Easy as pi
The other network I will discuss is the Pi network. Now that the basic L network is easily visualized, it can be seen that a Pi network is really nothing more than two L networks stacked together. While the T network is two L networks stacked where the shunt legs are next to each other, the Pi network can be thought of the combination where the shunt networks are opposite each other.
As a consequence of these arrangements, an "intermediate" impedance at the center of the network is mathematically created. This value is a virtual quantity; however, it is usually adjusted in order to meet a desired Q value, and by extension, bandwidth across the network. Note that in the Pi network this virtual impedance will be lower than either the generator or load impedance because the shunt legs of the two L networks are opposite each other. The reverse is true for the T network.
The math for the Pi network becomes a little messier as trigonometric functions are introduced. The existence of the trigonometric functions gives rise to the control of the phase shift across the network. Equations for the various legs of a Pi network for a purely resistive transformation become the following where "A" is the shunt leg at the generator side, "B" the shunt leg at the load side, and "C" the series element between the two:
The beta term introduced in these equations is the phase shift desired across the network, and is in terms of radians, not degrees. Converting from degrees to radians is accomplished by multiplying by the value of Pi, and then dividing by 180. Note also that these equations will break down if the sine of beta is zero. This occurs in cases where the phase shift would be zero degrees or plus or minus 180 degrees.
The choice of matching networks varies as much as the ranges of impedances over which transformations can be made. Not only can the basic L network be expanded into the T or Pi, but other exotic combinations as well through various methods of cascading. Each of the designs has their advantages and drawbacks including bandwidth considerations, cost, currents and voltages, and dc continuity, to name a few. In the end, though, even the greenest of broadcast engineers can become proficient enough with simple L, and by extension T networks, to restore an AM station to operation until additional help arrives. That is something that management will definitely appreciate.
Ruck is the principal engineer of Jeremy Ruck and Associates, Canton, IL.