A sequential circuit with two D Flip-Flops, A and B; two inputs, x and y; and one output, z, is specified by the following next-state and output equations:
Draw the circuit. [2]
List the state table for the sequential circuit [3]
Draw the corresponding state diagram.
The correct answer and explanation is :
In order to provide the correct answers for the sequential circuit, we need to understand the structure of the circuit based on the provided information: two D flip-flops (A and B), two inputs (x and y), and one output (z). Let’s break it down step by step:
Step 1: Draw the Circuit
The sequential circuit involves two D flip-flops, A and B. The D flip-flops will store the state of the system based on the inputs and previous states.
- Inputs: x and y
- Outputs: z
- State variables: A (stored in the first flip-flop) and B (stored in the second flip-flop)
- Next-state and output equations: These equations dictate the behavior of the flip-flops and the output, but they are not specified in your query. Assuming that these equations define how the flip-flops respond to the inputs, we would use them to build the transition logic.
Without the exact next-state and output equations, let’s assume typical behavior for D flip-flops:
- ( A(t+1) = D_A ) (next state of A is determined by ( D_A ), which is a function of x, y, A, and B)
- ( B(t+1) = D_B ) (next state of B is determined by ( D_B ), a function of x, y, A, and B)
Step 2: List the State Table
The state table lists all possible combinations of the current states and inputs, and shows the next state of the flip-flops (A and B) and the output z.
Let’s assume that we have the following next-state and output equations (for illustration purposes):
- ( A(t+1) = x \cdot y + A )
- ( B(t+1) = x \cdot A + y \cdot B )
- ( z = A \cdot B )
Now, we can construct the state table by evaluating the next states and output for all combinations of A, B, x, and y.
| Current State | x | y | A(t+1) | B(t+1) | z |
|---|---|---|---|---|---|
| 00 | 0 | 0 | 0 | 0 | 0 |
| 00 | 0 | 1 | 0 | 1 | 0 |
| 00 | 1 | 0 | 0 | 0 | 0 |
| 00 | 1 | 1 | 1 | 1 | 1 |
| 01 | 0 | 0 | 0 | 0 | 0 |
| 01 | 0 | 1 | 0 | 1 | 0 |
| 01 | 1 | 0 | 1 | 1 | 0 |
| 01 | 1 | 1 | 1 | 1 | 1 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 10 | 0 | 1 | 0 | 1 | 0 |
| 10 | 1 | 0 | 1 | 0 | 0 |
| 10 | 1 | 1 | 1 | 1 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 11 | 0 | 1 | 0 | 1 | 0 |
| 11 | 1 | 0 | 1 | 1 | 1 |
| 11 | 1 | 1 | 1 | 1 | 1 |
Step 3: Draw the State Diagram
The state diagram is a graphical representation of the states and the transitions between them. The states are represented by circles, and the transitions are represented by directed edges. The labels on the edges show the input conditions that cause the transitions.
Given the state table, the state diagram would look like this:
- State 00 (A=0, B=0) transitions to state 01 or 11 based on inputs x and y.
- State 01 (A=0, B=1) transitions to 00, 01, or 11.
- State 10 (A=1, B=0) transitions to 00, 01, or 11.
- State 11 (A=1, B=1) transitions to 11.
Explanation (300 words)
A sequential circuit’s behavior is determined by the states of its flip-flops, the inputs, and the next-state equations. In this case, the circuit has two D flip-flops, A and B, which hold the state information. The inputs, x and y, control the transitions from one state to another.
The next-state equations for A and B determine how the state changes at each clock cycle. For instance, the next state of A depends on the current value of A, the inputs x and y, and possibly the value of B. Similarly, the next state of B depends on both x, y, A, and B. These equations are fundamental to defining the behavior of the sequential circuit.
The state table lists all possible combinations of the current state and inputs, showing the corresponding next state of each flip-flop and the output. This table helps determine how the system behaves in response to different input combinations.
The state diagram provides a visual representation of the states and transitions. Each state is shown as a circle, and the transitions are represented by arrows. The labels on the arrows indicate the conditions (values of x and y) that cause the transitions. The state diagram is a useful tool for understanding the flow of control in the circuit, and it helps in visualizing the behavior over time.
In practice, the exact next-state and output equations would be provided in the problem description, and based on those equations, the state table and state diagram would be drawn accordingly.