Bartlett's bisection theorem
Bartlett's Bisection Theorem is an electrical theorem in network analysis attributed to Albert Charles Bartlett. The theorem shows that any symmetrical two-port network can be transformed into a lattice network. The theorem often appears in filter theory where the lattice network is sometimes known as a filter X-section following the common filter theory practice of naming sections after alphabetic letters to which they bear a resemblance.
The theorem as originally stated by Bartlett required the two halves of the network to be topologically symmetrical. The theorem was later extended by Wilhelm Cauer to apply to all networks which were electrically symmetrical. That is, the physical implementation of the network is not of any relevance. It is only required that its response in both halves are symmetrical.
Lattice topology filters are not very common. The reason for this is that they require more components (especially inductors) than other designs. Ladder topology is much more popular. However, they do have the property of being intrinsically balanced and a balanced version of another topology, such as T-sections, may actually end up using more inductors. One application is for all-pass phase correction filters on balanced telecommunication lines. The theorem also makes an appearance in the design of crystal filters at RF frequencies. Here ladder topologies have some undesirable properties, but a common design strategy is to start from a ladder implementation because of its simplicity. Bartlett's theorem is then used to transform the design to an intermediate stage as a step towards the final implementation (using a transformer to produce an unbalanced version of the lattice topology).
Start with a two-port network, N, with a plane of symmetry between the two ports. Next cut N through its plane of symmetry to form two new identical two-ports, ½N. Connect two identical voltage generators to the two ports of N. It is clear from the symmetry that no current is going to flow through any branch passing through the plane of symmetry. The impedance measured into a port of N under these circumstances will be the same as the impedance measured if all the branches passing through the plane of symmetry were open circuit. It is therefore the same impedance as the open circuit impedance of ½N. Let us call that impedance .
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