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## Synthesis of reactance 4-poles.

The box section is created by the application of a cross-pivot rotation to a trisection that has been synthesized within the overall coupling matrix for the filter. To transform the trisection into a basic box section, the rotation pivot is set to annihilate the second main-line coupling of the trisection in the coupling matrix. ie. pivot = [2,3] annihilating element M23 in the trisection 1-2-3 in the 4th degree example of Fig.13(a) and in its equivalent coupling and routing schematic in Fig.14(a). In the process of annihilating the main-line coupling M23, the coupling M24 is created (Fig.14(b)), and then, by ‘untwisting’ the network, the box section is formed (Fig.14(c)).

The box section is created by the application of a cross-pivot rotation to a trisection that has been synthesized within the overall coupling matrix for the filter. To transform the trisection into a basic box section, the rotation pivot is set to annihilate the second main-line coupling of the trisection in the coupling matrix. ie. pivot = [2,3] annihilating element M23 in the trisection 1-2-3 in the 4th degree example of Fig.13(a) and in its equivalent coupling and routing schematic in Fig.14(a). In the process of annihilating the main-line coupling M23, the coupling M24 is created (Fig.14(b)), and then, by ‘untwisting’ the network, the box section is formed (Fig.14(c)).

## On the Synthesis of Reactance 4-Poles.

OK, so there is actually a multi-way tie for first place in the speed rankings. Your daughter's question is what comes next? The actual answer is that it's a many-way tie between everything else. With enough energy, you can get anything moving at just under the speed of light, at least in principle. For example, subatomic particles such as electrons, protons, and even atomic nuclei are routinely accelerated to just under the speed of light in sophisticated machines around the world. The universe itself has natural accelerators that whip subatomic particles up until they are traveling so close to the speed of light that it's hard to tell that they aren't quite the speed of light. Nonetheless, I assure you that they're not. The speed of light is so special that nothing that has any mass at all can possibly travel at the speed of light. Only the ephemeral non-massive particles such as light particles (photons), gravity particles (gravitons), and strong force particles (gluons) can actually travel the speed of light. In fact, once photons, gravitons, and gluons begin to interact with matter, they don't travel at the speed of light either. It's sort of a guilt-by-association: as soon as these massless particles leave the essential emptiness of the vacuum and begin to interact with matter, even they can't travel at the speed of light anymore.

Fig. 9(a) is an example of a coupling and routing diagram for a 5th degree canonical filtering circuit. It shows clearly why this configuration is referred to as the ‘wheel.’ with the main-line couplings forming the (partially incomplete) rim and the cross-couplings and input/output coupling forming the spokes. Fig. 9(b) shows the corresponding coupling matrix where the cross-coupling elements are all in the last row and column, and together with the main line and self couplings on the main diagonals give the matrix the appearance of an arrow pointing downwards towards the lower right corner of the matrix. The arrow matrix may be synthesized from the canonical transversal matrix with a formal sequence of rotations, similar to that of the folded matrix.

## "Synthesis of reactance 4-poles which produce ..

The principal constituents of an atom are protons, neutrons, and electrons. These are three most important subatomic particles; the main building blocks of matter in the same way that wood, bricks, and steel are the major building blocks of houses. Each of these particles has a mass—the measure of their inertia—and two of them, electrons and protons, are electrically charged. Each electron has one unit of negative charge while each proton has one unit of positive charge. Because an atom is normally electrically neutral—its positive and negative charges must balance—it has an equal number of electrons and protons. The number of neutrons in an atom is somewhat flexible.

Fig. 9(a) is an example of a coupling and routing diagram for a 5th degree canonical filtering circuit. It shows clearly why this configuration is referred to as the ‘wheel.’ with the main-line couplings forming the (partially incomplete) rim and the cross-couplings and input/output coupling forming the spokes. Fig. 9(b) shows the corresponding coupling matrix where the cross-coupling elements are all in the last row and column, and together with the main line and self couplings on the main diagonals give the matrix the appearance of an arrow pointing downwards towards the lower right corner of the matrix. The arrow matrix may be synthesized from the canonical transversal matrix with a formal sequence of rotations, similar to that of the folded matrix.

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With the advent of the ‘digital revolution’ that has made possible services such as the world wide web, satellite broadcasting and mobile and trunk telephony, the finite RF spectrum allocated for terrestrial and satellite telecommunication systems is becoming increasingly crowded. This has impacted significantly upon the performance required from the microwave equipment that comprises these systems. In the case of microwave filters, greater in-band linearity to avoid signal distortion and out-of-band isolation to suppress interference are routinely specified, which can only be satisfied by advanced filtering characteristics. This article presents the coupling matrix approach to the synthesis of prototype filter networks, enabling the realization of the hardware embodying the enhanced performance needed by today’s high capacity systems.

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