As I've mentioned
previously, the main purpose of a capacitor bank for
the purposes of high power experiments is to deliver as much power as
possible as quickly as possible, and in order to do that the capacitors
have to be able to deliver very large currents. In order for this to be
possible the parasitic resistances and inductances of the connections
between capacitors need to be minimized. This means wide flat
conductors with large crosssections.
For any budgetminded
design the first step is probably going to be
locating a source of capacitors. In my mind, two categories of
capacitors stand out  electrolytic capacitors and pulse capacitors.
Pulse capacitors are typically large and have maximum voltage ratings
in the thousands of volts and are generally expensive. Electrolytics,
on the other hand, are limited to about 500V, are smaller, and
are reasonably cheap if you can get them as surplus. I went with
electrolytic capacitors, as I found a source for 128 Philips 3,600uF
350VDC 400VDC Surge rated capacitors that were recovered from old GE
medical imaging equipment on eBay for about $25 per 8 capacitors ($400
total).
Fig. 1: Capacitors
Luckily,
these came with
all of the necessary hardware, including bleeder resistors and mounting
braces. I used all of the original bleeder resistors, which worked out
to one 20kOhm resister per capacitor. The bleeder resistors are
also important, as the capacitors could otherwise retain a dangerous
level
of charge for long periods of time. With the bleeders, any remaining
charge will be slowly removed so that you don't get a nasty or
deadly shock if you forget to completely discharge them.
It's
important to also
know the time constant of the resistor /
capacitor combination. This tells you how quickly the capacitors will
discharge through the bleeder resistor. The time constant can be
calculated as
Eq.
1: Time Constant
The
time constant is
equal to the time it takes for the capacitor to discharge to 37% of its
original voltage, in this case about 72 seconds. As a rule of thumb, a
capacitor is generally considered discharged after 10 time constants,
or in this case, about 720 seconds (12 minutes.) Using a smaller value
resistor will decrease the time it takes the capacitor to discharge,
but will increase the amount of power that will be dissipated by the
resistor. The power dissipated by the resistor is given as
Eq.
2: Power Dissipated
in Resistor
In this case, each
bleeder resistor will consume about 6 Watts of power if the capacitors
are charged to 350V. Selecting a bleeder resistor is then a compromise
between the time for the capacitors to discharge (safer) and the amount
of power dissipated in the bleeder resistors (more efficient.)
The next
major decision that needs to be made concerns the conductors
that connect all of the capacitors together and something called skin
effect. Skin effect occurs whenever current through a conductor changes
abruptly, and limits higher frequency currents from penetrating further
into the conductor. So, if you have a large current and have selected a
conductor that would be appropriate for DC currents, you'll find that
at high AC frequencies the current will only travel through the portion
of the conductor close to the surface. This means that perhaps a
significant portion of your conductor is being wasted, and that the
conductor may heat much more than expected since the current is going
through an effectively smaller conductor. The skin depth is given as [1]
Eq.
3: Skin Depth
for good conductors where f is the frequency, mu is the
magnetic permeability of free space, and sigma is the conductivity of
the conductor.
To simplify this problem,
let's assume that the discharge of the capacitors' energy completes in
2.5ms, and that the current waveform approximates half a sine wave, and
that the conductor is aluminum. This corresponds to a frequency of
200Hz, with aluminum having a conductivity of 3.78x10^7[2].
Plugging these values into equation 3 yields a skin depth of roughly
5.8mm. Since the skin effect applies to the top and bottom surface of a
conductor, the maximum thickness of a plate of aluminum through which
this discharge will use then entire volume of the conductor is 1.16cm
(a little under a half an inch.) So there you have it, the aluminum
buss bars used to connect the capacitors in the capacitor bank should
not be more than about 1/2" thick, though they can be as wide as we
like to accommodate large currents.
The bleeder
resistance and the conductor thickness are the two
controlling factors that decide how to best construct the capacitor
bank. With both of these parameters defined, we can get to work. I
first assembled each set of four capacitors into a string using 1"x1/8"
aluminum bars (from Home Depot) with bleeder resistors connected
across the buss bars. Four of these strings of four capacitors were
then mounted
to a Lexan plate, and connected in parallel with 1.5"x1/8" aluminum
bars.
A total of 8 such plates were constructed, and mounted into two towers
of 64 capacitors, with the plates being connected with two runs of
1.5"x1/8" aluminum bars. The construction details are shown below.
When both stacks
are
connected in parallel they have a total capacitance of 0.46 Farads at
350V, with 128 20kOhm bleeder resistors connected in parallel
(equivalent to 156 Ohms.) This works out to 28,200 Joules of energy
stored at 350V, and 784 Watts dissipated in the bleeder resistors at
350V.
Warning  Electrocution Hazard 
Capacitors
can store lethal voltages for long
periods of time, or even accumulate lethal voltages over time. All
capacitors should have a bleeder resistor connected at all times to
prevent charge from building up and to safely discharge the capacitor
when not
in use. Even with these precautions, always treat capacitors as if they
are charged, and properly discharge them before working with them. You
are ultimately responsible for your own safety  so make sure you know
what you're doing before you do it!
Lowering
the
inductance of the connections within the capacitor bank is an important
factor in capacitor bank performance for discharge type experiments.
The wide flat conductors shown earlier help to reduce this inductance,
but there are other configurations that allow for lower total
inductance. In the previous design conductors are laid horizontal in
and edgetoedge configuration. The inductance can be lowered if the
conductors are slightly wider, and if they're set backtoback, as
shown below.
Fig. 3: (a)
Normal
Construction (b) Low Inductance Construction
The
low inductance
construction method shown in figure 3b is only necessary for extremely
low inductances in conjunction with low ESL capacitors. Otherwise, the
construction shown in figure 3a is sufficient.
References
[1] Fawwaz T. Ulaby, Fundamentals of
Applied Electromagnetics, Upper Saddle River: Pearson Prentice
Hall, 2004, pp. 277
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