Sia

To design a 3-bit synchronous counter using T flip-flops that counts in the sequence 0, 2, 4, 6, 0, we need to understand how T flip-flops work and how to set them up to count in the desired sequence. A T flip-flop toggles its output on every clock cycle if the T input (toggle input) is high (1), and it holds its current state if the T input is low (0).

Let's represent the 3 bits as $Q_2Q_1Q_0$, where $Q_2$ is the most significant bit (MSB) and $Q_0$ is the least significant bit (LSB). The sequence we want to achieve is:

$$\begin{align*}
0 & : 000 \\
2 & : 010 \\
4 & : 100 \\
6 & : 110 \\
\end{align*}$$

We will now create a state table that includes the current state, the next state, and the required T inputs to achieve the transition from the current state to the next state.

State Table:

$$\begin{array}{ccc|ccc}
Q_2 & Q_1 & Q_0 & T_2 & T_1 & T_0 \\
\hline
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 \\
\end{array}$$

From the state table, we can derive the logic for the T inputs:

$$\begin{align*}
T_2 & = Q_1 \cdot Q_0 \\
T_1 & = Q_0 \\
T_0 & = 0 \\
\end{align*}$$

Now, we can use this logic to design the counter using T flip-flops.