1. Adders

2. Full-Adder

3. The Binary Adder

4. Express Sum and Carry as a function of P, G, D Define 3 new variable which ONLY depend on A, B Generate (G) = AB Propagate (P) = A  B Delete =A B Can also derive expressions for S and C o based on D and P Propagate (P) = A  B Note that we will be sometimes using an alternate definition for

5. The Ripple-Carry Adder Worst case delay linear with the number of bits Goal: Make the fastest possible carry path circuit t d = O(N) t adder = (N-1)t carry + t sum

6. Complimentary Static CMOS Full Adder 28 Transistors

7. Inversion Property

8. Minimize Critical Path by Reducing Inverting Stages Exploit Inversion Property

9. A Better Structure: The Mirror Adder

10. Mirror Adder Stick Diagram

11. The Mirror Adder The NMOS and PMOS chains are completely symmetrical. A maximum of two series transistors can be observed in the carry- generation circuitry. When laying out the cell, the most critical issue is the minimization of the capacitance at node C o. The reduction of the diffusion capacitances is particularly important. The capacitance at node C o is composed of four diffusion capacitances, two internal gate capacitances, and six gate capacitances in the connecting adder cell. The transistors connected to C i are placed closest to the output. Only the transistors in the carry stage have to be optimized for optimal speed. All transistors in the sum stage can be minimal size.

12. Transmission Gate Full Adder

13. Manchester Carry Chain

15. Stick Diagram

16. Carry-Bypass Adder Also called Carry-Skip

17. Carry-Bypass Adder (cont.) t adder = t setup + M tcarry + (N/M-1)t bypass + (M-1)t carry + t sum

18. Carry Ripple versus Carry Bypass

19. Carry-Select Adder

20. Carry Select Adder: Critical Path

21. Linear Carry Select

22. Square Root Carry Select

23. Adder Delays - Comparison

24. LookAhead - Basic Idea

25. Look-Ahead: Topology Expanding Lookahead equations: All the way:

26. Logarithmic Look-Ahead Adder

27. Carry Lookahead Trees Can continue building the tree hierarchically.

28. Tree Adders 16-bit radix-2 Kogge-Stone tree

29. Tree Adders 16-bit radix-4 Kogge-Stone Tree

30. Sparse Trees 16-bit radix-2 sparse tree with sparseness of 2

31. Tree Adders Brent-Kung Tree

32. Example: Domino Adder PropagateGenerate

33. Example: Domino Adder PropagateGenerate

34. Example: Domino Sum