Designing a Three-Bit Counter with JK Flip-Flops to Count in a Specific Sequence

Introduction

Designing sequential circuits to count in specific sequences is a common task in digital electronics. This article discusses how to design a three-bit counter that counts in the sequence 0 2 4 6 using JK flip-flops. This counter will repeat the sequence 0 2 4 6 0 and so on.

Steps to Design the Counter

The design process involves several key steps, including creating a state table, determining state encoding, and deriving the flip-flop inputs to achieve the desired sequence.

State Table

The first step is to create a state table to show the transitions from one state to the next. The decimal numbers we are counting are mapped to binary as follows:

0 2 4 6 Present State Q2 Q1 Q0 Next State Qn 000 010 010 100 100 110 110 000

State Encoding

We will use three JK flip-flops, named Q2, Q1, and Q0, to encode the states, where Q2 is the most significant bit (MSB) and Q0 is the least significant bit (LSB).

Determine Flip-Flop Inputs

To achieve the desired sequence, we need to determine the required inputs for the JK flip-flops for each state transition. The JK flip-flop toggles its output if both J and K inputs are high. Let's analyze each transition:

Transition from 000 to 010

Q2: 0 to 0 J2 0, K2 X Q1: 0 to 1 J1 1, K1 X Q0: 0 to 0 J0 0, K0 X

Transition from 010 to 100

Q2: 0 to 1 J2 1, K2 X Q1: 1 to 0 J1 0, K1 1 Q0: 0 to 0 J0 0, K0 X

Transition from 100 to 110

Q2: 1 to 1 J2 X, K2 0 Q1: 0 to 1 J1 1, K1 X Q0: 0 to 0 J0 0, K0 X

Transition from 110 to 000

Q2: 1 to 0 J2 0, K2 1 Q1: 1 to 0 J1 0, K1 1 Q0: 0 to 0 J0 0, K0 X

Construct the Input Equations

We can derive the necessary J and K inputs for each flip-flop from the above transitions:

For Q2:

J2 Q1 Q0 Set when moving from 010 to 100 K2 Q2 Q1 Reset when moving from 110 to 000

For Q1:

J1 Q0 Set when moving from 000 to 010 or 100 to 110 K1 Q2 Reset when moving from 100 to 110 or 110 to 000

For Q0:

J0 0 Always 0 since Q0 does not change in any state K0 0 Always 0

Final JK Inputs

J2 Q1 Q0 K2 Q2 Q1 J1 Q0 K1 Q2 J0 0 K0 0

Circuit Implementation

Now, you can implement the circuit using JK flip-flops. Connect the J and K inputs of each flip-flop according to the logic equations derived. Use combinational logic AND, OR, and NOT gates to create the necessary connections based on the current state outputs Q2 Q1 Q0. Connect a clock signal to all flip-flops to synchronize state transitions.

Conclusion

This design will allow the three-bit counter to count in the desired sequence of 0 2 4 6 and then back to 0. The JK flip-flops will handle the state transitions according to the defined logic.