Formula Sheets: Combinational Logic | Digital Electronics - Electrical Engineering (EE) PDF Download

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Digital Electronics: Com binational Logic F orm ula
Sheet for Electrical GA TE
Basic Logic Gates
• AND : Y =A·B
• OR : Y =A+B
• NOT : Y =A
• NAND : Y =A·B
• NOR : Y =A+B
• X OR : Y =A?B =AB +AB
• XNOR : Y =A?B =AB +AB
Bo olean Algebra Applications
• De Morgan’s Theorems :
A+B =A·B, A·B =A+B
• Sum of Pro ducts (SOP) :
Y =
?
m
i
=ABC +ABC +...
where m
i
are min terms.
• Pro duct of Sums (POS) :
Y =
?
M
i
= (A+B +C)·(A+B +C)·...
where M
i
are maxterms.
Arithmetic Circuits
• Half A dder :
Sum =A?B, Carry =A·B
• F ull A dder :
Sum =A?B?C
in
, Carry =A·B +B·C
in
+A·C
in
• n-Bit A dder Propagation Dela y :
t
total
=n·t
F A
where t
F A
is full adder dela y .
• Binary Subtractor (2’s Complemen t) :
A-B =A+( 2’s Complemen t of B)
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Digital Electronics: Com binational Logic F orm ula
Sheet for Electrical GA TE
Basic Logic Gates
• AND : Y =A·B
• OR : Y =A+B
• NOT : Y =A
• NAND : Y =A·B
• NOR : Y =A+B
• X OR : Y =A?B =AB +AB
• XNOR : Y =A?B =AB +AB
Bo olean Algebra Applications
• De Morgan’s Theorems :
A+B =A·B, A·B =A+B
• Sum of Pro ducts (SOP) :
Y =
?
m
i
=ABC +ABC +...
where m
i
are min terms.
• Pro duct of Sums (POS) :
Y =
?
M
i
= (A+B +C)·(A+B +C)·...
where M
i
are maxterms.
Arithmetic Circuits
• Half A dder :
Sum =A?B, Carry =A·B
• F ull A dder :
Sum =A?B?C
in
, Carry =A·B +B·C
in
+A·C
in
• n-Bit A dder Propagation Dela y :
t
total
=n·t
F A
where t
F A
is full adder dela y .
• Binary Subtractor (2’s Complemen t) :
A-B =A+( 2’s Complemen t of B)
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Multiplexers (MUX)
• Output of 2
n
-to-1 MUX :
Y =
2
n
-1
?
i=0
(I
i
·S
i
)
where I
i
are inputs, S
i
is the select line com bination for input i .
• Select Lines: F or 2
n
inputs, n select lines are required.
• F unction Implemen tation :
Y =f(S
0
,S
1
,...,S
n-1
)
(MUX can implemen t an y n -v ariable logic function.)
Dem ultiplexers (DEMUX)
• Output of 1-to-2
n
DEMUX :
Y
i
=I·S
i
where I is input, S
i
is select line com bination for output i .
• Select Lines: n select lines for 2
n
outputs.
Enco ders and Deco ders
• Deco der (n-to-2
n
) :
Y
i
= 1 if input co de =i, Y
i
= 0 otherwise
• Binary Enco der (2
n
-to-n ) :
Output = Binary equiv alen t of activ e input
• Priorit y Enco der :
Y = Highest priorit y activ e input (in b inary)
Key Notes
• Com binational Logic : Output dep ends only on curren t inputs, no memory .
• GA TE F o cus : Design circuits using MUX, deco ders; simplify using K-Maps; analyze
adders/subtractors.
• MUX/DEMUX : Use for function implemen tation and data routing.
• Error A v oidance : Ensure all inputs are co v ered in truth tables; c hec k for glitc hes in
design.
• Units : Dela ys in ns , logic lev els in 0/1 .
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