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Adders

Adders. Full-Adder. The Binary Adder. 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. Å. Delete =. A. B. S. C. D and P. Can also derive expressions for. and. based on. o.

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Adders

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  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 Å Delete = A B S C D and P Can also derive expressions for and based on o Note that we will be sometimes using an alternate definition for + Propagate (P) = A B

  5. The Ripple-Carry Adder Worst case delay linear with the number of bits td = O(N) tadder = (N-1)tcarry + tsum Goal: Make the fastest possible carry path circuit

  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 Co. The reduction of the diffusion capacitances is particularly important. • The capacitance at node Co is composed of four diffusion capacitances, two internal gate capacitances, and six gate capacitances in the connecting adder cell . • The transistors connected to Ci 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

  14. Manchester Carry Chain

  15. Manchester Carry Chain Stick Diagram

  16. Carry-Bypass Adder Also called Carry-Skip

  17. Carry-Bypass Adder (cont.) tadder = tsetup + Mtcarry + (N/M-1)tbypass + (M-1)tcarry + tsum

  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 Propagate Generate

  33. Example: Domino Adder Propagate Generate

  34. Example: Domino Sum

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