Consider the circuit given below with initial state Q0 =1, Q1 = Q2 = 0. The…

1. Understanding the Circuit Equations

The circuit consists of three positive edge-triggered D flip-flops. The state of the circuit is represented as a 3-bit binary number (Q2 Q1 Q0)2, where its decimal value is given by:

State Value = 4Q2 + 2Q1 + Q0

By looking at the connections in the diagram, we can derive the next-state equations for each flip-flop:

• D0: The input to the first flip-flop comes from the output of the XOR gate. The inputs to this XOR gate are Q1 and Q2.

  D0 = Q1 ⊕ Q2

• D1: The input comes directly from the output of the previous flip-flop.

  D1 = Q0

• D2: The input comes directly from the output of the second flip-flop.

  D2 = Q1

Since for a D flip-flop, the next state Qnext = D, our state transition equations are:

• Q0(next) = Q1 ⊕ Q2

• Q1(next) = Q0

• Q2(next) = Q1

2. Step-by-Step State Transition (Solution)

We are given the initial state as Q0 = 1, Q1 = 0, Q2 = 0.

This means the starting binary state is 0012, which equals 1 in decimal.

Let's track the states for each clock cycle:

The equation 4Q2 + 2Q1 + Q0 is used to convert the multi-bit binary state of a sequential circuit (like a counter or a register) into a single, easily readable decimal number.

The generated decimal sequence is: 1, 2, 5, 3, 7, 6, 4 (and then it loops back to 1).