Bartlett's bisection theorem

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.

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.

Now consider the network N with two identical voltage generators connected to the ports but with opposite polarity.

Just as superposition of currents through the branches at the plane of symmetry must be zero in the previous case, by analogy and applying the principle of duality, superposition of voltages between nodes at the plane of symmetry must likewise be zero in this case.

[4] Consider the lattice network shown with identical generators, E, connected to each port.

giving a current in the loop of; and an input impedance of; as it is required to be for equivalence to the original two-port.

Similarly, reversing one of the generators results, by an identical argument, in a loop with an impedance of

were defined in the original two-port it is proved that the lattice is equivalent for those two cases.

However, unlike the examples above, the result is not always physically realisable with linear passive components.

Negative quantities can only be physically realised with active components present in the network.

In impedance matching networks, a usual design criterion is to maximise power transfer.

The output response is "the same shape" relative to the voltage of the theoretical ideal generator driving the input.

It is not the same relative to the actual input voltage which is delivered by the theoretical ideal generator via its load impedance.