The commonality of the 90-degree tee network is in some part due to the simplicity of its mathematics. The three equations that quantify the reactances in the three legs of the network are as follows:
In these equations, the subscripts 1, 2, and 3 correspond respectively to the input, output, and shunt legs. The Greek letter theta represents the phase shift across the network, and j is the imaginary operator.
In the 90-degree cases the equations collapse as a result of the trigonometric terms. The magnitude of the sine of 90 degrees is 1 and is positive if the phase angle is leading, negative if it is lagging. The cosine of 90 degrees is zero. So after substituting these values back in, we find that the reactance in each leg is nothing more than the square root of the product of the input and output resistance. If there is no reactance on either side of the network, then we are finished as long as we get the signs correct. In the leading case, as previously mentioned, the input and output legs are negative or capacitive, and the shunt is positive or inductive. For the lagging case, the situations are reversed.
We do not, however, live in a perfect world, and have to deal with load reactance. While the reactance on the input side can be eliminated if it is fed by transmission line, we find no such luxury with the load on the far side of the network. As a result, we have to correct for the reactance of the load impedance. This correction is made through the addition of an equal but opposite reactance to the value determined in the third equation above.
Put into practice
For an example, make the assumption that we have an antenna with an impedance of 72Ω+j50. This antenna will be matched to 50Ω transmission line through the use of a 90-degree tee network. It is desired to derive the most cost effective design for the network and for this example will ignore potential concerns with the shunt leg current.
Thinking about the topology of the network, we realize that the leading phase shift will require at a minimum two capacitors and three inductors if we limit our choices to micas. The lagging phase shift will also require three inductors, but will likely only need one capacitor in the shunt leg. The lagging case looks to be more cost effective.
The magnitude of each of the leg reactances will wind up being the square root of the product of the input and output resistance values. Numerically this is 60Ω. Since we have chosen the lagging network, the shunt leg will be a negative 60Ω, and the input leg will be a positive 60Ω. The output leg is also a positive 60Ω, but we need to subtract off the 50Ω of reactance inherent with the antenna. The result is the output leg has a positive reactance of 10Ω.
Once you have designed the tee network, it may require some adjustment to ensure that it is properly set up. A good technique is to go through and initially setup each of the legs based on the design, then move back to the input to the network, and bridge it when connected to the load. The impedance measured at the input should be very close to the design. If it needs to be trimmed, minor adjustments of the shunt and input legs should do the trick. Adjust the shunt leg to bring the resistance back to the desired value. This will skew the input reactance somewhat, which can then be trimmed out by adjustment of the input leg.
The tee network appears daunting at first, but with a little study is actually fairly simple to master and understand. Its wide versatility combined with ease of design and implementation, has made it a most ubiquitous choice for use in not only AM antennas but a myriad of other applications as well.
Ruck is a senior engineer with D.L. Markley and Associates, Peoria, IL.