Formula Sheets: Data Selectors & Multiplexers | Digital Electronics - Electrical Engineering (EE) PDF Download

 Page 1


Digital Electronics: Data Selectors & Multiplexers
F orm ula Sheet for Electrical GA TE
Multiplexer (MUX) Basics
• Definition : A 2
n
-to-1 MUX s elects one of 2
n
input lines based on n select lines.
• Output Expression :
Y =
2
n
-1
?
i=0
(I
i
·S
i
)
where I
i
is the i -th input, S
i
is the min term of select lines corresp onding to input i .
• Select Lines: F or 2
n
inputs, n select lines are required.
Num b er of se lect lines= n =log
2
( n um b er of inputs)
• Enable Input : When presen t, Y =0 if enable (E ) is lo w.
Y = E ·
(
2
n
-1
?
i=0
I
i
·S
i
)
F unction Implemen tation Using MUX
• Implemen ting n-v ariable F unction :
Y = f(S
0
,S
1
,...,S
n-1
)
Connect n v ariables to select lines, assign truth table outputs to MUX inputs.
• Example (3-v ariable function with 8-to-1 MUX) :
Y = I
0
S
2
S
1
S
0
+I
1
S
2
S
1
S
0
+···+I
7
S
2
S
1
S
0
where I
0
to I
7
are set to 0, 1, or a v ariable based on the function’s truth table.
• Using Smaller MUX for n-v ariables : Use n - 1 select lines, connect the n -th
v ariable or its complemen t to inputs.
Cascading Multiplexers
• Larger MUX from Smaller MUX :
F or 2
m
inputs using 2
k
-to-1 MUXes, m = k+log
2
( n um b er of MUXes)
• Example (16-to-1 MUX using 8-to-1 MUXes) :
Y = Y
1
·S
3
+Y
2
·S
3
where Y
1
, Y
2
are outputs of t w o 8-to-1 MUXes, S
3
is an additional select line.
1
Page 2


Digital Electronics: Data Selectors & Multiplexers
F orm ula Sheet for Electrical GA TE
Multiplexer (MUX) Basics
• Definition : A 2
n
-to-1 MUX s elects one of 2
n
input lines based on n select lines.
• Output Expression :
Y =
2
n
-1
?
i=0
(I
i
·S
i
)
where I
i
is the i -th input, S
i
is the min term of select lines corresp onding to input i .
• Select Lines: F or 2
n
inputs, n select lines are required.
Num b er of se lect lines= n =log
2
( n um b er of inputs)
• Enable Input : When presen t, Y =0 if enable (E ) is lo w.
Y = E ·
(
2
n
-1
?
i=0
I
i
·S
i
)
F unction Implemen tation Using MUX
• Implemen ting n-v ariable F unction :
Y = f(S
0
,S
1
,...,S
n-1
)
Connect n v ariables to select lines, assign truth table outputs to MUX inputs.
• Example (3-v ariable function with 8-to-1 MUX) :
Y = I
0
S
2
S
1
S
0
+I
1
S
2
S
1
S
0
+···+I
7
S
2
S
1
S
0
where I
0
to I
7
are set to 0, 1, or a v ariable based on the function’s truth table.
• Using Smaller MUX for n-v ariables : Use n - 1 select lines, connect the n -th
v ariable or its complemen t to inputs.
Cascading Multiplexers
• Larger MUX from Smaller MUX :
F or 2
m
inputs using 2
k
-to-1 MUXes, m = k+log
2
( n um b er of MUXes)
• Example (16-to-1 MUX using 8-to-1 MUXes) :
Y = Y
1
·S
3
+Y
2
·S
3
where Y
1
, Y
2
are outputs of t w o 8-to-1 MUXes, S
3
is an additional select line.
1
Data Selectors
• F unctionalit y : Data selector is synon ymous with MUX, selecting one input based on
select lines.
• T ruth T able for 2-to-1 MUX :
Y = I
0
·S +I
1
·S
• Applications : Data routing, function generation, parallel-to-serial con v ersion.
P erformance P arameters
• Propagation Dela y :
t
p
= t
gate
·( n um b er of gate lev els)
where t
gate
is the dela y p er gate (ns ).
• P o w er Dissipation (CMOS) :
P = C
L
V
2
DD
f
whereC
L
is load capacitance (pF ), V
DD
is supply v oltage (V ), f is switc hing frequency
(Hz ).
• F an-Out : Num b er of inputs a MUX output can driv e.
F an-Out=
I
OH( max)
I
IH( max)
Key Notes
– MUX T yp es : 2-to-1, 4-to-1, 8-to-1, 16-to-1 are common in GA TE problems.
– F unction Implemen tation : Use truth table outputs as MUX inputs or don’t
cares for optimization.
– Cascading : Com bine smaller MUXes for larger input coun ts; select lines increase
logarithmically .
– GA TE F o cus : Implemen t logic functions, analyze dela y , and design with minimal
MUXes.
– Units : Dela ys in ns , v oltages in V , p o w er in W .
2