Introduction The second type of digital circuitry are the sequential circuits. These circuitry are formed as a combination of combinational and the memory element. The memory element stores the state of the circuit. We start discussing on Latches and Flip-Flops. Latches Storage elements that operate with signal levels (i.e. level triggered of signal input) are termed as latches. A latch is a bi-stable element having two states Q and Q’. It is a sequential circuit that will continuously change its state as soon as the input changes without waiting for the clock signal. A latch usually has an enable input. When the enable is active the output changes as soon as there is a change in the inputs. Latches are not of practical

# Adder Circuits

Introduction In this article we will discuss the design of the arithmetic adder and subtracter circuits. These circuits will take input bits and produce two outputs sum and carry or difference and borrow. Half adder A half adder is a logic circuit that adds two bits at a time. These two bits may be the bits of two binary numbers or a bit of one number and a carry from previous stage. This circuit take two inputs and produces two outputs a Sum and a Carry. Truth table of a half adder The Sum output has two minterms and the carry output has one minterm. So the output equation in SOP form is Boolean equation are: Sum = A’B + AB’ Carry = AB The above equations do

# K-Map Redundant and Don’t Care

Introduction In Simplification using K-Map we used minterms or Maxterms to group 1's or 0's respectively to form pair, quad or octets. There may be situations where all the 1's which are part of one group are overlapped by other groups or certain input conditions may not be contributing in getting the minterms or maxterms of the output functions. In the following discussions, we will see how to account for such conditions. Redundant Groups A redundant group is one whose all the 1’s have been used or ovelapped by other groups. We can always eliminate such group. Let us take an example and understand how redundant group can be dropped thus not part of solution. Example: Simplify the SOP equation given by F(ABC) = ∑(2,3,5,7). Figure-1 There

# Simplification using K-Map

K-Map A K-Map is a pictorial representation of the Boolean expression. By entering the values of the minterm or Maxterm an SOP or POS equation can be very easily simplified. It is very easy tool for simplifying up to 5-variable Boolean equation, but as the variable increases, solution become tedious. To begin, we will learn how to draw a 2, 3 and 4 variable K-Map. Draw 2-Variable K-Map Draw 3-Variable K-Map Draw 4-Variable K-Map Simplification Using K-Map Simplifying 2-Variable SOP equation Ex-1: Simplifying F= A'B + AB Ex-2: Simplify A'B + AB + AB' Simplifying 3-Variable SOP equation Ex-3: Ex-4 Simplifying 4-Variable SOP Equation Ex-5 Simplify F(ABCD)=∑(m1, m3, m5, m7, m9, m10,m13, m15) Ans: D Practice Problem: Simplify F= A'B'+A'B+AB' Simplify F(WXY)=∑m(1,2,3,5,7) Simplify F(ABC)=∑m(2,3,4,5 ) Simplify F(ABC)=∑m(2,3,7) + d(1,5) Simplify F(ABCD)=∑m(2,3,5,7,10,15)+d(0, 4,9,13)

# Boolean Function

Boolean function A Boolean function is a Boolean algebraic equation derived generally from the word statement of the given problem or from the truth table. Such a function has two or more input variables. The function will produce a LOW or HIGH output when certain combinations of the input signal (binary value) is applied to the input variables. Two forms of Boolean functions Sum of Product Product of Sum Sum of Product: A sum of product (or SOP) equation is obtained by ORing the product or the minterms Minterm : A Minterm is also called as product term is one that produces a HIGH output when the values of the input variables are ANDed together. It is written for a ‘1’ or HIGH values in