Appendix A
Logic 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-variable
Karnaugh 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-segment
display 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 counter
that 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