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lecture12-140610
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Lecture 12 – Component Matching (6/10/14) Page 12-1
CMOS Analog Circuit Design © P.E. Allen - 2016
LECTURE 12 – COMPONENT MATCHING
LECTURE ORGANIZATION
Outline
• Introduction
• Electrical matching
• Physical matching
• Summary
CMOS Analog Circuit Design, 3
rd
Edition Reference
Pages 640-652 and new material
Lecture 12 – Component Matching (6/10/14) Page 12-2
CMOS Analog Circuit Design © P.E. Allen - 2016
INTRODUCTION
What is Accuracy and Matching?
The accuracy of a quantity specifies the difference between the actual value of the
quantity and the ideal or true value of the quantity.
The mismatch between two quantities is the difference between the actual ratio of the
quantities and the desired ratio of the two quantities.
Example:
x
1
= actual value of one quantity
x
2
= actual value of a second quantity
X
1
= desired value of the first quantity
X
2
= desired value of the second quantity
The accuracy of a quantity can be expressed as,
Accuracy =
x - X
X
=
X
X
The mismatch,
, can be expressed as,
=
x
2
x
1
-
X
2
X
1
X
2
X
1
=
X
1
x
2
X
2
x
1
- 1
Lecture 12 – Component Matching (6/10/14) Page 12-3
CMOS Analog Circuit Design © P.E. Allen - 2016
Relationship between Accuracy and Matching
Let:
X
1
= |x
1
- X
1
| → x
1
= X
1
±
X
1
and
X
2
= |x
2
– X
2
| → x
2
= X
2
±
X
2
Therefore, the mismatch can be expressed as,
=
X
1
(X
2
±
X
2
)
X
2
(X
1
±
X
1
)
– 1 =
1 ±
X
2
X
2
1 ±
X
1
X
1
– 1
1 ±
X
2
X
2
1
+
-
X
1
X
1
– 1
1 ±
X
2
X
2
+
-
X
1
X
1
– 1 = ±
X
2
X
2
+
-
X
1
X
1
Thus, the mismatch is approximately equal to the difference in the accuracies of x
1
and
x
2
assuming the deviations (
X) are small with respect to X.
Lecture 12 – Component Matching (6/10/14) Page 12-4
CMOS Analog Circuit Design © P.E. Allen - 2016
1
2
3
4
5
6
7
8
9
10
0 X
0 1 2 3 4 5 6 7 8 9 10 1211
Number of Samples
041005-01
1314
Characterization of the Mismatch
Mean of the mismatch for N samples-
m
=
1
N
i=1
N
i
Standard deviation of the mismatch for N samples-
s
=
=
1
N-1
i=1
N
(
i
- m
)
2
Example:
m
=
253
40
= 6.325 s
= 2.115
Lecture 12 – Component Matching (6/10/14) Page 12-5
CMOS Analog Circuit Design © P.E. Allen - 2016
Motivation for Matching of Components
The accuracy of analog signal processing is determined by the accuracy of gains and
time constants. These accuracies are dependent upon:
Gain Ratios of components or areas
Time constants Products of components or areas
Ratio Accuracy?
Actual Ratio =
X
1
±
X
1
X
2
±
X
2
=
X
1
X
2
1±
X
1
X
1
1±
X
2
X
2
≈
X
1
X
2
1±
X
1
X
1
1
+
-
X
2
X
2
≈
X
1
X
2
1±
X
1
X
1
+
-
X
2
X
2
If X
1
and X
2
match (
X
1
/X
1
≈
X
2
/X
2
), then the actual ratio becomes the ideal ratio.
Product Accuracy?
Product accuracy = (X
1
±
X
1
)(X
2
±
X
2
) = X
1
X
2
1±
X
1
X
1
1±
X
2
X
2
≈ X
1
X
2
1±
X
1
X
1
±
X
2
X
2
Unfortunately, the product cannot be accurately maintained in integrated circuits.
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