LECTURER NOTES
ON
EC6302-DIGITAL ELECTRONICS
II YEAR /III SEMESTER ECE
ACADEMIC YEAR 2014-2015
D.ANTONYPANDIARAJAN
ASSISTANT PROFESSOR
FMCET
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LECTURER NOTES
ON
EC6302-DIGITAL ELECTRONICS
II YEAR /III SEMESTER ECE
ACADEMIC YEAR 201 4 -201 5
D.ANTONYPANDIARAJAN
ASSISTANT PROFESSOR
FMCET
to Aristotle‘s system of logic. Boole wrote a treatise on the subject in 1854, titled An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities, which codified several rules of relationship between mathematical quantities limited to one of two possible values: true or false, 1 or 0. His mathematical system became known as Boolean algebra. All arithmetic operations performed with Boolean quantities have but one of two possible
Outcomes: either 1 or 0. There is no such thing as ‖2‖ or ‖-1‖ or ‖1/2‖ in the Boolean world. It is a world in which all other possibilities are invalid by fiat. As one might guess, this is not the kind of math you want to use when balancing a check book or calculating current through a resistor.
However, Claude Shannon of MIT fame recognized how Boolean algebra could be applied to on-and-off circuits, where all signals are characterized as either ‖high‖ (1) or ‖low‖ (0). His1938 thesis, titled A Symbolic Analysis of Relay and Switching Circuits, put Boole‘s theoretical work to use in a way Boole never could have imagined, giving us a powerful mathematical tool for designing and analyzing digital circuits.
Like ‖normal‖ algebra, Boolean algebra uses alphabetical letters to denote variables. Unlike ‖normal‖ algebra, though, Boolean variables are always CAPITAL letters, never lowercase.
Because they are allowed to possess only one of two possible values, either 1 or 0, each and every variable has a complement: the opposite of its value. For example, if variable ‖A‖ has a value of 0, then the complement of A has a value of 1. Boolean notation uses a bar above the variable character to denote complementation, like this:
In written form, the complement of ‖A‖ denoted as ‖A-not‖ or ‖A-bar‖. Sometimes a ‖prime‖ symbol is used to represent complementation. For example, A‘ would be the complement of A, much the same as using a prime symbol to denote differentiation in calculus rather than
Introduction:
The English mathematician George Boole (1815-1864) sought to give symbolic form
There is no such thing as subtraction in the realm of Boolean mathematics. Subtraction
Implies the existence of negative numbers: 5 - 3 is the same thing as 5 + (-3), and in Boolean algebra negative quantities are forbidden. There is no such thing as division in Boolean mathematics, either, since division is really nothing more than compounded subtraction, in the same way that multiplication is compounded addition.
Multiplication – AND Gate logic
Multiplication is valid in Boolean algebra, and thankfully it is the same as in real- number algebra: anything multiplied by 0 is 0, and anything multiplied by 1 remains unchanged:
0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1
This set of equations should also look familiar to you: it is the same pattern found in the truth table for an AND gate. In other words, Boolean multiplication corresponds to the logical function of an ‖AND‖ gate, as well as to series switch contacts:
Complementary Function – NOT gate Logic
Boolean complementation finds equivalency in the form of the NOT gate, or a normally closed switch or relay contact:
Boolean Algebraic Identities
value (x + x = 2x), but remember that there is no concept of ‖2‖ in the world of Boolean math, only 1 and 0, so we cannot say that A + A = 2A. Thus, when we add a Boolean quantity to itself, the sum is equal to the original quantity: 0 + 0 = 0, and 1 + 1 = 1.
Introducing the uniquely Boolean concept of complementation into an additive identity, we find an interesting effect. Since there must be one ‖1‖ value between any variable and its complement, and since the sum of any Boolean quantity and 1 is 1, the sum of a variable and its complement must be 1:
Four multiplicative identities: Ax0, Ax1, AxA, and AxA‘. Of these, the first two are no
different from their equivalent expressions in regular algebra:
The third multiplicative identity expresses the result of a Boolean quantity multiplied by
itself. In normal algebra, the product of a variable and itself is the square of that variable (3x 3 = 32 = 9). However, the concept of ‖square‖ implies a quantity of 2, which has no meaning
in Boolean algebra, so we cannot say that A x A = A2. Instead, we find that the product of a Boolean quantity and itself is the original quantity, since 0 x 0 = 0 and
1 x 1 = 1:
The fourth multiplicative identity has no equivalent in regular algebra because it uses the
complement of a variable, a concept unique to Boolean mathematics. Since there must be
one ‖0‖ value between any variable and its complement, and since the product of any Boolean quantity and 0 is 0, the product of a variable and its complement must be 0:
Principle of Duality:
It states that every algebraic expression is deducible from the postulates of Boolean algebra,and it remains valid if the operators & identity elements are interchanged. If the inputs of a NOR gate are inverted we get a AND equivalent circuit. Similarly when the inputs of a NAND gate are inverted, we get a OR equivalent circuit.This property is called DUALITY.
Theorems of Boolean algebra:
The theorems of Boolean algebra can be used to simplify many a complex Boolean expression and also to transform the given expression into a more useful and meaningful equivalent expression. The theorems are presented as pairs, with the two theorems in a given pair being the dual of each other. These theorems can be very easily verified by the method of ‗perfect induction‘. According to this method, the validity of the expression is tested for all possible combinations of values of the variables involved. Also, since the validity of the
Theorem 3(a) is a direct outcome of an AND gate operation, whereas theorem 3(b) represents an OR gate operation when all the inputs of the gate have been tied together. The scope of idempotent laws can be expanded further by considering X to be a term or an expression. For example, let us apply idempotent laws to simplify the following Boolean expression:
Theorem 4 (Complementation Law)
(a) X_X = 0 and (b) X+X = 1
According to this theorem, in general, any Boolean expression when ANDed to its complement yields a ‗0‘ and when ORed to its complement yields a ‗1‘, irrespective of the complexity of the expression:
Hence, theorem 4(a) is proved. Since theorem 4(b) is the dual of theorem 4(a), its proof is implied.
The example below further illustrates the application of complementation laws:
Theorem 5 (Commutative property)
Mathematical identity, called a ‖property‖ or a ‖law,‖ describes how differing
variables relate to each other in a system of numbers. One of these properties is known as
the commutative property, and it applies equally to addition and multiplication. In essence, the commutative property tells us we can reverse the order of variables that are either added together or multiplied together without changing the truth of the expression:
Commutative property of addition
A + B = B + A
Commutative property of multiplication
AB = BA
Theorem 6 (Associative Property)
The Associative Property, again applying equally well to addition and multiplication. This property tells us we can associate groups of added or multiplied variables together with parentheses without altering the truth of the equations.
Associative property of addition
A + (B + C) = (A + B) + C
Associative property of multiplication
A (BC) = (AB) C
Theorem 7 (Distributive Property)
The Distributive Property, illustrating how to expand a Boolean expression formed by the product of a sum, and in reverse shows us how terms may be factored out of Boolean sums-of-products:
Distributive property
A (B + C) = AB + AC
Theorem 8 (Absorption Law or Redundancy Law)
(a) X+X.Y = X and (b) X.(X+Y) = X
The proof of absorption law is straightforward:
X+X.Y = X. (1+Y) = X.1 = X
Theorem 8(b) is the dual of theorem 8(a) and hence stands proved.
The crux of this simplification theorem is that, if a smaller term appears in a larger term, then the larger term is redundant. The following examples further illustrate the underlying concept:
Demorgan‘s Theorem
De-Morgan was a great logician and mathematician. He had contributed much to logic. Among his contribution the following two theorems are important
De-Morgan‘s First Theorem
It States that ―The complement of the sum of the variables is equal to the product of the complement of each variable‖. This theorem may be expressed by the following Boolean expression.
Z=AB‘+A‘C+A‘B‘C‘
Canonical Form of Boolean Expressions
An expanded form of Boolean expression, where each term contains all Boolean variables in their true or complemented form, is also known as the canonical form of the expression. As an illustration, is a Boolean function of three variables expressed in canonical form. This function after simplification reduces to and loses its canonical form.
MIN TERMS AND MAX TERMS Any boolean expression may be expressed in terms of either minterms or maxterms. To do this we must first define the concept of a literal. A literal is a single variable within a term which may or may not be complemented. For an expression with N variables, minterms and maxterms are defined as follows :
- A minterm is the product of N distinct literals where each literal occurs exactly once.
- A maxterm is the sum of N distinct literals where each literal occurs exactly once.
Product-of-Sums Expressions
Standard Forms
A product-of-sums expression contains the product of different terms, with each term being either a single literal or a sum of more than one literal. It can be obtained from the truth table by considering those input combinations that produce a logic ‗0‘ at the output. Each such input combination gives a term, and the product of all such terms gives the expression. Different terms are obtained by taking the sum of the corresponding literals. Here, ‗0‘ and ‗1‘ respectively mean the uncomplemented and complemented variables, unlike sum-of-products expressions where ‗0‘ and ‗1‘ respectively mean complemented and uncomplemented variables.
Since each term in the case of the product-of-sums expression is going to be the sum of literals, this implies that it is going to be implemented using an OR operation. Now, an OR gate produces a logic ‗0‘ only when all its inputs are in the logic ‗0‘ state, which means that the first term corresponding to the second row of the truth table will be A+B+C. The product- of-sums Boolean expression for this truth table is given by Transforming the given product- of-sums expression into an equivalent sum-of-products expression is a straightforward process. Multiplying out the given expression and carrying out the obvious simplification provides the equivalent sum-of-products expression:
A given sum-of-products expression can be transformed into an equivalent product-of-sums expression by (a) taking the dual of the given expression, (b) multiplying out different terms to get the sum-of products form, (c) removing redundancy and (d) taking a dual to get the equivalent product-of-sums expression. As an illustration, let us find the equivalent product- of-sums expression of the sum-of products expression
The dual of the given expression =
The style of row identification need not be the same as that of column identification as long as it meets the basic requirement with respect to adjacent terms. It is, however, accepted practice to adopt a uniform style of row and column identification. Also, the style shown in the figure below is more commonly used. A similar discussion applies for maxterm Karnaugh maps. Having drawn the Karnaugh map, the next step is to form groups of 1s as per the following guidelines:
- Each square containing a ‗1‘ must be considered at least once, although it can be considered as often as desired.
- The objective should be to account for all the marked squares in the minimum number of groups.
- The number of squares in a group must always be a power of 2, i.e. groups can have 1,2, 4_ 8, 16, squares.
- Each group should be as large as possible, which means that a square should not be accounted for by itself if it can be accounted for by a group of two squares; a group of two squares should not be made if the involved squares can be included in a group of four squares and so on.
- ‗Don‘t care‘ entries can be used in accounting for all of 1-squares to make optimum groups. They are marked ‗X‘ in the corresponding squares. It is, however, not necessary to account for all ‗don‘t care‘ entries. Only such entries that can be used to advantage should be used.
Two variable K Map
Three variable K Map
Four variable K Map
Boolean function
To illustrate the process of forming groups and then writing the corresponding minimized Boolean expression, The below figures respectively show minterm and maxterm Karnaugh maps for the Boolean functions expressed by the below equations. The minimized expressions as deduced from Karnaugh maps in the two cases are given by Equation in the case of the minterm Karnaugh map and Equation in the case of the maxterm Karnaugh map:
Quine–McCluskey Tabular Method
The Quine–McCluskey tabular method of simplification is based on the complementation theorem, which says that
where X represents either a variable or a term or an expression and Y is a variable. This theorem implies that, if a Boolean expression contains two terms that differ only in one variable, then they can be combined together and replaced with a term that is smaller by one literal. The same procedure is applied for the other pairs of terms wherever such a reduction is possible. All these terms reduced by one literal are further examined to see if they can be reduced further. The process continues until the terms become irreducible. The irreducible terms are called prime implicants. An optimum set of prime implicants that can account for all the original terms then constitutes the minimized expression. The technique can be applied equally well for minimizing sum-of-products and product of-
sums expressions and is particularly useful for Boolean functions having more than six variables as it can be mechanized and run on a computer. On the other hand, the Karnaugh mapping method, to be discussed later, is a graphical method and becomes very cumbersome when the number of variables exceeds six. The step-by-step procedure for application of the tabular method for minimizing Boolean expressions,both sum-of-products and product-of- sums, is outlined as follows:
- The Boolean expression to be simplified is expanded if it is not in expanded form.
- Different terms in the expression are divided into groups depending upon the number of 1s they have.