Adder Circuits
Acknowledgement <ul><li>Slides taken from  http:// bwrc.eecs.berkeley.edu/IcBook/index.htm </li></ul><ul><li>which is the ...
Outline <ul><li>Background / Basics of Adders </li></ul><ul><li>Ripple Carry Adder </li></ul>
A Generic Digital Processor
Building Blocks for Digital Architectures Arithmetic unit - Bit-sliced datapath (adder, multiplier, shifter, comparator, e...
Bit-Sliced Design
Bit-Sliced Datapath
Itanium Integer Datapath Fetzer, Orton, ISSCC’02
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...
The Ripple-Carry Adder Worst case delay linear with the number of bits Goal: Make the fastest possible carry path circuit ...
Complimentary Static CMOS Full Adder 28 Transistors
Inversion Property
Minimize Critical Path by Reducing Inverting Stages Exploit Inversion Property
A Better Structure: The Mirror Adder
Mirror Adder Stick Diagram
The Mirror Adder <ul><ul><ul><li>The NMOS and PMOS chains are  completely symmetrical .  A maximum of two series transisto...
Transmission Gate Full Adder
Manchester Carry Chain
Manchester Carry Chain
Manchester Carry Chain Stick Diagram
Carry-Bypass Adder Also called  Carry-Skip
Carry-Bypass Adder (cont.) t adder  = t setup  + Mt carry  +  ( N/M -1) t bypass  + ( M -1) t carry  +  t sum
Carry Ripple versus Carry Bypass
Carry-Select Adder
Carry Select Adder: Critical Path
Linear Carry Select
Square Root Carry Select
Adder Delays - Comparison
LookAhead - Basic Idea
Look-Ahead: Topology Expanding Lookahead equations: All the way:
Logarithmic Look-Ahead Adder
Carry Lookahead Trees Can continue building the tree hierarchically.
Tree Adders 16-bit radix-2 Kogge-Stone tree
Tree Adders 16-bit radix-4 Kogge-Stone Tree
Sparse Trees 16-bit radix-2 sparse tree with sparseness of 2
Tree Adders Brent-Kung Tree
Example: Domino Adder Propagate Generate
Example: Domino Adder Propagate Generate
Example: Domino Sum
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L5 Adders

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L5 Adders

  1. 1. Adder Circuits
  2. 2. Acknowledgement <ul><li>Slides taken from http:// bwrc.eecs.berkeley.edu/IcBook/index.htm </li></ul><ul><li>which is the web-site of “Digital Integrated Circuit – A Design Perspective” by Rabaey, Chandrakasan, Nicolic </li></ul>
  3. 3. Outline <ul><li>Background / Basics of Adders </li></ul><ul><li>Ripple Carry Adder </li></ul>
  4. 4. A Generic Digital Processor
  5. 5. Building Blocks for Digital Architectures Arithmetic unit - Bit-sliced datapath (adder, multiplier, shifter, comparator, etc.) Memory - RAM, ROM, Buffers, Shift registers Control - Finite state machine (PLA, random logic.) - Counters Interconnect - Switches - Arbiters - Bus
  6. 6. Bit-Sliced Design
  7. 7. Bit-Sliced Datapath
  8. 8. Itanium Integer Datapath Fetzer, Orton, ISSCC’02
  9. 9. Full-Adder
  10. 10. The Binary Adder
  11. 11. 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
  12. 12. 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
  13. 13. Complimentary Static CMOS Full Adder 28 Transistors
  14. 14. Inversion Property
  15. 15. Minimize Critical Path by Reducing Inverting Stages Exploit Inversion Property
  16. 16. A Better Structure: The Mirror Adder
  17. 17. Mirror Adder Stick Diagram
  18. 18. The Mirror Adder <ul><ul><ul><li>The NMOS and PMOS chains are completely symmetrical . A maximum of two series transistors can be observed in the carry-generation circuitry. </li></ul></ul></ul><ul><ul><ul><li>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. </li></ul></ul></ul><ul><ul><ul><li>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 . </li></ul></ul></ul><ul><ul><ul><li>The transistors connected to C i are placed closest to the output. </li></ul></ul></ul><ul><ul><ul><li>Only the transistors in the carry stage have to be optimized for optimal speed. All transistors in the sum stage can be minimal size. </li></ul></ul></ul>
  19. 19. Transmission Gate Full Adder
  20. 20. Manchester Carry Chain
  21. 21. Manchester Carry Chain
  22. 22. Manchester Carry Chain Stick Diagram
  23. 23. Carry-Bypass Adder Also called Carry-Skip
  24. 24. Carry-Bypass Adder (cont.) t adder = t setup + Mt carry + ( N/M -1) t bypass + ( M -1) t carry + t sum
  25. 25. Carry Ripple versus Carry Bypass
  26. 26. Carry-Select Adder
  27. 27. Carry Select Adder: Critical Path
  28. 28. Linear Carry Select
  29. 29. Square Root Carry Select
  30. 30. Adder Delays - Comparison
  31. 31. LookAhead - Basic Idea
  32. 32. Look-Ahead: Topology Expanding Lookahead equations: All the way:
  33. 33. Logarithmic Look-Ahead Adder
  34. 34. Carry Lookahead Trees Can continue building the tree hierarchically.
  35. 35. Tree Adders 16-bit radix-2 Kogge-Stone tree
  36. 36. Tree Adders 16-bit radix-4 Kogge-Stone Tree
  37. 37. Sparse Trees 16-bit radix-2 sparse tree with sparseness of 2
  38. 38. Tree Adders Brent-Kung Tree
  39. 39. Example: Domino Adder Propagate Generate
  40. 40. Example: Domino Adder Propagate Generate
  41. 41. Example: Domino Sum
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