Electric circuits and the hydraulic analogy: capacitor behavior. To
understand how an electronic capacitor works, let's first look at the hydraulic
analogy. A hydraulic capacitor is a cylinder divided by a flexible rubber
sheet. I'm showing a cross-section view because it's easier for me to draw.
The volume of water stored is proportional to the pressure applied.
When I apply 100 kiloPascals pressure, I get 1 liter of storage; 200 kPa,
2 liters; 300 kPa, 3 liters, and so on. The volume is equal to
the capacitance times the pressure. The capacitance is the constant of
proportionality. In this case we have 3 liters equals 0.01 times 300 kPa.
An electronic capacitor is a set of parallel metal plates separated by an
insulator. The amount of charge stored is proportional to the voltage applied: zero
volts, zero coulombs; 100 volts, 1 coulomb; 200 volts, 2 coulombs; 300 volts, 3
coulombs and so on. The charge is equal to the capacitance times the voltage. For
example, we have 3 coulombs equals 0.01 times 300 volts. The units of
capacitance are coulombs per volt, also known as Farads. There's no flow under
constant pressure, and water flows only when the pressure changes. For example,
we'll apply increasing pressure on this capacitor: 0, 1, 2, 3. In 3
seconds we increased the pressure from 0 to 300 kPa and 3
liters of water flowed in. We saw before, volume equals capacitance times pressure.
That means the change in volume is equal to the capacitance times the change in
pressure. The change in volume is the flow rate. This notation, dP/dt, means the
change of pressure with the change in time. For example, when we increased the
pressure at a rate of 100 kPa per second,
we got an inflow one liter per second. If you reduce the pressure, you get the
opposite effect. If you decrease the pressure at a rate of 100 kPa
per second, you'll get a flow in the opposite direction of 1 liter per second.
That's minus 1 liter per second, and if you do not have any change of pressure
like we see right now, zero kPa per second change; you get no flow. Same
thing happens with the electronic capacitor: no flow under constant voltage,
current flows only when the voltage changes. For example we start now: 0,
1, 2, 3. In three seconds we increased the voltage from 0 to 300
volts and three coulombs of charge flowed in. Charge equals capacitance
times voltage. That means the change in charge equals the capacitance times the
change in voltage. The change in charge is the current, and this notation dv/dt
means change in voltage with the change in time. When we increase the voltage by
100 volts per second, we've got an inflow of 1 coulomb per second, or 1 amp. If we
decrease the voltage at a hundred volts per second, we'll get a negative flow of
1 coulomb per second, or minus one amp, and as we see here in the steady state,
with 0 volts per second change, we get 0 current. A bigger container has
more capacitance. For example, this capacitor stores twice as much water
with the same pressure, so it has twice the capacitance: 0.2 instead
of 0.1 liter per kPa. Same with electronic capacitors.
This capacitor has twice the area stores twice as much charge for the same
voltage, so it has 0.02 farad instead of 0.01 farad capacitance. The stored energy
is 1/2 the pressure times the volume for the hydraulic capacitor, and 1/2 times
voltage times charge for the electronic capacitor. Why only 1/2 the pressure
times the volume here? Because the pressure was ramped up from 0 to 300 to
push in this 3 liters. The average pressure is to push it in is 1/2 of 300.
That's where the 1/2 comes from. Same thing with the electronic capacitor.
This model is not quite right because it shows charge coming in seemingly from
nowhere. Charge is always conserved. let's look at another analogy: money. Money is
conserved. My twin brother and I are broke. We have no money.
Zero dollars on our net worth together is zero. My brother goes to the bank,
borrows $50, brings it back, and gives it to me. Now I have $50. My brother has
negative $50 -- that's the money borrowed and he still owes and our net
worth is together is still $0. Now my brother looks at me with envy, maybe
regrets giving me the money. He would like to have that money back to make
himself whole. He sees $100 difference between me and himself. OK, it's just
like that with a capacitor. We start with zero coulombs charge on both sides and a
net charge on the capacitor of zero coulombs. Connect a 100 volt battery;
if this is a 1 farad capacitor, the battery draws 50 coulombs from this side
and puts it on this side, leaving a deficit of 50 coulombs, negative 50
coulombs on this side, and positive 50 coulombs on this side. The net charge on this
capacitor is still 0, the sum of 50 and -50. If we take the battery away,
the 100 volt difference remains because the charge has no way of escaping from
here, and we have a difference of 100 coulombs between the
two sides. That's why we say we store 100 coulombs on this capacitor, and
there's an electric field between these two plates because the positive and
negative charges attract each other. The force of attraction holds that charge in
place. If we connect a wire between these two sides, there's a fast rush of
charge around here, which neutralizes everything, leaving 0 coulombs on both
sides again. In summary, volume is like charge, pressure is like voltage,
capacitance is liters per kPa for the hydraulic, or coulombs per volt
for the electric, also known as farads; and the energy is in joules in both
cases: kPa-liters for this type of capacitor, and volt-coulombs for this
type of capacitor. Let's take a look at inductor operation next.