Download Logic and Computer Design

Logic and Computer Design
Simon Petruc-Naum
CS 147 – Dr. S.M. Lee
Combinational Logic
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What is it?
What is it used for?
How is it Implemented?
Logical Gates
What is Combinational Logic?
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Computers make decisions based on a
combination of inputs.
A combinational Logic Unit takes Boolean
data as input and produces Boolean data as
output.
Uses for Combinational Logic
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Combinational Logic Units can be used to
perform many functions
Consider the binary adder function
Binary Adder
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Single Digit adder
How does it work?
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Boolean functions are created by combining
logical gates.
Examples of logical gates are NOT, AND,
OR, NAND, NOR, XOR…
In reality
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Logical gates are mathematical abstractions.
I reality they are implemented using
Transistor-Transistor Logic (TTL), or
Complementary Metal Oxide
Semiconductors (CMOS).
TTL for NAND gate
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If both A and B have a voltage than current
will flow to GND otherwise it will flow to Q.
NMOS Implementation of NAND
Similar to CMOS, NMOS uses only
N-type transistors.
Again we can see that if either
A or B has voltage, F will have
a current
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Positive vs. Negative Logic
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Assigning a logical 1 to a voltage is positive Logic
Assigning a logical 1 to a lack of voltage is negative logic
Positive Logic AND gate
Negative Logic OR gate
A
B
F
A
B
F
0
0
0
1
1
1
0
1
0
1
0
1
1
0
0
0
1
1
1
1
1
0
0
0
Combining Logical Gates
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Sum of Product form (SOP)
F=A’B’C+A’BC’+AB’C’+A’B’C’
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Each product term is a minterm because it
contains one of each variable.
Each minterm has a value of 1 for exactly 1
row in the truth table (and 0 for the others)
Combining Logical Gates
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Product of Sum form (POS)
F=(A’+B’+C)(A’+B+C’)(A+B’+C’)(A’+B’+C’)
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Each sum term is a maxterm because it
contains one of each variable.
Each maxterm has a value of 0 for exactly 1
row in the truth table (and 1 for the others)
Metrics for Combinational Logic
Gate Count: number of logic gates in the
logical unit.
 Gate Input Count: total number of inputs of
the combined gates.
 Circuit Depth: Maximum number of gates
along a path from inputs to outputs.
(e.g. 2 for as POS and SOP)
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Reduction of Combinational Logic
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Reducing the complexity of combinational
logic can save cost, and improve
performance.
Implicants
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A product term is an implicant of a Boolean
function if the function takes the value of 1
whenever the product term equals 1.
A prime implicant is an implicant that cannot
have any fewer literals. (i.e. cannot be
reduced)
Essential prime implicants cover the output of
a function that no other prime implicant can
cover.
Reducing Boolean Expressions
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Determine all prime implicants.
Select all essential prime implicants.
For non-essential prime implicants, select
those that include all remaining minterms
with the least amount of overlap withother
prime implicants.
Using the Karnaugh Map
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Prime implicants are rectancular groupings of
adjacent minterms. (there must be 2-to-the-n
minterms in each grouping).
Make the biggest possible groups to reduce
the number of literals in each term.
Avoid overlaps to reduce the total number of
terms.
Analog for Product of Sums
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Implicants are sum terms that have a value
of 0 whenever the function has a value of 0.
On the K-map, we place 0’s for the
maxterms.
We group the 0’s into essential and nonessential prime implicants just as we did with
the SOP K-map.
F=AB’C+A’BC’=(A+B’+C)(A’+B+C’)
Homogeneity of Logical Gates
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NAND and NOR can be combined to form
any Boolean function. (they are universal)
Using all NAND or all NOR gates can simplify
the production process for integrated circuits,
and reduce costs.
Converting a logical expression to all NAND
or NOR gates can however, increase circuit
depth.