1 An Introduction to An Introduction to Matching and Layout Matching and Layout Alan Hastings Alan Hastings Texas Instruments Texas Instruments
Mar 30, 2015
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An Introduction to An Introduction to Matching and LayoutMatching and Layout
Alan HastingsAlan HastingsTexas InstrumentsTexas Instruments
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Overview of Matching
Two devices with the same physical layout never have quite the same electrical properties. Variations between devices are called mismatches. Mismatches may have large impacts on certain
circuit parameters, for example common mode rejection ratio (CMRR).
By default, simulators such as SPICE do not model mismatches. The designer must deliberately insert mismatches to see their effects.
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Kinds of Mismatch
Mismatches may be either random or systematic, or a combination of both. Suppose two matched devices have parameters P1
and P2.
Then let the mismatch between the devices equal P = P2 – P1.
For a sample of units, measure P . Compute sample mean m(P ) and standard
deviation s(P ). m(P ) is a measure of systematic mismatch. s(P) is a measure of random mismatch.
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Random MismatchesRandom mismatches are usually due to
process variation. These process variations are usually manifestations
of statistical variation, for example in scattering of dopant atoms or defect sites.
Random mismatches cannot be eliminated, but they can be reduced by increasing device dimensions.
In a rectangular device with active dimensions W by L, an areal mismatch can be modeled as:
WL
kP P)(
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Random Mismatches (continued)
Random mismatches thus scales as the inverse square root of active device area. To reduce mismatch by a factor of two, increase area
by a factor of four. Precision matching requires large devices. Other performance criteria (such as speed) may
conflict with matching.
WL
kP P)(
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Systematic MismatchesSystematic mismatches may arise from
imperfect balancing in a circuit. Example: A mismatch VCE between the two bipolar
transistors of a differential pair generates an input offset voltage VBE equal to:
CEA
TBE V
V
VV
Simulations will readily show this source of systematic offset.
Usually, the circuit can be redesigned to minimize or even to completely eliminate this type of systematic offset.
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Gradients
Systematic mismatches may also arise from gradients. Certain physical parameters may vary gradually
across an integrated circuit, for example: Temperature Pressure Oxide thickness
These types of variations are usually treated as 2D fields, the gradients of which can (at least theoretically) be computed or measured.
Because of the way we mathematically treat these variations, they are called gradients.
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Gradients (continued)Even subtle gradients can produce large
effects. A 1C change in temperature produces a –2mV in
VBE, which equates to an 8% variation in IC.
Power devices on-board an integrated circuit can easily produce temperature differences of 10–20C.
Power device
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Analyzing GradientsFor a simplified analysis of gradients:
Make the following assumptions of linearity: The gradient is constant over the area of interest. Electrical parameters depend linearly upon physical
parameters. Although neither assumption is strictly true, they are
usually approximately true, at least for properly laid out devices.
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Analyzing Gradients (continued)Assuming linearity,
We can reduce each distributed device to a lumped device located at the centroid of the device area.
The magnitude of the mismatch equals the product of the distance between the centroids and the magnitude of the gradient along the axis of separation.
Therefore we can reduce the impact of the mismatch by reducing the separation of the centroids.
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How to Find a Centroid (Easily)Rules for finding a centroid (assuming
linearity): If a geometric figure has an axis of symmetry, then
the centroid lies on it. If a geometric figure has two or more axes of
symmetry, then the centroid must lie at their intersection.
Centroid Centroid
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The Centroid of an ArrayThe centroid of an array can be computed from
the centroids of its segments. If all of the segments of the array are of equal size,
then the location of the centroid of the array is the average of the centroids of the segments:
N
segmentarray dN
d1
Note that the centroid of an array does not have to fall within the active area of any of its segments.
This suggests that two properly constructed arrays could have the same centroid….
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Common-centroid ArraysArrays whose centroids coincide are called
common-centroid arrays. Theoretically, a common-centroid array should
entirely cancel systematic mismatches due to gradients.
In practice, this doesn’t happen because the assumptions of linearity are only approximately true.
Common-centroids don’t help random mismatches at all. Neither are they a cure for sloppy circuit design!
Virtually all precisely matched components in integrated circuits use common centroids.
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InterdigitationThe simplest sort of common-centroid array
consists of a series of devices arrayed in one dimension. One-dimensional common-centroid arrays are ideal
for long, thin devices, such as resistors. Since the segments of the matched devices are
slipped between one another to form the array, the process is often called interdigitation.
A
B
B
A
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Interdigitation (continued)Not all interdigitated arrays are made equal!
Certain arrays precisely align the centroids of the matched devices (A, C). These provide superior matching.
Other arrays only approximately align the centroids (B). These provide inferior matching.
Common axisof symmetry of device A
Axis of symmetry
A B B A A B A B
(A) (B) (C)
A AB
Common axis ofsymmetry
Axis of symmetryof device B
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2D Cross-coupled ArraysA more elaborate sort of common-centroid
array involves devices cross-coupled in a rectangular two-dimensional array. This type of array is ideal for roughly square devices,
such as capacitors and bipolars.
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2D Cross-coupled Arrays (cont’d) The simplest two-dimensional cross-coupled array
contains four segments. This type of array is called a cross-coupled pair. For many devices, particularly smaller ones, the
cross-coupled pair provides the best possible layout. More complicated 2D arrays containing more
segments provide better matching for large devices because they minimize the impact of nonlinearities.
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Practical Common-centroid ArraysOften, the design of a common-centroid array
is complicated by layout considerations. Sometimes certain devices can be merged, resulting
in a smaller overall array. Unfortunately, such mergers often constrain the
layout of the array. The proper design of a cross-coupled array is often
quite difficult, and a certain degree of experience is required to obtain good results.
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2:1:2 Bipolar ArrayThis array matches a 4X and a 1X bipolar
transistor using interdigitation.
1/2 Q Q 1/2 Q 2 1 2
S 1
S 2
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4:1:4 Bipolar ArrayThis array matches an 8X and a 1X bipolar
transistor using interdigitation.
1/2 Q
Q
1/2 Q 2
1
2
S 1
S 2
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Rules for Common CentroidsThe following rules summarize good design
practices: Coincidence: The centroids of the matched devices
should coincide, at least approximately. Ideally, the centroids should exactly coincide.
Symmetry: The array should be symmetric about both the X- and the Y-axes. Ideally, this symmetry should arise from the placement of the segments in the array, and not from the symmetry of the individual segments.
Dispersion: The array should exhibit the highest possible degree of dispersion; in other words, the segments of each device should be distributed throughout the array as uniformly as possible.
Compactness: The array should be as compact as possible. Ideally, it should be nearly square.