Digital Electronics And Logic Design

 Converting Logic Diagrams To NAND / NOR Logic Since, NAND logic and NOR logic are universal logic system, digital circuits which are first computed and converted to AOI logic may then be converted to either NAND logic or NOR logic depending on the choice. 1 NAND-NAND Logic A logic network can be converted into NAND-NAND gate network by going through following steps: Methodology: To Obtain NAND-NAND Gate Network Step 1: First draw the circuit in AOI logic i.e., using AND, OR and NOT gates. Step 2: Add a circle (bubble) at the output of each AND gate and at the inputs to all the OR gates. Step 3: Add an inverter on each line that received only one circle in steps 2, so that the polarity of signals on those lines remains unchanged from that of the original diagram. Step 4: Replace bubbled OR by NAND and each inverter by its  NAND equivalent. 2 NOR-NOR Logic The procedure of converting an AOI logic to NOR-NOR logic is same as above except steps 2 and 4. Methodology: To Obtain NOR-NOR Gate

Boolean Analysis of Logic Circuits A Boolean function may be transformed from an algebraic expression into a logic diagram using AND, OR and NOT gates. This is also referred to as AOI logic. Conversely, a logic circuit can be transformed into Boolean expressions for the analysis. 1 Converting Boolean Expressions to Logic Diagram 2 Converting Logic to Boolean Expressions Any logic circuit, no matter how complex it is, can be described using Boolean expressions. To derive the Boolean expression for a given logic circuit, start from the left-most input and work toward the final output, writing output for each gate. As an example, consider the logic diagram shown in Figure. We go through the following steps to get the Boolean expression.  REFERENCES 1.          Digital Design by M. Morris Mano, Michael D Ciletti, Pearson 2.          Digital Fundamentals by Thomas L. Floyd, R. P. Jain, Pearson 3.          Digital Circuits and Design by S Salivahanan, S Arivazhagan, Vikas Publishing House Pv

Alternate Logic-Gate Representations We have discussed the five basic logic gates (AND, OR, INVERTER, NAND, and NOR) and the standard symbols used to represent them in a logic circuit diagram. Most of the logic networks use standard symbols. But in some networks an alternative set of symbols is used in addition to the standard symbols. The table shows the alternate set of symbols for the five basic gates. Table: Alternate Logic Gate Representations To convert any normal symbol to its corresponding alternate symbol, the following steps are used: METHODOLOGY: TO CONVERT STANDARD SYMBOL TO ALTERNATE SYMBOL Step 1: Add bubbles (indication of inversion) at those input or output points where it is not present. Step 2: Remove all pre-existing bubbles of the normal symbol, if there is any at the point (only NOT, NAND and NOR gates) Step 3: If the existing normal logic symbol is AND, change it to OR, Similarly, if it is OR, then change it to AND. There is no change for the triangular symbol of

UNIVERSAL GATE NAND and NOR gates are known as universal gates because any of these two gates is capable of implementing all other gate functions. 1. NAND Gate as a Universal Gate The NAND gate can be used to implement the NOT function, AND function, the OR function and other functions also as explained below. The NOT Gate using NAND Gate An inverter can be made from a NAND gate by connecting all of the inputs together and creating, in effect, a single common input, as shown in Figure 2.10, for a two-input NAND gate. Algebraically, we may write The AND Gate Using NAND Gate To construct an AND gate from NAND gates, an inverter or a NOT gate is required to invert the output of a NAND gate. This inversion cancels out the first inverted operation of NAND gate and the final result will be AND function as depicted in Figure. Algebraically, The OR Gate using NAND Gate To construct OR function using only NAND gates, first we transform the OR function as follows. The above equat

  Truth Table A  truth table represents the relation between all inputs and possible outputs of any logic device or logic circuit in a tabular form. The number of inputs may vary from one to many depending upon the device or complexity of the circuit. Number of output also varies in this way and may be one or more. For different digital circuits, some of the examples of truth table are given below. Table: Examples of Truth Tables for 1-input, 2-input and 3-input Circuits REFERENCES 1.          Digital Design by M. Morris Mano, Michael D Ciletti, Pearson 2.          Digital Fundamentals by Thomas L. Floyd, R. P. Jain, Pearson 3.          Digital Circuits and Design by S Salivahanan, S Arivazhagan, Vikas Publishing House Pvt Ltd. 4.          Digital Systems by Ronald J. Tocci, Neal S. Widmer, Gregory L. Moss, Pearson 5.          Digital Electronics Principles And Integrated Circuits by Anil K. Maini 6.          Fundamentals of Digital Circuits by Anand Kumar 7.          Digital Electroni

Theorems of Boolean Algebra The theorems of Boolean algebra can be used to simplify many complex Boolean expression and also to transform the given expression into a more useful and meaningful equivalent expression. These theorems are discussed as below. 1 Complementation Laws The term complement implies to invert, i.e. to change 1’s to 0’s and 0’s to 1’s. The five laws of complementation are as follows: 1. The complement of 0 is 1, i.e. Ō = 1 2. The complement of 1 is 0, i.e. Ī = 0 3. If A = 0, then Ā = 1 4. If A = 1, then Ā  = 0 5. The double complementation does not change the function, i.e. 2 AND Laws The four AND laws are as follows: 3 OR Laws The four OR laws are as follows: 4 Commutative Laws Commutative law states that the order of the variable in OR and AND operations is not important. The two commutative laws are A + B = B + A A * B = B * A 5 Associative Laws Associative law states that the grouping of variables in AND or OR expression does not affect the result. There are tw

LOGIC GATES Logic gates are the fundamental building blocks of digital systems. Logic gates are electronic circuits that perform the most elementary Boolean operations. Before understanding the logic gates, we must understand the meaning of positive and negative logic. Types of Logic Gates Logic gates are electronic circuits with a number of inputs and one output. There are three basic logic gates, namely OR gate, AND gate, NOT gate Other logic gates that are derived from these basic gates are NAND gate, NOR gate, EXCLUSIVE-OR gate, EXCLUSIVE-NOR gate AND Gate An AND gate is a logic circuit with two or more inputs and one output that performs ANDing operation. The output of an AND gate is HIGH only when all of its inputs are in the HIGH state. In all other cases, the output is LOW. For a positive logic systems, it means that the output of the AND gate is a logic ‘1’ only when all of its inputs are in logic ‘1’ state. In all other cases, the output is logic ‘0’. The logic symbol and the

Logic Levels Boolean logic variable ‘0’ or ‘1’ is not used to represent actual numbers but it is used to represent the state of voltage variable called logical level. Commonly used representation of logic levels are shown in Table below. Table: Representation of Logic Levels for Boolean Variables Logic 0 Logic 1 False True Open switch Close switch Low High No Yes OFF ON Inputs and outputs of logic gates can occur only in two levels. These two levels are termed HIGH and LOW, or TRUE and FALSE, or ON and OFF, or simply 1 and 0. There are two different ways to assign a signal value to logic level such as positive logic and negative logic.  1. Positive Logic : If higher of the two voltage levels represents a logic ‘1’ and the lower of the two levels represents a logic ‘0’, then the logic system is referred to as a positive logic system. Figure 1 s