Binary Schemes

A six-stage binary scheme should normally be adequate to switch even very large banks. Yet this sequencing can be further modified to achieve a more particular sequencing pattern when required, as noted below. 

The binary scheme can be modified to suit any other requirement also. For example, for an induetion furnace, requiring very fast correction, the scheme may be modified, to have six stages in the ratio of 10 20 40 80 160 160 kVAr to make a total of 470 kVAr and correetions in steps of 10 kVAr each. 

For loads with fast variations, this type of a modified version too may be sluggish, due to step-by-step tuning. In such cases it is possible to employ two relays, one for coarse correction, such as through a serial relay (FIFO) which can quickly switch the large banks, and the other for fine tuning, which may be a binary relay. These two relays may be combined into one hybrid unit. The binary scheme can thus be modified to suit any required duty with prompt and finer corrections and least strain on a particular unit or on the system. 

Industrial Power Engineering and Applications Handbook Table 23.10 A six-stage modified binary scheme... 

This first example has approximately twice as many vertices in the new polygon as in the old. We call it a binary scheme. If there had been three times as many it would have been a ternary scheme, and such generalisations will be discussed in a few pages time. In principle at each refinement we can multiply the number of vertices by whatever we choose, and this number is called the arity and denoted by the letter a. It is also called the dilation factor, which stems from generating function usage. 

Both of the above examples approximately double the number of vertices in the polygon with each step of refinement. They are binary schemes. It is also possible to have schemes in which the number of vertices trebles or quadruples or is multiplied by a still higher factor. As mentioned above, we call that factor the arity, so that binary schemes have an arity of 2, ternary of 3, quaternary of 4 etc. Some of the mathematics applies to all arities, and in such cases we will denote the arity by the letter a. 

For example, the binary scheme whose mask is [1,3,3, l]/4 (the binary quadratic B-spline) has a support width of (4 — l)/2 = 3/2 spans on each side. 

The interesting property of an infinite matrix is that, lacking a top left hand corner, it doesn t have a principal diagonal. Any diagonal can be taken as principal. Because we have chosen a binary scheme for the example, all diagonals look the same anyway, so it makes no difference for our analysis which one we choose. 

What the choice of a diagonal does is to imply a labelling, giving a correspondence between a sequence of points of the old polygon and a sequence of the refined one. In particular it implies a mark point which is an abscissa value which maps into itself under the map from old abscissa values to new ones. In the case of a primal binary scheme, the mark point is at a point of both new and old polygons. In the case of a dual scheme the mark point is at a mid-edge in both old and new. 

This procedure can be applied to any primal binary scheme, although it may be necessary to imagine higher dimensions than 3 in order to keep applying the principle of suppressing successive dominant eigencomponents. 

A systematic procedure, which can be totally automated, for determining how many continuous derivatives the limit curve of a binary scheme has is as follows. 

For example, when the kernel is [2] of a binary scheme, the kernel of its quaternary square is [4], giving the matrix... 

We consider first binary schemes primal and dual need not be distinguished. If the symbol has d+ 1 factors of (1 + z)j2 then... 

The only solution is k = 1, so that the only binary schemes with piecewise polynomial limit curves are the B-splines. 

The kernel by definition has no further factors of a, but it can be expressed as a polynomial in a. In fact, because the kernel of a binary scheme always has an odd number of entries, its symmetric form can be expressed as a polynomial in a2. 

A slightly surprising result is that the limit curve of polygon P(z) under the binary scheme with mask 2ak(l + z2)/2z is identical to the limit curve of polygon aP(z) under the scheme whose mask is 2ak+i. It is a B-spline curve, but with a different control polygon. 

Quaternary and higher arities are significantly more complex. Quaternary schemes can, of course be created by squaring a binary scheme, but there are others which are not so created. Note that the square of a weighted mean of two binary schemes is not the same as the weighted mean of the squares of those schemes. 

We can look on any binary scheme as an affine combination of B-splines, and choosing the barycentric coordinates in such a combination is a very convenient viewpoint for designing a scheme to have specific properties. 

If the dimension is 1, a simple scanning and the plotting of graphs gives a complete enough guide, and we do this here for the case of binary schemes combining two successive B-splines of the same parity. 

Note that for dual binary schemes the distribution of control points gives an extra half-integer-worth of curve. 

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