L5 Adders

Acknowledgement Slides taken from http:// bwrc.eecs.berkeley.edu/IcBook/index.htm which is the web-site of “Digital Integrated Circuit – A Design Perspective” by Rabaey, Chandrakasan, Nicolic

Outline Background / Basics of Adders Ripple Carry Adder

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

Itanium Integer Datapath Fetzer, Orton, ISSCC’02

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

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

Complimentary Static CMOS Full Adder 28 Transistors

Minimize Critical Path by Reducing Inverting Stages Exploit Inversion Property

A Better Structure: The Mirror Adder

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.

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: Critical Path

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

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

Example: Domino Adder Propagate Generate

L5 Adders

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