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When an object at very high temperature is suddenly dropped in a cooler liquid and if it is assumed that the conductive resistance of the solid is very small in comparison to the surrounding convective resistance then the heat transfer analysis is called as lumped capacitance analysis (as shown in the figure given below). Here, we treat the system as a lump. In that case, we can assume that the rate of change of internal energy of lump will be equal to the heat interaction with the surrounding fluid. Mathematically, this can be written as − $$\mathrm{pcV\frac{\partial T}{\partial t} ... Read More

Before focusing on logical expression in SSOP (Standard Sum of Products) form and SPOS (Standard Product of Sum) form, let us have a brief introduction the "Sum of Products" and "Product of Sum" forms. SOP (Sum of Products) Form The SOP or Sum of Products form is a form of expressing a logical or Boolean expression. In SOP, different product terms of input variables are logically ORed together. Therefore, in the case of SOP form, we first logically AND the input variables, and then all these product terms are summed together with the help of logical OR operation. For example ... Read More

Electronic logic conventions are the sets of rules followed while designing a digital logic system or device. These conventions are adopted due to their different characteristics observed by several experiments. The use of electronic logic conventions, makes the implementation process of a digit system easy and smooth. Also, a standardization is achieved in the design. This tutorial is entirely meant for explaining different electronic logic conventions used in digital system implementations. As we know, the digital systems are implemented in binary number system due to some technical and economic reasons. The binary number system follows Boolean’s rules to perform arithmetic ... Read More

It is the most straightforward iterative strategy for tackling systems of linear equations shown below. $$\mathrm{a_{1, 1}\: x_{1} \: + \: a_{1, 2} \: x_{2} \: + \: \dotso\dotso \: + \: a_{1, n} \: x_{n} \: = \: b_{1}}$$ $$\mathrm{a_{2, 1} \: x_{1} \: + \: a_{2, 2} \: x_{2} \: + \: \dotso\dotso \: + \: a_{2, n} \: x_{n} \: = \: b_{2}}$$ $$\mathrm{\vdots}$$ $$\mathrm{a_{n, 1} \: x_{1} \: + \: a_{n, 2} \: x_{2} \: + \: \dotso\dotso \: + \: a_{n, n} \: x_{n} \: = \: b_{n}}$$ The fundamental concept is: each linear ... Read More

K-Map or Karnaugh Map is a graphical method of simplifying Boolean expression. A K-Map composed of an arrangement of adjacent squares or cells, where each cell represent a particular combination of variables in sum or product form. In the K-map method, there is a useful condition namely, Don’t Care Condition, which helps in simplifying a Boolean function. The don’t care condition makes the grouping of variables in K-map easy. In this tutorial, we will understand the "don’t care" concept in K-map reduction with the help of solved examples. Sometimes, in a Boolean expression for certain input combinations, the value of ... Read More

In Boolean algebra, several rules are defined to perform operations in digital logic circuits. Boolean algebra is a tool to perform operation on binary digits, i.e. 0 and 1. These two binary digits 0 and 1 are used to denote FALSE and TRUE states of a digital circuit at input and output ends. Boolean algebra, developed by George Boole, uses 0s and 1s to create truth tables and logic expressions of digital circuits like AND, OR, NOT, etc. which are used to analyze and simplify the complex circuits. There were another English mathematician Augustus De Morgan who explained the NAND ... Read More

A Boolean function can be expressed into two forms namely, Sum of Products (SOP) Form Product of Sums (SOP) Form The SOP (Sum of Products) form is one in which the Boolean function is expressed as the sum of product terms, while in the POS (Product of Sums) form, the Boolean function is expressed as the product of sum terms of the function. But, in the SOP and POS form, each term of the function may not contain all the variables. For example, consider a Boolean function in three variables, $$\mathrm{\mathit{f}\lgroup A, B, C\rgroup=A\overline{B}+\overline{B}C}$$ This is the ... Read More

When a Boolean expression is represented as a product of sum terms, it is called POS (Product of Sums) form. In POS form, each sum term of the expression may not contain all the variables. On the other hand, when the Boolean expression is represented as a product of sum terms, where each sum term contains all the variables of the function, it is called Standard Product of Sums (SPOS) form. In the Standard POS form, each sum term of the Boolean expression is called a maxterm. Now, let us discuss the expansion of a Boolean expression in POS form ... Read More

When a logical expression or Boolean function is expressed as a sum of minterms or as a product of maxterms, then it is called the canonical form of the expression or function. A canonical form of the Boolean expression is also known as standards form, i.e. Standard Sum of Products (SSOP) Form and Standard Product of Sums (SPOS) Form. The canonical form of a Boolean function involves minterms and maxterms. A minterm is a product term which contains all the variables of the Boolean function either in complemented or un-complemented form. A maxterm is a sum term which ... Read More

K-Map or Karnaugh Map is a simplification technique used to minimize a given complex Boolean function. K-Map or Karnaugh Map is a graph or chart which is composed of an arrangement of adjacent cells, where each cell of the K-Map represents a particular combination of variables in either sum or product form. The K-map can be used to simplify Boolean functions involving any number of variables. But, the simplification of a Boolean function using K-map becomes very complex for expressions involving five or more variables. Therefore, in actual practice, the K-map is limited to six variables. The number of cells ... Read More