Flip-Flop Conversion and Karnaugh Map Techniques

Posted on Feb 18, 2026 in Systems Computer Engineering

Flip-Flop Conversion: Steps and Examples

The goal of flip-flop conversion is to use an available flip-flop (the basic component you have) along with a combinational logic circuit to make the overall system behave like a required flip-flop (the target).

General Steps for Conversion

  • Identify Available and Required FFs:
    • Required FF (Target): The one you want to emulate (e.g., T-FF). Its inputs are the external inputs for the final circuit.
    • Available FF (Source): The one you are using (e.g., JK-FF). Its inputs (J and K) are the outputs of the combinational logic.
  • Conversion Table (The Core Step): Create a conversion table that combines:
    • The Characteristic Table of the Required FF (Inputs: Required FF inputs, Qn; Output: Qn+1).
    • The Excitation Table of the Available FF (Inputs: Qn, Qn+1; Output: Available FF inputs).
  • K-Map Simplification: Use the Karnaugh Map (K-Map) to simplify the expressions for the Available FF inputs (e.g., J and K) in terms of the Required FF inputs (e.g., T) and the Present State (Qn).
  • Draw the Logic Diagram: Implement the simplified Boolean expressions using the available flip-flop and basic logic gates (AND, OR, NOT, etc.).

Example 1: JK Flip-Flop to T Flip-Flop Conversion

Goal: Implement a T-FF (Required) using a JK-FF (Available).

1. Conversion Table

T (Required FF Input)Qn (Present State)Qn+1 (Next State)J K (Available FF Inputs)
0000 x
011x 0
1011 x
110x 1

2. K-Map Simplification (for J and K)

For J:

T \ Qn01
00x
11x

Expression for J: By grouping the 1 and the Don’t Care (‘x’), we find: J = T

For K:

T \ Qn01
0x0
1x1

Expression for K: By grouping the 1 and the Don’t Care (‘x’), we find: K = T

3. Logic Diagram

Connect the external input T directly to both the J and K inputs of the JK Flip-Flop.

Example 2: D Flip-Flop to JK Flip-Flop Conversion

Goal: Implement a JK-FF (Required) using a D-FF (Available).

1. Conversion Table

J K (Required FF Input)Qn (Present State)Qn+1 (Next State)D (Available FF Input)
0 0000
0 0111
0 1000
0 1100
1 0011
1 0111
1 1011
1 1100

2. K-Map Simplification (for D)

Since D has three inputs (J, K, Qn), a 3-variable K-Map is used for the D input. Expression for D: Grouping the 1s and simplifying leads to the equation:

3. Logic Diagram

The external inputs J and K are fed into an OR gate along with \(\bar{Q}_n\) and Qn to generate the D input of the D Flip-Flop.

The video below gives another perspective on the steps for converting flip-flops. [Convert JK Flip-Flop to T Flip-Flop | Truth Table, Excitation Table & Logic Explained]

Karnaugh Maps: Key Concepts and Rules

Karnaugh Maps (K-Maps) are a graphical method used to simplify Boolean algebra expressions. They are an essential tool for minimizing the number of logic gates and inputs required to implement a digital circuit. For your KUK syllabus (Unit II), understanding 3-variable and 4-variable K-Maps and the handling of Don’t Care conditions is a guaranteed area for problem-solving.

Key Concepts

1. Purpose

The primary goal of K-Maps is to obtain the minimal Sum-of-Products (SOP) or minimal Product-of-Sums (POS) expression, which leads to the simplest possible hardware realization.

2. K-Map Structure

  • 3-Variable K-Map: Has 23 = 8 cells, typically arranged as a 2 × 4 grid.
  • 4-Variable K-Map: Has 24 = 16 cells, arranged as a 4 × 4 grid.
  • Gray Code Labeling: The labels for rows and columns must follow the Gray Code sequence (only one bit changes between adjacent cells), e.g., 00, 01, 11, 10. This ensures that adjacent cells differ by only one variable, which is the basis of the simplification.

Rules for Grouping (Looping)

Simplification is achieved by forming groups (loops) of adjacent ‘1’s (for SOP) or ‘0’s (for POS) based on the following rules:

  • Group Size: Groups must contain 2n cells, where n = 0, 1, 2, 3, 4… (i.e., groups of 1, 2, 4, 8, or 16).
  • Adjacency: Groups must be formed by geometrically adjacent cells. Adjacency is defined by:
    • Side-by-side cells.
    • Top/Bottom cells (the map “wraps” around).
    • Left/Right cells (the map “wraps” around).
    • Corner cells (the four corners of a 4-variable map are adjacent).
  • Maximum Size: Always aim for the largest possible groups first.
  • Minimum Groups: Cover all the ‘1’s (or ‘0’s) using the minimum number of groups.
  • Overlap: Groups can overlap if it helps create a larger group or reduces the total number of groups.

Handling Don’t Care Conditions (d or X)

Don’t Care conditions are input combinations that will never occur in the real system, or whose output state (0 or 1) does not matter for the circuit’s function. In the K-Map, these are represented by ‘d’ or ‘X’.

Rule: Treat a Don’t Care cell as either a ‘1’ or a ‘0’ in order to make the grouping as large as possible.

  • Include X as a ‘1’: If including an X helps form a larger group of ‘1’s, making the overall expression simpler.
  • Treat X as a ‘0’ (Ignore): If including an X does not help form a larger group, or if it would require forming a new, smaller group that is not necessary to cover the existing ‘1’s.

Example Problem Structure

A typical exam problem would ask you to simplify a function like this:

Function (SOP):

  • Map: Place ‘1’s in the minterm cells (0, 2, 5, 7, 8, 10, 13, 15).
  • Don’t Cares: Place ‘d’s in the Don’t Care cells (1, 3).
  • Group:
    • Group the four corners (0, 2, 8, 10) to get a group of 4.
    • Group the ‘1’s (5, 7, 13, 15) to get a group of 4.
    • The Don’t Cares (1, 3) are already adjacent to a large group of 4 (‘1’s at 0 and 2) but are not needed to cover any remaining ‘1’s. However, if we group (1, 3) with (0, 2), we form a simpler term.

The most important takeaway is to be systematic: circle the largest possible groups of ‘1’s first, using the Don’t Cares to your advantage.

The video below demonstrates how to solve K-Map problems with Don’t Care conditions, which will be highly beneficial for practice. [Karnaugh Maps – Don’t Care Conditions]