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K-Maps, Timing Sequential Circuits: Latches & Flip-Flops Lecture 4 Digital Design and Computer Architecture Harris & Harris Morgan Kaufmann / Elsevier, 2007

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2.7 3-input K-map

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3-input K-map

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K-map Definitions Complement: variable with a bar over it Literal: variable or its complement Implicant: product of literals Prime implicant: implicant corresponding to the larges circle in the K-map

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K-map Rules Each circle must span a power of 2 (i.e. 1, 2, 4) squares in each direction Each circle must be as large as possible A circle may wrap around the edges of the K- map A one in a K-map may be circled multiple times A “don't care” (X) is circled only if it helps minimize the equation

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4-input K-map

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Don’t Cares

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Contention: X Not just 1’s and 0’s Contention: X

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Floating: Z Tri-state Buffer

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2.8 Building Blocks Multiplexer vs. Demultiplexer Decoder vs. Encoder Priority Encoder Adder

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Multiplexers

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Decoders

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2.9 Timing See chap08.ppt of last year.

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2.9.1 Propagation & Contamination Delay Propagation Delay: t pd = max delay from input to output Contamination Delay: t cd = min delay from input to output

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Critical (Long) and Short Path

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2.9.2 Glitches Glitch: when a single input change causes multiple output changes Glitches don’t cause problems because of synchronous design conventions (which we’ll talk about in a bit) But it’s important to recognize a glitch when you see one in timing diagrams

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Glitches: Example

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Ch.3 Sequential Circuits Circuits that: –give sequence to events –have memory (short-term) –use feedback from output to input

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3.2 State Elements State: information that determines future behavior of circuit State elements store state –Cross coupled inverter pair –SR Latch –D Latch –D Flip-flop

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Cross Coupled inverter pair

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3.2.1 SR Latch Bistable circuit

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3.2.2 D Latch

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3.2.3 D Flip-Flop How many transistors required for a D-FF ?

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3.2.4 Registers

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3.2.5x Enabled & Resettable FF

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Eg.3.2– D-FF vs. D Latch

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D Flip-Flop Input Timing Constraints Setup time: t setup = time before the clock edge that data must be stable (i.e. not changing) Setup time: t hold = time after the clock edge that data must be stable Aperture time: T o = time around clock edge that data must be stable

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D Flip-Flop Output Timing Propagation Delay: t pcq = time after clock edge that Q is guaranteed to be stable (i.e. stop changing) Contamination Delay: t ccq = time after clock edge that Q might be unstable (i.e. start changing)

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3.3 Synchronous Logic Design Some Problematic Circuits

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3.3.2 Sync. Seq. circuits

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3.4 Finite State Machines Moore & Mealy Machine

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3.4 Finite State Machines

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3.4.2 State Encoding Divide by N counter

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Next Time Finite State Machines

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