BASIC ELECTRICAL ENGG. NOTES
An Introduction to Digital Logic - Signals and Gates
What Are You Going To Learn?You are at: Elements - Logic Circuits - GatesLogic Signals And GatesAND
OR
NOT (Inverter)
NANDWiring A Gate
A NAND Simulation
An Example ProblemBoolean AlgebraProblems
Return to Table of Contents
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.
What Are You Going to Learn? There are at least two general areas you need to become familiar with.
- First, there's background you need to know - the basics of digital logic - things like zeros and ones (0s and 1s) and how you can represent signals as sequences of zeroes and ones. Eventually you will want to know how large arrays of zeroes and ones can be used in computer files to store information in pictures, documents, sounds and even movies and you'll want to learn about how information can be transmitted, between computers and digital signal sources.
- You will also need to know things about digital circuits - gates, flip-flops and memory elements and others - so that you can eventually design circuits to manipulate digital signals.
- Learn what logic signals look like
- Model logic signals
- Learn Boolean algebra for logic analysis
- Learn about gates that process logic signals
- Learn how to design some smaller logic circuits
- Learn about flip-flops and memory elements that store logic signals
Objectives For This Lesson Here's what we are after in this lesson - what you should be able to do.
Given a system that uses logic signals
Be able to specify what the output will be when the input is zero (0) and what the output will be when the input is one (1).
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.
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.
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 |
Let's imagine a physician prescribing two drugs. For some conditions drug A is prescribed, and for other conditions drug B is prescribed. Taken separately each drug is safe. When used together dangerous side effects are produced.
Let
- A = Truth of the statement "Drug 'A' is prescribed.".
- B = Truth of the statement "Drug 'B' is prescribed.".
- C = Truth of the statement "The patient is in danger.".
| 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.
- 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.
QuestionQ1. You have an AND gate. Both inputs are zero. What is the output?
We now have two ways of representing an AND gate, the truth table and the circuit diagram. However, there is a third way of representing this information - a symbolic way - that will take us toward Boolean algebra. Let us consider our variables, A, B and C to be algebraic variables, but algebraic variables that can only take on two values, 0 and 1. Then we represent the AND function symbolically in either of two ways.
Problems Assume you have an AND gate with two inputs, A and B. Determine the output,
C, for the following cases.
Once we introduce Boolean variables, we can rethink the concept of a truth table. In the truth table below, if A, B and C are truth tables and we have an AND gate with A and B as inputs and C as the output, the truth table would look like this.
| A | B | C |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
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 |
| A | B | C |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Problems Assume you have an OR gate with two inputs, A and B. Determine the output, C, for the following cases.
P5. A = 1, B = 0
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 |
Example Problem You need to control two pumps that supply two different concentrations of reactant to a chemical process. The strong reactant is used when pH is very far from the desired value, and the weak reactant when pH is close to desired.
You need to ensure that only one of the two pumps runs at any time. Each pump controller responds to standard logic signals, that is when the input to the pump controller is 1, the pump operates, and when that input is 0, the pump does not operate.
You have a bunch of two-input AND gates (IC chips), OR gates and Inverters, and you need to design a logic circuit to control the pumps. You can generate a signal that is 1 when Pump S is ON, and 0 when Pump W is ON. Can you design the circuit?
In order to solve the problem, consider that the pump controls should receive logical inverse signals. When one pump signal is one, the other is zero. Given that recognition this circuit should work. Here, if X is 1, Pump S pumps.
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.
| A | B | C |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Wiring a Quad-NAND Chip If you want to use gates, you will need to learn something about their physical characteristics. In this section we'll walk you through wiring a simple gate circuit using one specific integrated circuit (IC) the 7400 chip. It's a good introduction to some of the more complex logic chips that you'll probably be using later.
Here's a picture of the 7400 chip in a circuit board. This chip is actually an N74LS00P. The LS tells you that it is a low power Schottky chip. Every manufacturer will embed the 7400 or 74LS00 in other part numbers.
If you want to use an IC chip, then you will always need to know the pinout. That's electrical engineering lingo for describing the way the pins are connected to the internal circuitry of the chip. You need to know where the power supply is connected and where the gate inputs and outputs are connected. Here's the pinout for a 7400 chip.
Now you can connect the two inputs to one of the gates on the chip. You're going to put 5v on either of these inputs for a 1 and ground the input for a 0. There are two wires in the picture below that connect to pins 1 and 2 on the chip. Those pins are the inputs for one of the NAND gates on the chip.
Click here for an introductory laboratory on the 7400 chip.
QuestionQ1 In the picture above, (shown again here) is the power turned on for the chip power supply?
A NAND Gate Here is a photo of a NAND gate wired to display the input signals and output signals. In this simulation you can manipulate the inputs and see the inputs and outputs. Note the following.
- The input voltage can be set to either 5v or 0v (ground) for each input to one of the NAND gates on the chip. Five volts is a logical 1, and zero volts is a logical zero.
- Note how the push buttons move a connection from 5v to ground when the button is pushed.
- When an signal is a 1, there is an LED that lights to show that the input is 1. When the LED is not lit, the signal is 0.
- Note that there is a current limiting resistor in series with each LED. If the voltage at the output becomes 5v and the LED "saturates" around 1.8v, you need a current limiting resistor. These resistors look to be 1kW.
- The power supply connection and the ground connection to the chip are both shown. The vertical line of connection points on the circuit board is ground.
Example Problem Let's reconsider the pump problem. What happens if there are times when you don't want either pump to pump? Assume you have a digital signal that is 1 when one of the two pumps is to pump, and 0 when neither pump is to pump. For example, if the pH was very close to desired you wouldn't want to do anything at all so you wouldn't want either pump to turn on..
You still have the other signal that determines which pump is to pump whenever one of the pumps should pump.
Devise a circuit that will ensure that both pumps are OFF when the Pumpsignal is 0 and that the correct pump pumps when the Pump signal is 1.
The circuit you devise in this section will be simple enough that you can probably implement it with a few chips although you will need to look for chips with AND gates and inverters. You should be able to handle that now. Work through the solution in this lesson and try it out in lab if you can.
Example Solution
Let's look at this problem with a truth table. Here's the truth table.
| Pumps On 1 = ON | Pump Choice 0 = S 1 = W | Pump S | Pump W | |
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 |
| 2 | 1 | 0 | 0 | 1 |
| 3 | 1 | 1 | 1 | 0 |
If we examine it closely we see that there is exactly one term in each function. S is 1 only for choice 3, that is when you want PUMPS ONand you want the strong reactant. Similarly, W is 1 only for choice 2. Here's the truth table again. Note the following:
- We have defined Boolean variables here for the various signals, P, C, S, and W.
- We have indicated the inputs by shading them green, and the outputs by shading them orange.
| P | C | S | W | |
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 |
| 2 | 1 | 0 | 0 | 1 |
| 3 | 1 | 1 | 1 | 0 |
QuestionQ2 To check out the circuit you should what?
A QUICK QUESTION Within the simulated circuit, determine the part of the circuit that genrates a 1 when the pumps are ON, and a 0 when they both are OFF.
What If The Problem Isn't So Simple? Not all functions are as simple as this one, and certainly not all can be implemented with just a few gates. However, implementing this simple function gives us a clue how to implement more complex functions.
In the next lesson we'll look at a more general method for implementing functions - a method that uses only AND and OR gates and inverters - but a method which can also be implemented with only NAND gates. We hope that sounds intriguiging to you and that you are looking forward to the next lesson. Click here to go to the lesson on logic functions.
Boolean Algebra Clearly at this point we are entering a realm of a different kind of algebra. We have encountered some example terms in this algebra.
There are some simple things we need to establish before we can proceed.
- An AND gate has this truth table when the inputs are A and B, and the output is C:
| A | B | C |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
- So, clearly we have:
- 0·0 = 0, and
- 1·1 = 1, and
- 0·1 = 0
- An OR gate has this truth table when the inputs are A and B, and the output is C:
| A | B | C |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
- So, clearly we have:
- 0 + 0 = 0, and
- 1 + 1 = 1, and
- 0 + 1 = 1
We also have:
And - - - believe it or not, this result for A + A is very useful because it is a fundamental result that will let us build circuits with fewer gates. We'll come back to that later.
There are some interesting theorems that can be proved. Note the following:
- When we want to prove a theorem we will take the approach that we can prove the theorem by examining all possible combinations of the appropriate variables. We can do that because the possible combinations are finite.
| A | B |  |  |  |  |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
Theorem (de Morgan)
 |
The proof of this theorem is contained in the truth table above which lists every possible combination of A and B, and shows that this result is true.
One final note. There are some further simple facts that come in useful. Note the following:
Boolean Algebra can be a confusing and misleading business. De Morgan's theorem above seems almost trivial. However, there is a very interesting consequence of this theorem. Here it is:
- If you have a Boolean function that is a sum-of-products form it can be implemented using a two layer circuit with the first layer composed of AND gates, and the second layer composed of OR gates.
- Applying deMorgan's theorem to the function the circuit can be built using the same structure, but replacing every AND and OR gate with a NAND gate.
Problems
Links to Other Lessons on Digital Logic
Send your comments on these lessons.
No comments:
Post a Comment