Introduction - Why Do You Want to Learn This Material?
In this lesson you're going to be introduced to Digital Logic. There are lots of reasons to learn digital logic. Here are some of those reasons.
Digital logic is the foundation for digital computers. If you want to understand the innards of computers you need to know digital logic.
Digital logic has relations to other kinds of logic including:
Formal logic - as taught by many philosophy departments
Fuzzy logic - a tool used to design control systems and many other systems.
So, in learning digital logic you learn something that helps you elsewhere.
For many students, learning digital logic is fun.
Logic Signals
There are a number of different systems for representing binary information in physical systems. Here are a few.
A voltage signal with zero (0) corresponding to 0 volts and one (1) corresponding to five or three volts.
A sinusoidal signal with zero corresponding to some frequency, and one corresponding to some other frequency.
A current signal with zero corresponding to 4 milliamps and one corresponding to 20 milliamps.
And one last way is to use switches, OPEN for "0" and CLOSED for "1".
(And there are more ways!)
Characteristics of Logic Signals
We should note that all of these signals can and usually will change in time, so that we really are looking at dynamic situations. However, we will start by looking at these signals as though they were not changing in time.
We will pick a voltage signal as a working example. It can take on two values corresponding to 0 and 1.
We can associate a variable with that logic signal, and we can assign a symbol to represent that variable - like the symbol A.
Think Binary!
Let's examine a typical situation. You have some sort of device that generates a logic signal.
It could be a telephone that converts your voice signal into a sequence of zeros and ones.
It could be the thermostat on the wall that generates a 1 when the temperature is too low, and a 0 when the temperature is above the set point temperature.
The logic signal, A, takes on values of 0 (FALSE, OFF) or 1 (TRUE, ON). That signal might really be a voltage, a switch closure, etc. However, we want to think in terms of zeros and ones, not in terms of the values of the voltage.
Operations on Logic Signals
Once we have the concept of a logic signal we can talk about operations that can be performed on logic signals. Begin by assuming we have two logic signals, A and B. Then assume that those two signals form an input set to some circuit that takes two logic signals as inputs, and has an output that is also a logic signal. That situation is represented below.
The output, C, depends upon the inputs, A and B. There are many different ways that C could depend upon A and B. The output, C, is a function, - a logic function - of the inputs, A and B. IWe will examine a few basic logic functions - AND, OR and NOT functions and start learning the circuitry that you use to implement those functions.
Logic Gates
If we think of two signals, A and B, as representing a truth value of two different propositions, then A could be either TRUE (a logical 1) or FALSE (a logical 0). B can take on the same values. Now consider a situation in which the output, C, is TRUE only when both A is TRUE and B is TRUE. We can construct a truth table for this situation. In that truth table, we insert all of the possible combinations of inputs, A and B, and for every combination of A and B we list the output, C.
| A | B | C |
| False | False | False |
| False | True | False |
| True | False | False |
| True | True | True |
AND GATES
An AND function can be implemented electrically using a device known as an AND gate. You might imagine a system in which zero (0) is represented by zero (0) volts, and one (1) is represented by three (3) volts, for example. If we are going to use electrical devices we need some sort of symbolic representation. There is a standard symbol for an AND gate shown below.
Often in lab work it's helpful to use an LED to show when a signal is 0 or 1. Usually a 1 is indicated with an LED that is ON (i.e. glowing). You can use the buttons below to check out this AND gate (Note what an AND gate symbol looks like!) with a simulated LED. Note the following in the simulation (and you can use this in your lab experiments).
To get a logical zero, connect the input of the gate to ground to have zero (0) volts input.
To get a logical one, connect the input of the gate to a five (5) volts source to have five volts at the input.
Each button controls one switch (two buttons - two switches) so that you can control the individual inputs to the gate.
Each time you click a button, you toggle the switch to the opposite position.
OR Gates
Consider a case where a pressure can be high and a temperature can be high Let's assume we have two sensors that measure temperature and pressure.. The first sensor has an output, T, that is 1 when a temperature in a boiler is too high, and 0 otherwise. The second sensor produces an output, P, that is 1 when the pressure is too high, and 0 otherwise. Now, for the boiler, we have a dangerous situation when either the temperature or the pressure is too high. It only takes one. Let's construct a truth table for this situation. The output, D, is 1 when danger exists.
| T | P | D |
| False | False | False |
| False | True | True |
| True | False | True |
| True | True | True |
NOT Gates (Inverters)
A third important logical element is the inverter. An inverter does pretty much what it says. If the input is 0, the output is 1. Conversely, if the input is 1, the output is 0. The symbol for an inverter is shown below. Again, you can putter with this inverter with the simulated LEDs. X is the input to the inverter. The output is NOT-X represented as ~1 or:
| A | C |
| 0 | 1 |
| 1 | 0 |
NAND Gates
There is another important kind of gate, the NAND gate. Actually, the way to start thinking about a NAND gate is to think of it as an AND gate with an inverter on the output. That's shown below.
Actually, however, the symbol for a NAND gate compresses the inverter down to a dot at the output of the NAND gate as shown below.
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