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Berkeley ELENG 141 - Lecture 27 Adders

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EE1411EE1411EECS141EE141EE141--Fall 2006Fall 2006Digital Integrated Digital Integrated CircuitsCircuitsLecture 27Lecture 27AddersAddersEE1412EECS141AnnouncementsAnnouncements Homework 9 due today Homework 10 due next ThursdayEE1412EE1413EECS141Class MaterialClass Material Last lecture I/O Power distribution Intro to adders Today’s lecture Adders Reading Chapter 11EE1414EECS141AddersAddersEE1413EE1415EECS141The RippleThe Ripple--Carry AdderCarry AdderWorst case delay linear with the number of bitsGoal: Make the fastest possible carry path circuitFA FA FA FAA0B0S0A1B1S1A2B2S2A3B3S3Ci,0Co,0(= Ci,1)Co,1Co,2Co,3td= O(N)tadder= (N-1)tcarry+ tsumEE1416EECS141Complementary Static CMOS Full AdderComplementary Static CMOS Full Adder28 TransistorsABBACiCiAXVDDVDDA BCiBABVDDABCiCiABA CiBCoVDDSEE1414EE1417EECS141Inversion PropertyInversion PropertyABSCoCiFAABSCoCiFAEE1418EECS141Minimize Critical Path by Reducing Inverting StagesMinimize Critical Path by Reducing Inverting StagesExploit Inversion PropertyA3FA FA FAEven cell Odd cellFAA0B0S0A1B1S1A2B2S2B3S3Ci,0Co,0Co,1Co,3Co,2EE1415EE1419EECS141A Better Structure: The Mirror AdderA Better Structure: The Mirror AdderVDDCiABBABAABKillGenerate"1"-Propagate"0"-PropagateVDDCiABCiCiBACiABBAVDDSCo24 transistorsEE14110EECS141The Mirror AdderThe 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 Cois composed of four diffusion capacitances, two internal gate capacitances, and six gate capacitances in the connecting adder cell .•The transistors connected to Ciare placed closest to the output.•Only the transistors in the carry stage have to be optimized foroptimal speed. All transistors in the sum stage can be minimal size.EE1416EE14111EECS141Transmission Gate Full AdderTransmission Gate Full AdderABPCiVDDAAAVDDCiAPABVDDVDDCiCiCoSCiPPPPPSum GenerationCarry GenerationSetupEE14112EECS141Manchester Carry ChainManchester Carry ChainCoCiGiDiPiPiVDDCoCiGiPiVDDφφEE1417EE14113EECS141Manchester Carry ChainManchester Carry ChainG2φC3G3Ci,0P0G1VDDφG0P1P2P3C3C2C1C0EE14114EECS141Manchester Carry ChainManchester Carry ChainPi + 1Gi + 1φCiInverter/Sum RowPropagate/Generate RowPiGiφCi - 1Ci + 1VDDGNDStick DiagramEE1418EE14115EECS141CarryCarry--Bypass AdderBypass AdderFA FA FA FAP0G1P0G1P2G2P3G3Co,3Co,2Co,1Co,0Ci,0FA FA FA FAP0G1P0G1P2G2P3G3Co,2Co,1Co,0Ci,0Co,3MultiplexerBP=PoP1P2P3Idea: If (P0 and P1 and P2 and P3 = 1)then Co3 = C0, else “kill” or “generate”.Also called Carry-SkipEE14116EECS141CarryCarry--Bypass Adder (cont.)Bypass Adder (cont.)CarrypropagationSetupBit 0–3SumM bitstsetuptsumCarrypropagationSetupBit 4–7SumtbypassCarrypropagationSetupBit 8–11SumCarrypropagationSetupBit 12–15Sumtadder= tsetup+ Mtcarry+ (N/M-2)tbypass+ (M-1)tcarry+ tsumEE1419EE14117EECS141Carry Ripple versus Carry BypassCarry Ripple versus Carry BypassNtpripple adderbypass adder4..8EE14118EECS141CarryCarry--Select AdderSelect AdderSetup"0" Carry Propagation"1" Carry PropagationMultiplexerSum GenerationCo,k-1Co,k+3"0""1"P,GCarry VectorEE14110EE14119EECS141Carry Select Adder: Critical Path Carry Select Adder: Critical Path 01Sum GenerationMultiplexer1-Carry0-CarrySetupCi,0Co,3Co,7Co,11Co,15S0–3Bit 0–3 Bit 4–7 Bit 8–11 Bit 12–1501Sum GenerationMultiplexer1-Carry0-CarrySetupS4–701Sum GenerationMultiplexer1-Carry0-Carry 0-CarrySetupS8–1101Sum GenerationMultiplexer1-CarrySetupS12–15EE14120EECS141Linear Carry Select Linear Carry Select Setup"0" Carry "1" Carry MultiplexerSum Generation"0""1"Setup"0" Carry "1" Carry MultiplexerSum Generation"0""1"Setup"0" Carry "1" Carry MultiplexerSum Generation"0""1"Setup"0" Carry "1" Carry MultiplexerSum Generation"0""1"Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15S0-3S4-7S8-11S12-15Ci,0(1)(1)(5)(6) (7) (8)(9)(10)(5) (5) (5)(5)EE14111EE14121EECS141Square Root Carry Select Square Root Carry Select Setup"0" Carry "1" Carry MultiplexerSum Generation"0""1"Setup"0" Carry "1" Carry MultiplexerSum Generation"0""1"Setup"0" Carry "1" Carry MultiplexerSum Generation"0""1"Setup"0" Carry "1" Carry MultiplexerSum Generation"0""1"Bit 0-1 Bit 2-4 Bit 5-8 Bit 9-13S0-1S2-4S5-8S9-13Ci,0(4) (5) (6) (7)(1)(1)(3) (4) (5) (6)MuxSumS14-19(7)(8)Bit 14-19(9)(3)EE14122EECS141Adder Delays Adder Delays --Comparison Comparison Square root selectLinear selectRipple adder20 40Ntp(in unit delays)60010020304050EE14112EE14123EECS141LookaheadLookahead--Basic IdeaBasic IdeaCok,fAkBkCok,1–,,()GkPkCok 1–,+==AN-1, BN-1A1, B1P1S1••••••SN-1PN-1Ci, N-1S0P0Ci,0Ci,1A0, B0EE14124EECS141LookaheadLookahead: Topology: TopologyCok,GkPkGk1–Pk1–Cok 2–,+()+=Cok,GkPkGk1–Pk1–…P1G0P0Ci0,+()+()+()+=Expanding Lookahead equations:All the way:Co,3Ci,0VDDP0P1P2P3G0G1G2G3EE14113EE14125EECS141Logarithmic LookLogarithmic Look--Ahead AdderAhead AdderA7FA6A5A4A3A2A1A0A0A1A2A3A4A5A6A7Ftp∼ log2(N)tp∼ NEE14126EECS141Carry Carry LookaheadLookaheadTreesTreesCo0,G0P0Ci0,+=Co1,G1P1G0P1P0Ci0,++=Co2,G2P2G1P2P1G0P+2P1P0Ci0,++=G2P2G1+()=P2P1()G0P0Ci0,+()+G2:1P2:1Co0,+=Can continue building the tree hierarchically.EE14114EE14127EECS141Tree AddersTree Adders16-bit radix-2 Kogge-Stone tree(A0, B0)(A1, B1)(A2, B2)(A3, B3)(A4, B4)(A5, B5)(A6, B6)(A7, B7)(A8, B8)(A9, B9)(A10, B10)(A11, B11)(A12, B12)(A13, B13)(A14, B14)(A15, B15)S0S1S2S3S4S5S6S7S8S9S10S11S12S13S14S15EE14128EECS141Tree AddersTree Adders(a0, b0)(a1, b1)(a2, b2)(a3, b3)(a4, b4)(a5, b5)(a6, b6)(a7, b7)(a8, b8)(a9, b9)(a10, b10)(a11, b11)(a12, b12)(a13, b13)(a14, b14)(a15, b15)S0S1S2S3S4S5S6S7S8S9S10S11S12S13S14S1516-bit radix-4 Kogge-Stone TreeEE14115EE14129EECS141Sparse TreesSparse Trees(a0, b0)(a1, b1)(a2, b2)(a3, b3)(a4, b4)(a5, b5)(a6, b6)(a7, b7)(a8, b8)(a9, b9)(a10, b10)(a11, b11)(a12, b12)(a13, b13)(a14, b14)(a15, b15)S1S3S5S7S9S11S13S15S0S2S4S6S8S10S12S1416-bit radix-2 sparse tree with sparseness of 2EE14130EECS141Tree AddersTree Adders(A0, B0)(A1, B1)(A2, B2)(A3, B3)(A4, B4)(A5, B5)(A6, B6)(A7, B7)(A8, B8)(A9, B9)(A10, B10)(A11, B11)(A12, B12)(A13, B13)(A14, B14)(A15, B15)S0S1S2S3S4S5S6S7S8S9S10S11S12S13S14S15Brent-Kung TreeEE14116EE14131EECS141Example: Domino AdderExample: Domino AdderVDDClkPi= ai + biClkaibiVDDClkGi = aibiClkaibiPropagate GenerateEE14132EECS141Example: Domino


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Berkeley ELENG 141 - Lecture 27 Adders

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