PowerPoint PresentationOverview of MatchingKinds of MismatchRandom MismatchesRandom Mismatches (continued)Systematic MismatchesGradientsGradients (continued)Analyzing GradientsAnalyzing Gradients (continued)How to Find a Centroid (Easily)The Centroid of an ArrayCommon-centroid ArraysInterdigitationInterdigitation (continued)2D Cross-coupled Arrays2D Cross-coupled Arrays (cont’d)Practical Common-centroid Arrays2:1:2 Bipolar Array4:1:4 Bipolar ArrayRules for Common Centroids1An Introduction to An Introduction to Matching and LayoutMatching and LayoutAlan HastingsAlan HastingsTexas InstrumentsTexas Instruments2Overview 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.3Kinds 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.4Random 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:WLkPP)(5Random 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.WLkPP)(6Systematic 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: CEATBEVVVV 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.7Gradients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.8Gradients (continued)Even subtle gradients can produce large effects.A 1C 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 10–20C.Power device9Analyzing 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.10Analyzing 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.11How 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 Centroid12The 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:NsegmentarraydNd1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….13Common-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.14Interdigitation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.15Interdigitation (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 symmetryof device AAxis of symmetryAB BAA BAB(A) (B)(C)AABCommon axis ofsymmetryAxis of symmetryof device B162D 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.172D Cross-coupled Arrays (cont’d)The simplest two-dimensional cross-coupled array contains four segments. This