An Introduction to Digital Logic - Signals and Gates
What Are You Going To Learn?
Logic Signals And Gates
AND
OR
NOT (Inverter)
NAND

Wiring A Gate
A NAND Simulation

An Example Problem
Boolean Algebra
Problems
You are at:  Elements - Logic Circuits - Gates
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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.


What Are You Going to Learn?

        There are at least two general areas you need to become familiar with.

Here is a short list of the topics you will learn.
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).
   Given an AND, OR, NAND or NOT gate,
   Be able to determine the output of the gate given the input logic signals.
   Given a system that requires gates,
Be able to wire a chip correctly, and to  check that the chip is functioning properly.

Logic Signals

        There are a number of different systems for representing binary information in physical systems.  Here are a few.


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.


Think Binary!

        Let's examine a typical situation.  You have some sort of device that generates a logic signal.

        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

An AND Example

        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

Then, the truth table below shows when the patient is in danger.
 

A B C
False False False
False True False
True False False
True True True

Notice that C is TRUE when both A AND B are true and only then!


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).



Question

Q1.  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.

C = A·B  or C = AB

        Some will prefer always to insert the dot between the variables so that the AND operation is clearly indicated.  Many times, the  context will allow you just to use AB, without a dot between A and B, but if there is a variable named AB, then confusion can arise.


Problems

        Assume you have an AND gate with two inputs, A and B.  Determine the output,
C, for the following cases.

P1.  A = 1, B = 0

P2.  A = 0, B = 1

P3.  If either input is zero, what is the output?

P4.  A = 1, B = 1



        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

        What we have done is defined an OR gate.  An OR gate is a gate for which the output is 1 whenever one or more of the inputs is 1.  The output of an OR gate is 0 only when all inputs are 0.  Shown below is a schematic symbol for an OR gate, together with the simulated LEDs and input buttons so that you can explore OR gate behavior.

In terms of Boolean variables, the truth table for an OR gate looks like this.
 
 


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

P6.  A = 0, B = 1

P7.  If either input is one, what is the output?


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:

        The truth table for an inverter is pretty simple since there is only one input.  Call the input A, and the output C, and the truth table is:

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.

Notice the simple way we can use a switch and a five volt supply to produce a single logic signal that is ""0"" (ground) or 1 (5 volts).


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.

Here is a simulated NAND gate.  Check it out.



 
 


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.

        Notice that this chip has fourteen pins.

        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.

        The first step in wiring the 7400 is to connect the positive power supply.  Use a five volt (5v) power supply and don't turn it on yet.  Connect a lead to pin 14 as shown below, and connect the other end of that lead to a 5v supply.  Keep the power supply turned off until you have everything connected.  Here's what that looks like when the positive supply voltage to the chip is wired.

        The next step in wiring the 7400 is to connect the ground connection.  Connect a lead to pin 7 as shown below, and connect the other end of that lead to ground.

        Notice the pattern to this connection.  The power to this digital logic chip goes to the corners.  Remember, power to the corners for logic chips.

        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.

        Now you can connect the output of the gate.  You will need to connect this output to something like a voltmeter or an oscilloscope so that you can measure and observe the output of the gate.  (And the voltmeter or oscilloscope will also have to be connected to the ground.  You will measure output voltage with respect to ground.)  The output will be near 5v when the output is a 1 and near 0v when the output is a 0.

        Actually, you can often connect LEDs to give a visual indication of a 1 (LED lighted) or a 0 (LED dark).  Here some LEDs are shown, together with 1kW current limiting resistors.  If you connect LED indicators to your circuit remember that an LED is not the same in both directions, and you have to get the correct end connected to the resistor.  The other end of each LED is connected to ground (or just "grounded").  Here's the circuit to show the output of a NAND gate:

When the output of the gate is a 1, the output voltage will be five (5) volts.  Current will flow through the series combination of the resistor and the LED, so the LED will light.  When the output of the gate is a 0, the output voltage will be zero (0) volts and the LED will not be lit.  Thus, the LED lights up when the output is a 1, and doesn't light when the output is a 0.  You can use this indication scheme to show the status for any signal.  (It doesn't have to be the output of a gate.)

       Click here for an introductory laboratory on the 7400 chip.


Question

Q1        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.

Check out how the circuit works and note all of the connections that you need to make to ensure that the chip works as it is supposed to work.


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

        In English, we say to turn Pump S (Strong reactant) ON when the pumps are ON, and the strong reactant is chosen (Choice 3) and to turn Pump W (Weak reactant) ON when the pumps are ON and the weak reactant is chosen (Choice 2).  Otherwise, do nothing.

        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 ON and you want the strong reactant.  Similarly, W is 1 only for choice 2.  Here's the truth table again.  Note the following:


P C S W
0 0 0 0 0
1 0 1 0 0
2 1 0 0 1
3 1 1 1 0

        Looking at the statement "S is 1 . . . when you want Pumps ON AND you want the strong reactant" then you can generate a logic expression directly from the statement.

and also:

        Finally, realize that it doesn't take much to implement these functions.  Note you only need one inverter and two AND gates.  Here's the circuit that turns the pumps on at the proper time.  This is an interactive simulation of the circuit, so you can toggle the switches with the push buttons.  Check it out.


Question

Q2   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.

and:

        The algebra is unusual because the variables in the algebra (S, P, C and W in the example) can take only two values, 0 and 1.  In this section we will examine some of the properties of this algebra, and the implications of what we have already learned.

        There are some simple things we need to establish before we can proceed.


A B C
0 0 0
0 1 0
1 0 0
1 1 1
        Which may be exactly what you expected.

        We also need to consider an OR gate.


 

A B C
0 0 0
0 1 1
1 0 1
1 1 1
        Now, if you are taking a college course, and you write home that 1 + 1 = 1 is what you just learned, your parents may want your tuition refunded.

        Now, if you accepted what was claimed above, then you also have to accept the following:

A·A = A

Just let A be either zero or one and remember the truth table for an AND.

        We also have:

A + A = A

Again, just let A be either zero or one and remember the truth table for an OR.

        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:

These two little theorems will prove to be useful.  To prove these theorems you only need to know about the properties of AND, OR and NOT gates.  That is left as an exercise for you.

Here is a truth table.  It lists all possible combinations for two variables.
 
 

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

        This truth table proves the following theorem.

Theorem (de Morgan)
 
 


Proof
        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:


and:


        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:         First, you need to understand how to expand a function in terms of minterms. Click here to go to that lesson.
Problems

Links to Other Lessons on Digital Logic
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