Capacitors

A capacitor is analogous to a spring. It is capable of accumulating charge, and thus of storing energy as well. In fact, capacitors used to be called condensers or accumulators.

Earlier we described a hydraulic spring:

In this hydraulic spring, the idea of fluid volume "accumulating" can be seen in the figure. As the membrane distorts, and develops a pressure across it, there is an internally a "displaced" volume of fluid (shown dotted) which is the total accumulated flow that has gone down the pipe.The volume of fluid being stored in the spring is equal to the volume of fluid which has been displaced due to the stretch of the spring.

An electrical capacitor is of this kind. It consists of two metal plates separated by a thin insulator, just as is suggested by its schematic symbol:

Charge cannot cross the insulator, just as hydraulic fluid cannot cross the membrane. But positive charge can accumulate on one plate, exactly balanced by negative charge on the other. This physical separation of a non-trivial amount of charge is tolerable because the positive and negative charges remain in close proximity to each other, and they attract so they are reasonably happy about that. Still, as more charge gets segregated in this way, as current flow in from the left and out on the right, an increasing voltage develops across the capacitor. This is analogous to a pressure developing across the distended membrane in the hydraulic spring as fluid flow enters the left and exits the right, accumulating in the dotted "pouch".

The constitutive law of real capacitors is not typically as linear as that of resistors, but to the extent that it is approximately linear we use this constitutive law:
V = 1/C q
(1)
C is the capacitance, measured in farads. A 1 farad capacitor is a whopping big one. More typical are microfarads, 10-6 farad, abbreviated mF, and picofarad, 10-12 farad, abbreviated pF. You might expect mF to mean millifarad but it usually doesn't. It usually means microfarad as well, written by people who don't have Greek letters available. mmF means mF too. nF, or nanofarad, 10-9 farad, is somewhat unusual.

Really, it's hard to read capacitor values if you don't already have a rough idea of the answer based on the physical size and type. "222" on a tiny disk capacitor, for instance, is a cryptic 22 * 102 pF, encoded much the way the three color bands on a resistor are.

Recall, Charge is related to current just as volume is related to hydraulic flow:
qi dt 

i = q'

(2)
So, we can also write (1) as
or (3)

Parallel and Series

Similar to what we did for resistors, we can examine the effective capacitance of simple circuits of capacitors in series and an parallel. The same rules outlined before for elements in series and parallel hold. Here is an example of capacitors in parallel.


Energy

Suppose a capacitor of C farads presently contains a charge of q coulombs. Therefore the voltage across it is V=q/C. If we charge it some more, by injecting a current i, the power flowing into the capacitor is P=iV, just as it would be for any component (not just a capacitor). If we do this for a short interval of time dt (short enough that the voltage doesn't change significantly), the energy we have put into the capacitor is DE = P dt = V i dt.

In the same infinitesimal moment we have increased the charge of the capacitor from q to q+ i dt. The amount of charge we have added is dq = i dt. Therefore the amount of energy we have added is DE = V dq.

On a graph of the constitutive law of the capacitor, V dq has a geometric interpretation: it is the area under the curve, from q to q + dq. (This is so even if the constitutive law is not linear.)

In charging the capacitor from q0 to q, the total energy added to the capacitor is
E = q0q V dq
(4)
In the case of a linear capacitor (V = q/C), changing up all the way from from q0=0, we can perform the integral and find
E = q2 / 2C, or 

E = CV2/2

(5)
This is the energy stored in the capacitor. It can be recovered if the capacitor is discharged into a resistor, for instance. Capacitors have a maximum voltage rating, much like resistors have a maximum power rating. A capacitor can be charged up to only a certain voltage before the insulating material between the two plates begins to break down - the maximum voltage rating of the capacitor is usually noted on the capacitor itself.


 Recognizing a capacitor

Below are pictures of some capacitors. Capacitors are typically labelled with text which indicates their capacitance, and a voltage rating. However, the text is often not clearly marked and can be quite cryptic, as in the examples below. More pictures of capacitors here. To see inside a capacitor to see what it is made of, click here.