An Introduction to Matching and Layout

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1

An Introduction to

Matching and Layout


Alan Hastings

Texas Instruments


2

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.

3

Kinds of Mismatch


Mismatches may be either

random

or
systematic,

or a combination of both.


Suppose two matched devices have parameters
P
1

and
P
2
.


Then let the mismatch between the devices equal


P = P
2


P
1
.


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
.

4

Random Mismatches


Random 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:

5

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.

6

Systematic Mismatches


Systematic mismatches may arise from
imperfect balancing in a circuit.


Example:

A mismatch

V
CE

between the two bipolar
transistors of a differential pair generates an input
offset voltage

V
BE

equal to:



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.

7

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.

8

Gradients
(continued)


Even subtle gradients can produce large
effects.


A 1

C change in temperature produces a

2mV in
V
BE
,

which equates to an 8% variation in
I
C
.


Power devices on
-
board an integrated circuit can
easily produce temperature differences of 10

20

C
.

9

Analyzing Gradients


For 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.

10

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.

11

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.

12

The Centroid of an Array


The 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:


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….

13

Common
-
centroid Arrays


Arrays 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.

14

Interdigitation


The 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.

15

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.

16

2D Cross
-
coupled Arrays


A 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.

17

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.

18

Practical Common
-
centroid Arrays


Often, 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.

19

2:1:2 Bipolar Array


This array matches a 4X and a 1X bipolar
transistor using interdigitation.

20

4:1:4 Bipolar Array


This array matches an 8X and a 1X bipolar
transistor using interdigitation.

21

Rules for Common Centroids


The 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.