## Capacitance

Capacitors are interesting devices. The hydraulic analog to a capacitor is a storage device with two separate, nonconnected, sealed chambers (Figure 3-4). As a pump forces the water around the pipe, water flows into the top chamber and is pumped out of the lower chamber. At first, this happens with almost no impedance to the flow. But as the chamber begins to fill, the pressure in it builds up, pushing against the pressure of the pump and causing the flow of water to slow. When the chamber fills, the pressure at the chamber equals the pressure at the pump and the flow stops.

**Figure 3-4. Hydraulic model of a capacitor.**

A capacitor, in its simplest form, consists of two parallel plates placed closely together. Actual fabrication techniques are, of course, more complicated than this. But all capacitors, fundamentally, consist of two separate surfaces placed very close together with a thin material placed between them.

When a voltage is applied to a capacitor (perhaps by connecting a battery across its leads), electrons flow onto one of these plates. (Remember, the flow of electrons is current.) Electrons flow off the other plate and back around into the battery. This preserves the condition that current is the same everywhere in the loop. During the first instant, the flow is virtually unimpeded. But as the electrons begin to build up on the plate, they start building up a charge that begins to resist the continuing flow of more electrons. Thus the flow starts slowing down. At some point, the charge buildup is great enough to stop any further flow, and all current stops. The charge buildup is manifested by a voltage on the plate. The voltage increases as more charge builds, and current stops flowing when the voltage equals the initial voltage that was applied to the capacitor.

One key to understanding what happens with a capacitor is the recognition that what happens is *time dependent*. (This is *not* true with resistance.) That is, when current first starts flowing into a capacitor, there is no voltage across the plates
(i.e., across the capacitor). There can be no voltage across the capacitor until charge builds up on the plates. But charge
can't build up on the plates until charge (current) has actually flowed into the capacitor. Since current must flow before
a voltage appears, we say that current leads voltage. (Why this concept is important will become more apparent when, later
on, we see that the opposite is true with inductors; that is, current lags voltage in the case of an inductor.)

Now, consider the hydraulic analog and think about what happens if we turn the pump on in the forward direction for a moment and then reverse it, pumping in the opposite direction. Water will start to flow into the storage tank, but if we reverse the pump soon enough, negligible pressure will have built up. Then water will flow in the reverse direction. Now if we reverse the pump again, quickly enough, pressure will not build in the opposite direction either.

In this way, we can envision how it would be possible to have a pump that continually reverses direction fast enough that
water pressure cannot build up in the tank. The flow of water in the pipes will *look* as though the storage tank were not there. Of course it is, but the flow reverses so quickly that its effects are not seen.
We would see this same effect if the pump reversed more slowly, but the storage tank was larger. But if the pump reversal
was relatively slow, and the storage tank was relatively small, then we would see the flow of water speed up and slow down
in different parts of the pumping cycle as pressure built up and then decreased again.

So, the effect of the storage tank is a function of both the size of the tank and the frequency at which we alternate the pumping cycle. Water will flow as if the tank were not there if the tank size is very large or if the pump frequency is very high. Otherwise, water flow will slow down, and might even stop, during part of the cycle.

So it is with electrons and a capacitor. If a capacitor is very large, alternating current will flow virtually unimpeded,
as if the capacitor were not there. If the frequency is very high, current will flow as if the capacitor were not there. It
is not that *individual* electrons are flowing around the loop; they are not. Individual electrons cannot travel past the plate of the capacitor.
But the flow of electrons past any particular point will be the same as it would be if the capacitor were not present. On
the other hand, there will be significant impedance to current flow if (a) the frequency is relatively low, or (b) the capacitor
is relatively small.

Another way to think about this is to recognize how many electrons actually flow during a half-cycle of an AC current. A 1-MHz
waveform changes direction 10^{6} times each second. Therefore, for the same magnitude of peak current (charge/unit time), only one-millionth (10^{–6}) as many electrons flow during the half-cycle of a 1-MHz (10^{6}) waveform as flow during the half-cycle of a 1-Hz waveform. Therefore, the voltage buildup on a capacitor will be substantially
less for a half-waveform of a 1-MHz signal than for a 1-Hz signal. Table 3-1 summarizes the range of possibilities.

**Table 3-1. Capacitor size and frequency both affect the flow of current.**

Suppose we put two capacitors in parallel (Figure 3-5). If the value of one capacitor is very large, then it almost doesn't matter what the value of the other one is. Even if the other capacitor is very small, almost zero, the overall circuit will still behave as if there is a large capacitor in the flow loop. The flow will be determined, first, by the larger capacitor. The effective capacitance of the two will be at least as large as the larger of the two capacitors, and perhaps even more if the second capacitor is nearly the same size as the first.

**Figure 3-5. Hydraulic analog of two capacitors connected in parallel.**

In contrast, consider the case of two capacitors in series (Figure 3-6). Here, the flow will be restricted by the smaller capacitor. The smaller capacitor will charge up quickly (the smaller storage tank will fill up quickly) and flow will stop. No matter how large the larger capacitor is, the flow will never be greater than that allowed by the smaller capacitor. The total flow may be less if the other capacitor adds additional restriction (impedance to the flow), which would happen if it were nearly as small as the first one.

**Figure 3-6. Hydraulic analog of two capacitors connected in series.**

For the parallel combination of capacitors, Ceq is given by Equation 3-6:

Note that if, for example, C2 were very much larger than C1, the equivalent parallel combination would reduce to Equation 3-7:

For the series combination of capacitors, Ceq is given by Equation 3-8:

Note in Equation 3-8 that if C2, for example, were very large, the equation would reduce to Equation 3-9

This illustrates how the smaller capacitor becomes the limiting one in this case. Note that this is the opposite case from that of resistors (Equations 3-3 and 3-5).

The measure of capacitance is the farad (Michael Faraday). One farad of capacitance will have a force of one volt across its
plates when the capacitor stores one coulomb of charge. (Remember one coulomb is 6.25 × 10^{18} electrons.) Recall also that one amp is the flow of one coulomb of charge in one second. So, if one amp of current flows
for one second onto the plates of a one-farad capacitor, the voltage across the plates will rise one volt, or:

Perhaps of more relevance, what if 1.0 mA flows onto the plate of a 10.0 uF capacitor for 1.0 ms? Well, 1.0 mA (10^{–3}) for 1.0 ms (10^{–3}) would move 10^{–6} coulombs. A capacitance of 10.0 uF is 10^{–5} farads. So the voltage buildup would be: