Topic EC262 2: Introduction to Combinational Systems
Much of a computer's hardware is comprised of digital logic circuits. Digital logic circuits are built from just a handful of primitive elements, called logic gates, combined in various ways. In a digital logic circuit, only two values may be present. The values may be -5 and + 5 volts. Or the values may be 0.5 and 3.5 volts. Or the values may be... you get the picture. To allow consideration of all of these possibilities, we will say that digital logic circuits allow the presence of two logical values: 0 and 1. So, signals in a digital logic circuit take on the values of 0 or 1. Logic gates are devices which compute functions of these binary signals. The Three Basic Logic Gates AND Gate. Consider the circuit below which has 2 switches in series:
Switch S1
Switch S2
Light
When will the light turn on? We can represent the light's operation using a table: S1 open open shut shut S2 open shut open shut Light off off off on
Change the words open and off to 0 and the words shut and on to 1 and the table becomes S1 0 0 1 1 S2 0 1 0 1 Light 0 0 0 1 This is the truth table for the AND operation.
The simplest AND gate gas two inputs and one output. If both inputs are 1 then the output is 1. Otherwise, the output is 0. The symbol for the AND gate is:
We represent the AND operation by using either the multiplication symbol (i.e., " ") or by writing the inputs together. Thus, for the AND gate above, we would write the output c as c = a b or as c = ab. This would be pronounced: "C = a and b." 1
The truth table for the AND gate is shown below. The output c = ab is equal to 1 if and only if (iff) a is 1 and b is 1. Otherwise, the output is 0. a b c
(Some textbooks use the symbol for the AND operation; i.e., some textbooks would use the notation c= a b . We will not use this notation in this EC262, but you may see it in other classes in the future.) AND gates can have more than one input (however, an AND gate always has just a single output). Let's consider a three-input AND gate:
Write the expression for x : In words, how would you describe x ?
Let's write the truth table for the 3-input AND gate. Think: How many lines will be in a truth table with 3 inputs? Can you think of a way to present the table in an organized manner?
OR Gate. Now consider the circuit that has 2 switches in parallel.
Switch S1
Switch S2 Light
When will the light turn on? 2
We can represent the light's operation using a table S1 open open shut shut S2 open shut open shut Light off on on on
Changing the words open and off to 0 and the words shut and on to 1 and the table becomes S1 0 0 1 1 S2 0 1 0 1 Light 0 1 1 1 This is the truth table for the OR operation.
The simplest OR gate has two inputs and one output. If either or both inputs are 1, the output is 1. Otherwise, the output is 0. The symbol for the OR gate is:
We represent the OR operation by using the addition symbol. Thus, for the OR gate above, we would write the output c as c = a + b. This would be pronounced: "C = a or b." The truth table for the OR gate is shown below. The output is 1 if a is 1 or b is 1. Otherwise, the output is 0. a b c
(Some textbooks use the symbol for the OR operation; i.e., some textbooks would use the notation c= a b . We will not use this notation in this EC262, but you may see it in other classes in the future.) OR gates can have more than one input (however, an OR gate always has just a single output). Let's consider a three-input OR gate:
Write the expression for x : In words, how would you describe x ?
Let's write the truth table for the 3-input OR gate: T/F: OR gates can have any number of inputs. T/F: AND gates can have any number of inputs. 3
NOT gate (inverter) The NOT gate always has one input and one output. If the input is 1, the output is 0. If the input is 0, the output is 1.
P
P'
The math symbol for negation is an apostrophe. If the input to a NOT gate is P, the output, termed the complement, is denoted as P'. The symbol for a NOT gate is supposed to depict an amplifier followed by a bubble. Sometimes the NOT operation is represented by just the bubble, without the amplifier. We will see this later. (Some textbooks use the over-bar symbol or a preceding tilde symbol for the NOT operation; i.e., some textbooks would use the notation P or ~P. We will not use this notation in this EC262, but you may see it in other classes in the future.) Combinational Logic Circuits A combinational logic circuit is a digital logic circuit whose output is determined only by the current values of the inputs (with no dependence on past input values). A combinational circuit is built up by combining our three basic logic gates with the following provisos: The output of a gate may not eventually feed back to that same gate. The output lines from 2 different gates cannot be combined.
The following are not combinational circuits (Why not?):
A B
A
C D
B
E
It is very convenient to describe combinational logic circuits in terms of Boolean expressions, which can be manipulated using Boolean algebra. (Your textbook uses the term switching algebra, but most texts use the term Boolean algebra--we'll use both terms interchangeably!) You have already seen several Boolean expressions. For example, if the inputs to an OR gate are X and Y, the output is X + Y. The terms X, Y and X + Y are all Boolean expressions because they can only take on two values (0 or 1). Similarly, if the inputs to an AND gate are X and Y, the output is XY. The term XY is a Boolean expression. If X enters a NOT gate, the output is X'. X' is a Boolean expression. Note that since the OR operation uses the ordinary plus sign, the output of an OR gate--the term X + Y--is called a sum term. Similarly, since the AND operation uses the ordinary multiplication sign or no symbol at all, the output of an AND gate--the term XY or X Y --is called a product Our term. study of combinational circuits will involve five (very much related!) tasks (Task 1) Given a digital logic circuit, write the corresponding Boolean expression for the circuit. (Task 2) Given a Boolean expression for a digital logic circuit, construct the corresponding truth table 4
Example. Write the Boolean expression for the output of the logic circuit shown below (Task 1), and then construct the truth table for it (Task 2).
p q
The Boolean expression corresponding to a digital logic circuit can be determined by evaluating the effect of the logic gates on the input expressions. To develop the truth table for a complex statement we must consider each combination of truth values for the component statements. To write the truth table for __________ we start by making columns for p and q, considering all possible combinations of 1 and 0. How many rows will be in this truth table? p q
Now, use the definition of NOT to write a column for p'. Then, write a column for ________ using the definition of OR. p q
Note that sometimes it is easy to construct the truth table directly from the logic circuit (by tracing through the circuit for each possible combination of input values) without first deriving the Boolean expression. Example. Write the Boolean expression for the logic circuit shown below (Task 1), and then construct the truth table for it (Task 2).
p q
Evaluating the effect of the logic gates on the input expressions, we see:
Write a truth table for 5
p
q
Example. Write the Boolean expression for the logic circuit shown below (Task 1), and then construct the truth table for it (Task 2).
p q r
Evaluating the effect of the logic gates on the input expressions, we see: How many rows will be in this truth table? The truth table is
We mentioned that we have five (very much related!) tasks. Let's turn to Task 3. (Task 3) Given a Boolean expression, draw a digital logic circuit that represents this expression.
The basic approach is to write the expression on the right side of the page and then draw the circuit from right to left, by working in from the outermost part of the expression. Example: Construct a digital logic circuit that will implement the Boolean expression ( p q ') + p ' . At the highest level, we are
Now, let's continue to deconstruct this by looking at the top term,
6
Finally,
It is worth pausing to note the "order of precedence" for Boolean operations. If we have an expression with ANDs, ORs and NOTs, what is the correct order for performing the operations? To make this question more concrete, if we have the expression xy' does this correspond to the circuit
or the circuit The order of precedence is: Example: Draw a digital logic circuit that represents the Boolean expression x + ( y ( x z ') '
)
7
We mentioned that we have five (very much related!) tasks. Let's turn to Task 4. (Task 4) Given a truth table, construct a Boolean expression for the truth table.
We will first just mechanically run through the procedure. Then, once you appreciate the process involved, we will delve into why this procedure works. Suppose the inputs to the circuit are denoted x1 , x 2 ,... x n . Step 1. Identify the rows of the truth table that have an output of 1. Step 2. For each such row, form the Boolean expression. z1 z 2 ... z n where z i = x i if x i = 1 in this row, and zi = x i ' if x i = 0 in this row. Step 3. After step 2, we will have one Boolean expression corresponding to each row that had an output of 1. Take the OR of all these Boolean expressions. The result is the Boolean expression for the truth table under evaluation. We can then, if desired, form the digital logic circuit from this Boolean expression using the technique discussed in TASK 3 above. Example: Construct a logic circuit for the truth table shown. p 0 0 1 1 q 0 1 0 1 output 0 1 0 1
8
Example: Construct a logic circuit for the truth table shown. A B C Output
9
Recall our procedure from page 8 for constructing a Boolean expression from a truth table. We mentioned that we would first just mechanically run through the procedure, and then delve into why this procedure works. Think: Why does this procedure successfully construct a Boolean expression from a truth table?
If you think about it, we have covered how to perform the transitions outlined below:
It is quite possible to construct the truth table directly from the logic circuit (without first deriving the Boolean expression) by tracing through the circuit for each possible combination of input values, but, except for simple circuits, it is usually easier (and less prone to errors) to go via the Boolean expression as shown above. Terminology: Any appearance of a variable (or its complement) is called a literal. Example. How many literals are in this expression: ab' + bc'd + a'd + e
One or more literals connected by an AND operator is called a product term. Example. How many product terms are in this expression: ab' + bc'd + a'd + e
A minterm is a product term that includes all of the input variables of a problem (either complemented or uncomplemented). Example. Suppose our circuit has 4 inputs: a, b, c and d. Which of the following are minterms? abcd abc a'bcd' b'c'd'
A Sum of Products (SOP) consists of one or more product terms connected by an OR operator. A canonical sum is a SOP expression in which each individual product term is a minterm. In light of this terminology, or Task 4 (Given a truth table, construct a Boolean expression for the truth table) really involved deriving a canonical sum from the truth table. We mentioned that we have five (very much related!) tasks. Let's turn to Task 5. (Task 5) Given a digital logic circuit, design a simpler digital logic circuit that performs the equivalent function.
This topic is somewhat more involved...and will be our next topic! Reading: Marcovitz, pages 29-58 (skim) Homework: Marcovitz Chapter 2 Exercise 10 (a and b) 10