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An Introduction to Matching and Layout Alan Hastings Texas Instruments

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An Introduction to Matching and Layout Alan Hastings Texas Instruments

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## An Introduction to Matching and Layout Alan Hastings Texas Instruments

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An Introduction to Matching and Layout

Alan HastingsTexas Instruments

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.

Kinds of Mismatch

- Mismatches may be eitherrandomor 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.

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:

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.

Systematic Mismatches

- Systematic 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:

- 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.

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.

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.

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.

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.

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.

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….

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.

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.

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.

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.

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.

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.

2:1:2 Bipolar Array

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

4:1:4 Bipolar Array

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

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.

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