Consider the circuit, the next state \( Q^+ \) is
\[
\text{}
\]
Show Hint
Whenever the Set (\(S\)) and Reset (\(R\)) inputs of an SR flip-flop are completely complementary (\(R = \overline{S}\)), the invalid condition (\(SR=1\)) is physically impossible, and the next state equation reduces simply to \(Q^+ = S\)!
Understanding the Concept:
The fundamental characteristic behavior equation defining the next-state response (\(Q^+\)) of a standard SR (Set-Reset) flip-flop is expressed as:
\[
Q^+ = S + \overline{R}Q \quad \text{subject to the constraint } SR = 0
\]
By looking at the combinational gate inputs leading directly into the \(S\) and \(R\) terminals in the diagram, we can construct the precise algebraic representation for the circuit.
Step 1: Identify the gate and find the equations for S and R.
Looking at the digital schematic:
The logic gate shown at the input is an XNOR gate (Exclusive-NOR gate), indicated by the curved input backdrop and the output inversion bubble.
The two distinct input feeds entering this XNOR gate are:
• The external variable command line \(P\).
• The current flip-flop state feedback connection coming directly from output terminal \(Q\).
Therefore, the output expression of this gate is:
\[
P \odot Q = PQ + \overline{P}\overline{Q}
\]
Now look at how this XNOR output connects to the flip-flop inputs:
• The gate output connects directly to the Reset (\(R\)) pin. Thus, \(R = P \odot Q\).
• The gate output passes through an inverter (NOT gate bubble) before entering the Set (\(S\)) pin. Thus, \(S = \overline{P \odot Q} = P \oplus Q\).
Step 2: Express S and R in terms of Exclusive-OR components.
Using basic logical identities:
\[
S = P \oplus Q = \overline{P}Q + P\overline{Q}
\]
\[
R = P \odot Q = PQ + \overline{P}\overline{Q}
\]
Let us verify the complementary nature of these signals:
\[
\overline{R} = \overline{P \odot Q} = P \oplus Q = S
\]
Since \(S\) and \(R\) are exact logical complements of each other (\(R = \overline{S}\)), this circuit effectively acts exactly like a D flip-flop configuration where \(S = D\) and \(R = \overline{D}\).
Step 3: Substitute expressions into the SR characteristic next-state formula.
\[
Q^+ = S + \overline{R}Q
\]
Substitute \(S = P \oplus Q\) and \(\overline{R} = P \oplus Q\):
\[
Q^+ = (P \oplus Q) + (P \oplus Q)Q
\]
Using the Boolean absorption theorem (\(X + XY = X\)), where \(X = P \oplus Q\):
\[
Q^+ = P \oplus Q
\]
This mathematically demonstrates that the next state expression matches option (C).
