Logic gate 1

Logic gates
Logic gates are primarily implemented electronically using diodes or transistors, but can also be constructed using
electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements.
With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing
the construction of a physical model of all of Boolean logic, and therefore, all of the algorithms and mathematics that
can be described with Boolean logic.

A logic gate is an idealized or physical device implementing a Boolean function, that is, it performs a logical operation
on one or more logic inputs and produces a single logic output. Depending on the context, the term may refer to an
ideal logic gate, one that has for instance zero rise time and unlimited fan-out, or it may refer to a non-ideal physical
device.[1] (see Ideal and real op-amps for comparison)

Background
The simplest form of electronic logic is diode logic. This allows AND and OR gates to be built, but not inverters, and
so is an incomplete form of logic. Further, without some kind of amplification it is not possible to have such basic
logic operations cascaded as required for more complex logic functions. To build a functionally complete logic system,
relays, valves (vacuum tubes), or transistors can be used. The simplest family of logic gates using bipolar transistors
is called resistor-transistor logic (RTL). Unlike diode logic gates, RTL gates can be cascaded indefinitely to produce
more complex logic functions. These gates were used in early integrated circuits. For higher speed, the resistors used
in RTL were replaced by diodes, leading to diode-transistor logic (DTL). Transistor-transistor logic (TTL) then
supplanted DTL with the observation that one transistor could do the job of two diodes even more quickly, using only
half the space. In virtually every type of contemporary chip implementation of digital systems, the bipolar transistors
have been replaced by complementary field-effect transistors (MOSFETs) to reduce size and power consumption still
further, thereby resulting in complementary metal–oxide–semiconductor (CMOS) logic.
For small-scale logic, designers now use prefabricated logic gates from families of devices such as the TTL 7400 series
by Texas Instruments and the CMOS 4000 series by RCA, and their more recent descendants. Increasingly, these
fixed-function logic gates are being replaced by programmable logic devices, which allow designers to pack a large
number of mixed logic gates into a single integrated circuit. The field-programmable nature of programmable logic
devices such as FPGAs has removed the 'hard' property of hardware; it is now possible to change the logic design of
a hardware system by reprogramming some of its components, thus allowing the features or function of a hardware
implementation of a logic system to be changed.
Electronic logic gates differ significantly from their relay-and-switch equivalents. They are much faster, consume
much less power, and are much smaller (all by a factor of a million or more in most cases). Also, there is a fundamental
structural difference. The switch circuit creates a continuous metallic path for current to flow (in either direction)
between its input and its output. The semiconductor logic gate, on the other hand, acts as a high-gain voltage
amplifier, which sinks a tiny current at its input and produces a low-impedance voltage at its output. It is not possible
for current to flow between the output and the input of a semiconductor logic gate.
Another important advantage of standardised integrated circuit logic families, such as the 7400 and 4000 families, is
that they can be cascaded. This means that the output of one gate can be wired to the inputs of one or several other
gates, and so on. Systems with varying degrees of complexity can be built without great concern of the designer for
the internal workings of the gates, provided the limitations of each integrated circuit are considered.
The output of one gate can only drive a finite number of inputs to other gates, a number called the 'fanout limit'.
Also, there is always a delay, called the 'propagation delay', from a change in input of a gate to the corresponding
change in its output. When gates are cascaded, the total propagation delay is approximately the sum of the individual
delays, an effect which can become a problem in high-speed circuits. Additional delay can be caused when a large
Logic gate 2

number of inputs are connected to an output, due to the distributed capacitance of all the inputs and wiring and the
finite amount of current that each output can provide.

Logic gates
All other types of Boolean logic gates (i.e., AND, OR, NOT, XOR, XNOR)
can be created from a suitable network of NAND gates. Similarly all
gates can be created from a network of NOR gates. Historically, NAND
gates were easier to construct from MOS technology and thus NAND
gates served as the first pillar of Boolean logic in electronic
computation.

For an input of 2 variables, there are 16 possible boolean algebraic

functions. These 16 functions are enumerated below, together with
their outputs for each combination of inputs variables.

INPUT A 0 0 1 1 Meaning

B 0 1 0
1

OUTPUT FALSE 0 0 0 0 Whatever A and B, the output is false. Contradiction.

Logic gate 3

A AND B 0 0 0 1 Output is true if and only if (iff) both A and B are

true.

A B 0 0 1 0 A doesn't imply B. True iff A but not B.

A 0 0 1 1 True whenever A is true.

A B 0 1 0 0 A is not implied by B. True iff not A but B.

B 0 1 0 1 True whenever B is true.

A XOR B 0 1 1 0 True iff A is not equal to B.

A OR B 0 1 1 1 True iff A is true, or B is true, or both.

A NOR B 1 0 0 0 True iff neither A nor B.

A XNOR B 1 0 0 1 True iff A is equal to B.

NOT B 1 0 1 0 True iff B is false.

A B 1 0 1 1 A is implied by B. False if not A but B, otherwise true.

NOT A 1 1 0 0 True iff A is false.

A B 1 1 0 1 A implies B. False if A but not B, otherwise true.

A NAND B 1 1 1 0 A and B are not both true.

TRUE 1 1 1 1 Whatever A and B, the output is true. Tautology.

The four functions denoted by arrows are the logical implication functions. These functions are not usually
implemented as elementary circuits, but rather as combinations of a gate with an inverter at one input.

Symbols
There are two sets of symbols in common use, both now defined by
ANSI/IEEE Std 91-1984 and its supplement ANSI/IEEE Std 91a-1991. The
"distinctive shape" set, based on traditional schematics, is used for
simple drawings, and derives from MIL-STD-806 of the 1950s and 1960s.
It is sometimes unofficially described as "military", reflecting its origin.
The "rectangular shape" set, based on IEC 60617-12, has rectangular
outlines for all types of gate, and allows representation of a much wider
range of devices than is possible with the traditional symbols. The IEC's
system has been adopted by other standards, such as EN 60617-12:1999
in Europe and BS EN 60617-12:1999 in the United Kingdom.

The goal of IEEE Std 91-1984 was to provide a uniform method of

describing the complex logic functions of digital circuits with schematic
symbols. These functions were more complex than simple
AND and OR gates. They could be medium scale circuits such as a 4-bit counter to a large scale circuits such as a
microprocessor. IEC 617-12 and its successor IEC 60617-2 do not include the "distinctive shape" symbols. [2] These
are, however, included in ANSI/IEEE 91 (and 91a) with this note: "The distinctive-shape symbol is, according to IEC
Publication 617, Part 12, not preferred, but is not considered to be in contradiction to that standard." This
compromise was reached between the respective IEEE and IEC working groups to permit the IEEE and IEC standards
to be in mutual compliance with one another.
Logic gate 4

In the 1980s, schematics were the predominant method to design both circuit boards and custom ICs known as gate
arrays. Today custom ICs and the field-programmable gate array are typically designed with Hardware Description
Languages (HDL) such as Verilog or VHDL.
Type Distinctive shape Rectangular shape Boolean algebra Truth table
between A & B

AND

INPUT OUTPUT

A B A AND B

0 0 0

0 1 0

1 0 0

1 1 1

OR

INPUT OUTPUT

A B A OR B

0 0 0

0 1 1

1 0 1

1 1 1

NOT

INPUT OUTPUT

A NOT A

0 1

1 0

In electronics a NOT gate is more commonly called an inverter. The circle on the symbol is called a bubble, and i s used in logic diagrams to
indicate a logical inv ersion between the external logic state and the internal logic state(1 to 0 or vice versa). On a circuit diagram it must be
accompanied by a statement asserting that the logic convention or negative logic convention is being used (high voltage level = 1 or high
positive
voltage level = 0, res pectively). The wedge is used in circuit diagrams to directly indic ate an active-low (high voltage level = 0) input or output
without requiring a uniform convention throughout the circuit diagram. This is called Direct Polarity Indication. See IEEE Std 91/91A and IEC
60617-12. Both the bubble and the wedge can be on distinctive-shape and rectangular-shape symbols on circuit diagrams, depending on the
used
logic convention used. On pure logic diagrams, only t he bubble is meaningful.

NAND

INPUT OUTPUT
Logic gate 5

A B A NAND B

0 0 1

0 1 1

1 0 1

1 1 0

NOR

INPUT OUTPUT

A B A NOR B

0 0 1

0 1 0

1 0 0

1 1 0

XOR

INPUT OUTPUT

A B A XOR B

0 0 0

0 1 1

1 0 1

1 1 0

XNOR

INPUT OUTPUT

A B A XNOR B

0 0 1

0 1 0

1 0 0

1 1 1

Two more gates are the exclusive-OR or XOR function and its inverse, exclusive-NOR or XNOR. The two input
Exclusive-OR is true only when the two input values are different, false if they are equal, regardless of the value. If
there are more than two inputs, the gate generates a true at its output if the number of trues at its input is odd ([3]).
In practice, these gates are built from combinations of simpler logic gates.
Logic gate 6

Universal logic gates

Charles Sanders Peirce (winter of 1880–81) showed that NOR gates
alone (or alternatively NAND gates alone) can be used to reproduce the
functions of all the other logic gates, but his work on it was unpublished
until 1933.[4] The first published proof was by Henry M. Sheffer in 1913,
so the NAND logical operation is sometimes called

Sheffer stroke; the logical NOR is sometimes called Peirce's arrow.[5]

Consequently, these gates are sometimes called universal logic
gates.[6]

De Morgan equivalent symbols

By use of De Morgan's theorem, an AND gate can be turned into an OR
gate by inverting the sense of the logic at its inputs and outputs. This
leads to a separate set of symbols with inverted inputs and the opposite
core symbol. These symbols can make circuit diagrams for circuits using additional pins supply power (+5 V) and connect
active low signals much clearer and help to show accidental connection
of an active high output to an active low input or vice-versa.

Symbolically, a NAND gate can also be shown using the OR shape with bubbles on its inputs, and a NOR gate can be
shown as an AND gate with bubbles on its inputs. The bubble signifies a logic inversion. This reflects the equivalency
due to De Morgans law, but it also allows a diagram to be read more easily, or a circuit to be mapped onto available
physical gates in packages easily, since any circuit node that has bubbles at both ends can be replaced by a simple
bubble-less connection and a suitable change of gate. If the NAND is drawn as OR with input bubbles, and a NOR as
AND with input bubbles, this gate substitution occurs automatically in the diagram (effectively, bubbles "cancel").
This is commonly seen in real logic diagrams - thus the reader must not get into the habit of associating the shapes
exclusively as OR or AND shapes, but also take into account the bubbles at both inputs and outputs in order to
determine the "true" logic function indicated.
All logic relations can be realized by using NAND gates (this can also be done using NOR gates). De Morgan's theorem
is most commonly used to transform all logic gates to NAND gates or NOR gates. This is done mainly since it is easy
to buy logic gates in bulk and because many electronics labs stock only NAND and NOR gates.

Data storage
Logic gates can also be used to store data. A storage element can be constructed by connecting several gates in a
"latch" circuit. More complicated designs that use clock signals and that change only on a rising or falling edge of the
clock are called edge-triggered "flip-flops". The combination of multiple flip-flops in parallel, to store a multiple-bit
value, is known as a register. When using any of these gate setups the overall system has memory; it is then called a
sequential logic system since its output can be influenced by its previous state(s).
These logic circuits are known as computer memory. They vary in performance, based on factors of speed,
complexity, and reliability of storage, and many different types of designs are used based on the application.
Logic gate 7

Three-state logic gates

Three-state, or 3-state, logic gates are a
type of logic gates that have three states
of the output: high (H), low (L) and high-
impedance (Z). The high-impedance
state plays no role in the logic, which
remains strictly binary. These devices are
used on buses also known as the Data
Buses of the CPU to allow multiple chips to send data. A group of three-states driving a line with a suitable control
circuit is basically equivalent to a multiplexer, which may be physically distributed over separate devices or plug-in
cards.

In electronics, a high output would mean the output is sourcing current from the positive power terminal (positive
voltage). A low output would mean the output is sinking current to the negative power terminal (zero voltage). High
impedance would mean that the output is effectively disconnected from the circuit.
'Tri-state', a widely-used synonym of 'three-state', is a trademark of the National Semiconductor Corporation.

Miscellaneous
Logic circuits include such devices as multiplexers, registers, arithmetic logic units (ALUs), and computer memory, all
the way up through complete microprocessors, which may contain more than 100 million gates. In practice, the gates
are made from field-effect transistors (FETs), particularly MOSFETs (metal–oxide–semiconductor field-effect
transistors).
Compound logic gates AND-OR-Invert (AOI) and OR-AND-Invert (OAI) are often employed in circuit design because
their construction using MOSFET's is simpler and more efficient than the sum of the individual gates. [7] In reversible
logic, Toffoli gates are used.

History and development

In a 1886 letter, Charles Sanders Peirce described how logical operations could be carried out by electrical switching
circuits.[8] Starting in 1898, Nikola Tesla filed for patents of devices containing logic gate circuits (see List of Tesla
patents). Eventually, vacuum tubes replaced relays for logic operations. Lee De Forest's modification, in 1907, of the
Fleming valve can be used as AND logic gate. Ludwig Wittgenstein introduced a version of the 16-row truth table,
which is shown above, as proposition 5.101 of Tractatus Logico-Philosophicus (1921). Claude E. Shannon introduced
the use of Boolean algebra in the analysis and design of switching circuits in 1937. Walther Bothe, inventor of the
coincidence circuit, got part of the 1954 Nobel Prize in physics, for the first modern electronic AND gate in 1924.
Active research is taking place in molecular logic gates.

Implementations
Since the 1990s, most logic gates are made of CMOS transistors (i.e. NMOS and PMOS transistors are used). Often
millions of logic gates are packaged in a single integrated circuit.
There are several logic families with different characteristics ( power consumption, speed, cost, size) such as: RDL
(resistor-diode logic), RTL (resistor-transistor logic), DTL (diode-transistor logic), TTL (transistor-transistor logic) and
CMOS (complementary metal oxide semiconductor). There are also sub-variants, e.g. standard CMOS logic vs.
advanced types using still CMOS technology, but with some optimizations for avoiding loss of speed due to slower
PMOS transistors.
Logic gate 8

Many early electromechanical digital computers, such as the Harvard Mark I, were built from relay logic gates, using
electro-mechanical relays.
It is also possible to make logic gates out of pneumatic devices, such as the Sorteberg relayor mechanical logic gates,
including on a molecular scale.[9] Logic gates have been made out of DNA (see DNA nanotechnology)[10] and used to
create a computer called MAYA (see MAYA II).
Additionally, logic gates can be made from quantum mechanical effects (though quantum computing usually diverges
from boolean design).
It is also possible to make photonic logic gates using non-linear optical effects.

References
[1] Jaeger, Microelectronic Circuit Design, McGraw-Hill 1997, ISBN 0-07-032482-4, pp. 226-233
[2] http:/ / focus. ti. com/ lit/ ml/ sdyz001a/sdyz001a. pdf
[3] http:/ / www-inst. eecs.berkeley. edu/ ~cs61c/ resources/dg-BOOL-handout. pdf
[4] Peirce, C. S. (manuscript winter of 1880–81), "A Boolean Algebra with One Constant", published 1933 in Collected Papers v. 4,
paragraphs
12–20. Reprinted 1989 in Writings of Charles S. Peirce v. 4, pp. 218-21, Google Preview (http:/ / books. google. com/
books?id=E7ZUnx3FqrcC& q=378+Winter). See Roberts, Don D. (2009), The Existential Graphs of Charles S. Peirce, p. 131.
[5] Hans Kleine Büning; Theodor Lettmann (1999). Propositional logic: deduction and algorithms (http:/ / books. google. com/
books?id=3oJE9yczr3EC& pg=PA2). Cambridge University Press. p. 2. ISBN 9780521630177. .
[6] John Bird (2007). Engineering mathematics (http:// books.google. com/ books?id=1-fBmsEBNUoC& pg=PA532). Newnes. p. 532. ISBN
9780750685559. .
[7] Tinder, Richard F. (2000). Engineering digital design: Revised Second Edition (http:/ / books.google. com/ books?id=6x0pjjMKRh0C&
pg=PT347& lpg=PT347& dq=AOI+ gate& source=web& ots=t-wt6hoi1-& sig=dlnF_Kq9jYe27dbr7Rb5gThgM2Y& hl=en& sa=X&
oi=book_result& resnum=7& ct=result#PPT346,M1). pp. 317–319. ISBN 0126912955. . Retrieved 2008-07-04.
[8] Peirce, C. S., "Letter, Peirce to A. Marquand", dated 1886, Writings of Charles S. Peirce, v. 5, 1993, pp. 541–3. Google Preview (http:/
/ books.google. com/ books?id=DnvLHp919_wC& q=Marquand). See Burks, Arthur W., "Review: Charles S. Peirce, The new elements
of mathematics", Bulletin of the American Mathematical Society v. 84, n. 5 (1978), pp. 913–18, see 917. PDF Eprint (http:/ /
projecteuclid.org/ DPubS/ Repository/ 1. 0/ Disseminate?view=body& id=pdf_1& handle=euclid.bams/1183541145).
[9] Mechanical Logic gates (focused on molecular scale) (http:/ / www. zyvex. com/ nanotech/mechano. html)
[10] DNA Logic gates (https:/ / digamma. cs. unm. edu/wiki/ bin/view/ McogPublicWeb/ MolecularLogicGates)

• Symbols for logic gates (http://www.uq.edu.au/_School_Science_Lessons/ 39.4.01.GIF). Twenty First Century

Books, Breckenridge, CO.
• Wireless Remote Control and the Electronic Computer Logic logic elements (http://www.tfcbooks. com/
articles/control.htm)

Further reading
• Awschalom, D., D. Loss, and N. Samarth, Semiconductor Spintronics and Quantum Computation (2002), Springer-
Verlag, Berlin, Germany.
• Bostock, Geoff, Programmable Logic Devices. Technology and Applications (1988), McGraw-Hill, New York, NY.
• Brown, Stephen D. et al., Field-Programmable Gate Arrays (1992), Kluwer Academic Publishers, Boston, MA.

External links
• Digital Logic Simulator v0.4 - Brad-Ware Studios' free program that supports real-time edit and simulation, as well
as abstracting. Include entire scenes that you have created, as a single chip. (http:/ / bradwarestudios. com/
downloads/fun/Digital_Logic_Simulator)
• Using Logic Gates (http:/ / knol. google. com/ k/ max-iskram/ digital-electronic-design-for-beginners/
1f4zs8p9zgq0e/23)
• Online logic gate simulator (http://www.neuroproductions. be/logic-lab/index.php?id=52)
• Java applet of NOT gate (http:/ /www.phy.hk/wiki/englishhtm/NotGate.htm)
Logic gate 9

• LogicCircuit – is free educational software for designing and simulating digital logic circuits. (http:/ / www.
logiccircuit.org/)
• Logic Gate Simulator in Adobe Flex (http://joshblog.net/projects/logic-gate-simulator/Logicly.html)
• Redstone circuits (http://www.minecraftwiki.net/ wiki/Redstone_circuits) on the minecraft wiki (a specific type of
simulated logic circuitry).
 10

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