Latches, Flip-Flops (SR, JK, D, T), Conversions & Race-Around Condition
“Master-slave architectures, excitation tables, characteristic equations, and flip-flop conversion algorithms.”
1.The Bistable Multivibrator & Memory Dynamics
§ 4.1.A Latch is a level-sensitive bistable circuit that responds to input signals throughout the entire duration of an active clock level.
A Flip-Flop is an edge-triggered storage element that updates its state exclusively on the transition (rising 0->1 or falling 1->0) of the clock pulse.
The Race-Around Condition: In a level-triggered JK flip-flop with inputs J = K = 1, the output toggles every t_{pd} nanoseconds while the clock remains HIGH (t_p). If the clock pulse width t_p is greater than the propagation delay t_{pd}, the output will oscillate unpredictably between 0 and 1, leaving the final state undetermined when the clock drops LOW.
Solutions to Race-Around:
1. Clock narrowing: Ensure clock width t_p < t_{pd} (technologically impractical due to parasitic capacitances).
2. Master-Slave Architecture: The Master latch responds while clock is HIGH; the Slave latch is isolated. When clock falls LOW, Master is locked and Slave transfers the state to outputs.
3. Edge-Triggered Flip-Flops: Internal steering circuits lock inputs within a fraction of a nanosecond around the clock edge.
Conversion Methodology: To convert an existing Flip-Flop A into a target Flip-Flop B:
1. Write the truth table of target Flip-Flop B listing current state Q and target input.
2. Determine next state Q_{next} using target B's characteristic behavior.
3. Use the excitation table of source Flip-Flop A to determine required inputs of A for each transition Q \to Q_{next}.
4. Simplify the inputs of A using K-maps in terms of target inputs and Q.
Always use the TARGET flip-flop's truth table to find Q_next, and the AVAILABLE flip-flop's excitation table to find the input equations.
2. Worked Analysis: Converting a JK Flip-Flop to a T Flip-Flop
Ensure that Q is treated as an input variable alongside T when simplifying J and K.
Problem Statement (Modeled on GATE Pattern):
Derive the input equations to convert a JK flip-flop into a T flip-flop.
Step 1: Write Truth Table with Q, T and desired Q_next:
T | Q | Q_next
0 | 0 | 0 (T=0 holds state)
0 | 1 | 1 (T=0 holds state)
1 | 0 | 1 (T=1 toggles 0 -> 1)
1 | 1 | 0 (T=1 toggles 1 -> 0)
Step 2: Determine required J and K using JK Excitation Table:
Transition Q -> Q_next | Required J | Required K
0 -> 0 (Row 1): | 0 | X
1 -> 1 (Row 2): | X | 0
0 -> 1 (Row 3): | 1 | X
1 -> 0 (Row 4): | X | 1
Step 3: Combine with inputs (T, Q):
For J in terms of (T, Q):
Row 1 (T=0, Q=0): J = 0
Row 2 (T=0, Q=1): J = X
Row 3 (T=1, Q=0): J = 1
Row 4 (T=1, Q=1): J = X
K-map for J: Grouping cells with T=1 (giving J=1 and J=X) yields: J = T.
For K in terms of (T, Q):
Row 1 (T=0, Q=0): K = X
Row 2 (T=0, Q=1): K = 0
Row 3 (T=1, Q=0): K = X
Row 4 (T=1, Q=1): K = 1
K-map for K: Grouping cells with T=1 (giving K=X and K=1) yields: K = T.
Result: Connect both J and K together to input T: J = T and K = T.
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