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CSSE1000 Lecture 2
Intro to Logic Gates
School of Information Technology and Electrical EngineeringThe University of Queensland
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Admin Issues
Circuit Schematics
Boolean Algebra
If you have questions, ask them at anyme
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Digital circuits
Only two logical levels present (i.e. binary)0 usually small voltage (e.g. around 0 volts)
1 usually larger voltage (e.g. 1.5 to 5 volts, dependingon the logic family, i.e. type/size of transistors)
Logic gates are the building blocks of computers;
Each gate has one or more inputs
perform operations (or functions) e.g.: NOT, AND, OR, NAND, NOR gates
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Inputs and outputs can have only two states, 1 and 0 Can be called true and false
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NOT gate inverts the signal
If A is 0 X is 1
A X
This is thes y m b o l for
If A is 1, X is 0 Gates o eration can be re resented usin a
a NOT gate
truth table:A X
1 0Input Output
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Output is 1 if all inputs are 1, i.e. ifin ut A a n d in ut B
a n d input Ca n d areall 1, t he output is 1 Logic symbol
Truth Table
A B X
2-input AND gate The opposite is a
A X 0 1 0
1 0 0
NAND gate (output
is 0 if all inputs are 1)
1 1 1
NANDA
X
0 0 1
0 1 1A
X
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B
1 1 0B
Bubble m eans inversion
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Output is 1 if at least one of the inputs is 1 i.e. ifin ut A o r in ut B
o r input Co r is 1,the output is 1
Logic symbolru a e
A B X
0 0 0OR
2-input OR gate The opposite is a
0 1 11 0 1
1 1 1B
X
NORgate (output
is 0 if any input is 1)NOR
AXX
ORA
0 0 1
0 1 0
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B
Bubble m eans inversion
B1 1 0
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For a 2-input gate Out ut is 1 ifexactl one of the in uts is 1
Lo ic s mbolTruth Table
A
BX
0 0 0
0 1 1
1 1 10This entry is the
difference between For > 2 inputs
Output is 1 if an odd number of inputs is 1
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Sometimes called the odd function
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Logic functions can be expressed
variables, e.g. A B X
unc ons, e.g. + .Rules about how this works called
Boolean algebra
take on values 0 or 1
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Conventions well use: Inversion: overline
e.g. NOT(A) =
AND: dot(.) or implied (by adjacency)
e.g. AND(A,B) = AB = A.B OR: plus sign
e.g. OR(A,B,C) = A+B+C
Other examples:
NAND(A,B,C) = ABC
XOR(A,B) = A B = B + AB
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NOR(A,B) = A+B
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Summar of Lo ic FunctionRepresentations
There are four representations of logic
functions assume function ofn in uts Truth table
Lists output for all 2n combinations of inputs
Best to list inputs in a systematic way Boolean function (or equation)
output is 1
Lo ic Dia ram Combination of logic symbols joined by wires
Timing Diagram
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Inputs are conventionally on left Outputs are
conventionally
A
on right
BM
ElectricalConnection
No Connection Lines represent electricalwires. Can be at hi h or
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low potential (logic 1 orlogic 0 respectively).
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Gates onIntegrated Circuits (ICs)
14 13 12 11 10 9
VCC
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n spac ng s .
x 0.3; chip is about1 2 3 4 5 6 7
GND
74LS00 has four 2-input NAND gates
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. . ,
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More than a logic diagram, a circuit schematic tells
you how to build the circuit
ICs are labelled U1, U2 etc (Other components also labelled)T e of IC iven also e. . 74LS00 or 74HCT00
Outputs labelled
Connectionswhich arentpart of the
Pin numbers are iven
separately
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Inputs are labelled
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Lo ic FunctionImplementation
Any logic function can be implemented as
the OR f AND c mbinati ns f the in uts Called sum of products
Consider truth table
A B C
0 0 0M
0
,write down the AND combination
of in uts that ive that 1
0 1 0
0 1 1
0
1
OR these together 1 0 1
1 1 0
1
1
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1 1 1 1
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Lo ic FunctionImplementation
A B C
0 0 0
M
00 0 1
0 1 0
0 1 1
0
0
1
1 0 0
1 0 1
1 1 0
0
1
1
1 1 1 1
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Exam le cont.Equivalent Logic Diagram
A
A
B C A B C
M = ABC + ABC + ABC + ABC
A
A B C4
A B C5
M8B
2
C
A B C
6
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A B C7C
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Sum of products does not necessarily produce circuit
with minimum number of gatesfunction Use rules of Boolean algebra (next slide)
Example: Z = AB + AC = A(B+C)
A AB
A A(B + C)
B
AB + AC
C
B
B + CC AC
22Fewer gates
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= + =
Name AND form OR form
Null law
0A = 0
1 + A = 1
Inverse law
= =
AA = 0 A + A = 1
Commutative law
Associative law
AB = BA
(AB)C = A(BC)
A + B = B + A
(A + B) + C = A + (B + C)
Distributative law
Absorption law
A + BC = (A + B)(A + C)
A(A + B) = A A + AB = A
A(B + C) = AB + AC
23De Morgan's law AB = A + B A + B = AB
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Express Z = A(B+C(A + B)) as a sum
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r w u v
AND/OR can be interchanged if you invert
the inputs and outputsAB = A + B A + B = AB
(a) (b)
A + BAB = AB=A + B
(c) (d)
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Homework: Use truth tables to convinceyourself that these are valid
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All circuits can be constructed from NAND or NOR gates
These are called complete gates Exam les:
NOT AND OR
A + BAB
AB
B
A A
A+
B
BA
A
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Reason: Easier to build NAND and NOR gates fromtransistors
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A B XOR
0 0 0
A
B
0 1 1
1 0 1 A
On any wire, you can introduce a
1 1 0
A
B
A
B
AA
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BB
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Go through the course profile
Do the reading if you havent already Do the homework (due at Prac sessions)
Sign-up (via session change request form) for
rac sess on
Tutorial session
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