5 variable K map

Last Updated on July 31, 2023 by Mayank Dham

Karnaugh Maps, also known as K Maps, are a powerful graphical technique used in digital circuit design and simplifying Boolean algebra. They provide a systematic approach to simplifying Boolean expressions and condensing complex logic circuits to their simplest forms. When dealing with Boolean functions with up to five variables, K Maps come in handy. In this article, we will look at the concepts and techniques behind the 5 variable K Map in order to help you understand and use this useful tool in digital circuit design.

Basics of Karnaugh Maps

Karnaugh Maps are two-dimensional grids used to represent Boolean functions with multiple variables. Each cell in the grid corresponds to a unique combination of input variables, and the function’s output values are entered in these cells. The main advantage of using K Maps is that they allow us to identify and group adjacent cells that share the same output values. This grouping helps simplify the Boolean expression and leads to a more concise and efficient circuit implementation.

Creating 5 Variable K Map

To create a 5 variable K Map, we arrange the cells in a grid with 32 cells (2^5) since we have five variables. The variables’ values are represented in binary form, and the cell ordering follows the Gray code sequence. Each row and column header represents a unique combination of the five variables, and the function’s output values are placed in the corresponding cells.

The Gray code sequence is used to minimize errors when transitioning between binary representations of consecutive values. This feature is particularly important in K Maps to avoid potential errors and confusion when grouping cells.

Rules to be followed while creating 5 variable K map

• If a function is given in compact canonical SOP (Sum of Products) form, we write "1" in the corresponding cell numbers for each minterm (provided in the question). For example, for summation of (0, 1, 5, 7, 30, 31), we will write "1" for cell numbers (0, 1, 5, 7, 30, and 31).
• If a function is given in compact canonical POS (Product of Sums) form, we write "0" in the corresponding cell numbers for each maxterm (provided in the question). For example, for products of (0, 1, 5, 7, 30, 31), we will write "0" for cell numbers (0, 1, 5, 7, 30, and 31).

Steps to be followed while creating 5 variable K map

• In the K-Map, create the largest possible size subcube that covers all the marked 1’s in the case of SOP or all the marked 0’s in the case of POS. It should be noted that each subcube can only contain terms with powers of two. A subcube of 2^m cells is also possible if and only if each cell in that subcube has a "m" number of adjacent cells.

• All Essential Prime Implicants (EPIs) must be present in the minimal expressions.

Solving the SOP function

For a clear understanding, let us solve the example of SOP function minimization of5 variable K Map using the following expression : summation of (0, 2, 4, 7, 8, 10, 12, 16, 18, 20, 23, 24, 25, 26, 27, 28) In the above K-Map we have 4 subcubes:

• Subcube 1: The one marked in red comprises cells ( 0, 4, 8, 12, 16, 20, 24, 28)
• Subcube 2: The one marked in blue comprises cells (7, 23)
• Subcube 3: The one marked in pink comprises cells ( 0, 2, 8, 10, 16, 18, 24, 26)
• Subcube 4: The one marked in yellow comprises cells (24, 25, 26, 27)

Now, while writing the minimal expression of each of the subcubes, we will search for the literal that is common to all the cells present in that subcube.

Finally, the minimal expression of the given boolean Function can be expressed as follows:

Solving the POS function

Now, let us solve the example of POS function minimization of a5 variable K Map using the following expression: prod. of (0, 2, 4, 7, 8, 10, 12, 16, 18, 20, 23, 24, 25, 26, 27, 28) In the above K-Map we have 4 subcubes:

Now, while writing the minimal expression of each of the subcubes, we will search for the literal that is common to all the cells present in that subcube.

Finally, the minimal expression of the given boolean Function can be expressed as follows:

NOTE:

For the5 variable K Map, the Range of the cell numbers will be from 0 to 2^5 -1 i.e., 0 to 31.
The above-mentioned term “Adjacent Cells” means “any two cells that differ in only one variable”.

Grouping and Simplification

Grouping is a critical step in K-Map simplification. The goal is to identify adjacent cells with the same output value and form groups that cover as many cells as possible, preferably in powers of two (1, 2, 4, 8, etc.). The groups should be rectangular in shape and can wrap around the edges of the K-Map if necessary.

After forming groups, we express the simplified Boolean expression by combining the variables that remain constant within each group. We use the Boolean OR operation (+) to combine these groups, and the result represents the simplified expression.

Handling Don’t-Care Conditions

In practical digital circuit design, there are often instances where certain input combinations produce undefined or irrelevant output values. These undefined conditions are known as "Don’t-Care" conditions. In K Maps, we represent these Don’t-Care cells using the letter "X."

The Don’t-Care conditions provide additional flexibility in simplification, allowing us to optimize the Boolean expression further. When forming groups, we can include Don’t-Care cells as part of the groups, if doing so results in a more significant simplification.

Using 5-Variable K Maps in Circuit Design

The ultimate goal of K-Map simplification is to design more efficient digital circuits. The simplified Boolean expression obtained from the K-Map helps in creating a logic circuit with fewer gates and lower power consumption. This leads to more cost-effective and reliable designs. To implement the simplified expression, we use logic gates like AND, OR, and NOT gates to build the circuit. Utilizing the minimized expression reduces propagation delays and minimizes the chances of glitches or errors in the circuit’s output.

Conclusion
Five Variable K Maps are a valuable tool in digital circuit design, allowing designers to simplify complex Boolean expressions and create more efficient logic circuits. By understanding the principles of K Maps, how to create them, and how to group cells for simplification, you can effectively design digital circuits with fewer gates, lower power consumption, and improved reliability. Mastering K Maps is a crucial skill for digital circuit designers and engineers, as it enables them to optimize circuit designs and create high-performance systems. Through practice and application, you can leverage the power of K Maps to tackle even more substantial Boolean expressions and design innovative digital circuits for a wide range of applications.

Frequently Asked Questions (FAQs)

Here are some of the frequently asked questions on karnaugh map 5 variables

Q1: What is a 5-variable K-map?
A 5-variable Karnaugh Map (K-Map) is a graphical representation used to simplify Boolean functions with five variables. It consists of a two-dimensional grid with 32 cells (2^5) that represent all possible combinations of the five variables. The values in the cells correspond to the output of the Boolean function for each input combination. By grouping adjacent cells with the same output values, we can simplify the Boolean expression and design more efficient digital circuits.

Q2: How many cells are in a 5-variable K-map?
A 5-variable K-Map contains 32 cells, as there are 2^5 (2 raised to the power of 5) possible combinations of five variables. The cells are arranged in a 2x2x2x2x2 grid, with each axis representing one of the five variables.

Q3: What is a 6-variable K-map?
A 6-variable K-Map is a graphical representation used to simplify Boolean functions with six variables. It consists of a two-dimensional grid with 64 cells (2^6) representing all possible combinations of the six variables. The concept and methodology of the 6-variable K-Map are similar to those of the 5-variable K-Map, but they become more complex due to the increased number of cells.

Q4: How to do a 4-variable K-map?
To create a 4-variable Karnaugh Map (K-Map), follow these steps:

1. List all possible combinations of the four variables in binary form (00, 01, 10, 11) along the rows and columns of the K-Map.

2. Write down the corresponding output values (0 or 1) in each cell based on the given Boolean function.

3. Group adjacent cells with the same output values, forming groups with 1, 2, 4, or 8 cells (powers of 2). These groups should be rectangular in shape and can wrap around the edges of the K-Map if needed.
   K-Maps provide a systematic approach to simplifying Boolean expressions and reducing the complexity of logic circuits. By following these steps, you can effectively utilize a 4-variable K-Map to design optimized digital circuits.