Switching circuits •Composed of switching elements called “gates” that implement logical blocks or switching expressions •Positive logic convention (active high): – High voltage or H Boolean 1 – Low voltage or L Boolean 0 •Negative logic convention (active low): – Low voltage or L Boolean 1 – High voltage or H Boolean 0
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Switching circuits Composed of switching elements called gates that implement logical blocks or switching expressions Positive logic convention (active.
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Switching circuits
• Composed of switching elements called “gates” that implement logical blocks or switching expressions
• Positive logic convention (active high):– High voltage or H Boolean 1– Low voltage or L Boolean 0
• Negative logic convention (active low):– Low voltage or L Boolean 1– High voltage or H Boolean 0
Switching circuits
• Logic variables inputs/outputs “signals”
• Signals “asserted” when the voltage level assumes the corresponding “1” value– Positive logic asserted by H– Negative logic asserted by L
• Logic variables are written complemented when they are active low– Active high signals: a, b, c– Active low signals: ā, ē, ū
Logic gates
• Logic gates switching functions• Gate symbols – two sets
Logic gates
• Gate symbols – two sets
Logic gates
• The NAND logic function and gate
Logic gates
• The NAND gate can be used to implement all 3 elementary operations of switching algebra: AND, OR, NOT
Logic gates
• The set {AND, OR, NOT} implements any switching function (by definition): it is functionally complete
• Therefore, the “NAND” gate can be used to implement any switching function– It is functionally complete, or “primitive”
Logic gates
• The NOR logic function and gate
Logic gates
• The NOR function can be used to implement all 3 elementary operations of switching algebra: AND, OR, NOT– It is functionally complete too
Logic gates
• The NOR logic function and gate
Logic gates and equivalence
• CMOS is “inverting” logic– NOR and NAND are easier to implement than OR
and AND– They are implemented as NOR or NAND followed
by an inverter
• More than one representation is possible for the same switching function
• Different circuits of logic gates might perform the same switching function– Simpler networks are preferable– Need to analyze for equivalence and transform
Logic gates and equivalence
• Equivalent logic networks
Logic gates and equivalence
• Proving the equivalence
Digital circuits
• Analysis– Given a circuit, abstract the Boolean function it is
implementing and try to improve the implementation or verify the function• From gate diagrams• From timing diagrams
• Synthesis– Given a switching function, obtain the