Showing students that Karnaugh maps are really nothing more than truth tables in disguise helps them to more readily learn this powerful new tool. Question 5 Here is a truth table for a four-input logic circuit: If we translate this truth table into a Karnaugh map, we obtain the following result:
Make choices in cell selection to achieve a desired circuit result. Use manual and software methods for circuit minimisation. Karnaugh Maps offer a graphical method of reducing a digital circuit to its minimum number of gates. The map is a simple table containing 1s and 0s that can express a truth table or complex Boolean expression describing the operation of a digital circuit.
The map is then used to work out the minimum number of gates needed, by graphical means rather than by algebra.
Karnaugh maps can be used on small circuits having two or three inputs as an alternative to Boolean algebra, and on more complex circuits having up to 6 inputs, it can provide quicker and simpler minimisation than Boolean algebra. Constructing Karnaugh Maps Fig.
The map needs one cell for each possible binary word applied to the inputs. Notice that this edge numbering does not follow the normal binary counting sequence, but uses a Gray Code sequence where only one bit changes from one cell to the next. This is an important feature of Karnaugh maps; get the sequence wrong and the map will not work!
The input labels are written at the top left hand corner, divided by a diagonal line. The top and left edges of the map then represent all the possible input combinations for the inputs allocated to that edge.
For example, in the 3 input map b in Fig. Because example b in Fig. This map is therefore rectangular rather than square to cover the 8 possible combinations available from 3 inputs.
Using the Karnaugh Map The Karnaugh map can be populated with data from either a truth table or a Boolean equation. As an example, Table 2.
This results in a Boolean equation for the un-simplified circuit: The process is shown step by step in Fig. Step b In Table 2. Step c In Table 2. Step d In Table 2. Step e Finally, in Table 2. Later it will be shown that these blank cells can be useful when mapping larger circuits, but for now the map is ready for simplification.
Simplifying Karnaugh Maps Circuit simplification in any Karnaugh map is achieved by combining the cells containing 1 to make groups of cells.
In grouping the cells it is necessary to follow six rules. How these rules are applied is illustrated using a four input cell map shown in Fig. Karnaugh Map Rules 1. Groups can only contain 1, 2, 4, 8, 16 or This helps make smaller groups as large as possible, which is an advantage in finding the simplest solution.
There should be as few groups as possible. This will simplify the circuit being produced, but it is not optimum. Map b shows an improvement, still with 3 groups but they now contain 8, 4 and 4 cells. This map takes advantage of rule 5 by joining the 2 cells ringed in green in Map a with the top two cells in the blue group, see Map b to form a group of 4 ringed in cyan instead of a group of 2.
The map now conforms to all 6 rules. Sometimes however there may be a single cell that cannot be joined with other groups, as shown in map d. Rule 3 prohibits diagonal grouping so there is no alternative other than to leave a group of 1.Why Karnaugh Maps?
Karnaugh Maps offer a graphical method of reducing a digital circuit to its minimum number of gates. The map is a simple table containing 1s and 0s that can express a truth table or complex Boolean expression describing the operation of a digital circuit.
Logic circuit simplification (SOP and POS) This is an online Karnaugh map generator that makes a kmap, shows you how to group the terms, shows the simplified Boolean equation, and draws the circuit for up to 8 variables. A Karnaugh map is not the same thing as a Veitch diagram.
Veitch's diagram is used by virtually no one. I have the originals of both papers. The 2nd drawing on the right is . The Karnaugh map provides a simple and straight-forward method of minimising boolean expressions which represent combinational logic circuits.
A Karnaugh map is a pictorial method of grouping together expressions with common factors and then eliminating unwanted variables. 1 Elec 1 Karnaugh Maps Karnaugh Maps Objectives This section presents a technique for simplifying logical expressions.
It will: Define Karnaugh and establish the correspondence between Karnaugh maps and truth tables and logical expressions. The Karnaugh map can also be described as a special arrangement of a truth table.
The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem.