Formula sheet: Comparators | Digital Circuits - Electronics and Communication Engineering (ECE) PDF Download

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Comparators F orm ula Sheet for GA TE (Digital
Circuits)
Comparator Basics
• Definition : A comparator is a digital circuit that compares t w o input v oltages or
binary n um b ers and outputs a logic lev el indicating their relativ e magnitude.
• Output Logic :
Output =
{
1 ifA>B
0 ifA=B (for a greater-than comparator)
Similar logic applies forA<B orA =B .
• T yp es :
– Analog Comparators: Compare analog v oltages (e.g., op-amp based).
– Digital Comparators: Compare binary n um b ers (e.g., 1-bit, n-bit comparators).
Digital Comparators
• 1-Bit Comparator :
– Inputs: A ,B (single-bit bi nary inputs).
– Outputs: A>B ,A =B ,A<B .
– Logic Equations:
A =B?A·B +A·B =A?B
A>B?A·B
A<B?A·B
• n-Bit Comparator :
– Inputs: A =A
n-1
A
n-2
...A
0
,B =B
n-1
B
n-2
...B
0
.
– Equalit y Chec k:
A =B?
n-1
?
i=0
(A
i
?B
i
)
whereA
i
?B
i
= (A
i
·B
i
+A
i
·B
i
) .
– Greater Than:
A>B?A
n-1
·B
n-1
+(A
n-1
?B
n-1
)·(A
n-2
·B
n-2
+···+(A
1
?B
1
)·(A
0
·B
0
))
– Less Than:
A<B?A
n-1
·B
n-1
+(A
n-1
?B
n-1
)·(A
n-2
·B
n-2
+···+(A
1
?B
1
)·(A
0
·B
0
))
• Magnitude Comparator ICs : Common ICs lik e 7485 (4-bit comparator) implemen t
the ab o v e logic with cascadable inputs forA>B ,A =B ,A<B .
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Comparators F orm ula Sheet for GA TE (Digital
Circuits)
Comparator Basics
• Definition : A comparator is a digital circuit that compares t w o input v oltages or
binary n um b ers and outputs a logic lev el indicating their relativ e magnitude.
• Output Logic :
Output =
{
1 ifA>B
0 ifA=B (for a greater-than comparator)
Similar logic applies forA<B orA =B .
• T yp es :
– Analog Comparators: Compare analog v oltages (e.g., op-amp based).
– Digital Comparators: Compare binary n um b ers (e.g., 1-bit, n-bit comparators).
Digital Comparators
• 1-Bit Comparator :
– Inputs: A ,B (single-bit bi nary inputs).
– Outputs: A>B ,A =B ,A<B .
– Logic Equations:
A =B?A·B +A·B =A?B
A>B?A·B
A<B?A·B
• n-Bit Comparator :
– Inputs: A =A
n-1
A
n-2
...A
0
,B =B
n-1
B
n-2
...B
0
.
– Equalit y Chec k:
A =B?
n-1
?
i=0
(A
i
?B
i
)
whereA
i
?B
i
= (A
i
·B
i
+A
i
·B
i
) .
– Greater Than:
A>B?A
n-1
·B
n-1
+(A
n-1
?B
n-1
)·(A
n-2
·B
n-2
+···+(A
1
?B
1
)·(A
0
·B
0
))
– Less Than:
A<B?A
n-1
·B
n-1
+(A
n-1
?B
n-1
)·(A
n-2
·B
n-2
+···+(A
1
?B
1
)·(A
0
·B
0
))
• Magnitude Comparator ICs : Common ICs lik e 7485 (4-bit comparator) implemen t
the ab o v e logic with cascadable inputs forA>B ,A =B ,A<B .
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Analog Comparators
• Op eration : T ypically implemen ted using op erational amplifiers (op-amps).
V
out
=
{
V
+
sat
ifV
+
>V
-
V
-
sat
ifV
+
<V
-
where V
+
is the non-in v erting input, V
-
is the in v erting input, V
+
sat
and V
-
sat
are satu-
ration v oltages.
• Op en-Lo op Gain : High gain (A
v
˜ 10
5
-10
6
) ensures sharp transitions.
• Hysteresis (for Sc hmitt T rigger comparators) :
V
UT
=V
ref
+
R
1
R
1
+R
2
(V
+
sat
-V
ref
)
V
L T
=V
ref
-
R
1
R
1
+R
2
(V
-
sat
-V
ref
)
where V
UT
and V
L T
are upp er and lo w er threshold v oltages, R
1
and R
2
are feedbac k
resistors.
Design Considerations
• Propagation Dela y : Time tak en for the comparator output to resp ond to input
c hanges.
• Resolution : Minim um v oltage difference detectable b y an analog comparator.
• Common-Mo de V oltage Range : Range of input v oltages where the comparator
op erates correctly .
• P o w er Consumption : Critical for lo w-p o w er digital comparator designs.
Key Notes
• F or GA TE, fo cus on logic equations for 1-bit and n-bit digital comparators.
• Analog comparator problems ma y in v olv e op-amp c haracteristics or h ysteresis calcu-
lations.
• Use Bo olean algebra to simplify comparator logic designs.
• F or IC-based comparators (e.g., 7485), understand cascading for m ulti-bit compar-
isons.
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