EE210: Switching Systems
Lecture 4: Implementation AND, OR, NOT
Gates and Compliment
Prof. YingLi Tian
Sept. 10, 2012
Department of Electrical Engineering
The City College of New York
The City University of New York (CUNY)
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TA’s Email:
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Students who didn’t receive TA’s email,
please send an email to Mr. Zhang, by
putting subject: “EE210 email”
Mr. Chenyang Zhang
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czhang10@ccny.cuny.edu
Course website:
http://wwwee.ccny.cuny.edu/www/web/yltian/EE2100
.html
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Outlines
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Quick Review of the Last Lecture
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AND, OR, NOT Gates
Switching Algebra
Properties of Switching Algebra
Definitions of Algebraic Functions
Implementation AND, OR, NOT Gates
Complement (NOT)
Truth table to algebraic expressions
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Definition of Switching Algebra
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OR -- a + b (read a OR b)
AND -- a · b = ab (read a AND b)
NOT -- a´ (read NOT a)
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SOP and POS
A sum of products expression (often abbreviated SOP) is one or
more product terms connected by OR operators.
ab´ + bc´d + a´d + e´ ---- ?? terms, ?? literals
A product of sums expression (POS) is one or more sum terms
connected by AND operators.
SOP: x´y + xy´ + xyz
POS: (x + y´)(x´ + y)(x´ + z´)
A literal is the appearance of a variable or its complement.
A term is one or more literals connected by AND, OR, operators.
Gate Implementation
P2b: a(bc) = (ab) c
These three implementations are equal.
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Implementation of functions with
AND, OR, NOT Gates -- 1
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Given function: f= x´yz´ + x´yz + xy´z´ +
xy´z + xyz
Two-level circuit
(maximum number
of gates which a signal
must pass from the input
to the output)
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Implementation of functions with
AND, OR, NOT Gates -- 2
(1) x´yz´ + x´yz + xy´z´ + xy´z + xyz
(2) x´y + xy´ + xyz
(3) x´y + xy´ + xz
(4) x´y + xy´ + yz
Implementation of functions with
AND, OR, NOT Gates -- 3
Function: x´y + xy´ + xz,
when only use uncomplemented inputs:
Multi-level circuit
Function? (see Page50)
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Commonly used terms
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DIPs – dual in-line pin packages (chips)
ICs – integrated circuits
SSI – small-scale integration (a few gates)
MSI – medium-scale integration (~ 100
gates)
LSI -- large-scale integration
VLSI – very large-scale integration
GSI – giga-scale integration
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Examples
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Need a 3-input OR (or AND), and only 2input gates are available
Need a 2-input OR (or AND), and only 3input gates are available
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Positive and Negative Logic
Use 2 voltages to represent logic 0 and 1
For example:
Low: 0-1.4 Volt;
High: >2.1Volt;
Transition state: 1.4-2.1Volt
Positive logic: High voltage  1, Low voltage  0
Negative logic: Low voltage  1, High voltage  0
The Complement (NOT)
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DeMorgan:
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P11a: (a + b)´ = a´ b´
P11b: (ab)´ = a´ + b´
P11aa: (a + b + c …)´ = a´ b´ c´ …
P11bb: (abc…)´ = a´ + b´ + c´ + …
Note:
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(ab)´ ≠ a´ b´
(a + b)´ ≠ a´ + b´
ab + a´ b´ ≠ 1
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Find the complement of a given function
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Repeatedly apply DeMorgan’s theorem
1. Complement each variable (a to a´ or a´ to a)
2. Replace 0 by 1 and 1 by 0
3. Replace AND by OR, OR by AND, being
sure to preserve the order of operations
See Example 2.5 (Page53) and Example 2.6
(page 54).
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Example of Complement
f = wx´y + xy´ + wxz
f ´ = (wx´y + xy´ + wxz)´
= (wx´y)´(xy´)´(wxz)´
= (w´+x+y´)(x´+y)(w´+x´+z´)
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Truth Table to Algebraic Expressions
f is 1
if a = 0 AND b = 1 OR
if a = 1 AND b = 0 OR
if a = 1 AND b = 1
f is 1
if a´ = 1 AND b = 1 OR
if a = 1 AND b´ = 1 OR
if a = 1 AND b = 1
f is 1 if a´b = 1 OR if ab´ = 1 OR if
ab = 1
f = a´b + ab´ + ab = a + b (OR)
A standard product term, also minterm is a product term that
includes each variable of the problem, either uncomplemented
or complemented.
f
f´
0 1
1 0
1 0
1 0
To obtain f (A, B, C), add
all minterms with output
= 1 (SOP):
f (A, B, C) = ∑m(1, 2, 3, 4,5)
= A´B´C + A´BC´ + A´BC + AB´C´+ AB´C
f ´(A, B, C) = ∑m(0, 6, 7) = A´B´C´ + ABC´ + ABC
1 0
1 0
0 1
0 1
A standard sum term, also called a maxterm, is a sum term that
includes each variable of the problem, either uncomplemented
or complemented.
f
f´
0 1
1 0
1 0
1 0
1 0
1 0
0 1
POS:
f = (f ´ )´= (A + B + C)(A´+B´+C)(A´+B´+C´)
0 1
To simplify:
f (A, B, C) = A´B´C + A´BC´ + A´BC + AB´C´+ AB´C
= A´B´C + A´B + AB´ = A´(B´C + B) + AB´
= A´C + A´B + AB´
= B´C + A´B + AB´
P10a: B + C
f ´(A, B, C) = A´B´C´ + ABC´ + ABC
= A´B´C´ + AB
See page56 for details.
P8a: a (b + c) = ab + ac
P9a: ab + ab´ = a
P10a: a + a´ b = a + b
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Truth Table with don’t care
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Include them as a separate sum.
f (a, b, c) = ∑m(1, 2, 5) + ∑d(0, 3)
a
b
c
f
0
0
0
X
0
0
1
1
0
1
0
1
0
1
1
X
1
0
0
0
1
0
1
1
1
1
0
0
1
1
1
0
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Number of different functions of n variables
Announcement:
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Review Chapter 2.3-2.5
HW2 is out today, due on 9/12.
Next class (Chapter 2.6-2.7):
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NAND, NOR, Exclusive-OR (EOR) Gates
Simplification of Algebraic Expressions
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