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.
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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:
o Formal logic - as taught by many philosophy departments
o Fuzzy logic - a tool used to design control systems and many other
systems.
o 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.
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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.
Here is a short list of the topics you will learn.
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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).
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.
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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.
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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.
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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
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
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.".
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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).
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.
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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.
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.
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.
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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.
o 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.
o 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 1k.
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.
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 Pump Pump
0=S S
W
1=W
00
0
0
0
1 0
1
0
0
21
0
0
1
31
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:
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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
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.
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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
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1
1
So, clearly we have:
o 0·0 = 0, and
o 1·1 = 1, and
o 0·1 = 0
Which may be exactly what you expected.
We also need to consider an OR gate.
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An OR gate has this truth table when the inputs are A and B, and the output
is C:
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A
B
C
0
0
0
0
1
1
1
0
1
1
1
1
So, clearly we have:
o 0 + 0 = 0, and
o 1 + 1 = 1, and
o 0+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.
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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.
Here is a truth table. It lists all possible combinations for two variables.
AB
0 01
1
1
1
0 1 1
0
0
0
1 00
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:
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.
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First, you need to understand how to expand a function in terms of minterms.
Click here to go to that lesson.
Problems
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Problem Logic1P01 - Building a Three-Input AND.
Links to Other Lessons on Digital Logic
Gates
o Minterms
o Karnaugh Maps
o Flip-Flops
o
Counters
o Memory Elements
Digital Logic Laboratories
o
o