A low pass L network is then used to transform the complex feed point impedance to a high resistive impedance. And, the effect of the transmission line connecting the beamformer to the antenna must be taken into account, since it is unlikely that the feedpoint impedance will happen to just match the feedline characteristic impedance. These transformations are addressed in the page on transmission line equations. The choice of the resistive impedance is somewhat arbitrary, however, it should be higher than the magnitude of the feed point impedance, to insure that the capacitor is on the input side of the network. This allows the series reactance of the feed point (the jX component) to be absorbed by making the series L larger or smaller. The input C value is not critical, as it is later combined with the C of the phase shift network.
A pi network is then calculated to create the proper phase shift from the common feed point to the element feedpoing. The L network will have some phase shift, so the phase shift necessary in the pi network needs to take that additional shift into account. Finally, the input impedance to the pi network needs to be adjusted so that element receives the correct amount of power which is discussed in the page on power division. More information is available on pi and L network synthesis.
We now have a low pass pi network feeding a low pass L network. The output capacitor on the pi network and the input capacitor on the L network can be combined. All of the feed sides of the pi networks will be connected in parallel. So, we redraw the circuit to show it as a capacitor (at the feed end) feeding a lowpass T network which then feeds the antenna.
The next step is to transform the T network into an equivalent pi network, using a standard transformation which is also documented.
And, the final step is to pull the first shunt C of the pi out and combine it with the original pi network shunt C, leaving just an L network
Of course, in an actual system, some of the shunt X of the pi networks will turn out to be positive (i.e. inductive). However, since they wind up being combined in parallel with other shunt X's, some of which will be negative, the net effect is to require only one reactive shunt component for the entire beamformer input.
In practice, an iterative approach may be necessary to find the virtual R in the step where the L network is synthesized to transform the antenna/feedline to a resistive impedance, and some iteration will be necessary to find a set of phase shifts which produce element impedances of the correct signs for all networks. Recall that it is only the relative phases between element currents that are important, so adding a constant phase shift to all elements has no effect on the pattern.
Another approach to calculating the network component values is to create a matrix equation of the beamformer and the elements, then solve for the values of the variable elements, subject to suitable constraints. Some initial work has been done in this area, however, experimentally, I have found that the Excel 97 solver (which is essentially a gradient search solver) often does not converge to a solution.
Another thing to consider is that the adjustable networks have discrete values and are not continuously adjustable. Probably the best approach in the long run is to iteratively solve the matrix equation attempting to find a set of discrete element values that results in the element currents most closely matching the desired distribution. In fact, given the uncertainty in the initial mutual element impedance values (either by calculation or measurement) this approach, done in real time, will probably be the most successful.
The network synthesis and development described above, though, is useful, in that it shows that it is possible to create a passive beamforming network using a whole bunch of adjustable L networks, as available from LDG and others.