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When examining the power delivered to a load we can break the power down into three distinct quantities We have already seen one of these, the average power delivered to the load P= V0 I0 cos = Veff Ieff cos 2 (1017)

22

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CHAPTER 9:

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We have denoted it by P in this context Sometimes this is known as the real power It is related to two other measures of power known as the reactive power and the complex power This is done using a device known as an impedance triangle that relates the various components of the impedance to its magnitude Let s start by writing down the expression for impedance Z = R + jX (1018)

systems, which store some summaries in memory and store the base data and other summaries in a relational database, are called hybrid OLAP (HOLAP) systems Many OLAP systems are implemented as client server systems The server contains the relational database as well as any MOLAP data cubes Client systems obtain views of the data by communicating with the server A na ve way of computing the entire data cube (all groupings) on a relation is to use any standard algorithm for computing aggregate operations, one grouping at a time The na ve algorithm would require a large number of scans of the relation A simple optimization is to compute an aggregation on, say, (item-name, color) from an aggregation (item-name, color, size), instead of from the original relation For the standard SQL aggregate functions, we can compute an aggregate with grouping on a set of attributes A from an aggregate with grouping on a set of attributes B if A B; you can do so as an exercise (see Exercise 221), but note that to compute avg, we additionally need the count value (For some non-standard aggregate functions, such as median, aggregates cannot be computed as above; the optimization described here do not apply to such non-decomposable aggregate functions) The amount of data read drops signi cantly by computing an aggregate from another aggregate, instead of from the original relation Further improvements are possible; for instance, multiple groupings can be computed on a single scan of the data See the bibliographical notes for references to algorithms for ef ciently computing data cubes Early OLAP implementations precomputed and stored entire data cubes, that is, groupings on all subsets of the dimension attributes Precomputation allows OLAP queries to be answered within a few seconds, even on datasets that may contain millions of tuples adding up to gigabytes of data However, there are 2n groupings with n dimension attributes; hierarchies on attributes increase the number further As a result, the entire data cube is often larger than the original relation that formed the data cube and in many cases it is not feasible to store the entire data cube Instead of precomputing and storing all possible groupings, it makes sense to precompute and store some of the groupings, and to compute others on demand Instead of computing queries from the original relation, which may take a very long time, we can compute them from other precomputed queries For instance, suppose a query requires summaries by (item-name, color), which has not been precomputed The query result can be computed from summaries by (item-name, color, size), if that.

The McGraw Hill Companies, 2001

has been precomputed See the bibliographical notes for references on how to select a good set of groupings for precomputation, given limits on the storage available for precomputed results The data in a data cube cannot be generated by a single SQL query using the basic group by constructs, since aggregates are computed for several different groupings of the dimension attributes Section 2223 discusses SQL extensions to support OLAP functionality