GATE CS 1991 Q5-c — Sequential Circuits

In a standard synchronous binary counter with cascaded carry logic, the inputs to the successive flip-flops are generated using a chain of AND gates. For a 4-bit synchronous counter, the inputs to the most significant flip-flop ($FF_3$) are given by $J_3 = K_3 = Q_0 Q_1 Q_2$, which is implemented using two cascaded AND gates:

The critical path (longest delay path) starts from the clock edge triggering the first flip-flop ($FF_0$) and propagates through the two cascaded AND gates to reach the inputs of the last flip-flop ($FF_3$).

The minimum clock period $T_{\text{min}}$ required for stable operation is:

Given:

• Propagation delay of each flip-flop, $t_{pd, FF} = 10\text{ ns}$
• Propagation delay of each AND gate, $t_{pd, AND} = 10\text{ ns}$
• Setup time, $t_{\text{setup}} = 0\text{ ns}$

Substituting these values:

Thus, the maximum clock frequency $f_{\text{max}}$ is: