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Math  /  Discrete

Question5. Design a G-L flip-flop that behaves as follows: If G=0\mathrm{G}=0, the flip-flop does not change its output state. If G=1G=1, the next output state is equal to LL. a. Derive the characteristic equation of the G-L flip-flop. b. Convert J-K flip-flop to G-L flip-flop.

Studdy Solution

STEP 1

1. The G-L flip-flop is a type of flip-flop with inputs G and L.
2. The behavior of the G-L flip-flop is defined by the conditions given in the problem.
3. The J-K flip-flop is a known type of flip-flop with inputs J and K.

STEP 2

1. Derive the characteristic equation of the G-L flip-flop.
2. Convert the J-K flip-flop to a G-L flip-flop.

STEP 3

To derive the characteristic equation of the G-L flip-flop, analyze the behavior based on the given conditions:
- If G=0 G = 0 , the flip-flop retains its current state, Qnext=Q Q_{next} = Q . - If G=1 G = 1 , the next state is determined by the input L L , so Qnext=L Q_{next} = L .
Combine these conditions into a single equation:
Qnext=G⋅L+G‾⋅Q Q_{next} = G \cdot L + \overline{G} \cdot Q
This is the characteristic equation of the G-L flip-flop.

STEP 4

To convert a J-K flip-flop to a G-L flip-flop, use the characteristic equation derived in STEP_1 and the characteristic equation of the J-K flip-flop:
The characteristic equation of the J-K flip-flop is:
Qnext=J⋅Q‾+K‾⋅Q Q_{next} = J \cdot \overline{Q} + \overline{K} \cdot Q
To match the G-L flip-flop behavior, equate the two equations:
G⋅L+G‾⋅Q=J⋅Q‾+K‾⋅Q G \cdot L + \overline{G} \cdot Q = J \cdot \overline{Q} + \overline{K} \cdot Q
From this, determine the necessary conditions for J and K:
- When G=1 G = 1 , J=L J = L and K=L‾ K = \overline{L} . - When G=0 G = 0 , J=0 J = 0 and K=0 K = 0 .
Thus, the conversion is:
J=G⋅L J = G \cdot L K=G⋅L‾ K = G \cdot \overline{L}
The characteristic equation of the G-L flip-flop is:
Qnext=G⋅L+G‾⋅Q Q_{next} = G \cdot L + \overline{G} \cdot Q
And the conversion from J-K to G-L flip-flop is:
J=G⋅L J = G \cdot L K=G⋅L‾ K = G \cdot \overline{L}

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