Design of 3-bit Binary Synchronous Counter with JK Flip-Flops

A binary synchronous counter is a digital circuit that counts in binary sequence from 0 to a specified maximum count. In this case, we need to design a 3-bit synchronous counter that counts from 0 to 4. The counter will be implemented using JK flip-flops.

1. Understanding JK Flip-Flop

A JK flip-flop is a sequential logic device that has two inputs: J (set) and K (reset), and two outputs: Q (normal output) and Q' (complement output). The output state of the flip-flop depends on the current state and the input values.

2. State Diagram

To design the counter, we first need to create a state diagram that represents the desired sequence. In this case, the counter should go through the states 0, 1, 2, 3, 4 in a binary sequence.

State 0: Q2 Q1 Q0 = 0 0 0
State 1: Q2 Q1 Q0 = 0 0 1
State 2: Q2 Q1 Q0 = 0 1 0
State 3: Q2 Q1 Q0 = 0 1 1
State 4: Q2 Q1 Q0 = 1 0 0

3. Truth Table

Based on the state diagram, we can create a truth table that shows the required input values (J and K) for each flip-flop to transition from one state to another.

State 0: J2 J1 J0 = 0 0 0, K2 K1 K0 = X X X (don't care)
State 1: J2 J1 J0 = 0 0 1, K2 K1 K0 = X X X (don't care)
State 2: J2 J1 J0 = 0 1 0, K2 K1 K0 = X X X (don't care)
State 3: J2 J1 J0 = 0 1 1, K2 K1 K0 = X X X (don't care)
State 4: J2 J1 J0 = 1 0 0, K2 K1 K0 = X X X (don't care)

4. Karnaugh Maps

Next, we can simplify the truth table using Karnaugh maps to determine the required J and K values for each flip-flop.

For the first flip-flop (Q2):
- J2 = Q1'Q0
- K2 = Q1Q0

For the second flip-flop (Q1):
- J1 = Q2'Q0
- K1 = Q2Q0

For the third flip-flop (Q0):
- J0 = Q2'Q1'
- K0 = Q2Q1

5. Circuit Implementation

Based on the simplified expressions from the Karnaugh maps, we can implement the circuit using JK flip-flops. Connect the J and K inputs of each flip-flop according to the derived expressions.

Q2(t+1) = J2Q2'Q1' + K2Q2Q1