Appendix ALogic Circuits

Logic circuits

• Operate on binary variables that assume one of two distinct values, usually called 0 and 1

• Implement functions of logic variables

• Circuits have inputs and outputs

• Circuits are implemented using electronic logic gates

Standard logic gate symbols

Implementation of the XOR function using AND, OR, and NOT gates

Synthesis of logic functions

Synthesis is the process of designing and implementing a logic circuit defined by its functional specification.

The expression for f in the previous circuit is said to be in a sum-of-products form, because the OR and AND operations are sometimes called the sum and product functions.

Implementation of a logic function

Proving equivalence of expressions

Rules of binary logic

Minimization of logic expressions

As illustrated in the previous example, a logic function can be implemented with circuits of different complexities.

It is useful to minimize a logic expression to reduce the cost of the synthesized circuit.

Three-variable Karnaugh maps

Four-variableKarnaugh maps

Using don’t cares

NAND and NOR gates

Equivalence of NAND-NAND and AND-OR networks

Cascading of gates

Representation of logic values by voltage levels

Tri-state buffer

A basic latch implemented with NOR gates

Gated SR latch

Gated SR latch implemented with NAND gates

Gated D latch

Master-slave D flip-flop

A negative-edge-triggered D flip-flop

T flip-flop

JK flip-flop

Master-slave D flip-flop with Preset and Clear

Shift register

Parallel-access shift register

A 3-bit up-counter

A two-input to four-output decoder

A BCD-to-7-segmentdisplay decoder

A four-input multiplexer

Multiplexer implementation of a logic function

A block diagram for a PLD

Functional structure of a PLA

A simplified sketch of the previous PLA

An example of a PAL

Inclusion of a flip-flop in a PAL element

Organization of a CPLD

A conceptual block diagram of an FPGA

Sequential circuits

A logic circuit whose output is determined entirely by its present inputs is called a combinational circuit (e.g. decoders and multiplexers).

A logic circuit whose output depends on both the present inputs and the state of the circuit is called a sequential circuit (e.g. counters).

State diagram of a mod-4 up/down counterthat detects the count of 2

State table

State assignment table

The next-state expressions are:

The output expression is

Implementation of the up/down counter

Timing diagram for the designed counter

A formal model of a finite state machine