Question

The characteristic equation of SR flip-flop is

• A. \( Q(t+1) = SQ(t) + R \)
• B. \( Q(t+1) = S + RQ(t) \)
• C. \( Q(t+1) = \overline{S}Q(t) + \overline{R} Q(t) \)
• D. \( Q(t+1) = S + \overline{R} Q(t) \)

Hint:

The characteristic equation defines the next state \(Q(t+1)\) of a flip-flop in terms of its present state \(Q(t)\) and inputs.
For an SR flip-flop, assuming the condition \(SR=0\) (S and R are not simultaneously 1), the characteristic equation is \(Q(t+1) = S + \overline{R}Q(t)\).

Answer 1

The Correct Option is D

SR Flip-Flop Characteristic Equation 

The SR (Set-Reset) flip-flop is a basic sequential logic circuit that stores one bit of information. It has two inputs:

• S (Set) input, which sets the output to 1 when activated.
• R (Reset) input, which resets the output to 0 when activated.

The output \( Q \) is affected by these inputs, and the next state of the output \( Q(t+1) \) depends on the current input values of \( S \) and \( R \), as well as the current output \( Q(t) \).

The behavior of the SR flip-flop is summarized in the following truth table:

\[ \begin{array}{|c|c|c|c|} \hline S & R & Q(t+1) & \text{Comment} \\ \hline 0 & 0 & Q(t) & \text{No change (Hold)} \\ 0 & 1 & 0 & \text{Reset} \\ 1 & 0 & 1 & \text{Set} \\ 1 & 1 & - & \text{Forbidden} \\ \hline \end{array} \]

Where:

• When \( S = 1 \) and \( R = 0 \), the output \( Q \) is set to 1.
• When \( S = 0 \) and \( R = 1 \), the output \( Q \) is reset to 0.
• When both \( S = 0 \) and \( R = 0 \), the output \( Q \) remains unchanged (Hold condition).
• When both \( S = 1 \) and \( R = 1 \), it is a forbidden state, as the SR flip-flop cannot operate in this condition.

Deriving the Characteristic Equation

The characteristic equation for the SR flip-flop describes the next state of the output \( Q(t+1) \) as a function of the current state \( Q(t) \) and the inputs \( S \) and \( R \). From the truth table, we can derive the equation for the flip-flop’s behavior. We know: - When \( S = 1 \) and \( R = 0 \), the output is set to 1. - When \( S = 0 \) and \( R = 1 \), the output is reset to 0. - When \( S = 0 \) and \( R = 0 \), the output holds its previous state \( Q(t) \). The characteristic equation for the SR flip-flop can be written as: \[ Q(t+1) = S + \overline{R} Q(t) \] Where: - \( S \) is the Set input. - \( \overline{R} \) is the complement (NOT) of the Reset input \( R \). - \( Q(t) \) is the current state of the output.

Explanation of the Equation

The equation \( Q(t+1) = S + \overline{R} Q(t) \) can be explained as follows:

• When \( S = 1 \) and \( R = 0 \): The equation simplifies to \( Q(t+1) = 1 + 0 \cdot Q(t) = 1 \), which means the output is set to 1 (Set condition).
• When \( S = 0 \) and \( R = 1 \): The equation simplifies to \( Q(t+1) = 0 + \overline{1} \cdot Q(t) = 0 \), which means the output is reset to 0 (Reset condition).
• When \( S = 0 \) and \( R = 0 \): The equation simplifies to \( Q(t+1) = 0 + 1 \cdot Q(t) = Q(t) \), which means the output holds its previous state (Hold condition).
• When \( S = 1 \) and \( R = 1 \) (Forbidden state): The equation simplifies to \( Q(t+1) = 1 + 0 \cdot Q(t) = 1 \), which sets the output to 1, even though the state is forbidden.

Conclusion

The characteristic equation that describes the SR flip-flop's behavior is: \[ \boxed{Q(t+1) = S + \overline{R} Q(t)} \] This equation correctly describes the behavior of the SR flip-flop, including the Set, Reset, Hold, and Forbidden conditions.