Make Electronics

Switching Basics and More 71

Experiment 9: Time and Capacitors

theory

The time constant

You may be wondering if there’s a way to predict exactly
how much time it takes for various capacitors to charge,
when they are paired with various resistors. Is there a for-
mula to calculate this?

Of course, the answer is yes, but the way we measure it is a
bit tricky, because a capacitor doesn’t charge at a constant
rate. It accumulates the first volt very quickly, the second
volt not quite as quickly, the third volt even less quickly—
and so on. You can imagine the electrons accumulating on
the plate of a capacitor like people walking into an audito-
rium and looking for a place to sit. The fewer seats that are
left, the longer people take to find them.

The way we describe this is with something called a “time
constant.” The definition is very simple:

TC = R × C

where TC is the time constant, and a capacitor of C farads is
being charged through a resistor of R ohms.

Going back to the circuit you just tested, try using it again,
this time with a 1K resistor and the 1,000 μF capacitor. We
have to change those numbers to farads and ohms before
we can put them in the formula. Well, 1,000 μF is 0.001 far-
ads, and 1K is 1,000 ohms, so the formula looks like this:

TC = 1,000 × 0.001

In other words, TC = 1—a lesson that could not be much
easier to remember:

A 1K resistor in series with a 1,000 μF capacitor has a

time constant of 1.

Does this mean that the capacitor will be fully charged in 1

second? No, it’s not that simple. TC, the time constant, is the

time it takes for a capacitor to acquire 63% of the voltage

being supplied to it, if it starts with zero volts.

(Why 63%? The answer to that question is too complicated

for this book, and you’ll have to read about time constants

elsewhere if you want to know more. Be prepared for dif-

ferential equations.) Here’s a formal definition for future

reference:

TC, the time constant, is the time it takes for a capacitor

to acquire 63% of the difference between its current

charge and the voltage being applied to it. When TC=1,

the capacitor acquires 63% of its full charge in 1 sec-

ond. When TC=2, the capacitor acquires 63% of its full

charge in 2 seconds. And so on.

What happens if you continue to apply the voltage? History

repeats itself. The capacitor accumulates another 63% of the

remaining difference between its current charge, and the

voltage being applied to it.

Imagine someone eating a cake. In his first bite he’s raven-

ously hungry, and eats 63% of the cake in one second. In his

second bite, not wanting to seem too greedy, he takes just

another 63% of the cake that is left—and because he’s not

feeling so hungry anymore, he requires the same time to

eat it as he took to eat the first bite. In his third bite, he takes

63% of what still remains, and still takes the same amount

of time. And so on. He is behaving like a capacitor eating

electricity (Figure 2-81).

Figure 2-81. If our gourmet always eats just 63% of the cake still on the plate, he “charges up” his stomach in the same way that a
capacitor charges itself. No matter how long he keeps at it, his stomach is never completely filled.