If the presence of electricity
can be made visible
in any part of the circuit,
I see no reason why intelligence
may not be transmitted
instantaneously by electricity.
~Samuel Morse

This is the fourth part in my multi-part series on how computers work. Computers are thinking machines, but they can’t do this on their own. We need to teach them how to think. And for this, we need a language of logic. In the first part of the series, I introduced this language of logic, Boolean algebra. In the second part, I described how to formulate complex logical statements using Boolean algebra. In the third part, I described how some of the electronics in old-school computers worked.

This time, I finally get to the meat of things. I describe how you can use the circuits described in part 3 to actually implement Boolean logic electronically.

Voltages

The following discussion is going to depend heavily on an understanding of the triode vacuum tubes that I discussed last time. So if you haven’t already, you probably want to last week’s article before you read this one.

That said, some review is probably in order. First let’s review electric voltage. The salient points are this. Electrons want to flow from places in a circuit that are at low voltage to places that are at high voltage.  Voltage is measured in volts. We want to use voltage as a stand-in for the truth values described in parts one and two of my series on how computers work.

There are a number of ways we can use voltage to represent truth values. But let’s pick the most common way. Let’s say we have a circuit. We’ll connect a voltage-measuring device to a spot on our circuit—perhaps where two wires intersect—and say that, if it reads any voltage higher than +5 volts, then our circuit represents true at that spot. If it reads any lower, then our circuit represents false there.

(There’s an important detail I’m skipping here. Voltages are not absolute numbers. We know that electrons flow from low voltage to high voltage. But we need a baseline for what “zero volts” means. In every circuit, this “zero volts” value might be different, but we always have to set it and choose a value. We usually call “zero volts” ground.)

Vacuum Review

 Now let’s review the basics of a triode. As the name suggests, a triode is a circuit component with three terminals, called the cathode, anode, and filament, as shown below. There’s another piece called the called the heating element that we won’t discuss in much detail.

I discuss the triode in great detail in last week’s article.  But for now, here’s what you need to know. The filament controls the flow of electrons from the cathode to the anode. Suppose that I hook up the triode in a circuit so that the anode is at a high voltage, and the cathode is at a low voltage. Then if the filament is at a low voltage like the cathode, then no current will flow. If the filament is at a high voltage like the anode, then current will flow.

Fiddly Names

It’s worth noting that the pieces of a triode can have other names. The cathode is often called the plate and the filament is often called the grid. In this naming scheme, confusingly, the heating element is often called the filament.

It’s also worth noting that these days triodes have been almost completely replaced by transistors.But transistors have different names for these parts. In a transistor, the cathode is called the source, the anode is called the drain, and the filament (or grid) is called the gate. A transistor does not have a heating element. To avoid confusion, I’ll use vacuum tubes instead of transistors. And I’ll use the names in my diagram.

Boolean Circuity

Now let’s implement Boolean logic! To do this, let’s chain some triodes together, as shown below.

Here we keep the anodes at a very high voltage, much higher than the cathodes could possibly be at. We then connect both cathodes to a terminal that we call C. We name the filament terminals A and B respectively.

Now let’s manipulate the terminals A and B and see what terminal C does.If both A and B are at a low voltage (i.e., false), then current won’t flow through either triode. The result is that C stays at a low voltage, which we’ll call false. But now let’s put terminal A at a higher voltage, say true. (For now we’ll leave B as false.) Then current will flow through the left-most triode and raise the voltage at terminal C so that it becomes true. 

Similarly, if we keep A at a low voltage (i.e., false) but raise the voltage at B so that it becomes true, then current will flow through the right-most triode. This too will raise the voltage at terminal C and make it true. And of course, if both A and B are true, then current will flow and raise the voltage at C, making it true.

So if both A and B are false, then C is false.But if either A or B or both A and B are true, then C is true too. The astute among you might recognize C as the result of the logical operation A or B. We’ve just made the logical or work electronically! The circuit we just constructed is called an or gate.

More Gates

Can we do the same thing to make a logical and? Let’s try. We’ll still use two triodes. But now let’s connect them in a different way, as shown below. We put the anode of the first triode at a very high voltage. But we connect its cathode to the anode of the second triode. We label the filaments A and B respectively and the cathode of the second triode C.

Now current will be unable to flow unless both terminals A and B are at a high voltage. In other words, C will be true only if both A and B are true. You might then recognize C as the output of the logical A and B, which we’ve now implemented electronically! This circuit is called an and gate.

Gates in Practice

I showed you how to electronically implement the boolean operations and and or described in George Boole and the Language of Logic, Part 1. If we wanted to make composite statements as we did in part 2, then we’d also need the not operator. But that’s actually a bit trickier (though not impossible) to make electronically. So I won’t describe it.

Usually, in practice, people use nand and nor gates as opposed to and, or, and not. These are harder to make, however, so I won’t describe them. Hopefully the gates I did describe, however, give you a feel for how this works.

Electronic Caveats

In my description of the and and or gates, I claimed that when current flowed through a triode, the voltage at the cathode changed. In practice things are a lot more complicated. Voltage and current are related. In simple cases, the relationship is given by Ohm’s law, which many of you are probably familiar with. In more complicated cases, where the current changes quickly, you need to take more things into account, like the travel time of electrical signals.

That said, the description I gave gets the basic idea across and, if we added more circuit components (such as, capacitors, inductors, and resistors), the gates I described would work.

Next Time

Performing a logical calculation is useless unless you can record your findings. Next time, I’ll describe how to use logic gates to teach computers how to remember.

Further Reading

If you want to know more, here are some resources on how logic gates are implemented.

• +Olafur Jens Sigurdsson pointed me to this absolutely amazing MIT course that walks the student through the creation of a simple computer, from the hardware to the operating system to the program.
• HowStuffWorks has a nice article on how boolean logic works inside a computer, starting from the very beginning. You can find it here.
• This slideshow gently introduces how to build logic gates from transistors.

Related Reading

If you liked this article, you might also like the following articles I’ve written.

I am currently in the middle of a series on how computers work, from the ground up. Here are the previous articles:

• In this article, I describe the basics of Boolean logic, starting with truth tables.
• In this article, I describe how to make composite logical statements using Boolean logic.
• In this article, I introduce the triode vacuum tube and describe how it works.

And if you like articles on how things work. Here are some other articles I’ve written in the same vein.

• In this article, I describe one way to make a transistor.
• In this article, I describe how lasers work.
• In this article, I describe how to make a pulsed laser.