(a) Modulo-4 counter using 2 JK FFs. Q1 toggles on every clock pulse, Q2 toggles when Q1 is high. (b) Connect D FF output to JK FF input via inverter, next state is Q XOR Q. This circuit mimics JK FF functionality using a D FF.
Part (a): Analyzing the Counter Sequence
The given sequence, 0, 1, 2, 3, 0, 1..., indicates a modulo-4 counter, meaning it cycles through four distinct states (0, 1, 2, 3) before returning to its initial state (0). This suggests the need for two JK flip-flops (FFs) to represent the four states, with each flip-flop having two possible states (0 and 1).
State Transitions:
* State 0 (Q1 = 0, Q2 = 0): On the next clock pulse, Q1 should toggle to 1, and Q2 should remain 0.
* State 1 (Q1 = 1, Q2 = 0): On the next clock pulse, Q2 should toggle to 1, and Q1 should remain 1.
* State 2 (Q1 = 1, Q2 = 1): On the next clock pulse, both Q1 and Q2 should toggle to 0, resetting the counter.
Logic Circuit:
To achieve these transitions, we need to connect the J and K inputs of the JK FFs appropriately:
* Q1 JK inputs: J = 1 (always toggle), K = Q2 (toggle when Q2 is 1)
* Q2 JK inputs: J = Q1 (toggle when Q1 is 1), K = 1 (always toggle)
This logic ensures the desired state changes:
* At State 0, J1 = 1 and K1 = 0, making Q1 toggle to 1. Both J2 and K2 are 1, making Q2 toggle to 1.
* At State 1, J2 = 1 and K2 = 0, making Q2 toggle to 1. J1 = 1 and K1 = 1, making Q1 remain 1.
* At State 2, both J1 and K1 are 1, making Q1 toggle to 0. J2 = 1 and K2 = 1, making Q2 toggle to 0, resetting the counter.
Therefore, the logic circuit for this counter consists of two JK flip-flops with the J and K inputs connected as described above.
Part (b): Sketching and Expression for JK FF using D FF
Sketch:
1. Connect the D input of the D flip-flop to the output (Q) of the JK flip-flop.
2. Connect the clock input of the D flip-flop to the clock input of the JK flip-flop.
3. Connect an inverter to the output (Q) of the D flip-flop.
4. Connect the output of the inverter to the J and K inputs of the JK flip-flop.
Next State Expression:
The next state of the JK flip-flop (Q') is determined by the expression:`
In this case, J = Q' (from the inverter) and K = Q. Therefore, the next state expression becomes:`
Simplifying further:
This expression indicates that the next state of the JK flip-flop is simply the XOR of its current state and the inverted current state. This effectively implements the toggle functionality required for a JK flip-flop.
Note: This approach essentially converts the D flip-flop into a JK flip-flop by utilizing its feedback loop and the inverter. It demonstrates the versatility of logic gates and their ability to mimic other functionalities.