/d$ and the
more straight the line, the better our demonstration that $C \propto 1/d$.
We would expect the slope of this line to be $\E A$.

Our data also supports the relationship that $C \propto A$.  If we take the ratio
of the capacitnces
$$ { A_2 \over A_1 } = {  { (7) (0.2) (0.26) }\over {(\pi)( 0.048^2)} } = 50.3$$
and compare that to our measured capacitences
$$ { C_{2dry} \over C_{1dry} } = { 61.7 \over 1.21 } = 51.0
\hskip .5in \hbox{and} \hskip .5in
 { C_{2oil} \over C_{1oil} } = { 121.8 \over 2.15 } = 56.7 $$
The dry capacitors were only 1.39\% away from the expected increase in $C$, and
the wet capacitors were 12\% away from the expected increase
in $C$.  These results confirm the $C \propto A$ relationship.

The significant error we had showed up in the retro--calculations of
$\E_r$ for the oil dielectric.  I think the 12\% error for the change
in $A$ vs $C$ for the wet capacitor was because of the same reason that
gave the 10\% difference between the calculated $\E_r$ for the wet dielectric.
This difference may be attributed to a changing composition of dielectric
as we tightened the clamps.  The oil dielectric consisted of paper sheets
that were soaked in oil.  The oil, ``Diallyl,'' is supposed to have an $\E_r$
around 4 --- 6, and the $\E_r$ for paper is less than that.  As we tightened
the clamps, oil was squeezed out of the paper like water from a sponge.  This 
would decrease $\E_r$ for the combination of paper and oil.

We had no way to keep pressure constant between the two capacitors, so this is 
probably the source of the error.

\section { Going Further }

We were able to calculate $\E_r$ retroactively because we had a measured
$C$, and known $A$ and $d$.   Once we have a material with a known $\E_r$, 
our capacitor becomes a ``proof of principle'' device to measure
dielectric constants of materials.  An actual device would have to
keep track of area and distance, measure the capacitence, and then compute
an adjusted ratio of the capacitences to reveal the $\E_r$ for the
unknown material, as we did by hand for this analysis.


\end