# Fastest Reliable Frequency Calculation Let's break down the problem step by step. ## **Given Data** - **3-input gate delay:** 20-50 ps (assume worst case, **50 ps**) - **2-input gate delay:** 10-40 ps (assume worst case, **40 ps**) - **Flip-flop parameters:** - \( t_{pcq} = 25 \) ps - \( t_{ccq} = 15 \) ps - \( t_{setup} = 30 \) ps - \( t_{hold} = 17 \) ps ## **Step 1: Determine the Critical Path** The critical path is from the output of one flip-flop, through the combinational logic, to the D input of the next flip-flop. ### **Combinational Logic Delay** From the diagram, the longest path (worst-case) is likely: - From \( S_1 \) or \( S_0 \) output of the flip-flop, - Through a 3-input AND/OR gate (50 ps), - Then through a 2-input gate (40 ps), - To the D input of the flip-flop. **Total logic delay = 3-input gate + 2-input gate = 50 ps + 40 ps = 90 ps** ## **Step 2: Setup Time Constraint** The flip-flop's data must be stable at the D input at least \( t_{setup} \) before the clock edge. **Timing constraint:** \[ T_{clock} \geq t_{pcq} + t_{logic} + t_{setup} \] Where: - \( t_{pcq} \): Clock to Q delay (25 ps) - \( t_{logic} \): Logic delay (90 ps) - \( t_{setup} \): Setup time (30 ps) **Plug in the values:** \[ T_{clock} \geq 25\,\text{ps} + 90\,\text{ps} + 30\,\text{ps} = 145\,\text{ps} \] ## **Step 3: Compute Maximum Frequency** \[ f_{max} = \frac{1}{T_{clock}} \] \[ f_{max} = \frac{1}{145 \times 10^{-12}} \] \[ f_{max} \approx 6.90\,\text{GHz} \] ## **Step 4: Hold Time Check** The data must be held at the input for at least \( t_{hold} \) after the clock edge. The path is: - \( t_{ccq} \): Clock to Q contamination delay (15 ps) - \( t_{logic, min} \): Minimum logic delay (\( t_{cd} \) for each gate, so 20 ps + 10 ps = 30 ps) \[ t_{ccq} + t_{logic, min} = 15\,\text{ps} + 30\,\text{ps} = 45\,\text{ps} \] Since \( 45\,\text{ps} > t_{hold} = 17\,\text{ps} \), the hold time is satisfied. --- ## **Summary Table** | Parameter | Value | |----------------|----------| | \( t_{pcq} \) | 25 ps | | \( t_{logic} \)| 90 ps | | \( t_{setup} \)| 30 ps | | \( T_{clock} \)| 145 ps | | \( f_{max} \) | 6.90 GHz | --- ## **Final Answer** ### **The fastest frequency at which this circuit could run reliably is:** \[ \boxed{6.90\,\text{GHz}} \] --- **All timing constraints are met at this frequency.**