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Transcript
Digital Logic Circuits:
A Brief Introduction
(with Appendix p.25-33)
EETS8304/TC715-N
SMU/NTU
Lecture Scheduled Feb. 3, 2004
Digital Logic Devices
(print slides only, no notes pages)
Page 1
©1997-2004 R.Levine
Digital Hardware
• Exploits non-linear properties of
electronic devices
– Electro-mechanical relays, vacuum tubes were
used in the past
– Electronic, Dielectric and Magnetic devices
were also used historically
– Semiconductor electronic devices used
primarily today due to high component
density, rapid operation, and low cost of
integrated circuits
Page 2
©1997-2004 R.Levine
Semiconductor Devices
– Diodes
– Transistors
• Junction
• Field Effect (and Metal Oxide Silicon--MOS)
– Advantages:
• Small size
– small, high-functionality in integrated circuit package
• High reliability
– failure mechanisms studied, understood, and avoided
• Fast switching operation
– but faster switching requires higher power!
• Low power consumption vis-à-vis prior art
– No heated filament, as in vacuum tubes
• Low cost of major raw materials
– Silicon is readily available everywhere, some dopant
elements are scarce but are used in very small quantities
Page 3
©1997-2004 R.Levine
Linear Electric Circuit
vout = (R2•R3) •v1 + (R1•R3) •v2
R1•R2 + R2•R3 + R1•R3
R1
v1
R2
v2
R3
vout
vout = (2) •v1 + (2) •v2
4 + 2 +2
Page 4
©1997-2004 R.Levine
Non-linear Electric Circuit
Analysis using piece-wise linear model of
diode, illustrated on voltage-current graph
vout
Ideal voltmeter reading: 0 volts
vout
v1
Ideal, vout=v1
10 V
9.37 V
10 V
Actual reading: -10µA•10k= -0.1 V,
due to non-zero reverse current in
diode.
Page 5
©1997-2004 R.Levine
v1
One Stage Electronic Amplifiers
• Various different graphic symbols are
used to represent the amplifier
“ground return circuit”
for +5 V power
supply is customarily
omitted on drawings.
vout
6
Cutoff region
4
approximately
linear amplification
region of operation.
Note negative slope.
vout
2
vin
-2
Page 6
©1997-2004 R.Levine
-1
0
Vin =0.2 V is
-2
“edge” of
cutoff
-4
1
2
vin
Saturation region
Vin =0.9 V is
“edge” of
saturation
Amplifier Distinctions I
• Applications which use the approximately linear
region:
– Hi-fidelity audio amplifiers
– Radio frequency amplifiers (some types)
– Analog amplifiers in telephone transmission
• First used in early 20th century with vacuum tubes
• Linear region is not quite as linear as is needed
for a chain of numerous amplifier stages
– Cumulative distortion (“flattening” of the peaks of the
waveform) occurred in long distance analog telephone
conversations
• Negative feedback (invented by H.S. Black of Bell
Laboratories, ca. 1927) improves linearity, reduces
internal noise generation
Page 7
©1997-2004 R.Levine
Amplifier Distinctions II
• Non-linear applications intentionally use
primarily the cutoff and saturation regions
– Minimal power dissipated in these regions
• power (watts) is product of voltage with current
Power P= v•i
• In these operating regions, either v or I is low
– therefore low power dissipated in transistor
– design objective is typically to switch through the linear
region as rapidly as possible to minimize power
consumption
– Certain types of amplifiers (so-called Class C
and D, etc.) use non-linear properties for special
applications (not described in this course)
Page 8
©1997-2004 R.Levine
Digital Logic is the Main
Non-Linear Application
• Technology and design knowledge to
make digital logic equipment is
highly developed
• Fabrication in integrated circuit form
is well known, widely available
• Small devices switch between cutoff
and saturation states quickly
– thus minimizing power consumption
Page 9
©1997-2004 R.Levine
Boolean Algebra* Description
•Boolean Algebra is the symbolic “language” of digital
logic system design
– Computers and digital switching systems almost all use
hardware with 2 voltage levels
– Multi-level (e.g. 4 level) hardware devices announced by
Intel Corp. in 1998, but binary symbolism still used
•Theoretical invention by George Boole (British
mathematician ca. 1860)
– “Re-discovered” by Claude E. Shannon and applied to
relay logic circuit design in 1939 (master’s thesis).
•Two-valued variables
– Usually represented by the symbols 0 and 1
– Closely related to binary numbering system as well.
*from the name of the first Arabic book known in Europe on classic algebra, Al'jabr, by Abu Ja’far
Muhammed ibn Musa Al-Khwarizmi. Born approx. 780, he is credited with the introduction of the symbol for
zero into Europe, and his name led to the word “algorithm.” Historically, Al’jabr means “repair of broken
bones.” Today, in general, algebra means “rules for manipulation of symbols”; not limited to high school
algebra using decimal numbers.
Page 10
©1997-2004 R.Levine
Basic operations analogous
to ordinary arithmetic
• “Sum” analogous to addition, usually
represented by + or  sign
• “Product” analogous to multiplication,
usually represented by • (or * on
typewriter)
• Confusion is possible when binary
arithmetic is under discussion vs. binary
Boolean logical operations
– Read with care!
Page 11
©1997-2004 R.Levine
Binary Variables
• Two voltage levels are most widely used
– Called TTL (Transistor-transistor logic) levels
– Older equipment (and some recent Intel memory chips)
use more than two levels in some cases
• Typically zero (actually about 0.2) and 5 volts
– Lately, 3 or 1.5 volts is also used for the “high” level, for
faster switching and compatibility with battery voltage
(~1.5 V per cell)
• High level is symbolized by 1, low by 0
– Other symbols are occasionally used: T,F; H,L; ON, OFF,
etc.
– In some special cases, 0 is used for high and 1 for low.
This is sometimes called “negative” or “active low”
logic. We will not use this method here to avoid
confusion.
Page 12
©1997-2004 R.Levine
One-stage Amplifier forms an
Inverter (graphic symbols)
y
x
JEDEC (Joint
Electronic Defense
Department Experts
Commission) Symbol
y
x
The little circle is the true
graphic indicator of
inversion.
International
Electro-technical
Commission (IEC) symbol
x y
0 1
1 0
• Signal flow is normally left-to-right
• “Ground” wires are conventionally not drawn
• The small circle represents the negation (logical
negation changes 1 to 0; 0 to 1)
• The triangle symbol is also used to represent a
linear amplifier in other contexts.
– Read with care!
Page 13
©1997-2004 R.Levine
Two Chained Stages form a
non-inverting device
IEC Symbol
y
y
x
x
y
x
x y
0 0
1 1
JEDEC Symbol
• Logically, this does “nothing,” but in practical circuits it
allows greater “fan-out” and delay
• It produces greater delay
– However, in modern devices, the delay per stage is typically 20
to 50 nanoseconds
– Simple analysis ignores delay
– Intentional delay is desired in some cases to synchronize two
signal
paths ©1997-2004 R.Levine
Page
14
Boolean Symbolic
Operations
• Finite number of combinations, well
represented by a table
• Only 2-input forms shown here, but
multiple inputs are used in general
Logical sum
Logical product
(inclusive OR)
(AND)
x
y
z=x+y x0 y0 z0
w =x • y
0
0
0
1
1
Note, in this
context, that
1+1 is 1, not 2.
Page 15
1
0
1
1
1
1
©1997-2004 R.Levine
0
1
1
1
0
1
w
0
0
0
1
Logical OR Devices
• Multiple input non-inverting device with any one input
sufficient to raise the output
– Examples here use only 2 inputs, but multiple input
OR devices are widely used
• Several implementations:
–
–
–
–
Transistor with two controlling electrodes “side by side”
Junction transistor with two emitters
FET with two gates
single control electrode (emitter or gate), with multiple inputs
connected via resistors (used in past with discrete
components)
• Either controlling electrode alone can turn the anode
current ON, thus lowering Vout.
– Then a second chained inverting stage used to produce binary
1 output.
Page 16
©1997-2004 R.Levine
Logical AND Device
• Non-inverting device constructed so that
all inputs must go ON to switch the output
ON
– Examples here use only 2 inputs, but multiple
input AND devices are widely used
• Transistor constructed with multiple
controlling electrodes “in series”
– Any one electrode, if OFF, turns the device
anode current OFF
– Followed by chained inverting stage to get
binary 1 output
Page 17
©1997-2004 R.Levine
Other Implementations and Graphic
Symbols
&
x
JEDEC “shaped”
symbol. Bullet nose,
straight back
y
AND gate symbols
•
•
•
Hollow “box” graphic symbol
represents amplifier on page 6.
Via suitable resistor size choices,
the circuit above could act either as
an AND or an OR device (see pp.
30ff for details)
A multi-input device with special
input transistors can be an AND or
an OR device
Page 18
©1997-2004 R.Levine

JEDEC “shaped”
symbol. Bullet or
curved and pointed
nose, concave back
OR gate symbols
Basic Building Blocks
• In this course, we will use 3 basic building
block devices:
 AND
 OR
 NOT
• In actual system designs, the basic
electronic building block modules may be
NOR or NAND (that is NotOR, NotAND)
– Corresponds to one-stage output physical devices
– Less hardware and less signal delay in some cases
– Many abstract algebra structures have multiple choices
for their basic building blocks. Further exploration of
alternative building blocks is not done in this course.
Page 19
©1997-2004 R.Levine
Combinatorial Logical Design
• No “ingenuity” is required to make a base working system
– of course, further optimization is possible
• Desired logical functionality must first be stated in equation
or table form, relating all logical inputs to desired output(s)
• SUM OF PRODUCTS: Each row in the table which produces
binary 1 output can be related to a multi-input AND device,
and the outputs of all such devices can be used as input to
a multi-input OR device.
• PRODUCT OF SUMS: Alternatively, each row in the table
which produces a binary 0 can be related to a multi-input OR
device, and all these outputs can be used as input to a
multi-input AND device.
• Choice between these two methods based on smallest
component count or other criterion (fastest switching time,
minimization of the number of interconnections, etc.)
• Many other synthesis or design methods are also known,
but not described here
Page 20
©1997-2004 R.Levine
Example Logic Table and Formula
_
Negation of x denoted x in most documents. We write x’ due to typographic limitations.
row
0
1
2
3
4
5
6
7
x
0
0
0
0
1
1
1
1
y
0
0
1
1
0
0
1
1
Page 21
z
0
1
0
1
0
1
0
1
u
1
0
1
0
1
0
1
0
u = (x’ •y’ •z’) +
(x’• y • z’) +
(x • y’ • z’) +
(x • y • z’)
This is a “sum of products” formula.
An equally valid “product of sums”
formula can also be stated. Since this
example has 4 zero and 4 one values
for u, either method is equally “efficient”
using AND and OR building blocks.
©1997-2004 R.Levine
Simple Logic Synthesis
Note that • or T intersection in wiring graphics represents a connection.
&
x
Small hollow circles represent negation
at the input of a multi-input device.
Multi-input AND (&) output is 1
when all inputs are 1. Multi-input OR
is 1 when one (or more) input(s) is/are 1.
&
y

u
&
z
&
Page 22
©1997-2004 R.Levine
Example
corresponds to
previous page
sum of
products
This design can be simplified and
redesigned to operate faster (less
gates in the signal path) in different
ways depending on the optimization
criteria.
Many Standard Logic Devices
• Many logical functions are required in large
quantities, and are typically mass produced in
special purpose integrated circuit form.
– Hardware logic “works” fast, uses less chip area, has
other desirable properties (vis-à-vis software).
– Desirable way to implement a function if the quantity
needed is large and economically attractive.
• Often Called Application Specific Integrated
Circuits (ASICs). Examples:
– Error protection coding and decoding
– Encryption/decryption devices
– Devices which scan for a special binary bit pattern (e.g.,
for synchronization bit string within a bit stream)
Page 23
©1997-2004 R.Levine
“Wired” vs. Programmable Logic
• Logical operations performed by a programmable computer
usually require more time and power, since operations are
performed sequentially
• Because hardware design (and test) is a large “up-front” cost,
special hardware is usually made only for a large quantity
(mass produced) product
• ASICs often combine special hardware with a CPU for both the
flexibility of software and the speed of wired logic
• Field “Programmable” Logic Arrays (FPLAs) have numerous
multi-input gates on a chip, with interconnect “wiring” (a
surface metallic pattern of connections) applied by the
designer for a particular application. This allows the economy
of mass production with the flexibility of small production
quantities.
Page 24
©1997-2004 R.Levine
Appendix
Supplementary reading to do after the
lecture.
• Alternative “product of sums” logic
synthesis
• AND gate vs. OR gate by changing
resistors in the same basic circuit
design (from p. 18)
Page 25
©1997-2004 R.Levine
Alternative Logic Design
• On p. 21, a table describing an arbitrary
logical requirement is shown with a sum
of products Boolean equation
• Here is a product of sums equation with
the same resulting value:
u = (x +y +z’) •
(x+ y’ + z’) •
(x’ + y + z’) •
(x’ + y’ + z’)
Page 26
©1997-2004 R.Levine
Equation-Table Correspondence
• Equation on p. 21 corresponds to rows
0,2,4,6 of table (rows having value u=1)
• Equation on p. 26 corresponds to rows
1,3,5,7 of table (rows having value u=0)
• Diagram on next page corresponds to
these new Boolean equations from p. 26
Page 27
©1997-2004 R.Levine
Alternative Logic Circuit
product of sums; compare with p.22
x

y

Small circles represent negation
at the input of a multi-input device.
Multi-input AND (&) output is 0
when one or more of its 4 inputs is/are 0.
Output of multi-input OR
is 0 when all 3 inputs are 0.
&
u

z

Page 28
©1997-2004 R.Levine
This design can be simplified and
redesigned to operate faster (less
gates in the signal path) in different
ways depending on the optimization
criteria.
Circuit for AND or OR
• Diagram on p. 18 can be used to make
either an AND gate or an OR gate by
changing resistor sizes.
• For the inverter circuit on p. 6
– When vin0.9 V, output is zero (low).
– When vin 0.5V, output is 5V (high)
– The second chained inverter circuit on p.6 will
give overall non-inverting output in the final
design (see result on p. 32)
– Three equal value 1k resistors are shown on p.18.
Think about using other values for the two resistors on
inputs x and y, with third resistor still 1k
Page 29
©1997-2004 R.Levine
Chose Resistor Values
• Input circuit with 3 resistors (p.18) is topologically
equivalent to circuit with three resistors shown on p. 4. Use
same formula.
• R1 is like the x input resistor, v1 is like the x input
– R2 is like the y input resistor, v2 is like the y input
– R3 is like the third resistor (use 1k value)
• Design objective: Use equal value (called R) for R1 and R2 ,
so that specific combinations of v1 and v2 produce 0.9 or 0.5
volts at the amplifier input v
– Both input resistors x and y are made equal so the behavior of the
device will be symmetrical with respect to the two input voltages
– Small resistors produce an OR gate (one input alone can switch the
output). Larger resistors produce an AND gate (high voltage on both
inputs is needed to switch the output). But if the resistors are too big,
the output never will switch...
Page 30
©1997-2004 R.Levine
Logic Design Resistor Choices
vx
(V)
0 0
v=0
0 5
5 0
5 5
v=5/(2+R)
Required Conclufor AND sion for
AND
no conv0.5
clusion
v0.5
R8k
v=5/(2+R)
v0.5
R8k
v0.9
R3.55k
v=10/(2+R)
v0.9
R9.11k
v0.9
R9.11k
•
•
•
vy v at amp(V) lifier input
Required Conclufor OR
sion for
OR
no conv0.5
clusion
v0.9
R3.55k
0.5 and 0.9 volts are input voltage “edges” of cutoff and saturation
states of amplifier on page 6.
For AND gate, an R value in range 8kR9.11kwill be OK
For OR gate, an R3.55kwill be OK
– Note that OR gate on p.18 already has R=1k
Page 31
©1997-2004 R.Levine
AND gate with Resistor Input Circuit
x
8
y
• Same topology as p. 18, but larger
x,y resistors (8k instead of 1k)
• Internal construction of “non-inverting”
amplifier is usually two chained inverting
amplifiers like this example
Page 32
©1997-2004 R.Levine
Practical Component Notes
• Although an AND or OR gate can be made
using resistors at the input, the operation
is less reliable if the voltages or resistor
values are not precise. Multiple gate FETs
or multiple emitter BJTs are more reliable
over a range of power and signal voltage,
and require less semiconductor chip area.
• Many “real” logic designs use NOR (not
OR) or NAND (notAND), which do not
require a second inverting amplifier.
Details are omitted here for brevity.
Page 33
©1997-2004 R.Levine